Characterization and Measurement of Magnetic Materials (Electromagnetism)

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Measurement and Characterization of Magnetic Materials

A Volume in the Elsevier Series in Electromagnetism

Elsevier Science Internet Homepage - http://www.elsevier.com Consult the Elsevier homepage for full catalogue information on all books, journals and electronic products and services.

Elsevier Series in Electromagnetism (Series formerly known as Academic Press Series in Electromagnetism)

Edited by ISAAK MAYERGOYZ

Electromagnetism is a classical area of physics and engineering that still plays a very important role in the development of new technology. Electromagnetism often serves as a link between electrical engineers, material scientists, and applied physicists. This series presents volumes on those aspects of applied and theoretical electromagnetism that are becoming increasingly important in modern and rapidly development technology. Its objective is to meet the needs of researchers, students, and practicing engineers.

Books Published in the Series Giorgio Bertotti, Hysteresis in Magnetism: For Physicists, Material Scientists, and Engineers Scipione Bobbio, Electrodynamics of Materials: Forces, Stresses, and Energies in Solids and

Fluids Alain Bossavit, Computational Electromagnetism: Variational Formulations,

Complementarity, Edge Elements M.V.K. Chari and S.J. Salon, Numerical Methods in Electromagnetism Goran Engdahl, Handbook of Giant Magnetostrictive Materials Edward P. Furlani, Permanent Magnet and Electromechanical Devices Vadim Kuperman, Magnetic Resonance Imaging: Physical Principles and Applications John C. Mallinson, Magneto-Resistive Heads: Fundamentals and Applications Isaak Mayergoyz, Nonlinear Diffusion of Electromagnetic Fields Isaak Mayergoyz, Mathematical Models of Hysteresis and Their Applications Giovanni Miano and Antonio Maffucci, Transmission Lines and Lumped Circuits Shan X. Wang and Alexander M. Taratorin, Magnetic Information Storage Technology

Related Books John C. Mallinson, The Foundations of Magnetic Recording, Second Edition Reinaldo Perez, Handbook of Electromagnetic Compatibility

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Measurement and Characterization of Magnetic Materials

FAUSTO FIORILLO Istituto Elettrotecnico Nazionale Galileo Ferraris Strada delle Cacce 91 Torino, 10135 ITALY

2004

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A m s t e r d a m 9 B o s t o n 9 H e i d e l b e r g 9 L o n d o n 9 N e w York 9 O x f o r d Paris 9 San D i e g o 9 San F r a n c i s c o 9 S i n g a p o r e 9 S y d n e y 9 T o k y o

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To the memory of my beloved parents To my family

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Foreword

This volume in the Electromagnetism series presents a modern, in-depth, comprehensive and self-contained treatment of the characterization and measurement of magnetic materials. These materials are ubiquitous in numerous industrial applications that range from electric power generation, conversion and distribution to magnetic data storage. Currently, there does not exist any book that covers the physical properties of magnetic materials, their characterization and modern measurement techniques of various parameters of these materials in detail. This book represents the first successful attempt to give a synthetic exposition of all these issues within one volume. The author, Dr Fausto Fiorillo, is a well-known expert in the field. He has an extensive experience in the area of magnetic measurements as well as intimate and firsthand knowledge of the latest technological innovations and has thus managed to put together an extraordinary amount of technical information in one volume. The salient and unique features of the book are its scope and strong emphasis on the metrological aspects of magnetic measurements. The book also reflects the broad expertise and extensive knowledge accumulated over the years by the highly visible and respected research group of the IEN Galileo Ferraris Materials Department based in Turin, Italy. I maintain that this book will be a valuable reference for both experts and beginners in the field. Electrical engineers, material scientists, physicists, experienced researchers and graduate students will find this book to be a valuable source of new facts, novel measurement techniques and penetrating insights. Isaak Mayergoyz, Series Editor

vii

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Contents

Foreword

vii

Preface

xiii

Acknowledgments

xvii

Part I.

Properties of Magnetic Materials

1. Basic Phenomenology in Magnetic Materials 1.1 Magnetized Media 1.2 Demagnetizing Fields 1.3 Magnetization Process and Hysteresis

3 3 8 16

2. Soft 2.1 2.2 2.3

25 26 33 38 39 43 49 51 61 65 73

2.4 2.5 2.6 2.7

Magnetic Materials General Properties Pure Iron and Low-Carbon Steels Iron-Silicon Alloys 2.3.1 Non-oriented Fe-Si alloys 2.3.2 Grain-oriented Fe-Si alloys 2.3.3 Fe-(6.5 wt%)Si, Fe-A1 and Fe-Si-A1 alloys Amorphous and Nanocrystalline Alloys Nickel-Iron and Cobalt-Iron Alloys Soft Ferrites Soft Magnetic Thin Films

3. Operation of Permanent Magnets 3.1 Magnetic Circuit and Energy Product 3.2 Dynamic Recoil 3.3 Electrical Analogy and Numerical Modeling

89 90 95 98 ix

x

Contents

Part II.

Generation and Measurement of Magnetic Fields

103

4. Magnetic Field Sources 4.1 Filamentary Coils 4.1.1 Single current loop 4.1.2 Thin solenoids 4.1.3 Helmholtz coils 4.2 Thick Coils 4.3 AC and Pulsed Field Sources 4.4 Permanent Magnet Sources 4.5 Electromagnets

105 105 106 108 113 117 123 132 145

5. Measurement of Magnetic Fields 5.1 Fluxmetric Methods 5.1.1 Magnetic flux detection 5.1.2 Signal treatment and calibration of fluxmeters 5.2 Hall Effect and Magnetoresistance Methods 5.2.1 Physical mechanism of Hall effect and magnetoresistance 5.2.2 Measuring devices 5.3 Ferromagnetic Sensor Methods 5.3.1 Fluxgate magnetometers 5.3.2 Inductive magnetometers 5.3.3 Magnetostriction, magneto-optical, and microtorque magnetometers 5.4 Quantum Methods 5.4.1 Physical principles of NMR 5.4.2 NMR magnetometers 5.4.3 Electron spin resonance and optically pumped magnetometers 5.5 Magnetic Field Standards and Traceability

159 161 161 169 175

Part III.

Characterization of Magnetic Materials

175 185 196 197 202 209 217 218 227 251 262 279

6. Magnetic Circuits and General Measuring Problems 6.1 Closed Magnetic Circuits 6.2 Open Samples

281 282 295

7. Characterization of Soft Magnetic Materials 7.1 Bulk Samples, Laminations, and Ribbons: Test Specimens, Magnetizers, Measuring Standards

307

309

Contents 7.1.1 Bulk samples 7.1.2 Sheet, strip, and ribbon specimens 7.1.3 Anisotropic materials and two-dimensional testing 7.2 Measurement of the DC Magnetization Curves and the Related Parameters 7.2.1 Magnetometric methods 7.2.2 Inductive methods 7.3 AC Measurements 7.3.1 Low and power frequencies: basic measurements 7.3.2 Low and power frequencies: special measurements 7.3.3 Medium-to-high frequency measurements 7.3.4 Measurements at radiofrequencies

xi 309 315 326 336 337 340 362 364 385 409 432

8. Characterization of Hard Magnets 8.1 Closed Magnetic Circuit Measurements 8.2 Open Sample Measurements 8.2.1 Vibrating sample magnetometer 8.2.2 Alternating gradient force magnetometer 8.2.3 Extraction method 8.2.4 Pulsed field method

475 481 499 500 521 531 536

9. Measurement of Intrinsic Magnetic Properties of Ferromagnets 9.1 Spontaneous Magnetization and Curie Temperature 9.2 Magnetic Anisotropy

549 549 564

10. Uncertainty and Confidence in Measurements 10.1 Estimate of a Measurand Value and Measuring Uncertainty 10.2 Combined Uncertainty 10.3 Expanded Uncertainty and Confidence Level. Weighted Uncertainty 10.4 Traceability and Uncertainty in Magnetic Measurements 10.4.1 Calibration of a magnetic flux density standard 10.4.2 Determination of the DC polarization in a ferromagnetic alloy 10.4.3 Measurement of power losses in soft magnetic laminations

581

581 586 589 595 600 603 605

xii Appendix Appendix Appendix Appendix

Contents A: B: C: D:

The SI and the CGS Unit Systems in Magnetism Physical Constants Evaluation of Measuring Uncertainty Specifications of Magnetometers

613 621 623 629

List of Symbols

631

Subject Index

635

Preface

Magnets and measurements are everywhere. Magnetic materials are key pieces of a complex puzzle and are fundamental in satisfying basic demands of our society such as the generation, distribution, and conversion of energy, the storage and retrieval of information, many types of media and telecommunications. With their use in so many critical applications, these materials play a crucial role in our daily life and the present pace of research provides good reason for believing that their importance will continue to increase. With an annual global market valued at approximately EUR s the economic relevance of magnetic materials in industry is clear. Just as magnetic materials are important so accurate measurements are indispensable to science, industry, and commerce and are the prerequisite for any conceivable development in the production and trading of goods. They have relevant costs (about 5% of GNP in industrial countries) and require highly specialized organizations (such as the National Metrological Institutes) to develop and maintain the standards. Taken together these two elements are both scientifically and economically significant. This is the only book that takes that approach. Magnetism has a popular reputation of being a difficult subject. Part of this notion is as a result of the unfortunate duality of unit systems, which has greatly complicated life for students, researchers, and practitioners for many years. Nowadays, the SI system, recommended by the Conf&ence Poids et Mesures under the MKSA label since 1946, is establishing itself as the dominant system, despite resistance from many workers in the field. The SI system is preferentially adopted in most technical journals and recent books on the subject. There are plausible reasons for preferring the CGS system, not least the avoidance of redundant fields in free space, but diffusion of knowledge on magnetism and magnetic materials will certainly benefit from generalized adoption of the SI system. The topic of magnetic measurements is traditionally treated in textbooks as a branch of electrical measurement and the peculiar role of the materials and their physical properties are seldom emphasized. xiii

xiv

Preface

Textbooks on magnetic materials typically devote a chapter to experimental methods, but they obviously follow a concise approach to this matter, which is seen as a corollary to the treatment of physical topics. No modern treatise devoted to magnetic measurements and characterization of magnetic materials is therefore available nowadays. The standard text in the field is the two-volume book by H. Zijlstra Experimental Methods in Magnetism (North-Holland), which was published in 1967. Since the publication of that work there have been many changes such as the discovery of novel compositions and properties and the improved phenomenological understanding of the behavior of the materials. In addition, the digital revolution has brought about widespread changes in the way that measurements are taken both in research laboratories and in industry. Never has there been a greater need for a book that summarizes the principles and the present state of the art in the field of magnetic measurements. This book fills that need whilst bearing in mind materials scientists, the practical impact on everyday test activity, quality control in the laboratory and the education of scientists engaged in the basic characterization of materials. This is a consistent book drawn from the author's own long experience in the lab. It looks at measuring problems from a practical viewpoint and, by placing the treated topics within a clear physical f r a m e w o r k it will be useful both to those approaching the subject for the first time as well as to experienced researchers. It is intended for technicians in the lab and materials scientists in industry, university, and research centers. It aims at answering the basic questions and dilemmas people engaged in this field are faced with, enabling the reader to find straightforward answers without tiresome recourse to scattered literature. The various aspects of standardization of measurements are illustrated and constantly referred to. This goes hand in hand with a discussion on the metrological issues, which include intercomparison, traceability, and measuring uncertainty problems. The book is organized in three parts and 10 chapters. Part I is made of three introductive chapters. Chapter 1 illustrates the general physical concepts and introduces the quantities constantly referred to along the treatise. Chapter 2 consists in a synthetic presentation of soft magnetic materials and includes a description of the preparation methods and a discussion on their physical properties. Chapter 3 is focused on the operation of permanent magnets, the related energetic aspects, and the classical electrical analogy of the magnetic circuits. No attempt is made to delve into the specific physical properties of permanent magnets. Contrary to the case of soft magnets, where scant recent review literature exists, the reader can easily retrieve information

Preface

xv

on the physics of permanent magnets in a good number of comprehensive up-to-date books. Part II is devoted to the discussion on generation and measurement of magnetic fields, a necessary step in any characterization process, but one which also has value in different contexts, including environmental studies and medical applications. Generation techniques are presented in Chapter 4. Distinction is made there between coil-based sources (DC, AC, and pulsed fields) and generation by means of permanent magnets and electromagnets. It is stressed how the field generating capabilities of permanent magnet based sources can be strongly enhanced with the use of extra-hard rare-earth based compositions. Chapter 5 provides a comprehensive review of the physical principles exploited in the measurement of magnetic fields and of the solutions adopted in actual measuring devices. It is stressed that the basic problem of precise absolute measurement and traceability to the base SI units can be solved by use of quantum resonance magnetometers, where the determination of the field strength is reduced to a frequency measurement. The characterization of magnetic materials is discussed in Part III. After a preliminary introduction on general measuring problems and methodologies (Chapter 6), theory and practice in the measurement of the properties of soft and hard magnets are treated. Reference is made, whenever appropriate, to written measuring standards (e.g. IEC, ASTM, JIS standards). The discussion on the characterization of soft magnets is carried out by separately discussing the measurements under DC, low-frequency, medium-to-high frequency and radio frequency excitation (Chapter 7). In hard magnetic materials, distinction is made instead between closed magnetic circuit testing, where electromagnets are used at the same time as field sources and soft return paths for the magnetic flux, and open sample testing (Chapter 8). The latter methods often combine versatility with measuring sensitivity and are nowadays increasingly applied in the characterization of permanent magnets, besides being the natural choice for thin films and weak magnets. After a discussion in Chapter 9 regarding the measurement of intrinsic material parameters (Curie temperature, saturation magnetization, and magnetic anisotropy), Chapter 10 examines the very often neglected topic of measuring uncertainty and its crucial relationship with the metrological issues raised by intercomparisons and traceability to the relevant base and derived SI units. Specific examples regarding magnetic measurements are provided. The SI system of units has been adopted throughout the whole text. When data and graph scales expressed in CGS had to be taken from

xvi

Preface

the literature, the appropriate conversion to SI was made. Because of its persisting use, the measure of the magnetic moment has been provided also in e.m.u., in association with the corresponding SI unit (A m2). Conversion rules for translating SI equations in Gaussian equations and vice versa and a comprehensive conversion table are given in Appendix A. Whilst comprehensive this book is not meant to be exhaustive. A major part of it is devoted to methods for the determination of material properties having relevant interest for applications, i.e., the parameters associated with magnetic hysteresis. In this respect, the measuring written standards are constantly referred to guidelines. Magneto-optical, magnetostrictive, and superconductive effects are among the topics not discussed here. If the reader wishes to explore these further they can be found in many recent textbooks. For example, magnetostriction measurements are described to full extent in the E. du Tr6molet de Lacheisserie book Magnetostriction (CRC Press, 1993). Magneto-optical methods and phenomena are exhaustively discussed in the outstanding work, Magnetic Domains by A. Hubert and R. Sch/ifer (Springer, 1998).

Acknowledgments In preparing this book, I received contributions, suggestions, encouragement, and help from many friends and colleagues. I am indebted to all of them. I am especially grateful to Giorgio Bertotti, who assisted me in many ways, willingly engaging in many clarifying discussions, and allowing me to benefit from his deep knowledge of electromagnetism and magnetic phenomena. The series editor, Isaak Mayergoyz, fully supported my effort, fostering my confidence in the project and generously handling my outrageously delayed delivery of the manuscript. I would like to acknowledge that this project could only be pursued thanks to the special cooperating milieu, the broad expertise on the physics of magnetic materials, and the array of experimental researches developed by fellow scientists at the Materials Department of IEN. I found advice and support in all of them. I am also indebted to elder scientists in my lab who educated me in the early years of my careen The late Andrea Ferro Milone introduced me to materials science and Piero Mazzetti taught me the basic virtues of the experimental physicist. Aldo Stantero helped me in many ways and on innumerable occasions for more than twenty years. His untimely death was an untold loss to me and to the lab at IEN. Giorgio Bertotti, Vittorio Basso, Carlo Appino, and Alessandro Magni read substantial parts of the manuscript and Oriano Bottauscio provided me with crucial help by expressly performing numerous electromagnetic field computations. The field maps presented in Chapters 4 and 8 are due to him. Vittorio Basso, Cinzia Beatrice, Enzo Ferrara, and Eros Patroi kindly supplied me with their own experimental data and Marco Co'isson clarified to me specific aspects of magnetoimpedance measurements. Anna Maria Rietto carried out careful experiments to elucidate a few important details in the magnetic lamination testing with the Epstein test frame and Luciano Rocchino assisted me in assessing the problems related to reference field sources and their traceability to the base units. Sigfrido Leschiutta enhanced my sensitivity to metrological issues and the role of metrology in the physical sciences. I also need to thank all the many colleagues in Europe and elsewhere with whom I shared cooperative research activity and discussions on various scientific x-vii

xviii

Acknowledgments

matters. It is finally a pleasure to acknowledge the help provided by Christopher Greenwell and Sharon Brown at Elsevier, who assisted me in the various stages of the book production, and Lucia Bailo, Francesca Fia, and Emanuela Secinaro at the Publication Department of IEN, for their help in literature retrieval.

PART I

Properties of Magnetic Materials

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CHAPTER 1

Basic Phenomenology in Magnetic Materials

1.1 M A G N E T I Z E D

MEDIA

In September 1820 H. C. Oersted demonstrated that electrical currents and magnets displayed equivalent effects. In a matter of weeks, A. M. Amp6re, elaborating on Oersted's discovery and making his own experiments, boldly interpreted the magnetism of materials as electricity in motion, i.e. the result of hidden microscopic currents, circulating around "electrodynamic molecules". The pedestal of electromagnetism was built in those few weeks, to be crowned in less than 50 years by the towering achievement of Maxwell's equations. Nowadays, we know that these currents exist, but they are quantum-mechanical in nature. They naturally slip into the classical Maxwellian scenario through the concept of permanent magnetic moment and the useful intermixing of classical and quantum concepts in the description of their relationship with the electronic angular momenta. A material sample is fundamentally described, from the viewpoint of magnetic properties, as a collection of magnetic moments, resulting from the motion of the electrons. Classically, orbiting electrons generate microscopic currents and are endowed with a magnetic moment m = -(e/2me).L, if e and me are charge and mass of the electron, and L is the angular moment. Quantum mechanics makes the view of electronic magnetic moment physically consistent, besides providing the additional basic concept of magnetic moment associated with spin angular momentum. When writing the classical equations of electromagnetism and assigning a meaning to the value of the physical quantities involved, we look at the material as a continuum. This means that all atomic scale intricacies are lost. In particular, the internal currents of quantummechanical origin are retained as averages over elementary volumes AV, sufficiently small to be defined as local over the typical scale of the problem, but large enough with respect to the atomic scale. These currents

CHAPTER 1 Basic Phenomenology in Magnetic Materials (Amperian currents), resulting from electron trajectories at the atomic scale, do not convey any flow of charge across the body. Let us call jM(r) the associated current density. Because of its solenoidal character, jM(r) can be expressed as the curl of another vector function M(r) jM(r) = V X M(r).

(1.1)

Remarkably, it can be demonstrated that the vector function M(r) represents the magnetic moment per unit sample volume [1.1]. This quantity takes the name "magnetization". If the previous elementary volume contains a certain number of moment carriers, it is M ( r ) = ~imi/~V~where the summation runs over the moments contained in such a volume. Thanks to Eq. (1.1), we are in a position to describe the magnetic effects ensuing from steady external currents when media are involved. If such currents are made to circulate in the absence of media, the induction vector B (often called "B-field") is given by the Biot-Savart law, which is expressed in differential form as V x B =/~0je,

(1.2)

where je is the density of the supplied currents and ~ = 4Ir x 10 -7 N / A 2 is the magnetic constant (sometimes called permeability of vacuum). Analysis of the Biot-Savart law additionally shows that the B vector is solenoidal, i.e. it obeys the equation V.B = 0. It can be shown that this equation and Eq. (1.2) determine B uniquely for given je. We recall here that the operative definition of the vector B is provided by Lorentz's law, which describes the coupling of electrical and magnetic fields with electrical charges. It states that a charge q moving at velocity t is subjected to a force F = q(E + t x B).

(1.3)

In the presence of magnetic media, both the externally supplied currents and the internal microscopic averaged currents will contribute to the B-field. The two base equations will be V.B -- 0,

V x B --/~0(je + jM).

(1.4)

When dealing with experiments on magnetic materials and their applications, we endeavor to drive the magnetic state of the material by means of external currents, i.e. acting on the quantity je- We can single out je in Eq. (1.4), introducing through Eq. (1.1) the magnetization M, a quantity directly accessible to experiments, in place of the awkward internal currents jM. In this way we define the so-called H-field Vx

~- -M

=VXH--je.

(1.5)

1.1 MAGNETIZED MEDIA

5

H is the quantity conventionally defined as the magnetic field as it is the quantity susceptible to direct control by means of the external currents. On the other hand, B appears to be the ffmdamental field vector because it is characterized by the condition V.B = 0 everywhere, in the free space and inside the matter, and Lorentz's equation everywhere applies to it. According to Eq. (1.5), the general relationship connecting the vectors B, H, and M is B = p,oH + / z o M .

(1.6)

In the SI unit system the magnetization M is expressed, like the magnetic field, in A / m , putting in evidence the Amperian origin of the magnetic moment. In the absence of media, M = 0 and B = / ~ H , i.e. magnetic field and induction (i.e. H-field and B-field) are equivalent quantities, as they are related by the proportionality constant/z0. In many kinds of experiments, we exploit the Faraday-Maxwell law V x E = - O B / O t , where E is the electric field, in order to determine.the magnetic behavior of the material. We detect in this case the electromotive force generated in a linked search coil by the time variation of the induction. Normally, we wish to get rid of the term /z0H in Eq. (1.6), because we are only interested in the contribution/z0M deriving from the material. This contribution is called magnetic polarization, J =/z0M, a quantity having the same dimensions as B (tesla, T) and the same properties as M. We then write B =/z0H + J. This general relationship will be specialized to the magnetic properties of the investigated medium by means of some constitutive equation J(H) or B(H). In ferromagnetic materials, these relationships can be very complex and very difficult to predict. In some well-defined instances, it is meaningful to define the relationships B = / z H =/Zr/z0H and M -- xH, where/z is the permeability, /d, r is the relative permeability, and X is the susceptibility. The quantifies/zr and X are related by the equation ~r

m

1 + X.

(1.7)

Note that, in the old Gaussian system, the base Eq. (1.6) is written as B = H + 4rrM,

(1.8)

i.e. field, induction, and magnetization have all the same dimensions (though different names, oersted (Oe) for H and gauss (G) for 4rrM and B). In this book, the SI system will be used throughout and little reference will be made to the increasingly obsolete Gaussian units. A complete set of conversion formulae is nevertheless provided, together with a discussion on their logical foundation, in Appendix A.

CHAPTER 1 Basic Phenomenology in Magnetic Materials Ferromagnetic materials are characterized by hysteresis. A residual magnetization M is always left when the external field, associated with the presence of a current density je, is completely released having once attained a certain peak value. We can say that some internal mechanism, associated with the nature and structure of the material, preserves a non-zero curl of the microscopic average current jM (Eq. (1.1)). With je -- 0, we obtain from the previous Eqs. (1.4) and (1.5), that induction and H-field are described by the equations V.B = 0~

(1.9)

V X B -- /J~)jM

V.H = -V.M,

V x H = 0.

(1.10)

Equation (1.9) simply states that the B-field is now uniquely generated by the internal currents and it preserves its solenoidal character. The two equations (Eq. (1.10)) are instead formally equivalent to the equations for the electrostatic field V.E--p/8o and V x E = 0, with p the electric charge density and ~0 the electric constant. Thus, in the absence of external currents, the field H, whose divergence can by analogy be written as V.H - PM, with PM -- -V.M, can be considered as the gradient of a scalar magnetic potential H = - V ~ M. This potential satisfies the Poisson's equation V2(I)M----PM"The electrostatic analogy then permits us to introduce, in a purely fictitious way, magnetic charges of volume density PM acting as sources of the field H, whenever it occurs that V.M # 0. Although devoid of physical reality, the concept of magnetic charges is constantly applied in the investigation of magnetic materials and in magnetic measurements because of the simplifications it introduces in the description of many phenomena and in the calculations. It permits one, for example, to derive fields from scalar potential functions, which are solutions of Poisson's equation 1 ~ V.M(f) d3 f ~M(r)

=

- - 4---~

Ir -

el

'

(1.11)

thereby applying the conventional methods of electrostatics. Figure 1.1 provides a classical example where the role of magnetic charges can be invoked. It is the case of a cylindrical permanent magnet, where the magnetization M is uniform and axially directed. Since M suffers a discontinuity at the sample ends, the conditions are created for quasisingular behavior of the divergence V.M. It turns out that the potential function can be written as

1 ~ CrM(lJ) d2f, (I)M(r)-- ~ A Ir- rrl

(1.12)

1.1 MAGNETIZED MEDIA

I

(a)

7

lt[/f

(b)

(c)

FIGURE 1.1 Induction B and field H in a cylindrical permanent magnet in the absence of an external applied field (je = 0). It is assumed that the sample remanent magnetization M is uniform. The induction B =/~0H +/z0M is solenoidal (V.B = 0) and the field H satisfies the condition V x H = 0. This means that H can be expressed, in formal analogy with the electrostatic field, as the gradient of a scalar potential. In this respect, it is as if fictitious magnetic charges of equal densities and opposite signs were uniformly distributed over the top and bottom surfaces of the cylinder.

where the integration is performed over the total area A of the top and bottom surfaces. The quantity crM(1~) = n.M(r~), where n is the unit vector normal to these surfaces, plays the role of surface magnetic charge density. The correspondingly calculated magnetostatic field H(r) and the induction B(r) =/z0H(r) + / z 0 M are schematically s h o w n in Fig. 1.1b,c. Notice that within the sample, H(r) is directed in such a w a y as to oppose the magnetization. It is for this reason called a "demagnetizing field". If the magnetization is not uniform or the material is inhomogeneous, internal demagnetizing fields can also arise. In the free space, B(r) and H(r) (which takes the n a m e stray field) coincide (but for the proportionality factor/z0). Note further that the condition V.B = 0 implies that, on traversing the sample surface, the normal c o m p o n e n t B.n is preserved. The condition V x H - 0, however, implies that the same occurs to the tangential c o m p o n e n t of H.

CHAPTER 1 BasicPhenomenology in Magnetic Materials A magnet brought under the permanent condition shown in Fig. 1.1 is endowed with a certain magnetostatic energy content. Part of this energy is contained within the sample and part is associated with the stray field. Under very general terms, we can write the total energy as (1.13)

Et = -~ P,o H2dV,

where we have defined as H d the demagnetizing field and the integration extends all over the space. This is the energy that must be spent for the formation of the magnetic charges and it can be equivalently written as Et - - - ~ - / ~ v

(1.14)

Hd.M dV,

where integration is made over the sample volume.

1.2 D E M A G N E T I Z I N G

FIELDS

Demagnetizing effects are ubiquitous. Even in accurately closed specimens, (e.g. ring samples), one cannot get rid of them completely. This has fundamental consequences from the point of view of magnetic characterization and it requires measuring strategies aimed at minimizing and/or precisely controlling the demagnetizing fields. We shall discuss and clarify practical methods devised to this purpose, both in soft and hard magnets, in later chapters. In this section, we shall briefly discuss the basic problems connected with the prediction of the demagnetizing fields under different sample geometries. Calculations of demagnetizing fields date back to the 19th century. They were pursued by, among others, Maxwell [1.2], Lord Rayleigh [1.3], and Ewing [1.4]. One chief problem at that time involved a ship's magnetism and the correction to be made on the apparent declination of the magnetic compass to determine a ship's position. It was recognized that only in samples shaped as ellipsoids (or spheres) could the demagnetizing field be homogeneous and susceptible of full mathematical treatment. The general approach consists in determining the volume and surface charge densities pM(r~) = -V.M(r') and O'M(I d) -~ n.M(r') of the uniformly magnetized body and in correspondingly expressing the potential 1 f ~M(r)-- ~

pM(IJ) d3rI + 1 ; V I r - r'l

~

crM(r/) d2r/ A I r - r'l

(1 15) '

"

1.2 DEMAGNETIZING FIELDS

9

from which the demagnetizing field can be derived as the gradient Hd(r) -- --VCI)M(r). With M constant in modulus and direction everywhere inside the spheroidal sample, the volume charge density pM(r') is zero and the surface charge density crM(r~) is easily calculated. If M lies along one of the principal axes (a, b, c), the corresponding homogeneous demagnetizing field is N~ H d ___ _ X d M

= _ --u j,

(1.16)

/z0 where the proportionality factor Nd is called "demagnetizing coefficient". For a generic direction, we have H d -- -[[Ndl[M, where [[Nd[[ is a tensor having only the diagonal elements different from zero. These are the demagnetizing coefficients along the three principal axes Nda,, Ndb, Ndc. They obey the constraint Nda + Ndb + Ndc = 1. In the general ellipsoid, a # b # c and the demagnetizing factor Nda is obtained as the integral N d a _~ abc

2

jo[

(a 2 q-

~)

;(

a 2 q- O ( b 2 if- ~)(c 2 q- ~)

d~'.

(1.17)

which can be numerically calculated, together with Ndb and Ndc = 1 -- Nda -- Ndb. Results are reported in the literature (see, for instance, Ref. [1.5]). Closed expressions are found for ellipsoids of revolution (Fig. 1.2). In the limiting case of a sphere, we have, for reasons of synm~etry, Nda = Ndb = Ndr = 1/3. For a prolate spheroid, where a = b and the rotational symmetry axis c > a,b, the demagnetizing factors are given, for the defined ratio r = c/a -- c/b > 1, by the expressions Ndc

1 [ ~Jy2__r 1 /r+ r2 1/ 1]

r2 - 1

1

(1.18)

Nda -----Ndb = -~(1 -- Ndc ). z

For r >> 1, the approximation Ndc = In 2 r - 1/r 2 holds. If the same axis c < a, b (oblate spheroid, r < 1), we have

1[

r

Ndc-- 1 - r 2 1 -

~arcsin(x/1 x/1-r 2

1

-r2)], (1.19)

Nda -- Ndb ----- ~ ( 1 -- Ndc ). Z

Tables and graphs reporting the value of the coefficient Ndc, calculated for r varying over many decades, have been published over many years and can be found in several textbooks (see, for example, Refs. [1.6, 1.7]). It is

10

CHAPTER 1 Basic Phenomenology in Magnetic Materials

(a)

(b) FIGURE 1.2 Prolate (a = be c > a, b) and oblate (a = b, c < as b) spheroids, c is the rotational symmetry axis. The demagnetizing field lid is always uniform in ellipsoids if the magnetization M is uniform, whatever its direction. The demagnetizing coefficients calculated along the three symmetry axes, which completely define the problem, are given by Eqs. (1.18) (prolate spheroid) and (1.19) (oblate spheroid). They depend only on the ratio r = c/a = c/b. apparent here that the demagnetizing field does not depend on the sample volume, but only on its geometrical properties (the ratio r). Demagnetizing fields can never be ignored in measurements. To recover the intrinsic magnetic properties of the material under test, a correction is required, where the effective field H = Ha - H d ~ obtained as the difference between the applied field H a and the demagnetizing field, is calculated. The problem is apparent with bulk soft magnets having a relatively low aspect ratio in the direction of magnetization, but also there is heavy interference by demagnetizing effects with strips and ribbons. In a typical experiment, a 20 ~m thick, 200 m m long, and 10 m m wide high permeability amorphous ribbon is tested. It is found by hysteresis loop shearing analysis that this sample has what could be deemed a very low demagnetizing factor (Nd "" 1.3 X 10-5), corresponding to a value r---700 in a prolate ellipsoid. For peak polarization value, Jp--0.8 T, the demagnetizing field is, according to Eq. (1.16), Hd -~ 8.5 A / m , which is about eight times larger than the effective field H. In hard magnets, it is fortunately possible in principle to perform accurate measurements with open samples also, thanks to the very high fields intrinsically required for their magnetization and demagnetization. However, permanent magnets generally come as bulk specimens

1.2 DEMAGNETIZINGFIELDS

11

(cylinders, parallelepipeds, spheres), which have high demagnetizing coefficients. It may also happen that some kinds of tapes or thin films preferentially magnetize normal to their plane, thereby approaching the value Nd = 1, and the correction procedure may become very complex. Ellipsoidal test samples are seldom employed. A notable exception is represented by some open sample methods applied in permanent magnet testing (for example, with the vibrating sample magnetometer; see Section 8.2.1), where small spherical specimens (diameter around few mm) are adopted. Use of cylindrical, parallelepipedic, or disk-shaped test specimens is generally the rule, but these geometries engender significant complications in the definition and determination of the demagnetizing coefficient. In fact, even with homogeneous magnetization, the demagnetizing field is not homogeneous. In addition, it is also dependent on the material permeability. In soft magnets, this brings about notable complications, because permeability is high and, even if we disregard hysteresis, corrections become difficult and inaccurate. Large errors are expected to occur when, as is often the case, Ha and Hd have very close values. The problem of predicting demagnetizing fields in non-ellipsoidal bodies can be attacked in principle by computation of the integrals in Eq. (1.15), a relatively complex and time-consuming approach. Threedimensional and two-dimensional finite element calculations have been developed for this purpose. They are indispensable for treating complex micromagnetic problems and the analysis of specific domain patterns [1.8, 1.9]. For regular cylindrical or parallelepipedic shapes and structurally and magnetically homogeneous materials (no domains and constant susceptibility value X everywhere), calculations of the demagnetizing coefficient have been performed to various degrees of approximation. For uniformly magnetized samples and X--0, analytical approaches have been carried out and reasonable approximations can often be given, although, in a strict sense, they apply only to diamagnets, paramagnets, and saturated ferromagnets. For example, a uniformly magnetized cylinder has magnetic charge density r at its top and bottom surfaces. The demagnetizing field at the center of the cylinder is straightforwardly calculated by integrating the field generated by an infinitesimally thick annulus of surface charges and summing up the contributions of the top and bottom ends (details are given in Section 4.4). For a cylinder with height-to-diameter ratio r, such a field turns out to be

H d -= - N d M

-- - ( 1 -

1 )M. ~/1+ 1 / r 2

(1.20)

12

CHAPTER 1 Basic Phenomenology in Magnetic Materials

For r = 1, this provides Nd = 0.293, as compared with Nd = 0.333 in a sphere. The demagnetizing field at the center of a prism with square cross-section and height-to-side length ratio r, uniformly magnetized along the axial direction, is similarly obtained as NclM = --_2r arcsin ( l +1r

Hd ._

) M.

(1.21)

We see in Fig. 1.3 how the associated demagnetizing factor compares with that of an inscribed spheroid as a function of the ratio r. In particular, it turns out that, for a cube ( r - 1) Nd -- 1/3, exactly the same as for the sphere. Magnetic materials are normally characterized using regularly shaped samples, where either the measurement of the magnetic flux upon a well-defined cross-section or the determination of the total magnetic m o m e n t of the test specimen is performed and related to the material magnetization. At the same time, the concept of effective field must be given a practical meaning. In connection with these two measuring approaches, one talks of fluxmetric and magnetometric methods. When dealing with non-ellipsoidal open samples, it is therefore expedient to distinguish between fluxmetric (or ballistic) and magnetometric

1.0 0.8 0

.~

0.6

9

0.4

0.2

0.0

.

0

.

.

.

!

1

.

.

.

.

|

2

.

.

.

.

|

3

.

.

.

.

!

4

.

.

.

.

5

r=cla=clb

FIGURE 1.3 Behavior of the demagnetizing factor Nd = H d / M , with Hd the field at the center, in a uniformly magnetized prismatic sample with square crosssection and X = 0 (dashed line, Eq. (1.21)). Comparison is made with the same quantity calculated for an inscribed spheroid.

1.2 DEMAGNETIZING FIELDS

13

demagnetizing factors. Let us consider, as in the earlier example, a cylindrical or prismatic sample. The material is homogeneous, does not show hysteresis, and the susceptibility is isotropic and constant everywhere. A qualitative idea of the non-homogeneity of the demagnetizing field in connection with homogeneous magnetization is provided as a sketch in this case by Fig. 1.1. An example of quantitative derivation of the dependence of the axial component of the magnetization and the demagnetizing field for different susceptibility values in a cylindrical sample is illustrated in Fig. 1.4 [1.10]. We thus define the fluxmetric demagnetizing factor as the ratio of the average demagnetizing field to the average magnetization over the midplane perpendicular to the sample axis (z-direction)

Hd~dA N(df) =

MzdA

(1.22)

For reasons of symmetry, both demagnetizing field and magnetization at the midplane are directed along the cylinder axis. The magnetometric demagnetizing factor is defined, however, as the ratio of the volumeaveraged demagnetizing field to the volume-averaged magnetization _

fvHdzdv

1.23

N(df) and N~m) depend on the height-to-diameter ratio r and the effective susceptibility X of the material. This is defined as X = <M~.)/H and can be expressed in terms of the directly measured apparent susceptibility X~ = <M~)/Ha and the demagnetizing factor N~ ) Xa X = 1 - N 'd'-'''xa

(1.24)

Chen et al. have carried out a comprehensive calculation of fluxmetric and magnetometric demagnetizing factors in cylindrical samples for a very large range of aspect ratio r and magnetic susceptibility X values. For the case X-- 0, these authors use Brown's method [1.11], where the cylindrical sample is assimilated to a solenoid bearing the mantle of surface Amperian currents associated, according to Eqs. (1.1) and (1.9), with uniform magnetization. For long cylinders (r > 10) and X ~ O, N(df) and N (m) are analytically calculated by means of a unidimensional model, where the axial magnetization component M z is assumed uniform over each

14

CHAPTER 1 Basic Phenomenology in Magnetic Materials

I

'

O

~

x

"

= 10_3

0.8

=

0.6 0.4 0.2

z

T

0.0 0.0

(a)

0.2

0.4

0.6

0.8

1.0

z/h

0.0 -0.2,

-..• -0.4, t,4

-0.6 -0.8 -1.0

f

/

=10 5

0.0""" 012. . . . 014"'" 016. . . . 018. . . . 1.0

(b)

z/h

FIGURE 1.4 Behavior of magnetization and demagnetizing fields along the axis of a slender cylindrical sample (height-to-diameter ratio r = h/a = 20). Calculations have been made for different values of the effective susceptibility )6 With high X values, the magnetization tends to vanish at the sample ends and the demagnetizing field Hdz nearly completely compensates the applied field Ha (adapted from Ref. [1.10]).

cross-section of the cylinder. The model takes into account the presence of free poles both at the top and bottom ends and on the lateral surface of the sample. Figure 1.5b provides the results obtained with this m o d e l for X - 0, X - 100, and X - oo with r ranging between 10 and 1000. For short cylinders, where the previous assumption of tmiform Mz is untenable, a two-dimensional model, where the lateral surface of the

1.2 DEMAGNETIZING FIELDS 9 9

: :

; :

; :

: :

15 ; ;:;, ::::,

: :

; :

: :

: :

; ;;:i ::::,

:

: : ........

:

::::i

'~!:~':!:!:~.i:~:.:i:~~~i ::i~~: .~:i:!~:~~!!~i:~! i~: !:~:~~:~:~:!!~.~i~:!~:~!~:!~:~ .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

:

.

.

.

.

.

. . . . . . . .

. . . . . . . .

0 0

rN e-

0., "

iii'

E

9 .--i2~.-;i::!:i!:i "i ~ ....~...i..~,..i.:.~] ...... : ~ .:. ~.:..,::

..... iiiii} .........! .:i

0,01

........

!!!!~:--:i.....~---~-::i~i!i:i!i!,~!~!!!!i~!!!!!!~!!!~i!!i ~! .. ~0~ero,; 9

:

. . . . . . . .

0.01

(a)

.. i 0.01

L

:

0.1

:.'~.- .~.,,..

.! !:

. . . . . . .

i

r=h/a

~! i i~ .... v..,i

.

.

.

.

.

.

.

.

.

1

9

10

i....!-

~-.iii.!!

~

1E-3

9o .E

1E-4

I

"

'

~

"ILl

x~

c-

E

1E-5 9

(b)

10

:i.

i

:: '

100 r=h/a

:

.

i :

i.

'i:.

i

1000

FIGURE 1.5 Magnetometric N(dm) and fluxmetric N~f) demagnetizing factors calculated for cylindrical samples as a function of the height-to-diameter ratio r for different values of the susceptibility (adapted from Ref. [1.10]).

16

CHAPTER 1 BasicPhenomenology in Magnetic Materials

cylinder is subdivided into a suitable set of non-overlapping rings of uniform charge density, is numerically implemented. Selected behaviors of N(df) and N(dm) obtained with this model for 0.01 _< r _< 10 are shown, in comparison with the behavior of Nd in a spheroid, in Fig. 1.5a. The following general features of the predicted properties of the demagnetizing coefficients should be stressed: (1) For a given X value, both N~) and N(dm) decrease in slender samples. This is the obvious consequence of the crowding of the free poles at the sample ends. (2) For the same reason, it is always N(dm)> N~). (3) Whatever the value of r, N(dm) always decreases with increasing X- In contrast N(df) increases with // ' 7~T(f) increasing X if r > 1. The opposite occurs if r < 1.4. The ratio 7~ 9~T(m) d ~d increases with increasing r, for given X, and decreases with increasing X, for given r. The subject of analytical determination of N(df) and N~dm) in prisms of rectangular cross-section has been treated by Aharoni, who provided expressions for both of them under the assumption of saturated sample (X=0) [1.12, 1.13]. The results were applied to experiments on 2.35 ~m x 95 I~m x 250 p~m Ni80Fe20 thin films with easy axis directed along the major side. It was verified that a minimum applied field equal to the calculated fluxmetric demagnetizing field H d = -N(df)Ms (M~, saturation magnetization) was required to reverse the magnetization in the central part of the sample.

1.3 M A G N E T I Z A T I O N

PROCESS AND HYSTERESIS

We primarily identify the expression "magnetic characterization" with the experimental process by which we determine the constitutive law of a magnetic material. This is the functional dependence M(H) of the magnetization on the effective field or, according to Eq. (1.6), the equivalent laws J(H) and B(H). This book will be chiefly devoted to reviewing and discussing the measuring methods by which such laws can be experimentally derived. Diamagnetic and paramagnetic materials are described by simple single valued relationships between field and magnetization. This by no means implies that such relationships are easily accessible to experiments because the associated susceptibilities are very low and, consequently, it is difficult to discriminate between the response of the sample under test and that of the background. Ferromagnets (and ferrimagnets) display intrinsically large responses. The landmark property of these materials is that they are endowed with a molecular field of quantum-mechanical origin, which implies magnetic ordering, i.e. possible extraordinarily high values of the magnetic

1.3 MAGNETIZATION PROCESS AND HYSTERESIS

17

moment per unit volume. However, the phenomenology of the magnetization process in these materials has the traits of complexity, as embodied in the manifold manifestations of hysteresis. This appears as the macroscopic outcome of an intricate combination of microscopic processes, centered on the existence of domains (or, in limiting cases, single-domain particles), which lead to collective rearrangements of the magnetic moments under a changing applied magnetic field. The experimental investigation of magnetic hysteresis and the general testing of materials require that some kind of accessible reference state is defined. Magnetic saturation and demagnetized state are two such reference conditions. All trajectories in the H - M (or H-J) plane converge when reaching the saturated state. On recoiling from it, the system follows a unique trajectory. The demagnetized state acquires a similar distinctive character when the condition H - - 0 and M = 0 is attained. To this end, the sample is either brought beyond the Curie temperature and cooled in a zero-field environment or an alternating field of progressively and finely decreasing peak amplitude is applied, starting from saturation and ending at zero value. The demagnetized state should realize, from the microscopic viewpoint, that special arrangement of the internal magnetic structure corresponding to the condition of absolute minimum of the magnetic free energy. Indeed, an infinite number of trajectories can lead to sample demagnetization without leading to the demagnetized state. If, starting from such a state, the applied field amplitude is increased, a curve in the (H-J) or, equivalently, (H-M) plane is described. It is the initial magnetization curve. In current practice, thermal demagnetization is seldom adopted. When this is the case, it is not expected to lead to the same set of microscopic states obtained with the conventional demagnetization under an alternating field. The curve described after thermal demagnetization takes the name virgin curve and it can be slightly different from the initial curve. Figure 1.6 provides examples of initial magnetization curves in soft and hard magnets with both the q - H ) and (B-H) representations shown. These curves have been obtained using a closed magnetic circuit, where the applied field Ha ~-H. If open samples are tested or the flux closure is imperfect, Ha = H + (Nd/t2,o)J and the curves (J-Ha) and (B-Ha) will appear sheared with respect to the intrinsic curves (J-H) and (B-H). We see that in practical soft magnets there is a detectable difference between these two representations only on approaching the saturated state, where the term/z0H can be appreciated (Eq. (1.6)). In order to bring the material along the magnetization curve, we must spend energy. Let us assume that a field, slowly increasing with time, is applied to a sample forming a closed magnetic circuit

18

CHAPTER 1 Basic Phenomenology in Magnetic Materials

B(H) ~

/

"....,,,

2.0

/

4"

1.5 r

1.0

0.5

0.0 (a)

_. J i

i ~

100

i

Non-oriented Fe-Si i i iIinl

101

I

i i iiiiii

102

I

I i lllul

103

I

I I llllll

104

I I I II

10 5

H(A/m)

.~ /

s

"s

i

(Hp, Bp)

1

,,,

2

/

c~

"

1-

0

(b)

f

B(H) J(H)

,

/)

500

1000 H(Nm)

1500

FIGURE 1.6 Initial magnetization curves in non-oriented Fe-Si laminations and in a N d - F e - D y - A 1 - B sintered magnet. The soft magnetic laminations have been tested as strips forming a closed magnetic circuit within an Epstein frame. The permanent magnet has been demagnetized and magnetized as a parallelepipedic sample inserted between the pole faces of Ban electromagnet. The shaded area provides the energy per unit volume E = ~0p HdB to be spent for bringing the material from the demagnetized state to the peak induction value Bp.

1.3 MAGNETIZATION PROCESS AND HYSTERESIS

19

(e.g. a ring specimen) by use of a suitably linked winding supplied with a current i(t). At any instant of time, the supplied voltage is balanced by the resistive voltage drop of the winding Rwi(t) and the induced e.m.f, d ~ / d t UG(t) -- Rwi(t) + d~/dt.

(1.25)

Starting from the demagnetized state, a certain final state with induction value Bp is reached after a time interval to. The correspondingly supplied energy E = ~0~uG(t)i(t)dt is partly dissipated by Joule heating in the conductor and partly delivered to the magnetic system

E=

Rwia(t)dt 4-

X:

NwAi(t)-dTdt

(1.26)

where Nw is the number of turns of the winding and A is the crosssectional area of the sample. We are interested in the second term on the right-hand side of this equation. If, to simplify the matter, we consider a ring sample of average circumferential length lm, we can write i(t)= (lm/Nw)H(t) and we find that the energy delivered by the field-supplying external system in bringing the magnet to the final state is

dB dt = V fSp HdB. U - V f~ H(t) --~

(1.27)

0

If we refer to the graphical representation of the initial magnetization curve in Fig. 1.6, we conclude that the energy per unit volume to be supplied in order to reach the induction value Bp (or, equivalently, the polarization value Jp), is given by the area between the B(H) curve and the ordinate axis. Introducing Eq. (1.6) in Eq. (1.27), we obtain

U= V

ia,oHdH + V

HdJ,

(1.28)

and we can distinguish between the energy stored in the magnetic field and the energy stored in the material. Part of the latter is expected to be lost during the process. Except in somewhat unusual cases, the magnetization process is always associated with measurable energy dissipation. This is mirrored in hysteresis, the phenomenon of output lagging behind input. Figure 1.7 provides two examples of hysteresis loops in the previous soft and hard magnets, as obtained by cycling the field between two symmetric peak values +Hp. If the integration in

20

CHAPTER 1 Basic Phenomenology in Magnetic Materials 1.5

Jr, er 1.0 0.5 0.0 -0.5

-1.5

Non-oriented Fe-Si

m i

-1.0

I

.

.

.

.

-1000

I

.

.

.

-500

.

.

.

.

.

I,

0

(a)

,,,~

,

500

,

I

1000

H(Nm) Nd-Fe-Dy-AI-B

Br, Jr

""4HI

-2 -3 "''

I

. . . . .

-1500

(b)

I

. . . . .

I

. . . . . . . . . .

-1000 -500

~ ~'""

0 500 H(Nm)

'"1

. . . . .

1000

I ' ' "

1500

FIGURE 1.7 Examples of hysteresis loops in soft and hard magnets. In the soft Fe-Si laminations, there is no detectable difference between the B(H) and J(H) curves for magnetizations and fields of technical interest. The difference is instead apparent in permanent magnets. It leads to two different definitions of coercive field: HcB is the field required to bring the induction to zero value starting from the saturated state and Hcj is required to reduce to zero the polarization J (i.e. magnetization M). It is always Hcl > HcB (courtesy of E. Patroi).

1.3 MAGNETIZATION PROCESS AND HYSTERESIS

21

Eq. (1.28) is carried out over a full cycle, the energy balance is obtained

W= ylzoHdH4- yHdJ= yHdJ= ~HdB.

(1.29)

The quantity W is the energy lost per unit volume in the sample. In fact, the purely reactive term ~ la,oHdH averages out to zero over a cycle. It then turns out that the area of the B(H) loop is equal to the area of the J(H) loop. Notice in the loops represented in Fig. 1.7 the remanence point, where Jr = Br, and the distinction existing between the coercive fields HcB and Hcj, the latter being the field value where the material is demagnetized (indeed, a demagnetized state far removed from the one previously discussed). The phenomenology of magnetic hysteresis is extremely complex but endowed with a certain mathematical regularity, which has attracted relevant modeling efforts starting from the milestone approach of Preisach [1.14]. A full account of the mathematical aspects of hysteresis is found in Ref. [1.15], while the physics of magnetic hysteresis is treated with deep insight in Ref. [1.16]. We only remark here that, if we limit ourselves to material testing at low field strengths, we can fully describe initial curve and symmetric hysteresis loops by means of a defined function, the Rayleigh Law. Its discovery was a triumph of physical intuition and experimental skill in measurements [1.17], which is paralleled by the physical explanation advanced by N6el [1.18], based on the statistical description of the reversible and irreversible displacements of the domain walls. An example of loops determined in the Rayleigh region is reported in Fig. 1.8, where it is observed that the material polarization follows a quadratic dependence on the magnetic field. In particular, any hysteresis loop determined between the peak field values +Hp has ascending and descending branches described by the equations

b (Hp - H2), I(H) = (a + bHp)H -T- -~

(1.30)

where a and b are called reversible and irreversible Rayleigh constants. The tip points of the loops (Hp, Jp) describe the initial magnetization curve (also called normal magnetization curve), and follow the law

Jp = aHp 4- bH2.

(1.31)

In the limit of very low fields, the magnetization curve becomes linear (as is apparent in Fig. 1.8) with the constant a proportional to the initial

22

CHAPTER 1 Basic Phenomenology in Magnetic Materials (Hp, J p ) ~ . .

Pure Ni foil

0.06

/

5/

0.040.02-

aH

o.oo -0.02

J

-0.04 -0.06 " '

I

'

-300

'

' "

I

"

-200

'

'

'

I

.

-100

.

.

.

.

.

.

.

I

. . . .

0 100 H(A/m)

I

. . . .

200

I

r ,

300

FIGURE 1.8 Experimental hysteresis loops in the Rayleigh region. The ascending and descending branches of the loops and, a fortiori, the initial magnetization curve connecting the tip points of the loops, follow the quadratic law (1.30). The energy per unit volume dissipated upon completing a full cycle is given by the equation W = -~bH~, where b is a structure dependent coefficient and Hp is the peak field value (courtesy of C. Beatrice). susceptibility Xi lim Jp Sp--.0 Hpp "-- [d'OXi"

a--

(1.32)

Integration of the loop area provides the hysteresis loss per unit volume

W=~4 blip,

(1.33)

where, as expected, the reversible coefficient a has disappeared. Equivalently, we can express the energy loss for a given peak polarization value Jp in the Rayleigh domain as

1[

W = -~

-a +

2 + 4blp

]3 .

(1.34)

Remarkably, N6el's theory predicts that the quantities aHc and bile~a, where Hc is the coercive field, as obtained with a major loop, are independent of the structure of the material. This conclusion derives from

1.3 MAGNETIZATIONPROCESS AND HYSTERESIS

23

the assumption of scale invariance of the equations describing the interaction of the Bloch walls with the pinning defects. Our discussion has so far highlighted some essential facts concerning the so-called "quasi-static" magnetic behavior of the materials, which is observed when the rate of change of the magnetization is so low that dynamic viscous-type effects do not interfere with the magnetization process. When this condition is no longer fulfilled, we observe ratedependent hysteresis effects. In metallic materials, they are almost exclusively due to eddy currents, whose circulation in the sample takes patterns depending on, besides the magnetization rate, the material resistivity, sample geometry, and domain structure. We shall consider dynamic loss behavior in connection with the properties of soft magnetic materials (Chapter 2) and the related measurements (Chapter 7). We only remark here that the fundamental consequence of the viscous effects associated with long-range eddy currents is the increase of the energy dissipated in any cycle, as manifest in the observed broadening of the hysteresis loop with increasing magnetizing frequency (Fig. 1.9).

'

0.2

Co71Fe4B15Silo '

'

'

I

'

'

'

'

I

.

.

.

.

amorphousribbon

0.1

f

o.o

//~~_,flO0k5~150 kHz

-0.1

300kHz

-0.2 .

-60

1 kHzi

/

.

.

|

I

-40

,

,

,

,

I

-20

.

.

.

.

|

0

,

H(Nm)

|

,

I

20

.

,

.

.

I

40

,

,

,

,

60

FIGURE 1.9 Broadening of the hysteresis loops with increasing magnetizing frequency in a soft magnetic alloy. The measurements refer to an amorphous ribbon (thickness 20 ~m), wound into a many layer ring sample and tested under controlled sinusoidal induction waveform.

24

CHAPTER 1 Basic Phenomenology in Magnetic Materials

The whole phenomenology can be assessed in most cases by applying the concept of loss separation [1.16].

aefeyences 1.1. L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media (Oxford: Pergamon Press, 1989). 1.2. J.C. Maxwell, A Treatise on Electricity and Magnetism (London: Clarendon, 1891, 3rd edition. Reprinted by Dover, New York, 1954), Vol. 2, p. 59. 1.3. J.W. Strutt (Lord Rayleigh), "Notes on magnetism. On the energy of magnetized iron," Philos. Mag., 22 (1886), 175-183. 1.4. J.A. Ewing, "Experimental researches in magnetism," Philos. Trans. Roy. Soc. London, 176 (1885), 523-640. 1.5. A. Hubert and R. Sch/ifer, Magnetic Domains (Berlin: Springer, 1998), p. 184. 1.6. B.D. Cullity, Introduction to Magnetic Materials (Reading, MA: AddisonWesley, 1972), p. 58. 1.7. D. Jiles, Introduction to Magnetism and Magnetic Materials (London: Chapman & Hall, 1991), p. 39. 1.8. S.W.Yuan and H.N. Bertram, "Fast adaptive algorithms for micromagnetics," IEEE. Trans. Magn., 28 (1992), 2031-2036. 1.9. D.V. Berkov, K. Ramst6ck, and A. Hubert, "Solving micromagnetic problemsmtowards an optimal numerical method," Phys. Status Solidi, A-137 (1993), 207-225. 1.10. D.X. Chen, J.A. Brug, and R.B. Goldfarb, "Demagnetizing factors for cylinders," IEEE. Trans. Magn., 27 (1991), 3601-3619. 1.11. W.F. Brown, "Single domain particles: new uses of old theorems," Am. I. Phys., 28 (1960), 542-551. 1.12. A. Aharoni, "Demagnetizing factors for rectangular ferromagnetic prisms," J. Appl. Phys., 83 (1998), 3432-3434. 1.13. A. Aharoni, L. Pust, and M. Kief, "Comparing theoretical demagnetizing factors with the observed saturation process in rectangular shields," J. Appl. Phys., 87 (2000), 6564-6566. 1.14. F. Preisach, "Uber die magnetische Nachwirkung," Z. Phys., 94 (1935), 277-302. 1.15. I.D. Mayergoyz, Mathematical Models of Hysteresis (New York: SpringerVerlag, 1991). 1.16. G. Bertotti, Hysteresis in Magnetism, (San Diego: Academic Press, 1998). 1.17. J.W. Strutt (Lord Rayleigh), "On the behaviour of iron and steel under the operation of feeble magnetic forces," Philos. Mag., 23 (1887), 225-245. 1.18. L. N6el, "Th6orie des lois d'aimantation de Lord Rayleigh. I. Les d6placements d'une paroi isol6e," Cahiers de Physique, 12 (1942), 1-20.

CHAPTER 2

Soft Magnetic Materials

A magnetic material is considered "soft" when its coercivity is of the order of, or lower than, the earth's magnetic field. A soft magnetic material (SMM) can be employed as an efficient flux multiplier in a large variety of devices, including transformers, generators, and motors, to be used in the generation and distribution of electrical energy; and in a wide array of apparatus, from household appliances to scientific equipment. With a market around s billion/year, SMMs are today an ever more important industrial product, offering challenging issues in properties understanding, preparation and characterization. SMMs have been at the core of the development of the early industrial applications of electricity. Steel production was sufficiently developed at the turn of the century to satisfy the increasing need of mild steel for the electrical machine cores. In 1900, Hadfield, Barrett, and Brown proved that, by adding around 2% in weight Si to the conventional magnetic steels, one could achieve an increase of permeability and a decrease of energy losses [2.1]. Fe-Si alloys were more expensive and more difficult to produce and gained slow acceptance. In addition, the poor control of the C content was to mask the prospective performance of this product compared with mild steels. It took more than two decades, characterized by a gradual improvement of the metallurgical processes, for Fe-Si to become the material of choice for transformers. An empirical attitude towards research in magnetic materials was prevalent at the time and applications came well before theoretical understanding. This was the case for the Goss process, developed in the early 1930s, by which the first grain-oriented Fe-Si laminations could be produced industrially [2.2]. In the years 1915-1923, Elmen and co-workers at the Bell Telephone Laboratories systematically investigated alloys made of Fe and Ni, discovering the excellent properties of the extra-soft permalloys (78% Ni) [2.3]. Snoek is credited for the successful industrial development of ferrites in the 1940s [2.4], following attempts dating back to the first decade of the century. The discovery in 1967 of the soft magnetic amorphous alloys again occurred nearly by chance [2.5], but it provided a fertile field for technologists and 25

26

CHAPTER 2 SoftMagnetic Materials

theorists. The discovery enriched the landscape of applicative magnetic materials, while straining existing theories on magnetic ordering.

2.1 GENERAL PROPERTIES The rough attribution of magnetic softness, based on the value of coercivity, is completed and made useful by the q,H) hysteresis loop. Figure 2.1 provides a comparison of major DC hysteresis loops in a number of representative soft magnetic alloys. All measurements have been performed under closed-flux conditions. The Ni-Fe (Mumetal type) and the Co-based amorphous alloys reach the highest values of permeability and the lowest coercivities, but their saturation magnetization is somewhat reduced with respect to the Fe-Si and Fe-based amorphous alloys. The ferrites do not display prominent soft DC properties but, being nonconductive, become the best choice at frequencies in the MHz range. The actual and the prospective applications of magnetic materials have thus to be evaluated against a number of parameters, such as initial and peak permeabilities, coercive field, remanence, AC energy losses, squareness ratio, etc., which are the result of both compositional and structural properties. The composition determines the values of the so-called intrinsic magnetic parameters, such as the saturation magnetization, the magnetic anisotropy constants and the magnetostriction constants, which, in turn, affect the magnetization process in a way related to the material structure

1.0 [ 0.5 ~-

er

]

~,

!.0 0.5

0.0

~" 0.0

-0.5

-0.5

-1.0

-1.0

-1.5 -150-100-50 0 50 100 150 H (A/m)

-1.5 -20

............

-10

0 H (A/m)

10

20

FIGURE 2.1 Representative DC hysteresis loops in different soft magnetic alloys. (a) Grain-oriented Fe-(3 wt%)Si, non-oriented Fe-(3 wt%)Si, MnZn ferrites. (b) Fe78B13Si9 annealed amorphous ribbons (1), Co71Fe4B15Si10as-quenched amorphous ribbons (2), Fe-Ni (Mumetal type) alloys (3).

2.1 GENERAL PROPERTIES

27

(e.g. crystallographic texture, grain size, foreign phases, lattice defects, etc.). By a proper choice of composition and suitable metallurgical and thermal treatments, extra-soft magnets are obtained, where the coercive field and the relative permeability attain values of the order of 0.1 A / m and 106, respectively. However, it should be stressed that a number of additional properties, such as thermal and structural stability, stress sensitivity of the magnetic parameters, mechanical properties and machinability, and thermal conductivity, have to be considered. The final acceptance of a material in applications will result from a cost-benefit evaluation of all these properties. The magnetization process in an SMM occurs by means of two microscopic mechanisms: motion of the domain walls and uniform rotation of the magnetization inside the magnetic domains. Rotations require high field strengths in the conventional Fe-based crystalline alloys because the Zeeman energy EH = -I-I.Js (with H and Js the applied field and the saturation polarization, respectively) must balance a magnetocrystalline anisotropy energy term FK roughly of the order of the anisotropy constant K1. Given that K1 is of the order of few 104 J / m 3, fields in the 10 3 A / m range must be applied to achieve substantial rotations. A soft magnetic behavior can possibly be achieved in these materials only through displacements of the domain walls. Frictional forces, inevitable in real defective materials, resist these displacements. The coercive field measures the typical field strengths at which the domain walls are unpinned from defects and a substantial part of the magnetization is reversed. Energy is lost in this process, and magnetic hysteresis is accordingly observed. The subject of coercivity and hysteresis is classically treated by theorizing the motion of a domain wall, assumed either as a rigid [2.6, 2.7] or a flexible [2.8] object, in a perturbed medium. Basically, it is assumed that the structural perturbation generates a random wall energy profile, whose spatial derivative represents the pressure to be applied by a field in order to achieve wall motion. The hindering effect of the structural defects is chiefly controlled by the value of the anisotropy constant K1, implying that the value of coercivity tends to increase with the strength of the anisotropy effects. However, even with the relatively high values attained by K1, very soft magnetic behavior can be achieved in Fe and Fe-Si when the microstructure is suitably controlled. In practice, this means having the least content of precipitates, voids, dislocations, and point defects, together with large and favorably oriented grains. By having the applied field directed as far as possible alongside one of the (100) easy axes, i.e. in the plane of the main 180~ domain walls, one gets an obvious directional advantage for the wall displacements, as remarkably demonstrated by the grain-oriented Fe-Si

28

CHAPTER 2 Soft Magnetic Materials

laminations. The role of the microstructural defects is clearly observed in Fe. Here, one can reach coercive fields as low as Hc-" 1 A / m upon prolonged purification and annealing treatments [2.9], leading to very low dislocation densities and C and N concentrations (some 10-20 parts in 106 (ppm)). Coercivities of a few hundred A / m can be found instead when these concentrations are in the 100 p p m range [2.7]. C and N are basically insoluble in Fe and tend to form carbides and nitrides, which act as strong pinning centers for the domain walls. With much higher C content (say around I wt%), graphite precipitates and martensitic domains are additionally formed, and Hc can reach values typical of hard magnets (several 104 A/m). Soft and extra-soft magnetic properties are naturally associated with very low values of the magnetic anisotropy (say with K in the range of a few tens J / m 3 and less). This is the case, for example, of F e - N i alloys, with composition around Fe20-Nis0. K1 is positive in ~-Fe (bcc cell) and negative in Ni (fcc cell) and it passes through zero on the high Ni side in the F e - N i alloys. Vanishing anisotropy can equally be obtained in amorphous and nanocrystalline alloys because the structural order in these materials is extended over limited distances, from a few atomic spacings to a few nanometers. The characteristic length L controlling the magnetization process, represented by the domain wall thickness, encompasses a large number of the local ordered structural units, so that the magnetocrystalline anisotropy is effectively averaged out. The residual anisotropy is calculated to be [2.10]

K4 6 K0-"

A3 ,

(2.1)

where A is the exchange stiffness constant and 3 is the average size of the structural units. By taking 3 = 10 -9 m, K1 = 4.8 x 104 J/m 3 (as in Fe) and A--- 10 -11 J/m, we find the negligible value K0 = 5.3 x 10 -3 J/m 3. With the elimination of the magnetocrystalline effects, other sources of magnetic anisotropy can be brought to light in these materials. For instance, in the highly magnetostrictive Fe-based amorphous alloys, a substantial anisotropy can result from the magnetoelastic coupling between frozen-in or applied stresses and the magnetization. In the typical alloy of composition, Fe78B13Si9, the saturation magnetostriction is As "~ 30 x 10 -6 and the long-range internal stresses generated by the rapid solidification process are of the order of 50-100 MPa [2.11]. Induced anisotropies K r 3As~r, in the range of several hundred J / m 3, can therefore arise and annealing treatments are required in order to achieve excellent soft magnetic behavior. Co-based amorphous alloys (like

2.1 GENERAL PROPERTIES

29

Co71Fe4B15Si10 or Co66Fe5Cr4B15Si10) and F e - N i alloys of the permalloy type have vanishing magnetostriction (in the 10-8-10 -7 range) and, lacking also the magnetocrystalline anisotrop36 attain the lowest coercivities and record values of permeability. In addition, their properties can be adjusted by inducing calibrated uniaxial anisotropies through annealing treatments under saturating fields. In a saturated F e - N i alloy, the magnetization interacts with the Fe-Fe and N i - N i atomic pairs in such a way that, if the temperature is sufficiently high, they tend to distribute preferentially along the field direction, although the alloy preserves its character of random solid solution. In the amorphous alloys, anisotropic atomic rearrangements in the local ordered units, with symmetry influenced by the direction of the magnetization, are expected to occur [2.12]. In all cases, dramatic changes of the magnetization curve can be obtained by means of field annealing, as illustrated for a Co-based amorphous alloy by the example shown in Fig. 2.2. It should be noted that

0.8

Amorphous alloy C071Fe4B15Si10

1

0.4 Field annealed, H• 0.0

-0.4

\

-0.8 -20

J Field annealed, H// I

-10

I

0 H (A/m)

,

I

10

,

20

FIGURE 2.2 DC hysteresis loops in amorphous ribbons of composition Co71Fe4_ Bt5Sil0 after annealing under a saturating magnetic field. A rectangular loop with Hc "--0.5 A / m and peak relative permeability /Zr "~ 106 is obtained after longitudinal field annealing (HII, 1000 s at 340 ~ Transverse field annealing (H,, T - 260 ~ leads to linearization of the loops, as shown here after 600 and 3000 s long treatments. The correspondingly induced transverse anisotropies are Ku "" 5 J/m 3 (2) and Ku "~ 25 J/m 3 (3).

30

CHAPTER 2 Soft Magnetic Materials

the hysteresis loops in these amorphous ribbons can be linearized by means of a transverse induced anisotropy around some 10 J/m 3. In this case the rotation of magnetization becomes an easy process, leading to quite high initial permeabilities (of the order of s o m e 104), a welcome property in many applications. SMMs find most applications in magnetic cores of AC apparatus, from 50 Hz to several MHz. The quantity of basic technical interest is in this case the energy loss. Eddy currents are generated by time-varying magnetic flux, leading both to shielding of the core by the associated counterfields (skin effect) and generation of heat by Joule effect. Excluding ferrites, which are basically insulating materials, all soft magnets have to be used in sheet form in order to minimize skin effect and losses. The theoretical assessment of these effects is by no means a simple one because it is seldom possible to treat the material as a continuum, characterized by a given magnetic permeability, and apply to it the Maxwell equations. The magnetic structure is made of domains and domain walls and the distribution of eddy currents can be extraordinarily complex and nonuniform. Williams, Shockley and Kittel were the first to take such a complexity at face value by investigating the dynamic behavior of a 180~ Bloch wall in single crystals of Fe-Si. They theorized it through a balance equation involving, on the one hand, the applied magnetic field and, on the other hand, the structural pinning field and the eddy current counterfield (WSK model) [2.13]. Pry and Bean generalized the WSK model to a system of 180~ walls, emulating the real domain structure of a grain-oriented Fe-Si lamination [2.14]. The whole problem has in recent times received a complete assessment by Bertotti, who has shown that the complexity of the dynamic magnetization process in real structures can be properly described by means of statistical methods [2.15]. The general phenomenology of losses in magnetic laminations is summarized in Fig. 2.3, illustrating the dependence of the hysteresis loop shape and area (i.e. energy loss) on the magnetizing frequency in a grain-oriented Fe-Si alloy. Bertotti's theory provides a physically based demonstration of the concept of loss separation, as expressed by the equation: W = Wh q- Wcl q- Wexc.

(2.2)

The energy loss per cycle W is taken, at a given magnetizing frequency f and peak polarization ]p, as the sum of three components, having the following meaning. Wh, called hysteresis loss, is the residual energy dissipated in the limit f--, O. Wd is the loss component calculated by applying the Maxwell equations to the material when it is assumed completely homogeneous from the magnetic viewpoint (absence of domains). By summing up Nh and Wd, one falls short of the measured

2.1 GENERAL PROPERTIES

1.5

.....

.

j----

31 ......

i

9 .~

30

!

-0.5 -1.0

(a)

-60

,i..

. . . . . .

,...,

. . . . . . . .

J,......

. . . .

,. | . ,

,~ 20

0.0

-1.5

j

GO Fe-(3wt%)Si ..~ Jp=1.32T ~ w l

1.0 0.5

. . . .

i i,/

...............

-40

-20

lO

2;2 z

i 0

H (A/m)

~~

E

20 40 60

. . . . . . . . .

(b)

0

,..,

50

. . . . . . . .

, . . . . . . . . . .

1O0

f(Hz)

,..,..,~

150

. . . . .

2(10

FIGURE 2.3 Hysteresis loops (a) and specific energy loss per cycle (b) vs. magnetizing frequency in grain-oriented Fe-Si laminations (thickness 0.29 mm) under sinusoidal time dependence of the polarization (Jp = 1.32 T). The energy loss in (b) is proportional, at a given frequency f, to the area of the corresponding (J,H) hysteresis loop. The loss separation concept formulated in Eq. (2.2) is illustrated in (b). value. The remainder, We• is called excess loss. The three loss components are associated with different eddy current mechanisms and different space-time scales of the magnetization process. The classical loss Wd is a sort of background, always present and independent of any structural feature. For a lamination of thickness, d, conductivity ~r and density 3, one finds, under sinusoidal time dependence of the magnetic polarization, the classical loss per unit mass: Wcl-

~T2

o-d2j2f

6 T '

(2.3)

where 3 is the density of the material and it is assumed that complete flux penetration in the lamination cross-section occurs. Figure 2.3 demonstrates how an elementary approach to losses based on the classical approximation largely underestimates the measured loss in grain-oriented laminations at 50-60 Hz, the frequencies at which the large part of electrical energy is generated and produced. In addition, the dependence of W o n f is non-linear, in contrast with the prediction on Wd (Eq. (2.3)). The components Wh and We• derive from the heterogeneous nature of the magnetic structure of the material and the inherently discrete behavior of the magnetization process. Flux reversal is concentrated at moving domain walls and, even under quasi-static excitation, eddy currents arise because the domain wall displacements, hindered by the pinning centers, occur in

32

CHAPTER 2 Soft Magnetic Materials

jerky fashion (Barkhausen effect). Intense current pulses, with lifetimes around 10 -9 s, are generated, localized around the jumping wall segments. In this way, the hysteresis loss Wh is generated and, since the time constant of the eddy current pulses is always many orders of magnitude smaller than the typical magnetization period T = l / f , it is concluded that Wh is independent of frequency. For a given value Jp, Wh gives a measure of the coercive field. This is totally consistent with the fact that the Barkhausen mechanism is a volume effect, independent, as the coercivity should be, of the material conductivity. The excess loss Wex c is associated with the largescale motion of the domain walls. The theory shows that, in most SMMs, the following expression holds:

kexc~-~J~/2

Wexc =

,

(2.4)

where kexc is a parameter related to the type of existing domain structure and its relationship with the material structure. Very broadly, it can be stated that the larger is kexc the more discrete is the magnetization process. Very large domains are therefore not desirable from this viewpoint and, as discussed below, methods have sometimes to be devised to increase the number of domain walls in the material. In conclusion, if minimization of the AC energy losses is sought, not only the lamination thickness and 10

'

I

'

7

E~

/

I

'

I

'

I

Low-carbon steel

I

=SOHz .. 9

\._

(/) O t..

O

-o~ 9 ~

Grain-oriented Fe-Si

O Q.

'

Jp=IT

"\.\

Fe-Si

'

0.1

/

f

~O--o._,

Amorphous FeBSi "

1880

,

I

1900

,

I

1920

=

I

1940

,

l

1960

,

I

1980

,

2000

Year

FIGURE 2.4 Record loss figures over a century in soft magnetic laminations for transformer cores.

33

2.2 PURE IRON AND LOW-CARBON STEELS

TABLE 2.1 Representative soft magnetic materials and typical values of some basic magnetic parameters H c (A/m) Js (T)

Composition

/d, m a x

Fe NO Fe-Si GO Fe-Si Fe-Si 6.5% Sintered powders Permalloy Permendur Ferrites

Fel00 Fe(>96)-Si(<4) Fe97-Si3 Fe93.5-Si6.5 Fe99.5-P0.5 Fe16-Ni79-Mo5 Fe49-Co49-V2 (Mn,Zn)O.Fe203

1-100 3-50 X 103 3-10 X 103 30-80 20-80 X 103 4-15 5-30 X 103 10-40 0.2-2 X 103 100-500 5 x 105 0.4 2 X 103 100 3 X 103 20-80

2.16 1.98-2.12 2.03 1.80 1.65-1.95 0.80 2.4 0.2-0.5

Sendust

Fe85-Si9.5-A15.5

50 x 103

5

1.70

2 0.5 0.5

1.56 0.86 1.2

Amorphous (Fe based) Fe78B13Si9 105 Amorphous (Co based) Co71Fe4B15Si10 5 x 105 Nanocrystalline Fe73.5CulNb3Si13.sB9 105

/d, m a x , maximum DC relative permeability; Hc, coercive field; Js, saturation polarization at room temperature. The composition is given in wt%, but for the amorphous and nanocrystalline alloys, where it is expressed in at.%.

the material conductivity need to be reduced, as suggested by the classical approach (Eq. (2.3)), but also the microstructure must be controlled in order to minimize both Wh (i.e. the coercive field) and Wexc. This emphasizes the role of metallurgical processing, whose continuous refinement over the years has produced increasingly better control of the various structural parameters (e.g. impurities, defects, grain size, and crystallographic texture) and clear progress of the magnetic properties of the materials (see Fig. 2.4). A representative list of different types of SMMs is given in Table 2.1. Most of these materials are produced and traded under defined specifications. The measurement of their properties is therefore subjected to acknowledged standards issued by national and international bodies. The International Electrotechnical Commission (IEC) Standards of the 60404 Series cover specifications and measurements of commercial SMMs.

2.2 P U R E I R O N A N D L O W - C A R B O N STEELS Iron is referred to as "high purity" when the total concentration of impurities (typically C, N, O, P, S, Si, and A1) does not exceed a few hundred ppm. It is otherwise called low-carbon steel or non-alloyed steel. When soluble elements like Si and A1 are deliberately introduced,

34

CHAPTER 2 Soft Magnetic Materials

typically in the range of a few percent, it is appropriate to speak of "silicon steels". Very pure iron is rarely used in applications, but the study of its properties is of basic physical interest. The main practical drawbacks of pure Fe are its relatively high electrical conductivity, which makes it unsuitable for AC applications, its poor mechanical properties and its cost. Low-cost low-carbon steels (C < 0.1 wt%) are largely applied in a multitude of small electrical machines (for instance, fractional horse power motors) and devices where efficiency is not of primary concern. Together with the silicon steels, they cover about 80% of the world tonnage of SMMs. More efficient low-carbon steels are increasingly developed today, under the pressure of rising energy costs and environmental concerns. Higher grades are therefore now available, where improved magnetic properties are obtained chiefly by introducing a small amount of Si (~ 1 wt%) and decreasing the content of impurities (especially sulfides, carbides, and nit-rides). A practical method to obtain high-purity iron is to start with commercially pure iron (e.g. of the ARMCO type) and refine it by suitable methods. These include prolonged annealing in pure H2 at temperatures not far from the melting point (for instance, 48 h at 1480 ~ zone melting and levitation melting. By means of these methods, some 20-30 p p m maximum total impurity content can be reached, with C and N less than 10ppm. Record permeabilities /~r "" 106 and coercivities Hc = 1-2 A / m have been obtained in highly purified iron samples. Some common iron grades are listed in Table 2.2. Low-carbon steels to be used in magnetic cores are generally produced as sheets, through a sequence of hot and cold rolling passes and thermal treatments, as shown schematically in Table 2.3. The specifications for these steels are provided in the IEC Standard 604048-3 [2.16]. In the case of low-cost materials, heat and mechanical treatments are limited to those necessary to reach the final sheet thickness, in the range 0.50-0.85 mm. To improve the magnetic performance, TABLE 2.2 Typical impurities and their concentrations (wt ppm) in several grades of iron and in low-carbon steel Iron type

C

N

ARMCO Electrolytic H2 treated Zone refined Low-carbon

150 40 30 7 50-1000

20 150 100 100 10 30 < 10 2 30-200 20-100

steel

O

Mn

P

S

280 50 250 15 20 30 280 40 < 30 0.5 < 0.1 0.2 5 x 103 200-1000 50-300

Si

Cu

Ni

30 30

150 40

10

103-104

1.5 100

0.50

2.2 PURE IRON AND LOW-CARBON STEELS

35

TABLE 2.3 Scheme of low-carbon steel sheet processing Melting, degassing, continuous casting of slabs Re-heating (1000-1250 ~ and hot rolling to thickness 2-2.5 mm Pickling and cold rolling to final thickness Intermediate annealing for re-crystallization Temper rolling (reduction 3-5%) Punching Final annealing (decarburization, grain growth, controlled surface oxidation) Core assemblage

the laminations must be decarburized. It is a final treatment, where the laminations are annealed in a wet hydrogen atmosphere, at temperatures around 800 ~ By this process the carbon concentration can be typically reduced to less than 50 ppm. The main detrimental effect of C is magnetic aging, i.e. the increase of coercivity over time consequent to the formation of cementite precipitates. These give rise to substantial domain wall pinning. Aging may represent a real threat in actual magnetic cores, where operating temperatures of 50-100 ~ are common. Figure 2.5 shows that, in low-carbon steels with C concentrations as low as 45 ppm, a potential for aging still exists [2.17]. Nitrogen can equally induce aging, but it can be partly stabilized by the formation of A1N precipitates. These, however, may adversely affect the grain texture during re-crystallization annealing by favoring the growth of unfavorable {111} planes. Such an effect can be largely avoided by controlled addition of either B (---30 ppm) or Zr (0.07 wt%) [2.18]. Reduction of the C, N, and S concentrations in the range of 20-30 p p m can also be obtained, in high-quality steels, by vacuum degassing of the melt. This makes unnecessary the final decarburization anneal, with beneficial effects on production costs. The e~--*~ transition takes place in pure Fe at 911 ~ and the final thermal treatments are thus preferably made at lower temperatures, which may limit the range of attainable grain sizes and crystallographic textures. Low-carbon steels are generally delivered as semi-processed products because they need to be in a cold worked state before punching and cutting. The necessary mechanical hardness is imparted by means of temper rolling, a 3-5% cold reduction. Once punched, the laminations are subjected to decarburization and grain growth annealing, eventually followed by controlled surface oxidation ("bluing") to ensure acceptable interlaminar insulation in the core. One notable consequence of temper rolling is a somewhat exaggerated grain growth upon final annealing, which overcomes, to some extent, the limitations imposed on the upper treatment temperatures by the ~ --* ~ transition.

36

CHAPTER 2 Soft Magnetic Materials . . . . . . . .

I

. . . . . . . .

I

''

. . . . . . .

I

0.6 - Low-carbon steel

T=150~

156 ppm

0.5 b

0.4 o

57 ppm ,,o.-o-=~

0.3

45 ppm ...

~'~ 0.2 0.1

21ppm

0.0 -0.1

. . . . . . . .

10 3

I

10 4

. . . . . . . .

I

. . . . . . . .

105

I

10 6

,

.

,

|,..,I

10 7

|

|

|

,..

108

ta (s)

FIGURE 2.5 Relative increase of power losses (f= 50 Hz, Jp = 1.5 T) in a lowcarbon steel lamination (Si = 0.3 wt%) after aging at 150 ~ up to 600 days [2.17]. Different curves refer to different C concentrations, ranging between 156 and 21 ppm. Low-carbon steel performance is best described in terms of AC magnetic properties at 50-60 Hz. In the absence of purification treatments and significant Si content, AC losses at 60 Hz and 1.5 T can reach some 1 5 W / k g in 0.65 m m thick laminations, with relative permeability /a,r -- 500-1000. The addition of ---lwt% Si, together with better composition control, may contribute to lowering the power loss to less than 8 W / k g , with/d, r ~ " 2000. However, the introduction of Si decreases the saturation magnetization, which may be somewhat detrimental to permeability. Therefore, it is preferable, whenever possible, to improve loss performance by the use of clean materials. In particular, with the extensive use of vacuum degassing, the development of clean ultralow-carbon steels is possible. Coated semi-processed materials are available, which combine improved loss and permeability behavior ( P ~ - 4 W / k g and / d , r " ~ 3000 at 1.5T and 5 0 H z in 0.50mm thick laminations) with excellent punching performance [2.17]. Pure Fe and non-alloyed steels find applications as cores of DC electromagnets, where one exploits their high saturation magnetization in producing strong fields. Typical AC applications are relays, lamp ballasts, fractional horse power motors and small transformers, where

2.2 PURE IRON AND LOW-CARBON STEELS

37

performance is needed at low cost. It is known that, in small motors (power less than 1-2 kW), where the limited size imposes high induction values in the stator teeth, copper losses tend to predominate over iron losses. The solution offered by non-alloyed steels, with their high values of permeability at high inductions and affordable cost, represents a good compromise between the requirements of costs and machine efficiency. Optimal product performance is eventually obtained through proper design considerations. For applications in the kHz range, soft iron powder cores are often used [2.19]. They are obtained initially from iron particles of 50-100 ~m size, prepared by milling or electrolytic deposition and insulated by surface treatment. The particles are compressed as cores of the desired shape, which are heat-treated and coated by protective painting. Alloying with P or Si is often carried out during the sintering process. Being finely subdivided and, consequentl~ characterized by a distributed air gap, the iron powder cores exhibit a sheared hysteresis loop and relatively low values of permeability (/~r "" few hundred). The lack of any large-scale electrical conductivity gives reduced dynamic losses with respect to the bulk samples,, as well as with respect to lowcarbon steel laminated cores (Fig. 2.6). A nearly frequency-independent

0.6

0.4

0.2

0.0

0

200

f(Hz)

400

600

FIGURE 2.6 Energy loss per cycle vs. magnetizing frequency in: (1) low-carbon steel laminations, 0.64 mm thick (LCS); (2) non-oriented Fe-Si laminations (NO), 0.61 mm thick; (3) Pressed iron powder core (PC). Data taken from Ref. [2.19].

38

CHAPTER 2 Soft Magnetic Materials

permeability is generally obtained up to 10-100 kHz. Applications of iron powder cores include loading coils, pulse transformers, inductors in switching mode power supplies and small high-speed motors. They are the material of choice when complex core shapes are needed.

2.3 I R O N - S I L I C O N ALLOYS The addition of few atomic percent Si brings about notable changes in the physical, mechanical, and magnetic properties of Fe. The most notable effect regards the electrical resistivity, which increases at a rate around 5 x 10-8 f~ m per soluted atomic percent. This engenders a remarkable benefit in terms of decreased AC losses, as demonstrated in Eqs. (2.3) and (2.4). There are further properties that take advantage of Si alloying. The magnetocrystalline anisotropy constant K1 decreases with increasing Si, which is reflected in lower coercivity. The yield strength increases, which favors material handling and machining. In addition, inspection of the Fe-Si phase diagram (see Fig. 2.7) shows that, above about 2 wt% Si, the ~ --* transition (bcc to fcc structure) no longer takes place and the previously remarked restrictions on the final annealing temperatures in the low-carbon steels no longer exist. The chief factors against substantial addition of Si are

1400

1300

••

~, bcc

1392 ~

1200 0o F-

,,,Co

1100

1000

900 0.0

.

-1.86

911 ~

0.5

1.0 1.5 Si (wt%)

2.0

FIGURE 2.7 Fe-Si phase diagram: the ~/loop.

2.3 IRON-SILICON ALLOYS 5 0 ,~.

9

39

,

9

,

.

,

.

,

,:,

9

1o o

p

40

80

20

9

,,

10

60

~"

40

~o,..

20

0

0

2.2

,

,

,

,

.

,

,

,

,

(~y

-

400

2.0

300 &.

1.8

200 >,

1.6 1.4

100

,

0

I

2

,

I

,

4

I

6

,

I

8

,

t0 0

10

si (vvt%)

FIGURE 2.8 Magnetocrystalline anisotropy constant K1, electrical resistivity saturation magnetization Js, and yield stress Cryvs. Si concentration in Fe-Si alloys. the reduction of the saturation magnetization and the fact that there is no practical way of achieving laminations with more than about 4 wt% Si by conventional rolling processes. The heterogeneous formation of FeSi and FeBSi ordered phases leads, in fact, to severe material embrittlement. The behaviors of resistivity, anisotropy constant, saturation magnetization and yield stress vs. Si concentration are shown in Fig. 2.8.

2.3.1 Non-oriented Fe-Si alloys Non-oriented (NO) Fe-Si alloys are SMMs with an approximately isotropic grain texture. They cover the medium and high quality range of SMMs for applications in electrical rotating machines, where good isotropic magnetic properties are required. They come in a variety of

40

CHAPTER 2 Soft Magnetic Materials

grades, the higher ones being associated with higher Si content. Specifications for NO Si steels are provided in the IEC Standards 604048-2 [2.20] and 60404-8-4 [2.21]. The Si concentration can actually vary between I and 3.7 wt% and some percentage of A1 (0.2-0.8 wt%) and Mn (0.1-0.3 wt%) is usually added, by which the alloy resistivity is further increased without impairing the mechanical properties. A1 also prevents aging by N precipitation, by stabilizing it through the formation of A1N second phases. The lower grade NO laminations (up to ---2 wt% Si) are produced and delivered in the semi-processed state and follow the same thermomechanical history of low-carbon steels (Table 2.3), with final thickness ranging between 0.65 and 0.50 mm. The higher grades are instead fully processed materials. They are obtained according to the procedure outlined in Table 2.4. The hot rolled sheets (thickness 2.31.8 mm) are cold rolled to intermediate gauge, annealed at 750-900 ~ reduced to the final gauge of 0.65-0.35 mm, and subjected to a recrystallization and decarburization anneal at 830-900 ~ and a final grain-growth anneal at 850-1100 ~ A single-stage cold reduction is a basic variant of this process. A phosphate-based or chromate-based coating is then applied, which not only provides the necessary interlaminar insulation, but also ensures good lamination punchability. The latter property is important because of the strict tolerances required in rotating machine cores and the need to avoid edge burrs, a possible cause of interlaminar short circuits in the assembled cores. Contrary to the case of semi-finished products, no stress relief treatment is in general applied after sheet punching. By developing the methods of composition and preparation, and with the help of improved understanding of the role of the structural parameters on the loss and permeability behavior, a wide range of NO steels with variable quality have been made available to TABLE 2.4 Preparation stages in fully processed non-oriented Fe-Si alloys Composition (wt%): Si (0.9-3.7), A1 (0.2-0.8), Mn (0.1-0.3) Melting, degassing, continuous casting of slabs Re-heating (1000-1250 ~ and hot rolling to thickness 1.8-2.3 mm Pickling and cold rolling to intermediate gauge Intermediate annealing (750-900 ~ Cold rolling to final gauge (0.65-0.35 mm) Decarburization and re-crystallization annealing (830-900 ~ Final grain-growth annealing (850-1100 ~ Coating Punching Core assemblage

2.3 IRON-SILICON ALLOYS .0

9 ,

41 9 ,

9 ,

,

,

9 ,

NO Fe-(3wt%)Si Jp= 1.5 T

2.8

f = 50 H z S

~ 2.6 v

m

ffl o t_

r

2.4

o 13_

2.2

o

2 . 0

,

0

i

10

,

i

20

,

i

30

,

i

,

40

i

50

,

i

60

,

70

S, O, N (wt p p m )

FIGURE 2.9 Role of O, S, N impurities on the power losses in 0.35 mm thick nonoriented Fe-Si laminations [2.22]. the users. They are never isotropic, exhibiting in general some 10-20% variation of the loss figure along different directions in the lamination plane. The top commercial grades have around 4wt% (Si + A1) concentration and, with a gauge of 0.35-0.50 mm, they exhibit a loss figure W15/50 of 2.10-2.30 W / k g at 1.5 T and 50 Hz, reaching an induction B25-- 1.55 T at 2500 A / m . Non-oriented Fe-Si alloys are preferentially employed in medium and high power rotating machines whereas, as previously stressed, low-carbon steel laminations are preferred in small apparatus. The highest efficiency (>95%) is sought in big electrical machines, not only to save energy, but also to avoid overheating and shortened machine lifespan. The development of improved non-oriented alloys is related to the control of a number of structural parameters, namely impurities, grain size, crystallographic texture, surface state, residual and applied stresses. A few tens p p m concentrations of impurities like C, N, S, O tend to increase coercivity and losses (see Figs. 2.5 and 2.9) [2.22]. They can do this directly, by forming precipitates that act as pinning centers for the domain walls, and indirectly, by adversely affecting grain growth and texture. The role of grain size <s>is illustrated in Fig. 2.10, where it is observed that the optimal <s> value is, depending on the composition, around 100-200 ~m, where

42

CHAPTER 2 Soft Magnetic Materials

the total power loss attains a m i n i m u m value [2.23]. This is understood in terms of the opposite dependencies on grain size exhibited by the hysteresis loss component Wh, decreasing as (s) -n, with n = 0.5-1, and by the excess loss We• approximately increasing a s (S)1/2. A low impurity content is mandatory in order to achieve this optimal grain size because precipitates tend to hinder grain growth. In addition, some particles, like MnS and A1N, favor the establishment of a detrimental texture, rich in {111} planes. On the other hand, there are soluted impurities, like Sb and Sn, that can induce selective growth of those re-crystallized grains that have orientations close to the ideal random cubic texture {100}(0vw). A similar texture can be approached, in two-stage reduced alloys, by increasing the A1 concentration up to 1.1 wt% [2.22] or even 1.8 wt% [2.24], which permits one to achieve W15/so"" 2 W / k g . The troublesome aspect of an increased A1 content, besides the cost, is an increased tendency to surface and subsurface oxidation, occurring especially during the wet H2 decarburization. In this case it is expedient to achieve the desired non-aging properties by decarburization and denitrogenization of the melt through v a c u u m degassing. The punching operation generates localized internal stresses

O}

0

4

~ ~ ( 4 )

3

(5)

t,.,

0 0

n

2 Jp= 1.5T f= 50 Hz 0

10

i

NO Fe-Si i,

i

i

i

i

i

i

I

,

i

lOO

Average grain size (l.l,m)

FIGURE 2.10 Power loss vs. grain size in 0.50 mm thick NO Fe-Si laminations. The curves (1)-(5) correspond to different (Si + A1) concentrations: (1) Si 0.01wt%; (2) Si 0.3wt%; (3) Si 0.8wt% +A1 0.2wt%; (4) Si 1.1wt% +A1 0.2 wt%; (5) Si 3 wt% + A1 I wt% [2.23].

2.3 IRON-SILICON ALLOYS

43

and, consequently, it might affect the loss figure in fully processed materials where, in general, stress relief annealing is not performed. In large machines, however, a much larger effect on magnetic losses is expected to derive from the stresses permanently introduced by stacking and assembling the laminations in the core.

2.3.2 Grain-oriented Fe-Si alloys Fe single crystals exhibit minimum coercivity and maximum permeability when magnetized along one of the (001) axes. This property has fundamental implications on a theoretical level and outstanding practical consequences. In fact, most transformer cores are built today with grainoriented (GO) Fe-Si laminations, where the crystallites have their [001] easy axis close to the rolling direction (RD) and their (110) plane nearly parallel to the lamination surface (Fig. 2.11). This is the so-called (110)[001] texture or Goss texture, after Goss, the first to develop such materials [2.2]. The remarkable texture of the GO alloys, together with a large grain size (from a few millimeters to a few centimeters) and a small impurity content, leads to coercive fields as low as 4-10 A / m and a maximum

[o~o]

[x

.,

[001] .

.

.

.

5 mm

.

v RD

FIGURE 2.11 Domains in HGO Fe-Si laminations. They are oriented along the [001] axis and tend to multiply upon scribing the sheet surface.

44

CI-~_PTER 2 Soft Magnetic Materials

permeability around 5 x 104. These figures differ by about an order of magnitude from those typically found in NO alloys. Single-phase transformer cores can be made either by rolling up a long lamination or by stacking and suitably joining separate sheet pieces at the comers, so that the magnetic-flux path is everywhere aligned to RD. Three-phase cores are always of the stacked type and cover the high power range (starting from some 50 kV A). GO laminations are subdivided in two main classes: conventional grain oriented (CGO) and high permeability (HGO) alloys, characterized by a dispersion of the [001] axes of the crystallites around RD of the order of ---7 and ---3~ respectively. Specifications are provided in the IEC Standard 60404-8-7 [2.25]. The CGO materials, although lower performing than the HGO ones, cover about 80% of the market. As illustrated,in Table 2.5, commercial products are offered with thicknesses ranging between 0.35 and 0.23 m m and a loss figure W17/50 = 1.40-0.80 W / k g at 1.7T and 50Hz. More than 1 million ton/year of GO alloys are produced worldwide, with a market value estimated around (~1.5 billion. Following the original method of Goss, a number of patented processes for the production of GO sheets have been developed, based on various complex thermomechanical sequences [2.26-2.28]. Table 2.6 offers a schematic illustration of the chief processing methods. In the case of CGO materials, the main preparation steps can be summarized as follows: (1) Melting in the arc furnace, vacuum degassing

TABLE 2.5 Typical specifications for conventional and highpermeability grain-oriented Fe-(3 wt%)Si alloys Thickness (mm)

Specific power losses lp = 1.7 T, f = 50 Hz (W/kg)

Polarization at 800 A/m (T)

Conventional (CGO) 0.35 0.30 0.27 0.23

1.40 1.30 1.21 1.15

1.82 1.83 1.84 1.84

High permeability (HGO) 0.30 0.27 0.23 0.23 (scribed)

1.06 0.99 0.92 0.80

1.91 1.92 1.92 1.90

2.3 IRON-SILICON ALLOYS

45

TABLE 2.6 Summary of industrial processing of grain-oriented silicon steel sheets CGO

HGO-1

HGO-2

HGO-3

2.9-3.3, Si 0.03, A1 0.07, Mn 0.03, S 0.015, N 0.05-0.07, C Balance, Fe

2.9-3, Si 0.05, Mn 0.02, Se 0.04, Sb 0.03-0.07, C Balance, Fe

3.1-3.3, Si 0.02, Mn 0.02, S 0.001, B 0.005, N 0.03-0.05, C Balance, Fe

MnS + A1N

MnSe + Sb

B+ N + S

Composition (wt%) 3-3.2, Si 0.04-0.1, Mn 0.02, S 0.03, C Balance, Fe

h~hibitors MnS

Melting, vacuum degassing and continuous casting of slabs Re-heating-hot rolling 1320 ~

1360 ~

1320 ~

1250 ~

1100-1150 ~

900 ~

870-1020 ~

87%

60-70%

80%

Annealing 900-1100 ~

Cold reduction 70%

Annealing 800-1000 ~

800-1000 ~

Cold reduction 65%

55%

Decarburization 800-850 ~ (wet H2 atmosphere)

MgO coating and coiling Box annealing (secondary recrystallization ) 1200 ~

1200 ~

820-900 ~ + 1200 ~

1200 ~

Phosphate coating and thermal flattening CGO denotes the conventional grain-oriented steels. HGO refers to the high permeability steels, of which three processing methods are presented. CGO and HGO-2 processes adopt a two-stage cold reduction. HGO-1 and HGO-3 reach the final gauge in a single step. The inhibitors are: MnS precipitates (CGO); M n S + A1N particles (HGO-1); MnSe particles + solute Sb (HGO-2); solute B + N + S in HGO-3.

46

CHAPTER 2 Soft Magnetic Materials

and continuous casting. Besides Si, ranging in concentration between 2.9 and 3.2 wt%, the following impurities are usually present: Mn (0.040.1 wt%), S (0.02 wt%), C (0.03 wt%). (2) Slab re-heating at 1300-1350 ~ followed by hot rolling to the thickness of 2mm, annealing at a temperature of 900-1100 ~ and rapid cooling. Times and temperatures through this stage are finely adjusted in order to achieve a homogeneous distribution of precipitates, namely MnS particles of around 10-20 nm. (3) Two-stage cold rolling (---70% plus ---55%) to the final gauge, with intervening annealing treatment at 800-1000 ~ (4) Decarburizing anneal in wet H2 atmosphere at 800-850 ~ Since a huge amount of deformation is accumulated by the previous cold reduction, complete primary re-crystallization takes place at this stage. However, the newly formed grains are strongly inhibited in their growth because of the presence of the finely precipitated MnS impurities. (5) MgO coating, coiling and 48 h box annealing at a temperature of 1200 ~ During this final annealing treatment, secondary re-crystallization takes place, where abnormally large and sharply (110)[001] oriented grains grow within the precipitate stabilized primary matrix and eventually cover the whole sheet. This is thought to occur because these grains have a boundary mobility much larger than the great majority of the primary grains, whose prevalent texture is around {111}(110) and {111}(112). It is estimated that one in about 106 primary grains is (110)[001] oriented, which explains the final large secondary-grain size. At the end of box annealing, the precipitates are completely dissolved and harmful effects on the magnetization process are avoided. (6) Phosphate coating and thermal flattening. The HGO laminations are obtained with some variants to the previous sequence, which lead to a sharper Goss texture. Three basic industrial HGO processes are employed today, as summarized in Table 2.6. In a first process, where a single-stage 87% cold reduction is adopted, the MnS inhibitors are reinforced with A1N precipitates [2.26]. A partial austenitic transformation, during hot rolling and successive annealing at 11001150 ~ is achieved by adjusting the C concentration around 700 ppm. With 3.25 wt% Si and 700 ppm C, the ~/-phase fraction at 1150 ~ amounts to 40-50%, which leads to optimally sized A1N particles. The second method is a two-step cold rolling process which exploits the combined inhibiting action of MnSe precipitates and Sb solute atoms, segregated at grain boundaries [2.27]. In the third process, solute B, N, and S atoms act as inhibitors and a single-stage cold reduction is again performed [2.28]. Commercial HGO laminations come into the thickness range 0.300.23 mm. To achieve lower gauges while maintaining a comparable textural quality is more difficult. It has been shown, however, that high permeability laminations in wide gauge range (0.18-0.50 mm) can be

2.3 IRON-SILICON ALLOYS

47

produced with the process HGO-1 (87% single-stage cold reduction), by forming the A1N inhibitor at the end of the decarburizing stage, through N2 injection (acquired inhibitor method) [2.29]. The beneficial effects on permeability and coercivity produced by an improved Goss texture are directly related to the morphology of the magnetic domains, which, for a perfect (110)[001] grain, form a bar-like array, with the 180~ Bloch walls running along the [001] direction. This ideal structure is approached in the HGO sheets, which exhibit the lowest coercivity and the highest permeability among the Fe-Si alloys, thanks to the combined absence of DW pinning centers and the presence of large well-oriented grains. Figure 2.11 shows a typical domain structure in an HGO lamination. One can notice the presence of spike-shaped supplementary domains. These domains form when the [001] axis does not lie in the plane of the lamination. Their role is to create closed paths for the magnetic flux in order to reduce the magnetostatic energy of the system. In a material with few, wide bar-like domains, like the one in Fig. 2.11, the magnetization reversal, localized at each instant of time at the moving walls, is highly non-homogeneous and the excess loss component We• is accordingly large (see Eq. (2.4) and related discussion). In HGO materials with a sharp (110)[001] texture, this feature has detrimental consequences on power losses because the excess loss can be quite high, wiping out the beneficial effects on permeability and DC coercivity introduced by the excellent textural properties of the material. In order to optimize the material behavior in terms of both permeability and losses, it is expedient to apply, both in CGO and HGO laminations, a coating (thickness -~2.5 l,m) capable of exerting a tensile stress of 2-10 MPa. This stress leads, via magnetoelastic interaction, to partial or complete disappearance of the flux closing domains, which have a high magnetoelastic energy cost. Under these conditions, the system can reduce its magnetostatic energy by reducing the size of the main bar-like domains, leading to a more homogeneous magnetization process and reduced dynamic losses. However, the energy balance is such that stress-induced domain refinement hardly occurs when the angle made by the [001] axis with the plane of the lamination is 13 ~< 1.5~ For this reason, in the best HGO materials, significant domain multiplication is achieved through a combination of magnetoelastic and magnetostatic effects, by scribing patterns on the lamination surface (Fig. 2.11). An array of scribing techniques (e.g. mechanical scratching, laser irradiation, plasma jet scribing, etch pitting) have been devised and are employed in the production of the highest HGO grades. The joint effects of tensile stress and surface scribing on power losses can be appreciated by experiments carried out in single crystals characterized by different values of the angle

48

CHAPTER 2 Soft Magnetic Materials

/3 [2.30]. As s h o w n in Fig. 2.12, the 50 Hz p o w e r loss is drastically reduced by scribing w h e n / 3 <~ 1~ and further benefit is introduced by the tensile stress. This occurs with little detrimental effects on permeability. For applications at m e d i u m frequencies (0.4-10 kHz), thinned GO laminations are commercially available, which are p r o d u c e d by rolling standard laminations to a reduced gauge of 0.15-0.10 m m and carrying out an appropriate heat treatment. As these alloys present inferior texture and permeability, new processes have been devised in the laboratory in order to achieve highly oriented extra-thin materials [2.31]. Excellent performances have been obtained in laminations with thickness ranging between 10 and 100 ~ m by adopting, after cold reduction to the final gauge, a special sequence of annealing treatments. These induce a sharply defined (110)[001] texture through a tertiary re-crystallization process

1.4

I

'

I

'

I

'

I

'

I

Jp = 1.5T

1.2 \ A

f=50 Hz (1)

'

I

'

A

a

z~

"~/

I

/~"

9

1.0

:// 0.6

m 0.4 I

0

(4)

1

2

3

(deg)

4

5

6

7

FIGURE 2.12 Power loss P17/50 measured at 50 Hz and 1.7 T on 0.20 mm thick Fe-(3 wt%)Si single crystal strips, as a function of the misorientation angle /3 made by the [001] direction with respect to the strip plane. Different tensile stresses and scribing conditions are considered: (1) or= 0, before scribing; (2) or = 0, scribed; (3) or = 15 MPa, before scribing; (4) or-- 15 MPa, scribed. Tensile stress and scribing of the strip surface both produce a decrease of the power loss, due to a decrease of the excess loss component. Combination of texture perfection with scribing is the route to combined maximum permeabilities and minimum losses. Data taken from Ref. [2.30].

2.3 IRON-SILICON ALLOYS

49

[2.32], which occurs because the (110) crystallographic planes are those having minimum energy when exposed to the lamination surface. This principle is also the basis of the preparation of cube-on-face (100)[001] textured Fe-(3 wt%)Si laminations by secondary re-crystallization in a slightly oxidizing atmosphere [2.33]. These materials are, in principle, attractive because the biaxial symmetry introduced by the two easy axes in the lamination plane has potential for applications both in transformers and rotating machines. However, they have failed to attract commercial interest and their production was discontinued many years ago.

2.3.3 Fe-(6.5 wt%)Si, Fe-AI and Fe-Si-AI alloys Fe-(6.5 wt%)Si alloys are a prospective route to low loss materials. When compared with the conventional Fe-(3 wt%)Si alloys, they exhibit a favorable combination of lowered anisotropy (K = 2.1 x 104 J / m 3 instead of K = 3 . 6 x l 0 4 j / m 3) and increased resistivity ( p = 8 0 x l 0 - S f ~ m instead of p = 45 x 10 -812 m), which compound with vanishing magnetostriction (A100= - 0 . 5 x 10 -6,/~111 ~--" 2 x 10 -6, see Fig. 2.13) to provide a potentially excellent soft magnetic alloy for applications at power and

! O"

Fe-AI ~ , . . ~ /

60

/

p" /"

,,o .o"

40 9.~ 0 " " " 0 ~ .r-

20 Fe-Si

\

1 Fe-AI

-20

,

0

....

,;, . . . . %. CA, (wt%)

;''

,

8

FIGURE 2.13 Magnetostriction constants A100and ~111 in Fe-Si and Fe-A1 alloys and their evolution with the concentrations of Si and A1. The magnetostriction in Fe-Si becomes vanishingly small for Csi "" 6.5 wt%.

CHAPTER 2 Soft Magnetic Materials

50

medium frequencies. Fe-(6.5 wt%)Si alloys cannot be prepared by cold rolling because the heterogeneous formation of ordered FeSi (B2) and Fe3Si (DO3) phases during cooling makes them hard and brittle. To avoid ordering, cooling rates greater than about 103 ~ in the temperature interval 800-500 ~ are needed, a condition that can be satisfied by rapid quenching from the melt. By planar flow casting (PFC), a method where a molten metal stream is ejected onto a rotating metallic drum (Fig. 2.14), ductile Fe-(6.5 wt%)Si ribbons are obtained, with thickness typically ranging between 30 and 100 ~m. Once annealed in vacuum at 11001200 ~ they exhibit a columnar grain structure ((s) --- 100-500 t~m) with a prevalent texture (100)(0vw) (random cube-on-face). The final ribbons may show, at the price of a somewhat reduced ductility with respect to the as-quenched state, a coercivity lower than 10 A / m and a maximum relative permeability larger than 104 and constant up to the kHz region. At I kHz and 1 T, the power loss can be lower than 15 W / k g in 50 I~m thick ribbons, a value that compares favorably with the typical loss figure of thinned GO Fe-(3 wt%)Si laminations. Fe-(6.5 wt%)Si alloys can equally be prepared by Si enrichment of standard NO and GO laminations. Classically, Si enrichment is carried out by means of a chemical vapor

---4---

i .......

i ..........

\ Planar Flow Casting

\ Chill Block Melt Spinning

FIGURE 2.14 Preparation of soft magnetic ribbons by means of rapid quenching from the melt. The peripheral velocity of the metallic wheel (e.g. copper, iron, steel) typically ranges between 10 and 40 m/s. Casting can be performed either in air, inert gas, or vacuum.

2.4 AMORPHOUS AND NANOCRYSTALLINE ALLOYS

51

deposition (CVD) process, where SIC14 is used as the donor phase, according to the reaction SIC14 + 5Fe ~ FeBSi + 2FeC12. This process has been industrially applied to NO laminations, prepared with thickness ranging between 0.10 and 0.30 m m [2.34], which exhibit good workability. For peak induction values below 1 T and frequencies higher than a few hundred Hz, the reported loss figures are better than those found in GO laminations of the same thickness. At the same time, the reduced magnetostriction yields a definite decrease of the acoustic noise level under operating conditions. Solute A1 and Si atoms affect the physical properties of Fe in a similar way [2.35], but Si is preferred in magnetic alloys because it is less prone than A1 to reaction with oxygen and is less expensive. On the other hand, Fe-A1 alloys are ductile, even when partial ordering occurs (above 7 8 wt% A1 concentration). The magnetostriction constant &00 suffers an approximately fivefold growth on increasing A1 up to 10wt% (from 20 x 10 -6 to ---90 x 10-6), thus following an opposite and unfavorable trend with respect to Si addition (see Fig. 2.13), and eventually drops around zero for A1 concentrations ---16 wt%. At the same time, the anisotropy constant follows a monotonic decrease and, depending on the degree of ordering, passes through zero around A1 = 11-14 wt%. Two compositions have applicative relevance. The Fe-(13 wt%)A1 alloy combines high magnetostriction with low anisotropy and is of interest for magnetoelastic transducers. The Fe-(17 wt%)A1 alloy is characterized by mechanical hardness and high permeability and is used in magnetic heads. The ternary Fe-(9.6 wt%)Si-(5.5 wt%)A1 alloy, known as Sendust, is characterized by an extremely soft magnetic behavior because the constants K1, ,~100,and ~111 all approach the zero value simultaneously. The coercive field can reach values around 1-2 A / m and the relative permeability is of the order of 105. Sendust alloys are extremely brittle and are therefore used in cast form for DC applications and as powder cores in AC devices.

2.4 A M O R P H O U S

AND NANOCRYSTALLINE

ALLOYS

Amorphous SMMs can be obtained as thin laminations by means of rapid solidification. Today, the predominantly employed rapid solidification technique is PFC, by which one can prepare ribbons of variable width (up to 100-200 mm) and thickness usually ranging between 10 and 40 ~m. In a typical PFC setup (Fig. 2.14), a quartz crucible, which holds the liquefied master alloy, is placed nearly in contact with the surface of a rotating metallic drum, which drags the liquid at a velocity of 10-40 m / s . This ensures a cooling rate of the order of 105-106 ~ sufficient to undercool

52

CHAPTER 2 Soft Magnetic Materials

the alloy through the glass transition temperature Tg, where it achieves the typical viscosity of a solid, retaining, at the same time, the disordered atomic arrangement typical of a liquid. Narrow ribbons can also be produced by means of the chill block melt spinning technique, of which PFC is a derivation (Fig. 2.14). With a more complex method, the so-called in-water-quenching technique, amorphous wires of 50-100 ~m diameter are prepared [2.36]. In this case, the molten metal jet is plunged into rotating water. The general composition of soft magnetic amorphous alloys is T70-80M30-20, where T stands for one or more of the transition metals Fe, Co, and Ni, and M is a combination of metalloids (e.g. B, Si, P, C). The metalloid atoms, with their radius much smaller than that of Fe, Co, and Ni, play an indispensable role as glass formers besides providing the eutectic composition required for the safe achievement of the amorphous state. Lack of crystalline order does not prevent the formation of ferromagnetic order. The existence of a large-scale magnetic moment is basically unaffected by disorder, although its strength is reduced by the presence of the metalloids [2.37]. The Ni atoms do not apparently contribute to the total magnetic moment and it is usual to classify magnetic glasses as Fe based or Co based. The Curie temperature Tc of an amorphous alloy is found to be lower than that of the corresponding crystalline alloy, with the remarkable property that, in the Fe-based materials, it decreases when the proportion of Fe is increased, pointing to an extrapolated minimum Tc value for hypothetical pure amorphous Fe. Figure 2.15 shows that the saturation magnetization decreases by a noticeable amount when passing from the Fe-based to the Co-based composition in the representative alloy Fe/Co80-xB20 [2.37]. The mean free path of conduction electrons in glassy metals is of the order of the interatomic distance and the electrical resistivity is consequently increased by a factor 2 - 3 with respect to their crystalline counterpart, and is quite independent of temperature. At the same time, plastic slip is highly limited, because no dislocation glide is possible and a very high fracture stress err ~" 2800MPa is obtained, with cumulated strain 8 r " ~ " 2.5%. The ratio E/or, where E is Young's modulus, attains a value 50, which is typical of high-strength materials and makes amorphous alloys interesting for structural applications. As summarized by the parameters presented in Table 2.7, the disordered atomic structure provides a unique combination of mechanical hardness and magnetic softness, the latter coming from a subtle mechanism which involves the competition between the exchange interaction responsible for ferromagnetic order and the local anisotropies. As mentioned in Section 2.2, one may describe an amorphous alloy as a random ensemble of structural units, each extending over a distance 3 equal to few atomic spacings, that

2.4 AMORPHOUS AND NANOCRYSTALLINE ALLOYS 40

53

.......................................

30 i

2

.o" - -0.

")1

~-- 2O o v

10

\

0

Fe8o.xCOxB2o -10

P

...................................... 0 20 40

|

60

80

Xco (at%)

FIGURE 2.15 Saturation polarization Js and saturation magnetostriction constant As vs. Co atomic concentration in Fes0-xCoxB20 amorphous alloys. When Xco "-" 75%, zero magnetostriction is achieved.

TABLE 2.7 Physical and magnetic properties of Fe78B13Si9 amorphous alloys and grain-oriented Fe-(3 wt%)Si laminations

Density (kg/m 3) Young's modulus (GPa) Yield stress (MPa) Fracture stress (MPa) Fracture strain (%) Vicker's hardness Electrical resistivity (12 m) Lamination factor (%) Curie temperature (~ Saturation polarization (T) Saturation magnetostriction DC coercive field (A/m) Maximum relative permeability 50 Hz power loss at 1.4 T (W/kg)

Amorphous ribbon Fe78B13Si9 (thickness 0.025 mm)

GO Fe-(3 wt%)Si (thickness 0.23 mm)

7.2 x 103 150 > 700 2800 2.5 800 135 x 10 -8 <90 410 1.55 32 x 10 -6 2 (after annealing) 2 x 105 (after annealing) 0.25 (after annealing)

7.65 x 103 120 < 300 350 25 180 45 x 10 -8 95 740 2.03 25 x 10 -6 (Aloo) 5 8 x 104 0.60

54

CHAPTER 2 Soft Magnetic Materials

have definite symmetry properties and are therefore characterized by a local magnetocrystalline anisotropy of streng4th K~ oriented at random. In 3d-based alloys, where K is of the order of 10 -10 J / m 3, the anisotropy strength is not large enough to force the local alignment of the magnetization to the easy axis of the individual structural units. Because of the exchange interaction, the magnetization maintains the same orientation over a correlation length L )> 3. The effective magnetocrystalline anisotropy of the material results from the average of the local anisotropies over distances of the order of L, which leads to a very low final value K0. According to Eq. (2.1) we obtain

Ko ~ K(3/L) 6,

(2.5)

where L = x/-A/K. For a structural wavelength 3--- 10-9m and a value A---10 -11 J/m, we can estimate an effective anisotropy K0--" 10 -610 -1 J / m 3, depending on the value of K. This negligible value of the average magnetocrystalline anisotropy is the key to the soft magnetic properties of the amorphous alloys. In fact, under these conditions, coercivity and permeability are due only to residual anisotropies of magnetoelastic origin, or induced anisotropies created by suitable treatments. A stress cr causes a uniaxial anisotropy Kr = (3/2)AsCr in a material characterized by the saturation magnetostricfion constant As. This value provides a sort of "a priori" indicator of the achievable ultimate soft magnetic properties of a given amorphous alloy. Figure 2.15 shows that, in the representative composition Fes0-xCoxB20, As strongly depends on the relative proportions of Fe and Co. It ranges from positive to negative values (from ---30 x 10 -6 to ---- 3 x 10 -6) on passing from the Fe-rich to the Co-rich side and intersecting the value As "" 0 at Fe concentrations around 5-8 at.%. In the highly magnetostrictive Fe-rich alloys, the random distribution of internal stresses introduced during the rapid solidification, typically of the order of 50-100 MPa, is the source of complex anisotropy patterns, with K values in the range of some 102103 J / m 3, and, consequently, of coercivity [2.38]. These stresses can never be completely relieved by annealing, as the treatment temperatures are in any case limited by the necessity to avoid the slightest precipitation of crystalline phases. Even after carefully controlled annealing under a saturating longitudinal field, the Fe-based ribbons reach, at best, coercive fields of 2-3 A/m. The influence of stress anisotropies becomes negligible in the vanishing magnetostriction Co-rich alloys. Accordingly, these materials exhibit the lowest energy losses and the highest permeability at all frequencies. In addition, their properties can be tuned to specific needs by suitable thermal treatments under a saturating magnetic field.

2.4 AMORPHOUS AND NANOCRYSTALLINEALLOYS

55

These can induce a large-scale anisotropy, Ku, as a consequence of localized atomic rearrangements having a definite directional order. Being the only form of anisotropy present in the material, Ku fully governs coercivity, permeability and loop shapes. Figure 2.2 provides an example of the magnetic softness and versatility of the near-zero-magnetostrictive Co-based alloys, which, prepared as very thin ribbons (8-15 ~m), favorably compete with ferrite and Fe-Ni cores up to the MHz region, where they can display initial relative permeabilities approaching 104 [2.39]. With a dominant transverse induced anisotropy Ku, the rotation of the magnetic moments is the chief magnetization mechanism and the associated permeability is

/d'r/d'0-

2Ku'

(2.6)

where/~0 is the vacuum permeability. With the domain wall processes basically suppressed, the loss is minimized and the best high frequency properties are obtained. The extra-soft magnetic properties of the Co-rich compositions are obtained at the expense of a substantial reduction of the saturation polarization with respect to the Fe-based alloys (0.9-0.5 vs. 1.3-1.6 T). This compounds with the obvious cost problems associated with the use of Co so that the related alloys are reserved for specialized applications. Amorphous wires prepared by the in-water-quenching technique exhibit a bistable magnetic behavior, regardless of the sign of magnetostriction. This property derives from the special domain structure that is formed in the wire, typically made of an active longitudinal core, reversing its magnetization with a single Barkhausen jump, and an outer shell, having either radial or circumferential domains. The origin of such a structure is to be found in the anisotropies induced by the large stresses frozen-in during the solidification process, in association with the anisotropy of magnetostatic origin (shape anisotropy). The switching-like behavior of the magnetization reversal in amorphous wires can be exploited in a number of applications, such as jitter-free pulse generators, digitizing tablets, speed and position sensors, and antitheft devices. A further remarkable property of amorphous wires is that their reactance at MHz frequencies can change to a large extent upon application of a DC field (giant magnetoimpedance effect). For instance, variations &X/X-~ 0.1-1 under an applied field of 100 A / m can be found in Co-based amorphous wires [2.40]. This effect originates in the strong variation of skin depth with the variation of the domain

56

CHAPTER 2 Soft Magnetic Materials

structure imposed by the DC field and has potential for many types of magnetic field sensors. Table 2.8 summarizes the behavior of the main physical parameters in a number of common amorphous alloys. Fe-based alloys are used in applications like the distribution transformer cores, where they can often replace the high permeability GO Fe-Si laminations. A total loss reduction by a factor 2-3 can be obtained at 50 Hz on passing from GO to amorphous Fe78B13Si9 laminations (Fig. 2.16). The loss analysis, schematically illustrated in Fig. 2.16b, demonstrates that this is due to a drastic reduction of the excess and the classical loss components brought about, according to Eqs. (2.3) and (2.4), by the combination of low ribbon thickness and high material resistivity. In recent years, increasing emphasis on energy saving has favored the introduction by electrical utilities of distribution transformers made of amorphous alloy cores, especially in the single-phase low power range (10-50 kVA). These devices are characterized by reduced total ownership costs, regarding both purchasing and operating costs through the device lifetime, as well as exhibiting good stability over time [2.41]. Less favorable economic conditions are attached to applications in three-phase power transformers. Significant use of amorphous alloys is made in electronics [2.42]. For instance, Co-based alloys are ideal as cores of inductive components to be employed up to frequencies of the order of I MHz, as found, for instance, in the switched-mode power supplies and in digital telecommunication circuits. Their low Js value is not a disadvantage in these cases, where, in order to limit core heating, the working induction is always kept small. The unique combination of high elastic limit and high magnetostriction in the Fe-based materials is exploited in high-sensitivity sensors and transducers. Further applications include electromagnetic interference filtering, magnetic heads, various types of magnetic shielding and ground fault interrupters. Amorphous alloys tend to crystallize heterogeneously upon annealing, with scattered nucleation and growth of microcrystals taking place at temperatures well below the bulk transition to the crystalline state. This has detrimental consequences on the soft magnetic behavior of the material, besides being associated with drastic mechanical embrittlement. Fe-based alloys actually need stress-relief annealing in order to achieve optimized magnetic properties, but if the treatment temperature is brought to values of the order of 400 ~ a sharp increase of the coercive field is observed due to the heterogeneous formation of microcrystals of 0.1-1 ~m size [2.43]. In Fe-Si-B alloys, however, it is possible to achieve both homogeneous accelerated grain nucleation and restrained grain coarsening by the addition of Cu and Nb [2.44].

o

~o

0

0

0

L'b 0 0

t~

O 0

0

e

"

I.~

o

0

0 O0 O'b ~'--~

t'xl 0 0

~. b~ ~.o o~b~ o"1

~ ~C-,~

eeeee

0

~ ~ d c 5 c 5 ~

~

0

t'xl P'~, I..~ O0 " ~ t'~l O~ O"b 0 r

Cxl txl ~--~

~ ~ . c 5 c 5 c 5

o~,

o ~o

o

~o~

un

~o~ o~

9

~o

o o .n9 ".n

2.4 AMORPHOUS A N D NANOCRYSTALLINE ALLOYS

o

,.-.-i

o ,.~ o

..~ t~

r~

I-i

t~

~~,,,i

v

o

57

58

CHAPTER 2 Soft Magnetic Materials .

"

.

.

!

.

.

.

.

!

.

.

.

.

!

[i; s0.z]

1.5

C3~

.

.

.

.

.

o /

/

Fe-(6.5 wt Yo)Si

1.0

v O~ O)

._o n O

0.5 ,,,

,,, ,, "

............. 0.0 0.75

.

.

.

.

i

.

.

.

AmorphousFe78B13Si9

.

i

1.00

(a) 1.50 r

.

,

,

1.25 Jp (T)

.

I

.

1.50

.

.

.

1.75

f= 50 Hz J_ = 1.25 T

P

1.25 "~ 1.00 ffJ

._o 0.75 O

O

a. 0.50 0.25 0.00 (b)

Ill NO (0.35 mm)

Fe-(6.5wt%)Si (30 mm)

GO

Amorphous

(0.30 mm)

(20 mm)

FIGURE 2.16 Power loss at 50 Hz as a function of peak polarization Jp (a) and its decomposition at Jp = 1.25 T (b) in a number of representative soft magnetic alloys: NO Fe-(3 wt%)Si laminations, 0.35 and 0.50 mm thick; GO Fe-(3 wt%)Si laminations, 0.30 and 0.23 mm thick; Fe-(6.5 wt%)Si rapidly quenched ribbons, 30 ~m thick; Fe78B13Si9 amorphous ribbons, 20 ~m thick. The excess loss component Pexr is the largest one in the GO laminations, whereas the classical component Pd is negligibly small in the high Si alloys and in the amorphous ribbons.

2.4 AMORPHOUS AND NANOCRYSTALLINEALLOYS

59

In particular, by treating amorphous ribbons with composition Fe73.5a homogeneous nanocrystalline structure is obtained, composed of oL-Fe-(---20 at.%)Si grains, having dimensions of the order of 10 nm, embedded in a residual amorphous matrix. The crystallites occupy about 70% of the material volume and are separated by amorphous layers 1-2 nm thick. Quite a similar structure can be obtained in alloys with composition Fe91_84(Zr, NB)7B2_9, the crystalline phase now being made of ~-Fe grains [2.45]. Since the grain size 3 is smaller than the correlation length L = x/A/K and the intervening amorphous phase ensures grain-to-grain exchange coupling, conditions similar to those found in amorphous structures are created, leading to vanishing crystalline anisotropy. In particular, with 3 - 10 nm, K = 104 J/m 3, and L = 50 nm, Eq. (2.5) yields a value of the average magnetocrystalline anisotropy K0-" 0.5 J/m 3. This feature is accompanied by magnetostrictive anisotropies averaging out to vanishing values. In fact, there is a balance in the material between the negative magnetostriction of the crystalline phase and the positive magnetostriction of the amorphous phase. Any applied or residual stress may generate anisotropies at the nanometer scale, having directions dictated by the nature of the stress (tensile/compressive) and the sign of the magnetostriction constant, but, again, the exchange interaction acts to suppress any mesoscopic and macroscopic anisotropy. Equation (2.5) suggests that the coercive field, which can be roughly estimated to be proportional to the average anisotropy constant K0, increases with the sixth power of the grain size 3. Experiments show that this relationship is verified to a good approximation up to 3---100 nm [2.46], a limit beyond which the coercive field starts decreasing with increasing 3. Febased nanocrystalline alloys emulate the properties of the amorphous Co-based alloys, with the advantage that one can deal with inexpensive raw materials while achieving higher saturation magnetization (e.g. 1.24 T in Fe73.5CulNb3B9Si13.5 and 1.63 T in Fe91Zr7B2 vs. 0.61 T in Co67Fe4B14.5Si14.5 and 0.86 T in Co71Fe4B15Si10) and improved thermal stability. The hysteresis loop of nanocrystalline alloys is sensitive to field annealing, although the ordering mechanism, investing the crystalline Fe-Si phase, is less effective than in the Co-based amorphous alloys. In any case, nearly linear, low-remanence hysteresis loops can be achieved by suitable treatments under transverse field, which yield low power losses and high permeabilities up to frequencies of several hundred kHz. An example of loss and permeability behaviors up to the MHz range in nanocrystalline Fe73.5CulNb3B9Si13.5 ribbons is provided in Fig. 2.17. They compare favorably with the properties of other types of soft magnetic alloys prepared for medium-to-high frequency applications.

CulNb3B9Si13.5 at a temperature around 550~

60

CHAPTER 2

103

.

dp= 0.2 T

.

.

.

.

.

i

s"

. . . . .

"

,',.';

G" .."; /

_.

,./" ~k,/'"

z~ S

\

aanocrystalline:

Z

/ ~/.;/'Permalloy ,,. , . ) i " ~F~based

ffJ

L 0

,",, ~" /, "

o ] 01 a.

S

9 , " ~ Co-based , amorphous alloy

t 10 0

.

.

.

.

10 4

105

'

.

.

.

.

o. 104 ! ._= .__.

I

.

.

105 Frequency (Hz)

. . . . . . .

I

~

..Q

alloy

amorphous alloy

f ,'/,

! ~ ~"

E

"./

9 ./.

Mn-Znferrite~

--~ 102

O

Soft Magnetic Materials

.

.

.

.

.

.

.

.

.

.

.

.

I

.

.

'

106 9

9

r---'-- Nanocrystalline

Co!based "":-.~~ / "'.."~

amorphous alloy

I

O

,, ~

Permall~

,_

-12

0

Mn-Zn ferrite

nr"

Fe-based amorphous alloy

103 103

~~~~

.

.

.

.

.

.

.

.

I

" "

I Hp= 0.4 A/m I .

.

.

.

.

.

.

.

n

104 105 Frequency (Hz)

|

FIGURE 2.17 Power losses (peak magnetization Jp = 0.2 T) and relative initial permeability (Hp = 0.4 A/m) in 18 ~m thick nanocrystalline Fe73.sCulNb3BgSi13.5 ribbons (solid lines). The ribbons have been suitably annealed under a saturating transverse magnetic field. Their properties are compared with those of Fe-based and Co-based amorphous ribbons and of Fe-Ni tapes of Permalloy type, all having comparable thickness. Data from a M n - Z n ferrite are also reported (adapted from Ref. [2.44]).

2.5 NICKEL-IRON AND COBALT-IRON ALLOYS

61

2.5 N I C K E L - I R O N A N D C O B A L T - I R O N ALLOYS Nickel-iron alloys display a broad range of magnetic properties and a well-defined structure in the range 35%-< N i - 80%. A stable random fcc solid solution (~-phase) is obtained above 35% Ni by a suitable choice of annealing temperatures, cooling rates, and the possible addition of elements like Mo, Cu, and Cr. In fact, the ~--* ~ phase transition on cooling from high temperatures occurs at T G 500 ~ and, because of the low diffusion rates, it can consequently be easily restrained, together with the formation of the ordered NiBFe phase. Structural stability and homogeneity are conducive to good mechanical properties and ease in cold rolling, down to thicknesses in the 5-10 ~m range. The variety of magnetic behaviors achieved in the final F e - N i laminations are rooted in the remarkable evolution of the intrinsic magnetic parameters with composition and treatment (see Fig. 2.18). It is noticed, in particular, that both the first magnetocrystalline anisotropy constant K1 and the magnetostriction constants ~100 and ~111 pass through the zero value in the Ni-rich side, with a positive K1 value (i.e. a (100) easy axis) co-existing with ~-phase up to ---75 wt% Ni. Compositions with Ni < 30 wt% are illdefined from the structural point of view and bear little interest as magnetic materials. The Fe70-Ni30 alloy is characterized by a singular drop of the Curie temperature Tc, which becomes of the order of room temperature. It increases in a near-linear fashion in the range 30-35 wt% Ni, a property that is sometimes exploited in magnetic shunt devices. At 36 wt% Ni concentration, Tc has already reached the value of 230 ~ the thermal expansion coefficient is extremely low (---1 x 10 -6 K -1, the socalled Invar behavior) and the resistivity is quite high (75 x 10-8 f~ m). This latter feature is conducive to low losses at high frequencies and makes the Fe64-Ni36 tapes interesting for applications like radar pulse transformers. The Fe50-Ni50 alloys are characterized by a high saturation polarization of 1.6 T. They can be prepared as strongly (100)[001] textured sheets by means of severe cold rolling to the final thickness (>95%) and primary re-crystallization annealing around 1000 ~ The favorable directional feature provided by the texture can be reinforced by the anisotropy Ku induced by means of a longitudinal magnetic annealing at a temperature T---450 ~ (To "" 500 ~ and followed by slow cooling. In this way, a squared hysteresis loop is achieved, to be exploited, for instance, in magnetic amplifiers and saturable reactor cores. By increasing the Ni content towards 55-60 wt%, the induced anisotropy Ku, of the order of 300-400 J / m 3, is not far from the value of the magnetocrystalline anisotropy K1. By annealing under a transverse

62

CHAPTER 2 Soft Magnetic Materials 2.0

,..

.,.

,,.

600

.,..

s s

,.,.

500

1.5

400 A

I-v~.aco 1.0

300

bY

200

0.5

/ I

100

i

0.0 30

~111

2O o, o

10

o

-10 -20

,

|

,

|

|

2 K 1 disordered A O3

E

,

0

..

,.,.

.

.

.

.

.

.

.

.

, . .

,

,,,

o

2

K 1 orde

~F-2

', I

t t

-3

e t

! t

-4

3o

Ni (wt %)

e

8o

FIGURE 2.18 Dependence of the magnetic intrinsic parameters on the Ni concentration in Ni-Fe alloys. Js, saturation polarization; To, Curie temperature; K1, magnetocrystalline anisotropy constant; A100 and Am, magnetostriction constants. Tc approaches the room temperature for Ni--- 30 wt% and interesting magnetic properties are observed only at higher Ni concentrations. The anisotropy constant depends, besides the composition, on the degree of structural ordering, associated with the formation of the Ni3Fe phase. It is thereby related to the annealing temperature and the cooling rate. A uniaxial anisotropy Ku is induced by cooling below Tc under a saturating magnetic field.

2.5 NICKEL-IRON AND COBALT-IRON ALLOYS

63

saturating field, one can therefore achieve a sheared low remanence hysteresis loop characterized by a large unipolar swing (0.9-1.2 T). This feature is welcome in devices like unipolar pulse transformers or ground fault interrupters. The highest permeabilities and lowest coercivities are obtained around 75-80 wt% Ni concentration because it is possible to approach vanishing values for both magnetostriction and anisotropy. Figure 2.18 shows that it is not actually possible to simultaneously achieve zero values for K1, &00, and Am. It is therefore expedient to make calibrated additions of elements like Mo, Cu, and Cr by which one can achieve, at the same time, an isotropic magnetostriction constant As ~" 0 and a good control of the FegNi phase ordering through annealing and cooling. Since the anisotropy constant K1 depends on ordering, it is possible to devise a thermal treatment leading to K1 "-" 0 [2.47]. A further advantage introduced by the additives is a substantial increase of resistivity (e.g. from 20 x 10 -8 to 62 x 10 -8 12 m by introducing 5 wt% Mo in 78 wt% Ni alloys) at the cost of a certain reduction of Is. As reported in Table 2.9, a coercive field lower than 1 A / m and relative initial permeabilities higher than 105 can be obtained in these alloys, generally known under the trade name of permalloys. A typical DC hysteresis loop in a permalloy tape is given in Fig. 2.19a, illustrating, through comparison with the loop of a GO Fe-Si lamination, a somewhat extreme example of magnetic softness. By field annealing at temperatures ranging between 250 and 380 ~ a substantial manipulation of the hysteresis loop can be obtained because the magnetocrystalline anisotropy can be overcome by the anisotropy Ku induced by magnetic ordering (less than 1 0 0 J / m 3, see Fig. 2.18). On then passing from longitudinal to transverse field annealing, the hysteresis loop shape may undergo a change as shown in Fig. 2.19b. This is associated with a change of the mechanism of the magnetization process, which becomes dominated by the rotation of the magnetic moments. It should be TABLE 2.9 Properties of some basic Fe-Ni and Fe-Co alloys

Fe64-Ni36 Fe50-Ni50 Fe15-Ni80-Mo5 Fe14-Ni77-Mo4-Cu5 Fe49-Co49-V2

Js (T)

Tc (~

p (10-812 m)

Hc (A/m)

/~i (103)

1.30 1.60 0.80 0.78 2.35

230 490 400 400 930

75 45 60 60 27

40 7 0.4 1.5 100

2 15 150 40 2

Js, saturation polarization; Tc, Curie temperature; p, electrical resistivity; Hc, coercive field;/~, relative initial permeability. Compositions are given in wt%.

64

CHAPTER 2 Soft Magnetic Materials 1.5 / (T) ......... ; "7"'~ s-

GOFe-SI

/'f

""" "- "--

.

t

;' 1.0~-

,

s

;'

9

"

~, / . /

0. !

I[1

~ Fe15-Ni80-Mo5

:

"

/i

.,,."/

--.1-: . . . . . . . . . . . .

H (A/m)

: _t.., -'.i --i ."5

0.75 Fe15-Ni80-Mo5

I

/

J (T)

~..

"Jll

.i

i'1/

Long. f i e l d ~ "i anneal ~ ' ~ ~I / / -10 -5 [,, i; !/

----

Transverse field anneal _ 5 10 H (Nm)

FIGURE 2.19 (a) DC hysteresis loops in GO Fe-(3 wt%)Si laminations and in permalloy (Fe15-Ni80-Mo5) tapes. (b) Loop shearing in permalloy by means of annealing under a transverse saturating magnetic field. remarked that K1 increases with decrease in temperature. Consequently, the treatments should be calibrated for the temperature at which the magnetic core is eventually employed. For instance, in permalloys for cryogenic applications, the annealing temperature and the cooling rate are conveniently reduced in order to achieve K1 "" 0 at such temperatures (i.e. K1 < 0 at room temperature) [2.48].

2.6 SOFT FERRITES

65

Cobalt-iron alloys do not display outstanding soft magnetic properties, but represent a unique solution in terms of Curie temperature and saturation polarization, both remarkably higher than in pure Fe. In the classical Fe50-Co50 alloy, we have, for instance, Tc = 980 ~ and Js = 2.40 T. This is useful for a number of applications where volume reduction and high working temperature may be required, as in the case of onboard high-speed generators for aircraft and spacecraft, without concern for the cost of Co. The Fe50-Co50 alloy transforms from fcc (~/) to bcc (o0 solid solution at 1000 ~ on cooling and undergoes rapid long-range ordering below 730~ The ~-~/ transformation can to some extent limit the re-crystallization process and solid state refining, while ordering detrimentally affects the mechanical properties, leading to a brittle material. Ordering can, however, be retarded by the addition of 2 wt% V and rapid quenching, so that Fe49-Co49-V2 alloys can eventually be prepared as thin sheets by cold rolling, with the further benefit of a large increase of resistivity with respect to Fe50-Co50 (27 x 10 -8 vs. 7 x 10-811 m). By adjusting the cooling rate, one can also dramatically affect the anisotropy constant, which can be made to approach the zero value, but the magnetostriction always remains extremely high, with A100 "" 150 x 10 -6 and '~111 "" 30 X 10 -6, which hinders the achievement of a really soft magnetic behavior. The value of the coercive field in regular Fe49Co49-V2 alloys (Permendur) is around 100A/m, with a relative permeability ---2 x 103. A substantial property improvement can be obtained by very careful control of the material purity and magnetic field annealing. The high purity alloy, called Supermendur, can exhibit Hc --" 10 A / m a n d / z --- 8 x 104.

2.6 S O F T FERRITES Soft spinel ferrites are largely applied at frequencies above the audio range, up to a few hundred MHz, because of their non-metallic character. They have the general composition MO.Fe203, where M is a divalent metal ion such as Fe 2+ , Mn2 + , Ni 2+ , Zn 2 + , Mg 2 + . Typical applications include pulse and wide-band transformers for television and telecommunications, inductor cores in switched-mode power supplies, antenna rods, cores for electromagnetic interference suppression, and magnetic heads. For frequencies in the range 500 MHz-500 GHz, the so-called microwave ferrites are employed. Some types of spinel ferrites, hexagonal ferrites (like BaFe12019), and garnets (like Y3Fe5012) belong to this class of materials. They are used in a variety of devices, such as waveguide

66

CHAPTER 2 Soft Magnetic Materials

isolators, gyrators, and modulators, to control the transmission or absorption of electromagnetic waves. The magnetic properties of ferrites are due to the magnetic moments of the metal ions. The i o n - i o n interaction is antiferromagnetic in nature and leads to the distinctive temperature dependence of the inverse of susceptibility shown in Fig. 2.20 [2.49]. The oxygen ions in spinel ferrites are arranged in a close-packed face-centered cubic structure and the small metal ions slip into interstitial positions, at either tetrahedral (A) or octahedral (B) sites, which are surrounded by four and six oxygen ions, respectively (Fig. 2.21). In a unit cell, which contains eight formula units (i.e. 32 0 2- ions, 16 Fe 3+ ions, and 8 M 2+ ions), 8 of the available A sites and 16 of the available B sites are occupied by the metal ions. When the M 2+ ions and the Fe 3+ ions are in the A and B sites, respectively, we have the so-called normal spinel structure. The inverse spinel structure is obtained when the 16 Fe 3+ ions are equally subdivided between the A

1/X

,so'* ,Is,,,~

(a) 0.75

(b)

T(K)

...................

0.50

0.00

T.

C~

0

25O

~l=e O

500

75O 1000 T(K) -

-

FIGURE 2.20 (a) Predicted non-linear temperature dependence of the inverse of susceptibility in a ferrimagnetic material [2.49]. The paramagnetic transition occurs at the N6el temperature TN. (b) Saturation polarization vs. temperature in a number of cubic ferrites [2.50].

2.6 SOFT FERRITES

67

.... @'%, B %

....-"

9 9%

~

%o

s

so

osoS~ oS oS / e s -,r.,..--

/

FIGURE 2.21 Portion (one-eighth) of a unit cell of a cubic spinel. The 0 2 - ions (dark) are arranged in an fcc structure. The metal ions (white) are interstitially arranged in tetrahedral (A) and octahedral (B) sites. and B sites, the latter being shared with the M 2+ ions. However, intermediate cases are very frequent. The spontaneous magnetization of ferrites and its temperature dependence were explained by N6el [2.49], by assuming that the spin moments of the metal ions in the A and B sublattices are antiferromagnetically coupled through indirect exchange interaction. Actually, since the cations are separated by the oxygen anions, direct exchange interaction between their 3d electron spins is negligible and an indirect coupling mechanism, the superexchange [2.50], is expected to take place. This involves the spins of the two extra 2p electrons in the 0 2- ion, which interact by direct exchange with the 3d spins of two neighboring metal cations. The mediating effect of the oxygen spins, which are oppositely directed, is such that, if the two cations have five or more 3d electrons (half-full or more than half-full 3d shell), their total magnetic moments are bound, according to Hund's rule, to antiparallel directions. This is the case of the common ferrite ions Mn 2 + , Fe 2 + , Co 2 + , Ni 2 + . The strength of the superexchange interaction is the greatest when a straight line connects the cations through the 0 2+ ion. The A - B coupling, which is associated to an A - O - B angle around 125~ is then much stronger than the A - A and B-B couplings, where the angles are 90 and 80 ~ respectively. At the end, one is left with a system made of two coupled arrays of antiparallel magnetic moments of unequal magnitude, which results in a net magnetic moment. This uncompensated antiferromagnetic behavior is called ferrimagnetism and the resulting magnetic moment per unit cell can be calculated through N6el's hypothesis. These calculations are in good agreement with the experimental values of

68

CHAPTER 2 Soft Magnetic Materials

the saturation magnetization and are s u p p o r t e d b y neutron diffraction experiments. A l t h o u g h the magnetic m o m e n t per formula unit m a y be very large in terms of n u m b e r of Bohr magnetons, the saturation polarization Js of spinel ferrites is low (typically a r o u n d or below 0.5 T at room temperature), because of the low density of the u n c o m p e n s a t e d magnetic ions. In addition, the temperature d e p e n d e n c e of Js, which results from the composition of the temperature variations of the magnetization of the individual sublattices, m a y give rise, according to N6el's theory; to a variety of behaviors [2.51]. Most magnetic spinel ferrites, like FeFe204, NiFe204, and CoFe204, are of the inverse type. In this case, the magnetic m o m e n t per formula unit equals that of the M 2+ ion because the Fe B+ ions are evenly distributed a m o n g the A and B sublattices. The ZnFe204 ferrite is of the n o r m a l type, but, since the Z n 2+ ion has a closed 3d shell and zero magnetic moment, it is paramagnetic at r o o m temperature. MnFe204 is an example of partly n o r m a l and partly inverse spinel structure, where the M n 2+ and Fe 3+ share in certain proportions the A and B sites. Table 2.10 provides a few examples of ion and m o m e n t distribution a m o n g sites in a few types of spinel ferrites. General properties are s h o w n in Table 2.11. The differences observed TABLE 2.10 Cation occupancy and magnetic moments in different types of spinel ferrites. (1) Inverse ferrite NiFe204 (Fe3+ cations in the tetrahedral A sites). (2) Mostly normal ferrite 1V[nFe204. (3) Normal ferrite ZnFe204 (no net magnetic moment). (4) Mixed ferrite ZnxMn(1-x)Fe204.The addition of the non-magnetic Zn 2+ ion increases the magnetic moment per formula unit. (5) The same occurs with the mixed ferrite ZnxNi(1-x)Fe204 Ferrite

Tetrahedral sites A

Octahedral sites B

Bohr magnetons per formula unit

NiFe204

Fe3+ ~ 5/~B

Ni 2+ 1"2/~B Fe3+ T5~B

2~B

MnFe204

Mn 2+ 10.8 x 5/~B Fe3+ ~ 0.2 X 5/~B

Mn 2+ T0.2 x 5/~B Fe3+ T0.8 x 5/~B Fe 3+ 1"5/~B

5/~B

ZnFe204

Zn 2+ ~ 0/U,B

Fe3+ T5/~B Fe3+ ~ 5/~B

0~B

ZnxMno_x)Fe204 Zn 2+ ~ 0/J,B Fe3+ T5/J,B Mn 2+ ~ (1 - x) x 5/~B Fe3+ T5/~B ZnxNi(1-x)Fe204

Zn 2+ ~ 0/d,B Fe3+ 1 (1 - x) • 5/.sB

(1 + x) x 5/~B

Ni 2+ T (1 - x) x 2/J,B (1 q- 4x) X 2p,B Fe3+ T5/J,B Fe3+ Tx x 5/~B

2.6 SOFT FERRITES

69

TABLE 2.11 Properties of some basic spinel ferrites

r/B,th r/B,exp Js (T) FeFe204 NiFe204

4 2 CoFe204 3 MgFe204 1 MnFe204 5

4.1 2.3 3.7 1.0 4.6

0.603 0.340 0.534 0.151 0.503

Zc (~

(103kg/m 3) (f~m)

/~

p

K1

As

585 585 520 440 330

5.24 5.38 5.29 4.52 5.00

10-5 102 105 105 102

- 12 - 7 200 -4 -- 4

40 - 26 - 110 -6 -- 5

(103j/m 3) (10 -6)

FeFe204, NiFe204, and CoFe204 are inverse spinel ferrites, MgFe204 is mostly inverse (90% of A sites occupied by Fe3+, 10% by Mg2+), MnFe204 is mostly normal (80% of A sites occupied by Mn 2+, 20% by FEB+). r/B,th and r/B,exp are the calculated and experimental magnetic moments at 0 K per formula unit (Bohr magnetons). Js, saturation polarization at room temperature; Tc, Curie temperature; 3, density; p, electrical resistivity; K1, anisotropy constant; As, saturation magnetostricfion. between predicted and measured magnetic moments are ascribed to a number of factors, such as imperfect quenching of the orbital magnetic moments, changes in the ion valence, and fluctuations in the cation distribution between the A and B sites. Such differences are small, in general, except for CoFe204, where the orbital contribution is important. Ferrites, being ionic compounds, are insulators in principle and display in practice a wide range of resistivity values, always orders of magnitude higher than in typical Fe-Si or amorphous alloys. The most important conduction mechanism is the transfer of electrons between Fe 2+ and Fe B+ ions in the octahedral sites. Magnetite (FeFe204) therefore exhibits a nearly metallic behavior, with resistivity p--- 10-5 f~ m. Most technical spinel ferrites are of the mixed type, where the presence of two or more metal ions M 2+, often introduced in non-stoichiometric proportions, can provide great versatility in the magnetic properties. M n - Z n and N i - Z n ferrites are the two basic families of mixed soft ferrites, where, by tuning the relative concentrations of the metal ions and making suitable additions and thermal treatments, material tailoring to specific applications can be achieved. Although, as previously stressed, normal ZnFe204 has zero magnetic moment, its addition to the inverse MnFe204 or NiFe204 ferrites leads to an increase of the global saturation magnetization at 0 K. This can be understood, in terms of N6el's theory, as due to the parallel alignment of the magnetic moments of the Fe 3+ ions in the B sites of ZnFe204, which is enforced by antiferromagnetic coupling with the ion moments in the A sites (see Table 2.10). The price one has to pay for mixing is a progressive decrease of the Curie temperature with

70

CHAPTER 2 Soft Magnetic Materials

increase of the ZnFe204 proportion. This is due to the weakening of the A - B coupling, as summarized for the Mnl-xZnxFe204 and Nil-xZnx Fe204 ferrites in Fig. 2.22 [2.52, 2.53]. The simple spinel ferrites, having cubic symmetry, generally display a negative value of the anisotropy constant K1 ((111) easy axis). This negative anisotropy derives, according to the single ion model, from the sum of the opposite contributions of the Fe 3+ ion moments occupying

,.,% 9

1.0

9

9

i

-

-

-

,

-

-

9

i

9

-

9

|

9

",'7

'

~l

0.8 ~,,,

0.6

~~ 1 ~ - ~

" 9,~

II I

0.4 0.2 0.0

0

200

(a)

400

600

800

600

800

T(K)

1.0 0.8

0.6 0.4 0.2 0.0 (b)

0

200

400 T (K)

FIGURE 2.22 Effect of Zn substitution on the saturation polarization and its temperature dependence in Mnl-xZnxFe204 (a) and Nil-xZnxFe204 (b) ferrites (from Ref. [2.53]).

2.6 SOFT FERRITES

71

the A and B sites, respectively, where the negative KIB term eventually prevails over the positive term K1A. This occurs because the orbital angular moment in the octahedral sites is not fully quenched by the crystal field. In CoFe204, however, K1 is large and positive ((100) easy axis), because the large spin-orbit coupling of the Co 2+ ions predominates (see Table 2.11). By acting on both the starting composition and the processing method, mixed M n - Z n and N i - Z n ferrites can be prepared having very low anisotropy values in a range of temperatures suitable for applications (20-100 ~ It has been shown that Zn substitution in Mn and Ni ferrites leads to weakening of the exchange field acting on the octahedral (B) Fe ions and, consequently, to weakening of the negative KIB constant on approaching the room temperature [2.54]. If, in addition, calibrated replacement of divalent cations with Fe 2+ or Co 2+ ions is made, one can combine the related positive anisotropy with the negative K1 value of the host in such a way that the resultant anisotropy constant K1 will cross the zero value around a convenient temperature (e.g. room temperature). Full anisotropy compensation is, for example, obtained at 300 K with the composition Mn0.s3Zn0.40Feo.07Fe2 2+ 3+04. As remarked in previous sections, a small anisotropy value straightforwardly leads to soft magnetic behavior. It can be stated, in fact, that the coercivity and the initial susceptibility approximately follow the relationships: Hc oc

K1/2 Js.<S> '

~i ~

j2

9 oc -K '

dw

oc

J2"<s> K1/2

(2.7)

'

where ~i ~ and Xdw are the contributions to the initial susceptibility deriving from coherent rotation of the magnetization and domain wall displacements, respectively, and <s> is the average grain size [2.55]. Besides the crystalline structure, applied and residual stresses and magnetic ordering induced by field annealing may contribute to the magnetic anisotropy. The effect of these various terms is summarized by the constant K in Eq. (2.7). According to this equation, the temperature stability of permeability, which is important in many applications, is determined by the dependence of Js and K on temperature. This can be controlled by acting on the addition of Fe 2+ (mainly in the M n - Z n ferrites) or Co 2+ (mainly in the N i - Z n ferrite). The highest permeabilities are reached in the M n - Z n ferrites, whereas somewhat lower values are obtained in the N i - Z n ferrites. However, the latter display much higher resistivities (some 107 vs. 10 -2 to ---10 f~ m, depending on the amount of doping with Fe 2+ in M n - Z n ferrites). The near-insulating character of ferrites is conducive to a nearly constant value of the initial susceptibility over many frequency decades, typically up to the MHz region in M n - Z n

72

CHAPTER 2 Soft Magnetic Materials 104

........

,

........

,

........

,

........

MnZnFe204

.jz" 1oa "

-d.

NiZnFe204

102

10= 0.1

.................... 1

, .............. 10

100

1000

f(MHz)

FIGURE 2.23 Dependence of the real (/~) and imaginary (/z') components of the initial permeability in selected commercial Mn-Zn and Ni-Zn ferrite samples (from Ref. [2.56]). ferrites and the 100 MHz region in N i - Z n ferrites, respectively. This is illustrated in Fig. 2.23, where the behavior vs. frequency of the real and imaginary parts of the relative initial permeability / z - / z ~ -i/z" are presented for selected industrial products [2.56]. In the low induction regimes and at sufficiently high frequencies, the dissipation of energy can be related to the phase shift 3 between J(t) and H(t) according to the equations: tan 3 (2.8) /~,, tan 3 - /z" W = "rrJpHp ~/1 + tan 23 , where W is the energy loss per unit volume and the field and the induction have peak amplitude Hp and Bp, respectively. Besides possible damping effects by eddy currents, which in anisotropy-compensated M n - Z n ferrites critically depend on the addition of Fe 2+ ions, resonant absorption of energy is generally invoked for the high frequency losses. In any case, whatever the predominant magnetization mechanism, either coherent magnetization rotation or domain wall displacements, it appears that high susceptibility and high limiting frequency of operation are conflicting requirements (see Fig. 2.23). By denoting with f0 the relaxation frequency, it is predicted, in particular, that for spin rotations f0x[ ~ oc J~ and for domain wall processes f0x~iWoc J2/(s)[2.57]. This suggests that

2.7 SOFT MAGNETIC THIN FILMS

73

the dispersion of the susceptibility is shifted towards higher frequencies in small-grained ferrites. The accurate control of the intrinsic and structural properties of spinel ferrites (e.g. ion valence, stoichiometry, grain size, and porosity) during material preparation is accomplished through the well-established routes of powder metallurgy. The conventional production process starts with the preparation of the base oxides, typically by calcination of suitable iron salts, and their mixing by prolonged wet grinding. This leads to a homogeneously fine powder, where the dimension of the single granules is around or less than I ~m. The resulting mixture is then dried and prefired in air at 900-1200 ~ During this stage, the spinel ferrite is formed by solid state reaction of Fe203 with the other metal oxides present (MO or M203). The so-prepared powders are then compacted, either by die-punching or hydrostatic pressing, and pieces of the desired shape are obtained. The filling factor of the so-obtained assembly of particles is around 50-60%. In the final main step, the pieces are brought to a temperature of 1200-1400 ~ in an oxidizing atmosphere, with or without application of external pressure. The desired final magnetic and structural properties of the material are thus achieved through: (i) particle bonding by interdiffusion and grain growth; (ii) densification, by elimination of the interparticle voids, up to -~95-98% filling factor; and (iii) chemical homogenization, by completion of unfinished reactions. The resulting product is hard and brittle and, if required, it is eventually machined with precision abrasive tools in order to meet the final tolerances.

2.7 S O F T M A G N E T I C

THIN FILMS

Trends towards miniaturization of components and devices and the need for soft magnets for high frequency applications are placing increasing emphasis on the preparation and the properties of thin soft magnetic films. Three applicative areas, in particular, benefit from the use of soft film cores: (1) magnetic recording heads; (2) sensors and actuators; and (3) high-frequency inductors. While with the thin-film geometry one cannot attain the extra-soft behavior of bulk materials, novel structures can be achieved and novel properties can be demonstrated. Thin magnetic films are prepared by a number of chemical or physical methods. Electroplating and CVD are the most frequently used chemical methods. Electroplating applies to conducting materials and requires metallic substrates. It can provide very high growth rates (up to --- 1 ~m/s), depending on the current density. The CVD method is based on the transport of the constituents of the film, usually as vapors of a halide

74

CHAPTER 2 Soft Magnetic Materials

compound, over the surface of the substrate. Ar is normally employed as the carrier of the compound. The plate to be coated, which is kept at a conveniently high temperature, is placed in a continually renewed gaseous environment, with which it reacts. Garnet single-crystal films are often prepared by this technique. The thin-film preparation method which is by far the most prevalent in the laboratory and in the industrial milieu today is the sputtering deposition. It is a flexible technique, which can be applied to a wide range of materials, both metals and insulators. It is characterized by good control and ease of change of the deposition parameters. Sputtering is based on a process of ejection of particles (normally atoms) from the surface of a target bombarded by ions and their deposition on a substrate. Figure 2.24 offers a schematic representation of a DC sputtering setup [2.58]. An inert gas (usually Ar), kept at a partial pressure of 0.1-1 Pa in a vacuum chamber (10-4-10 -6 Pa), is ionized in a strong electrical field and brought to a regime of self-sustained glow discharge. The positively charged ions are attracted by the target, made of the material to be deposited, where they extract by m o m e n t u m transfer either single atoms or atom clusters. Electrons generated by a heated filament are useful in assisting the glow discharge and a magnetic field can additionally confine the plasma (magnetic bottle), increasing the efficiency of the available

I coil I

VT

Vs

Target Ar

stmte

Vacuum pump

Th. emitter

FIGURE 2.24 Schematic representation of a DC bias sputtering setup for deposition of thin magnetic films. The target voltage VT is high and negative (around I kV or higher). The substrate voltage Vs is slightly negative with respect to the anode (---50-100 V). The glow discharge column, thermionically and magnetically assisted, is sketched at the center of the vacuum chamber (from Ref. [2.58]).

2.7 SOFT MAGNETIC THIN FILMS

75

electrons and the rate of ionic bombardment of the target. In a DC sputtering setup, the target voltage is high and negative (typically around I kV). The substrate voltage is generally kept slightly negative with respect to the plasma (typically - 5 0 to - 1 0 0 V) in order to favor light ionic bombardment of the deposit and the ensuing removal of impurities. The DC method cannot be used with insulating materials because the Ar ions would rapidly charge the target to a positive potential, repelling further incoming Ar ions. It is expedient in this case to resort to the RF sputtering technique. This is based on the application of a radio frequency voltage ( f = 13.56 MHz) to the target, which is then bombarded for a fraction of a period by the positive ions and for the remaining fraction by the electrons. The latter neutralize the positive charge left by the ions and the extraction process from the target can continue. Vacuum evaporation is widely employed as a thin-film deposition method. The material to be deposited is heated in a crucible and the vapor, resulting from sublimation or melt evaporation, is condensed on a substrate at a rate generally varying between I and 100 n m / s . Heating is realized in several ways, including the passage of a current in a refractory metal crucible (Ta, W, Mo) and the bombardment by an electron gun of the material, held in a cooled crucible. For a practical growth rate, the partial pressure of the obtained vapor must be in the range of some 10 Pa. For Fe this means, for example, a temperature around 1650 ~ A vacuum is required for two reasons. First, the mean free path of evaporated atoms must be high enough to permit their flight to the substrate without collisions. Second, contamination of the source and of the newly formed deposit must be avoided. A special variant of vacuum evaporation is the molecular beam epitaxy (MBE) method. By this term is generally meant a system with a base pressure of 10-9-10 -7 Pa and an "in situ" monitoring, layer by layer, of film growth. The reflection high energy electron diffraction (RHEED) technique, based on the real time analysis of the diffraction of an electron beam at a grazing angle trajectory, is used for this purpose because it does not interfere with the deposition process. MBE is especially suited for the preparation of epitaxial ultrathin films and multilayers. The term "epitaxy" defines a growth process where the crystallographic properties of the deposit and the substrate are tightly related (for instance, growth of a monocrystalline film on a monocrystalline substrate). Artificial lattices, a few atomic layers thick, with unique magnetic properties, rapidly varying with the number of atomic layers, can be created by epitaxial growth (e.g. Fe on Au and W, Ni on Cu and W, Co on Au and Pd) [2.59]. Using either MBE (for single crystals) or sputtering (for polycrystalline samples), complex and interesting structures, made of a sequence of

76

CHAPTER 2 Soft Magnetic Materials

few lattice spacing thick ferromagnetic layers separated by metallic non-magnetic or antiferromagnetic layers, can be prepared. The same exchange or superexchange interaction effects occurring between magnetic moments in bulk materials give rise to coupling between layers. Coupling can be conveniently modulated by changing the nature, the thickness, and the number of layers, subject to the requirement of excellent crystallographic quality of the deposited layers and their interfaces. It has been observed that the exchange interaction between two ferromagnetic transition metal ultrathin films separated by a spacer made of a non-magnetic transition metal (e.g. Mo, Ru, Pd) or a noble metal (e.g. Au, Ag, Cu) oscillates between antiferromagnetic and ferromagnetic coupling as a function of the spacer thickness. The example reported in Fig. 2.25a, regarding the behavior of the exchange coupling energy in Nis0Co20/Ru multilayers vs. the Ru spacer thickness, shows that the associated wavelength is in the nanometer range [2.60]. The coupling mechanism is assumed to be strictly similar to the indirect exchange interaction of magnetic impurities in a metallic host (RKKY coupling), which oscillates with distance between ferromagnetic and antiferromagnetic coupling. The spin polarized conduction electrons in the metallic non-magnetic spacer act as carriers of the interaction between the magnetic moments of the ferromagnetic layers. The antiferromagnetically coupled layers can have their magnetization forced into parallelism by a suitably high applied field. When this occurs, the resistance R of the multilayered structure is reduced to a considerable extent with respect to the antiparallel magnetization state, as first shown by Baibich et al. in [Fe (3 nm)/Cr (0.9 nm)]60 structures [2.61]. This is the so-called giant magnetoresistance (GMR) effect, where relative resistance changes AR/R around 100% and more can be achieved upon the field induced antiparallel-parallel transition of the magnetization in the layers. This is a much stronger variation than the classical anisotropic magnetoresistance (AMR) can provide, for example, by permalloy, whose resistivity is 2-3% lower along a direction normal to the magnetization direction than along the parallel direction. The current interpretation of GMR is based on the appraisal of the different role of the spin T and spin I conduction electrons in ferromagnets [2.62]. According to Mott's model, two conductivity channels are associated with these electrons and the total conductivity is cr = cr T +or 1. The conduction electrons can be scattered into localized d states; the higher the density of such states the larger the scattering probability. In Fe, Co, and Ni, the density of spin T states at the Fermi level is definitely lower than the density of the spin I states. The spin T channel is thus associated with low or negligible s - d scattering and consequently cr T >>or1. In a multilayer structure subjected

2.7 SOFT MAGNETIC THIN FILMS

77

0.15

,, !

NisoCo2o/Ru

i

!

0.10.

io

E E

I

|

!

i

i

I

!

c-

0.05,

o) cO O O O~ c0~

multilayer

i

!

antiferromagnetic

9 |

!

|

! I |

!

! I

c= 0.00

0 X I.U

oi

a, i i

_~

_

-,,O_.o.-~

-~o q,

-.

ferromagnetic

% . . . I"

i

. ,~"D",Q,

O'~,,

u

-0.05

|

0

(a)

.

.

.

.

1

u

u

2

3

.

.

.

- tRu (nm) 30

Ni81Fe19/Cu multilayer T=4.2K

.i I. II.

20

i i i i i

v

10

<1

~ , , '

i

!

n !

! !

!

§

i

i

l,,e.,., fi

i

0

(b)

. . . .

!

;

. . . .

I

i

i

~...~.~,..~-e -e.-o

;

. . . .

tcu (nm)

FIGURE 2.25 (a) Oscillating behavior of the exchange coupling in Ni80Co20/Ru multilayers as a function of thickness tRu of the Ru spacer. The films have been deposited by high-vacuum sputtering on a (100) Si substrate. The line is obtained by fitting the data points with the prediction of the RKKY model [2.60]. (b) Oscillating saturation magnetoresistance as a function of the Cu spacer thickness tcu in the multilayers Si/Ni81Fe19(0.5 nm) / [Ni81Fe19(1.5 nm)/Cu(tcu)]14/Ru(2.5 run). The peaks of &R/R correlate with peak values of the antiferromagnetic coupling constant (from Ref. [2.63]).

78

CHAPTER 2 Soft Magnetic Materials

to a high field, all layers are in the parallel state and the spin T conduction electrons form a low resistivity channel, while the spin I electrons suffer a strong scattering effect. At zero field, the magnetization is arranged antiparallel in the successive layers and the scattering asymmetry between spin T and spin1 electrons disappears because the low resistivity channel in one layer becomes a high resistivity channel in the next, and the final resistance increases. This mechanism evidently requires that the mean free path of the electrons is sufficiently high for the electrons to sample more than one magnetic layer. By changing the thickness of the spacer and, consequently, making the magnetic exchange interaction oscillate, the amplitude of the magnetoresistance effect oscillates, as shown in Fig. 2.25b for the structure [Ni81Fe19(1.5nm)/ Cu(tcu)]14. It is observed, in particular, that the saturation value AR/R is related to the strength of the antiferromagnetic coupling and is negligible for ferromagnetic coupling [2.63]. However, the requirement of strong antiferromagnetic coupling for GMR implies that large applied fields are involved (typically in the 105-106 A / m range), which is inconvenient from the viewpoint of many applications, like read magnetic heads and low field detection. Solutions have actually been devised in order to achieve GMR at (relatively) low fields (from a few 102 to a few 103 A/m), of which the so-called spin-valve is a widely investigated one [2.64]. A spin-valve multilayer has the general structure F1/S/F2/AF, where F1 and F2 stand for soft ferromagnetic layers, S is a non-magnetic metallic spacer, and AF is an antiferromagnetic layer (e.g. Fe-Mn, Pt-Mn, Tb-Co). F2 is strongly coupled to AF, while the spacer S is sufficiently thick to make F1 only loosely coupled with F2. The strong coupling between AF and F2 is known as exchangeanisotropy coupling and is a function of the anisotropy of the antiferromagnetic material. Its landmark feature is the biasing of the q, H) loop of F2 on the H-axis, with the bias field in the range of a few 104 A/m. The magnetization in the weakly coupled F1 layer is instead free to follow the direction of applied fields having relatively low amplitudes and accomplishes, in particular, the parallel-antiparallel magnetization transition that generates GMR. An example regarding the multilayer structure Ni81Fe19(5.3nm)/Co(0.3 nm)/Cu(3.2 nm)/Co(0.3 nm)/ Ni81Fe19(2.2 nm)/FeMn(9 nm)/Cu(0.1 nm) is illustrated in Fig. 2.26. Thin soft magnetic films are largely and increasingly employed in magnetic heads because they provide an excellent solution to highdensity and high-frequency magnetic recording. When used in inductive magnetic heads (both for writing and reading), they have to meet a number of requirements regarding their physical and magnetic properties: (1) high permeability in a wide frequency range; (2) low and controllable magnetic anisotropy; (3) low magnetostriction; (4) high

2.7 SOFT MAGNETIC THIN FILMS

~~

79

Ni81Fe19/FeMnspinvalve " ~ T=300K

C\/Ne,Fe,9,I

~4 o

."

v

2 -

-

-

-2000 -1000 0 1000 2O00 H (A/m) i

.

.

.

.

i

.

.

.

.

i

.

.

.

.

i

.

.

.

.

i

FIGURE 2.26 Room temperature relative resistance variation with applied field of a Si/NislFe19(5.3 nm)/Co(0.3 nm)/Cu(3.2 nm)/Co(0.3 nm)/Nis1Fe19(2.2 nm)/ FeMn(9 nm)/Cu(0.1 nm) exchange-biased multilayer. The shift of the AR/R loop indicates weak ferromagnetic coupling between the two Ni-Fe layers. The 0.3 nm thick Co layer inserted between the Ni-Fe layers and the spacer has the role of enhancing the magnetoresistance effect.

resistivity; and (5) high resistance to wear and corrosion. They should also fit easily into the standard integrated circuit technology. A schematic view of a single-turn thin-film inductive head is provided in Fig. 2.27a. Typical dimensions are: 1-4 ~m for the film thickness, 5-20 ~m for the gap width (i.e. track width), and 0.2 ~m for the gap thickness. In most cases, NislFe19 (permalloy type) thin films are employed as head material. By preparing them under a sufficiently high and constant magnetic field, applied in the plane of the film, a uniaxial anisotropy Ku is induced parallel to the gap length, typically around some 102 J / m 3. Given the small value of the magnetocrystalline anisotropy constant (see Section 2.5), an effective easy axis for the magnetization is obtained along a direction transverse to the flux path and the fields generated by the substrate will favor rotations with respect to domain wall displacements. From Eq. (2.6), it is expected that the related permeability/d, r is in the range of a few 103. By limiting the role of domain walls, the eddy current losses are reduced toward the classical limit (see Section 2.1), and a constant/d, r value up to the 10 MHz region can be achieved (with complete flux penetration). A drawback of these permalloy type heads is their poor wear and corrosion resistance. Alternative materials are Sendust (Section 2.3) and Co-Zr-based amorphous thin films. Amorphous alloys have a significantly higher

80

CHAPTER 2 Soft Magnetic Materials Coil

Ni-Fe thin film

(a)

=ecor0,n

I

(b)

FIGURE 2.27 (a) Scheme of a single-turn inductive thin-film magnetic head. (b) Magnetoresistive (MR) thin-film read-head. The combined effects of the uniaxial anisotropy Ku and of the bias field Hb drive the direction of the magnetization Js in the quiescent state at about 45~ to the sense current i. This direction is associated with the greatest magnetoresistive sensitivity. The bias field can be provided by a soft layer adjacent to the MR film, magnetized by i. The magnetoresistive signal originates from the passage of the MR film over the charged domain boundaries, emanating the field He. The stabilizing field Hs, applied along the easy axis, removes most of the domain walls, eventually suppressing the Barkhausen noise.

resistivity than permalloys (see Tables 2.8 and 2.9) and can therefore represent a good solution for working frequencies around 100 MHz and more. Layered structures, where magnetic and non-magnetic insulating layers alternate, both to eliminate flux closing domains, detrimental to the rotational process, and to counter skin effect have also been developed for the high frequencies. Examples of permeability behaviors vs. magnetizing frequency in different thin films for magnetic heads are given in Fig. 2.28 [2.65-2.67]. Magnetoresistive thin-film read-heads are especially suited to highdensity recording because of their compact structure, not requiring turns, their high sensitivity and their independence of the speed of the recording medium. They can be conveniently integrated with inductive write-heads. Permalloy films are typically employed as sensing material. Besides

2.7 SOFT MAGNETIC THIN FILMS 10000

..................

81 .................

4

: ~ ..Q (D d)

1000

> .m

,oa 1

rr

I O0 O. I

................................... I I0

I O0

1000

Frequency (MHz)

F I G U R E 2.28 Frequency dependence of the relative permeability in thin films for inductive heads. (1) Sendust layered structure (12 layers, 3 ~m thick) [2.65]. (2) Fe77Ga8Si15 film, 2 ~m thick [2.66]. (3) Sharply textured microcrystalline Co-FeSi-B film, 1 ~m thick, magnetized along the hard axis [2.67]. (4) Layered structure Ni-Fe(0.18 ~m)/SiO2(0.01 ~m), total thickness 0.72 ~m [2.65].

displaying a few percent anisotropic magnetoresistance, they have small magnetostriction and develop uniaxial anisotropy when deposited under a field. A schematic view of a magnetoresistive thin-film head is given in Fig. 2.27b. Typical dimensions of the MR sensing element are: thickness 10-50 nm, width 2-10 ~m, and height 0.5-2 ~m. Shape and induced anisotropy provide an easy axis along the track width and a stabilization field Hs removes the domain walls, thereby suppressing the Barkhausen noise. A bias field, usually provided by a soft magnetic layer magnetized by i, adjacent to the MR element (not shown in the figure), brings the magnetization Js in the quiescent state along a direction making an angle of about 45 ~ with respect to the sense current. This ensures the highest sensitivity when Js rotates under the action of the field He, emanating from the charged domain boundaries in the recording substrate. The discovery of GMR has opened novel perspectives in the field of magnetoresistive reading, prompting the industrial development of spinvalve heads [2.68]. These devices have the general structure of the exchange-biased multilayer shown in Fig. 2.26. In a typical configuration, the magnetization in the pinned layer F2 is directed normal to the substrate, while the magnetization in the free layer F1 is parallel to the substrate in the quiescent state. The passage of the head over the charged boundaries in the medium makes the magnetization in F1 flip up and

82

CHAPTER 2 Soft Magnetic Materials

down, according to the sign of the field He, therefore realizing the transition sketched in Fig. 2.26. Spin-valve thin films are basically on-off devices and their use as memory elements can be envisaged. It has been shown that spin-valve arrays can be prepared by means of standard lithographic techniques to form non-volatile magnetic random access memories (MRAMs). They are expected to be competitive in terms of performances, costs, and reliability with the semiconductor RAMs [2.69]. AMR and GMR in thin films and multilayers can be exploited in a variety of field sensors [2.70, 2.71]. Magnetoresistive sensors have some distinctive properties: good sensitivity, low source resistance, low stress sensitivity, and a wide range of AC and DC magnetic fields. A welldesigned AMR field sensor can have a sensitivity around 500 ~ V / ( A / m) and good linear behavior [2.58]. GMR sensors with similar and superior sensitivities can be developed for field measurements in the range 10-104 A / m [2.72, 2.73]. It can be said, in general, that, with the chance offered by thin-film technology to create novel heterogeneous structures, new possibilities arise across the whole field of magnetic sensors. For instance, trilayer structures of the type F / M / F , with F a soft magnet and M a high-conductivity non-magnetic metal, have been shown to exhibit a giant magnetoimpedance (GMI) effect [2.74]. Similar to the case of amorphous wires (Section 2.4), this effect consists in a very large variation (even four- or fivefold) of the impedance of the thin-film structure at a frequency 1-10 MHz when a DC field is applied. An example of GMI effect in a Co-Si-B(2 ~m)/Cu(3 ~ m ) / C o - S i - B ( 2 ~m) sandwich is reported in Fig. 2.29. The two amorphous soft ferromagnetic layers have transverse easy axis. The transverse permeability/-Lt, associated with the AC field generated by the current /AC, is small because the domain wall motion is damped in the MHz range and the impedance of the sandwich structure is nearly coincident with the resistance of the Cu layer. If a longitudinal field He, of value close to the anisotropy field, is applied, the magnetization is set free to rotate under the AC field and/d, t can increase remarkably, giving rise to a large increase in the impedance of the sandwich structure. It should be stressed that the low resistance value of the Cu layer is instrumental in achieving such an effect, which, contrary to the case of GMI in wires, is not related to the skin effect. Soft magnetic films in integrated circuits represent an excellent solution for small area inductors with a high quality factor in place of the traditional air-core spirals. Ferrite and permalloy films are typically employed for this purpose, but, with frequencies approaching the GHz range as found in mobile phone systems, stringent requirements

2.7 SOFT MAGNETIC THIN FILMS

83

&c

g

0.8 ;I

! !!

0.6-

..,-

.,'

"

i !!

J

i i i

.,

,"

0.4

,J "~,

eS

0.2.

0.0

\

"-,

o/

"o~ )1

~,,~s.o S

....o S 9

N

6

i

! :

i

~e~

"o~,,

""'.o,

Co-Si-B/Cu/Co-Si-B f = 1 MHz i^C,peak - 135 mA

9

!

.

.

-2000

.

.

i

.

.

-1000

.

.

i

0

.

.

.

.

i

.

.

1000

.

.

i

'

2000

H (A/m)

FIGURE 2.29 Impedance at 1MHz of a Co-Si-B(2~m)/Cu(3 I~m)/Co-SiB(2 t~m) trilayer as a function of the applied field H (from Ref. [2.74]).

regarding material composition (imposing, for instance, resistivities in excess of 1 0 - 6 ~ m) and inductor design must be taken into account. Amorphous (Fe-B-Si, C o - N b - Z r ) , granular Fe-A1-O, polycrystalline F e - N thin films, typically deposited on Si wafers, with thickness 0.11 I~m, are among the considered materials [2.75]. They should preferably magnetize by the rotational process, which can be obtained by applying the radio frequency field perpendicular to an easy axis defined by induced/shape anisotropy. Besides eddy current losses, ferromagnetic resonance (FMR) can be a limiting factor at 1 GHz and above. Since the resonance frequency/FMR ~ ('YlZo/2"rr)Hk, with Y the gyromagnetic ratio and Hk the anisotropy field, FMR can be displaced towards higher frequencies by increasing the anisotropy, in proper balance with the corresponding reduction of the permeability (i.e. the quality factor).

84

CHAPTER 2 Soft Magnetic Materials

References 2.1. W.F. Barrett, W. Brown, and R.A. Hadfield, "Electrical conductivity and magnetic permeability of various alloys of Fe," Sci. Trans. Roy. Dublin Soc., 7 (1900), 67-126. 2.2. Goss, N.P., Electrical sheet and method and apparatus for its manufacture and test, US Patent No. 1,965,559 (1933). 2.3. Elmen, G.W., Magnetic material, Can. Patent No. 180,359 (1916). 2.4. Snoek, J.L., Magnetic material and core, US Patent No. 2,452,531 (1948). 2.5. P.W. Duwez and S.C.H. Lin, "Amorphous ferromagnetic phase in ironcarbon-phosphorous alloys," J. Appl. Phys., 38 (1967), 4096-4097. 2.6. L. N6el, "Bases d'une nouvelle th6orie g6nerale du champ coercitif," Ann. Univ. Grenoble, 22 (1946), 299-319. 2.7. L.J. Dijkstra and C. Wert, "Effect of inclusions on coercive force of iron," Phys. Rev., 79 (1950), 979-985. 2.8. H.R. Hilzinger and H. Kronmtiller, "Statistical theory of the pinning of Bloch walls by randomly distributed defects," J. Magn. Magn. Mater., 2 (1976), 11-17. 2.9. A. Ferro and F. Fiorillo, Grain size dependence of the coercive force in very pure iron (Cardiff: The University College, Proc. 7th Soft Magn. Mater. Conf., 1985), 75-77. 2.10. R. Alben, J.J. Becker, and M.C. Chin, "Random anisotropy in amorphous ferromagnets," J. Appl. Phys., 49 (1978), 1653-1658. 2.11. C. Appino and F. Fiorillo, "A model for the reversible magnetization in amorphous alloys," J. Appl. Phys., 76 (1994), 5371-5379. 2.12. H. Morita, Y. Obi, and H. Fujimori, "Magnetic anisotropy of (Fe, Co, Ni)78Si10B12 alloy system," in Rapidly Quenched Metals (S. Steeb and R. Warlimont, eds., Amsterdam: North-Holland, 1985), 1283-1286. 2.13. H.J. Williams, W. Shockley, and C. Kittel, "Studies on the propagation velocity of a ferromagnetic domain boundary," Phys. Rev., 80 (1950), 1090-1094. 2.14. R.H. Pry and C.P. Bean, "Calculation of energy losses in magnetic sheet material using a domain model," J. Appl. Phys., 29 (1958), 532-533. 2.15. G. Bertotti, "Physical interpretation of eddy current losses in ferromagnetic materials," J. Appl. Phys., 57 (1985), 2110-2126. 2.16. IEC Standard Publication 60404-8-3, Specifications for Individual Materials--

Cold-Rolled Electrical Non-alloyed Steel Sheet and Strip Delivered in the Semiprocessed State (Geneva: IEC Central Office, 1998). 2.17. K. Ueno, I. Tachino, and T. Kubota, "Advantages of vacuum degassing of non-oriented electrical steels," in Metallurgy of Vacuum Degassed Steel

REFERENCES

85

Products (R. Pradham, ed., Warrendale, PA: Minerals, Metals and Materials Society, 1990), 347-350. 2.18. G. Lyudkovsky and P.K. Rastogi, "Effects of boron and zirconium on microstructure and magnetic properties of batch annealed Al-killed low carbon steels," IEEE Trans. Magn., 21 (1985), 1912-1914. 2.19. J.H. Bularzik, R.F. Krause, and R. Kokal, "Properties of a new soft magnetic material for AC and DC motor applications," J. Phys. IV (France), 8-Pr2 (1998), 747-753. 2.20. IEC Standard Publication 60404-8-2, Specifications for Individual Materials-Cold-Rolled Electrical Alloyed Steel Sheet and Strip Delivered in the Semiprocessed State (Geneva: IEC Central Office, 1998). 2.21. IEC Standard Publication 60404-8-4, Specifications for Individual Materials-Cold-Rolled Electrical Non-alloyed Steel Sheet and Strip Delivered in the Fully Processed State (Geneva: IEC Central Office, 1998). 2.22. H. Shimanaka, Y. Ito, T. Irie, K. Matsumura, E. Nakamura, and Y. Shono, "Non-oriented Fe-Si steels useful for energy efficient electrical apparatus," in Energy Efficient Electrical Steels (H.R. Marden and E.T. Stephenson, eds., Warrendale, PA: The Metallurgical Society of AIME, 1980), 193-204. 2.23. M. Shiozaki and Y. Kurosaki, "The effects of grain size on the magnetic properties of nonoriented electrical steel sheets," J. Mater. Eng., 11 (1989), 37-43. 2.24. P. Brissonneau, J. Quenin, and J. Verdun, A new sheet with a pseudo-cubic texture for application to large rotating machines (Cardiff: The University College, Proc. 7th Soft Magn. Mater. Conf., 1985), 209-211. 2.25. IEC Standard Publication 60404-8-7, Specifications for Individual Materials-Cold-Rolled Grain-Oriented Electrical Steel Sheet and Strip Delivered in the Fully Processed State (Geneva: IEC Central Office, 1998). 2.26. T. Yamamoto, S. Taguchi, A. Sakakura, and T. Nozawa, "Magnetic properties of grain-oriented silicon steels with high permeability Orientcore HI-B," IEEE Trans. Magn., 8 (1972), 677-681. 2.27. I. Goto, I. Matoba, T. Imanaka, T. Gotoh, and T. Kan, Development of a new grain-oriented silicon steels "RG-H" with high permeability (Cardiff: The University College, Proc. 2nd Soft Magn. Mater. Conf., 1975), 262-268. 2.28. H.C. Fiedler, "A new high induction grain-oriented 3% silicon-iron," IEEE Trans. Magn., 13 (1977), 1433-1436. 2.29. N. Takahashi, Y. Suga, and H. Kobayashi, "Recent developments in grainoriented silicon-steel," J. Magn. Magn. Mater., 160 (1996), 98-101. 2.30. T. Nozawa, M. Yabumoto, and Y. Matsuo, Studies of domain refining of grainoriented silicon steel (Cardiff: The University College, Proc. 7th Soft Magn. Mater. Conf., 1985), 131-136.

86

CHAPTER 2 Soft Magnetic Materials

2.31. F. Fiorillo, "Advances in Fe-Si properties and their interpretation," J. Magn. Magn. Mater., 157-158 (1996), 428-431. 2.32. K.I. Arai, K. Ishiyama, and H. Mogi, "Iron loss of tertiary re-crystallized silicon-steel," IEEE Trans. Magn., 25 (1989), 3949-3954. 2.33. D. Kohler, "Promotion of cubic grain growth in 3% silicon iron by control of annealing atmosphere composition," J. Appl. Phys., 31 (1960), 408S-409S. 2.34. T. Yamaji, M. Abe, Y. Takada, K. Okada, and T. Hiratani, "Magnetic properties and workability of 6.5% silicon steel sheet manufactured in continuous CVD siliconizing line," J. Magn. Magn. Mater., 133 (1994), 187-189. 2.35. J.C. Perrier and P. Brissonneau, "Some physical and mechanical properties of SiA1Fe alloys," J. Magn. Magn. Mater., 26 (1982), 79-82. 2.36. T. Masumoto, I. Ohnaka, A. Inoue, and M. Hajiwara, "Production of P d Cu-Si amorphous wires by melt spinning method using rotating water," Scripta Met., 15 (1981), 293-296. 2.37. R.C. O'Handley, "Fundamental magnetic properties," in Amorphous Metallic Alloys (EE. Luborsky, ed., London: Butterworths, 1983), 257-282. 2.38. H. Kronm~iller, N. Moser, and T. Reininger, "Magnetization processes domain patterns and microstructure in amorphous alloys," Anal. Fis., B86 (1990), 1-6. 2.39. M. Yagi, T. Sato, Y. Sakaki, T. Sawa, and K. Inomata, "Very low loss ultrathin Co-based amorphous ribbon cores," J. Appl. Phys., 64 (1988), 6050-6052. 2.40. L.V. Panina and K. Mohri, "Effect of magnetic structure on giant magnetoimpedance in Co-rich amorphous alloys," J. Magn. Magn. Mater., 1 5 7 - 1 5 8 (1996), 137-140. 2.41. V.R.V. Ramanan, "Metallic glasses in distribution transformer applications: an update," J. Mater. Eng., 13 (1991), 119-127. 2.42. C.H. Smith, "Applications of rapidly solidified soft magnetic alloys," in Rapidly Solidified Alloys (H.H. Liebermann, ed., New York: Marcel Dekker, 1993), 617-663. 2.43. G. Bertotti, E. Ferrara, E Fiorillo, and P. Tiberto, "Magnetic properties of rapidly quenched soft magnetic materials," Mater. Sci. Eng., A 2 2 6 - 2 2 8 (1997), 603-613. 2.44. Y. Yoshizawa, S. Oguma, and K. Yamauchi, "New Fe-based magnetic alloy composed of ultrafine grain structure," J. Appl. Phys., 64 (1988), 6044-6046. 2.45. A. Makino, A. Inoue, and T. Masumoto, "Nanocrystalline soft magnetic FeM-B (M--Zr, Hf, Nb) alloys produced by crystallization of amorphous phase," Mater. Trans. JIM, 36 (1995), 924-938. 2.46. G. Herzer, "Grain size dependence of coercivity and permeability in nanocrystalline ferromagnets," IEEE Trans. Magn., 26 (1990), 1397-1402.

REFERENCES

87

2.47. G. Couderchon and J.F. Thiers, "Some aspects of magnetic properties of N i Fe and Co-Fe alloys," J. Magn. Magn. Mater., 26 (1982), 196-214. 2.48. G.Y. Chin, "Review of magnetic properties of Fe-Ni alloys," IEEE Trans. Magn., 7 (1971), 102-113. 2.49. L. N6el, "Propriet6s magn6tiques des ferrites: ferrimagn6tisme et antiferromagn6tisme," Ann. Phys., 3 (1948), 137-198. 2.50. P.W. Anderson, "Antiferromagnetism. Theory of superexchange interaction, " Phys. Rev., 79 (1950), 350-356. 2.51. J. Smit and H.P.J. Wijn, Ferrites (New York: Wiley, 1959). 2.52. C. Guillaud, "Propriet6s magn6tiques des ferrites," J. Phys. Radium, 12 (1951), 239-248. 2.53. R. Pauthenet, "Aimantation spontan6e des ferrites," Ann. Phys., 7 (1952), 710-745. 2.54. A. Broese Van Groenou, J.A. Schulkes, and D.A. Annis, "Magnetic anisotropy of some nickel zinc ferrite crystals," J. Appl. Phys., 38 (1967), 1133-1134. 2.55. A. Globus, Magnetization mechanisms and specific polycrystalline properties in soft magnetic materials (Cardiff: The University College, Proc. 2nd Soft Magn. Mater. Conf., 1975), 233-248. 2.56. D. Stoppels, "Developments in soft magnetic power ferrites," J. Magn. Magn. Mater., 160 (1996), 323-329. 2.57. J. Gieraltowski and A. Globus, "Domain wall size and magnetic losses in frequency spectra of ferrites and garnets," IEEE Trans. Magn., 13 (1977), 1357-1359. 2.58. H. Hauser, J. Hochreiter, G. Stangl, R. Chabicovsky, M. Janiba, and K. Riedling, "Anisotropic magnetoresistance effect field sensors," J. Magn. Magn. Mater., 215-216 (2000), 788-791. 2.59. U. Gradmann, "Magnetism in ultrathin transition metal films," in Handbook of Magnetic Materials (K.H.J. Buschow, ed., Amsterdam: North-Holland, 1993), 1-96. 2.60. S.S.P. Parkin and D. Mauri, "Spin engineering: direct determination of the Ruderman-Kittel-Yasuda-Yosida far-field range function in ruthenium," Phys. Rev. B, 44 (1991), 7131-7134. 2.61. M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friedrich, and J. Chazelas, "Giant magnetoresistance of (001) Fe/(001)Cr magnetic superlattices," Phys. Rev. Lett., 61 (1988), 2472-2474. 2.62. A. Fert and P. Bruno, "Interlayer coupling and magnetoresistance in multilayers," in Ultrathin Magnetic Films (B. Heinrich and J.A.C. Bland, eds., Berlin: Springer, 1994), 148-186.

88

CHAPTER 2 Soft Magnetic Materials

2.63. S.S.P. Parkin, "Giant magnetoresistance and oscillatory interlayer coupling in polycrystalline transition metal multilayers," in Ultrathin Magnetic Fihns (B. Heinrich and J.A.C. Bland, eds., Berlin: Springer, 1994), 148-186. 2.64. B. Dieny, "Giant magnetoresistance in spin-valve multilayers," J. Magn. Magn. Mater., 136 (1994), 355-359. 2.65. T. Jagielinski, "Materials for future high performance magnetic recording," MRS Bull., 15 (1990), 36-44. 2.66. K. Hayashi, M. Hayakawa, W. Ishikawa, Y. Ochiai, H. Matsuda, Y. Iwasaki, and K. Aso, "New crystalline soft magnetic alloy with high saturation magnetization," J. Appl. Phys., 61 (1987), 3514-3519. 2.67. EW.A. Dime and M. Brouha, "Soft magnetic properties of microcrystalline Co-Fe-Si-B alloys prepared by sputtering," IEEE Trans. Magn., 24 (1988), 1862-1864. 2.68. C. Tsang, R.E. Fontana, T. Lin, D.E. Helm, V.S. Speriosu, B.A. Gurney, and M.L. Williams, "Design, fabrication, and testing of spin valve read heads for high density recording," IEEE Trans. Magn., 30 (1994), 3801-3806. 2.69. S. Tehrani, J.M. Slaughter, E. Chen, M. Durlam, J. Shi, and M. DeHerrera, "Progress and outlook for MRAM technology," IEEE Trans. Magn., 35 (1999), 2814-2819. 2.70. D.J. Mapps, "Magnetoresistive sensors," Sens. Actuators A, 59 (1997), 9-19. 2.71. P.P. Freitas, F. Silva, N.J. Oliveira, L.V. Melo, L. Costa, and N. Almeida, "Spin valve sensors," Sens. Actuators, 81 (2000), 2-8. 2.72. J.M. Daughton and Y.J. Chen, "GMR materials for low field applications," IEEE Trans. Magn., 26 (1993), 2705-2710. 2.73. L. Vieux-Rochaz, R. Cuchet, and M.H. Vaudaine, "A new GMR sensor based on NiFe/Ag multilayers," Sens. Actuators, 81 (2000), 53-56. 2.74. T. Morikawa, Y. Nishibe, H. Yamadera, Y. Nonomura, M. Takeuchi, and Y. Taga, "Giant magneto-impedance in layered thin films," IEEE Trans. Magn., 33 (1997), 4367-4372. 2.75. V. Korenivski, "GHz magnetic film inductors," J. Magn. Magn. Mater., 215-216 (2000), 800-806.

CHAPTER 3

Operation of Permanent Magnets

Hard magnetic materials are useful because of their ability to retain firmly their magnetized state against external fields. This property can be used either for storing information or for producing permanent fields in a suitable region of space. We talk about magnetic recording media in the first case and of permanent magnets in the second. There is no basic conceptual difference between these two functions, were if not for the scale involved. Recording media can be seen as a vast assembly of microscopic permanent magnets, which are made to operate independently. Permanent magnets operate as a whole. There is also an obvious difference regarding the strengths of the fields involved. Recording requires writing and erasing, i.e. reasonable coercive fields. They are typically in the range 1042 x 10 A / m . With modern rare-earth based permanent magnets, coercive fields as high as 2 x 106 A / m can be achieved in technical products. What chiefly matters in permanent magnets is the energy density available in the useful region of space. This depends both on the intrinsic properties of the magnet and on the geometry of the magnetic circuit. In this chapter we shall not attempt to discuss the physical properties of hard magnetic materials. We will limit ourselves to summarizing the basic problems encountered when performing operations with permanent magnets because of the relationship they have with the methods of magnetic characterization and magnetic field generation. They include general energetic considerations and the concept of the magnetic circuit. In recent times, the review literature regarding the physical properties of permanent magnets has grown considerably and a number of excellent specialized books are now available. This state of affairs contrasts somewhat with a lack of recent comprehensive treatises on soft magnetic materials. The different approach to the discussion on soft and hard magnets followed in this book is therefore justified in the light of the different situation in the pertaining literature. For a comprehensive treatment of properties, preparation, and applications of permanent 89

90

CHAPTER 3 Operation of Permanent Magnets

magnets and magnetic recording media, the reader might refer to the following textbooks and monographs [3.1-3.12].

3.1 MAGNETIC CIRCUIT A N D ENERGY PRODUCT The ability of a permanent magnet to provide a defined and stable magnetic field in a certain region of space depends, besides the material properties, on the specific features of the magnetic circuit employed. This means that the shape of the magnet must be optimized for the required application, often in combination with flux-guiding soft magnetic parts. Since the magnet operates as an open magnetic circuit, its basic features can be illustrated by representing it as a gapped ring of constant crosssectional area Sm (Fig. 3.1a), where the rather strong approximation of flux uniformity along the whole circuit is made. This simplifies the whole

(a)

4-----'-.... #m .-.-,...,-.~

(b) FIGURE 3.1 (a) Schematic representation of a permanent magnet as an open ring with homogeneous field in the gap. (b) Example of a practical magnetic circuit, where the flux generated by a cylindrical (or parallelepipedic) permanent magnet of length l m is conveyed in the gap through soft pole pieces.

3.1 MAGNETIC CIRCUIT AND ENERGY PRODUCT

91

treatment because the magnetic quantities involved can be treated as scalars. In particular, the continuity equation for the flux cI) imposes the same value of the magnetic induction in the material and in the gap B = BI -- Bg. On the other hand, the line integral of the field along the average circumferential length l = Im at- lg is, according to Amp6re's law, equal to zero, which means to say that Gig

-- - H m l m

(3.1)

if H m and Hg are the fields in the material and in the gap, respectively. It is understood that the ring is sufficiently slender to ensure a meaning to the value of the magnetic path length l. Hm is the demagnetizing field, directed against the polarization Jm in the material, and, for the made assumption of uniform induction, the related demagnetizing coefficient Nd is a constant Hm = - Nd Jm. /-to

(3.2)

The field in the gap is thus a function of the polarization Jm and the circuit geometry. In fact, by equating Bm and Bg and using Eq. (3.1), we obtain lg Bm - Bg =/a, oHg --/~oHm q- Jm -- -/zoHg ~m q- Jm

(3.3)

and Hg --

1 Im/lg Jm. t2,o 1 + lm/lg

(3.4)

The smaller the gap, the closer the free poles of opposite sign on the facing ends of the ring. This has a compensating effect on the demagnetizing field Hm __ _ Jm. lg /-t0 Ig+lm which decreases and brings about an increase of the field in the gap. In the narrow slit limit Im/lg >> 1, Hg eventually saturates to Jm

Hg ~ - - , /.t0

(3.5)

t

the maximum value for the fi~ld in the gap. If Eq. (3.2) is introduced in Eq. (3.3), Hg can be expressed equivalently as a function of the demagnetizing coefficient |

~=

1 - Nd

/.to

lm

(3.6)

92

CHAPTER 3 Operation of Permanent Magnets

with Nd related to the magnet geometry as

1 N d = 1 + t /ttm/tg

(3.7)

A gapped ring is an ideal device, a practical one being more like the circuit shown in Fig. 3.1b, where the flux generated by a cylindrical or parallelepipedic permanent magnet is conveyed in the gap by a pair of soft pole pieces. Normally, these poles are made of very soft Fe and act, thanks to their high saturation polarization and high (ideally infinite) permeability, as virtual short-circuits for the flux. The previous equations still appl)~ with the provision that, if the cross-sectional area of the gap Sg is different from that of the magnet Sm, Eq. (3.4) modifies in the following

1 lm/lg tzo 1 + Sglm/Smlg Ira"

Hg

(3.8)

The basic aim of a permanent magnet circuit is clearly one of delivering the maximum energy in the gap for a given magnet volume (i.e. material cost) or, equivalently, devising the configuration which minimizes such a volume for a given energy in the gap. This means finding the optimal working point P(Hm, Bm) on the magnetization curve in the second quadrant (see Fig. 3.2). Such a point is obtained at the intersection of the curve with the load line, as determined by Eqs. (3.2) and (3.3) 1 -- 1 )Hm. Bm -- --/.l,O(~d

(3.9)

The slope of this line is often referred to as the permeance coefficient. To find the energy stored in the field outside the magnet we start by considering the whole magnetostatic self-energy of the system. By treating the magnet as an ellipsoid of volume Vm magnetized along one of its principal axes, we obtain Eros ----"- 2I f v~ Hm'JmdV=

~-~oNdJ2mVm.

(3.10)

Eros can be regarded as the sum of the field energy stored inside the magnet

1

E m - ~/z0

f

Vm

H2dV = 2-~-~ 1 ~/'2r2 t~djmwm

(3.11)

3.1 MAGNETIC CIRCUIT AND ENERGY PRODUCT B, J ,

I

93

/Br=Jr

,-i7 " ~

i,p ~ 0

s

Hca//

'NHm

BH

FIGURE 3.2 Return curve in the second quadrant and related energy product (BH) (conventionally taken as positive) in a permanent magnet. The working point of the magnet P(Bm, Hm) is obtained at the intersection of the curve with the load line of equation Brn =--/~0((1/Nd)--1)Hm, where Nd is the demagnetizing coefficient. The shaded area represents the energy stored in the gap divided by the magnet volume E g / V m (see Eq. (3.13)). The dash-dot line of equation B m H m = const, identifies the point P0 where the energy product is maximum. and the energy stored outside it, possibly confined to a good extent within the gap. This external energy is then given by the difference

1 j 2 ( N d _ N~)Vm. Eg-- Ems - Em - 2~o

(3.12)

If we pose, according to Eq. (3.3), Jm = B m - p,0Hm in this equation, we can equivalently write 1 Eg ---- - ~ (BmHm)Wm.

(3.13)

The larger the quantity (BmHm) ~ called the energy product, the smaller the volume of the magnet required to achieve a given energy Eg in the gap. As shown in Fig. 3.2, the energy product passes through a m a x i m u m value (BH)max (conventionally taken as positive) along the return curve. It is expedient to find the optimal working point P0 on this curve by looking for the intercepts with the family of hyperbolic functions (BH) - c o n s t . In the ideal magnet, the q , H ) loop is rectangular (i.e. J~ = Jr~ with Jr the remanent polarization), and the coercive field IHc/i > Jr/i~o. The (B,H) curve in the second quadrant reduces then to a straight line of slope/~0

94

CHAPTER 3 Operation of Permanent Magnets

Br=4 ~._1B

H

\

-Br/21.1,o

o

BH

FIGURE 3.3 Return curve and energy product in an ideal permanent magnet. If the material is brought to the saturated state (Jr - Js), the maximum theoretical energy product (BH)max = j2s/4lZo is attained at the working point P0.

and HcB---Jr/I.to (see Fig. 3.3). Equations (3.4)-(3.6) then provide the field in the gap as function of Jr. Equation (3.12) provides the maximum Eg value for Nd = 1/2, which corresponds to a load line of slope equal to -/.to, with intercept at the point Po(-Br/2t.to, Br/2 ). Notice that the condition B m / H m -- ~ at P0 implies, according to Eq. (3.7), that lm = lg. For a cylindrical magnet, a magnetometric demagnetizing factor N ~ d 1/2 corresponds to a height-to-diameter ratio r = 0.45 [3.13]. To work near its (BH)max point, this magnet must withstand quite high demagnetizing fields, which is possible with m o d e m materials. For example, a N d - F e - B magnet has a typical intrinsic coercive field Hcj "" 106 A/m, while with Jr "-1.2 T and Nd = 1/2 it is subjected around the (BH)max point to a counterfield of the order of 0.5 x 106 A / m . An intrinsic upper limit to the energy product is envisaged for the ideal magnet. In fact, when Jr = Js and Nd = 1/2, the working point is Po(-Js/21zo,Js/2) and, consequently, (BH)max -

4/z0

.

(3.14)

For a hypothetical material with Js = 2 T, this upper limit is around 800 k J / m ~ In actual magnetic circuits, allowance should be made for flux leakage and non-ideal soft magnetic properties of the pole pieces. This can be taken into account phenomenologically, for example, by introducing a leakage factor/3 and a reluctance factor a, such that the flux continuity

3.2 DYNAMIC RECOIL

95

equation and the Amp6re's law can be re-stated as

Bg = [3Bm~

Hglg = -crHml m.

(3.15)

Severe flux spreading around the gap can actually take place and the factor fl can be as low as 0.1. On the other hand, conventional design can keep the factor c~around 0.7-0.9. On account of this, the expression for the field in the gap is re-written as

1 Odm/lg Hg = ~o 1 4- odm/[3lg Jrn.

3.2 D Y N A M I C

(3.16)

RECOIL

A standard requirement of many applications involving permanent magnets is that a stable and high magnetic field is available in a certain region of space, either uniform or with a defined spatial profile, so that the working point P(Bm, Hm) is fixed. Other applications, however, including, for example, holding devices, loudspeakers, and motors, require movable soft magnetic parts or variable fields; which implies that the position of the working point changes cyclically with time. One speaks in this case of dynamic recoil conditions. An example is shown in Fig. 3.4, illustrating dynamic recoil in a lifting device. When the movable armature is far removed from the magnet, the load line, determined according to Eq. (3.9), identifies the working point P on the return curve. If the armature is made to approach the pole pieces, the flux divides between the gaps gl and g2, the demagnetizing field decreases and the working point correspondingly recoils towards H m --0 (the ordinate axis), which is reached at point Q when the armature comes in close contact with the magnet. The recoil path does not follow the return curve in reverse. This is obviously related to the properties of hysteresis. In the special case of ideal magnets, however, where the return curve in the second quadrant is linear (no associated variation of magnetization), there is only one (Bm~Hm) linear path, of slope/~0, to be followed (see Fig. 3.3). In a real magnet the recoil takes place, as shown in Fig. 3.4, along the lower branch of a very narrow minor loop, which is closed on removing the armature again (or at least upon a few stabilization cycles). It is normally assumed that the effect of hysteresis is not relevant and the recoil path is taken as a straight line (PQ in Fig. 3.4), whose slope is called recoil permeability. Under these conditions, the reversible conversion of magnetic energy into mechanical energy and vice versa can be assessed following

96

CHAPTER 3 Operation of Permanent Magnets

P"/.."- 7 <.2-. ~- ' ~~ ~ , , , ' \ ' ,. \. . ,,.,- . i! /ii "-..:--.. ..":. x..."... ~ ii ""-.. " "I

H

E

O'

FIGURE 3.4 Dynamic recoil in a permanent magnet and example of application to the behavior of a lifting device. Here the flux divides between the gaps gl and g2 according to the distance of the mobile armature from the pole pieces. The working point A correspondingly moves along the reversible recoil line PQ. The magnetic energy converted into mechanical work (per unit volume of the magnet) on going from P to A is given by the area of the triangle OPA. The useful recoil energy, equal to half the area of the shaded rectangle AA1A2A3, is maximum when the load line OA bisects PQ. In an ideal magnet the recoil line has slope/x0 and coincides with the return B(H) curve.

the displacement of the working point A on the recoil line. Let us therefore consider an infinitesimal displacement of the armature and the related mechanical work dW. Such a displacement has the effect of changing the flux closing conditions through gl and g2 and the working point on the recoil line undergoes a small shift (for example from P to P' in Fig. 3.4). For a non-ideal magnet, this implies a variation dJm of the polarization and, consequently, of the relevant internal energy terms (i.e. anisotropy and magnetoelastic energy). At the same time, the magnetostatic self-energy (Eq. (3.10)) is affected. Making reference to a unit volume of the magnet, we write the internal energy variation as dEi = HmdJm, while the magnetostatic term is expressed, according to Eq. (3.10), as 1 ) 1 1 d E m s - - d - ~Hm.Jm -- - ~HmdJm - ~JmdH m.

(3.17)

The total variation of the magnetic energy is then 1

dE = dE i + dEms = ~- (H mdJm - JmdHm)"

(3.18)

3.2 DYNAMIC RECOIL

97

However, Jm = Bm -/~0Hm and we eventually obtain dE-

1 -;(HmdBm - BmdHm). z

(3.19)

If we refer, for example, to the elementary displacement PP~ in Fig. 3.4, by simple geometrical arguments we find that the free energy variation dE, which is converted into the mechanical work d W, is equal to the area of the triangle POP ~ [3.14]. The total mechanical work per unit volume provided by the magnet on attracting the armature from a distant position to full contact with the pole faces is thus the area of the triangle POQ. The theoretically highest possible value of the work is then predicted for the ideal magnet 1 1 2 W -- -~ HcBB r - ~ Js, (3.20) where Jr = Js and the recoil permeability is/,to, when P coincides with the coercivity point HcB. This condition is not accessible, however, because it would imply a demagnetizing coefficient Nd -- 1. For a generic working point A on the recoil line, the (BH) product is the sum of two contributions, one associated with the leakage flux OA 2 in the gap gl and the other with the useful flux A2A 1 in the gap g2. The latter is given by the area of the rectangle AA1A2A3 (shaded region in Fig. 3.4), whose half value represents the useful energy available in g2. It is easily verified by geometrical considerations that this area is a maximum when the point A is halfway between P and Q. Under these conditions, it is, with the working point P(Hm, Bm), a a I - - 1H m and AIA2 = I O Q . In the ideal magnet it is A1 A2 -- 89 Jr and the useful maximum energy in the gap g2 per unit volume of the magnet is Eg,max - -~H m J r. A maximum force is exerted on the armature when the circuit is closed (point Q). If, under these conditions, the armature is subjected to an infinitesimal displacement dl, the work made dW - F dl is balanced by the creation in the gap of volume Vg of the magnetic energy dEg = ~ 1 B g d V g (for two pole faces of cross-sectional we obtain the force per unit surface

area

1 ~ B2gSm2d l 2/1, Sm).

2/2,0 '

By equating dW and dEg,

3"21

which is of the order of 0.4 MPa when Bg = 1 T. The force decreases by moving the armature further away, in a way that can be estimated by first

98

CHAPTER 3 Operation of Permanent Magnets

relating the evolution of the load line OA with the gap length lg according to Eq. (3.7) and then calculating the derivative with respect to lg of the area AOQ [3.15].

3.3 E L E C T R I C A L A N A L O G Y

NUMERICAL

AND

MODELING

A popular way to treat magnetic circuits is to assimilate them to electrical circuits. This method is justified on the grounds of the analogy between the line integrals of the magnetic field in the absence of external currents and of the electrostatic field in the absence of time-dependent magnetic fields ~ E.dl = 0. The continuity equation regarding the electrical current i is paralleled by that of the total magnetic flux ~. A permanent magnet, like a battery, provides a contribution to the line integral of the field opposite to that of the rest of the circuit and can be considered as the source of a magnetomotive force. The analogy is then completed with the introduction of the reluctance, defined as the ratio between the potential drop qOi "-" f li H.dl and the flux ~i over a piece of circuit of length l i. Assuming qOi -" N i l i and Id,i - - c Y P i / S i H i (and considering B(H) a singlevalued function), we can express the reluctance of the ith portion of the circuit of length li, cross-sectional area Si and permeability/zi as ~i-

qOi (i)i

1 li [a'i Si

(3.22)

In many cases the linear approximation (/d,i constant) is adequate and the solution of the electrical analogue is carried out immediately by applying Ohm's law to a network of reluctances having a priori defined values. In general, however,/d, i depends on the actual value of ~i and an iterative approach is required. Solving magnetic circuits by the conventional methods of the electrical circuits associates simplicity with obvious limitations because one cannot envisage a perfectly insulating surrounding medium. One can, however, approximately account for this effect by introducing leakage reluctances in the equivalent circuit. Figure 3.5 illustrates how the electrical analogy can be pursued in a simple magnet with soft pole pieces. The magnet, of length Im and cross-sectional area Sin, generates a magnetomotive force Fm and has internal reluctance ~tm = (1/la, m)(lm/Sm), where ~ = / . L 0 / d , r is the permeability at the working point. In the ideal magnet, /Zm--/z0 (see Fig. 3.3). Fm coincides with the magnetomotive force appearing between the points 1 and 2 when the external reluctance is infinite.

3.3 ELECTRICAL ANALOGY AND NUMERICAL MODELING

99

1

o

~m

1.. li I j

!

T

- - - 2 . . . . 1 .....

FIGURE 3.5 Permanent magnet with soft limbs and equivalent circuit. The magnet is described as a source of magnetomotive force Fm with internal reluctance 9tm. The reluctance 9tl, in parallel with the gap reluctance 9tg, accounts for magnetic flux leakage. For a leakage factor/3 = (I)m/(I)g ~ w i t h (I) m and ~g the flux in the magnet and in the gap, respectively, it turns out 911 = 03/1 - ]3)91g.

This limit is approached w h e n the m a g n e t has such a lateral extension that N d = 1 and the working point is P(Hm--HcB, B m - - 0 ) , leading to F m - - H c B l m - - B r l m / i a , rla,O. In the ideal m a g n e t it is then Fm -

Brlm

.

(3.23)

~0 Let us denote by 9tg and 9tl the gap and the leakage reluctances, respectively, and let us assume that the reluctance of the soft limbs is negligible. If the effect of leakage is lumped, according to Eq. (3.15), into a factor/3 < 1, such that the flux in the gap (I)g = ~(I)m~ with (I)m the flux in the magnet, ~g and 9tl are related by the expression ~tl -

1 - ]3 ~g~

(3.24)

where 9tg is given by Eq. (3.22). The flux in the gap is then obtained by applying O h m ' s law Fm (I)g = ]~ ~}~mq-]3r

~

(3.25)

from which it is easy to derive the field in the gap Hg-

~g /~0Sg

(3.26)

and the related energy density Eg-- 89 in terms of the circuit parameters. It is immediately possible to verify that, by posing:/3 = 1,

100

CHAPTER 3 Operation of Permanent Magnets

Fm--Brlm/lXrlXO, ~m'~(1/la~31a,r)(Im/Sm), and ~g--(1/~o)(lg/Sg ) in Eq. (3.26), the previous Eqs. (3.4) (for S m = Sg) and (3.8) (for Sm # Sg) are recovered. Modeling the behavior of actual devices by means of circuits with lumped parameters and uniform fluxes is an excellent simplification for preliminary design and testing. However, when cosily steps in device prototyping and building are required, it is often appropriate to look for an accurate prediction of field and flux distribution. Closed form solutions of the electromagnetic equations are only possible in a few simple cases. Numerical techniques have therefore been developed, where Maxwell's equations are solved with the appropriate boundary conditions over a reasonably large number of points in space. Magnetic field and induction in systems composed of soft and hard magnets, operating under static or quasi-static excitation (no eddy currents), obey Maxwell's equations V x H = jext,

V-B = 0,

(3.27)

where jext is the density of the current provided by external sources. B and H are related by the constitutive equations B =/~0H, B =/~0H +/-LoM = ~0P,r H, B =/~/~r H q- P,0Mr

(3.28)

in air, soft magnets, and permanent magnets, respectively9 Mr being the remanent magnetization. It is assumed in this equation that soft magnets can be described by means of a single valued B(H) function (normal magnetization curve) and that the permanent magnet is used in the second quadrant. Notice that the equation for the permanent magnet is inclusive of the other two equations, which are particular cases with Mr = 0 (soft magnet) and Mr = 0,/d,r = 1 (air). The magnetostatic problem is completely defined by the set of Eqs. (3.27) and (3.28), but their direct solution is difficult to handle because the fields show discontinuities at the boundaries between different media. It is therefore preferred in numerical methods to find a solution in terms of a scalar or vector potential function, from which the field is obtained by derivation. In treating a problem with current sources, the appropriate function is the vector potential A, related to the magnetic induction by the equation V x A = B.

(3.29)

Uniqueness of the function A, which is defined by Eq. (3.29) to within a function deriving from the gradient of a scalar, requires a supplementary condition, which is given by the equation V.A = 0. While, in general, one needs to find a solution for each of the three components of A, it is

3.3 ELECTRICAL ANALOGY AND NUMERICAL MODELING

101

remarkable that in many practical instances two-dimensional fields (let us say in the x - y plane) are involved. This largely simplifies the problem because the vector potential reduces to the A~ component, orthogonal to the x - y plane, and it can be treated as a scalar. Let us therefore combine the condition (3.27) on the H field with the constitutive equation (3.28), written as 1 _ H + ~ M r.

B /d0/dr

(3.30)

/dr

If we apply the curl on both sides of Eq. (3.30) and we introduce the vector potential, according to Eq. (3.29), we get Vx

(1

) +1 V x A -- jext

/d0/dr

V X Mr.

(3.31)

/dr

In a bi-dimensional problem B = Bxi + Byj, A = Azk, and ]ext-" jext,zk with i, j, and k the unit vectors along the coordinate axes, and Eq. (3.31) reduces to

0 10Az) 0(

0---X /d0/dr

0X

-[- ~ y

1 -/d0/dr -

0Az ~ 3y ]

1

- - jext,z q- m V X M r. /dr

(3.32)

Equation (3.32) can be solved numerically by a number of techniques, the most widely used being the finite difference method (FDM) and the finite element method (FEM). Once Az is found, its curl will provide Bx and By. In the FDM the partial derivatives are expressed as differences, taken over a grid of regularly spaced points distributed over the required portion of the x - y plane, where jext,z, Mr, and /dr are defined. The FEM is based on the formulation of the differential equation for the vector potential in terms of an energy functional, expressed in integral form and defined in such a way that the condition of its minimization yields the starting partial differential equation for the potential. The functional is evaluated by subdividing the region of interest into finite elements, normally of triangular shape and variable size, and assigning values to the potential at the nodes. Like the FDM, the FEM requires an adequate number of iterations to yield the numerical solution of the problem. Nowadays it is increasingly applied because of its flexibility, the capability to handle non-linear problems and the accuracy and stability of the solutions. Given their vast domain of application, the numerical methods in the solution of electromagnetic problems in circuits and devices are abundantly treated in the literature. A full discussion can be found, for example, in Refs. [3.16, 3.17].

102

CHAPTER 3 Operation of Permanent Magnets

References 3.1. R.A. McCurrie, Ferromagnetic Materials: Structure and Properties (New York: Academic Press, 1994). 3.2. M. McCaig and A.G. Clegg, Permanent Magnets in Theory and Practice (London: Pentech Press, 1987). 3.3. R.J. Parker, Advances in Permanent Magnetism (New York: Wiley, 1990). 3.4. P. Campbell, Permanent Magnet Materials and their Applications (Cambridge: Cambridge University Press, 1994). 3.5. Rare Earth Iron Permanent Magnets (J.M.D. Coey, ed., Oxford: Clarendon Press, 1996). 3.6. M. Rossignol and J.P. Yonnet, "Les aimants permanents," in Magn~tisme. II. Mat&iaux et Applications (E. du Tr6molet de Lacheisserie, ed., Grenoble: Presses Universitaires, 1999). 3.7. R. Skomski and J.M.D. Coey, Permanent Magnetism (Bristol: Institute of Physics Publishing, 1999). 3.8. R.C. O'Handley, Modern Magnetic Materials (New York: Wiley, 2000). 3.9. J. Mallinson, Foundations of Magnetic Recording (San Diego: Academic Press, 1987). 3.10. H.N. Bertram, Theory of Magnetic Recording (Cambridge: Cambridge University Press, 1994). 3.11. Magnetic Recording Handbook (C.D. Mee and E.D. Daniel, eds., New York: IEEE Press, 1995). 3.12. R.L. Comstock, Introduction to Magnetism and Magnetic Recording (New York: Wiley, 1999). 3.13. D. Cheng, J.A. Brug, and R.B. Goldfarb, "Demagnetizing factors for cylinders," IEEE Trans. Magn., 27 (1991), 3601-3619. 3.14. R.A. McCurrie, IEEE Trans. Magn., 27 (1991), 197. 3.15. R.J. Parker, IEEE Trans. Magn., 27 (1991), 135. 3.16. A. Bossavit, Computational Electromagnetism (San Diego: Academic Press, 1998). 3.17. M.V.K. Chari and S.J. Salon, Numerical Methods in Electromagnetism (San Diego: Academic Press, 2000).

PART II

Generation and Mesurement of Magnetic Fields

This Page Intentionally Left Blank

CHAPTER 4

Magnetic Field Sources

The characterization of technical magnetic materials requires that magnetic fields are generated and controlled over a range of amplitudes covering several orders of magnitude. The classical field sources obtained by making DC or AC currents flowing in suitably shaped windings are universally applied for the measurement of the properties of soft magnets. For the characterization of recording materials and permanent magnets, which require field amplitudes in excess of 104-105 A / m , the use of conventional air-cored windings is normally envisaged only under pulsed regimes, because of obvious heat dissipation problems. For the generation of high-amplitude DC or very slowly varying fields, either electromagnets or superconducting coils are employed, with the latter covering the upper range of field values, up to some 107 A / m . Suitably arranged rare-earth based permanent magnets can provide, thanks to their resistance to demagnetization, DC fields in excess of those achieved by means of electromagnets. The natural limitation to static fields can be overcome if practical techniques for putting the magnets in motion one with respect to another are devised. The subject of magnetic field generation has quite general relevance, exceeding the area of materials characterization. A comprehensive treatment of the whole matter would, however, go far beyond the aim of this book and the following discussion will focus on those field sources having significant interest for measurements on technical materials.

4.1 FILAMENTARY COILS A straightforward calculation of the magnetic field generated by currents circulating in coils is based on the use of the Biot-Savart's law in differential form, which provides the elementary contribution dH generated at distance r by a current i flowing in a filamentary wire of 105

106

CHAPTER 4 Magnetic Field Sources

infinitesimal length dl dH=

idl x r 4wr3

(4.1)

and on its integration by numerical methods over the coil length. Alternatively, the field generated by a system of current loops can be expressed, in the regions not occupied by the conductors, as the negative gradient of a conservative potential V. This is obtained as a unique solution of Laplace's equation AV = 0, under the set of boundary conditions provided by the given distribution of current loops. Each loop can in fact be equivalently represented by a magnetic dipole shell, with density uniformly distributed over a generic surface bounded by the current circuit. The analytical approach to the calculation of coilgenerated magnetic fields can be pursued to some extent only under suitable symmetry conditions, which, however, are often achieved with the typical configurations adopted to test magnetic materials. Much emphasis was given in the past to analytical methods, but nowadays a wealth of results can be obtained quickly and exhaustively, also with complex current systems, by means of numerical techniques. We shall therefore limit our discussion on the exact derivation of the field to some basic examples of filamentary coils. It will then be shown how these results can be generalized to the case of practical coils, having nonnegligible cross-sectional area. It should be remarked that highly uniform fields are desirable, but not always required in the characterization of soft and hard magnets. For instance, in the determination of the magnetic polarization or the hysteresis loop, a ---1% field inhomogeneity over the sample volume is often acceptable. On the other hand, if calibrated field sources, traceable to primary standards, are to be developed, uniformity must be pursued to an extent significant for the specific kind of calibration required. For example, for an NMR field probe to be locked in the produced field, a maximum relative variation around 10 -3 m m -1 is typically required, which not only calls for suitable geometry of the field sources, but also for elimination or compensation of various spurious effects (e.g. earth magnetic field).

4.1.1 Single current loop The field generated by a current i flowing in a circular filamentary loop is endowed with obvious symmetry properties. It is independent of the azimuthal angle (axial symmetry) and it is the same in the two half-spaces delimited by the loop plane (mirror symmetry). With reference to Fig. 4.1, we can express the field generated along the axial direction in a point P of

4.1 FILAMENTARY COILS

107 x

#'#

p

T X

,4~,,~, ~,!, ,! .... ! ,', '~,,i',,',':':~~77:'~{/'J/J ;',!.,~:i}i,D ,,1! ili !lilit ", \ '.,"'--.'----_.---"./1/~,/,

ll~,

\ "'..".._Y_.L. ....

.,"

FIGURE 4.1 Magnetic field generated by a circular filamentary coil. The field lines are calculated by numerical integration of the Biot-Savart's law (4.1). The lines are shown on a portion of a symmetry plane normal to the loop plane. cylindrical coordinates (p, x, @) by the loop of radius a centered at x = 0 through the equation (see Ref. [4.1])

a2-p2-x2) i 1 ( Hx(p, x) - 2w x/(a + p)2 + X2 K(k) + (a - p)2 + x 2 E(k) ,

(4.2)

where K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively [4.2], with

k2 =

4ap (a + p)2 + x 2"

(4.3)

For the points belonging to the loop axis Eq. (4.2) provides the closed solution i Hx(O , x) =

-~

a2 (a 2 + x2)3/2

(4.4)

which at the origin reduces to H ( 0 , 0 ) = i/2a and at distances x >>a takes the expected 1/x 3 dipole behavior. The symmetry of the problem imposes that Hx(O,x) is an even function, which, expressed as a series

108

CHAPTER 4 Magnetic Field Sources

development, can be written as i( 3 X2 1 5 X 4 - - 1 0 5 X6 + . . . ) Hx(0,x)=~-~ 1 - ~ - X + ~ a 4 48a6 .

(4.5)

The second-order term leads to a relatively rapid decrease of the field at short distances from the loop plane. For x = a/lO one finds Hx(O,x)/ Hx(O, O) ~ 0.985. There are cases in measurements and in applications, where either square or rectangular loop sources are used [4.1, 4.3, 4.4]. This occurs, for example, when large sizes are required, making inconvenient the realization of circular coils. Again, the field can be calculated by integrating Eq. (4.1). The axial field at a distance x from the plane of a rectangular loop of sides 2a I and 2a 2 is given by the equation (see Ref. [4.41)

H~(O, x) = i

ala2

(

1

~r" ~a 2 + a 2 + X2" a 2 +

X2

1 q- a 2 +

) X2

'

(4.6)

which, for a square loop of side 2a becomes

Hx(O x ) - - '

i a2 2 77" ~/2a 2 if_ x 2 a 2 q- x 2"

(4.7)

The field at the origin is now H(0, 0) -- (i/2a)(2x/2/~r), about 10% smaller than that of an inscribed circular coil of radius a, and it shows a nearly identical decay with distance from the loop plane [4.1]. 4.1.2

Thin

solenoids

The practical characterization of materials, namely soft magnets, often requires uniform fields over extended lengths. A usual and convenient way to achieve such fields is by means of a thin solenoid, which is a sheet of current realized in most cases by uniformly winding a coil on a cylindrical former. For an infinitely long solenoid there is no need to calculate the field, which is perfectly uniform and directly given by any path integral enclosing the current as H - noi, where no is the number of turns per unit length and i is the current circulating in the wire. This idealized condition can be emulated by means of a toroidally shaped solenoid, but here the magnetic path length is obviously increasing with the radius r of the field lines and the magnetic field is correspondingly decreasing as 1/r on going from the inner to the outer diameter of the ring.

4.1 FILAMENTARY COILS

109 !

:5~:5:v:.:::~7::~C;L.IL2:,:.:!::

~-~.,,I~-.2,1;Z,Z,;"!d,'Z;~2~;

:ZL Z:~ 5

I :- LLZZLZ;:!7|3~LZL-......':~;.Z~k~.kL~.~..,'.....~,.....~

.:~ ~ ~ 5CZ.L.L

~. ~2:~iZ Li~'~7 C~.L.-2GL~.CU.~:Z~ ......

:

1

I

! !

,- - - - - . e , - - .... ~~- o ! k ....................... .. : ................L o~ .................. 9

.... "

.....

D

I

I

....................

,

!

~

L

0.9 d ,' / ~

~

...................

\

;

08

',

r

0.7

I

',

i

,'

"

i i

I t

i

i

0.6 0.5 -04

-02

00

02

04

x/L

FIGURE 4.2 Model thin solenoid of length L and diameter of the circular crosssection D. The behavior of the reduced axial field H/Ho, with H0 = noi, is shown as a function of the reduced distance from the center x/L. The dashed line is the calculated dependence of the same quantity when the ends of the solenoid are overwound, in order to compensate for the second order term in the H(x/L) power series expansion around x/L = O.

For a straight solenoid of finite length L a n d circular cross-section of d i a m e t e r D = 2a (see Fig. 4.2), the axial field is easily calculated starting f r o m Eq. (4.4). A ring slice of infinitesimal w i d t h dx located at a distance xl f r o m the center of the solenoid p r o v i d e s on a point on the axis of coordinate x the contribution

dHx(Xl)-

noi a2 2 (a 2 q-(x 1 -- x)2)3/2 dxl'

(4.8)

110

CHAPTER 4 Magnetic Field Sources

which, integrated upon the whole length L, provides for the field on the axis the expression

noi [ Hx(x) = --2

L + ax

L - ax

]

(D 2 q- (L + 2x)2) 1/2 q- (D 2 q- (L - 2x)2) 1/2 "

(4.9)

The m a x i m u m field value is obtained at the center of the solenoid L

Hx(O) = noi (D 2 q- L2)1/2

(4.10)

and is reduced by a factor around 2 at the solenoid ends L

Hx(L/2) = noi (D 2 if- 4L2)1/2 .

(4.11)

Equation (4.9) can alternatively be written as

noi Hx(x) = - ~ [cos 0l(x)

if- COS 02(X)],

(4.12)

with 01(x) and 02(x) the angles made by the lines connecting the coordinate x with the solenoid ends. Notice that the helical winding of the coil produces a uniform longitudinal hollow tube of currents. This, however, can produce a field only outside the solenoid. Figure 4.2 provides an example of axial field behavior in a thin solenoid with D/L -- 0.2. For x = a/lO one finds now Hx(x)/Hx(O) - 0.99998. The general problem of field uniformity with an axially symmetric coil configurations can be conveniently treated by expanding the axial field Hx(x) as a power series around x = 0 and concentrating on those coil systems leading to elimination or minimization of successive low order derivatives dnHx(x)/dx '' at the origin, as well as mutual compensation over convenient distances of terms of different order. As already shown by Maxwell [4.5] and amply discussed by Garrett [4.6], one can expand the solution of the Laplace equation A V = 0 for the potential V associated with an axi-symmetric distribution of currents, additionally endowed with symmetry (or antisymmetry) about the equatorial plane, in the form of an infinite series of spherical harmonics. It turns out that, within a spherical region free from conductors centered at the origin of the system (central field region), the field can be everywhere expressed in terms of the successive derivatives of the axial field at the origin. In a solenoid we are chiefly interested in the behavior of the field component parallel to the axis. If the center of the solenoid is taken as the origin, the central field region is a sphere of m a x i m u m diameter D. In a generic point within the sphere defined by the polar

4.1 FILAMENTARYCOILS

111

coordinates (R, O, cI)), with x = R cos O, the x-field component can be written as

Hx(R, O ) - yoo -~.l[ d"Hx(x'O) n=O

R"P,,(cos O),

(4.13)

x=O

where Pn(cos O) is the n-th order Legendre polynomial. Because of the mirror symmetry, only even order terms in the series development are retained. It is therefore apparent that the radial variation of the field strength can be obtained from knowledge of the axial variation and that this eventually depends on the values of the successive derivatives of the axial field at the origin. Equation (4.13) provides for the axial dependence of Hx about the origin

x2[

Hx(x , 0) -- Hx(0 , 0) 3- -~-

dx 2

]

x=0 z=0

X4[ d4Hr(x'O)] 3- 2-4

dx 4

3-...

(4.14)

x=0 z=0

and for the radial dependence

1 z2[d2gx(x,O)] Hx(O,z) - Hx(O,O)- -~--~ dx 2 x=0 z=0

3 z 4 [ d4Hx(x,0) ] +... 3- ~ ~ dx 4 x=O '

(4.15)

z--0

if z is the radial coordinate. It is remarkable that the two second-order terms have opposite signs in Eqs. (4.14) and (4.15). Since

[d2Hx(x'O)]
x=0 Z--0

this implies that the field in the radial direction around the origin has to increase. In order to improve the uniformity of the field about the origin, one can resort to double winding of the outer ends of the solenoid, by which compensation of the second-order term in Eq. (4.14) can be achieved [4.7]. Figure 4.2 shows an example of improved uniformity of Hx(x,O) in the central region upon end overwinding. In this case the cancellation

112

CHAPTER 4 Magnetic Field Sources

of the second-order term is associated with a positive value of the fourth-order term. Conditions for the simultaneous cancellation of the second and the fourth-order terms with end corrections can be found, as discussed by Garrett [4.6]. Solenoids with rectangular cross-section are frequently needed in experiments. They are the natural choice when testing soft magnetic laminations (Epstein, single strip, single sheet) and they guarantee a better field uniformity with respect to solenoids with circular cross-section. To calculate this field we must integrate Eq. (4.6), written for a rectangular ring of infinitesimal width dx and sides 2al and 2a2, in the same way as we did with Eq. (4.8). Figure 4.3 shows the result of such a calculation for one of the solenoids employed in the Epstein test frame (L = 190 mm, 2al = 32 ram, 2 a 2 - - 5 mm). It is observed that this solenoid advantageously compares as to the axial field uniformity with a solenoid of equal length L having circular cross-section of diameter D = 2 a 1.

0.9

,

Rectangular cross-section

,

.//,

I I

0.8

0.7

'

Circular cross-section

ll/ T llt

/ |

]i

I

I

,,"

",

[~----~' 2al ~

2a 2

I

', ',

~. I' #

'

"

0.6

0.5 -0.4

-0.2

0.0

0.2

0.4

x/L

FIGURE 4.3 Dependence of the axial field, normalized to the field value H 0 -- noi , on the reduced distance x/L in two solenoids of equal length L having circular and rectangular cross-sections, respectively. The latter emulates one of the four arms of the Epstein frame (L -- 190 mm, 2al = 32 mm, 2/12 - - 5 m m ) and the former has diameter D = 2al.

4.1 FILAMENTARY COILS 4.1.3

Helmholtz

113

coils

Let x be the coordinate of the axis of a spherical region of radius R. If the whole surface of the sphere is homogeneously wired between + R, so that the current per unit length of the x-axis di/dx is constant, the axial field within the sphere is uniform. The solution of the Laplace equation for the potential at a point within the sphere reduces to a first-degree harmonic and its derivative is

H(x)-

1N/

3 R '

(4.16)

for a current i circulating in an N-turn winding. We find here a perfect analogy with the behavior of the magnetic flux density in a homogeneously magnetized sphere. A spherical winding is obviously difficult to realize and in any case it is of little use, because it is rather inaccessible. Maxwell [4.5] pointed out that by using two or more filamentary windings coinciding with parallel circles of the sphere, symmetrically placed with respect to the equatorial plane, one could make the field uniform within a reasonably wide portion of the sphere. This occurs because, on the one hand, the mirror symmetry guarantees the vanishing of the coefficients of the odd order terms in the series development (4.13) and, on the other hand, some of the coefficients of the even order terms can be made to disappear for a suitable choice of the axial location of the windings. The axial field generated by two symmetrically located N-turn filamentary circular coils of radius a (see Fig. 4.4) can be written, besides its general form (4.14), as

Hx(x,O)=Ni(sin20)~.(x)2n .......r

r

co

P/2n+l= ~. Hx,a,,(x),

lZ--0

(4.17)

ll----0

where (r, 0) and (r, r 0) are the spherical coordinates of the two loops, r = a/sin 0, and the prime on the Legendre polynomial symbol indicates derivative with respect to cos 0 [4.8]. The second term in the series development H x 2(x, 0) '

(sinS0)(/)

--

Ni 7

r

2

P~3 = Ni

(sinS0)(/) r

r

23

(5 C O S

2 0

--

1) (4.18)

becomes zero for cos20 = 1/5 (0 = 63.43~ a condition realized by placing the coils at a distance d = a. This is the so-called Helmholez configuration. An example of field lines distribution around a Helmholtz pair is provided in Fig. 4.4b. A closed form expression for the axial field can be easily obtained in this case by summing up the contributions of the two

114

CHAPTER 4

Magnetic Field Sources

a

(a)

"'" ~\ \ ,I / //,::-'.~:--~i ......:~\\.~\ '

/ ' / "

...... ". . . . . . . . .

" .....

~ ',, " ~ ' ' '.'." ~ ) : , I I / \ \ \,\, ~,:...:~~.: f.-.,::~:..~ .... . .I . I.

'Z

\ ", .','-",.L-'-~. :-T.:. :..- : _ ~

I

~i; 'V ......

.~

x" /"

..-

. ,.

a ~--J-.-:~..-~..-:-:-:.-~,:.,,,....,,,

,

, , ,,\,,'~~-~~CC//~///)

k \\~%-:~:=-==;::v:~./// \

(b)

\

\ \ T L ~

Z. . . . . . . . . . . . . L . ~ _ . , ~ / , ,

,

I

1

]

FIGURE 4.4 A couple of filamentary coils coinciding with parallel circles belonging to a sphere of radius r centered in O generate in the whole region occupied by the sphere a field that can be expressed in terms of the successive derivatives of the axial field at the origin (Eqs. (4.13) and (4.14)). The axial field Hx(x, 0) in this central region is given, as a function of the coil coordinates, by the series development (4.17). The field lines are shown in the (x,z) plane for the Helmholtz coil configuration, where the distance between the loops is equal to their radius.

coils, as given by Eq. (4.4). It turns out

Hx(x,O)__4Nia2 [

1 1 ](4.19) (4a2 4- (2x 4- a)2) 3/2 4- (4a2 4- (2x - a)2) 3/2

which, in agreement with Eq. (4.17), at the origin reduces to N/ Hx(0, 0) = 0.7155 ~a.

(4.20)

4.1 FILAMENTARY COILS

115

Figure

4.5 illustrates the d e p e n d e n c e of the r e d u c e d axial field the r e d u c e d distance x/a, as calculated by m e a n s of Eq. (4.19). An excellent uniformity of the predicted field is achieved in the n e i g h b o r h o o d of the origin. One finds, for instance, that Hx(x, 0)/ Hx(0, 0) -- 0.99989 for x/a = 0.1. The radial d e p e n d e n c e of the field can be evaluated using Eq. (4.15), in conjunction with Eqs. (4.14) a n d (4.17). It turns out, in particular, that Hx(0, z)/Hx(0, 0) = 0.99995 for z/a = 0.1. It is interesting to r e m a r k that one can s o m e w h a t extend the b r e a d t h of

Hx(x, O)/Hx(O,O) on

0.9999 0

.

.I--2

0.9998 0

0.9997

0.9996

0.9995 . -0.2

. -0.1

.

0.0

x/a

.

O. 1

0.2

O2

FIGURE 4.5 Behavior of the reduced axial field Hx(x,0)/Hr(0, 0) generated by filamentary pairs vs. the reduced distance from the origin x/a, where a is the radius of the two loops forming the standard Helmholtz pair (d = a). The field behavior of the standard pair is shown by curve 1. Curve 2 is obtained with slightly increased pair separation (d = 1.01a). Curve 3 shows that increased field uniformity is achieved with the double Helmholtz configuration, where the two series-connected pairs are characterized by the coordinates 01 = 40.09 ~ and 02 = 73.43~ and the turn ratio N1/N2 = 0.68212.

116

CHAPTER 4 Magnetic Field Sources

the central region having stated field deviation if the distance between the two loops is made slightly greater than a. This occurs, as illustrated in Fig. 4.5 for the case d - 1.01a, at the cost of a slightly depressed field value at the origin. In fact, while the value of the fourth-order derivative at the origin has decreased, the second-order derivative, previously made to vanish through the proper choice of the 0 value, has become slightly positive. Equation (4.17) provides for the fourth-order term the expression

Hx.4(x~O)-- Nisin20(x) 4 r r PIs =Nisin20(x)418 r ~ ~(21

c o s 4 0 - 14 cos20+ 1),

(4.21)

01--40.09

~ and 02 = 73.43 ~ Two which has two roots for the angles fourth-order loop pairs belonging to symmetric parallel circles of the sphere of radius r can correspondingly be identified, having radii al r sin 01 and a 2 -- r sin 02, respectively (double Helmholtz coil, see Fig. 4.5). The radii of the pairs are then in the ratio al/a2 = sin 01/sin 02 = 0.67189 and their distances dl/d 2 ---cos 01/cos 02 = 2.685. If the loops are series connected and their turns are in the ratio N1/N2 = 0.68212, the secondorder term is additionally suppressed, because it receives opposite and equal contributions from the two pairs. It can be shown that the elimination of Hx, 2 brings about, under the condition that Hx,4 --O, the further elimination of the sixth-order term Hx, 6 [4.6]. It turns out that the field uniformity is largely increased with respect to the case of a single Helmholtz pair, as illustrated by behavior of curve 3 in Fig. 4.5. If the coils of a pair are connected in series opposition, an axial field of reasonably uniform gradient around the origin can be obtained. This antisymmetrical configuration imposes zero field value at the origin and the presence of odd order terms only in the series development of the field. For a distance d between the loops of radius a, the field gradient at the center is calculated as [4.9] G(o,o)

=

3 ~Ni

a2d d2 5/2'

(4.22)

(l/2q- -~--)

which attains a m a x i m u m value

G(0, 0)max-- 0 . 8 5 8 7Ni a~2

(4.23)

4.2 THICK COILS

117

for the Helmholtz distance d = a. The linearity of the field can be improved by placing the coils at the distance d = ~/3a, which engenders the suppression of the third-order term. The field gradient, having the value at the origin N/ H~(0, 0) = 0.6413 a-T,

(4.24)

is therefore characterized by fourth and higher order inhomogeneities only. The use of square filamentary loops instead of circular loops leads to similar results. For square coils of side 2a placed at a distance d = ka, one finds through Eq. (4.7) the following expression for the axial field Hx(x, O) - 2 Nia2 [ 1 " rr L (a2 q- (X q- ka)2)~(aa 2 q- (x q- k//) 2)

1

]

q- (a2 q- (x - ka)2)x/(2a 2 q- (x - ka)2) '

(4.25)

which satisfies the Helmholtz condition, where the second-order derivative of the field at the origin is zero, when d = 0.544506a. The field at the origin correspondingly takes the value Ni Hx(0, 0) -- 0.6481 ~ . a

(4.26)

Though somewhat lower at the origin, Hx(x,O) exhibits a slightly improved uniformity with respect to the field generated by a circular Helmholtz pair [4.1].

4.2 T H I C K C O I L S There are instances where multilayered solenoids and coils are needed for the characterization of materials or the creation of reference sources providing adequately high fields. Increasing the number of windings is indeed preferable to the increase of the driving current in a filamentary coil, because in the latter case the power dissipation displays a quadratic law of increase with the field amplitude. There are no difficulties in principle for extrapolating the results achieved with thin solenoids and filamentary coils to thick coils, for example by numerically integrating the Biot-Savart law with respect to the novel coordinate. In practice, the optimization of the field source involves several non-independent

118

CHAPTER 4 Magnetic Field Sources

requirements like power efficiency, volume of the source, accessible volume, field intensity and homogeneity, etc. which complicate to a good extent the design of practical coils. Let us consider a thick uniformly w o u n d multilayered solenoid of length L and n u m b e r of turns per unit length no, having inner and outer radii R1 and a2, respectively (see Fig. 4.6a) with a circulating current i. The axial field generated in a point of coordinate x is obtained starting from the elementary contribution of the loop of infinitesimal area dA = drdx

I i r

t (a)

|

L

~

I

I

,

(b)

X

(c)

FIGURE 4.6 (a) Thick solenoid with inner and outer radii R1 and R2, respectively. (b) Piled-up disks in a Bitter magnet. (c) Helmholtz pair with coils of finite rectangular cross-sectional area. The shaded square around the center puts in evidence the region associated with the field behaviors reported in Fig. 4.7.

4.2 THICK COILS

119

di=(noi/(R2-R1))drdx.

and radius r, where it flows a current Integration provides

G(x)

=

n~ 4(R2

I(L 4- 2x) In 2R2 + R1) L

+ ( L - 2 x ) ln

~4R 24- (L + 2x) 2

2R14- ~ 4Ri 4- (n 4- 2x) 2

2R2 + ~4R 2 + (L __ _ - . -- 2x)22x)21 2R 14- ~4R~ q-(L

(4.27)

The field at the center of the solenoid is

HK(O) --

noiL

In

2(R2 _ R1)

2R2 + ff4R22+ L2

~

,

(4.28)

2R14- ~4R 24- L2

which reduces to Eq. (4.10) in the limit R1 --'*R2. The heat that can be dissipated usually limits the maximum field value. When cooling is made by natural air convection the typical maximum current density is around 1 A / m m 2. Such a density can be raised by more than an order of magnitude by forcing oil or water to circulate in intimate contact with the current bearing conductors. A variety of methods have been devised, which include the use of hollow conductors, cooled by purified water running in their interior, or watercooled copper disk spacers, electrically insulated and suitably interleaved with the winding conductors. In designing the solenoid it is useful to connect the generated field with both the geometrical parameters and the power consumption. For a winding of resistance R the latter is W - Ri 2. To calculate R we consider the elementary loop of infinitesimal area dA and radius r shown in Fig. 4.6 and, assuming that the winding is characterized by a turn density dN dA

-

N (R2 - R1)L

(4.29)

and a filling factor & < 1 (because of the necessary space occupied by cooling and insulation), we express its resistance as 2 vrr 21rp N2 dR -- p &(dA/dN) dN -- & (R2 _ R1)2L2 r dr dx,

(4.30)

120

CHAPTER 4 Magnetic Field Sources

if p is the electrical resistivity of the conductor. Integration over the solenoid radius and length eventually gives the total winding resistance R=

2 rrpN 2 R2 -Jr-R ,~L R2 - R-------~"

(4.31)

The current i = ( W / R ) 1/2 is related to the field at the origin Hx(O) through Eq. (4.28) and we can therefore express the latter in terms of the dissipated power

1( Hx(O)- 2x/~

)1J2 2R 2 q- ~4R 2 q- a 2 R2 - R2

-- G( W,~ 1/2 ~7)

1/2

In 2R1 + ~/,iR~ + 7(WApR~I)

'

(4.32)

where the dimensionless factor G can be written in terms of the ratios a - - R2/R 1 and 13 = L/2R1 as 1

13

) 1/21nCr _]_~/Or _}_]~2

(4.33)

This factor attains a m a x i m u m value G - 0.142 for R2/R 1 ---3 and L/R1 --- 4 (full tables and graphical analysis of G(~, ]3) are given in Refs. [4.10-4.12]). Let us suppose, for example, that a field Hx(O) = 105 A / m is to be generated by means of a thick solenoid having length L - I m, and radii R 1 - 50 m m and R 2 --150 mm. The solenoid is obtained by winding enameled copper wire with cooling spacers and the ensuing filling factor is estimated to be ~ - 0.75. From Eq. (4.33) we find G = 6.18 x 10 -2 and from Eq. (4.32) we obtain that the power consumption is W - 2.84 x 103 W. Notice that this result is independent of the chosen wire diameter. A certain increase of efficiency can be obtained if the solenoid is made by superposing thin disk-shaped copper plates with a central hole (Bitter coil [4.13]), as schematically shown in Fig. 4.6b. The disks are provided with radial slits and are superposed in such a way that successive slits are rotated by about 20 ~ with respect to one another. They are insulated, except for the narrow region between the slits, where the current jumps from one disk to the adjacent one, thereby spiraling in a manner similar to that of a conventional solenoid. The resulting current density is, however, inversely proportional to the radius r. The calculation of the power consumption, made as previously shown for the wirewound solenoid, leads to the very same equation (4.32), with the factor G now given by

4.2 THICK COILS

121

the expression 1 ( 1 ) G

-

22 ~

1/2 c~(]3 q- x/1 q- ]32) ~2 In ~ff-~/Cr 2 q/31na

"

(4.34)

The m a x i m u m value G = 0.166 is achieved for R2/R1 ~" 6 and L/R1 "--"4. For a same dissipated power, the Bitter coil, taking advantage of the radial current distribution, provides about 17% higher still than the uniform current density solenoid. Gaume showed that still higher efficiency can be obtained by imposing a profile to the current density along the coil length, with its m a x i m u m value at the center [4.14]. To this end, the disk thickness is varied accordingly, with the thinnest disks at the center and the thickest at the ends of the solenoid. The generation of uniform fields over a wide accessible region may require the realization of Helmholtz coils having finite winding area. In this case it is desirable to know how one can emulate the conditions of field homogeneity predicted for the filamentary coils. To this end, let us consider the coil pair in Fig. 4.6c, having rectangular cross-sectional winding area of height h and width w, mean radius a and mean separation d. It can be shown, according to Franzen [4.8], that the condition leading to the elimination of the second-order term in the expansion of the axial field at the origin can still be achieved, similar to the filamentary pair, when d = a. It is required in this case that the sides of the winding area are in the ratio h/w = 1.0776, provided h and w are small compared to a. If this is not the case, the Helmholtz condition can equally be obtained by increasing the value of h/w. For w/a varying between 0.1 and 0.36, it has been found, in particular, 1.0776---h/w <1.118 [4.15]. It can then confidently be stated that Helmholtz pairs with finite winding area can provide as homogenous a field as filamentary pairs. Similar conclusions can be d r a w n for the case of double Helmholtz coils. Franzen has shown that a sixth-order coil can be obtained by adding to the conditions on the mean radii and distances between the pairs already demonstrated for the filamentary coils (al/a 2 --- 0.67189 and dl/d 2 --2.685) the requirement that the coil thicknesses are in the ratio hl/h 2 --0.672 [4.8]. If the same conductor is used for the two coil pairs and the filling factor is the same, then N1/hlwl = N 2 / h 2 w 2 , which implies that Wl/W 2 "--1.015. It should be stressed that dimensional tolerances due to coil manufacturing might represent a substantial limiting factor in the field homogeneity. An example is provided in Fig. 4.7a, where the effect on the behavior of the axial field generated by a pair with finite winding area (h/w = 1.06) of a + 0.5% fluctuation of the mean distance between the coils is shown. Notice in Fig. 4.7b that

122

CHAPTER 4

Magnetic Field Sources

1.0000-

/

0.9999 O

d ~'~ 0.9998-

/

/

O

0.9997 "'d= 1.005a 0.9996

/

He/mh010t~pair

0.9995 l -0.15 -0.10 -0.05

|

0.00

0.05

.....

0.10

0.15

x/a 1.001

zA

xo=0 o

1.ooo

._?

0.998 h/w = 1.06 -0.15

-0.10

x0/a = 0.17 -0.05

0.00

0.05

0.10

0.15

z/a FIGURE 4.7 (a) Effect on the axial field uniformity of a + 0.5% fluctuation of the mean distance d between the coils of a thick Helmholtz pair of mean radius a and winding area hw (see Fig. 4.6c). (b) Predicted behavior, for d -- a, of the axial field Hx(x0, z) in the radial direction. Its dependence on the radial coordinate z is shown for three different values of the reduced coordinate xo/a.

4.3 AC AND PULSED FIELD SOURCES

~'

123

" ~i

I~i

-i

FIGURE 4.8 Helmholtz coil used as a standard field source in the range 200 A/m-< H-< 2.8 X 10 4 A/m. The coil constant kH is determined, via NMR measurement, with expanded relative uncertainty U/kH = 8 X 10-s. The coil is placed inside a large triaxial Helmholtz setup supplied by three independent current sources, providing active cancellation of the external fields. the radial dependence of the axial field Hx(xo,z) predicted for the same winding by means of a numerical technique, exhibits an increase in the radial direction around z - 0, as it was anticipated in analyzing the structure of the series development of Eq. (4.15). The case represented here is that of a pair used at IEN Galileo Ferraris for calibration purposes (see Fig. 4.8). It has the following dimensions: inner radius R1 = 90 mm, h-53mm, w=50mm, d--a-117mm, wire diameter l m m . It provides, with a current i--- 1.6 A, a sufficiently high field at the origin to be measured by means of an NMR magnetometer (H20 probe). One can see that, within the + 0.5% uncertainty in the position of the coils, the relative field variation over a region of the dimensions of the probe (---5 mm), is at most 10 -4, small enough to ensure a narrow absorption line and locking of the oscillator.

4.3 AC A N D PULSED FIELD SOURCES Soft magnetic materials are ubiquitously employed in AC apparatus and generally need to be characterized under defined AC exciting conditions. The generation of time varying fields does not present special conceptual difficulties and the methods illustrated in Sections 4.1.1-4.1.3 still apply.

124

CHAPTER 4 MagneticField Sources

Obvious limitations, however, arise and the higher is the frequency the more numerous become the constraints on the field generating setups. For one thing, the increase with frequency of the impedance of the field winding calls for a correspondingly increasing power demand. For the largest part it is reactive power, with the additional contribution of core losses and copper losses. The control of the time dependence of the magnetizing current might also be required and at frequencies higher than few hundred Hz the role of stray capacitances and inductances should be taken into account. At very high frequencies, materials are characterized under weak fields only (e.g. initial permeability measurements) and the methods of impedance analysis are very often adopted. Figure 4.9 illustrates a typical AC field generating setup, producing in a coupled magnetic core a well-defined time dependence of the flux. The magnetizing current is generated by combination of a variable frequency and amplitude oscillator (G) and a power amplifier (A2). The latter can be, for instance, a DC coupled audio amplifier, characterized by very low output impedance. If, as is often the case, sinusoidal time dependence of the magnetic induction in the core is required, a sinusoidal voltage of suitable peak amplitude is provided by G. This, however, results in a non-sinusoidal B(t) function, because of the non-linear B - H behavior of the core and the non-ideal inductive nature of the circuit. The signal VB(t) detected on a secondary winding is then fed into a differential amplifier (A1), where it is compared with the oscillator voltage VG(t). With the gain of A1 set to very high value,

T -

-

-

,

" VH(t)

R

-

FIGURE 4.9 AC field generation with feedback, providing sinusoidal B(t) waveform in the ferromagnetic core. The distortion on the i(t) waveform reflects the nonlinearity of the B(H) relationship in the material.

4.3 AC AND PULSED FIELD SOURCES

125

the prescribed sinusoidal B(t) behavior can be recovered, as measured by the deviation of the form factor of the secondary voltage VB(t) from the prescribed value FF = 1.1107. It is therefore possible to test under defined induction waveshape (not necessarily sinusoidal) even pretty small samples at low frequencies, far from the conditions of highly inductive primary circuit. Digital methods are nowadays increasingly favored in the control of the flux waveform, because of obvious advantages in terms of stability, flexibility and reproducibility (see Section 7.2). The previous discussion on thick coils and the example reported there demonstrate the objective limits imposed by heat consumption on the generation of high fields. According to Eqs. (4.32)-(4.34), we expect a quadratic dependence of the dissipated power on the field amplitude, which implies that few hundred kW are to be spent in order to approach the 106 A / m range. Especially designed Bitter coils, where magnetic flux densities of 10-30 T can be sustained for a few seconds while dissipating several MW power, have indeed been developed [4.16], but the characterization of technical materials calls for relatively inexpensive and simple techniques of high field production. A significant demand exists, for example, for quick and accurate testing of rare-earth based hard magnets, having coercive fields around 106 A / m or higher. The classical test method based on the use of electromagnets is limited by the saturation magnetization of the core and is generally impossible to bring the permanent magnet into the saturated state on site before starting the determination of the magnetization curve in the second quadrant. For example, the fields required to approach the saturation in the high coercivity sintered or bonded N d - F e - B can be greater than 5 x 10 6 A/m, while maximum fields of the order of 2.5 x 106 A / m can be achieved in Co-capped electromagnets (see Section 4.5). Remarkably, saturating transient fields can be obtained, where a relatively modest quantity of energy (5-30 kJ) is stored in a bank of capacitors and subsequently released by impulse discharge through an air-cored coil. Pulsed field magnets have indeed been developed since many years for the generation of extra-high fields, either non-destructively (up to about 60 T) or by destructive electromagnetic flux compression (above 60 T) [4.17], but their rather extreme features, requiring stored energies of several MJ, are redundant for practical material testing. A pulse magnetizer with electrostatic energy storage, useful for saturating and testing high-energy product permanent magnets, is schematically represented in Fig. 4.10a. The capacitor bank is charged to high voltage (V0 > I kV) then discharged through the solenoid of inductance L and internal resistance R. Let us consider first the case where the switch $2 is open. The current

126

CHAPTER 4 Magnetic Field Sources

i(t)

. A

vo

m

L

m

/

1

(a)

4000,1~ . . ~

S2

closed

ooo1!//V

ooOOV

-4000

(b)

L = 0.98mH

0

10

20 Time(ms)

30

FIGURE 4.10 Pulsed field generated by discharge of a capacitor bank through a solenoid of inductance L. With the shown values of R, L, and C (R2C/4L = 2 • 10 -3) the condition is obtained for the occurrence of an oscillating damped discharge, with time constant ~'0 = 2L/R (switch $2 open). If $2 is closed, the diode prevents the capacitor from charging with reversed polarity and the current decays from its maximum value with time constant rl = L/R. transient is described b y the equation

Oi(t) 1 i(t) = 0 L 02i(t) Ot2 + R O - - ~ - C

(4.35)

w h i c h has a general solution of the type

i(t) = A~ exp(ml t) + A2 exp(m2t),

(4.36)

4.3 AC AND PULSED FIELD SOURCES

127

with m l , 2 - -(R/2L) +_~/R2/4L 2 - 1/LC

(4.37)

and A~ and A2 constants to be determined by imposing the initial conditions. According to whether the quantity under square root is negative (RaC/4L < 1), equal to zero or positive (RaC/4L > 1), an oscillatory damped, a critically damped, or an overdamped i(t) behavior is predicted. Under the condition that at time t = 0 the capacitor is charged at a voltage V0 and the current i = 0, the oscillatory damped solution is --L-- exp - ~-~t

sin wt

,

(4.38)

with

~o= ~I /LC - R 2/4L 2

(4.39)

and time constant z = 2L/R. An example of oscillating decay of i(t) is provided in Fig. 4.10b, where the circuit parameters provide R2C/4L = 2 x 10 -3. The overdamped solution is analogously found to be

i(t)= Vo exp(

-E-

R ) sinh kt k

(4.40) ,

with

k - ~R2/4L 2 - 1/LC.

(4.41)

The oscillatory condition is normally adopted for efficient conversion of the electrostatic energy into magnetic energy. However, it eventually brings the magnetic sample in the demagnetized state and, in order to maintain the sample in the remanent state, the switch $2 is closed at t -- 0. Thus, after the current has reached its maximum value im at time t -- t m and the capacitor is completely discharged, the diode, acting as a short circuit, prevents C from charging with opposite polarity. The current decays then exponentially, with the time constant of the LR circuit Zl = L/R. From Eq. (4.36) and its time derivative we obtain that at the time 1

t m - --.atan(wz) o9

(4.42)

128

CHAPTER 4 Magnetic Field Sources

the current achieves its maximum value

(atan(o~z))

~-~ im = V0 .exp -

oJ~

"

(4.43)

The conversion of the electrostatic energy E c - 89 into magnetic energy EL -- 1 Li 2 peaks at time t = tm. Part of Ec is dissipated as heat in the leads and the internal resistance of the coil. The efficiency of the conversion is obtained from Eq. (4.43), (1/2)Li2m (atan(oJ~') ) 7 q - (1/2)CV2 = exp - 2 - - ~ 0 r "

(4.44)

The efficiency increases with increasing the quantity

roz =

~4L ~T~ - 1.

(4.45)

In a similar way, one finds for the overdamped case Li 2 ( atanh(k~') ) 7q .- C V 2 - exp - 2 kr '

(4.46)

with k'r =

4L R2 C

(4.47)

The efficiency is obviously decreasing with increasing damping, as illustrated in Fig. 4.11 by the dependence, calculated through Eqs. (4.44) and (4.46), of ~/ on the damping coefficient R2C/4L. The advantage of using the circuit of Fig. 4.10a ($2 closed) for the generation of a non-oscillating current transient instead of adopting an overdamped configuration is clear. To be stressed that, for a given stored energy Ec = 89 2, better efficiency is achieved using high voltage and low capacitance. The pulsed field setups employed in permanent magnet testing are generally required to produce peak fields of the order of 5 x 106 A / m or higher with rise time tm > I ms. They should be reasonably uniform over volumes of several cubic centimeters, thereby sufficient to accommodate samples of technical size [4.18]. The magnetizing coil is normally

4.3 AC AND PULSED FIELD SOURCES

1.0-

129

Underdamped

',

0.8-

~

0.6-

0.4-

0.2-

0.0 1 E-4

. . . . . . . .

I

1E-3

0.01

0.1

I

1

10

R2C/4L FIGURE 4.11 Efficiency of the conversion of the capacitively stored energy into magnetic energy in a pulsed field setup as a function of the damping factor

R2C/4L.

a multilayered solenoid, though Bitter type coils can also be used, and the flow of current in the copper conductors is normally not affected by the skin effect. Limited increase of the coil temperature by Joule heating and good mechanical strength are two important factors to be considered in designing the magnetizer [4.19]. Because of the short duration of the current pulse, the coil is expected to behave adiabatically during the transient and nearly all of the energy delivered by the capacitor is retained in the copper winding. If the mass of the copper is m, an upper limit for the temperature increase AT of the coil upon a field pulse is therefore obtained as AT=

(1/2)CV~ Cpm

(4.48)

if Cp is the specific heat of copper. For energies up to 30-50 kJ, no special provisions for cooling are required, unless the system is subjected to high repetition rates, as it often occurs in industrial environments. Stresses in the coil arise because the conductors are subjected in part to the very same field they produce. As schematically illustrated for the case of a thick solenoid in Fig. 4.12, the longitudinal field

130

CHAPTER 4 Magnetic Field Sources

'r

__.z__....

......... L................................. ~i""

,J

T

~

H,

......

Z

'i

FIGURE 4.12 Stress distribution in a solenoid. The axial field component Hx is the source of a radial stress err which is equilibrated by a tangential tension cre in the conductor. The radial field component Fir generates in turn an axially directed stress crx pointing towards the equatorial plane zz.

component Hx generates by Lorentz force an outwardly directed radial stress err which tends to expand the solenoid diameter, err is equilibrated in each point of the conductor by a tangential stress or0. The radial field component Hr is the source of an axially directed stress Or r pointing towards the equatorial plane, which tends to shorten the solenoid. A Bitter type solenoid has good mechanical strength, while reinforcement of the structure of the solenoid is required for wire-wound and tapew o u n d coils. These can be made rigid by impregnation with suitable resins and by fitting them into circumferential strengthening rings made of some hard material. It is imperative, in any case, that the yield strength of copper is not exceeded anywhere in the coil. We can make an estimate of the m a x i m u m stress to be endured by the coil by equating the field pressure, which is compensated by the stresses in the conductor, with the energy density of the magnetic field E = (1/2)/z0 H2 [4.1]. A field of 5.106 A / m corresponds, for example, to a stress of 16 MPa, safely below the yield stress of hard d r a w n Cu (~ry = 350 MPa). When the field to be produced is so high that the ensuing stresses overcome the yield strength of the conductor in the coil, one can still carry out the experiment, provided the pulse duration is shortened

4.3 AC AND PULSED FIELD SOURCES

131

to the point that it is extinguished before a substantial a m o u n t of kinetic energy can be transferred to the material. The example reported in Fig. 4.10 refers to the case where a pulsed field is p r o d u c e d by a solenoid having the following dimensions (see Fig. 4.6a): L = 102 mm, R 1 = 25 mm, R 2 -- 36 mm. The solenoid is m a d e of 10 layers of h a r d - d r a w n copper wire of rectangular cross-section 5 m m x I m m and its inductance is L = 0.98 mH. The generated axial field can be calculated t h r o u g h Eq. (4.27). The solenoid constant takes at the center the value Hx(O)/i = 1667 m -1. Table 4.1 provides an overview of the main features of this pulse generator, where the storage of energy is accomplished by charging a capacitor bank C - - 8 0 0 ~F at a voltage V0 = 6000 V. A second typical setup, as reported in the literature [4.21], is also considered. Peak field amplitude suitable for saturating high coercivity p e r m a n e n t magnets is obtained with good efficiency in both cases, while keeping the rise of coil temperature and stresses within comfortably low limits.

TABLE 4.1 Significant quantities in pulsed field setups. System 1 is described in Fig. 4.10. System 2 is described in Ref. [4.21] Quantity Inside diameter of the solenoid (R1) Outside diameter of the solenoid (R2) Length of the solenoid (L) Cross-sectional area of the wire (Sw) Resistance (R) Inductance (L) Capacity (C) Voltage (V0) Stored energy (Ec) Damping coefficient (R 2C/4L) Time constant (~-) Rise time (tm) Maximum current value (/m) Maximum field value (Hm) Efficiency (7/) Temperature increase (AT) Maximum stress (o-)

System I

System 2

25 mm

13.5 mm

36 mm

32.2 mm

102 mm 5 mm 2

96.2 mm

0.13 f~ 0.98 mH 800 ~F 6000 V 14.4 kJ 3.3 x 10 -3 15 ms 1.24 ms 4960 A 8.27 X 10 6 A / m 0.84 21 ~ 40 MPa

10 m m 2

0.036 f~ 0.24 mH 8000 ~F 2000 V 16 kJ 1.08 x 10 -2 13.3 ms 2.05 ms 9900 A 12.3 X 10 6 A / m 0.74 40 ~ 170 MPa

132

CHAPTER 4 Magnetic Field Sources

4.4 PERMANENT MAGNET SOURCES The very first magnetic phenomena observed by man had to do with fields generated by minerals like magnetite and other oxides having the character of permanent magnets. The objective historical importance of permanent magnet field sources is actually associated for the most part with the useful mechanical effects they inexhaustibly display once they are magnetized. They have generally played a minor role in the magnetic characterization of materials, where the flexibility of the current based sources, with or without soft iron cores, has revealed indispensable, for example, in the experimental approach to all the phenomena related to magnetization processes and hysteresis. Advances in the properties of materials, namely the development of the high coercivity, high remanence rare-earth based hard magnets, have changed somewhat this state of affairs and enriched the landscape of field sources useful in measurements. A classical permanent magnet source can be schematically represented by the gapped ring discussed in Section 3.1 and shown in Fig. 3.1. It was shown there that, under the approximation of uniform induction in the whole magnetic circuit, the useful field in the gap Hg is related to the value of the polarization Jm in the material by the equation Hg --

1 Im/Ig Jm, ~o 1 + lm/lg

(4.49)

with lm and lg the lengths of the magnet and the gap, respectively. Equation (4.49) provides in a narrow slit (lm/lg >> 1) the maximum field value Hg ----Jm/p,o. An upper limit is then predicted for the induction in the gap, equal to the saturation polarization Js of the material. As we see in Fig. 4.13b, the idea of complete flux channeling, assumed in the circuit of Fig. 3.1 and in Eq. (4.49), is much too often a rough approximation. On the other hand, the magnetization M of an ideal magnet is rigidly fixed along the axial direction and is constant in modulus. This implies that no volume free charges are present in the material and manageable determination of the gap field by analytical means can be envisaged. Let us therefore consider in some detail this calculation in the model pole pieces of circular cross-section shown in Fig. 4.13a. These cylindrical pole pieces are assumed to be a portion either of a closed circuit or of very long rods. A practical circuit would actually be made of relatively short rods connected by a soft iron return path. In any case, the gap field is attributed to the free charges, of

4.4 PERMANENT MAGNET SOURCES

t

"

_1 ..... ~

?

r

""-,,

M

[

'

i \,

--..

i

Ig

~'

i

/

\,, "'.

--

i

\

..

M

/

'

'-~

..

9

ii X l,ql--, 1/

+1'<,<,

(a)

,"

,,

4._'.~_ . . . . . i I

++J +It-_-

~

I

i

O,/>j,,,

..............................

I /

133

t.>

-

~

,/

!

+.'/~",, ,,............. , /"'..

~

.\ ~

=_~__-----~:= =.i:7~i . . . - .... . ._. : -: . Z--T_A

- .....

... ."

-+"~

.

"

_-

. . . . . . .

.

""

. : :-.

.~---

-2~--~_~-=:---.

.......

_ _-~~.~-_ _._

--)- 7i :: ............. .

"

, ".--'-_-;---~__-=_.--__~.

9=-r._-_~-_~..-._-_---~ _

/

~ l

,,

...... _-_-_:. ~ -.___._~..:--._-_=_ _.:. _ :

"~ {

-

........ .

.

_

/ \:;-: ---L>:;, ....

,

] I

~, ,,,

',,

\

,,

j

t/

i

!

\

,

,<,

(b)

FIGURE 4.13 (a) Cross-sectional view of permanent magnet cylindrical pole pieces of semi-infinite length separated by a gap. Ideal magnets are assumed, where the polarization Jm -- t-t0M is rigidly oriented along the magnet axis and is not affected by the demagnetizing field. (b) Corresponding behavior of the flux lines, as obtained by numerical modeling [4.22]. density or= M = Jm/l-to, appearing at the gap surfaces reference to the left pole piece in Fig. 4.13a, we calculate the contribution d~ol(x) to the magnetic potential in a point of coordinate x which is provided by an annular portion of having radius r, area dS -- 2frr dr and charge d q - - tr dS as dq01(x)

1 or dS 4~r ~(lg/2 4- X ) 2

1 -t-

M r dr

r 2 -- ~ v~/(ig/2 4- X ) 2

only. With elementary the axis of the surface

(4.50) q--

r 2'

where lg is the gap length. The corresponding axial field is obtained by differentiation 0 M (lg/2 + x)r dr (4.51) At4 r~,~..gl,X, = -- 0--xdq~ = 2 [lg/2 + x) 2 + 1"2]3/2.

134

CHAPTER 4 Magnetic Field Sources

The total contribution of the pole face is then given by the integral

M(Ig/2 4- x) f~ Hgl(x) =

2

r dr [lg/2 4- X]2 4- F2]3/2'

(4.52)

which provides

Hgl(x)--- -~- 1 -

(lg/2 4- x) ) M ~(ig/2 4- x)2 4- r~ = -~- (1 - cos 01(x)).

(4.53)

The contribution Hg2(X)from the second pole face is found in the same way/ leading to the total axial field in the gap Hg(x)-- Hgl(X) 4- Hg2(X ) : M(1 - cos 01(x)+ cos 02(x) )

2 The value of Hg(x) at the center is then

Hg(0) = M ( 1 -

1)

I/1 4- (2rg/lg) 2

(4.54)

(4.55)

and at the pole surfaces is

(

1 )

Hg(lg/2) -- M 1 - 2~1 + (rg/lg) 2 "

(4.56)

We find again, from Eqs. (4.55) and (4.56), that the maximum predicted field in the gap is equal, in the limit lg/rg---,0~ to the saturation magnetization Ms. Figure 4.13b, obtained by numerical modeling, illustrates the distribution of the flux lines within and around the air gap when lg -- 2rg [4.22]. Flux fringing is observed and, according to Eq. (4.55), we have in this case Hg(0)= 0.30M. To be remarked that the same results could have been obtained by aonsidering, instead of the free charges at the pole surfaces, the mantle of amperian currents, having density per unit length M, which make the magnets equivalent to two semi-infinite solenoids of radius rg. Another method consists in summing up the contributions coming from the dipole ensemble making up the whole magnet [4.23], which

4.4 PERMANENT MAGNET SOURCES

135

is advantageously applied whenever the condition of parallelism of the magnetic moments must be relaxed. As discussed in Section 4.1 for the case of filamentary windings, the symmetry of the problem permits one to find a solution where we can write the axial field Hg(x) and the radial field Hg(z) as Taylor developments around ( x - 0, z = 0), with coefficients related as shown in Eqs. (4.14) and (4.15). The behavior of the reduced axial field in the gap Hg/M, calculated by means of Eq. (4.54), is shown in Fig. 4.14. It appears that, in order to have excellent uniformity of the field, the magnet diameter must be at least few times the gap width. For example, for a diameter of the magnet 2rg--51g, we find that the relative variation of Hg is less than 10 -4 within an interval around x = 0 of the order of 0.10lg. This uniformity is required when, for instance, Hg is to be measured for calibration purposes using an NMR probe. With typical probe size (>-5 mm), a magnet diameter larger than 1 2 0 m m would then be recommended in this case. The natural limiting value of the gap field Hg = Ms predicted for this kind of source can be overcome in several ways. The classical

1.0

In= 0.05 0.8---

0.4

In= 0.2

fl//~.//

"

0.2

0.0

.

.0

.

.

.

|

-0.s

.

.

.

.

0.0

0.a

~0

x /(Id2) FIGURE 4.14 Reduced gap field Hg/M vs. reduced distance

x/(Ig/2) in the gap of the permanent magnet field source shown in Fig. 4.13. The curves have been calculated by means of Eq. (4.54) for different values of the reduced gap width In = ls/2rg. The limiting curve for In---*0 is the constant line H~/M = 1, implying that the maximum possible field value is equal to the saturat~ion magnetization (Hg)max -- Ms.

CHAPTER 4 Magnetic Field Sources

136

t,

x

iz i,J

!~i

r

i

l

.....

l

I -

o .

.... ~-~-I'TI~} ~- ......._~i~ x~__ .... r

/ /

j

!

!--

I

.

.

.

.

.

=: ..............

~

q

\

FIGURE 4.15 Field generation in the gap of permanent magnet polar pieces with tronco-conical pole tips. For given radius rg of the flat pole and gap width Ig, the maximum axial field is obtained with a taper angle 13= 54.74~ method consists in tapering the polar pieces, while leaving enough room for the air gap (Fig. 4.15). To find the optimal taper angle ]3, we consider the axial field generated by an elementary charged ring of radius r and surface area dS = (2~Tr/sin/3)dr belonging to the conical surface (shaded area in Fig. 4.15). Since the charge density on the ring surface is or= M sin/3, we have the elementary charge dq = crdS = 2~Mr dr and, according to Eqs. (4.50) and (4.51), the elementary axial contribution at a distance x0 from the ring plane is

M xor dr dHgl(X0) = -~- [x~ + 7"2]3/2.

(4.57)

dHgl(X0) passes through a m a x i m u m value when x0 = r/~/-2. Consequently, in the special case where it is assumed the taper angle 1 3 - 54.74 ~ it happens that all the rings belonging to the conical surface produce their m a x i m u m field in the same point, which coincides with the cone apex. Each infinitesimal element of radius r of the conical surface provides, for/3 = 54.74 ~ a contribution to the axial field in this point equal to M

dr

(dHgl)rnax = ~3~/~ - -r .

(4.58)

Given the divergence displayed by this equation, one can conclude that whatever field amplitude can in principle be generated, depending on the value of the flat pole surface radius rg. If we consider, as in Fig. 4.15, two symmetrically placed polar pieces with / / - 54.74 ~ and coinciding apexes, integration between rg and r0 of Eq. (4.58) on both

4.4 PERMANENT MAGNET SOURCES

137

tronco-conical surfaces provides 2M In r0

(Hg)max- 3~3

(4.59)

rg

at the center (x - 0). There are obvious practical limits to the achievable value of (Hg)max. First, the generated field eventually affects the magnetization itself. Second, reasonable gap volume and field uniformity are required, which imposes suitable values of the ratio ro/rg. The general expression for the axial field generated by the left polar piece in a point of coordinate x is obtained by summing up the contributions of the flat surface Hg1FP(x) and the conical surface HTgC(x). If we take, with reference to Fig. 4.15, the origin at midpoint in the gap, we can write Xo = x + lg/2 + (r - rg)/tan/3 in Eq. (4.57) and we obtain, with the aid of Eq. (4.53),

/

FP TC M Hg 1(x) = Hg I (x) q- Sg I (x) -- -~- 1 -

(lg/2 + x)

)

~(lg/2 + x) 2 + r~

M ~,o r[x q- lg/2 4- (r - rg)/tan j3] dr, q- -2- rg [r2 q- Ix q- Ig/2 q- (r - rg)/tan ]~]213/2

(4.60)

which is combined with the equal contribution of the second polar piece to provide the total axial field Hg(x) = Hgl(X) 4- Hg2(X). The field at the center Hg(0) takes a particularly simple form when the cone apexes coincide. In this case lg/2 = rg/tan/3 and Eq. (4.60) provides

[

r0]

Hg(0) = M (1 - cos/3) + sin2]3 cos j3 ln~-g .

(4.61)

Crowding of the flux lines in the gap due to tapering is apparent in the maps shown in Fig. 4.16a, where the prediction of numerical calculations, carried out with the FEM method mentioned in Section 3.3, is shown. The behavior of the reduced gap field Hg(x)/M vs. the reduced distance x/(lg/2), calculated by means of Eq. (4.60), is shown for a pair of cylindrical pole pieces with tronco-conical tips in Fig. 4.17 (dashed lines). These have the optimal taper angle/3 -- 54.74 ~ and coinciding apexes. The behavior of the gap field in flat-tip pole pieces having the same values of r0 and lg is also shown for comparison (solid lines). It is apparent that tapered poles can lead to Hg values higher than the magnetization M in

138

CHAPTER 4 Magnetic Field Sources "" "t '~"x

_: L--..: _- -.-.

, ---_~.;':_---- ..--_-----(7;" "":. 3~;__~. .----

-_-L_C--C--I7_ ..-3L_--_Z-L[

- -

--C7Z_2_~-7 ;_ -~ '" ' - L 7___C_ . . . . . .

~ - = " : ' ~ d / ~

_---. . . . . . .

....i,..... "', ~,---~.............

-'-.;~" k - " f " ; l e : :

..-------. 2 - - - - _

....

(a)

~- ~ d

\ t

- - o

"

"

t!

. . . . .

8

(b)

FIGURE 4.16 (a) Magnetic flux lines in cylindrical permanent magnet pole pieces with same radius and gap length. The tapering angle is 13- 54.74~ with coinciding cone apexes. (b) Behavior of the magnetic field created by the surface free charges. The dimensions of the arrows are proportional to the field strength. Because the free charges at the opposing gap surfaces have opposite signs, their fields add within the gap and subtract within the material. The computations have been made by means of a finite element method technique [4.22].

suitably narrow gaps. The price one has to pay for such an increase of the gap field strength is the reduction of the spatial uniformity and the working volume for a same magnet volume. The extreme magnetic hardness of rare-earth based magnets, based on the very high value of the anisotropy energy, makes them pretty close to ideal magnets. They can withstand their own demagnetizing field and the fields created by neighboring magnets because of their near-zero longitudinal and transverse susceptibility. It is therefore possible to realize high performance magnetic circuits by suitably arranging blocks magnetized in different directions, often in combination with soft magnet flux guides, which provide zero reluctance and equipotential return

4.4 PERMANENT MAGNET SOURCES

1.0 0.8

139

--,q-

/I, , o=021 "

" "

" "

"--

,--

_.,

.

.

.

.

.

.

.

"N

0.6 ~

0.4

0.2 0.0

.

.0

.

.

.

|

-0.5

.

0.0 x /(~g/ 2)

0.5

1.0

FIGURE 4.17 Reduced axial field Hg(x)/M vs. reduced distance x/(Ig/2) in the gap of cylindrical pole pairs. Solid lines and dashed lines refer to flat pole tips and tronco-conical pole tips, respectively. The pairs have same radii r0 and gap lengths lg and the tronco-conical tips have taper angle/3 = 54.74~ and coinciding apexes. The curves refer to two different values of the ratio lg/r O. If this ratio is sufficiently low, tapering leads to a gap field larger than the magnetization M in the material.

paths. A number of beautiful applications have been realized, ranging from the generation of fields as high as 3.4 x 10 6 A / m (B ~ 4.3 T) [4.24] to the creation of variable flux sources [4.25]. It is possible, for example, to successfully address the problem of field homogeneity in the gap of the horseshoe-like hard magnet shown in Fig. 4.13 by applying the principle of cladding [4.26]. Figure 4.18 illustrates such a case, where the two cylindrical polar pieces of length lm are provided with a soft return path and are completed with a magnetically hard radially magnetized structure, by which they are completely enclosed. The purpose of such a structure is one of creating, by virtue of the rigidity of its magnetization, a well-defined supplementary flux pattern, which combines with the flux produced by the horseshoe magnet and eventually leads to homogeneous field in the gap. In particular, if the whole outer surface of the magnet is made equipotential, no field can exist between any two points outside it. The surface of the zero reluctance soft yoke is equipotential by definition and, by cladding, the magnet outer surface is brought to the same potential. With reference to the cross-sectional view of the magnet shown in Fig. 4.18, this means, for example, that between point A (still belonging to the soft

140

CHAPTER 4 Magnetic Field Sources

D

z~ ! I i

A

..... ",~'~.... B

i

A"

A1

i ...... ~

.............

tO'

A2

x

i

FIGURE 4.18 Generation of uniform flux in the gap of a horseshoe permanent magnet by means of cladding. The polar pieces of length lm are connected by means of a zero reluctance soft yoke and are completely enclosed by the cladding magnets. Flux confinement calls for the conditions tan a = ~g/(2~ m + ~}~g),with ~tg and ~tm the reluctances of the air-gap and the polar piece, respectively. (Adapted from Ref. [4.26].)

yoke) and point C on the cladding surface there is no potential difference and the line integral of the field along any path connecting A and C is zero. Let us choose the path ABC and express the potential difference ~AC as ~AC = q0AB -+- qOBC =

Hm xl q- Hm.L zl -- O.

(4.62)

We assume that flux confinement in the tube AAIA2A3 has been achieved and we look for the specific geometrical properties of the cladding structure consistent with this. If we take the equivalent circuit of the horseshoe magnet and we define by Fm the total magnetomotive force, we can write, according to the discussion in Section 3.3, the flux 9 as (I) =

fm

2~m + ~g'

(4.63)

where ~m and 9tg are the reluctances of the pole pieces and the gap, respectively. According to Eq. (3.23), we can also pose, making the assumption of ideal magnet (i.e. /~ =/~0), Fm - 2(Brlm/la'o), where Br is the remanent induction and the factor 2 takes into account the fact that there are two identical core magnets of length lm. The potential difference qOAB= Hm xj is obtained by considering the equivalent circuit of a portion x ~ of the magnet. To this length we associate the magnetomotive force Fm(x/) -(Br/la, o)X! and the reluctance ~I(X/) ---(X///m)~m, SO that,

4.4 PERMANENT MAGNET SOURCES

141

according to Eq. (4.63), we obtain q~

~m(X/)CI)--B--Lr ( /0z 2~ m~g+9tg ) x~"

(4.64)

Because of the assumptions made, no radial flux can exist and the potential difference ~PBCupon the distance z~is qoBC-- Fm(zl) -- - ~Br z/. /z0

(4.65)

By introducing Eqs. (4.64) and (4.65) in Eq. (4.62), we obtain that, in order to satisfy the equipotential condition and achieve the desired flux confinement, the angle a at the base of the wedge-shaped cladding magnet must satisfy the relation tan c~-

Z/ ~g x/ 2~Rm q- ~g"

(4.66)

The tip of the wedge (point D) is reached at the pole face plane and the following portion, ending at mid-gap, automatically satisfies the equipotential property along the line DO. With the previous assumptions, we see from Eq. (4.65) that q~BCtends to zero value at point O and the same q0AB does. In fact, the potential inside the gap linearly decreases from the value qo(-lg/2) -- Hmlm (the potential at point A being taken as zero) to ~o(0) = H m l m - Hgl~;/2 = 0 at the midplane OCY. The structure is then completed beyond OO'as required by the symmetry of the problem, which imposes that the polarization in the cladding blocks is reversed. Any practical application of this structure is possible only through openings into the gap cladding, resulting in residual flux leakage. The principle of cladding is applied, generally in combination with suitably arranged flux guiding soft yokes, to obtain confinement of high and uniform fields in a number of structures of interest in applications. An example is represented by the permanent magnet solenoid. As discussed by Leupold et al. [4.27], in this device the hard magnet has the shape of a longitudinally magnetized hollow cylinder, which is closed at the ends by a couple of soft disks. These soft pieces guide the flux longitudinally in the interior of the cylinder, where complete confinement is ensured by tronco-conical cladding pieces, fit on the tube like A D N in Fig. 4.18. A great variety of rare-earth based permanent magnet structures have actually been devised, with or without soft yokes, which are capable of providing uniform fields in accessible and often very large regions, one main objective being the application of such structures in magnetic resonance imaging for medical diagnostics [4.28]. The hollow cylindrical

142

CHAPTER 4 Magnetic Field Sources

structures, obtained as an assembly of elongated rare-earth blocks magnetized transversally with respect to their axis, are examples of all-magnet structures, which can provide fields largely in excess of the material saturation. Figure 4.19 shows a couple of such cylindrical devices, with octagonal and square cross-sectional profiles, respectively. The building blocs of these devices emulate the function of the thin prismatic slices shown in the cylindrical shell of Fig. 4.20b (Halbach's cylinder). Each slice can be treated as a dipole line, for which we know that the generated field has constant modulus and always lies in a plane normal to the line. In particular, a dipole line having moment per unit length A generates in a point of polar coordinates (r, (9, q~) a field of components A

A

Hr - 2"/Tr2 COS (9,

H~ = 2"/W2 s i n (9,

Hq~ = 0

(4.67)

and modulus IHI = A/2vrr2 independent of O. We see, from Eq. (4.67) and Fig. 4.20a, that H makes an angle ~ / - 2(9 with the direction of the magnetic moment. It is apparent that the two diametrically placed dipole lines highlighted in Fig. 4.20b generate in the cavity a field having the same direction of the magnetic moments. If we imagine now that this couple of thin blocks is rotated by an angle (9 around the cylinder axis, the field in the cavity remains the same in amplitude and direction if, according to Eq. (4.67), the direction of the magnetization inside the blocks rotates in turn by the angle 3' = 2(9. It is concluded that with a ring shell made of a nearly continuous distribution of sectors magnetized in accordance with this rule, all the dipole line fields constructively add within the cavity. The calculations show that the so generated field has

/ /

(a)

\ t4

L/

(b)

FIGURE 4.19 Permanent magnets generating transverse uniform magnetic fields in a cylindrical chamber. The blocks in the assemblies have directions consistent with the distribution sketched in Fig. 4.20.

4.4 PERMANENT MAGNET SOURCES

143

7

" -.

(a)

i

~../-"

(b)

I

~

FIGURE 4.20 A magnetic dipole line generates a field H of constant modulus lying in a plane orthogonal to the line. The angle ~,made by H with the direction of the magnetic moment in a point of polar coordinates (r, O,q~) is ~, = 20. The thin longitudinal sector of the annular shell shown in the figure can be assimilated to a dipole line. Since the magnetization there is radially directed, the field generated on the cylinder axis is co-linear with the magnetic moment. One can imagine filling uniformly the ring with identical blocks, by respecting the condition ~/- 20 between the angle made by the direction of the magnetization inside each block and the polar angle. The field generated on the cylinder axis by the segments from all positions of the ring is consequently always pointing along the same direction (gray arrow).

amplitude H -- Br In r0 /z0 rg

(4.68)

if Br is the remanent induction of the employed material and r0 and rg are the cylinder and the cavity radii, respectively. It is stressed that the generated field is totally confined within the cylinder. We find again, like in the tapered horseshoe magnet, a logarithmic divergence of the field and again one should keep in mind that practical limitations to the actually available field strength exist, as imposed by the finite values of coercivity and anisotropy fields. On the other hand, the minimum value of rg is dictated by the applications specifically envisaged for the device and r0 can increase only to the point where the gain associated with the logarithmic dependence of the field strength on r0 makes sense compared with the ~ increase of the volume (and cost). For a material with Br - 1 T, cavity radius rg -- 20 m m and cylinder radius r0 = 100 mm, Eq. (4.68) provides the respectably high value H = 1.28x106 A / m (B = 1.61 T). The practical realization of this device shown in Fig. 4.19a is obtained by assembling trapezoidal segments into an octagonal

144

CHAPTER 4 Magnetic Field Sources

structure. The calculations provide a field strength value about 90% of the one predicted by Eq. (4.68) [4.26]. The square cross-section cylinder shown in Fig. 4.19b is stray-field free only when the side of the cavity Ig and the thickness of the wall t are in the ratio lg/t = 2/(x/2 - 1) [4.25]. The field correspondingly generated in the cavity is H -- 0.293Br//Z 0. It is a remarkable feature of these rare-earth magnet structures that also variable fields can be produced [4.29]. By nesting, for example, two Halbach's cylinders having the same ratio ro/rg, we generate fields of equal strength H0 in the cavity, which add one to another. If, by means of a suitable mechanism, the two cylinders are synchronously rotated in opposite directions by an angle + a around their longitudinal axis, a resulting field having fixed direction and amplitude varying as H ( a ) 2H0 cos a is obtained in the cavity (see Fig. 4.21a). By this method, the realization of compact variable field sources can be envisaged. This has been demonstrated, for example, by Cugat et al., which have built a vibrating-sample magnetometer around a source made of two nested N d - F e - B octagonal cylinders [4.30]. This device, with an outside diameter 108 mm, a cavity diameter 26 m m and height 115 mm, can produce a m a x i m u m field around 106 A / m , with homogeneity within the usable volume of about + 3 x 103 A / m . A practical solution to the generation of variable fields is schematically illustrated in Fig. 4.21b, which can be understood as a simplification of the base cylinder structure

a ....

(a)

(b)

FIGURE 4.21 Two synchronously counter-rotating Halbach's cylinders (approximated here by octagonal devices) having the same ratio between outside and inside diameters produce a field of amplitude continuously variable between + 2H0 and fixed orientation. In a simplified realization of this device, four identical cylindrical rods, magnetized in the transverse direction, are made to counterrotate as shown in (b).

4.5 ELECTROMAGNETS

145

of Fig. 4.19b. Here four parallel rods, magnetized transverse to their axis, are made to alternately rotate clockwise and counter-clockwise in a synchronous fashion, thereby generating a field fixed in direction and variable in amplitude. With the magnetization M in the rods directed as shown in Fig. 4.21b, the maximum field strength at the center H m a x is obtained. For rod radius a and distance between the rod axes d, H m a x is calculated by adding the equal contributions provided by the four equivalent dipole lines of linear density A = "rra2M. It turns out, according to Eq. (4.67), Hmax - - 4 2 /vr(d 2~

~2

= 4 ~- M.

(4.69)

For a complete 360 ~ rotation of the rods, the field will continuously vary between + Hmax and, based on this method, alternating field sources can be built with working frequencies up to few Hz [4.31]. For a ratio a / d = 0.25, permitting convenient access to the central region, and N d - F e - B magnets with remanence Br --1.2 T, we get Hmax = 2 X 10s A/m. It is easily realized, always having in mind the ~/= 20 rule illustrated in Fig. 4.19, that, if the magnetization direction in the lower (upper) rod couple in Fig. 4.21b is inverted, a uniform magnetic field gradient is produced along the horizontal axis passing through the center, where the field value is zero. It should be stressed once again that crucial to the performance of the permanent magnet sources here discussed is that the employed materials are magnetically so hard that they are substantially transparent to the generated fields. The rare-earth based magnets excellently comply with such a requirement.

4.5 E L E C T R O M A G N E T S The use of soft iron cores as field-producing devices is a classical solution in many magnetic testing instances, where the requirement of high field strengths often comes with demand for flexible control. Though permanent magnets can be adapted, by means of mechanical systems, to the generation of high-amplitude variable fields, soft-cored electromagnets still cover most of the laboratory needs in this respect, namely for hard magnet testing, sometimes in conjunction with pulse magnetizers or superconducting coils. The obvious drawback of electromagnets is the natural limit posed by the saturation magnetization of iron, which compounds with the usually large mass of the employed cores and the continuous expenditure of energy; with the related dissipation

146

CHAPTER 4 Magnetic Field Sources

and cooling problems. Optimal electromagnet design is a difficult task and since the cost of the final device may be pretty high, a careful theoretical treatment of the whole matter, including the coil properties and related cooling system, is required. To this end, one can take advantage, as mentioned in Section 3.3, of advanced numerical techniques for electromagnetic field calculations, by which detailed and precise information on the field and flux patterns in the useful region and in the core can be achieved. The basic structure of an electromagnet is simple: a useful field of variable strength is generated in the gap of a soft magnet ring, channeling and amplifying the field generated by an exciting coil. A practical realization appears as the variable gap device schematically shown in Fig. 4.22. This amounts, in a first approximation, to consider the gapped ring of uniform cross-section shown in Fig. 3.1, with the addition of an N-turn winding with a circulating current i. Under the strongly simplifying assumption of uniform induction in the whole circuit, we apply, in strict analogy with the discussion made in Section 3.1, the continuity equation for the flux and Amp6re's law for the field, all quantifies being taken as scalars. By denoting with Bm and Bg the induction in the core (of length /m) and in the gap (of length lg), respectively, we write Bm = Bg,

Hglg -Jr-H m l m -- Ni.

(4.70)

The continuity condition relates the field in the gap Hg and that in the core H m with the polarization Jm /z0Hm + Jm--/z0Hg

(4.71)

and, by introducing Eq. (4.70), we obtain the flux density in the gap as a function of core geometry, material polarization and magnetomotive

FIGURE 4.22 Schematic view of a variable-gap H-frame electromagnet.

4.5 ELECTROMAGNETS

147

force Ni

tL~

=

la,oNi/Ig + Jmlm/Ig I q- Im/lg "

(4.72)

Equation (4.72) provides an upper limit for ~0Hg in the case of small gap

(lm ~ lg) (4.73)

(/d,oHg)max ~ p,oNi/lm + Jm,

where we can recognize the separate contributions of the windings and the material. Contrary to previously treated case of ideal or nearly ideal permanent magnet field sources, the polarization Jm is affected by the demagnetizing field and is consequently related to the geometrical parameters of the core and the material permeability/~ =/~0/~r. We can conveniently express Jm, always within the limits of the scalar model of the magnetic circuit, as a function of these quantities. To this end, we can resort to the electrical analogy introduced in Section 3.3 and describe the electromagnet by means of the circuit shown in Fig. 3.5, where we pose the magnetomotive force Fm = Ni and we take into account flux fringing at the gap by introducing the leakage reluctance 9tl. We express the reluctance of the yoke 91m and the reluctance of the gap 91g by means of Eq. (3.22) and the leakage reluctance through Eq. (3.24), with the leakage factor ]3 = (Bg/Bm)< 1. By solving the circuit, we obtain, according to Eq. (3.25), the flux in the gap ~g = ]~(I)m, w i t h (I) m the flux in the yoke, as

Ni (I)g =/d,0HgSg--/~ 91m -}-]~91g

(4.74)

and by means of Eq. (3.22) we get the induction in the gap

Ni I~r Bg =/~0Hg --/.L0 lg /d'r..}___1

(4.75) Im

"

fl Ig We see from this equation that if the iron core is far from saturation (i.e. ~r is high) and the gap has practical size, so that/d, r )) (1/~)(lm/lg), most of the magnetomotive force appears at the gap and

Ni /z0Hg ~ ~0 Ig

(4.76)

These approximate relationships are very useful when assessing in general terms the performances of the electromagnet, but we generally

148

CHAPTER 4 Magnetic Field Sources

need to be more inquisitive regarding the flux distribution in the gap and the yoke. The classical approach to the analytical calculation of Hg basically calls for the permanent magnet approximation, where the core magnetization is everywhere fixed in amplitude and direction and the magnetizing coils provide an additional contribution to Hg. The relevant aspects of this approach have been discussed in Section 4.4, where we have, in particular, provided expressions for the axial field provided by cylindrical permanent magnet poles, either with flat or tapered tips (see Figs. 4.13-4.17). Pole tapering, with its predicted ln(ro/rg) dependence of the axial gap field on the ratio between base and top radii of the troncoconical pole tips (Fig. 4.15), permits one to overcome the natural limit to the gap field strength provided by the saturation magnetization of the material. The optimal taper angle is found to be ~ -- 54.74 ~ If we stick to this modeling framework in soft core electromagnets, as done in the less recent literature [4.11, 4.12, 4.23], we have, however, to rely on substantial approximations, because we can only make reasonable assumptions regarding the relationship between magnetomotive force and magnetization in the material. This is expectedly non-uniform, as it should be in non-ellipsoidal non-linear magnetically soft open cores, which, in addition, are excited by localized magnetizing coils. A matter of this kind falls nowadays in the general domain of problems that can be successfully investigated by means of numerical methods, where the Maxwell's equations with the appropriate boundary conditions are solved according to the principles outlined in Section 3.3. It is a magnetostatic problem defined by the equations (3.27) and the constitutive equations B =/~0H 4- ~0M =/a,0/~rH (in the soft material, disregarding hysteresis) and B =/~0H (in air). In order to find the behavior of B, it is expedient to resort to the vector potential function, applying the general procedure sketched in Section 3.3. Figure 4.23 illustrates the result of a finite element method computation of the magnetic induction in a couple of identical soft iron C-cores, one tapered (with 13 = 54.74 ~ and one untapered, having circular cross-section of diameter 250 m m and gap length of 35.4 mm. The cores are supplied with the same magnetomotive force Ni -- 25 x 103 A. The calculations show that we achieve in both cases the same flux density in the gap/-~0Hg ~ 0.88 T, in spite of a dramatically different induction level in the back core. This is illustrated in Fig. 4.24a by the computed behavior of the axial flux density in the gap. It is easy to understand that tapering is normally of little avail for achieving, with a given Ni value, higher gap fields when the magnetization in the core is far from saturation. In fact, under these conditions the relative permeabilit~ of commercial purity Fe is expected to range between I x 10~ and 5 x 10~,

4.5 ELECTROMAGNETS

149

,;/' ,". ':

i-.:--":. ---!:

--.---.--- ----:::"-::

, ,

","

i,;',, ',,':....-~::~--.----:-.--.... ----~-----:"--.:..: " :"','"~ ." "i I.".,'.'.""",.." ....... ,-- : ........

,I~,i?, 't,l+~,',

.' "..-." ' '. ,

'I! ' +I.'.Ill ,t..:,.,, ",I [.i', ' !I I' I . 11[ '.< ,, ,i,~::'.t"i ,",,",

IiI .'~,,",, ', ',. ,,

i'l" ,.I.''I+,'II:, ,;.' I II , , I 1+,I ,I. ,I I+ ,,.,I ,! 'it(:',

'

,:

',

I,,.., I,,,,,,.,,.,',,.',..,':I i! ,,i p ', , ,, v .. ~,', ...,?..t____

'.,

L,", ,,,' ,.! J , ', J,

I~~,, ..,..".-: ...... --............ I i'~ ~,'~," ,:.: 7-: "-':::-:::-.::::-:1 \ '" " "-..":-:7----' ",'.,

22"

:

IiI lli,~" i ~ ' t ,;l,','i i' 1

__

.

-.-

.':

= - J': ,';..":" /' ",] ,' -'-" ' ",'., ' ,,'[ .'~,I

. ._..

-. -. ...... I;~

:- :: ,":

:'.'.; I.t'~

', - "_:-:::::--. :-::--:-::: 7--:--..::--._L- " ""

. ~_

I ....................... (

I ,

~,'i,,,I

~~,",":.','J

,,

"~

,,,'ljJ

/

"

" ----~'

', '\,5,<:....:,_~ :I..L:_:_-.......... :-::4: \

I

/i I

'

/ ,' ,". ,".'L,"i ."i ; _.-:-:..:.:'//I,:/

'. ...... L .": ..... :--:. ! ~_-7- --: :.-: .... .:~k:: .... :7

;.-'.

/

,;

FIGURE 4.23 FEM calculated lines of magnetic induction in identical soft iron C-cores of circular cross-section and ratio Ig/ro = 0.28. The tapering angle is /3= 54.74~ Both cores are supplied with the same magnetomotive force Ni = 25 x 103A and provide approximately the same flux density in the gap /.LoHg -- 0.88 T.

the yoke reluctance becomes negligible and Eq. (4.76) applies. All the magnetomotive force appears at the gap, irrespective of the specific shape of the pole tips. For Ni = 25 x 103A and Ig--35.4 mm, we obtain from Eq. (4.76) /z0Hg = 0.87T, nearly the same result provided by FEM computation. Figure 4.24b, showing the calculated radial dependence of /-~0Hg, puts in evidence the obvious limitations in field homogeneity introduced by tapering, which, consequently, does not appear a convenient solution in the intermediate and low induction range. However, distributed air gaps are inevitable in actual electromagnets and some

150

CHAPTER 4 Magnetic Field Sources 2.5

2.0 G" 1.5

\

/

Tapered

/

Untapered

Ni = 25 103 A

./

o 1.0 :::L

0.5 0.0

Ni= 125 103A

/

Soft Fe C-core Js= 2.16T

_

. . . . . . . . .

.0

i

. . . . . . . . .

i

-0.5

. . . . . . . . .

0.0

|

. . . . . . . . .

0.5

1.0

x/(Ig/2)

(a)

2.5 Tapered......._,~ G" 2.0

..

..-

i N i = 125.103 A

:

Untapered

/

i

,

~.5 i

,4--

0.5 0.0 (b)

N i = 25 103 A

:

1.0

/ '

4,

_ ....

i .........

-6

i .........

-4

Soft Fe C-cor~ Js = 2.16T | ....

-2

~ ....

i ....

0

|

....

| .........

2

i .........

4

| .....

6

z/(Ig/2)

FIGURE 4.24 Axial flux density /~H g in the gap of the tapered (/3--54.74 ~ dashed lines) and the untapered (solid lines) soft Fe C-cores of Fig. 4.23, as computed by means of an FEM technique for two different values of the applied magnetomotive force Ni. The conical tips have coinciding apexes. (a) and (b) refer to the dependence of /~0Hg on the axial and the radial coordinates x and z, respectively. The gap length and the core radius are in the ratio l~/ro = 0.28.

small advantage m a y be associated with tapering also at low fields (Fig. 4.25) [4.32]. Notice that the results here discussed are invariant if all the dimensions of the core and the m a g n e t o m o t i v e force are changed by the same factor (Kelvin's rule). W h e n the m a g n e t o m o t i v e force is increased to such an extent that the saturation magnetization of the material is approached and a substantial potential drop across the yoke appears, the gain in the field gap strength associated with pole tapering

4.5 ELECTROMAGNETS

151

Untapered 2.5

......

....

Tapered,,, ~ , , " ~ ~ " ' ~ " ; Ig=2Omm

2.0

/

#=1.5

-

!

f

.

~"

I /

,.

"

1.0 0.5 0.0

20

40

60 80 I(A)

1O0

120

140

FIGURE 4.25 Flux density as a function of the supply current in the center of the gap of length lg of a commercial H-type electromagnet. The cylindrical poles have diameter 2r0 = 250 mm. Continuous lines: fiat pole faces (rg = r0 = 125 mm). Dashed lines: tapered poles (rg = 50 mm) [4.32].

comes out clearly. In the example reported in Fig. 4.24, where we suppose to supply the model Fe C-yokes of Fig. 4.23 with Ni -- 125 • 103 A,/-~0Hg becomes pretty larger than Js if the poles are skewed (/~0Hg(0) = 2.53 T), while remaining below Js(/~0Hg(0)= 1.78 T) if they are not. It should be stressed that a prediction made using the permanent magnet approximation (Eq. (4.61)) with the addition of the coil field contribution is rather inaccurate, because for reasonable Ni values the magnetization is not uniform. Figure 4.26 compares the prediction of the axial field strength made through Eq. (4.61), where the Fe core is assimilated to a hypothetical permanent magnet with Js = 2.16 T, with the exact FEM calculation, after subtraction of the field directly contributed by the current circulating in the windings. Such a contribution can be estimated, for the given magnetomotive force, by means of Eq. (4.27). The permanent magnet approximation overestimates the gap field with flat pole faces, while it falls short of the correct FEM prediction with optimally tapered pole tips. This is the natural consequence of having assumed the magnetization M uniform and constrained to be parallel to the pole axis everywhere. Figure 4.27 offers a detailed view of the space distribution of the induction

CHAPTER 4 Magnetic Field Sources

152 2.5 - Tapered 54.74 ~ ........ ~ 2.0 ~, 1.5

......................

JS-

.,o

FEM _

.........................

- FEM

~,,~-'

T6,

~

0.5

........

Flat pole faces

Soft Fe C-core

Ni= 125 103A

Hcoim

0.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -" .0 -0.5 0.0

x/qj2)

'. . . . . . . . . . . . . . . . . 0.5 1.0

FIGURE 4.26 The FEM calculated induction in the gap /z0Hg, presented in Fig. 4.24 for the tapered and untapered electromagnet Fe C-cores of Fig. 4.23 (Ni--125 x 10 3 A), is shown here (dashed lines) after subtraction of the field directly applied by the current in the windings (short-dash line). It is compared with the prediction made through Eq. (4.61), where the core is assimilated to a hypothetical identical permanent magnet of same Js (solid lines).

B within and in the neighborhood of the air gap (a) and (b) and of the magnetic polarization J in the material (c) for Ni = 125 x 103 A. Contrary to the case of the hard magnet (Fig. 4.16), in soft Fe (assumed isotropic and without hysteresis) direction and amplitude of J follow direction and strength of the effective field H, sum of the field generated by the current in the windings and that generated by the free charges. B, H, and M are co-linear everywhere (the material is assumed isotropic) and it is understood that the opposite trends of the induction lines in the untapered and tapered cores (spreading vs. crowding towards the axis, respectively, on approaching the gap) are associated with opposite variations of M in these two cases. This is put in evidence by the isointensity contour maps of J in Fig. 4.27c. It was previously recalled that the field generated by the magnet in the gap could be assumed as due to the ensemble of magnetic dipoles constituting the magnet, a method allowing in principle for field prediction also in presence of volume free charges. Any dipole provides a field at the center of the gap, which depends on its orientation. In particular, for a dipole of moment m located at a point of polar coordinates (r, 0), whose orientation makes an angle qJ with the pole

4.5 ELECTROMAGNETS

153

_Ezz::3]_XL_L~

i::-::-~'-:-::-~

(b)

(a)

..

L__.

,,, -: x, --...... :;~.;:7

Z

t

.9' : . " ,

2 - : ~

~ t,. ..

'

' ,,

,..'

.:~

.~..~K., ~ ;:4~. .... .r' :"'4,.;; ~ $#.'a~ ~.4

~,

(c) ~. ~ ,~.,~

,~

J

,:"..~;: 3T

'-|

;: ! .... ;...... :,c

\

,,

r

1.5T

,....

\

17_TZi_IZ21 t

0

FIGURE 4.27 FEM computed lines (a) and contour maps (b) of magnetic induction within and around the gap of the representative C-cored soft Fe electromagnets shown in Fig. 4.23 and discussed in the text. The condition here represented is obtained by applying a magnetomotive force Ni = 125 • 103 A (the coil size is not represented in scale in this figure). The associated gap field behavior is shown in Fig. 4.24. The contour map of the polarization in the material is shown in (c). The inset describes the position of a generic dipole of moment m with respect to the pole axis. (Courtesy of O. Bottauscio).

axis, this field is m a x i m u m w h e n qb satisfies the condition tan q~ =

3 sin 0 cos 0 3 C O S 219 - - 1 "

(4.77)

This implies that an o p t i m u m distribution of the orientations of the magnetic m o m e n t s in the material exists, w h i c h does not c o r r e s p o n d to a state of u n i f o r m m a g n e t i z a t i o n M, parallel to the pole axis. It can

154

CHAPTER 4 Magnetic Field Sources

be shown that optimally arranged dipoles should lie along the field lines generated by a horizontal dipole (or an equivalent coil) placed at the center of the gap [4.33]. While attaining this distribution, which bundles towards the center, is not practical, it is clear, looking at the flux lines in Fig. 4.27, that it is approximated, at least to some extent, by the soft tapered poles arrangement. In terms of dipole field, this arrangement can then be seen not only, like in the permanent magnet poles, as a means to cut away the dipoles negatively contributing to the gap field (i.e. the portion of the iron core containing such dipoles), but also as a condition closer to the ideal dipole orientation distribution. This justifies the result of the FEM calculation shown in Fig. 4.26. The design of an electromagnet is a complex procedure, which involves a large number of parameters and is subjected to many constraints, the most demanding one being eventually the cost of the device. When it is to be used for the characterization of magnetic materials, a variable gap type electromagnet is preferred, very often provided with removable Fe49Co49V2 tronco-conical caps. This alloy is characterized by a saturation polarization Js = 2.35 T (Table 2.9) and is therefore conducive to high gap fields. ~ H g can actually be obtained higher than 3 T with reasonably dimensioned supply coils and gap lengths in the range 5-10 mm [4.34]. In the general case, the starting requirement regards stated field uniformity and maximum field value over a certain volume of the gap. For example, magnetic resonance experiments require very high field uniformity, which, in turn, call for large pole diameters. The solution discussed here, making use of flat polar faces of diameter 250mm and with gap length 35.4mm ensures, according to the calculations reported in Fig. 4.24, a field uniformity, for /a,0Hg -- 1.79 T, better than 10 -4 over a centered sphere of diameter about 20 ram. Application of Fe (or Fe-Co) pole caps while maintaining the same gap length will obviously increase the gap field, at the cost of a severe reduction of uniformity. The example given in Fig. 4.24, with 54.74 ~ tapering, provides uniformity better than 10 -2 for p~Hg = 2.53 T over a cylindrical region of height 25 mm and diameter 15 ram, acceptable for the testing of hard magnet samples. The core diameter 2r0 should in any case be the minimum compatible with the stated requirements of field amplitude and uniformity, because the volume of the magnet and its cost approximately depend on ~. In order to reduce the drop of magnetomotive force over the yoke, the use of an "Hframe", schematically shown in the Fig. 4.22, where the return flux divides between two equal branches, is appropriate. The yoke diameter is also important for the role the eddy currents can play in shielding

REFERENCES

155

the inner region and leading to non-uniformity of the generated field when the current in the supply coil is slowly varying with time. Such kind of supply is required, for example, when testing hard magnets with the hysteresisgraph method (see Section 8.1 and the IEC-60404-5 standard). Flux penetration can be calculated with standard methods [4.35] or by numerical FEM solution of the appropriate Maxwell's equations. For the model Fe C-yoke treated here (2r0 = 250 mm), with resistivity of Fe p = 1 0 - 7 ~ m and estimated apparent relative permeability ~a ~ 80~ we find about 5% decrease of the magnetic induction on reaching the core axis from periphery under cyclic excitation at frequency f = 10-2 Hz. A further basic aspect of electromagnet design is represented by coil dimensioning and heat dissipation. The coils are placed, for obvious reasons, as close as possible to the gap and have a natural size constraint imposed by the size of the core. The power consumption can be estimated with the aid of Eqs. (4.32) and (4.33) and it turns out to range, for normal operating conditions, between 1 and 20 kW. Heat removal requires specific strategies in winding construction. The use of conventional wirewound coils, possibly cooled by forced ventilation is suitable only for low powers, because the thermal conductivity of the assembly of insulated wires, with their poor filling factor, is very low. Efficient cooling is obtained by using insulated copper tape and segmenting the coil into narrow portions, separated by water-cooled copper disks, with which they are kept in good thermal contact (pancake coils). Tube wound coils with water flowing in their interior are also employed in very large electromagnets. As a whole, optimal solution to the problem of heat dissipation in high power coils requires a quantitative approach to the phenomena of heat conduction in the metal, the transmission between the metal and the cooling liquid, and the transport by means of the liquid itself (laminar vs. turbulent regime, water vs. other liquids, etc.). A discussion on this subject, going beyond the scope of this treatise, can be found in [4.10-4.12]. References 4.1. W.M. Frix, G.G. Karady, and B.A. Venetz, "Comparison of calibration systems for magnetic field measuring equipment," IEEE Trans. Power Delivery, 9 (1994), 100-106.

4.2. I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products (New York: Academic Press, 1980). 4.3. A. Firester, "Design of square Helrnholtz coil systems," Rev. Sci. Instrum., 37 (1966), 1264-1265.

156

CHAPTER 4 Magnetic Field Sources

4.4. M. Misiakian, "Equations for the magnetic field produced by one or more rectangular loops of wire in the same plane," J. Res. Natl Inst. Stand. Technol., 105 (2000), 557-564. 4.5. J.C. Maxwell, A Treatise on Electricity and Magnetism, (London: Clarendon, 1891, third edition. Reprinted by Dover: New York, 1954), Vol. 2, p. 331. 4.6. M.W. Garrett, "Axially symmetric systems for generating and measuring magnetic fields," J. Appl. Phys., 22 (1951), 1091-1107. 4.7. M.E. Gardner, J.A. Jungerman, P.G. Lichtenstein, and C.G. Patten, "Production of a uniform magnetic field by means of an end-corrected solenoid," Rev. Sci. Instrum., 31 (1960), 929-934. 4.8. W. Franzen, "Generation of uniform magnetic fields by means of air-core coils," Rev. Sci. Instrum., 33 (1962), 933-938. 4.9. P.N. Murgatroyd and B.E. Bernard, "Inverse Helmholtz pairs," Rev. Sci. Instrum., 54 (1983), 1736-1738. 4.10. H. Zijlstra, Experimental methods in magnetism (Amsterdam: North-Holland,, 1967), vol. 1, p. 191. 4.11. D.J. Kroon, Laboratory magnets (Eindhoven: Philips Technical Library, 1968), p. 29. 4.12. D. De Klerk, The construction of high-field electromagnets (Oxford: Newport Instruments, 1965), p. 42. 4.13. F. Bitter, "The design of powerful electromagnets. Part Ih the magnetizing coil," Rev. Sci. Instrum., 7 (1936), 482-488. 4.14. F. Gaume, "New solenoid magnets," Proc. Int. Conf. on High Magnetic Fields (H. Kolm, B. Lax, F. Bitter, and R. Mills, eds., MIT Press and Wiley, Cambridge, MA, 1962), 27-38. 4.15. K. Kaminishi and S. Nawata, "Practical method for improving the uniformity of magnetic fields generated by single and double Helmholtz coils," Rev. Sci. Instrum., 52 (1981), 447-453. 4.16. L.J. Campbell, H.J. Boenig, D.G. Rickel, J.B. Schilling, J.R. Sims, and H.J. Schneider-Muntau, "Status of the NHFML 60 Tesla quasi-continuous magnet," IEEE Trans. Magn., 32 (1996), 2454-2457. 4.17. F. Herlach and M. von Ortenberg, "Pulsed magnets for strong and ultrastrong fields," IEEE Trans. Magn., 32 (1996), 2438-2443. 4.18. R. Gr6ssinger, E. Witfig, M. Kfipferling, M. Taraba, G. Reyne, C. Golovanov, B. Enzberg-Mahlke, W. Fernengel, P. Lethuillier, and J. Dudding, "Large bore pulsed field magnetometer for characterizing permanent magnets," IEEE Trans. Magn., 35 (1999), 3971-3973. 4.19. G. Dworshack, F. Haberey, P. Hildebrand, E. Kneller, and D. Schreiber, "Production of pulse magnetic fields with a flat pulse top of 440 kOe and 1 ms duration," Rev. Sci. Instrum., 45 (1974), 243-249.

REFERENCES

157

4.20. H.P. Furth, M.A. Levine, and R.W. Waniek, "Production and use of high transient magnetic fields. II," Rev. Sci. Instrum., 28 (1957), 949-958. 4.21. R. Gr6ssinger, Ch. Gigler, A. Keresztes, and H. Fillunger, "A pulsed field magnetometer for the characterization of hard magnetic materials," IEEE Trans. Magn., 24 (1988), 970-973. 4.22. O. Bottauscio, private communication. 4.23. D.B. Montgomery, "Iron magnet design," Proc. Int. Conf. on High Magnetic Fields (H. Kolm, B. Lax, F. Bitter, R. Mills, eds., MIT Press and Wiley, Cambridge, MA, 1962), 180-193. 4.24. F. Bloch, O. Cugat, G. Meunier, and J.C. Toussaint, "Innovating approaches to the generation of intense magnetic fields: design and optimization of a 4 Tesla permanent magnet flux sources," IEEE Trans. Magn., 34 (1998), 2465-2468. 4.25. R. Skomski and J.M.D. Coey, Permanent magnetism (London: IOP, 1999), p. 303. 4.26. H.A. Leupold, "Static applications," in Rare-earth iron permanent magnets (J.M.D. Coey, ed., Oxford: Oxford University Press, 1996), p. 381. 4.27. H.A. Leupold, E. Potenziani II, D.J. Basarab, and A. Tilak, "Magnetic field source for bichambered electro-beam devices," J. Appl. Phys., 67 (1990), 4650-4652. 4.28. M.G. Abele, Structures of permanent magnets (New York: Wiley, 1993). 4.29. H.A. Leupold, E. Potenziani, II, and M.G. Abele, "Applications of yokeless flux confinement," J. Appl. Phys., 63 (1988), 5994-5996. 4.30. O. Cugat, R. Byme, J. McCaulay, and J.M.D. Coey, "A compact vibratingsample magnetometer with variable permanent magnet flux source," Rev. Sci. Instrum., 65 (1994), 3570-3573. 4.31. O. Cugat, P. Hansson, and J.M.D. Coey, "Permanent magnet variable flux sources," IEEE Trans. Magn., 30 (1994), 4602-4604. 4.32. http://www.lakeshore.com/magnetics. 4.33. F. Bitter, "The design of powerful electromagnets. Part I: the use of iron," Rev. Sci. Instrum., 7 (1936), 479-481. 4.34. http://www.walkerscientific.com/Electromagnets. 4.35. R.M. Bozorth, Ferromagnetism (New York: Van Nostrand, 1951), p. 769.

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CHAPTER 5

Measurement of Magnetic Fields

The measurement of magnetic fields is needed at all points in the experimental characterization of magnetic materials. Be they applied, generated by the material itself, or present in the environment, no meaningful material testing can be performed without their precise determination. The old method of field measurement by suspended current-carrying coil and galvanometer has given way to a host of techniques, which cover today more than ten orders of magnitude in field strength and satisfy most present requirements of science and technology. The objective importance of field measurements goes beyond the domain of materials testing, being connected with such diverse areas as medicine, geology, astrophysics, and environment. Entering these formidably vast issues would be outside the scope of this treatise, but it appears nevertheless appropriate to look beyond the relatively narrow subject of magnetic materials characterization by keeping track of some remarkable devices used in these areas and by providing a comprehensive approach to some basic methodological aspects of field measurement (e.g. calibration schemes, uncertainties, etc.). A classical way to reveal and measure a magnetic field is by means of a closed-loop winding, where an electromotive force is detected, according to the Faraday-Maxwell law of electromagnetic induction, proportional to the time derivative of the linked flux. Detecting DC fields by this method requires either on-off switching of the field or some mechanical action on the winding (removal, rotation, flipping) in order to achieve a controlled flux variation. Linearity and accuracy, wide frequency and field strength ranges, ruggedness and low cost of apparatus make the fluxmetric method ideal for AC field measurements. There are instances, however, where direct sensing of the field is preferred. Galvanomagnetic sensors, exploiting the effect of the Lorentz force on the transport properties of metals or semiconductors, can provide a signal directly related to the field strength. They are mainly Hall-effect 159

160

CHAPTER 5 Measurement of Magnetic Fields

devices, semiconducting plates where a circulating DC current is affected by an external field in such a way that a potential difference proportional to the magnetic field strength is generated. Hall sensors combine good sensitivity with simplicity of use and high flexibility. They consequently find pervasive applications and nowadays dominate the commercial landscape of field measuring devices. It is clearly possible to exploit ferromagnetic cores for field measurements. The most obvious way to do so, by determination of the torque (i.e. the oscillation period) of a freely suspended macroscopic permanent magnet dipole subjected to a DC field, is now relegated to history. In the typical soft magnet-based magnetometers, the field to be sensed modifies the shape of the magnetization curve in a detectable way. For example, in the fluxgate magnetometer, the most conspicuous of these devices, an external DC field is revealed by the asymmetry it introduces in the hysteresis loop. If the core is subjected to AC magnetization at the same time, this asymmetry gives rise to even harmonics in the secondary voltage, which can be quantitatively related to the strength of the field under testing. Magnetoelastic and magneto-optic effects are also frequently exploited for the sake of field sensing. The development of magnetostrictive magnetometers, in particular, has been propelled in recent years by the advent of the high magnetostriction soft amorphous ribbons, often incorporated in fiber-optic interferometers. For all their practical importance, these measuring methods call for some kind of connection with either the geometrical features of the probes or the structural properties of the materials employed in them, which is not totally convenient from the viewpoint of accuracy and stability of the measuring apparatus. Remarkably, however, instrument calibration is always possible, at least in principle, in terms of precisely known fundamental physical constants. The key to this property is the phenomenon of nuclear magnetic resonance (NMR), by which the magnetic flux density can be expressed in terms of an accurately determined frequency value and the nuclear gyromagnetic ratio. There is indeed a variety of field measuring methods based on resonance phenomena in quantum systems, like the free proton precession, electron spin resonance, atomic resonance in optically pumped gases, and the Overhauser effect. Altogether they cover with the utmost precision a range of flux density values from less than 10-6T to more than 10 T. Another quantum device, the SQUID, which is a superconductive ring containing one or more weakly conductive links, provides the ultimate sensitivity in the measurement of flux. More precisely, a SQUID is always coupled to a magnetic flux multiple of the quantum ~0 = h/2e = 2.07 x 10 -15 Wb, where h is the Planck constant

5.1 FLUXMETRIC METHODS

161

and e is the electronic charge. Flux densities lower than 10-12 T can be m e a s u r e d with SQUID magnetometers.

5.1 F L U X M E T R I C M E T H O D S 5.1.1 M a g n e t i c flux d e t e c t i o n A fiat coil, of area A and n u m b e r of turns N placed in a region where a magnetic flux density B --/~0 H is present, is linked with a flux

= N f /z0H.n dA, dA

(5.1)

where n is the unit vector normal to the elementary area dA. If 9 is timedependent, either due to a time variation of B or some mechanical action on the winding, an electromotive force (e.m.f.) e =-

d~ d-T

(5.2)

is generated. The flux variation taking place between two instants of time tl and t2 is therefore obtained by integrating the e.m.f. q~(t2) - q~(tl) = -

e dt.

(5.3)

tl

Suppose that the coil, initially i m m e r s e d in a h o m o g e n e o u s field H m a k i n g an angle a with its axis (see Fig. 5.1), is b r o u g h t to some distant field-immune position and the resulting e.m.f, is integrated over the pertaining time interval according to Eq. (5.3). The field strength is obtained as t2 e dt H =

-

tl

p,oNA cos a

9

(5.4)

A n u m b e r of variants to the coil removal m e t h o d can be devised. If, for instance, it is not convenient or possible to displace the coil to a zero field region, it can be flipped over by 180 ~ or even rotated at constant angular speed to a r o u n d its axis on the plane (generator fashion). In the latter case, an o u t p u t sinusoidal voltage e(t) = oOlxoNAH sin ~ot

(5.5)

is observed and the integration becomes unnecessary to find H. Nonuniformity of the flux density is signaled by the distortion of the e(t)

162

CHAPTER 5 Measurement of Magnetic Fields

x

a

ffII

H

FIGURE 5.1 A flat search coil of area A and number of turns N, immersed in a homogeneous magnetic field H making an angle a with its axis, is linked with a flux 9 = ~oNAH cos a. The field strength can be determined by inducing a convenient variation cI). Coil removal, flipping, or rotation at constant speed are possible ways to produce such a variation.

waveform. If, on the contrary, the coil position is fixed, one can measure the flux variation either u p o n switching on or switching off the field source and applying Eq. (5.4). If the coil is not flat, the enclosed flux is proportional to the volume average of the axial component of H. Let us consider in a thin-walled cylindrical coil of length h0, turn density n, and cross-sectional area A, and elementary ring of thickness dh. The flux linked with this ring can be written, according to Eq. (5.1) as

dcI) = n dh ~A/-~0H(h).n dA

(5.6)

and the total flux enclosed by the coil

~ = nl~o~1,o ~aHx(h) dA dh,

(5.7)

with Hx the local axial field component. For uniform field H intercepted by a straight cylindrical coil, this equation reduces to

= NAtzoH cos

a,

(5.8)

with N = nh, equivalent to Eq. (5.4). Stable and accurate search coils, realized using a rigid insulating former and copper wire, are commercially available. They can be calibrated with reference to their m e a s u r e d dimensions, ensuring traceability to the standard of the SI unit of length [5.1]. Their area-turns product NA can in this case be k n o w n with

5.1 FLUXMETRIC METHODS

163

an uncertainty at best around some 10 -3 . In general, calibration is made by reference to standard field sources (see Section 5.5). Here the area-turns product NA is determined by supplying a reference solenoid or Helmholtz pair with a precisely known current and placing the coil in a region with stated field homogeneity. The flux is measured either by extraction, inversion of the DC current or by using an AC current source. The calibration is therefore traceable to the quantum standards of voltage, resistance, and frequency. An alternative method consists in comparing the coil with a standard mutual inductance using an AC bridge. Typical commercial search coils are offered with uncertainties (l~r) ranging between a few 10 -3 to a few 10 -2, while extended uncertainties (---2o') of the order of lower than 10 -3 are declared by metrological laboratories [5.2]. Several factors may lead to some change with time of the area-turns value and periodic re-calibration is recommended. Depending on frequency, stated accuracy and space resolution, measurement by search coils can cover a ver[r wide range of field strengths, approximately extending from some 1 0 - ' T (---0.1 A / m ) [5.3] to 50T (---4 x 107 A / m ) [5.4], thanks to the intrinsic linearity of the induction law. The use of conventional multiturn coils beyond the 20-100 kHz range (as well as in the case of comparably fast field transients) is generally limited by the presence of stray capacitances in the winding and in the connecting cables. Few-turn single-layer windings, with short and low-capacitance connecting coaxial cables and wide-band signal acquisition devices permit one to attain the MHz range with reasonably good accuracy. Integrated setups based on inductive search coils are largely employed to monitor the environmental electromagnetic fields in the ELF (3 H z - 2 kHz) and VLF (2-40 kHz) bands. A popular application of search coils is related to the mapping of the field generated by electromagnets, magnets assemblies, and other sources. For instance, with large particle accelerating machines there is a stringent need for precise mapping of the field generated by the beam handling magnets, which is generally satisfied by means of ad hoc developed rotating coil and moving coil setups [5.5, 5.6]. When high space resolution is required, the size of the coil must be reduced in order to carry out the averaging of the field in as small a region as possible. This poses obvious limits to the measuring sensitivity. It is, however, possible to carry out a point-like determination of the field by using a conveniently large spherical coil, provided the turns are uniformly distributed along an axial direction (the fluxball [5.7]). If the turns are so distributed, we obviously achieve, as in the previous case of the cylindrical coil, the volume average of the axial field Hx. However, this quantity is equal to the value of Hx at the center of the sphere, a property of any harmonic function, as can be shown by solving Laplace's equation for the associated potential and taking its

164

CHAPTER 5 Measurement of Magnetic Fields

space derivative [5.8]. The maximum value of the enclosed flux is evidently obtained when the winding axis coincides with the field direction in the center of the sphere. It should be noticed that, as mentioned in Section 4.1.3, when the spherical coil is used as a primary, it produces a homogeneous axial field within the sphere volume. The construction of the fluxball is quite complex and we might content ourselves with approximating its properties. For example, by using a cylindrical coil and by keeping its height-to-diameter ratio to the value 0.866, one can measure the value of Hx at the coil center to third-order accuracy [5.8]. A sensing coil made to vibrate by a small motor or a piezoelectric transducer is sensitive to the space gradient of the magnetic field. A magnetized open sample generates a non-uniform field in its neighborhood, whose presence can be detected by means of a vibrating search coil. This provides a simple method to check the magnetized state of the material and, in particular, to determine its coercive field [5.9]. If the sample is put in a uniform magnetizing field provided by a solenoid and this is increased to the point that the demagnetized state is attained, the signal detected by the coil becomes zero (or at least passes through a minimum value) because the stray field generated by the sample disappears. The value of the coercive field is then obtained by knowledge of the solenoid constant and the supply current (Eq. (4.10)). With miniature coil and vibration frequencies around I kHz, high-resolution field mapping can be carried out. Let us consider a N-turn rectangular flat coil of length l and width w, immersed in a field with component Hz normal to the coil plane. The coil harmonically oscillates along its major side and its coordinate varies as x(t)--Xo sin(27rft). If the field gradient OHz/OX ~ O, the enclosed flux correspondingly oscillates. With small oscillations, OHz/Ox is constant over the swept distance x0 and it can be determined by measuring the voltage induced in the coil OHz V(t) = 2 7rf Nwlt.~o. --~x .XoCOS(2rrf t).

(5.9)

The field evolution over a certain swept area can thus be obtained by integrating V(t), but for a constant additional term, which can be measured with one of the methods previously described. We can directly obtain the total field profile over limited areas by using Mende's method, where only the tip of a harmonically oscillating coil is placed over the region of interest [5.10]. Figure 5.2 illustrates an application of this method to the determination of the stray field generated by a small permalloy plate. The tip of a long rectangular coil of width w is placed over the sheet edge while the opposite end is located in a zero or constant field region. During the back-and-forth harmonic oscillation of peak amplitude

5.1 FLUXMETRIC METHODS &

165

0.04 mm 0.02

Bz (T)

H

-266

-~b6

" " " 0 ....

~66

" " 260

X (p,m)

FIGURE 5.2 (a) Measurement of the stray flux density Bz leaving a magnetized permalloy sheet of dimensions 35 mm x 8 mm x 0.04 mm normally to the edge. The sensing coil is a single turn rectangular loop 0.13 mm wide and 10 mm long, which is made to harmonically oscillate back and forth along its long side, reaching a peak displacement x0 = + 12 ~m. (b) Correspondingly obtained crossprofile of Bz (from Ref. [5.10]). ---x• the enclosed flux varies as

O(t) = ~o + xowBz(x, y)sin(2~rft),

(5.10)

where (I)0 is constant and Bz(x, y) =/~0Hz(x, y) is the normal component of the flux density. It is assumed here that the swept +x0 region is sufficiently small for considering B,(x, y) constant over it. Bz(x, y) is then provided directly by the detected voltage

V(t) -- - d O / d t = -2~rx0wfB~(x, y)cos(2zrft).

(5.11)

The stray field profile reported in Fig. 5.2 is obtained by actuating the coil vibration (+ 12 t~m) by two piezoelectric plates at a frequency of 1.9 kHz. It should be noticed that field mapping of soft and extra-soft magnet surfaces by this intrinsically passive method ensures no interference with the field distribution. There are circumstances where we need to know experimentally the spatial behavior of the magnetic potential in magnetic circuits and assemblies or simply determine the average value of the field strength over a substantial length. This occurs, for example, when poor knowledge of the magnetic path length in soft lamination testing setups makes

166

CHAPTER 5 Measurement of Magnetic Fields

the calculation of the field based on a current m e a s u r e m e n t imprecise or w h e n it is desired to know to what extent an impressed magnetomotive force distributes among various portions of a magnetic circuit. A simple w a y to meet this d e m a n d is by use of a Chattock coil (magnetic potentiometer) [5.11]. It is based on the property that any closed path integral of the magnetic field not containing current sources is zero. In its c o m m o n form, a magnetic potentiometer consists in a long multiturn coil uniformly w o u n d on a non-magnetic insulating former and bent in such a w a y that its endfaces are placed in close contact with a plane. Figure 5.3a illustrates the representative case of a Chattock coil placed on a magnetic lamination, by which one can make a determination of the effective field inside the sheet. Let us consider the two integration paths (dashed lines) connecting points 1 and 2 on the lamination surface and let us calculate the flux enclosed by the bent coil. We assume that the field at the surface

z

, L'

'~

(a)

T z

Ha Xk

....

L '' 'H''r-"

(b) FIGURE 5.3 A Chattock coil is linked with a flux proportional to the difference of the magnetic potential between its ends. When it is placed over a magnetized sheet and its end surfaces are at a distance L, it provides the quantity V(L) - ~L Hs.dx, with Hs the effective field at the sheet surface. This quantity can equally be measured by means of a uniformly wound fiat coil.

5.1 FLUXMETRIC METHODS

167

I-~ is tangentially directed and homogeneous. Due to the absence of currents, the line integral of the field between points 1 and 2 is the same along any path, in particular the paths running immediately beneath the surface and immediately above it (L), and the path running along the coil axis (L') L Hi'dx = ~L Hs'dx - JL' H.dl,

(5.12)

that is H i L - Yr' H.dl.

(5.13)

Let us now assume that the cross-sectional area A of the coil is so small that the field value is constant over any given cross-section along the path (although its direction and modulus vary along L~). From Eq. (5.7), we obtain that the flux enclosed by the whole Chattock coil, of uniform turn density n, is proportional to the line integral of H, that is the magnetic potential difference between points 1 and 2 on the lamination surface f

ap - tzonA |

d LI

H.dl.

(5.14)

By virtue of Eq. (5.13), we then have that the effective field in the material (or, in any case, its tangential component) is related to the flux r measured with the Chattock coil Hi - ~ / p , onAL.

(5.15)

Thus, any uniformly wound coil of defined product nA, making whatever path between two fixed points, always links with the same flux q~. It is immediately obvious that the same principle applies, under the condition that the field Hi is homogeneous, if the coil-cross-sectional area has a finite value because we can imagine obtaining this coil by bundling together a number of the small-area coils satisfying Eq. (5.15) [5.12]. The integration path L is now equal to the distance between the centers of the contact areas. As an alternative to the Chattock coil, a flat H-winding can be placed between the measuring points 1 and 2, as shown in Fig. 5.3b. Sometimes the latter solution, which is simpler, is preferred for the determination of Hi, but the actual finite thickness of the coil may introduce an error because the field value tends to change rapidly with the distance from the surface. This has practical relevance, for instance, in the case of single sheet testing of soft magnetic laminations [5.13]. It has been verified that, under the usual geometrical arrangement suggested by

168

CHAPTER 5 Measurement of Magnetic Fields

the measuring standards [5.14], the error in the measured surface field can be strongly reduced by superposing two coils (for instance, at distances of 1.5 and 5 m m from the surface) and by linearly extrapolating their results to zero distance [5.15] (for a detailed discussion of the single sheet testing method see Chapter 7). Magnetic potential differences can be measured under AC excitation by placing the sensing coil in the desired position and making conventional signal acquisition and processing. DC testing requires applied field switching or coil extraction. The latter is indispensable when permanent magnets or final magnetic circuits containing permanent magnet sources are tested. In this case, a practical approach is based on the use of the Steingroever's coil, a long multiturn rod-like probe with small cross-sectional area (typical lengths 100-800 m m and diameters 3 10 mm) [5.16]. One end of the probe is put in contact with the desired point of the circuit while the other end is kept in a zero-field region. By displacing the sensing end to another point of the circuit, a flux variation is detected, which is proportional, according to Eq. (5.14), to the field potential difference between the two points. Figure 5.4 illustrates the case where the potential drop Vab - - f b H.dl in a soft branch of a permanent magnet source is detected by this method, showing the extent to which the soft magnet behaves as a short circuit for the flux. By repeating this operation between the ends of the magnet, the field strength at the working point can be found. It should finally be pointed out that in most instances where the extraction method is applied, the final reference condition is one where,

b

FIGURE 5.4 Use of the field potentiometer to evaluate the potential drop Vab = jb H.dl in the soft branch of a permanent magnet circuit. The line integral of the field over the circuit abc is zero and, consequently, jb H.dl = y~H . d l - ~ H.dl. This quantity is proportional to the flux variation detected by moving the straight multiturn potential coil between the two symmetric positions shown in the figure (from Ref. [5.16]).

5.1 FLUXMETRIC METHODS

169

instead of having zero field, the earth magnetic field exists. It is therefore important in these cases always to keep the orientation of the probe transverse to the direction of the earth magnetic field. 5.1.2 Signal treatment and calibration of fluxmeters The search coils provide a signal proportional to the flux derivative and we generally need to integrate it in order to recover the flux density value (or, more exactly, the variation of the flux density determined by an external action or the displacement of the coil). This operation is carried out either by means of analog electronic integrators or by digital methods. The venerable electromechanical integration method, centered on the measurement of the d a m p e d motion of a moving coil connected in series with the search coil, is no longer in use today and has only historical value [5.17]. The classical electronic fluxmeter is a compact instrument of general use in the laboratory, sometimes available as a handheld device, which is based on the principle of integration by means of an operational amplifier, used in the inverting-integrating mode. The base circuit is shown in Fig. 5.5. The ideal operational amplifier is a DC-coupled amplifier with the following chief properties: (1) The intrinsic gain

I Reset

d~ dt

i ilu

'~

_ -

Display

Uo

-

+

IIIIIIII

J,

-

FIGURE 5.5 Electronic integration of the signal detected by a search coil by means of an inverting operational amplifier. If the electromotive force is e(t) = - d ~ / d t , the output voltage is uo(t) -- ~ f~ e(t)dt and can be displayed by means of an analog or digital readout.

170

CHAPTER 5 Measurement of Magnetic Fields

A (open-loop gain) is infinite; (2) The input resistance is infinite and the output resistance is zero; (3) The bandwidth is infinite. Although impossible to achieve, these characteristics are reasonably approached with present-day operational amplifiers [5.18]. Let us consider now the input-output relationship in the circuit of Fig. 5.5 in the ideal approximation by noting first that, since the input resistance is infinite, the current is provided by the search coil flows entirely through the feedback condenser C. The error signal at the input is then v~ = 0 and the inverting input terminal behaves as a virtual ground. We can therefore write for the input current is = vi/Ri. On the other hand, if we write the equation for the loop from the output to ground, passing through the condenser C and the input terminals, we find that the output voltage v0 is v0 - -Vc + v~ = -Vc where Vc is the voltage on the condenser. Since vc(t) = ~

0 is dt,

(5.16)

we eventually obtain vo(t) = ~iC

vi dt.

(5.17)

A troublesome aspect of real operational amplifiers is the inevitable presence of an offset signal, originating from small bias currents flowing in the input terminals and thermal e.m.f.s generated in the search coil circuit. The integrating device will respond with a drift, which can be compensated as shown in Fig. 5.5, where a suitable DC voltage is regulated via a potentiometer and added to the input signal via the summing resistor Rb. Commercial electronic fluxmeters can operate from DC to about 100 kHz, with full scale ranges typically included between 10 -5 and 10 -1 Wb, the different scales being obtained by different selection of the Ri and C values. With Ri values of the order of 104 ~ (or higher), the error introduced by the finite value of the coil resistance is by and large negligible. Appropriate correction factors are introduced when the search coils have unusually high resistance values. Integration by digital methods is increasingly employed nowadays, both in commercially available devices [5.4] and in laboratory-developed setups. Figure 5.6 offers a possible configuration of a digital fluxmeter where the signal, provided to a Sample & Hold device by a low-noise precision amplifier, is digitized and transferred to a computing unit which, besides presiding over the whole measurement process, performs the numerical integration and all the required calculations. Normally, the Sample & Hold and analog-to-digital converter circuits are integrated in

5.1 FLUXMETRIC METHODS

171

FIGURE 5.6 Setup for signal acquisition and conditioning and field calculation. The measurement is coordinated and controlled by means of a PC, which drives the acquisition card, with its sample and hold and ADC conversion units. These two units are often integrated in a digital voltmeter or digital oscilloscope. The signal integration and all other mathematical operations are conveniently performed via software. the same setup, which can be, for example, an acquisition card, a digital voltmeter or a digital oscilloscope. For most practical purposes, a 12-bit AD converter, with sampling rates around 10s-10 6 samples -1, is appropriate. For sampling rates higher than 10 7 sample s -1, the standard 8-bit conversion is generally offered. Very small AC signals may require, besides very low-noise amplifiers, some treatment for noise reduction. Band-pass filtering is an obvious signal cleaning measure, but with too narrow a band some information can be lost because the signal is not necessarily sinusoidal and it may contain some kind of modulation. For a stationary signal, synchronous averaging, a feature commonly available with digital oscilloscopes, is effective given the random nature of the noise signal. Signal triggering is performed via a reference signal having the same frequency of the signal to be measured. With N successive samples, the root mean square (r.m.s.) value o- of the noise signal is reduced to ~rN = ~r/V~ and the signal-tonoise ratio is correspondingly increased. Synchronous detection (or phase-sensitive detection) is a method, realized by a device called "lock-in amplifier", by which the signal waveform is extracted from the noise using a reference signal with which it has a definite phase and frequency relationship [5.19]. The reference signal can be, for example, a square wave generated by the lock-in amplifier itself, which is multiplied by the input signal, thereby implementing the synchronous detection (Fig. 5.7). If the signal to be extracted is a sinusoid, its product with the square wave produces a full-wave rectified sinusoid, which, once passed through a low-pass filter, provides a DC voltage proportional to the r.m.s. value of the in-phase component of the input signal. The low-pass filter then rejects all the noise frequencies higher than its cut-off frequency.

172

CHAPTER 5 Measurement of Magnetic Fields

/ s multiplier input~(~

low-pass

Vout

F,-

referencelI I FIGURE 5.7 Block diagram of a lock-in amplifier. Multiplication of the input signal by a reference signal generates a full-wave rectified waveform. A successive low-pass filter provides a DC voltage proportional to the r.m.s value of the inphase component of the input signal and rejects the noise frequencies higher than its cut-off frequency.

This frequency determines the settling time, which must be sufficiently low to allow for any imposed modulation of the input signal. Integrating fluxmeters are calibrated using or emulating precisely known flux variations. Among the possible methods for achieving such variations, the following are in general use: (1) Search coil of known areaturns and calibrated magnetic flux density source; (2) calibrated mutual inductor; (3) volt-second generator. With method (1) the flux variation is produced either by extraction of the pickup coil from (or introduction into) the interior of a source or by producing a variation of the supply current in a solenoid or Helmholtz pair. The extraction (introduction) method is preferably adopted when using a reference magnet as the field source. Shielded and stabilized Alnico magnets are usually employed to this end, over a range of flux densities going from some 10 -2 T to about 1 T. They are offered with stated relative uncertainties around 0.5-1.0%. The flux variation measured by the fluxmeter upon the extraction (introduction) of the search coil is (I)12 = GNAI.~oH12, where H12 is the difference of the field values inside and outside the source and G is the gain of the fluxmeter. The outside contribution is normally due to the earth magnetic field. Let us therefore make an estimate of the calibration uncertainty over the abovementioned range of flux densities. We assume that temperature and humidity are in the recommended ranges, the electromagnetic interferences are minimized, and the effects of drift are controlled and corrected. We also assume that the input resistance of the fluxmeter is known and that the possible correction for the finite value of the search coil resistance is made. With reference to the methods discussed in Chapter 10, which are based on the ISO guide [5.20],

5.1 FLUXMETRIC METHODS

173

we express the relative measuring uncertainty as

Uc((~)12) (~)12

--

I U2((~)12) u2(NA) u2(H12) (~)~2 + (NA)2 + H22 }

U2B(G) ,

~

G2

(5.18)

where with (I)12 we have denoted the best estimate of (I)12, as obtained by repeated experiments, and the suffixes indicate uncertainties obtained by Type A or Type B method. In a typical case, the Type A uncertainty is evaluated, for an adequate number of repetitions under normal conditions, as (UA((~)12)/(~)12) = 5 X 10 -4. Normally, commercial search coils come with a declaration of accuracy of the area-turns product ranging between 10 -3 and 2 x 10 -2. We can reasonably adapt these accuracy figures to the requirements of Eq. (5.18) by taking them as 2odeviations. We consequently estimate the range (UB(NA)/NA)= 5 x 10 -4-10 -2. We can similarly evaluate the uncertainty range associated with the reference field strengths provided by the permanent magnet sources. Given that the minimum field involved in this example is more than two orders of magnitude higher than the earth magnetic field, we might disregard the uncertainty on the correction for the field outside the magnet. The extraction method may not be very practical, however, when using solenoids and Helmholtz pairs as reference field sources because these generate far lower fields than permanent magnets. In this case, a suitable approach consists in imposing a rapid change of the supply current while keeping the search coil fixed at the center of the source. To minimize the effect of the earth magnetic field, the axis of the windings is placed along the East-West direction and two readings are made for the two current polarities. For the present case, we assume a reasonable range of uncertainty values (UB(H12)/H12) -- 2.5 x 1 0 - 3 - 5 • 10 -3. We then regroup the uncertainty on the previously mentioned corrections of the fluxmeter parameters in a contribution (UB(G)/G)= 1 x 10 -3. By introducing the Type A and Type B uncertainty terms in Eq. (5.18), we eventually obtain that a fluxmeter calibration performed by using permanent magnet field sources and search coils is likely characterized by a combined uncertainty (Uc((~)12)/(~)12) = 2.7 X 10-3-1.1 X 10 -2. There are cases where this uncertainty appears too high and it is desirable to exploit more precisely defined magnetic flux variations. Mutual inductors are largely employed as standard flux sources. They are rugged and very stable devices. If they are carefully realized (for instance, using marble stone or fused silica as a former for the windings) and stored, they can be used in the laboratory for decades. The mutual induction coefficient is defined as M = c~2/il, where (I)2 is the flux linked with the secondary winding and iI is the primary current. The typical

174

CHAPTER 5 Measurement of Magnetic Fields

relative uncertainty associated with the declared value of M in commercially available or laboratory-built standard mutual inductors ranges between 10 -3 and 10 -4. The temperature dependence of M is also and can be disregarded under normal very low (AM/M ~ 10-5/~ conditions. The resistance of the secondary winding for typical M values in the mH range can be of the order of a few f~, which is negligible with respect to the input resistance of the fluxmeter. In a typical experiment, schematically represented in Fig. 5.8, a stabilized and precisely known DC current il flowing in the primary winding is inverted by means of a switch, generating a variation ~ = 2Mil of the flux linked with the secondary winding. This is connected with the input of the integrating fluxmeter, whose reading is recorded, together with the reading of i 1. The latter quantity is obtained as the ratio il = V~/R1, where R1 is a standard resistor connected in series with the primary winding and V1 is the related voltage drop. The experiment is repeated a convenient number of times, under controlled ambient conditions, and both the reference best estimate of the flux variation (~)2 -~- 2MV1/R1 and the best estimate ~ provided by the fluxmeter readouts are calculated. The combined uncertainty of the reference flux variation is

2 Uc((~)2) __ [ U--((~2) ~ u2(M) (~)2 ~ (~)22 M2

} Ul~(Vl ) ~ u~(R1) . V~ R2

(5.19)

The Type A uncertainty, associated with the measurement repeatability, is evaluated as (UA((~)2)/(~)2) -- 2.5 x 1 0 -5. The calibration certificates of the mutual inductor and of the standard resistor provide the 2~r

i1

T_

Fluxmeter

Digital voltmeter

FIGURE 5.8 Calibration of a fluxmeter by a DC source, standard shunt resistor R1, and standard mutual inductor.

5.2 HALL EFFECT AND MAGNETORESISTANCE METHODS

175

uncertainties 5 x 10 -4 and 5 x 10 -5, respectively. It is consequently (uB(M)/M) = 2.5 x 10 -4 and (UB(R1)/R1) -~ 2.5 x 10 -5. From the voltmeter specifications provided by the manufacturer, we obtain (see the example reported in Appendix C) (UB(V1)/V1)= 1.5 X 10 -5. W e see here that the mutual inductor provides the major contribution to the calibrating flux uncertainty. It turns then out from Eq. (5.19) that (Uc((~)2)/(~:)2): 2.53 x 10 -4. The fluxmeter estimate comes with its own Type A and Type B uncertainties (UA(Cb~)/cb~)and (UB(G)/G), respectively. It is then concluded that the reading of the fluxmeter should be corrected by the difference (~2 -- (~)2 and that the associated relative measuring uncertainty is

Uc((:~2) -- I U~(([D~) ug(G) u2((:~2) (~)~ (~)~2 q- G 2 J (~:)2 "

(5.20)

The calibrated flux variation can be conveniently emulated by means of a rectangular voltage pulse, with accurately known amplitude and time duration. Volt-second generators are commercially available, with declared accuracy of the pulse area around 0.1-0.5%. A major source of uncertainty in the calibration by narrow rectangular pulses is, besides residual offset, the rounding-off of the pulse due to bandwidth limitations. It has been shown that such limitations can be largely overcome by using digitally synthesized trapezoidal pulses, with slew rates of the order 2 x 103 V/s [5.21].

5.2 H A L L E F F E C T A N D M A G N E T O R E S I S T A N C E METHODS

5.2.1 Physical mechanism of Hall effect and magnetoresistance A magnetic field can be revealed by the force it exerts on moving charges. This can be directly appreciated, for example, by observing the bizarre effects induced by a magnet on the screen of a cathode ray tube or through the attractive or repulsive actions wires and coils with circulating currents exert between themselves. A flat rigid N-turn coil of area A with a circulating current i immersed in uniform field B =/z0H (see Fig. 5.1) is subjected to a torque i~i = tzoiNAiH x nO,if n is the unit vector normal to the coil surface. A restoring torque imposed by a spring of known constant k will balance T for a given rotation angle c~= k/~-. This angle will then provide a measure of the field strength. By measuring the force

176

CHAPTER 5 Measurement of Magnetic Fields

on a coil we can, in general, make an absolute determination of the field, traceable to the base SI unit standards [5.22]. However, we can exploit the Lorentz force in subtler ways by considering the effect of magnetic fields on the transport properties of metals and semiconductors. The deflecting action of the magnetic field on the steady course followed by the carriers is the source of two basic effects on the electrical conductivity: generation of Hall voltage and apparent increase of resistance (magnetoresistance). It is expedient, in discussing these effects, to consider the practical case of a conducting rectangular plate, where, as illustrated in Fig. 5.9, the circulating current ix flows along the major side and the magnetic field Bz is directed normal to the plate surface. In a metal, the current ix is generated by the drift of the free electrons under the action of the applied electrical field Ex. The force applied by Ex on each electron of wavevector k is Fx = - e E x = h d k / d t (with e the elementary charge) and is balanced, under stationary conditions, by the frictional force provided by the scattering centers. At steady state, the Fermi sphere in the k-space is shifted in the kx direction, reflecting the existence of a drift velocity Vx, and all the wave vectors are augmented by the quantity 8kx = G'r/h,, where ~" is the time of flight between scattering events. By denoting with m e the electron mass, we can write the m o m e n t u m increment as ttSkx - meV x and

Tz y

G

o

FIGURE 5.9 Principle of Hall voltage generation in a conducting plate. The free electrons and the holes, drifting at the velocity Vx, are subjected to the Lorentz force by the magnetic field Bz, thereby tending to move on one side of the plate. The cumulating charge gives rise to the transverse field EH, eventually balancing the Lorentz force at steady state. In this example the electrons are the majority carriers. The deflection of the charges from the direction of the current ix is additionally responsible for an increase of the apparent resistance Rx = Vx/ix (magnetoresistance effect).

5.2 HALL EFFECT AND MAGNETORESISTANCE METHODS

177

we obtain Vx m

Fx'r me

eEx'r me

(5.21)

With n e electrons per unit volume, the current density is jx ~

-- Yle e V x

(5.22)

and the electrical conductivity ~. - j x / E x is eventually obtained as n e e2 'r o-x - -

_

n e e P , e~

(5.23)

me

with the mobility [Vx[

eT -

[d,e m

Ex

me"

(5.24)

The application of the magnetic field Bz adds the Lorentz term F L -- e v x Bz to the coulombian force. It is orthogonally directed with respect to both the drift velocity and the field and it tends to deflect laterally the free electrons (see Fig. 5.9). The equations of motion for the x and y axis become meVx/'r-- -eEx - evyBz, meVy/'r-- -eEy - evxBz.

(5.25)

The deflection of the electrons takes place immediately after application of the magnetic field, for a time comparable with T (around 10 - 1 4 10 -12 S), and soon stops because of charge buildup at the edges of the plate. No current can flow along the y direction and the related counterfield EH, the Hall field, is obtained as the value taken by Ey w h e n vy = 0 in Eq. (5.25) e~EH = -- ~ E x B z . me

(5.26)

It is measured by detecting the Hall voltage VH = EHW over two points symmetrically placed across the plate width w. It is usually assumed that, by virtue of the balance between Lorentz force and Hall field, the carrier trajectories are straightened out and there is no magnetoresistance (i.e. rise of the sample resistance along the x direction after application of Bz due to lengthening of the electron paths [5.23]). A measure of the Hall effect in a given material is provided by the ratio EH a H -- jxBz

~

(5.27)

178

CHAPTER 5 Measurement of Magnetic Fields

which is called Hall coefficient. For a plate of thickness d, the Hall voltage is

RHixBz d

VH =

(5.28)

and by combination of the previous equations we get RH __

1

(5.29)

= --~e

nee

or

(with or the conductivity scalar defined in Eq. (5.23)). R H is thus a negative quantity w h e n the involved charge carriers are electrons. The inverse proportionality of RH with ne occurs because, with smaller carrier densities, higher drift velocities are required for a same current, which implies larger deflection by the magnetic field. Examples of experimental values of the Hall coefficient in metals and semiconductors are given in Table 5.1. The Hall resistivity is often introduced, defined as EH

OH-- jx

-

RHBz.

(5.30)

The value of the Hall coefficient predicted by Eq. (5.29) is in satisfactory agreement with the experiments in monovalent metals, but a correction TABLE 5.1 Experimental Hall coefficients in metals, the

semimetals Bi and Sb and n-type semiconductors at room temperature. The reported values of RH and the temperature coefficient (1/RH)(dRH/dT) in semiconductors are indicative, for any material, of a range of values. They vary in fact with concentration and mobility of the carriers and are therefore sensitive to impurity levels and preparation methods 1 dRH dT (K-l)

RH ( m 3 / C )

RH Au

- 7.2 x 1 0 - n

_

Na

- 2 . 3 x I0 - w

-

Ag

-9

Be

2.4 x I 0 - I ~

-

Sb

- 2 x 10 -9

Bi

- 6 x 10 -7

5 x 10 -3

Si InAs GaAs

- 1 x 10 -5 - 1 x 10 -4 - 2 x 10-4

- 1 x 10 -3 1 X 10 -4 3 X 10 -4

InSb

x I0 - l l

- 7 . 5 x 10 -4

-

8 x 10 - 4

5.2 HALL EFFECT AND MAGNETORESISTANCE METHODS

179

factor is generally required and we have

R H - - - r--c~ nee

(5.31)

The factor r0 mainly depends on the statistical character of the electron scattering process and the band structure and can even be negative in some cases. It is a function of the applied field and it approaches unity in the strong field regime [5.24]. In practical Hall plates, Eq. (5.31) is multiplied by a further geometrical correction factor g < 1, taking into account the finite value of the ratio between length and width [5.25]. Two types of carriers, electrons and holes, are involved in semiconductors and the conductivity receives contributions from different bands. In particular, if we denote the density and the mobility of the holes with n h and/d, h and the drift velocities of electrons and holes with Vx- and Vx+ (Vx- < 0, Vx+ > 0, according to Fig. 5.9), respectively, we obtain that the current density is jx -- jxe + jxh = - - n e e V x - + nheVx+

(5.32)

and the conductivity Crx

Ex

ere + crh -- neela'e + nhela'h"

(5.33)

Notice that, in defining the conductivity of semiconductors, the effective mass for electrons and holes must be considered. The application of Bz makes the holes and the electrons, drifting in opposite directions, deflect towards the same side of the plate. The Lorentz forces, however, are not the same, because hole and electron mobilities, and hence velocities, are different. The equations of motion (5.25) must now be written for both types of carriers, imposing the steady-state condition of net zero transverse current jy "~ jye + jyh -" - n e e v y _

+ nhevy+ = O.

(5.34)

In compact notation, the process can be described by expressing the electric field as [5.26] E -- Ex~ q- EH~ = ---lje O'e + ~eeejeX Bz-- ----1 Orh j h -

~ j h X Bz

(5.35)

with ~ and ~ the unit vectors in the reference system of Fig. 5.9. The hole mobility ~tl,h is defined as in Eq. (5.24). Equation (5.35) can be solved for je and Jh in terms of E and Bz and the total current density j = je + jh, directed along the x-axis, is obtained. The relationships between fields

180

CHAPTER 5 Measurement of Magnetic Fields

and currents with the two-carrier conduction mechanism are summarized in Fig. 5.10. While the zero charge transfer constraint for a metal implies zero transverse drift velocity of the free electrons, in semiconductors it allows for transverse motion of the carriers because the Hall field cannot compensate for both hole and electron lateral drift at the same time. By introducing the condition (5.34) in the equations of motion for the holes and the electrons and by assuming the weak field approximation (which is largely satisfied with the fields ordinarily applied to materials), we obtain the following expression for the Hall coefficient (see for instance Ref. [5.26]) RH -- 1 nh/~ -- he/J,2 e (nh/~h -ff ne~e) 2"

(5.36)

It is apparent that in semiconductors both mobility and concentration of the charge carriers play a role in determining the strength of the Hall coefficient. Minority carriers can actually exert a remarkable effect when they have large mobilities. In all cases, R H is orders of magnitude larger in semiconductors than in metals, which is explainable on account of similarly large differences in the carrier concentration. It is clear that R H

'<

;"

I

jhN,~ ............

Jh/ (Yh ~ ~ / ~ _ l l h

t -

.----'-

Bz

/(~h).jh XBz

FIGURE 5.10 Currents and electrical field with the two-carrier conduction process in a magnetic field (Eqs. (5.32)-(5.35) and Ref. [5.26]). The vector composition of electron and hole currents, allowing for lateral drift of the carriers and, consequently, path lengthening, is conducive to an increase of the apparent resistivity p = Ex/jx (magnetoresistance effect). The electrical field E is the vector combination of the applied field Ex and the Hall field EH.

5.2 HALL EFFECT AND MAGNETORESISTANCE METHODS

181

must depend on temperature in semiconductors because this reflects the corresponding evolution of the carrier concentrations and mobilities. In n-doped semiconductors, R H is negative and the temperature coefficient dRH/dt is positive. In heavily doped materials, the majority carriers are nearly all in the conduction band at room temperature and dRH/dt is correspondingly low. Values of dRH/dt in materials typically employed in Hall sensing devices are given in Table 5.1. In order to make a practical estimate of the Hall voltage, let us consider an InAs plate of thickness 100 ~m immersed in a field Bz = 0.01 T. We obtain from Eq. (5.28) VH = 1 mV with a supply a current ix = 0.1 A. The fact that a conduction process involving two (or more) carrier types allows for transverse drift in the presence of a magnetic field implies that the net current, the vector sum of the components associated to the different carriers, is correspondingly reduced. The apparent resistance of the material is thereby increased because the carriers follow curved paths. If we consider the solution for the current density j provided by Eq. (5.35), we can calculate the resistivity under the magnetic field as p = (E.j)/j 2

(5.37)

and compare it with the zero magnetic field resistivity P0 - 1/(ere + to find [5.26] Ap P

P - P0 P0

1 e (or e

OreO'h(//, e -}-/J,h)2B 2

+ Oh)2 q--(/d, eO"e

--

/ZhO'h)2B 2"

O'h)~

(5.38)

We predict by this equation that the magnetoresistance is positive, it has a quadratic dependence on the field at low field values and tends to saturate at high fields. Note that the Hall resistivity PH (Eq. (5.30)) has instead a linear dependence on Bz. We call it transverse magnetoresistance to distinguish it from the weaker longitudinal magnetoresistance, which occurs when the magnetic field is applied in the direction of the current. If we assume, as a first approximation, that both types of carriers have the same value of T, we obtain from Eqs. (5.24) and (5.38) that &p/p goes like (Bz~')2. This is equivalent to say that, to the lowest order, Ap/p oc (Bz/p) 2, a property known as Kohler's rule. The experiments also show that metals can display a very small magnetoresistance, which is justified in terms of mobility fluctuations of the free electrons. A large magnetoresistance effect can be found in the semimetal Bi, depending on its degree of purity. Very pure monocrystalline Bi thin films can exhibit a giant, though strongly temperature-dependent, increase of resistivity under a field greater than about 0.5 T [5.27].

182

CHAPTER 5 Measurement of Magnetic Fields

Magnetic materials customarily exhibit important galvanomagnetic effects because the field interacting with the charge carriers is B /~01-I+/~0M, which is strongly reinforced with respect to the external field. This is conventionally taken into account by splitting both the Hall resistivity and the magnetoresistance of ferromagnets in two terms, the "ordinary" and the "spontaneous" contributions, with the former, independent of the magnetic order, related to tz0I-I only and the latter related to/~0M. The spontaneous magnetoresistance is attributed to the scattering of the conduction electrons by the magnetic moments through spin-spin and spin-orbit interactions. With perfect magnetic order, the spin-spin interaction is zero because the exchange coupling of the conduction s-electrons with the ordered d-electron spin system forms a periodic potential. This is what occurs in the limit of zero temperature. By increasing the temperature, the potential periodicity is disrupted and a spin disorder contribution to the resistivity arises, which saturates at the Curie temperature. The spin-orbit interaction is thought to be the source of the anisotropy of the spontaneous magnetoresistance (AMR), that is, the dependence of the material resistivity on the angle made by the current with the magnetization. Thus, if we disregard the ordinary magnetoresistance, the evolution of the resistivity with temperature in a ferromagnet can be interpreted as due to the composition of the impurity, phonon, and magnetic scattering contributions, as schematically depicted in Fig. 5.11. We notice here that the magnetic contribution is lower when the sensing current is perpendicular to the direction of the magnetization. This shows that the mechanism of ordinary magnetoresistance, where the Lorentz force deflects the carrier path from the direction of the applied voltage, does not apply to AMR because in this case one would instead expect Pll K P• A classical example of anisotropy of the spontaneous magnetoresistance, measured in polycrystalline Ni as a function of the applied field, is provided in Fig. 5.12a [5.28]. The evolution from the multidomain near-isotropic condition in the demagnetized state towards the monodomain state beyond technical saturation is associated with increasing splitting of the values of the resistivities Pll and p• These quantities both decrease at very high fields because forced spin ordering sets in. The domain structure in the demagnetized state can be far from isotropic. The magnetoresistance and its dependence on the field strength are correspondingly affected. This is nicely demonstrated by the example reported in Fig. 5.12b (taken from Ref. [5.28]), where the distribution of the occupied easy axes in an Fe-Ni polycrystalline alloy is modified by an applied tensile stress o" in a way depending on the sign of the magnetostriction. The resistance, measured in the direction of the stress, is correspondingly increased

5.2 HALL EFFECT AND MAGNETORESISTANCE METHODS

183

rr

............

i L............ Pi

Temperature

FIGURE 5.11 The chief contributions to the resistivity of a magnetic single crystal and their dependence on temperature are schematically represented. Pi is the residual contribution due to impurities, Pc is the lattice contribution, associated with phonon scattering p• and Pll are the resistivities, measured when the current is perpendicular or parallel to the magnetization, respectively. The ordinary magnetoresistive effect is disregarded in this scheme. Tc is the Curie temperature (adapted from Ref. [5.29]).

or decreased according to whether the magnetostriction is positive or negative. A general phenomenological description of the role of the magnetization on the electrical conduction process in a ferromagnet can be provided by the vector generalization of O h m ' s law E -- Ilpllj, with Ilpll the resistivity tensor. To simplify the matter, we assume that we are dealing with an isotropic material, with the magnetization M directed along the z-axis and the current density j making an angle 0 with M. We consider only the spontaneous magnetoresistance and Hall effects and we write, from s y m m e t r y considerations (see for example Ref. [5.29]), O h m ' s law as

/ x/t ~ / Ey Ez

=

PH

P•

0

jy

0

0

Pll

kjz

.

(5.39)

From this equation, we can derive the dependence of the resistivity on the angle 0. In fact, introducing Eq. (5.39) in Eq. (5.37) we obtain

p(O)- E.Jj2 - - p•

+ j'~) } PllJ j2 2 ,

(5.40)

184

CHAPTER 5 Measurement of Magnetic Fields

1.0

ap///p

Zs>0

"

~ 0.5

~. 0.0, -0"5"1 ~

Polycrystalline Ni

_,.o.

I

-1.5. (a)

0.0

_ 0.5

1.0

,% H(T)

1.5

2.0

89 Permalloy ~s < 0

""-~"~ 0

.

.

.

.

(b)

20 |

9

.

,

"4~0 . . . .

I

60

o"(aPa)

FIGURE 5.12 (a) Resistivity change in Ni along directions parallel and transverse to the applied field. The material is brought from the demagnetized state deep into the forced magnetization regime. Once the technical saturation is reached, the resistivity decreases, irrespective of the field direction, because spin disorder is reduced by forced alignment. (b) A tensile stress makes the direction of the domain magnetization prevalently distributed along the stress axis or perpendicular to it, according to whether the magnetostriction is positive or negative. The resistance, measured in the stress direction, is accordingly increase or decreased (from Ref. [5.28]). that is, taking into account that (j2x + j~)/j2 = sin20 and j2z/j2 -- cos2(9~

p(O) = p L + (Pll -- PL )c082 O. Equation (5.41) can also be written as p(O) = Pll + 2p• + (Pll - P• 3

(

(5.41)

1)

COs2 0 -- ~

(5.42)

an equation bearing a good similarity with the classical isotropic magnetostriction equation. This is consistent with the result on the stress dependence of magnetoresistance reported in Fig. 5.12b. A significant aspect of conductivity in 3d ferromagnets is the different role played by the spin T and spin I conduction electrons. Since at temperatures well below the Curie temperature the spin direction of these electrons is conserved upon scattering, one can assume that the conduction involves two channels, having spin T and spin I electrons, respectively. In a strong ferromagnet, the majority spin T 3d band is nearly full and the spin T conduction electrons find little available free

5.2 HALL EFFECTAND MAGNETORESISTANCE METHODS

185

levels in which they can be scattered. This does not occur to the spin 1 conduction electrons, which can be scattered in the partially occupied minority band levels. The spin T channel therefore acts as a virtual shortcircuit for the magnetic dependent part of the conduction process. It is possible, however, to modulate the roles of the two conductivity channels by using artificial structures like the thin-film multilayers. Here, the layer thickness is of the order of a few atomic spacings and is comparable to or lower than the mean free path of the conduction electrons. The interaction exchange between different layers can then be made to oscillate between ferromagnetic and antiferromagnetic by modulating the thickness of a non-magnetic spacer layer (see the examples sketched in Section 2.7). We find that under the condition of antiferromagnetic coupling, where the successive layers are magnetized along opposite directions, both the spin T and spin I conduction electrons are subjected to scattering by their minority 3d levels and there is no low resistance path. The current here flows in the plane of the film and the electrons can sample different layers by thermal drift (mean free path greater than the layer thickness). If we apply a sufficiently strong field the magnetization becomes parallel to it everywhere. The privileged role of the spin T conductivity channel is then recovered and the thin-film resistance is decreased. An illustration of this mechanism is given in Fig. 5.13, where the classical result of Baibich et al. [5.30] in Fe-Cr multilayers is reported, together with a scheme of the equivalent resistance arrangement of the process we have just described. The associated change of resistance is remarkable if compared with the conventional AMR, and for this reason it has been aptly defined as a giant magnetoresistance effect (GMR). The required fields are nevertheless quite important and in applications a figure of merit F = (AR/R)/Hsa t is customarily defined, where the relative resistance change at saturation is normalized to the correspondingly required field.

5.2.2 Measuring devices Hall magnetometers are perhaps the most popular kind of field measuring devices. They find widespread applications in such diverse areas as magnetic materials characterization, environmental measurements, magnetic field mapping in electrical and magnetic setups such as motors, generators, electromagnets and permanent magnets, magnetic recording heads and disks, MRI magnet assemblies, magnetic shields, and particle accelerators, to name only some of them. The wide acceptance of the Hall magnetometers, both in the research laboratories and in the industrial milieu, is justified by the combination of many merits and few drawbacks: good accuracy and linearity, wide measuring range,

CHAPTER 5 Measurement of Magnetic Fields

186

1.0-

R 1"

R,I.

R?

R* /

0.9.

I

R 1"

\

R

~: 0.8 0.7 (Fe 3 nm / Cr 0.9 nm)6o\

r

0.6 0.5

i

-2

. . . .

i

-1

. . . .

i

0

. . . .

#oH(T)

i

1

. . . .

i

2

FIGURE 5.13 Reduced resistance R(H)/R(O) as a function of the applied field in an Fe-Cr multilayer at T = 4 K (from Ref. [5.30]). The insets schematically show the equivalent resistance circuits corresponding to the two conditions achieved under zero field (antiparallel alignment of the magnetization of the Fe layers) and high fields (parallel alignment). With antiparallel alignment both spin I and spin T electrons are scattered (R = (R T+R 1)/2 with two magnetic layers). With parallel alignment the spin T conduction channel acts as a low resistance path and the total resistance of the device decreases (R = 2(R T .R ~)/(R T +R ~)).

small probe size, direct field reading and ease of use, availability of handheld devices, and sensitivity to DC and AC fields. On the other hand, the Hall voltage depends on temperature (Table 5.1), it deviates from linear behavior at high fields, the low-field sensitivity is somewhat limited and laborious zeroing of the reading is often required. The Hall sensing element is usually made of one of the semiconductors listed in Table 5.1 (with R H in increasing order): Bi (which is actually a semimetal), Si, InAs, GaAs, and InSb. It comes either as a plate, sliced from a crystal and conveniently machined and lapped, or as a thin film, obtained by vacuum evaporation or sputtering. In Fig. 5.14, we see schematic examples of bulk and thin-film sensors. Plates and films are affixed to an insulating and rigid substrate, like aluminum oxide, and four ohmic contacts (usually gold or silver) are made. They are usually mounted on a flat bar (transverse probe) or on top of a rod (axial probe) provided with suitable leads. In some cases, two or three plates are placed together along orthogonal planes, thereby allowing for vector field measurement. Mechanical strength and protection from environmental hazards can be achieved by encapsulating the probe in a fiberglass or aluminum stem but

5.2 HALL EFFECT AND MAGNETORESISTANCE METHODS

L. I"

i

.,

(a)

2 mm

-

~1"'

"q.

mm

J/

1/

187

.q

(b)

~

B

(c)

(d) FIGURE 5.14 Examples of Hall plates and probes. (a) Hall plate cut from a semiconductor crystal, connected with current pads at the shorter sides and Hall voltage pads at the center of the longer sides. The dashed circle defines the active area. The conducting pads are deposited on a ceramic substrate. (b) Thin film cloverleaf sensor with active area at the intersection of the lobes (diameter ---5 ~m). (c) Transverse Hall probe. (d) Axial Hall probe.

the latter is not recommended for AC use. GaAs and InAs sensors are the standard choice for general-purpose industrial magnetometers, where reproducibility, absolute accuracy, ease of calibration, and wide measuring range are required over a substantial range of temperatures. While the recommended operating temperatures are typically contained in the interval - 4 0 to 100 ~ cryogenic probes are commercially available for use d o w n to 4 K. InAs, with its very small temperature dependence of the Hall coefficient, is best suited for applications in the 4-300 K range. The example reported in Fig. 5.15, adapted from Ref. [5.31], illustrates the excellently linear and weakly temperature-dependent Hall response of an InAs thin film. GaAs can have a high Hall coefficient [5.32], but it also has high resistivity, which makes it relatively noisy and sensitive to electromagnetic interferences. It has an intrinsic advantage for hightemperature applications (up to about 250 ~ because of its wide bandgap (1.42 eV), which makes intrinsic conduction relevant at high temperatures only. The same can be said of Si, which, in spite of its low and appreciably temperature-dependent R H value, is sometimes favored because of its long-term stability and, above all, for the possibility it offers for easy integration in a well-known and sophisticated technology [5.33].

188

CHAPTER 5 Measurement of Magnetic Fields 600

250

200 T

E

v

~ ~oo 50

300

/ /

200

/ 0

4 #oH(T)

6

t:300<

/ /

100

I '-'~ 2

_

400

_.

~" 150 v

T

500

!nsb thin film i= 0.5 mA

y 0

.

.

.

.

|

2

9

, ,

,

,

|

4

.

.

.

.

!

6

9

,

/XoH(T)

FIGURE 5.15 Hall voltage VH vS. applied field at liquid helium and room temperature in 1 ~m thick InAs and InSb thin films. The much larger Hall sensitivity of the InSb samples is associated with stronger temperature dependence and non-linear field behavior of VH at low temperatures (from Ref. [5.31]).

InSb exhibits the largest Hall coefficient, but, as illustrated in Fig. 5.15, it also shows substantial temperature sensitivity. Because of their high sensitivity and the low control currents (typically of the order of I mA or lower), implying low fields generated by the probe itself, thin-film InSb Hall sensors with active area in the micrometer range are often employed as micro-sized detectors for precise mapping of fields and domains in soft and hard magnetic materials and flux profiles in superconductors [5.34]. Recent developments in high-resolution surface field measurements integrate the Hall sensing probe (a GaAs/A1GaAs heterostructure) in a scanning tunneling microscope for simultaneous detection of magnetic field and surface topography [5.35]. A Hall magnetometer (gaussmeter) can be represented, in essence, by the circuit shown in Fig. 5.16. A stabilized source provides the constant control current ic and the Hall voltage VH, detected on the transversally placed contacts, is fed into a precision digital voltmeter, possibly through a pre-amplification stage. The role of the load resistance R c is one of counteracting, when required, the possible non-linear behavior of VH, in balance with the variation of the internal resistance of the Hall elements as measured between the Hall contacts. Note that the Hall semiconducting plate electrically connects the current source circuit and the output circuit. Consequently, no other direct connections can be tolerated without

5.2 HALL EFFECT AND MAGNETORESISTANCE METHODS

189

Digital

RL

voltmeter

FIGURE 5.16 Schematic view of a Hall generator circuit with DC control current ic. The value of the load resistance RL is chosen in order to linearize, if required, the Hall voltage in a wide field range.

disrupting the potentials at the four contacts on the Hall plate. In particular, no common ground can exist between input and output circuits. Deviations from perfect material homogeneity, slight asymmetries of the plate and misalignment of the contacts, thermal electromotive forces and the signal conditioning circuitry itself can give rise to offset voltages. These are normally compensated for preliminary to any measurement, either by acting on a potentiometer of a zero voltage balance circuit or, more frequently in recent commercial products, by exploiting automatic self-zeroing of the setup itself. Because of the presence of the earth magnetic field, it is expedient to make zeroing with the Hall probe inserted in a null chamber. This is a double Mumetal shield, accurately annealed and demagnetized, by which the earth field can be attenuated by more than one order of magnitude. While this is sufficient for the appropriate use of most practical gaussmeters, a better zero-field approximation might be required with the increasingly available lowfield probes. Active field cancellation via a triaxial Helmholtz coil system (see Fig. 4.8) can lead to effective suppression of the environmental field d o w n to 10-1 nT [5.36]. Besides offset voltage and non-linearity, further sources of uncertainty in the field measurement by a Hall gaussmeter come from the temperature dependence of RH and the uncertainty in the Hall plate orientation. The thermal variation of the Hall voltage is kept around and below + 0.05%/K in high-quality commercial devices using InAs sensing plates. In any case, it can be taken into account by calibrating the setup at a defined temperature and making a correction for the actual measuring temperature. The Hall plate is placed, during the measurement, normal to the field direction, a position associated with the m a x i m u m instrumental

190

CHAPTER 5 Measurement of Magnetic Fields

reading. With an angular uncertainty of + 1~ on this position, we obtain around 0.01% uncertainty on the field reading. When the field has a strong planar component, the so-called planar Hall effect introduces a small contribution to VH. It is actually an effect deriving from the anisotropy of magnetoresistance and is proportional to the difference Pll - P• It can be disregarded under normal measuring conditions. For low field measurements, an AC stable control current ic is frequently used. In this case, VH has the same frequency and phase of the current and it can be conveniently treated by means of a lock-in amplifier. One advantage of the AC current method is that spurious thermoelectric and thermomagnetic effects are eliminated and the measuring sensitivity is consequently enhanced. The main sources of error now become the inductive pickup at the line frequency and the current frequency (100 H z - 1 kHz) and the contact misalignment voltages. AC fields can equally be measured by means of a Hall magnetometer. In principle, there should be no frequency limitations, up to the THz range, but inductive and capacitive effects of the Hall probe and the leads limit the actual working frequencies to around 10-50 kHz. Commercial Hall effect gaussmeters are nowadays available for measurements in a range of field strengths from a few ~T to some 2050 T. The stated uncertainties for DC field reading can be around 0.1% in the best laboratory models, decreasing around 1-5% in the handheld instruments. The AC field accuracy is generally worse. To remark that the appropriate sensitivity in the lowest field ranges is generally achieved using modified probes, where the flux multiplying effect of small concentrators (typically made of Mumetal), placed near the sensing semiconducting plate, is exploited. The space resolution of the field measurement is dramatically reduced with the use of concentrators because of their substantial size and the perturbation they impose to the field line pattern. Hall probes and magnetometers are quickly calibrated by means of reference permanent sources when moderate accuracy (e.g. + 1%) is required. For best high-field measurement accuracy, we must resort either to a stable superconducting source, an electromagnet, or to a permanent magnet and make a simultaneous measurement with an NMR magnetometer. At medium and low fields (below about 0.40 T), calibrated solenoids and Helmholtz coils are preferable. The ordinary magnetoresistance effect described by Eq. (5.38) is, as previously mentioned, especially relevant in high-purity Bi single crystal and polycrystalline thin films. This property derives from the combination of long mean free path and small effective mass of the charge carriers. Experiments in electrodeposited Bi thin films (thickness 10-20 ~m) show an approximately quadratic field dependence of the magnetoresistance

5.2 HALL EFFECT AND MAGNETORESISTANCE METHODS

191

up to around 1 T, where the resistivity is increased about 20% at room temperature [5.27]. This dependence becomes approximately linear at higher fields and an increase of the order of 200% is found with fields perpendicular to the film surface of the order of 5 T. Even larger magnetoresistance effects, with the same quadratic behavior at lowmedium field strengths, are found in InSb thin films, which are currently employed in commercially available field sensors [5.37]. Both Bi and InSb exhibit a large temperature coefficient of magnetoresistivity caused by the temperature dependence of the mobility of the charge carriers. The anisotropy of magnetoresistance displayed by permalloy is largely exploited in the development of magnetic field sensors. A very soft material, basically lacking magnetocrystalline anisotropy and magnetostriction and prone to convenient incorporation in the conventional Sibased integrated circuit technology, permalloy displays a 2-3% relative decrease of the resistivity p on passing from parallel to perpendicular orientation of the magnetization M with respect to the sense current i~. Equation (5.41) shows that such a decrease is predicted to follow a cos2O dependence on the angle 0 between M and ic. Permalloy thin films, deposited under a magnetic field of a few thousand A / m , develop a uniaxial anisotropy of the order of 50-100 J / m 3, thereby identifying an easy axis (EA) in the plane of the film. If, for example, the EA is made to coincide with the direction of i~, we can envisage a decrease of p, following Eq. (5.41), when an applied field makes M rotate away from ic. In the idealized case of a large-area film, where a field Hy is directed normal to i~ and no domain wall displacements take place, the magnetization component M y - Ms sin 0 increases with Hy in a linear fashion. The equation of the magnetization curve is in fact My -- (Ms/Hk)Hy, where H k is the anisotropy field

Hk = 2Ku/Js

(5.43)

(Ku induced anisotropy constant, Js saturation polarization). Typical values of Hk in thin permalloy films are around 100-200 A/re. If we identify Hy with the field to be measured, we obtain from Eq. (5.41) that the corresponding magnetoresistance variation Zkp = p(O)- p• is related to its strength by the parabolic equation z~p = (Pll- p•

-

(Hy/Hk)2).

(5.44)

In the actual narrow pattern sensors, the rotation of the magnetization is not uniform across the film width and it turns out that the idealistic parabolic behavior of Ap/APmax , w i t h A P m a x = Pll - - P_L, given by Eq. (5.44) modifies as shown in Fig. 5.17a (solid line). The inflexion point, with

192

CHAPTER 5 Measurement of Magnetic Fields

HTeT

i .....

~.=

H/H~:,

(a) x

y

o

=,=

1

ic v

G

(b)

H

,cb (c)

li

C

(d)

FIGURE 5.17 (a) Relative variation of the magnetoresistance with the hard-axis field Hy in an AMR thin film. The maximum sensitivity is obtained at the inflexion point, approximately corresponding to an angle 0 - 45~ between magnetization Ms and sense current ic. The dashed line represents the idealized quadratic response of a large area film. (b) Rotation of Ms to 45~degree equilibrium direction under the effect of the bias field HB. The sensed field he is co-linear with Hb. (c) Realization of a thin-film AMR magnetometer with enhanced vector sensitivity, exploiting permalloy flux concentrators (adapted from Ref. [5.39]). (d) Barber pole sensor, with shorting bars placed at 45~ to the easy axis (EA).

a roughly linear region a r o u n d it, provides an optimal working point, approximately corresponding to an angle 0 = 45 ~ where the greatest magnetoresistance sensitivity is obtained. This point can be reached, as s h o w n in Fig. 5.17b, by application of a bias field Hb of convenient strength, orthogonal to ic (i.e. to the EA), keeping M oriented along the desired 45 ~ angle. To estimate the voltage response of a strip of length s to a hard-axis oriented field he, small with respect to HB, we linearize Eq. (5.44) around the bias point 8p(he)---2Apmax Hb he H2at

(5.45)

to obtain, for a sense current of density jc, ~Vc(he)---2jcs

Hb he , H2at

(5.46)

5.2 HALL EFFECT AND MAGNETORESISTANCE METHODS

193

where we have substituted Hk with an effective saturation field Hsat, accounting for the actual geometrical arrangement of the AMR sensor. A definite domain structure, mainly directed along the EA, exists in the thinfilm sensor [5.38]. This is the source of Barkhausen noise during operation and, in order to suppress it via a monodomain condition, a further bias field, directed along the EA, can be applied. Figure 5.17c schematically illustrates a realization of a high-sensitivity magnetoresistive thin-film magnetometer [5.39]. A 35 ~m x 1260 ~m permalloy AMR thin film is sputtered to 25 nm thickness on a SiO2 coated alumina substrate in the presence of a longitudinally directed anisotropy-inducing field of about 6000 A / m . A flux concentrator structure is further sputtered by alternating a number of 1 ~m thick Ni-Fe layers with SiO2 spacers. The role of the flux-concentrator is twofold: to magnify the y-directed flux density component and to shield the x-directed one, thereby ensuring excellent vector sensitivity. It also conveys towards the hard-axis the bias field Hb produced by a current ib circulating in a surrounding few-turn winding. With a sense current i~ = 15 mA and a resistance R0 = 400 f~ (power drain ---100 mW), the intrinsic sensitivity of the device is around 3.5 m V / ( A / m ) with a flat response from DC to about 50 MHz. Several schemes for AMR field sensing have been envisaged, which is justified by the objective relevance this method has, besides the measuring aspects, in the field of magnetic read heads. In a typical magnetometric configuration, a permalloy or N i - F e - C o meandered thin-film pattern is obtained by deposition and etching, with the narrow conducting track (thickness ---0.1 ~m, width ---10 ~m) folded into a series of parallel strips, leading to a total film resistance of the order of several kf~ (and ensuing low power dissipation) over a sensing area of few m m 2 [5.40]. In this case, the EA is directed at 45 ~ to strip length and the bias field is applied either along with or transverse to the EA. Since the sensing field direction is orthogonal to the bias field, vector performance can be achieved. Field biasing is dispensed with by adopting the so-called "barber pole" geometry, shown in Fig. 5.17d. Here, the canting of the current direction with respect to the EA is obtained by covering the magnetoresistive film with regularly spaced 45 ~ slanted conducting bars (made, for example, by deposition of Au or Ag) [5.41]. The current in the magnetic film, taking the shortest path between the conductors, is forced to flow at 45 ~ with respect to 1VI~. Good sensitivity and temperature independence of the measured resistance change, which, in the case of AMR, may be less than a 2% modulation of the sensing current, are typically achieved with a Wheatstone bridge configuration, employing four equal sensing resistors. The required unbalance of the bridge is obtained by making one pair of

194

CHAPTER 5 Measurement of Magnetic Fields I

Ro-SR

c

jvMs

Ro+6R

Vo

O

Vout

Ro+SR

JvMs

Ro-SR

FIGURE 5.18 Full bridge barber pole AMR sensor. For a leg resistance variation 8R generated by the applied field he, the output voltage is Vout = GVo 8R/Ro.

opposite legs respond in the opposite way to the applied field with respect to the other pair. For instance, one pair can be shielded or the two pairs can be differently biased. A possible scheme is shown in Fig. 5.18, where the zero field resistance R0 of the four barber pole sensors is made to either decrease or increase by a quantity 8R upon the application of an external field he according to whether the magnetization is correspondingly rotated away from or towards the direction of the sense current. The voltage obtained at the output of the differential amplifier is, for a given input voltage and amplifier gain G, given by Vou t "- G V 0

8R/Ro.

(5.47)

The sensitivity of a bridge setup is usually expressed in terms of output voltage per unit input voltage and unit field (e.g. in ~ V / V / ( A / m ) ) . A typical sensitivity is 10 ~ V / V / ( A / m ) . With an input voltage V0 - 5 V and an amplifier gain G = 200, we obtain Vout = 1 V for h e = 100 A/m. Giant magnetoresistance multilayers offer excellent perspectives for magnetometers and sensors because the large effect of the magnetic field on resistance is associated with a large dynamic range. As shown in Fig. 5.13, saturation can in fact occur at very high fields. On the other hand, the nature and strength of the exchange coupling can be modulated by acting on composition and thickness of the layers. Consequently, devices with properties appropriate to either low- or high-field measurements can be developed. For example, substantially soft multilayers can be obtained by operating on the second antiferromagnetic peak (see Fig. 2.25). A reduced GMR ratio is the obvious price one has to pay for

5.2 HALL EFFECT AND MAGNETORESISTANCE METHODS

195

sensitivity at low fields. Multilayers, characterized by a combination of 5-20% magnetoresistance and 2.5-15 k A / m saturation field, have been demonstrated in N i F e / A g and C o N i F e / C o F e / A g C u / C o F e / C o N i F e multilayers [5.42, 5.43]. An example is shown in Fig. 5.19. Soft GMR behavior is also exhibited by the spin-valve multilayers (see Section 2.7 and Fig. 2.26). A major point in GMR magnetometers, as discussed above for the AMR devices, is the method used to obtain a differential output signal in the Wheatstone bridge. One method consists in shielding the field across two opposite legs, so that a bridge unbalance is created by the GMR response of the other two resistors. The low-field sensitivity of this device can be enhanced if the shielding elements are simultaneously exploited as flux concentrators for the two active resistors [5.44]. Another solution calls for the application of a bias field to the four GMR multilayer elements. This field can be provided, for instance, by a patterned array of thin-film magnets, set at one half the saturation field and applied in opposite directions upon opposite legs. According to the GMR behavior shown in Fig. 5.19, an external field will make the resistance of the

1

1lOO

8

rr

Hbiast[]-~ f

1050

lOOO (NIFe 2,nm / Ag 1 1 nm)21 950 -2x104

-lx104

0 H (A/m)

:I Hsat !

i/

lx104

2x104

FIGURE 5.19 Low field GMR response of a NiFe/Ag multilayer (from Ref. [5.42]). The application of a bias field Hbias halfway between the saturation field/-/sat and zero brings the working point within a region of near-linear GMR behavior, where the resistance R0 increases or decreases by a quantity 8R according to the polarity of the external field. With two opposite legs of the bridge subjected to Hbias and the other two to -Hbias, the same situation represented in Fig. (5.18) is obtained and the output voltage is given again by Eq. (5.46).

196

CHAPTER 5 Measurement of Magnetic Fields

lool > v

E

Gr /IR

50" 0 -50 T = 100 o c / / / -100

~ ~~ 9

i

.

.

-8000

.

.

T = 24 oC i

.

.

-4000

.

.

.

.

0

.

.

H (A/m)

i

'-

4000

-

9

9

i

'-

8000

FIGURE 5.20 Response of a NiFe/Ag multilayer GMR bridge sensor (Ref. [5.42]) as a function of the applied field compared with the response of a NislFe19 AMR bridge (Ref. [5.45]). The curves are normalized to the same input voltage. multilayer decrease or increase according to its polarity and the polarity of the bias, and the output voltage will be given, as for the AMR bridge of Fig. 5.18, by Eq. (5.47). Contrary to the field-shunting device, the field bias bridge can thus reveal the polarity of the sensed field. Linearity, large dynamic range, and lack of hysteresis are distinctive features of the GMR magnetic sensors, as illustrated by the output voltage curve of the biased NiFe/Ag multilayer bridge, shown in Fig. 5.20 in comparison with the response of a high-performance Ni81Fe19 AMR bridge [5.45].

5.3 F E R R O M A G N E T I C S E N S O R M E T H O D S A natural way of increasing the sensitivity of the fluxmetric field detection method consists in filling the sensing coil area with a soft magnetic core in order to exploit the flux multiplying properties of the material. There are no special difficulties in the direct use of a soft ferromagnetic material as a simple flux multiplier. However, phenomena like magnetic saturation, eddy currents, non-linearity and hysteresis are detrimental to the measuring accuracy and more indirect methods, not necessarily based on the behavior of the J-H curve, have been developed. Altogether, they belong to the wide domain of magnetic field sensing technology and their applications, ranging from geophysical surveys to space exploration, are in many cases only loosely related to the problem of magnetic materials

5.3 FERROMAGNETIC SENSOR METHODS

197

characterization. Consequently, they will be discussed here to a somewhat limited extent, a more detailed analysis being available in recent literature reviews [5.46, 5.47].

5.3.1 Fluxgate magnetometers A way to get rid of the intricacies of hysteresis loops, while exploiting the symmetry properties of the magnetization curve, consists in imposing an alternating exciting field, strong enough to cyclically drive the soft sensing core to deep saturation, and in analyzing the modifications occurring in the secondary voltage when the core is exposed to the field under measurement. This is the working principle of the fluxgate magnetometers, summarily sketched in Fig. 5.21. We see in this figure how an external DC field Hs, combining with the AC exciting field Hexc(t), brings about hysteresis loop asymmetry and leads to modified periodic behavior of the secondary voltage. The lost symmetry property leads to the presence of even harmonics in this voltage, which suggests that the measurement of the external field can be accomplished through the determination of the amplitude of the second harmonic. This can be appreciated by means of an approximate argument, where the role of e

A ou

11

Time

Hexc+H s

Hexc

~--,

Yout'

9

v

Time

Time

FIGURE 5.21 Generation of the secondary voltage in a fluxgate core, described by means of a simplified parallelogram-shaped hysteresis loop, under a triangular driving field He• If the core is exposed to a DC external field Hs the symmetry of the described B(H) curve is lost and the secondary voltage V1 contains even harmonics.

198

CHAPTER 5 Measurement of Magnetic Fields

the time-varying permeability of the core on the secondary voltage is considered [5.48, 5.49]. Let us thus assume that the core, having crosssectional area A, is subjected to a field H which is the sum of a largeamplitude alternating driving field Hexc(t)~ cyclically saturating the material, and a DC external field Hs. The latter is sufficiently small to Its effect is taken into account be considered as a perturbation on He• by writing the voltage u ( t ) = - N A d B / d t induced on the N-turn secondary winding as a series development truncated to the first order dB dH

u(t) = - N A d---H dt

~ -NA

dHexc dt

((/d'd)ne•

-}- (di~d/dH)HexcHs)'

(5.48)

where the apparent differential permeability /d,d can be related to the effective differential permeability ~e(t) via the sample demagnetizing coefficient Nd /~e (~e)"

~d=

(5.49)

lff-N d ~ - - 1 Equation (5.48) contains, besides the usual odd symmetry term in the output voltage, the even symmetry term dHexc d/d,d dt dH

d/~d dt ~

having the periodicity of the permeability. By revealing the second harmonic through filtering, one can obtain, according to this equation, a measure of Hs. For a magnetic core excited as in Fig. 5.21, this means, for example, locking to the second harmonic of the output voltage Uout(t) using phase sensitive detection. By adopting the two-core sensing configuration shown in Fig. 5.22 (known as Vacquier-type sensor), one can automatically filter out the odd-harmonic components. Here, the exciting field, whose frequencyftypically ranges between I and 10 kHz, is applied to the two identical cores along opposite directions. In the absence of any other field, the voltage Uo~t induced in the secondary winding is zero. With an external field Hs~ the working point on the hysteresis loop is displaced differently in the two cores and a resulting instantaneous flux ~ s ( t ) ~ 2NAi~d(t)Hs at the frequency 2f becomes linked with the secondary winding (small Hs values). The sensor reveals in principle only the component of the external field parallel to the probe axis and it therefore works as a vector magnetometer. Its sensitivity, proportional to d/~d/dt, and its dynamic range both increase with increasing the peak amplitude of Hexc(t). It also increases with a decrease of the demagnetizing coefficient, somewhat at the expense of the linearity of

5.3 FERROMAGNETIC SENSOR METHODS

199 B

lexc

-Hexc +Hs .,.~ |

-.&

._..~-.'-

.._1 ~

_.~--

_.,i ~

I z'~/',#''

/__t'

~

/3

+G

"4 FIGURE 5.22 Two-core fluxgate sensor (Vacquier type). The two identical cores, which can be obtained by using either soft ferrite rods, permalloy strips or amorphous ribbons, are subjected to the AC exciting field Hexc (frequency in the kHz range) along opposite directions and are exposed to the same DC or lowfrequency external field I-I~.The corresponding working points on the hysteresis loop are asymmetrically displaced by I-I~. This implies that the voltage Vout detected by a secondary winding contains only even order harmonics and it is, according to Eq. (5.48), proportional to Hs. the response. On the other hand, all spurious contributions due to imperfectly balanced cores and residual second harmonic in the input are enhanced with stronger exciting fields, and the Barkhausen effect, which is the main source of noise in fluxgate devices, is known to increase with the apparent differential permeability /.4,d [5.50]. In order to increase linearity and measuring range, the closed-loop feedback technique is often adopted. In this case, the magnetometer is used as a null-sensing device, where a supplementary coil is added and a field exactly balancing the external field Hs is provided. As core materials, Mumetal or permalloy wires or strips are used in most cases because of their obvious extra-soft magnetic behavior, indispensable for low-field sensitivity, vanishing magnetostriction and very low Barkhausen noise. Since the 1980s, Co-based amorphous ribbons, exhibiting near-zero magnetostriction (see Section 2.4), have been increasingly employed [5.51]. Ribbon annealing under transverse field or stress annealing, both leading to uniaxial transverse anisotropy (i.e. favoring homogeneous moment rotations vs. domain wall displacements during the magnetization process) are exploited in order to minimize the noise. Besides the sensor presented in Fig. 5.22, several realizations of the fluxgate principle are reported in the literature and employed in practical setups. Figure 5.23 shows two examples. The ring-core device, which can be considered as derived from the Vacquier-type sensor by bending and

200

CHAPTER 5 Measurement of Magnetic Fields

Vout

I.

(a)

(b)

FIGURE 5.23 (a) Ring-core fluxgate sensor. It can be considered as a derivation of the Vacquier-type sensor, where the two straight cores are bent and connected at their ends. The output voltage Vout is different from zero only in the presence of the external field Hs, which creates a flux unbalance between the two half-cores. (b) Miniature mixed parallel-orthogonal fluxgate. A circular AC exciting field is generated by the current iexcflowing along the narrow amorphous ribbon, which i * . . . . . s characterized by 45o directed helical anlsotropy (indicated by Ku at the surface). The field H s breaks the symmetry of the magnetization process in the two ribbons and a second harmonic output is generated (from Ref. [5.54]).

connecting the limbs at their ends, is very popular and extensively applied because it offers some intrinsic advantages. It can be fabricated and wound with a high degree of uniformity, and the magnetization in the material is, in the absence of the external field, the same in all crosssections. In addition, it can be rotated within its sensing solenoid in order to minimize offset and other spurious effects due to any residual geometrical imperfection. On the other hand, it is more sensitive to crossfields than the highly anisotropic Vacquier-type sensor, although always faring much better in this respect than magnetoresistance magnetometers. It is typically obtained by wrapping a few turns of a permalloy or amorphous ribbon on a non-magnetic former, but, in recent developments of miniaturized sensors, single foil rings have been prepared by photolithographic techniques and chemical etching and incorporated in suitably prepared printed circuit boards [5.52]. There is no basic difference in the working principle of the Vacquier-type and ring-core fluxgate sensors, both belonging to the category of the so-called "parallelgating" fluxgates, the parallelism referred to being the exciting and the sensing fields. Orthogonal and mixed orthogonal-parallel gated devices have also been developed and patented [5.53]. Figure 5.23b schematically illustrates an example of an orthogonal-parallel fluxgate, the miniature hairpin sensor developed by Nielsen et al. [5.54]. It makes use of two identical near-zero magnetostriction ribbon pieces (composition

5.3 FERROMAGNETIC SENSOR METHODS

201

Fe3.sCo66.sSi12B18) I m m wide and 23 ~m thick, with minimum length 8 mm, characterized by helical anisotropy Ku ~ 50 J/m 3 directed at 45 ~ to the ribbon length. Anisotropy is induced by subjecting the ribbon to torsional stress-annealing at temperatures between 300 and 400 ~ The two ribbon pieces are series-connected as shown in the figure and a 15 kHz exciting current is made to circulate in them. This generates a circular field, which gives rise to a net longitudinal flux in each piece, but no net flux in the sense coil wrapped around the two pieces. The longitudinal external field Hs breaks the symmetric condition so obtained and even harmonics are induced in the sense coil. Detection methods other than revealing the second harmonic of the output voltage are frequently adopted in fluxgates. For example, with the classical "peaking strip" technique one measures the deviation that the external field Hs imposes on the symmetry of the position of the voltage pulses generated each half-period by the saturating exciting field when traversing the hysteresis loop (see Fig. 5.21). A variant of this method consists in the determination of the zero passage of the hysteresis loop, which is displaced by the quantity Hs [5.55]. In the self-oscillating type magnetometer of Takeuchii and Harada [5.56], making use of a single-core amorphous sensor, the generated square wave has a duty ratio depending on Hs, which is revealed as a DC output voltage equal to the mean value of the waveform. In the multivibrator-type magnetometer of Mohri et al. [5.57], two identical sensing cores made of short amorphous ribbon pieces (length 5-50 mm) are used as the arms of a bridge, which generates a DC voltage upon unbalance of the core voltages created by the external field. Conventional fluxgate sensors are bulky objects with mass and volume typically in the 0.1 kg and 100cm ~ range, respectively, and power consumption might represent a problem. Several attempts have been made in recent times to overcome the size problem, either by trying to integrate the magnetic core preparation via thin-film deposition or electroplating with the Si technology [5.58] or by combining millimetersized cores, etched from amorphous ribbons, with planar coils and printed circuit boards [5.59]. At the present time, the miniature fluxgates do not reach the performance of the conventional bulk devices because thin films lack the soft magnetic quality available in sheets and ribbons, the noise increases and it is difficult to drive the core into saturation [5.60]. Fluxgate magnetometers are largely employed for DC and low frequency (typically up to few hundred Hz) field measurements because of their high sensitivity, linearity, directional properties, wide range of operating temperatures, stability, reliability, and ruggedness. They have

202

CHAPTER 5 Measurement of Magnetic Fields

105 10 4

AMR ""'------~_....~....._

N

-r

03

. FG3

v

a

m < 102 O '""

-

o 101

z

S ~

100 ....

6'1

FG2

.......

i

. . . . . . .

~0

Frequency (Hz) FIGURE 5.24 Amplitude spectral density (ASD) of the noise in different types of magnetic field sensors. FGI: Race-track fluxgate; FG2: ring-core fluxgate. FG3: miniature fluxgate; AMR: permalloy magnetoresistor (from Ref. [5.61]). ME: magnetostrictive magnetometer (from Ref. [5.62]). FO: fiber-optic coupled magnetostrictive magnetometer (from Ref. [5.63]). been widely applied over the past several decades in areas like geomagnetic studies, airborne detection, and space magnetometry and in other instances, like medical investigations, where extremely low field strengths, in the nT range, need to be reliably determined. Commercial setups covering the range I nT-1 mT are offered. The limit to fluxgate sensitivity is chiefly due to the inevitable presence of hysteresis and magnetization noise, while an upper field limit is found when non-linear behavior sets in or, in any case, the field Hs definitely brings the magnetic probe into saturation. Fluxgates are nowadays second only to SQUIDs for sensitivity, with incomparably lower complexity and running costs. One can appreciate their performance when comparing their noise behavior with that of other sensitive magnetometers, as in the example reported in Fig. 5.24 [5.61-63].

5.3.2 Inductive magnetometers There is no clear line between fluxgates and generally inductive magnetometers. The latter will be considered here as those devices not necessarily working under a deeply saturating AC exciting field. In many cases, these cores are made of deposited or electroplated permalloy-type

5.3 FERROMAGNETIC SENSOR METHODS

203

thin films, having a defined uniaxial anisotropy Ku. This is obtained by applying a saturating field in the desired direction during the film preparation. It permits one to identify an EA and, orthogonal to it, a hard axis (HA) in the plane of the film. The magnetization process can consequently be made to occur in such a way that an external magnetic field can be revealed and measured with little or no interference from hysteresis and its memory effects. A thin film is evidently characterized by very low in-plane and very high out-of-plane demagnetizing coefficients. This ensures that there is little difference between in-plane applied and effective fields and that there is no out-of-plane magnetization component. Since the obtained magnetization variations and the cross-sectional areas of the thin film cores are generally small, the exciting field frequency is high, from a few hundred kHz to several MHz, in order to obtain usable voltages across the inductor. The domain configuration in a field-annealed permalloy film generally consists, in the demagnetized state, in a simple balanced array of 180 ~ slab-like antiparallel domains directed along the EA. Such a structure remains unchanged under an AC exciting field Hexc of frequencyfdirected along the HA, because, as shown in Fig. 5.25, the only effect of Hexc is to instigate the homogeneous rotation

v

Hm

....

~ i~'~-~

i

,

Vout

(a)

(b)

FIGURE 5.25 Examples of thin-film inductive magnetic field sensors. The domain structure and the oscillating magnetic moments are schematically depicted. Preparing the permalloy films under a saturating magnetic field creates an EA with anisotropy Ku. An exciting field I~xc at radiofrequencyfis applied along the hard axis (HA) and the DC/low-frequency field FI~under measurement is directed along EA. In the sensor (a) a modulating field Hm of frequency fro <
204

CHAPTER 5 Measurement of Magnetic Fields

of the magnetic moments in opposite senses within adjacent domains (we use here the scalar notation for the fields since their direction is understood). An external DC magnetic field applied along EA creates an unbalance in the antiparallel domain structure and the signal detected either along EA or HA is consequently affected. In the sensor depicted in Fig. 5.25a, no voltage is induced in the sense coil if the external field Hs and the modulating field H m are both zero [5.64]. The DC field Hs generates a net flux directed along EA and the output voltage Vout consists, in the absence of Hm, of a rectified waveform of frequency 2f. This waveform attains a maximum value at each zero passage of Hexc and becomes zero when Hexc = Hk, Hk being the anisotropy field, where the rotation of the magnetization towards HA is concluded. At low field strengths, the displacement of the domain walls is approximately proportional to the value of Hs and Vout directly provides a measure of Hs. In a real material, characterized by distributed anisotropies and nonnegligible hysteresis, the additional presence of the modulating field Hm of frequency fm <
(5.50)

S k + Hb + I--Is '

where Hb is a DC bias field, normally larger than Hs, which has the role of providing a stable operating point by providing an equivalent anisotropy field Hk + HB. The susceptibility (i.e. the sensor inductance) changes then linearly with Hs for small values of the ratio H s / ( H k + Hb), with a rate of change depending on the strength and sign of Hb. A practical magnetometer circuit can be realized by the use of two oppositely biased identical transducers used as inductances in a Colpitt's oscillator, where their developed radiofrequency voltages are rectified and summed with opposing DC polarity [5.65]. The output voltage is then different from zero only if Hs ~a 0 and it will go negative or positive according to the sign of Hs. In order to improve the signal-to-noise ratio, a feedback system with an additional winding maintaining zero external field can be employed.

5.3 FERROMAGNETIC SENSOR METHODS

205

Thin-film inductance magnetometers can have very high resolution (down to less than I nT) and the measuring range generally extends to about I mT. They are typically used for the measurement of DC environmental fields, but AC fields can also be determined provided their frequency is at least one or two orders of magnitude lower than the frequency of the exciting field Hexc. The development of novel materials, such as amorphous ribbons and wires and thin-film multilayers (see Sections 2.4 and 2.7), has opened new perspectives in the domain of magneto-inductive field sensors. Basically, with these materials one can realize devices where the magnetization process, dominated by the presence of induced uniaxial anisotropies, is driven by an AC field generated by a current circulating in the ferromagnetic sample itself and has features dependent on the presence of an external DC field. Such a field can affect the AC permeability and it can therefore be revealed by taking the sample as a circuit element and by measuring the variation of the voltage drop through it. This effect is especially relevant in the near-zero magnetostriction Co-based amorphous wires (having, for example, composition Co69Fe4B15Si12 and small negative saturation magnetostriction ~s "~-10-7), which display a domain structure in the demagnetized state as shown in Fig. 5.26a. This structure, which is determined by the anisotropies due to the stresses frozen in during the rapid solidification and the condition of free poles avoidance in the material, consists of outer shell antiparallel domains, where the magnetization is circumferentially directed, and axially directed domains at the wire core [5.66]. The circular field generated by an AC current of frequency f flowing in the wire causes the circumferential domain walls to oscillate and gives rise to the hysteresis loop (B~, H~), as shown in the example of Fig. 5.26b [5.67]. Given the soft behavior of these walls, characterized by coercivities around some 1050 A / m , peak current values of the order 10-50 mA are sufficient to complete most of the reversal (with typical wire diameters 30-130 txm). At low frequencies (i.e. 1-10 kHz) the voltage across the wire sample, normally a few m m long, consists of an ohmic contribution V0 - RDC/exc, where RDC is the DC resistance, and an inductive contribution eL ---dq~c/dt, ~c being the circumferential flux. By subtracting the resistive part, using for instance a bridge circuit, the inductive term eL can be singled out [5.68]. It appears as a periodic sequence of pulses associated with the time of passage through the coercive field. The HA-oriented DC field H s does interfere with this process because it tends to align the magnetization in each domain with the wire axis, thereby weakening the circumferential flux. If we assume negligible anisotropy dispersion and we define a circumferential anisotropy K~, the angle 0 by which

206

CHAPTER 5 Measurement of Magnetic Fields

H~

(a)

Hs=O ___-- a s i n g

Hs

(b)

FIGURE 5.26 (a) The domain structure of a Co-based amorphous wire is made of an outer shell, where the magnetization is circumferentially directed, and a core with longitudinal domains. The domain walls in the shell, driven by the field H~ generated by the AC current iexc circulating in the wire, give rise to a circumferential magnetization reversal whose amplitude decreases with the increase of the strength of the external DC field Hs. The corresponding evolution of the AC circumferential hysteresis loops (B,~,H,p) is schematically shown in (b) (adapted from Ref. [5.67]).

the magnetization rotates under the action of Hs is 0 = sin-l(Hs/Hk),

(5.51)

with Hk = 2K,~/Js. The m a x i m u m possible reversal that can be p r o d u c e d by d o m a i n wall displacements under the AC circular field H , then decreases with increasing Hs as Jcmax) = Js cos 0 = Js ~/1 - qsHs/2K,~) 2

(5.52)

and it tends to zero, at least in principle, for Hs = Hk. The hysteresis loop (B~,H~) consequently evolves with Hs as s h o w n in Fig. 5.26b and the corresponding decrease of the value of the inductive voltage eL can then be taken as a measure of Hs. Magnetic field sensors based on this m e c h a n i s m are called magnetoinductive magnetometers. Their behavior can actually be interpreted in terms of the change imposed by the external

5.3 FERROMAGNETIC SENSOR METHODS

207

field on the wire inductance L =/-~,s where /~ and s are the circumferential differential permeability and the wire length, respectively. If the exciting frequency is increased to the point that the skin effect becomes important, the external field will act to change both the resistive and the inductive contributions to the sample voltage and a large effect is consequently expected. At such frequencies, one gets quasi-linear magnetic behavior and it is reasonable to apply the concept of sample impedance. Under the condition of penetration depth 3 = ~ / p (where p is the material resistivity), much lower than the wire "radius a, the impedance can be written as a

Z - 2x/~ RDC(1 + j)~2~'flz~.

(5.53)

1/2

This equation predicts the same ~ dependence for the resistive and reactive parts of Z [5.69]. Figure 5.27 provides an example of

FeCoBSiwirel / ~

,•10

MHz

f= 3 MHz

o!. -800

f= 1 MHz '

'-4bo'

'

'

o

'

'

'46o'

Externalfield(A/m)

'

'8oo

FIGURE 5.27 Reduced voltage amplitude Vz/Vo vs. external field, applied colinear with the wire axis, measured across a n Fe4.3Co68Si12.5B15 wire (length 5 ram, diameter 30 ~m, As ~ - 1 • 10-7) at three different frequencies of the exciting current (iexc = 15 mA). The wire was annealed under a tensile stress of 20 MPa, in order to achieve a well-defined circumferential anisotropy (HK ~ 80 A/m). The increase of Vz/Vo at low fields is due to a corresponding increase of the circumferential permeability, engendered by the rotation processes. These are enhanced at higher frequencies (adapted from Ref. [5.69]).

208

CHAPTER 5 Measurement of Magnetic Fields

the experimental dependence of the reduced voltage Vz/Vo, where V0 = RDcie• is the amplitude of the voltage drop under purely resistive conditions, on the external field [5.69]. One can notice the very large variation of this quantity, which is the more important the higher the frequency f. This effect is aptly called giant magnetoimpedance (GMI). Vz/Vo passes through a maximum at low fields (Hs <-- Hk). This effect becomes more pronounced with increasing f. It derives from a correspondingly increased damping of the domain wall motion, whose role must be taken up by the rotational processes. The contribution of the latter to the permeability/~ and, in force of Eq. (5.53), to the impedance is maximum when Hs ~ Hk and the magnetization direction approaches the wire axis. As already stressed in Section 2.7, thin-film multilayers can be prepared which exhibit GMI. Their typical structure is of the type F/ M / F , with M a non-magnetic metallic layer and F a soft magnetic layer (e.g. permalloy or amorphous alloy). An example provided by the C o - S i - B ( 2 ~ m ) / C u ( 3 ~ m ) / C o - S i - B ( 2 ~ m ) sandwich is reported in Fig. 2.29 [2.74]. It is shown that in this case the GMI is not strictly related to the skin effect, but to the possibility of changing the permeability by setting free the rotations in the anisotropic amorphous layer through the application of an external field orthogonal to the EA. Skin-effect dominated GMI in thin-film multilayers (for example Ni-Fe/SiO2) can still be obtained, but with working frequencies moved towards the GHz range [5.70]. Co-based amorphous ribbons have also been proposed as GMI elements. Their use can be envisaged under the condition both of transverse domains, resulting from the anisotropy induced by transverse field annealing, and of longitudinal domains. The first condition leads to a phenomenology similar to the one exhibited by the Co-based amorphous wires, while in the second case only rotations can take place, the magnetization being aligned with the current at the start. The EA directed field Hs hinders the rotations and /~ is correspondingly decreased. Whatever the domain structure, the resulting effect is weaker than in Co-based amorphous wires and requires higher Hs values [5.67]. Magnetometric applications of the GMI sensors are best accomplished over a region of the impedance vs. field characteristic approximating a linear behavior. Having to deal with curves like the ones shown in Fig. 5.27, one can find, similar to the case of GMR devices shown in Fig. 5.19, an optimal working point by suitable field biasing. This should bring the Hs = 0 operating point to the left or to the right of the maximum. Several practical circuits have been developed in order to achieve linear GMI response, looking at the same time for sensor miniaturization,

5.3 FERROMAGNETIC SENSOR METHODS

209

quick response, and small power consumption, as summarized by Mohri et al. [5.71].

5.3.3 Magnetostriction, magneto-optical, and microtorque magnetometers Any field changing the magnetization state of a ferromagnetic material brings about dimensional changes. Conversely, a ferromagnetic sample exposed to an external field does change its magnetization when strained. Under certain conditions, both these effects can be exploited in order to reveal and measure DC/low-frequency fields to excellent sensitivity levels. The development of the Fe-based amorphous alloys, with their unique combination of high magnetostriction (typical values of the magnetostriction constant hs around 30x 10 -6) and soft magnetic behavior (coercive field of few A/m), has propelled the creation of magnetostrictive magnetometers based on amorphous ribbons. In general, the ribbon sample is prepared by annealing and cooling under a transverse field so that it shows a substantial transverse anisotropy Ku and responds to a longitudinal field Hs mainly by homogeneous rotation of the magnetization. At equilibrium, the rotation angle 0 is given by Eq. (5.51) and the angle ~ -- vr/2 - 0 made by the magnetization with Hs is related to the magnetostrictive strain 8~(Hs) suffered by the ribbon starting from the demagnetized state according to 3

3

H2

8h(Hs) = ~ '~s COs2q~ = ~ As H---~~

(5.54)

with Hk = 2Ku/Js. 8,~(Hs) is therefore predicted to have a quadratic dependence on the external field and is expected to saturate at high magnetization levels. The basic approach to the measurement of 8A(Hs) is by mechanical coupling of the magnetostrictive ribbon to a single-mode optical fiber, where the length variation of the sample is linearly transformed into a variation of the optical path length and interferometrically detected with very high resolution as a phase shift (down to some 10 -6 rad) [5.72]. The main drawback of the fiber-optic technique is the interference by the low-frequency thermal and mechanical perturbations generated by the environment, which degrade the signal-to-noise ratio in the direct measurement of DC and low frequency (f < 20 Hz) fields. In order to overcome this difficulty, it is expedient to apply an additional dither field, typically at a frequencyfd of few kHz, by which the DC/lowfrequency external field is determined through its influence on the magnetostrictive response of the material at the frequencyfd. If we denote in fact by Hs(t) =/-/so cos COstand Hd(t) = Hdo cos COdt the low-frequency

210

CHAPTER 5 Measurement of Magnetic Fields

and the high-frequency fields, respectively, and we substitute Hs in Eq. (5.54) with their sum, we obtain for the magnetostrictive strain 3 As 8'~(ros~ rOd) -- 2 H~ [Hs2~COS2rOst+ H~~ COS2rOdt + HsoHdo(COS(rOs+ rod)t + cos(rOs -- rOd)t)].

(5.55)

We see that, by applying phase-sensitive detection, one can achieve at the interferometer output a signal proportional to the external low-frequency field/-/so at the sideband frequencies fd + fs, where the noise is largely reduced. For a DC field of amplitude Hs, Eq. (5.55) reduces to As .[H2 + H~~ cos2rOdt + 2HsHdo cosrOdt] 8A(0~ rOd) = ~3 H2

(5.56)

and Hs is recovered by detection of the optical phase shift at the dither frequency fd. To this end, the optical signal is converted by a photodiode and synchronously recovered by a lock-in amplifier. Closed-loop operation is often adopted in fiber-optic magnetometers in order to eliminate hysteresis effects and achieving maximum linearity. In this case, the output signal of the lock-in amplifier is integrated and used as a current source for a supplementary winding, which provides a bias field equal and opposite to Hs. Thus, the magnitude of the external field is obtained from the value of the current supplying the feedback solenoid. Some degradation of the magnetometer performance might derive, however, from the 2nd-harmonic strain (the cos2rOdt term in Eq. (5.56)), which is not compensated by this operation. Coupling of the sensing ribbon to a piezoelectric transducer in place of the optical fiber appears to provide a relatively simple and robust alternative to the fiber-optic technique [5.73]. A piezoelectric system has been used by Mermelstein in order to apply an oscillating stress to a magnetostrictive amorphous ribbon, endowed with transverse anisotropy Ku, and to induce oscillations of the magnetization under an external field (inverse magnetostrictive effect) [5.74]. Figure 5.28 provides a schematic view of the measuring setup. The ribbon is epoxied at one end to a finely positioned piezoelectric transducer (PZT) and clamped at the other end to a support structure. The PZT is excited at its resonance frequency f0 (around 100 kHz) and a lock-in amplifier tuned at the PZT driving frequency detects the signal induced on the pickup coil. The method is based on the idea that the oscillating stress-induced anisotropy Kr 3hsg(t) modulates the anisotropy Ku, thereby modulating the equilibrium rotation angle of the magnetization in the presence of an external field. If we assume that

5.3 FERROMAGNETIC SENSOR METHODS

211

PZT

s,.'~

Lock-in

G

~~s

O

Vout

s S TQ

FIGURE 5.28 Example of magnetostriction magnetometer. An Fe-based amorphous ribbon, characterized by prevalent transverse anisotropy, is fixed at one end and glued to a piezoelectric transducer (PZT) at the other end. The transducer applies an oscillating stress ~r to the ribbon, whose magnetization oscillates when it is exposed to an external field Hs. In the ideal case shown in the figure this is accomplished by means of rotational processes. A signal at the transducer frequency (---100 kHz) e(t)oc Hs is induced in the coil and retrieved by means of phase sensitive detection (adapted from Ref. [5.74]). Ku >> Kr and that the magnetization process is chiefly due to rotations, we obtain the following relationship between the ribbon magnetization and the external field Hs Hs

l(t) -- Js Hku + Hk~(t)

~js gs ( ~ 1

Sk~(t)) Hku '

(5.57)

w h e r e the t i m e - d e p e n d e n t stress anisotropy field is H k r 3Xsa(t)/Js. The second term on the right-hand side oscillates with the applied stress and, if r - ~r0 cos toot, we obtain for the signal induced on an N-turn coil 3•s -

e(t) - NAtooO'o H2----~ Hs sin toot,

(5.58)

where A is the ribbon cross-sectional area. It should be r e m a r k e d that in this and in the previous examples, Hs stands for the effective field.

212

CHAPTER 5 Measurement of Magnetic Fields

It can be related to the applied field by the usual correction for the demagnetizing factor. The elastic modulus of a magnetostrictive material depends on the orientation distribution of the magnetization. An applied magnetic field, by changing this distribution, either by domain rearrangements or rotational processes, can change the modulus and, consequently, the speed by which acoustic waves are transmitted in the material. For example, a shear wave of frequency f propagating along a strip of shear modulus /~ suffers a change of its velocity v = x/~/3, where 3 is the material density, upon application of a field Hs. This means that between two points of the strip at a distance s there exists a phase difference AqJ that is a function of H~. It can be shown, in particular, that such a phase difference is proportional to the fractional change of the shear modulus per unit applied field according to the expression Aq~= -~--~~-s( ~1r ' C a /f~ sH. !

(5.59)

Squire and Gibbs have developed a shear wave magnetometer making use of a 25 cm long Fe-based amorphous strip, excited at a frequency f = 20 MHz, having a sensitivity of the order of 2.5 nT between DC and 1.6 Hz [5.75]. Again, direct determination of the DC field is made difficult by different noise sources and a modulation method is used by applying a modulating supplementary field at I kHz. Magnetostriction-based magnetometers have excellent field sensitivity, comparable to that of fluxgates (see Fig. 5.24) over the same DC/lowfrequency measuring range. They do not lend themselves easily to miniaturization, which can possibly be achieved by the development of good quality amorphous magnetoelastic thin films [5.76]. A plane polarized light beam passing through an isotropic material invested by a magnetic field suffers a rotation of its plane of polarization proportional to the field strength. As schematically illustrated in Fig. 5.29a, this effect results from the interaction of the magnetic field with the electrons oscillating under the action of the electromagnetic wave, which generates a Lorentz force at a right angle to both oscillation and field directions. The angle of rotation (Faraday's rotation angle) is proportional to the length s of the path followed by the light in the material, according to the conventional expression 0F-- VIJ~oH-r where the proportionality constant V is called Verdet constant. The value of V for the visible light in ordinary flint glass is around 0.03 r a d / T / m m (i.e. about 180~rotation in a 10 mm thick plate exposed to a field of 10 T). Being associated with the activity of deep-seated electron levels, it is rather

5.3 FERROMAGNETIC SENSOR METHODS

213 Laser

1 PBS X

X

I (a)

(b)

PD

PD

I

FIGURE 5.29 (a) Mechanism of Faraday rotation in garnets (polar effect). The interaction with the magnetization Ms of the electron oscillations caused by the plane-polarized electromagnetic wave generates a Lorentz force FL, which imposes an additional vibration in a direction perpendicular to both E and Ms. This causes a rotation by an angle OFof the plane of polarization of the emerging beam. The angle OFcan be measured by splitting the beam E~and recovering the two components along the orthogonal coordinates x and y. (b) Schematic example of a magneto-optical field sensing device: P, polarizer; G, garnet plate; PBS, polarizing beam splitter; PD, photodetector.

insensitive to the environmental conditions, which makes possible, as proposed many years ago, the accurate measurement of strong continuous and pulsated magnetic fields [5.77]. The Verdet constant becomes orders of magnitude higher in the presence of magnetic order and transparent ferrimagnetic compounds, such as the iron garnets, can be used in low and medium strength field measurement using magnetooptical techniques. A scheme of the principle of the measuring setup is given in Fig. 5.29b. The linearly polarized light beam supplied by the laser has its plane of polarization rotated by an angle OF, proportional to the net magnetization of the garnet plate G. To measure OF, the emerging beam is split in two orthogonally polarized beams by the splitter PBS and the two components are separately determined via the voltages provided by the two photodetectors (PD). Figure 5.30 provides an example of Faraday response in an YgFe4.53Ga0.47012 single-crystal disk at the wavelength 1.3 p~m [5.78]. The disk (diameter 3 mm, height 2.48 mm) is magnetized and illuminated in the axial direction and OFshows a wide linear response region. Linearity results from the combination of low coercive field and large demagnetizing factor Nd, which makes the applied field Ha ~ NdM if M is the sample magnetization. The sensitivity of the ferrimagnetic

214

CHAPTER 5 Measurement of Magnetic Fields 40 20

Y3 o v

0

-20

-40 -60

i

. . . .

i

-40

. . . .

i

-20

. . . .

i

0

. . . .

i

20

. . . .

i

40

~''~

9 ~

60

,Uo/-/a(mT)

FIGURE 5.30 Faraday rotation angle vs. applied field in a 2.48 mm thick circular Y3Fe4.sBGa0.47012 sample of diameter 3 mm (from Ref. [5.78]).

element S = d0F/dHa is therefore written as ~l(sat)

S -~ vF

(5.60)

NdMs ~

where O!sat)Fis the Faraday's rotation angle in the saturated material (see Fig. 5.30) and Nd is the magnetometric demagnetizing coefficient of the disk sample. The sensitivity can be increased by acting either on the composition, in order to increase the ratio ~(sat)/]~,~ vF /~v.s~or the sample aspect ratio. For example, ions in the base yttrium iron garnet Y3FesO12 can be partially replaced by other ions in order to change e i t h e r ~Fsat) o r M s. Substitution of Fe with Ga leads to decrease of Ms, while substituting Y ions with Bi ions increases 0(Fsat). On the other hand, using a number of superposed crystal plates can decrease the demagnetized coefficient. Greater flexibility in composition tuning is achieved with garnet thin films, which can be grown with either in-plane or normal-to-the-surface EA. With in-plane anisotropy, the rotation of the polarization plane of the incoming beam occurs when a field normal to the film plane provides a net magnetization via the rotational processes [5.79]. Rotations can equally be excited and exploited in place of domain wall displacements in the out-of-plane anisotropy films if the magnetic field and the propagating light beam are both applied in the film plane [5.80].

5.3 FERROMAGNETIC SENSOR METHODS

215

Linearity, wide range of measuring field strengths and frequencies (from DC to several hundred MHz), sensitivity and small size are typical properties of the magneto-optical field sensors. They find specific and wide applications in current metering under hostile environmental conditions (for example, in high-voltage power line monitoring), where limited accuracy (not better than 1%) is acceptable. In this case, remote sensing is obtained by connecting the garnet plate to the light source and the analyzer by optical fibers. Garnet thin films can also be used as stray field visualizers at the surface of superconductors, hard magnets and recording materials, provided they are of the in-plane anisotropy type and thus immune from interference by domain activity [5.79]. In all cases, the response of the rare-earth iron garnets exhibits a substantial temperature dependence, the sensitivity S suffering about 10% variation from 300 to 400 K. It has been shown that in the mixed rare-earth iron garnets, TbxYl-xFe5012, this variation is decreased to 1% for x = 0.2 [5.81]. The increasing trend towards miniaturization and integration of complex systems, often merging electrical, mechanical, and magnetic microfabrication technologies, poses a demand for magnetic field measurements at very small scale. This raises non-trivial problems in dimensional scaling regarding both the practical realization of the microsized sensors and the sensitivity of the devices. For example, the sensitivity of search coil and inductive probe magnetometers scales with area. Hall-effect sensors lack overall sensitivity, while magneto-resistive sensors may require excessive power consumption and SQUID systems are slow, cumbersome, and expensive. The old technique of torque magnetometry has therefore been revived and fast and sensitive microsized torque systems have been realized. They exploit advances in the preparation and manipulation of micro-mechanical devices through the use of the silicon micromachining techniques. Typically, a cantilever structure is prepared, with dimensions around a few hundred microns, which suffers a torque from a small permanent magnet fixed on it when it is immersed in a magnetic field with a non-zero component orthogonal to the moment direction. In the setup by Cowburn et al. [5.82], schematically shown in Fig. 5.31, an aggregate of aligned acicular ~/-Fe203 single-domain particles, carrying a total moment m---- 10-9A m 2, is stuck at the tip of a V-shaped Si3N4 microcantilaver, whose displacement is measured by a reflecting laser beam. The moment m, directed along the major axis of the acicular particles, is orthogonal to the cantilever plane and a field I-Is applied along the cantilever length exerts a torque on it. If 0 is the angle by which I-Is rotates the magnetization Ms away from the EA and ANd is the difference between

216

CHAPTER 5 Measurement of Magnetic Fields

m = 10 - 9 A - m 2

1E-7

1E-8 E

v

=

.o

1E-9

0

D

1E-10

1E-11 .j.,...l" 1E-12

1E-5

...

. . . . . . . . ........ . . . . . . . . . . . . . . . . . , . . . . . . . . , 1E-4 1E-3 0.01 0.1 1

...

,uo/-/s ( m T )

FIGURE 5.31 Micromechanical magnetometer. The magnetic moment m of a small aggregate of ~/-Fe203 single-domain particles, attached at the tip of a V-shaped Si3N4 microcantilever, interacts with the external field I-~. The ensuing cantilever deflection is measured by a laser beam reflected from the back of the cantilever. The graph provides the dependence of the cantilever deflection on the external field. Scheme and data are adapted from Ref. [5.82]. the minor and major axis demagnetizing coefficients, the torque, calculated by the minimization of the s u m of Z e e m a n and anisotropy energies E = E H q- E a =

-I~oH~M~sin

tz~ ANdM2 sin 2 O, 0 + -~-

(5.61)

5.4 QUANTUM METHODS

217

turns out to be n s2 'r = m p, oHs

1 -

A N 2 M2s .

(5.62)

The torque on the cantilever is therefore proportional to the external field H s as far as this is much lower than the anisotropy field A N d M s . The graph in Fig. 5.31 shows that the cantilever deflection follows a quite linear dependence on the amplitude of Hs over many decades, with a maximum resolution around 10 nT. Both AC and DC fields can be measured, but in the latter case the average between two measurements with m flipped through 180~is made to eliminate the drift in the optical signal. Rossel et al. report a similar detecting sensitivity and linear response up to 10 mT for a Si torsion cantilever using capacity transducers and large moment N d Fe-B sensing element (m--- 10-3A m 2) [5.83]. Eyre et al. have demonstrated a micromechanical field sensor dispensing with magnets and solely exploiting the Lorentz force exerted by the field on a microsized current-bearing loop [5.84]. An AC current is circulated in a single-turn microcoil (---1 mm x 0.5 mm), obtained by the aluminum metallization, on a SiO2 platform suspended on an etched cavity. The rectangular coil hinges on two SiO2 torsion beams connected to the substrate at mid-point on the long side. The torque on the beams ensuing from the application of the field is sensed by two polysilicon piezoresistors deposited on the beams. By running the sensor at the resonant mechanical frequency, the sensitivity is enhanced by the Q factor.

5.4 Q U A N T U M

METHODS

Electron and nuclear magnetic moments in a substance subjected to a magnetic field are characterized by a natural frequency of precession and they can be attuned to it by investing them with a suitable radiofrequency (r.f.) or microwave field. This is the phenomenon of magnetic resonance, upon which powerful experimental methods on the investigation of the structure of the matter at atomic level have been developed. It is possible to determine with high precision the strength of a magnetic field when it is applied to a simple and exactly defined system by simply determining the resonance frequency. In this way, we have direct access to the intrinsic properties of the elementary magnetic moments via the gyromagnetic ratio, a fundamental physical constant, and an absolute determination of the magnetic field is consequently possible. We shall introduce here the resonance methods that are most

218

CHAPTER 5 Measurement of Magnetic Fields

frequently applied in the precise measurement of DC or low-frequency (e.g. with slew rate around 1%/s or lower) magnetic fields. NMR magnetometers are the most widely used kind of resonance-based magnetometers and many commercial setups are nowadays available. The conventional water-probe devices are normally limited to fields higher than about 50 mT. To cover lower field strengths, several solutions exist: free-induction decay, flowing water NMR, optical pumping, electron spin resonance (ESR), and Overhauser effect. They' will all shortly be illustrated in the following. Quantum effects are also directly exploited in the superconducting quantum interference devices (SQUIDs), by which the highest possible measuring sensitivity, that of the flux quantum ~0 = h/2e = 2.07 x 10 -15 Wb, where h is the Planck's constant and e is the electron charge, can be reached. ~0 approximately corresponds to the flux of the earth field linked to a loop of 7 ~m diameter. A SQUID magnetometer can be viewed as a quantum flux counter, providing results in terms of fundamental physical constants.

5.4.1 Physical principles of NMR Nuclei are endowed with spin and, being the seat of a magnetic moment, they can display a paramagnetic susceptibility. The nuclear magnetic moments are, however, much smaller than atomic moments. The nuclear magneton of hydrogen is, for example, eh /~N -- 2mp

(5.63)

where h = h/2~r./-~Nis smaller than the Bohr magneton by a factor 1836, equal to the ratio between the electron and proton masses and the nuclear paramagnetic susceptibility, as provided by the Langevin function, is correspondingly affected. For all its smallness, the nuclear magnetism can be revealed and exploited, either as a tool for investigating the structure of the matter and of living organisms or for the accurate measurement of an external field, if the ensemble of nuclear spins is brought to its natural resonance frequency (the Larmor frequency), taking advantage of the ensuing selective absorption of energy. The essential features of NMR can be grasped by considering the two-level system description of an ensemble of nuclear moments with quantum number I = 1/2 in thermal equilibrium under an external field H0. Because of the interaction of the magnetic moments with the field, the spins distribute, according to Boltzmann statistics, between the two

5.4 QUANTUM METHODS

219

levels, separated by the energy gap

AE = 2gtzNIlzoH o = h?/z0H 0,

(5.64)

where the g factor is g = 5.586 and ? is the nuclear gyromagnetic ratio (see Fig. 5.32a). For the bare proton it is ~/= 2.6752221 x 108 T -1 s -1 (see Appendix B). The spin populations N~- and N O in the two levels are, at temperature T, in the ratio No~N-~ = e x p ( - h E / k T ) , with k the Boltzmann constant, and the system correspondingly exhibits a net magnetization along the field direction M0 = XoHo 9At room temperature, the proton susceptibility X0 is about 4 x 10 -9 and is overcome by electronic diamagnetism and paramagnetism, if this is present. The establishment of the equilibrium magnetization after application of H0 requires a time T1, called the "spin-lattice relaxation time" or "longitudinal relaxation time" (since it applies to the conventionally defined longitudinal magnetization component Mz). In other words, the steady-state population difference n o between up and d o w n spins is attained according to an exponential rise to equilibrium n = n0(1 - exp(-t/T1)).

(5.65)

T 1 is related to the microscopic interaction mechanisms of the spins with the lattice of the solid or liquid of which they make part. It is dependent on temperature and can vary between very wide limits. In pure water at room temperature, it is T 1 --- 3.5 s, while in ice at 80 K it becomes T 1 --104 s. T1 can be strongly reduced by even minute concentrations of

TZ m i = -1/2

I = 1/2

= hT#o H o

I Ho (a)

I"

mi= 1/2

% = raoG[

(b)

FIGURE 5.32 (a) Energy level splitting of the nuclear magnetic moment I = 1/2 in a proton subjected to an external field H0. Notice that the proton magnetic and angular moments point in the same direction. (b) Precession of the magnetization vector Mo around the direction of H0 (weak rotating radiofrequency field H1L).

220

CHAPTER 5 Measurement of Magnetic Fields

paramagnetic ions, which, with their high magnetic moment, favor the heat contact between the nuclear spins and their surroundings. By choosing, for example, the appropriate concentration of a paramagnetic salt (e.g. GdC13, CuSO4, FeNO3) in water, one can tailor the relaxation time to specific NMR field measuring experiments. For example, T1 in water is reduced to around I ms in a saturated solution of CuSO4 [5.85]. Let us now suppose that an alternating field H1 is applied, whose frequency exactly matches the resonance condition h~o0 -- AE. It is known from the timedependent perturbation theory that the transitions between the two states induced by this field have the same probability in unit time p in both directions. In the presence of a higher spin population in the lower energy level, as brought about by the static field H0, there will be a prevalence of transitions from the lower to the higher state, tending to re-establish equal populations. Energy will be absorbed in this process, which is evidently contrasted, over the relaxation time T1, by the effect of H0. The rate equation for the population difference n turns then out to be [5.86] dn d--t = - 2 p n +

no - n T~

'

(5.66)

which has the steady-state solution n - n0/(1 + 2pT1). As long as 2pT1 << 1, n ~ no and the thermal equilibrium populations are little affected by the energy injected by the alternating field H1. A continuous process of energy absorption dE

p

dt - nPh~~176 = noh~o I + 2pT~ '

(5.67)

which increases with increasing the amplitude of/-/1, occurs. This is the condition generally met when using NMR for the precise measurement of magnetic fields. If, on the contrary, p and T1 are so large that 2pT1 >> 1, the absorbed power becomes independent of/-/1 and saturates, a condition generally avoided in such a measurement. In the theoretical treatment of NMR, we are basically interested in finding out the evolution of the nuclear magnetization M with time. In quantum mechanical terms, this amounts to obtaining the equation for the expectation value of M. This does not require solving the appropriate time-dependent Schr6dinger equation because the expectation value follows in its time dependence the classical equations of motion. We can therefore write, for the time dependence of M in the absence of relaxation effects, the classical gyroscopic equation dM dt - 7M x/z0H0.

(5.68)

5.4 QUANTUM METHODS

221

This equation describes the free precession of the magnetization M around the longitudinally directed field H0 (see Fig. 5.32b). Equation (5.68), written for the three components as dM~9 dt - )'MY/z~176

dMy _ - ~/Mx/z0H0, dt

d]VIz - 0 dt

(5.69)

provides Mx = m cos(a~ot) ,

My = - m sin(oJot) ,

Mz = Mo,

(5.70)

as expected for a vector M precessing at the Larmor frequency ~o0 = ~'/z0H0. Free precession of M is not self-sustaining in real systems because the moments are subjected, besides the field H0, to internal random fields, which introduce randomizing effects in the precession frequency of each magnetic moment. Such fields are assumed to derive from the interaction by neighboring nuclear spins, but also the spin-lattice interaction, associated with the longitudinal decay, is expected to contribute to moment de-phasing. Whatever the mechanism, it results in a transverse relaxation time T2, with Mx and My eventually decaying to zero. In nonviscous liquids it is generally found T1 ~ 1.5T2 [5.85]. As discussed at the start, steady-state resonance is predicted to occur through the absorption of the energy provided by an alternating field H1, matching the frequency condition o)0 = AE/h. Restated in classical terms, this amounts to saying that, under these conditions, a persistent component of the magnetization vector M (or, equivalently, of its expectation value) is achieved, which rotates at the frequency ~o0. This comes out from the previous gyroscopic Eq. (5.69) by rewriting it through the introduction of the total field H -H0 + H1 and of additional terms phenomenologically accounting for the rate of variation of Mx and My due to transverse relaxation and that of Mz due to longitudinal relaxation. We obtain the "Bloch equations" [5.87] dMx dt = ~//z0(MyHz - MzHy)

Mx T2

dMy = ~/Izo(MzHx - MxHz) dt

My T2

(5.71)

dMz (Mo - Mz) dt - ~/tzo(Mxt-Iy - MyH,.) + T1 ' where the magnetization rates associated with the transverse and longitudinal relaxations are described by the terms (Mx/T2,My/T2) and (Mo - Mz)/T1, respectively. It is expected that, being the rate of exchange of energy between an AC field I-Ii(t) and the precessing magnetization

222

CHAPTER 5 Measurement of Magnetic Fields

M(t) given by the product izoHl(t).dM(t)/dt , this field should be mostly effective when applied in a plane normal to H0. This is what is done in experiments, where the field H1 (t) = 2H1 cos(cot) can actually be thought of as the superposition of two circularly polarized fields HiL(t) and H1R(t) of same peak amplitude H1, counter-rotating in the same plane. This is a natural way to demonstrate the normal modes of the process. Only the field HIL(t), rotating in the same sense as the precession (see Fig. 5.32b), with components

Hx(t) = H1 cos(~ot),

Hy(t) = - H 1 sin(oJt),

(5.72)

is effective and, at least around the resonance frequency, the other rotating component can be disregarded. In Eq. (5.72), o~ has been conventionally taken as positive for H1L(t). A relatively simple way to find solutions to the Bloch equations is to move to a novel frame of reference (x~,y', z'), with z ' = z, rotating about the z-axis at the frequency a~ In this frame, the circular field of amplitude H1 is fixed. If we make the further assumption of weak circular r.f. field (H1 ~ H0), low enough to avoid saturation (so that we can pose Mz ~" Mo, Mx,My K<Mo and, in particular, (T/z0H1) 2 ~ (1/T1)(1/T2)), the following solutions of the Bloch equations in the rotating frame are obtained [5.88] (~o0 - co)T2 M,~ = X0co0T2 1 q- (a~0 - o~)2T22H1, 1 M~ = Xoo~oT21 + (oJ0 - oj)2T22H1

(5.73)

M~, = M0, with, as previously stressed, X0 - Mo/Ho and ~o0 the resonance frequency. These solutions show that M~ and My are fixed in the rotating frame because H1 is fixed and so is the transverse magnetization component m, given by their composition. In the laboratory frame, m, having constant modulus m = ~M2~ + M~ = ~/1

X~176176 H1, + (~0 o))2T2

(5.74)

rotates at the frequency o4 which suggests that it can be observed, as shown later, by detecting the electromotive force induced in a surrounding coil. We see in Eq. (5.74) that at resonance, where ~o = ~o0, m attains the maximum value m--Xo~oT2H1. Such a value is proportional to the transverse relaxation time T2, which is natural because this quantity provides a measure of the degree of precession coherence of the individual moments. The vector m lags behind the rotating field H1L(t) by the angle ~, which is found by

5.4 QUANTUM METHODS

223

relating the two components of m Mx = m cos(cot - ~),

My = - m sin(cot - ~o)

(5.75)

to the rotating-frame counterparts M.,~ and My, in Eq. (5.73). We find [5.89] cos ~ =

(coo- co)T2 ~1 + (coo -

,

sin ~0=

1

.

(5.76)

~1 4- (coo - co)2T22

co) 2 T 2

If we take the in-phase and out-of-phase components (m cos ~p and m sin ~p, respectively), we can correspondingly define the real and imaginary susceptibilities ~ and &Y~ = m cos ~ = Xo cooT2 (coo - co)T2 1 4- (co0 - 09) 2 T~' 2H1 2 (5.77) &/, = m sin r = X0 co0T2 1 2H1 2 1 4- (co0 - co)2T2" )~ is an odd function of (coo - co) while ~ achieves a m a x i m u m value ~ -x0 2 cooT2 for co = coo as illustrated in Fig. 5.33. ~ and ~ represent the dispersion and absorption parts of the susceptibility. They are often associated with the term "Lorentzian lines". At resonance, only the out-of-phase susceptibility survives and m lags behind the rotating field by the angle ~ = 7r/2. We can write ,f'(~oo) = xo ~o 2 Aco'

(5.78)

where Aco = 1/T2 is the half-width of the absorption line and co0/Aco = Q,, is the related quality factor. From Eq. (5.74), we obtain, recalling that coo-7/z0H0 = 7/z0M0/X0 coo

m(coo) = XoH1 ~

= 2AJ'(co0)H1 = ~/MzT2tzoH1,

(5.79)

having posed Mz ~ M0. ~ is directly related to the power P absorbed per unit volume by the system of nuclear spins subjected to the r.f. field Hi(t) = 2H1 cos(cot) in the presence of the static field H0. In fact, the instantaneous power P, equal to the average power because of the stationary conditions, can be expressed as P =/zoH1L.dm/dt. By the use of the previous Eqs. (5.72)(5.77), we obtain P(co) = 2~~

= P'~176176 1 + (coo1- co)2T2 coH~"

(5.80)

The resonance condition can then be recognized either by detecting the m a x i m u m of the voltage induced in a detecting coil by the rotating transverse

224

CHAPTER 5

Measurement of Magnetic Fields

1.0

0.5

%

0.0 Aco = I l T 2 "--0.5

"

-1.0 '

'~'I

-4

'

'

'

.

I

-2

'

.

.

.

.

.

0

(r162

'

9

I

2

9

,

9

,

I

4

'

'

T2

FIGURE 5.33 Normalized absorption ~! and dispersion ~ susceptibility components, obtained by dividing Eqs. (5.77) by the quantity (Xo/2)~oT2, as a function of (oJ0 -co)T2.~0 is the resonance frequency, X0 is the nuclear paramagnetic susceptibility, and T2 is the transverse relaxation time. The absorbed power is, according to Eq. (5.80), proportional to ~/I.

magnetization m or the maximum value of the absorbed energy per unit volume E(~o) = ;ff P(~o). We have E(oJ0) = 2 ,rlzoXo oo T2H2 .

(5.81)

Once the resonance frequency to0 = h/AE is measured, an operation that can be done with great accuracy, the field strength B0 =/z0H0 is obtained, according to Eq. (5.64), as B0 = Wo. Y

(5.82)

The measurement of the magnetic field is then reduced to the measurement of the resonant frequency. NMR thus provides a means for the absolute determination of a magnetic field with the highest accuracy. If we use pure water probes, the value of the proton gyromagnetic ratio appearing in Eq. (5.82) is slightly different from the value given above for the bare proton because there is a small diamagnetic shielding effect by the enveloping electron cloud and the effective field acting on the nuclei is slightly smaller than the external field. One talks of a "shielded proton gyromagnetic ratio", whose recommended value is (for a spherical container)

5.4 QUANTUM METHODS

225

3/p = 2.67515341 x 108 T -1 s -1 (relative uncertainty 4.2 x 10 -8) [5.90]. The eventually achieved field measuring uncertainty is around 2 - 5 x 10 -6 [5.3]. With a field B0 = 1 T, we have a Larmor frequency f0 = to0/2vr = 42.57639 MHz. For fields higher than a few Tesla, nuclei with a lower value of ~/may be preferred in order to decrease the working frequencies. With deuterons we get, for instance, 3i= 4.10605 x 107 T -1 s -1 and for B0 = 1 T the resonance frequency becomes f0 = 6.53498 MHz. The NMR magnetometers often operate under a weak r.f. field H1, that is, far from saturation. On increasing H1, the condition can in fact be approached where, as remarked in Eq. (5.67), the transition rate from the lower to the higher spin energy level is large enough to overcome the rate at which the energy is relaxed to the lattice (time T1), so that the nuclear populations in the two states tend to equalize. Consequently, the magnetization Mz, associated with the population unbalance created by the field H0, decreases and eventually becomes zero at saturation. This is the result provided by the Bloch equations (5.71) when the condition (T/z0H1)2 ~ (1/T1)(1/T2) is not satisfied (it is understood here and in the following that the notation ~/stands for the effective gyromagnetic ratio). It turns out in this case that, at resonance,

Mz(tOo) =

Mo 1 + (T/zoH1)2T1T2, (5.83)

XooJoT2 1 ~' ( o~o) = 2 1 + (~,/z0H1)2T1 T2 and m(oJ0)= 2X'(w0)H1 is consequently affected. While Mz(oJ0) is a monotonically decreasing function of H1, m(o~0) attains a maximum value [m(w0)]max -- Mo~/T2/T 1

(5.84)

for Hi = 1/')'lZo~/TiT2, to eventually vanish at high H 1 values, together with Mz(oJ0) (Fig. 5.34). On the other hand, the absorbed energy E(oo) = 4~'/Zo~'(wo)H~ = 2rrlZoXoooH~

T2 1 + (-y/zoH1)2T1T2

(5.85)

saturates for H1 >> 1/'yp, o~/TIT2. The resonance conditions can be searched for in experiments either by keeping the field H0 fixed and sweeping the frequency till passing through r 0 = ~//z0H0 = 3'B0 or by changing the field at a fixed frequency. If we associate any field value H with a frequency o~ - ~B = wz0H, we can

226

CHAPTER 5 Measurement of Magnetic Fields 1.0 0.8 0.6 0.4 0.2 0.0

!

,

,

,

o

~

.

.

.

.

.

i.

2 7~toH1 ( T1 T2) 1/2

FIGURE 5.34 The longitudinal magnetization Mz/Mo, with M0 = XoHo the thermal equilibrium magnetization, decreases when increasing the strength of the r.f. field H1. At the same time, the transverse component m(roo) passes through a maximum value for H1 = 1/'ytzox/-T~lT2. express, based on Eq. (5.80), the specific energy absorption line as

E(B) = 4~q~oAY'(w)H2 = 2r

T2

1 + y2(B o - B)2T2

H2

"

(5.86)

Figure 5.35 shows the behavior of the reduced energy absorption line E(B)/E(Bo), together with its derivative E'(B/Bo). It appears here that, in order to make precise determination of the resonance field B0 for a given frequency co0, it is convenient to rely on E'(B/Bo), looking for its zero passage. This can be accomplished in practice, as shown in the figure, by sweeping the field B amplitude through the line and superposing to it a small modulating field Bm (smaller than the line width) at some suitably low frequency, by which the slope of the line E'(B/Bo) can be retrieved. Since B/Bo = offro0 and E(B)/E(Bo)= E(ra)/E(roo), we can, in a perfectly analogous way, apply the modulation to the fixed field B0 and find the zero passage of E'(w/ro o) by scanning the frequency of the r.f. field H1. An alternative technique consists in applying a sufficiently large field Bm and sweeping it back and forth through the resonance condition, which is then revealed by a signal at twice the modulation frequency.

5.4 QUANTUM METHODS

227

_

-x

o

LU

Il

I

/

I

!

I

t

E'

I I v,

0.8

'

1.'0

I

'

B/B o

1.2

FIGURE 5.35 Solid curve: reduced NMR energy absorption line E(B)/E(Bo), with B = oJ/~/. If the field B is made to sweep through the line while amplitude modulated by a low-frequency field smaller than the line half-width value, a signal proportional to the derivative E' is obtained (dashed line). The resonance condition is thus determined with precision in correspondence of the passage of E' through the zero value. An equivalent result is obtained by sweeping the energy absorption line E(~)/E(~o).

5.4.2 N M R magnetometers 5.4.2.1 Continuous-wave magnetometers. The early successful NMR field measurements by Bloch et al. hinged on the determination of the signal generated at resonance by the rotating transverse magnetization m (Eq. (5.74) in a water-filled spherical sample (volume 1.46 cm3)) [5.91]. Figure 5.36 schematically illustrates an NMR field measuring setup based on Bloch's method. The DC field under measurement B0 is amplitude modulated by a co-linear low-frequency field (e.g. f m - - 6 0 Hz) of peak amplitude Bin, provided by a pair of supplementary coils. Normal to Bm and B0, the alternating field Hi is applied using two transmitter coils. The measurement of B0 is performed by sweeping the frequency of H1, thereby scanning the absorption line and detecting the signal Ur induced by the rotating transverse magnetization vector m in a coil wound around the sample, perpendicular to both H1 and B0 (the y-axis in Fig. 5.36). Since the transmitting and receiving windings are placed exactly at right angles, only a very small fraction of the signal generated by the Hi windings can leak directly to the Ur winding. At resonance, the induced r.f. signal Ur is

228

CHAPTER 5 Measurement of Magnetic Fields

sweep generat~ I H~

]

Bo

' y

Bo ~

x

2

I FIGURE 5.36 Schematic view of an NMR magnetometer based on the nuclear induction method [5.91]. The water-filled sample (e.g. small cylinder or sphere) is immersed in the uniform DC field B0 under measurement, which is modulated at the low-frequencyfm by the co-linear field Bm, having sufficient peak amplitude to sweep back and forth through the resonance condition. The r.f. field H1 is applied in a direction normal to B0 and a signal ur, induced by the precessing magnetization, is detected along the third direction by means of a coil wound around the sample. The r.f. signal Ur, modulated at the frequency 2fm, is r.f. amplified, demodulated and audio-amplified. The resonance frequency ro0 = ~,B0 can then be obtained by search of optimum signal on the scope and direct measure of the generated r.f. frequency by means of a frequency counter.

modulated in amplitude at a frequency 2J:m because of the periodic passage of the total field B0 + Bm through the center of the resonance line. Usually, a shunt condenser is used to resonate the coil and to increase the signal-to-noise ratio by the related Q factor. Notice that the small residual leakage coupling of transmitter and receiver coils provides a reference phase for the signal. After r.f. amplification, demodulation and lowfrequency amplification, the resulting signal of frequency 2fm can be sent to the y-axis of a scope, while the signal of frequency fm from the sweep generator is fed into the x-axis. Very fine frequency tuning is achieved by looking for a precisely centered and symmetric pattern on the screen graticule. The B0 field value is then obtained, according to Eq. (5.82), by determining the frequency of the corresponding field H1 with a frequency counter, a measurement which can be done with great accuracy. One major source of error, with this and other NMR methods, resides in the inhomogeneity of the field B0 over the NMR probe region, which leads to line broadening and flattening. Being the typical NMR probe size

5.4 QUANTUM METHODS

229

of 5-10 mm, the experimentally tolerated field gradient is around some 10 -4 cm -1. The water sample is usually contained in a glass, alumina, or plexiglas holder and special precautions must be taken to ensure the greatest rigidity of the assembly in order to avoid fluctuating signal leakage from transmitter to receiver coils and related microphonic effects. Notice that, given the intrinsically isotropic nature of NMR, a certain tolerance exists regarding the exact orientation of the measuring head with respect to the direction of the field B0. Most NMR magnetometers are nowadays based on the determination of the energy absorption at resonance, a feat that can be accomplished with great sensitivity by incorporating the resonator, made of a single coil inductive head, and the tuning shunt capacitor, into a marginal oscillator circuit. This is an amplifier with positive feedback loop kept at a gain around unity, that is, on the verge of oscillation. Under these conditions, the oscillation level becomes a very sensitive function of the Q factor of the resonator and amplifies the relatively small dip of it ensuing from the absorption of the energy E(Bo)at resonance (Eq. (5.86)). In practice, a large change of the oscillator voltage is obtained with a small change of Q. By frequency scanning the resonance line E(~/Wo)with the r.f. field H1 (or the E(B/Bo) line with the field B) and by superposing to the measuring field B0, a low-frequency modulating field (fm < I kHz) of amplitude Bm smaller than the half-width of the absorption line, we obtain that the oscillator voltage is modulated in amplitude at the frequency fro, with depth following the derivative E~(~/~o).Figure 5.37 provides an example of a setup implementing the power absorption method in the NMR measurement [5.85, 5.92]. The resonator is attuned to the precession frequency w0 by means of a couple of varactors (voltage-driven variable capacitances), controlled by the signal provided by the frequency control circuit. In order to cover an adequate field (i.e. frequency) interval (---50 m T - 1 T in typical commercial setups), in general several coils with different numbers of turns are used. The marginal oscillator, which is endowed with JFET input stage and low-noise wide-band r.f. amplification, feeds back its output current via the conductance Gf. The voltage leaving the oscillator is demodulated, amplified, and fed into the phasesensitive detector. Thus, if the previous condition of small Bm amplitude is satisfied, phase-sensitive detection using the reference signal provided by the 1.f. generator (frequencyfm) permits one to run through the derivative of the absorption line E~(w/wo) while scanning it with the oscillator frequency. The zero passage of E~(w/~o) is detected by means of the resonance discriminator, whose output can be used to lock the oscillator frequency onto the resonance condition. Frequency readout by means of a counter provides, via Eq. (5.82), the value of the field B0.

230

CHAPTER 5 Measurement of Magnetic Fields

.. J f.back J _ . frequency FLI" ] circuit ~ counter

eo+ ~

r~

~~]dem~176

-- ~~,

~ ,,,~,, J frequencyJ ~ / t) CvI,-4-1 control I-q-ldiscri~natorl---~l~l 1/ ~ I circuit I L.. J i i

~~ 'I nIf J IphaseJ J ge eratorfml = I shifter I '

~"

PSD

i_.-~

FIGURE 5.37 Scheme of an NMR magnetometer based on the detection of the power absorption at resonance and the use of a marginal oscillator circuit (adapted from Refs. [5.85, 5.92]). The resonating circuit is tuned over the appropriate frequency range (-2-50 MHz in advanced commercial devices) using different detector coils and varactor diodes as tuning capacitors. The input stage of the r.f. amplifier in the marginal oscillator circuit employs a couple of JFETs. The derivative of the absorption line is obtained by field modulation at the frequencyfm < I kHz, demodulation of the oscillator voltage and phase-sensitive detection (PSD). NMR field measurements outside the 50 mT-1 T standard interval offer a number of problems that are dealt with by resorting to specific solutions. Measuring increasingly high fields implies correspondingly increasing frequencies and when these approach the 50-100 MHz range the transmission of the signal through conventional coaxial lines may be hampered because the transmission line becomes part of the tuning circuit. Classically, either deuteron, 7Li (f0--16.547 MHz/T), or 27A1 (f0---11.1119 MHz/T) samples in place of proton assemblies are employed in order to decrease the value of ~0- There are cases, however, where the combination of high measuring fields (B0 > 1 T) and large distances between the sample coil and the oscillator tank circuit cannot be avoided. This may happen, for example, in the large superconducting dipole magnets used in particle accelerators, where distances around a few meters may compound with fields larger than 10 T. In such cases, resonator tuning must be carried out by taking into account the length of the connecting cable and the characteristic impedance of the coaxial line [5.93]. Let the input admittance of the ensemble made of the NMR coil and the coaxial cable be, according

5.4 QUANTUM METHODS

231

to the theory of lossless transmission lines, Yi -

1 Z0Ys cos(/3g) 4- j sin(13g) Z0 cos(13g) + jZoYs sin(/3g) '

(5.87)

where Z0 is the characteristic impedance of the coaxial line, Ys the admittance of the NMR coil, g the cable length, and 13 the propagation constant. The NMR coil behaves in practice as a pure reactance and it is usually assumed Ys = 1/j0)Ls. Having assumed that the leakage conductance and the resistance per unit length of the line are negligible, it is 13 = wx/LcCc, with Lc and Cc the series inductance and the leakage capacitance per unit length of the line, respectively. For the same reason, Yi is an imaginary quantity. The resonance frequency is thus obtained by imposing

jYi = 0)0Cv

(5.88)

with Cv the variable tuning capacitance of the oscillator tank circuit. This condition has periodic solutions (oscillation modes). From Eq. (5.87), we obtain the input admittance Yi as a periodic function of 0) 1

Wi m_ .--~-cot(13e + 0), ]L0

(5.89)

where 0 = tan -1

1

jZoYs

- tan -1 ~0)Ls .

Zo

(5.90)

A line of length g terminated in an inductance Ls acts, as far as the input admittance Yi is concerned, like a short-circuited line of length g 4-d, with d -- 0//3. If s is equal to a half-wavelength A/2 (A = 2~rv~/0), with v,p = 1/x/LcCc. the velocity of the electromagnetic wave along the cable), the input admittance Yi is equal to the admittance of the NMR coil. In this case/3g = zr and from Eqs. (5.89) and (5.90) we obtain Yi -- Ys -- 1/jwLs. Everything goes as if no connecting line were present. For a resonant frequency f0 = 50 MHz and v, -- 2 x 10 s m / s we have A/2 = 2 m. In practice, real lines suffer signal attenuation due to losses. Consequently, they are used as short as permitted by the structure of the measuring environment, which requires tuning of the capacitor Cv with the admittance Yi ~ Ys. The tuning condition (5.88) has solutions as long as Yi - 0. For a given length of the coaxial line and sample inductance, the fundamental mode can then be established only up to a maximum frequency O ) o l , provided by the condition Yi -- 0 (i.e. cot(/?~ + 0) - 0) in Eq. (5.89). This amounts to pose ~oolL~L~cC~g,+ 0 = zr/2 or, equivalently, g + d = A/4. To make a numerical example, we calculate the limiting

232

CHAPTER 5 Measurement of Magnetic Fields

frequency value fol when a conventional NMR probe of inductance Ls = 0.2 ~H is connected to the tuning tank with a 1.5 m long coaxial cable of characteristic impedance Z0 = 50 f~. Since 1/x/LcCc -- v~, we find through Eq. (5.90)fol ~ 23 MHz. With a proton probe, this corresponds, using the fundamental mode, to a field of the order of 0.5 T, rising to about 2 T if a 27A1 probe is used. It has been suggested to overcome this limitation by suppressing the fundamental mode and exciting the upper modes, exploiting the function of a suppressing network inserted between the tuning capacitor and the connecting line [5.93]. The NMR measurements at field levels below some 100 mT are naturally associated with sensitivity problems because of the interference by the background fluctuating fields and the decrease of the frequency resolution. The r.f. field H1 has correspondingly reduced strengths. We can see how the signal decreases with H1, if the weak r.f. field condition H1 << H0 is respected, from Eq. (5.79) (nuclear induction method) and Eq. (5.81) (energy absorption method). Actually, the condition of maximum transverse magnetization re(co0) (Fig. 5.34) may be pursued by imposing the r.f. field amplitude H I ~ 1/~//.~0r

2 (with T1 ~ 1.5T2). This relation-

ship provides the condition on the transverse relaxation time T2 for maximum energy absorption, as can be verified by taking the derivative dE(~o)/dT2, with E(wo) provided by Eq. (5.85). For a given measuring field B0, a reasonable absorption linewidth AB ~-2/7T2, ensuring a sufficiently accurate determination of the resonance frequency, is around 10-3B0 . This means, as an example, AB ~- 2 p~T for B0 = 1 mT, that is T2 4 ms. To obtain this value for T2, a water probe with suitable addition of a paramagnetic salt can be used. The proton precession line can be slightly shifted by the presence of the additives and the related magnetic moments. For example, a relative shift around 10 -6 is introduced by addition of CuSO4 with a concentration 0.5 g / d i n 3 [5.94]. In the field measurement method based on the energy absorption and the use of the marginal oscillator (Fig. 5.37), the change at resonance of the conductance Gs of the circuit made of sample and tuning capacitors is detected. For all its sensitivity, this method is in general limited to the measurement of fields larger than about 50 mT because too small a change AGs is otherwise obtained. Weyand has, however, achieved a dramatic sensitivity improvement by using a marginal oscillator where the NMR signal due to energy absorption is enlarged by making the feedback conductance Gf depend directly on the modulated value of the conductance Gs [5.85]. In this way, the lower limit for the measuring field has been brought down to -~ 0.5 mT. Figure 5.38, adapted from Ref. [5.85], provides an example of correspondingly obtained differentiated absorption line

b.4 QU AN 1 UM ME 1HODS

233

Bo= 15mT

1

DV

FIGURE 5.38 Differentiated proton absorption line U(B/Bo) obtained under a measuring field of 15 mT by means of a high sensitivity NMR magnetometer using a marginal oscillator with modulation of the feedback conductance. The probe sample, having volume around 2 cm 3, is filled with a CuSO4 aqueous solution (after Weyand [5.85]).

U(B/Bo) at a field B0 = 15 mT. In this case, a cylindrical sample of diameter 12 m m and height 18 m m (volume about 2 cm 3) filled with a CuSO4 0.5 g / d m 3 aqueous solution is used. The detector coil is made of 120 turns and the modulating field Bm -- 1 p~T is applied at the frequency fm = 43 Hz. The typical size of the measuring NMR head, which is around 1 - 2 cm, m a y be shown to be inconvenient in certain applications, for example those pertaining to NMR spectroscopy and imaging. Efforts have been made in recent times to overcome this limitation by the realization of miniature probes. Boero et al. have presented an integrated device based on CMOS technology, which makes use of an amorphous solid polymer sensing head (cis-polyisoprene) of volume around I rnm 3, by which the range 0.7-7 T is covered with somewhat reduced accuracy with respect to the conventional water probes [5.95]. 5.4.2.2 Free-precession and pulse magnetometers.

Very low strength DC fields, of the order of the earth magnetic field and smaller, can still be measured with NMR, but the continuous wave methods discussed so far must be abandoned in favor of other methods. The classical approach to precise measurements of fields lower than about I mT consists in observing the nuclear induction signal generated by the free precession of

234

CHAPTER 5 Measurement of Magnetic Fields

the proton moments. As experimentally verified for the first time by Packard and Varian [5.96], a transient condition can be achieved where the coherent precession of the nuclear magnetic moments around the measuring field B0 is established without applying any r.f. field. In the experiment, first a polarizing field Bp much larger than B0 and orthogonal to it is applied for a time t >> T1 so that an equilibrium magnetization M = XoHp (Hp -- Bp//.~0) is obtained. Then Bp is abruptly switched off. If the switching time is sufficiently short, M remains unchanged and a state is constructed where it starts precessing around B0, the only remaining macroscopic field, at the frequency f0 = ~Bo/2~'. M is much larger than M0, the equilibrium magnetization pertaining to the field B0, and the expected NMR signal strength is accordingly larger than the one achievable with the continuous wave resonance method, with the additional benefit of increased measuring sensitivity because there is no disturbance due to the exciting signal. The condition on the switching time can be defined more precisely with an example. If B0 is the earth magnetic field, that is B0 "" 50 ~T, an orthogonal polarizing field Bp -10 mT is appropriate. The resultant field is basically coincident with Bp and it also remains so after the sudden decrease of Bp down to a value B!p~ a few times larger than B0 (say around 250 ~T). To preserve the magnetization value M, it is then required that the time interval At1 needed in order to pass from Bp to Bp be much lower than the spin-lattice ! relaxation time T1. The further decrease from Bp to zero must instead occur in a time At2 KK1~fo, if f0 is the precession frequency, so that M has no time meanwhile to re-orient in a field different from B0. For the specific case here considered, f0 ~ 2130 Hz (pure water sample), so that At2 KK50 ~s. The sensor generally has a considerable inductance and the energy stored in it is equally significant. It is notable that most of it can be dissipated along a reasonably long time At1. A schematic view of a low-field measuring setup exploiting nuclear free induction decay is provided in Fig. 5.39. In this circuit, the same winding is used both to apply the polarizing field and detect, after switch-off, the nuclear induction signal eN. The sensor can have cylindrical, spherical, or toroidal shape. The latter has the advantage of being omni-directional because, whatever the direction of B0, there is always a portion of the winding perpendicular to the measuring field [5.97]. With the other probes, we instead have a dead cone of orientations of B0 around the coil axis, where the perpendicular component of B0 is too small to be detected and the signal drowns into the noise. The electronic switching circuit provides the necessarily fast transition between the polarizing and the measuring configurations. If an N-turn solenoid of length s is used on a sample of volume V, the peak value reached by the induced

235

5.4 QUANTUM METHODS

~ ~

_~

frequency counter

switching electronics

Bo DCcurrent source

v

~

~

PC

(a)

00 r

r'~

I C~ co0

rv Z (b)

0

1

2 Time (s)

3

FIGURE 5.39 (a) Measurement of magnetic field B0 by free induction decay. The water-filled sample is first subjected for a time t >> T1 to the polarizing field Bp, perpendicular to and much larger than Bo. Bp is then switched-off in a very short time and the nuclear magnetization M startsto precess around the direction of B0 at the frequency to0 = 3'B0. The signal correspondingly induced in the winding (normally the same winding used to polarize the sample, switched between the DC source and the tuned amplifying circuit) after switching decays with time constant determined by transverse relaxation and radiation damping. (b) Example of time decay of the NMR induced signal for Bp = 5 mT and B0 = 20 ~T. The oscillation period results from beating of the induced e.m.f, with a reference signal (from Ref. [5.94]).

signal immediately after switching is calculated from the F a r a d a y Maxwell law as

N2ip eNp = ~0X0c~ s

V~lk,

(5.91)

236

CHAPTER 5 Measurement of Magnetic Fields

where ip is the peak value of the polarizing current, ~/< 1 a factor taking into account the non-homogeneity of the field over the sample volume due to the finite length of the solenoid, and k the volume fraction of the solenoid occupied by the water [5.98]. Some other proton-rich fluid, like kerosene or ethanol, can be used in place of water. In this case, a small correction to the value of 3' for protons should be applied in order to account for the so-called "chemical shift" (equivalent in these cases to about I nT in the earth's magnetic field). In fact, the magnetic resonance of the proton may occur at different frequencies in different compounds because the specific chemical environment can affect the diamagnetic shielding. Incidentally, it is noticed that chemical shift, making it possible to distinguish between different molecular environments, is a precious tool of biochemical research. The value of ep is very small and it can reach, in accurately designed probes, the microvolt range, with a signal-to-noise ratio around a few hundred (for B0 "" 50 ~T). On the other hand, there are obvious limitations to the values of N, ip~ and V. The number of turns and current are limited by the available power, a critical point in portable instruments, and a large probe volume may be associated with field inhomogeneity, which causes line broadening and decrease of the s p i n - s p i n relaxation time T2. In a typical setup one can find V = 50-300 cm3~ N = 1000-3000, ip = 1 - 2 A. A sufficiently long decay time ~'F is actually needed for the accurate measurement of the precession frequency. Under transient phenomena, rF is related to the intrinsic spin relaxation mechanisms and the damping of magnetic resonance brought about by the tuned electrical circuit. The latter effect, called radiation damping, is associated with the dissipation by Joule effect of the energy provided to the circuit by the precessing spins, the sole source of energy during the transient. According to Bloombergen and Pound [5.99], the time constant for radiation damping is given by ~R = 2/%~oBpQ~,where Q is the quality factor of the coil when connected to the amplifier input. It turns out that, under certain circumstances, ~'R is comparable with T2 and the resultant decay time ~'F =

~RT2 ~R+T2

(5.92)

can be reduced with respect to T2. For instance, with Bp = 10 mT, Q = 50, and 77= 0.7, we obtain ~R = 5.2 S. With water samples, where T2 = 2.4 s, this leads to ~'F -- 1.64 s. Free precession proton magnetometers find relevant applications in geophysical and environmental surveying thanks to their combination

5.4 QUANTUM METHODS

237

of accuracy, sensitivity to low fields, and the absolute character of their measurement. Commercial portable setups are available, whose typical specifications provide a measuring range 20-120 ~T and a resolution of InT. With laboratory instruments, developed and used in a tight metrological environment, a range 10 ~T-2 mT is covered with relative measuring uncertainty varying between 10 -4 and 10 -6 [5.94]. Note that, in detecting local perturbations of the earth magnetic field, a gradiometer configuration is often adopted, where two identical sensors are placed a distance apart and the difference of the local fields is read as a difference in the frequency readings. A transient resonant state useful for the purpose of field measurement can also be obtained by applying the r.f. field as a single pulse of convenient duration. Strict metrological applications are a somewhat minor subject in the vast and fertile area of pulsed NMR, which has led to outstanding progress in materials science, chemistry, biology and medicine, the latter field having benefited enormously, for example, from the development of the magnetic resonance imaging techniques. The method of field measurement by pulsed NMR consists, in principle, in applying to the sample probe, immersed in the field B0, a r.f. pulse of amplitude 2H1 and convenient duration t I along a direction perpendicular to B0 instead of the steady-state r.f. field applied in the conventional continuous-wave method. If, starting from a condition of equilibrium where the system is endowed with the magnetization M0--XoHo = xoBo/~o directed along the z-axis, the r.f. pulse of frequency equal to the resonance frequency f0 is applied for a time t 1 -- (~/2)(1/~/p, oH1), the vector M 0 is tipped down into the x - y plane, describing a 90 ~ angle (Fig. 5.40). This can be understood if, as previously discussed for the solutions (5.73) of the Bloch equations, we move to the frame (x/,y~,z ~) rotating with the angular velocity w0. It can be shown that, in this frame, the field B0 -- 0 and only the field HIL ~fixed in the direction of the x~-axis, remains [5.88]. It then turns out that at the time t -- 0 the magnetization M0 finds, in this frame, the field HIL only and it starts precessing around it at the angular velocity wl -- ~//~0H1. After the time interval tl has elapsed, M0 lies along the yCaxis and there it remains if the r.f. field is switched-off at that instant of time. Going back to the laboratory frame, we eventually find the vector M0 precessing around B0 at the frequency f0, after having suffered transient nutation along the time tl. The signal correspondingly induced in the x- (or y-) directed receiver coil, which can be the same winding used for launching the r.f. pulse, is initially proportional to M0, that is, far higher than the signal induced with the conventional continuous-wave weak r.f. field method, where it is proportional to the transverse magnetization m. A time decay will be observed with

238

CHAPTER 5 Measurement of Magnetic Fields

I Z'_----Z I JL..

I ~,

Ho

y'

~o

..........

....

',",";'; H1 L ~"....t~,~

..-'"" (a)

Y.~,

x'

""

(b)

d

Mo - ~ tlN--(c)

FIGURE 5.40 Pulsed NMR. (a) In the frame (x~,b/, z'), rotating at resonance with angular velocity to0 -- ~//~0H0,only the rotating r.f. field of amplitude H1 is left. If H1 is applied as an r.f. pulse of duration tl = (fr/2)(1/'),~H1), the magnetization M0 is tipped down in the plane x'-y j (90~ pulse). (b) In the laboratory frame (x, y, z) M0 is observed precessing around H0. The rotating spins fan out because of transverse relaxation and the signal induced in a receiver coil placed along the x- or y-axis decays with time. If longitudinal relaxation is appreciably involved (time T1), the length of the vectors precessing in the plane x - y is progressively shortened. (c) r.f. field pulse of time duration tl and decay of the nuclear induction signal. the progressive fanning out of the rotating spins (transverse relaxation) and recovery of the longitudinal equilibrium magnetization (time T1). Pulsed NMR magnetometers have been developed, which are based on the determination of the frequency of the time-decaying free induction signal. Their notable advantage with respect to the continuous-wave magnetometers is that they do not require field modulation and the related supplementary windings, which are sometimes incompatible with specific measuring configurations. This is the case, for instance, with precise field mapping in the superconducting magnets employed in particle accelerators, where simultaneous NMR decay frequency readings are made on a large number of probes suitably located at different points of the beam pipe [5.100]. These probes satisfy the demanding requirements on spatial resolution and physical restrictions on the probe volume. Since the circuit for pulse generation and signal analysis is normally connected to the probe by a long coaxial cable, the optimum conditions for signal propagation should be satisfied [5.101]. This means, in particular, providing for a cable length s equal to a multiple of the half-wavelength ~./2 = v~/2fo. For a field B0 = 2 T, we have that a length s = 2.34 m is equal to the full wavelength A. The basic drawback of the NMR pulse methods is that a complex coupling scheme between transmitter, probe, and receiver must be realized. In fact, during the time interval where

5.4 QUANTUM METHODS

239

the strong r.f. pulse is applied to the probe, the receiver must be protected from overload (ringing), while it is required that, in a short time after the end of the pulse, the energy conveyed by the transmitted pulse is dissipated and the receiver starts amplifying the small time-decaying nuclear resonance signal. To this end, several coupling schemes have been developed in the literature, which aim at damping the probe for a convenient time interval. This basically implies a decrease of the Q factor of the tuning circuit, which must possibly be obtained with little deterioration of the signal-to-noise ratio [5.102].

5.4.2.3 Flowing-water magnetometers. The measurement of low fields by means of the free-induction magnetometer has a basic limitation in its non-continuous nature, which prevents its application in the active control of magnetic fields besides requiring an often inconveniently big sensing head. It is, however, possible to retain the basic principle of the free-precession method, which is one of forcing a large out-of-equilibrium magnetization in the sensing sample and make it resonate in the low measuring field, while maintaining a continuouswave approach. This is accomplished with the flowing-water NMR method, where the operation of polarization, r.f. excitation, and signal detection are performed over spatially separated regions while maintaining sufficiently short time intervals between subsequent measuring steps to avoid important signal loss due to the relaxation mechanisms. Originally developed by Sherman [5.103], following an idea of E. M. Purcell, in order to measure the magnetic field with high precision over an extended region in space, and further assessed by Pendlebury et al. [5.104], the flowing-water technique can, in principle, span a very large measuring field range, from a few ~T to several T. This is extremely appealing, both from the viewpoint of establishing flux density standards in the laboratory and of achieving a general-purpose, easy-to-use device of superior accuracy and stability. The working principle of the flowing-water NMR magnetometer can be understood by making reference to the setup developed by Kim et al., schematically shown in Fig. 5.41. With this circuit, excellent signal-to-noise ratio down to about a measuring field strength B0 ~ 100 p~T has been achieved [5.105]. The water is pumped at a rate of a few ten cm3/s through a baffled polarizing chamber, where it is subjected to a large field Hp, of the order of some hundred roT, and it spends a time ~-p generally larger than the longitudinal relaxation time T1 ('-"3.5 s in pure water). It thereby acquires a magnetization Mpo -- M0(1 - e -TP/T1 ) close to the equilibrium nuclear magnetization M0 = XoHp. The pipe brings it to the region subjected to the measuring field H0, to which the magnetization becomes

240

CHAPTER 5 Measurement of Magnetic Fields

|

,~rator

~mpl.

y,,

~

--z~

S PC

~x

Jfllllj L:~

C

pump

FIGURE 5.41 Flowing-water NMR magnetometer. The water is pumped at a rate of a few ten cm3/s through a polarizing chamber, inserted between the pole faces of a permanent magnet, where a field Hp is applied, and spends there time enough for the magnetization Mp to approach the equilibrium value. It then enters the region subjected to the measuring field Ho, where it receives first a transversely directed r.f. field pulse of amplitude 2H1 and duration tl, to eventually pass in the detecting region. Here the signal induced by the precessing magnetization vector in a sensing coil is collected. Such a signal is proportional to product HpHo (adapted from Ref. [5.105]). aligned, and, after a traveling time 71, the water is made to traverse a short region, where it is irradiated by a transversely directed oscillatory field H 1 (of peak magnitude 2/-/1). This corresponds to receiving a r.f. pulse, whose duration corresponds to the time of passage beneath the irradiating coils. The detecting region immediately follows, where a multiturn sensing coil with axis perpendicular to both H 1 and H0 is used to sense the precessing magnetization. Note that the directional change of the field from Hp to H 0 occurs over a sufficiently long time, much longer than the resonating period (of the order of 1/~//~0H0). The conditions are thus respected for the so-called "adiabatic variation" and the magnetization always sticks to the external field when it changes its orientation [5.106]. Because of longitudinal relaxation, the magnetization intensity, which lacks any transverse component, decays along the travel of the water from the polarizer to the sensing region. If 71 is the traveling time, we have that the magnetization arriving at the entrance of the exciting region is Mp = M0(1 - e -~'p/T1)e - r l / T ~ . Here, the effect of the r.f.

5.4 QUANTUM METHODS

241

pulse is one of producing non-adiabatic re-orientation of the magnetization vector Mp~ eventually leaving it precessing a r o u n d H0 with a canting angle 0 determined by the m a g n i t u d e of H 1 and the pulse duration t 1. There is a close analogy between this process and the classical molecular beam experiment, where nuclear and rotational magnetic m o m e n t s are m e a s u r e d by subjecting the traveling atoms to a perturbing localized transverse oscillating field. Beam defocusing is obtained in this case because of the ensuing non-adiabatic re-orientation of the magnetic m o m e n t s [5.107]. As previously remarked, the analysis of transient situations is simplified by m o v i n g to a frame (x~,y~, z ~) solidly rotating with the r.f. field H I L of m o d u l u s H I L = H1 (Fig. 5.42). If co is the angular velocity of H I L } o n e finds that in this new frame the field H0 is substituted by a field H 8 = H0 - ~o//~0~/[5.86], directed along z ~ = z, and, at the entrance of the r.f. tract, Mp starts precessing with angular velocity f~ around the effective field Her -- H 0 - oo//.~03/if- HIL. W e have evidently

AZ'-=Z I I

!i

A~ Z

TM "'~\

Ho [ ..............

,,'"

~'~

. [[ ~/'i~,'/"

a~~,,O?,~,,,,He,,r[[f]7"'-.,k I1__;"-. II M

"'n lL

(a)

Ho " ......................

:, . . . . -~,""

).,,'

. . . . _,~. X'

../,~"

Io

/'

H

1L

%"

~[~r~(~p'}

.. x'

, ' (b)

"

HIL

X

(c)

FIGURE 5.42 Precession of the magnetization Mp in flowing water, as occurring in the region irradiated by the r.f. coils. (a) In the frame (x~,yJ,z~), rotating with angular velocity to, precession occurs around the effective field H e r - H 0 oo/H,0~/q-H1L. (b) At resonance (to = r , the precession angle is 0* = Ir/2, the precessional angular velocity is fl = ~/H,0H1L and the angle cKtl) covered by Mp from t --- 0 (entrance) to t = tl (exit) is cKtl) -- W-l,0H1ctl. (C) If we come back to the laboratory frame (x, y, z), we find that the rotating transverse component Mpt -Mp sin cr either trails or leads the rotating field H1L by 90~according to whether cr < "a"or or(t1) > 7. A signal e(t) = b~ooM P sin a(tl) sin(oJ0t), with b a constant depending on coil geometry, is subsequently induced in a sensing coil perpendicular to both Ho and H1.

242

CHAPTER 5 Measurement of Magnetic Fields

-- ~//~0Her. Let us assume that at time t = 0 Mp is aligned with H0 (i.e. the z-axis) and that the fluid leaves the r.f. irradiated region after a time interval tl. With elementary geometrical considerations as in Fig. 5.42a we find that the angle a(t~) described by Mp from t = 0 to t = tl satisfies the equation cos

a(tl)= 1 - 2 sin20 * sin 2 f~tl 2 '

(5.93)

if 0 * is the precession angle around Her. It is evident from Fig. 5.42a that 0* = t a n - 1

H1 H0 - to//.t03"

(5.94)

Notably, the correspondence between classical variables and quantummechanical expectation values permits one to write Eq. (5.93) also as cos a ( t l ) = 1 - 2 p ( r o , t l ) , where p(to,tl) is the transition probability P1/2,-1/2 between the two spin q u a n t u m states with mz = + 1 / 2 [5.108]. At resonance, the angular velocity of H1L is ro0 = T/z0H0 and the field H~ = 0. The effective field reduces then to Her -- H1L and the situation in the rotating frame becomes the one shown in Fig. 5.42b, where at time t - 0 the magnetization vector Mp, directed along the z-axis, starts its precessional motion in the plane (z',b/) around HIL (0" = ~r/2, see Eq. (5.94)). Since the angular velocity of Mp in the rotating frame is f~ = 7/~0H1, the angle covered from entrance to exit of the r.f. region is

a(tl) = f~tl = 7/~0H1tl.

(5.95)

Thus, as an example, the time tl required to tip Mp by 90 ~ in an r.f. field of peak value 2H1 = 2 A / m is tl = 4 . 6 7 x 1 0 -3 s. Returning to the laboratory flame (Fig. 5.42c), we eventually find at the time t = tl the magnetization vector Mp precessing with angular velocity ~o0 around H 0. The longitudinal magnetization is now Mpz---Mp cos a(t 1) and the transverse component is Mpt = Mp sin c~(tl). If the transit time t I is regulated in such a way that cr < r Mpt trails the rotating field HIL by 90 ~ If 7r < a(tl) < 2~r, it leads HIL by 90 ~ On leaving the r.f. coils at t - tl, Mp retains its direction in the rotating frame because of the rapid non-adiabatic removal of H1. This means that it continues its precessional motion around the z-axis in the laboratory frame, subjected to longitudinal decay (T1) and dephasing (T2). It immediately enters the detection region (a 3 cm diameter spherical sample in the apparatus of Kim et al. [5.105]), where a signal is induced in a multiturn coil directed perpendicular to both H0 and H1. The signal is e(t)=-d~/dt, where

5.4 QUANTUM METHODS

243

the flux linked with the coil at resonance is ~(t) = bMpt cos(co0t) (as one can easily induce from Fig. 5.42c) and b is a constant accounting for the geometrical parameters of the coil. We get

e(t) = bcooMp sin a(tl) sin(co0t) bTt~,oXoHoHp sin(7/~0Hlt~) sin(~0t),

(5.96)

which is m a x i m u m for ~ ( t l ) = Ir/2, that is, for a 90 ~ r.f. pulse. We see here that e(t) is proportional to the product HoHp. With the conventional NMR continuous-wave method in still water, we would have obtained (using, for example, the expression (5.79) for the transverse magnetization re(co0)) e(t)oc H 2. This implies a sensitivity advantage of the flowing-water method of the order of Hp/Ho, justifying the special interest attached to it in low-field measurement. Tuning and phasesensitive detection of e(t) can eventually provide the power absorption line, examples of which are shown in Fig. 5.43a [5.105]. One can see in this figure that the absorption peak can pass from positive to negative on increasing the magnitude of the r.f. field Hi. This occurs because the angle ~(tl) becomes greater than 180 ~ and the transverse magnetization component Mpt leads HIL. The energy balance is preserved because it involves both the r.f. circuit and the water pump. The technique originally developed by Pendlebury et al. [5.104] differs from the one schematically shown in Fig. 5.41 because in it the detection of the signal is carried out, after excitation by the r.f. pulse under the measuring field I-I0, in a conventional NMR setup. This is adapted for use with the water duct in place of the measuring head and is tuned to the Larmor frequency cood = '~/d'0Hdet. Hdet is a suitably high field provided, for example, by a permanent magnet or an electromagnet. Let us thus assume that the r.f. field Hi is applied. If ~'1 and ~'2 are the times taken by the water, pre-polarized to the magnetization level Mp, to flow between the polarizer and the r.f. coil and from the r.f. coil to the NMR detector, respectively, and t I is the time spent beneath the r.f. coil, we have that the longitudinal magnetization of the water at the entrance of the detector is Mpz = M0(1 - e-~p/T1)e-~l+~2)/rlcos a(tl).

(5.97)

By entering the resonating detector, Mpz suffers a further magnitude change, the mechanism for it being the same as the one that occurred before under the previous r.f. coil. Such a change ~ p z OCCURSunder the DC field Hdet and requires that energy be supplied to the detector coil, which, for a volume flow rate 17of the water through the coil, is in unit time of the order of ~r~/pzHde t. In the limiting case where ~ p z -- 2Mpz [5.104],

244

0

LO

0

(b)~ soo

E

o

0

CHAPTER 5

~0

L E

~u..

O ~

~_o

(s~,!un "qJe)leuB!s uo!;dJosqv

~i cu~

~~

I-i

~

N

~~-

x

~

xA

~§ E

~

Measurement of Magnetic Fields

cS

.-I

(o-

U_

~

5.4 QUANTUM METHODS

245

we thus get, through Eq. (5.97), the expression for the absorbed power P = 2W~0HdetM0(1 -

e-%/T1)e-(Zl+z2)/TIcosor(t1).

(5.98)

By sweeping the frequency of the r.f. field, we find, according to Eq. (5.93), a m i n i m u m of cos a(tl) at 00 = 000 (see Fig. 5.43b) and the magnitude of the field H0 is correspondingly determined as H0 = Wo/~'i~o.Remarkably, Eq. (5.98) shows that P depends on the product of the fields in the detector and the polarizer field. The signal strength being thus quite independent of H0, this method can be used to cover a very wide range of measuring field strengths. Accurate measurements down to around 2 ~T have been demonstrated, for example, by Woo et al. [5.109]. Commercial flowingwater magnetometers are available today by which the range of measuring fields 1.4 ~T-23 T can be covered [5.110].

5.4.2.40verhauser magnetometers. The notable low-field measuring capability exhibited by the free-induction and the flowing-water magnetometer is based on the creation of a magnetization value Mp much stronger than the equilibrium value M0 = XoHo in order to achieve a greatly increased measuring signal. This feature is obtained by applying a conveniently strong polarizing field far from the measuring region (flowing water) or at a different time (free-induction). There is, however, a subtler way to increase Mp beyond equilibrium that does not require any polarization field. It is based on a powerful physical idea by Overhauser, who boldly predicted that the saturation of the electron spin resonance in a metal, brought about by a r.f. field, could produce an enormous increase in the nuclear polarization [5.111]. He stated, in particular, that the steady-state nuclear polarization would be augmented by the amount expected for an increase of the nuclear gyromagnetic ratio to the value of the electron gyromagnetic ratio (~/e -- 21/zel/h -- 1.76085979 x 1011 T -1 s -1 for the bare electron). In short, it would be as if the nuclei were partially to take up, under such conditions, the equilibrium magnetization of the electrons, which had disappeared because of saturation and restoration of equally populated levels. Carver and Slichter, working on Li 7, provided an experimental verification of Overhauser's proposal [5.112]. They additionally showed, by working with a solution of Na in ammonia, that this effect did not require a metal, but, basically, the presence of unpaired electrons. Working with pure hydrofluoric acid, Solomon demonstrated that polarization transfer could also occur between different nuclei [5.113]. The physical mechanism lying behind the Overhauser effect (also called "dynamic nuclear polarization" (DNP)) is the coupling between

246

CHAPTER 5

Measurement

of M a g n e t i c F i e l d s

the nuclear and electronic spins, occurring either by hyperfine interaction (in metals) or dipolar interaction, the latter process also being responsible for the nucleus-nucleus transfer of polarization. This coupling is a route by which the nuclear spin-lattice relaxation processes, which tend to restore the conditions of thermodynamical equilibrium, can take place. It requires that each nuclear spin flip be associated with a simultaneous electron spin flip. Let us therefore consider a system endowed with dominant nuclear spin relaxation via coupling to electrons, where, under the applied field H0, electron spin resonance at the frequency f0 = 7e/~0H0/2vr (f0-28.02495GHz for /~0H0 = 1 T) is sustained by means of a transverse r.f. field H1. For a system of this kind, Overhauser's analysis provides, per unit volume, a rate equation for the difference D = N e - N + between down and up electron spin populations in relation to the nuclear spin population difference A = N + - Nn. For nuclear spin, dD/dt-

(D O - D)/T~i)e + (A0 - A)/T~in) ,

(5.99)

where Do and A0 are the population differences at equilibrium under the field H0. In Eq. (5.99), as in the following, to simplify matters we assume, I = 1/2. T(i) ~le and T(i) ~ln are the longitudinal electronic and nuclear relaxation times arising from the hyperfine or dipolar interaction only. We do not consider, for the time being, other relaxation processes. Since the interaction keeps the total spin momentum constant, we have d D / d t - d A / d t . This implies that, once the steady-state conditions are attained and the electronic resonance is saturated, d A / d t - - 0 . We consequently obtain from Eq. (5.99) that the nuclear spin population difference is

T(i)

A = A o + ~ in7s D 0 ,

(5.100)

where we have introduced the saturation factor s = 1 - D/Do. s = 1 when the two electronic spin populations are equal. With hyperfine interaction we have that, at temperature T,

TIn( i ) i

T(i) le

2 TF 3 T'

(5.101)

where TF is the Fermi temperature [5.112]. The equilibrium spin population differences are related to the susceptibilities and the applied field H0 Do

= XPauliH0/ld, e,

A 0 -" ,]f'0H0/ld, n,

(5.102)

5.4 QUANTUM METHODS

247

where XPauli is the Pauli paramagnetic susceptibility of the conduction electrons, X0 is the nuclear susceptibility, and/d, e and/z~ are the electronic and nuclear magnetic moments, respectively. Using the known expressions for the susceptibilities 3 XPauli - - ~/-i'0

Ne/Ze 2 kTF

'

Xo -

/z0 Nn/z2n I + 1 3 kT I '

(5.103)

we obtain from Eq. (5.102), posing Ne = Nn, Do

=

A0 /~e 23 T--~" T

(5.104)

By introducing Eqs. (5.101) and (5.104) in Eq. (5.100) and recalling that /Ze//Zn = ~/e/~'n (same spin quantum number), we eventually obtain A = A0(1 + s ~/e ). Tn

(5.105)

It then turns out that the saturation of the electron resonance ( s - 1) brings about an enhancement by a factor ye/~/n of the nuclear magnetization with respect to the equilibrium magnetization M 0 /zeA0. With proton nuclei, this factor is around 660 and it is reduced when other interactions, besides electron-nucleus spin coupling, can provide nuclear relaxation. For example, Carver and Slichter find that the theoretical enhancement factor ~'e/~'n is more than 80% reduced in their experiments on Li 7 [5.112]. An example of the evolution of the nuclear resonance signal with the strength of the r.f. field H 1 is shown, for this specific case, in Fig. 5.44. Practical Overhauser magnetometers are generally based on the use of liquid samples, where a free radical, playing the role of electron donor, is diluted in a proton-rich solvent. We deal in this case with dipolar coupling, as presented for the first time by Beljers et al. in the free radical diphenyl-picrylhydrazyl (DPPH) [5.114]. This is an organic salt with one free electron (g factor, g = - 2 . 0 0 3 6 + 0.0002) and a very sharp resonant lineshape, a property deriving from an effect called "exchange narrowing" [5.115]. Nitroxide free radicals are currently applied in DNP magnetometers [5.116]. The nitroxide has a free electron associated with the nitrogen atom, dwelling in the relatively large magnetic field, of the order of 2 mT, provided by the nitrogen nucleus [5.117]. This is extremely interesting for low-field (e.g. earth field) measurements because the ensuing hyperfine splitting of the energy levels (zero field splitting) makes available to the electrons a low energy state, which is then crowded by a spin population much larger than the one expected under

248

CHAPTER 5 Measurement of Magnetic Fields

100

50

.

0

.

.

.

.

.

.

.

1()0

!

200

.

.

.

.

300

H 1 (A/m)

FIGURE 5.44 Overhauser effect in Li7. Electron spin resonance (ESR) is obtained by applying a r.f. field H1 at a frequency around 100 MHz and nuclear resonance at 50 kHz is simultaneously produced and observed. By increasing the amplitude of Hi and approaching the ESR saturation, the NMR absorption signal is largely increased. The obtained enhancement factor A is larger than 100 (from Ref. [5.112]).

the low-strength measuring field only. By flooding the sample with the saturating r.f. field and, consequently, restoring the electronic spin population balance, a correspondingly larger polarization is transferred to the proton nuclei and the DNP gain can attain, for earth field measurements, the order of a few thousand. Figure 5.45 illustrates this case, where the hyperfine interval factor a corresponds to a shift of the resonance frequency to f0 = 2 x 10 -3 Te/2~r of about 60 MHz. The upper level suffers further Zeeman splitting upon application of the external field. Of the two allowed transitions, the lower one (1), enriches the population of the lower nuclear level, bringing about positive DNP gain on saturation. Transition (2) has the opposite effect, leading to negative DNP amplification. The magnetometer setup developed by Kernevez and G16nat [5.117] is schematically shown in Fig. 5.45. It employs two DNP probes in a bridge configuration. The probes are placed in a resonator, producing the saturation of electronic resonance at the radiofrequencyf0, and are, at the same time, excited at the low NMR frequency. A special design of the coils permits one to always find a part of the sensor where H 1 and H0 are perpendicular one to another and consequently eliminate

5.4 QUANTUM METHODS

,'

i

,

',

"

,I

a/4+e.

;

a/4

DNP probe

a/4-e r.f.

a

,, I ; "

,~,

249

generator 1

2

I

-3a/4

Ho=O Ho>O

-" 1

o i% i I

O!

1

!

DNP probe ____-c___,i !

If1 1

! readout I

FIGURE 5.45 A practical Overhauser magnetometer is often based on the use of a nitroxide free radical as a source of unpaired electrons and a proton-rich solvent. The electrons reside in a relatively strong nuclear field and suffer hyperfine splitting (zero field splitting). The upper level is further split by the Zeeman interaction with the measuring field H0. The two possible electronic transitions lead to DNP gains of opposite sign. Thanks to the accurate choice of the solvents, they are separately obtained at the same r.f. frequency in the two probes employed in the bridge circuit. It is therefore possible to reject the external interference signal, which is instead symmetrically detected. The background noise spectral density turns then out to be lower than 10 pT Hz -1/2 (adapted from Ref. [5.117]). the signal extinction zones. Two different solvents are used with the nitroxide free radical (methanol and dimethoxyethane), which produce different chemical shifts. They are calibrated in such a way that, at the same frequency, f0, the DNP gain is positive in the first probe (proton polarization parallel to the external field H0) and negative in the second (proton polarization antiparallel to H0). It then turns out that, using the bridge circuit, the NMR resonant signals are amplified and the symmetrically detected disturbances are eliminated. The sensitivity claimed for this type of continuous-wave magnetometer is better than 10 pT H z -1/2. It is also possible to design an Overhauser magnetometer operating under transient conditions, like a free-precession magnetometer. In this case, the r.f. field is applied for a time interval sufficient to establish electron saturation and the ensuing proton polarization. After ringing has subsided, a short DC current pulse in the pickup coil aligns the proton moments perpendicularly to the measuring field, around which they are left to freely precess and generate the time-decaying signal in the sensing coil. Commercial Overhauser magnetometers display a typical measuring range of 20-120 ~T and a sensitivity around 0.1 nT. They are prevalently employed in the measurement of the environmental fields, especially the terrestrial magnetic field and its variations due to geophysical

250

CHAPTER 5 Measurement of Magnetic Fields

phenomena and various man-generated disturbances. To this end, they are generally made portable and respond excellently to the ensuing requirement of low power consumption, requiring typically 1-2 W. In this respect, they favorably compete with the free-induction magnetometers. Power is chiefly required for saturating the electronic resonance and, in order to minimize it, a narrow absorption line would be required. The typical resonance linewidth of nitroxides is, however, fairly broad, being around 100 ~T in the earth magnetic field, but it can be reduced to about 20 ~T by substituting hydrogen atoms with deuterium atoms in the compound [5.118]. An alternative free radical with 2.5 ~T linewidth has been proposed, which can reach saturation with far less power than the perdeuterated nitroxide [5.119]. This compound does not display zero field splitting and, in order to have high DNP gain, separate polarization in a homogeneous high field is normally provided. The NMR magnetometers are assumed to provide absolute measurements. Equation (5.82) shows that the value of the field expressed in T is obtained from knowledge of the fundamental constant 3/ and the determination of the resonance frequency. For the conversion of T in A / m , division is made by the magnetic constant/~0 = 47r x 10 -7 N / A 2 exact by definition. The measurement is therefore traceable to the national standards of time. These are maintained today with an uncertainty of the order of 10 -13 and the generally available frequency counters, calibrated against these standards, have a time base stability better than 10 -6. With pure water probes, the shielded proton gyromagnetic ratio ~/~ should be adopted. There is a long history of ~/~ determinations, which have been carried out in the last 50 years in different national metrology laboratories. The motivation for such experiments is, on the one hand, the obvious desire for more precise magnetic field measurements and, on the other hand, the control of the practical unit of current in the laboratories. Basically, ~/~ is obtained by applying Eq. (5.82) in reverse, where the field is measured with the highest possible accuracy with a force method (high fields) or produced by means of an accurately calculated and realized single-layer solenoid (low fields). Mohr and Taylor provide a detailed critical account of the latest experiments in their comprehensive report on the CODATA recommended values of the fundamental constants [5.90]. The 1998 adjustment provides "}/p = 2.67515341 x 10s T -1 s -1 with relative uncertainty 4.2 x 10 -s. If, instead of pure water, proton-rich substances are used, the related chemical shifts must be taken into account. This correction can be the source of substantial uncertainty, typically some 10 -6 . Further uncertainty contributions can arise. For example, traces of magnetic impurities might remain in the elements of the sensing head and some detrimental effect could be associated with AC

5.4 QUANTUM METHODS

2~1

environmental fields. The precise measurement of the precession frequency in the decaying free-induction signal at low fields might be a problem because of the low frequency, the noise, and the limited time available for the measurement. The adoption of digital methods and specific algorithms, including spectral analysis by fast Fourier transform, can be instrumental in achieving the best measuring accuracy.

5.4.3 Electron spin resonance and optically pumped magnetometers In the Overhauser magnetometers, the resonant absorption of a r.f. signal by the electrons is exploited indirectly, the field measurement solidly relying on the proton resonance. It is, however, possible to make direct use of electron spin resonance, the basic difference with respect to NMR being that the involved frequencies are multiplied by a factor 3'e/~'~ ~ 660. This means that with fields higher than about 10 mTwe fall into the microwave region. ESR resonance magnetometers have been principally developed for low-field measurements, where the high value of ~/e provides an advantage with respect to NMR in terms of signal-to-noise ratio (S/N). However, the electronic relaxation times are generally much smaller than the nuclear ones and the resonance linewidth A~o0 = 2/T2, in particular, is much enlarged with respect to the proton linewidth. Interactions between electron spins are much stronger than between nuclear spins and the coherence of the spin precession is in most cases rapidly destroyed. Since the sensitivity of a resonance magnetometer, that is, the smallest detectable field change 8H0, can be written as [5.120] 8Ho - &tOo(S/N)_I/2,

(5.106)

it is concluded that optimal tradeoff between linewidth and signalto-noise ratio is required for ESR to compare advantageously with NMR. In practice, ESR magnetometers only employ sensing materials (e.g. free radicals) where the unpaired electrons are well separated and the spinspin interactions are minimized. The organic salt DPPH is a classical ESR compound, endowed with a nearly free electron ( g - - - 2 . 0 0 3 6 vs. g - -2.0023193 of the bare electron) and an exchange-narrowed linewidth A~o0/~/e ~ 0.27 mT [5.115]. Other narrow linewidth materials, typically radical cations and anions, have been developed and applied. An example is the fluoranthene radical cation salt (FA)2PF6, where, thanks to an important motional narrowing effect, Aco0/Ye ~ 1.5 ~T is obtained [5.121]. A highly accurate magnetometer based on its use in a small-sized (around 100 m m 3 or less) probe head has been developed by Gebhardt

252

CHAPTER 5 Measurement of Magnetic Fields

and Dormann for measurements between 50 ~T and 10 mT [5.122]. There are no basic differences in the electronic design of NMR and low-field ESR magnetometers, but for the increase of the resonance frequencies in the latter. ESR obviously displays much larger (S/N) value and it might be preferred for continuous measurements of the earth magnetic field, as required, for example, in land-surveying and defense activities. The associated measuring resolution can be of the order of 10 nT over an angle of + 85 ~ around the measuring axis [5.121]. They do in general not provide absolute measurements because the electron gyromagnetic ratio, subjected to a variety of electronic interactions, can change over different materials. A calibration in a known field by comparison with a NMR device is therefore recommended. The ESR magnetometers, like the NMR ones, are intrinsically scalar devices, providing a measure of the modulus of the field. They can, however, be adapted, at the cost of a somewhat reduced resolution, to vector field measurements. Duret et al. [5.121] combine a known static internally generated field Hi with the measuring field H0. If 0 is the angle made by H i with H 0 and the condition Hi >> H0 is satisfied, the modulus of the resulting field H m -- H i q- H0, which is the quantity measured by the magnetometer, is given to good accuracy by Hm = Hi + H0 cos 0. The auxiliary field Hi, being so much higher than H0, must obviously be very stable in order to achieve the desired measuring resolution. If such stability cannot be achieved, it is expedient to make a double measurement, with the sign of H i reversed. As shown in Ref. [5.121], the two measured dispersion resonant signals can be combined to provide H0 cos 0, independent of Hi and its fluctuations. The ESR magnetometers exploit the resonant behavior of unpaired electrons in some specific compounds, where they act almost like free spins in thermal equilibrium. The magnetization level associated with a given measuring field is the one we expect from Boltzmann statistics. It might be asked whether we can overcome, as already obtained with nuclear magnetism, the thermal equilibrium limitation and correspondingly increase the sensitivity of the field-measuring device via increased non-equilibrium magnetization. The method of optical pumping provides an affirmative answer to this question. This method, while having an importance going far beyond the relatively narrow subject of low-field magnetometry (being the basis, for example, of atomic clocks), provides a practical and widely applied route to measurements in the ~T and nT range. The physical basis of the optically p u m p e d magnetometers can be understood by making reference to the atomic energy level diagrams of two commonly employed sensing elements: He 4 and Rb s7. The first is shown in Fig. 5.46a, the second in Fig. 5.47a. The He 4 atom has no nuclear moment and the two electrons in the ground state 1~S0 have antiparallel

5.4 QUANTUM METHODS

253 r.f.

m j= 0 i

23P~ ; 23 P1

. . . . . . .

,

:

generator I He lamp ( )

.......

3,= 1083 pm 23S1~ ,,~

,, !-I

interference filter

lens

i i__

m j= 1

J

circular polarizer ~

m j= 0

1 He cell

/

/,"

mj= -1

rf

IHo (a)

(b)

i r. detector I

~

output

m

FIGURE 5.46 Optical pumping in He 4. (a) Atoms in a gas cell are raised from the ground state I 1 So to the first excited state 23 $1 of the triplet system by means of a r.f. discharge. Circularly polarized light (Do line at 1.083 ~m) is selectively absorbed by one of the m} = + 1 Zeeman sublevels and the corresponding atoms reach the excited state 23P0. They radiatively decay in a very short time and with the same probability into the three 23S1 sublevels. The absorbing level, whose depopulation is signaled by cell transparency, can then be repopulated by the action of an AC field I-I1 at the Larmor frequency ~0, applied orthogonal to the external field H0. (b) Schematic diagram of the servomode magnetometer. The r.f. field I-I1is slightly modulated in frequency. When the center frequency is equal to ~0, the output signal is made of even harmonics only of the sweep frequency. Phase detection and feedback are used to lock the frequency of H1 to oJ0.

spins Sz = + 1/2 [5.123]. He 4 atoms can be excited from the g r o u n d state to high energy levels or even ionized by r.f. generated electron collisions in a discharge tube, where the gas is held at a pressure of some 102 Pa. A n u m b e r of the excited atoms can decay back to the first excited parallel spin state 2381 (orbital m o m e n t u m L = 0, total angular m o m e n t u m J = total spin m o m e n t u m S = 1), whose radiative decay to the g r o u n d state has a forbidden character, since it w o u l d involve the reversal of a spin. Consequently, this state is sufficiently long-lived (10-20 ms) to be considered metastable. U n d e r steady-state conditions, the singlet 11S0 g r o u n d state population can then be considered as a buffer gas (parahelium) for the triplet 23S1 atoms (orthohelium). The next higherlevel triplet state (L = 1) is 23p, which is s u b d i v i d e d in the three substates 23po, 23p1, 23p2, corresponding to the three values of the total m o m e n t u m

254

CHAPTER 5 Measurement of Magnetic Fields

I R o7

2 ......

/

A

N

(,

, I

x

i~

I ! , I

\

'

,'~ m~=-2 i! i !

~ i

'

! l !

!

: I I l

I i I

9

I]

i I

I

1

~I

i

'

,, I ,I

i

I I

i I

15,

I

i

,,

_

"-

, ,'i" . :~G,' =

i

/r"-,.?

'

)?

_______mF=.____~ 2

lo,

mF= ' 1

!.~.//-1

\

mF=--4~ oo

A =hf o

2 = 2a

F=4

5.

i I ~.~1r

i

5 2 S1/2

0

a.

I'1.4 ! ! 1 I I I

,'

Z = 0.7948/zm

"

N.N

, I

,

mF=4

I l

; ~ 1

__

'~AmF-+I

~

J

6S1/2

"~.o 20. o"~

AmF=O,• '

C s 133

25.

0.2

0.4 0.6 0.8 Light intensity (a.u.)

1.0

(b)

m F- -2

J= 1/2 ~

/=3/2

F=I

(a) FIGURE 5.47 (a) Relevant energy levels and optical pumping in Rb 87. Atoms in the hyperfine levels F -- I and F -- 2 of the ground state 52S1/2 can be raised to the first excited state 52p1/2by circularly polarized light of wavelength A - 0.7948 I~m. Transitions from the mF -- +2 Zeeman sublevel are forbidden by the selection rule AmF -- -t-1. Decay from the excited state is equally likely to occur in all ground state Zeeman sublevels. The light-absorbing levels consequently depopulate and the cell becomes transparent. Equal populations, restoring light absorption, can again be obtained by applying, at right angle to the external field H0, a resonating r.f. field H1, whose frequency provides a measure of H0. (b) Example of evolution with the intensity of the pumping light of Zeeman sublevel pp~ulations. They belong to the hyperfine level F - 4 of the ground state 6S1/2 of Cs 1~~(adapted from Ref. [5.130]).

J = 0, 1, 2. These substates are slightly separated in energy, their separation being m u c h smaller than their distance from the first excited state 23S1 9In the presence of an external field Ho, the state 23S1 splits into three Z e e m a n sublevels, corresponding to the q u a n t u m n u m b e r s

5.4 QUANTUM METHODS

255

mI -- - 1 , 0, + 1. The Zeeman energy separation is AE -- hco0 = g/xB/x0H0,

(5.107)

where/XB is the Bohr magneton and the g factor is that of the free electron. This is a good approximation because the optical electron is basically decoupled from the atom and behaves as in a vacuum (g = -2.0023193). The field-frequency conversion factor of about 28 kHz/~xT obviously provides a specific advantage in the measurement of very low fields. Let us now suppose that the gas cell containing the mixture of parahelium and orthohelium is invested by a circularly polarized light beam, with optical axis z coincident with the direction of H0. Such a beam can be generated either by a He lamp or a laser. In all cases, the light is tuned to the infrared wavelength A = 1.083 Ixm, corresponding to the Do spectral line, which connects the states 23S1 and 23p0. The atoms, initially assumed to be equally distributed among the available sublevels, selectively absorb the polarized photons because only the Zeeman sublevels endowed with the appropriate mj value can make the transition to the excited state 23p0. Thus, if the absorbed photons transfer, depending on the sense of the circular polarization, the axial angular momentum mj = +1, only the atoms with quantum numbers mj = -T-1 can be excited, respectively, to the mj = 0 state 23p0. Unequal distribution between the mj = 1 and m j - - 1 sublevels amounts to a net magnetization Mz, directed either with or against the field H0. The optically excited atoms are short-lived, having a lifetime around 10 -4 s, during which mixing of the P states induced by the relatively high pressure buffer gas is likely to occur. It is therefore assumed that the optically excited atoms tend to relax with equal probabilities into the different 23S1 sublevels, regardless of the level from which they originated. Under the counteracting effects of optical pumping and relaxation, with their characteristic times ~'p and ~'R,a higher than thermodynamic equilibrium value of the magnetization Mz is reached in a time ~-given by 1 1 1 - -t , "r"

"rp

"rR

(5.108)

of the order of some 10 -4 s [5.124]. With the absorbing state mj - - 1 (or, equivalently, the state mj = +1) depopulated, the polarized light can propagate through the gas cell, which becomes transparent. In order to recover a homogenous distribution of atomic states, a transition between the Zeeman sublevels can be induced by means of an AC field H: applied at right angle (or at least with a component at right angle) to the external field, having a frequency equal to the Larmor frequency to0 = AE/h. If

256

CHAPTER 5 Measurement of Magnetic Fields

the AC field frequency is swept through the resonance frequency COo,the cell opacity is recovered when CO= COo, with a corresponding dip in the transmitted light intensity. At steady state, the behavior of the magnetization M, precessing around the direction of H0, is the one predicted by Bloch's equations. A typical scheme employed in optically p u m p e d He 4 magnetometers is shown in Fig. 5.46b. The He lamp generates the Do, D1, and D2 infrared spectral lines. D1 and D2 connect the state 23S1 with 23p1 and 23p2, respectively, and can be suppressed by filtering. Alternatively, a laser tuned to the Do line can be used, which has the advantage of a largely increased pumping efficiency [5.125]. With this scheme a servoed magnetometer is realized, where the r.f. field HI is modulated with a small swing +ACOaround a center frequency. If the sweep frequency is f/ and the center frequency coincides with COo,the output signal is made of even harmonics of 1~ only. Any deviation of the center frequency from COo results in the appearance of a fundamental component, which is phasedetected and used in a feedback loop to lock the frequency of H1 at COo, which then provides the value of the field H0 (Eq. (5.107)). The He 4 magnetometer is characterized by a resonance linewidth ACO/2~r of the order of few kHz, corresponding to ACO0/~/~ 0.1 ~T. Field tracking speed is therefore limited by the modulation frequency f~, which is of the order of a few hundred Hz. The sensitivity can be better than 100 pT and is predictably improved, even by two orders of magnitude, by the use of increased cell size and laser pumping in place of the He lamp [5.125]. Like all the Larmor resonance devices, the He 4 magnetometer provides a measurement of the modulus of the external field, whatever the angle 0 made by H0 with the optical axis. The signal amplitude, however, depends on such an angle as cos20 [5.124] and the progressive degradation of the signal on approaching 0 = 90 ~ eventually results in a dead zone, where the noise can be too high for the magnetometer to comply with defined specifications. Methods for automatic orientation of the optical axis with the external field and of the r.f. field perpendicular to it have been devised [5.126]. Alkali elements have a single electron in their outer shell, which is available for magnetic resonance. Na 23, K 39, Rb 87, Rb 8s, and CS 133 have all been used to realize optically p u m p e d magnetometers. The optimal alkali vapor pressure, of the order of 10 -4 Pa, is maintained in a glass cell with a buffer inert gas at a pressure around some 103 Pa. The temperature required to obtain the appropriate vapor pressure ranges between 126 ~ in Na 23 and 23 ~ in Cs 133. The physical mechanism of optical pumping in alkali elements is analogous to the one already7 described in He 4 and is schematically illustrated for the case of Rb 8 , an element frequently

5.4 QUANTUM METHODS

257

employed in devices [5.127], in Fig. 5.47a. Transitions in Rb 87 can be induced from the ground state 5281/2 to the first excited state 52p1/2, lying 3.7725 x 1014 Hz above it by absorption of circularly polarized light of wavelength A = 0.7948 ~m. The nucleus of Rb 87 and the optical electron have total momentum number I = 3/2 and J = 1/2, respectively, and their interaction gives rise to hyperfine splitting of the 5281/2 a n d 52p1/2 states. I and J couple in a manner which is analogous to the previously sketched L-S coupling in the He 4 atom and produce a total angular momentum with quantum number F. This runs from I + J through i I - Ji, thereby attaining, in this specific case, the values F = 1 and F = 2 for b o t h 5281/2 and 52P1/2 states. The interaction energy between nuclear and electron magnetic dipole moments can be written as [5.128] a WF = ~(F(F + 1) - I(I 4- 1) - lq + 1)), (5.109) where the constant a is the so-called interval factor of hyperfine structure. We have WF = 3 a and WF = -- 88 a for F -- 2 and F -- 1, respectively, so that the hyperfine energy splitting is &W12 = 2a. This amounts to the frequency /12 -'- AW]2/h -- 6.8347 x 109 Hz for the split ground state 5281/2 and f12 = 8.18 x 108 Hz for the state 52p1/2 (vs. the much higher dipole radiation frequency f = 3.7725 x 1014 Hz connecting these two states). In presence of the external field I-I0, Zeeman splitting occurs, corresponding to the possible orientations of the vector F with respect to H0. If m F is the associated magnetic quantum number, five (m F -- 2, 2, 1, 0, 1, 2) and three (m F = 1, 0, 1) sublevels within the hyperfine levels F = 2 and F -- 1, respectively, are generated. At high fields, when the Zeeman splitting becomes larger than the zero-field hyperfine splitting AWl2 , I and J become decoupled and quantize independently in the direction of H 0 (Paschen-Back effect), so that no pumping will take place. The energy behavior in the Zeeman and intermediate field amplitude domains can be quantitatively predicted, for the present case of electronic angular momentum J = 1/2, by means of the Breit-Rabi formula. It provides for the energy in the field H0 when the momenta are mF, J = 1/2, I

WI+l/2'mF

a

4 [

aWl2

2

4mFgJi~'BIJ'oHo (gJ[d'B[Ld'oHo)2]1/2

X 1 4- 21 +'------~ •W12

4-

AWl2

,

(5.110)

where gj is the Land6 atomic g factor. The dependence of WI+l/2,mFo n H0 at low and intermediate field values is shown in sketch form in Fig. 5.47a. It is noticed that this formula can predict the transition to the Paschen-Back

258

CHAPTER 5 Measurement of Magnetic Fields

regime at high fields, where the two mj levels mj = + 1/2 are each split in the four sublevels with quantum numbers m1 = 3/2, 1/2, - 1/2, - 3/2. We are interested in the transitions between sublevels in t h e 281/2 state, for which gl is equal to the g factor and AW12 = 2a. For very low values of H0 (typically H0 < 10 p~T in the alkali atoms), the energy difference between :tifferent adjacent sublevels (AmF -- +__1) is the same and it is provided by the Breit-Rabi formula in the limit gj/~B/~0H0 ~ AW12. This limit amounts to the condition that the energy of the whole atom in the external magnetic field is negligible with respect to the hyperfine structure separation (the distance between the two F levels). Under these conditions, Eq. (5.110) reduces to the linear law

mF

H0),

WI+l/2,mF = _ a-'4 + -- a 1 + -~a gl/~,B~0

(5.111)

where it has been posed I = 3/2. The separation between the sublevels turns out to be 1

bE = ~gjj/,BIU,0H0 = h30/zoHo,

(5.112)

with ~ the effective gyromagnetic ratio. ~ is then reduced by a factor four with respect to the gyromagnetic ratio of the free electron and the corresponding frequency-field conversion factor of Rb s7 is then 7.006 kHz/~T. The same value is obtained for N a 23 and K 39, while it is 4.668 k H z / ~ T in Rb 85 and 3.50 k H z / p X in CS 133. The curvature of the level lines on increasing H0, predicted by Eq. (5.110), is apparent in Fig. 5.47a. It leads to different AE values between adjacent sublevels and thereby to a spectrum of resonance lines. As for the previously discussed case of H e 4, the pumping in Rb 87 from the 5 2$1/2 to the 52p1/2 state with circularly polarized light and the successive radiative decay will lead to unequally populated sublevels in the ground state [5.129]. For one thing, the transitions from the ground state to the excited state occur with different probabilities for different Zeeman sublevels. It is easily understood, for example, that the atoms in the ground state sublevel mF - - + 2 cannot absorb photons if the sense of rotation of the polarization imposes the selection rule A m F = +1. If the sense of the circular polarization is the opposite, it is the transition from the sublevel mF -- - 2 that is inhibited. Because of the relatively high pressure buffer gas contained in the cell, fast mixing of the Zeeman sublevels occurs in the excited state and it can be safely assumed that all sublevels of the ground state have an equal probability of being repopulated in the relaxing atoms. The result is depletion of the absorbing levels, which implies larger than equilibrium magnetization and cell transparency. Figure 5.47b provides an example of

5.4 QUANTUM METHODS

259

evolution of the Zeeman level populations in the ground state of C s 133 upon optical pumping [5.130]. C s 133 has nuclear and electron angular moments I = 7/2 and J - - 1/2, respectively, and is characterized by two hyperfine energy levels in the ground state with F -- 3 and F -- 4. We can see how, following optical pumping with AmF = +1, the non-absorbing sublevel mF --4 in the hyperfine F = 4 becomes increasingly populated with the increase of the pumping light intensity. Again, redistribution of the populations amongst the different Zeeman sublevels in the ground state can be obtained by means of a resonating r.f. field I-I1, with at least a component orthogonal to H0, but, contrary to the case of He 4, now we have, depending on the H0 value (Eq. (5.110)), as many resonant lines as possible transitions between sublevels. In some cases, the lines can be resolved and, according to the Breit-Rabi formula, a very precise field measurement can be done. This has been demonstrated with C s 133 for fields larger than about I mT [5.130] and is realized in potassium magnetometers in the range of the earth magnetic field [5.116]. In the latter case, a resonant linewidth A~0/~/around 100 pT (A~0 ~ 5 s -1) and sensitivity better than 0.1 pT are reported. The obvious limitation of a magnetometer exploiting narrow line response is its relative inability to track rapidly changing fields (i.e. over times shorter than about 1/A~0). In many practical instances, the individual resonance lines are broad enough to overlap and merge in a single unresolved structure. The intrinsic linewidths A~0/~ can be as narrow as I nT, but they become much larger under experimental conditions. Broad lines can actually provide an advantage in terms of the tracking capability of time-varying fields, but unresolved structures can lead to problems of measuring accuracy. The unresolved line shape is not symmetrical and, in particular, the position of its peak is a function of the sensor orientation with respect to the measuring field. This is the so-called "heading error", which is due to the fact that the amplitudes of the individual overlapping lines change in a different way when the component of H 0 along the optical axis is changed. In the limit where the direction of H0 is reversed, the symmetry is also reversed because this amounts to interchanging the signs of the magnetic quantum numbers. Alkali magnetometers are generally realized using the so-called autooscillating configuration, where resonance is revealed for its modulating effect on the transmission through the cell of an auxiliary polarized light beam directed orthogonal to the pumping beam. Let us suppose that H0 is aligned with the optical axis and consider the resonant precession around z of the magnetization M induced by the r.f. field H 1. The transverse magnetization component m (see Fig. 5.32) rotates at the Larmor frequency in the x - y plane and, in doing so, it periodically changes the angle it makes

260

CHAPTER 5 Measurement of Magnetic Fields r.f.

generator Rb lamp (~

I

) interference filter

lens

J

circular

polarizer",~

0=45 ~ photo detector hase ~er

output

FIGURE 5.48 Scheme of principle of the alkali-based autooscillating optically pumped magnetometer. At resonance the light beam intensity is modulated at the Larmor frequency. The signal at the output of the photodetector is re-injected into the r.f. coil after 90~ phase shift and the oscillation is self-sustained, provided the feedback loop gain is unity and the total phase shift is zero. The output signal amplitude varies approximately as the product sin 0 cos 0.

with the transversally directed light beam. Since light is absorbed to an extent depending on the angle made by the precessing moments with the light wave normal, a modulation of the transmitted light intensity at the resonance frequency occurs. Practical devices do not generally make use of two independent orthogonally directed light beams and, as shown in the schematic representation of Fig. 5.48, the same beam directed at 45 ~ to H0 is used for both pumping and monitoring. The signal collected by the photodetector at resonance, modulated at the Larmor frequency, is amplified, 90 ~ phase-shifted and re-injected in the r.f. coil. The 90 ~ phase shift is required because, at resonance, the transverse magnetization m lags 90 ~ behind the active rotating component of the r.f. field. Consequently, if the gain of the loop is unity and the sum of all phase shifts is zero, the system oscillation is self-sustained. While in the previously discussed servo-type magnetometer the output signal amplitude depends on

5.4 QUANTUM METHODS

261

the angle 0 made by H 0 with the pumping light propagation direction according to cos 20, in the single-beam autooscillating-type magnetometer it varies as the product sin 0 cos 0 because the secular pumping process varies as cos 0 and the Larmor frequency modulation varies as sin 0. The maximum signal is therefore achieved for either 0 -- 45 ~or 0 = 135 ~ In the latter case, however, wiring of the feedback loop must be interchanged because the product sin 0 cos 0 changes sign and the phase shift must correspondingly change sign. An alternative solution consists in making use of a dual cell system, placed back-to-back with a single light source at the center, which, however, requires strict geometrical tolerances [5.131]. Given this angular dependence of the signal, the response of the autooscillating magnetometer is affected by two dead zones, corresponding to field orientations close to the optical axis or a direction perpendicular to it. Either servoed or autooscillating, optically p u m p e d magnetometers represent, in summary, an ideal response to the need for accurate measurement and tracking of weak and very weak magnetic fields, like those of geological origin or the ones encountered in deep space. They can display sensitivities of the order of I pT and a fast response to transients, the latter being limited either by the transverse relaxation time T2 of the precessing spins or by the amplifier bandwidths. Their use in the laboratory is mainly associated with the generation of reference fields in the geomagnetic range [5.130, 5.132]. We have previously remarked how optical pumping in He 4 required the creation of a certain proportion of atoms in the metastable state 23S1, whose distribution among the three available Zeeman sublevels becomes inhomogeneous after radiative decay from the state 23p0. A very similar process can occur in the He 3 isotope, whose use in the previously described optically p u m p e d magnetometers can then be envisaged. However, the fact that He~ endowed with nuclear momentum (I = 1/2) implies not only hyperfine splitting of the atomic levels and, consequently, different Zeeman resonance frequencies with respect to He 4, but also the existence of an atomic ground state 11S0 forming a Zeeman doublet. Each metastable atom can exchange its metastability with a ground state atom by collision, under the condition of conservation of the angular momentum. In He 4, this has no special consequences because this atom has zero-momentum ground state and such an exchange leaves the incident and the emerging metastable atoms with the same momentum. In He 3 it may occur, instead, that incident and emerging ground state atoms have their magnetic quantum numbers differing by + 1, with the change u 1 simultaneously affecting the metastable atoms. Momentum transfer from the metastable to the ground state atoms can create a very large fraction of oriented nuclei and, at a pressure of 10-100 Pa, the net

262

CHAPTER 5 Measurement of Magnetic Fields

oriented population can amount to 20-40% [5.133]. They are driven to resonance by a r.f. field at the Larmor frequency of 32.45 MHz/T, with resonance detected either by optical absorption monitoring or conventional nuclear induction methods. The two additional Zeeman resonances concerning the 23S1 metastable state can also be observed. In spite of the low density of the gas with respect to water (number of He 3 atoms per unit volume N h -~ 8.7 • 1022 m -3 at room temperature with a pressure of 200Pa vs. density of protons in water ---6.7x 1028m-3), the latter technique can provide an easily detectable signal. For example, for a 20% fraction of net oriented nuclei, which is easily obtained under geomagnetic fields, we calculate a magnetization M -- 0.2Nh/~h, where the J/T, of about helion (He 3 nucleus) magnetic moment is /d,h = - 1 . 0 7 4 6 7 x 10 -5 A/re. This is approximately the magnetization we would obtain in water under a field H0 ~- 2 x 104 A/re. Given the gaseous nature of He 3 and the weak interatomic interaction effects, very long spin-lattice and s p i n - s p i n relaxation times T1 and T2 are expected. They can vary in practice between very wide limits, depending on factors like the presence of field gradients, radiation damping and interaction with the cell walls, but values ranging between 103 and 105 s can be achieved [5.134]. A He 3 nuclear free-precession magnetometer with limiting transverse relaxation times ranging between 1 and 24 h and sensitivity of 0.1 nT has been demonstrated [5.134]. With the resonance only indirectly coupled to the pumping light, the problem of resonance frequency shifts due to various asymmetries and effects of the pumping light are largely avoided. Outstandingly accurate measurements of the nuclear free-induction decay in He 3 have been recently performed in order to determine the ratio between the helion and the shielded proton magnetic moments /d,h//d,~ [5.135]. NMR measurements in He 3 and pure water have been carried out, in particular, under the same field of 0.1 T, using the very same 25 m m diameter spherical cell for the two elements. These are interchanged without removing the cell from the magnet. The final result is ~h//d,~ -- -0.761786131, known with an accuracy of 4 parts in 109.

5.5 MAGNETIC FIELD S T A N D A R D S A N D TRACEABILITY Any magnetic field measurement has a meaning when it can be traced to the relevant base and derived SI units. Traceability requires the action of the National Metrological Institutes (NMIs), which have the mandate of developing, maintaining, and retaining custody of the standards

5.5 MAGNETIC FIELD STANDARDS AND TRACEABILITY

263

of measurements. Standards are traceable with stated uncertainties to the SI units, which can then be disseminated for general measurement and testing activities. The end users can thus relate their measurements to the SI units through an unbroken flow of calibrations originating in the NMI laboratories. These, in turn, engage in mutual comparisons with other NMIs, under supervision by the Bureau International Poids et Mesures (BIPM), by periodically reproducing the units and maintaining sound quality system principles [5.136]. Magnetic field (or, equivalently, magnetic flux density) standards were classically realized in the past as physical artifacts, that is, accurately designed and built solenoids or Helmholtz pairs, which were made traceable to the SI units of length, resistance, and voltage. Nowadays, when talking of a field standard, reference is usually made to a system combining a field source with a nuclear or atomic resonance device, provided with all the auxiliary setups required to stabilize the values of the involved physical quantities and minimize the effect of external interferences. The resonance devices play the role of intrinsic standard, realizing traceability to the SI unit of time via the resonance equation (5.82). The uncertainty of the whole standard is therefore made of two components, one associated with the consensus value of the gyromagnetic ratio ~, and the other associated with the practical realization of the standard. The latter is largely dominant and may be calculated according to the example reported in Section 10.4. An illustrative discussion on the realization of magnetic field standards by an NMI laboratory for the sake of dissemination of the SI field unit is given by Weyand [5.94]. This author shows, in particular, how NMR-based standards developed at the Physikalisch Technische Bundesanstalt (PTB) can cover a range of magnetic field values stretching between 10 ~T and 2 T, with relative uncertainties ranging between 10 -6 and 10 -4 (see Fig. 5.49). Various coil types (e.g. solenoids and Helmholtz pairs), characterized by accurate realization and low thermal coefficient, are used as field sources in the lower field range, up to about 100 mT. Electromagnets are used in the upper field range. In order to achieve the measurement and calibration capabilities described in Fig. 5.49, full use is made of specifically developed continuous-wave NMR magnetometers, based on the marginal oscillator technique, by which the lower measuring limit of present-day commercial NMR magnetometers of about 40 mT is extended down to 0.5 mT [5.85]. As remarked in Section 5.4.2.1, the sensing probe does not in this case contain pure water but a dilute CuSO4 aqueous solution, which implies a small shift of the resonance frequency. The amplitude of the modulating field is I ~T. The standard field sources used to cover the lower field range are calibrated by means of a free-precession magnetometer, pre-polarized in a conveniently strong field (e.g. 5 mT for

264

CHAPTER 5 Measurement of Magnetic Fields

1E-4

nti~8~NMR

>, r

o

r

free inductiondecay

1E-5

> 0 rV

1E-6 #oH

(T)

.....

o'.1

. . . . . . .

'

FIGURE 5.49 Field range covered by the NMR-based magnetic field standards maintained at Physikalisch Technische Bundesanstalt and associated relative uncertainty. Solenoids and Helmholtz pairs are used as field sources up to about 100roT, while electromagnets cover the upper range of field strengths. Continuous-wave and free-induction decay NMR magnetometers are applied in the upper and lower field range, respectively. A laboratory-developed continuouswave NMR setup based on a marginal oscillator method permits one to overcome the low-field limitations of commercial devices (around 40 mT) and is applied down to about 0.5 mT [5.85] (adapted from Ref. [5.94]).

a measuring field/~0H--20 I~T). It is seen in Fig. 5.49 that the relative uncertainty associated with the free-precession based standard increases rapidly with decreasing of the field amplitude. This is due, on the one hand, to natural weakening of the signal-to-noise ratio and, on the other hand, to the decrease of the resonance frequency with the decrease of the measuring field strength, which implies progressively lower accuracy in its determination. The observation time is in fact limited by the transient nature of the experiment. The decay time is related to transverse relaxation and radiation damping (see Eq. (5.92)) and a reasonable measuring time (for example, a few seconds) requires sufficient homogeneity of the field across the probe. Possible alternatives to the free-precession NMR magnetometers for the calibration of low field standards are provided, for example, by optically p u m p e d and flowing water NMR magnetometers. Recent developments in the commercially available flowing water magnetometers have been announced [5.110]. With a very small sensing head (---10 m m 3) and a measuring range

5.5 MAGNETIC FIELD STANDARDS AND TRACEABILITY

265

extending from 1.4 ~T to 2.1 T, this instrument holds promise for improved and more generally available traceability to the SI units. In order to develop magnetic field standards with the very low uncertainties allowed for by the NMR techniques, it is imperative to achieve, especially at low and medium field strengths, shielding from the earth magnetic field [5.137]. To this end, calibration is performed by surrounding the coil with a large triaxial Helmholtz setup, supplied by three independent current sources, regulated in such a way to suppress the earth field components. The value of these components is around 40 ~T (vertical direction), 4 ~T (East-West direction), and 30 ~T (NorthSouth direction). With the setup shown in Fig. 4.8 (diameter 1.2 m), which displays a central spherical region of 10cm diameter with field homogeneity better than 5 x 10 -5, one can obtain, for example, cancellation of the earth field components down to about 20 nT by direct control of the supply currents. Earth field, however, is subjected to diurnal variations (of the order of 10-20 nT) and other uncontrolled environmental field sources are usually present. Consequently, for certain demanding low-field calibration requirements like those sometimes required with geomagnetometers, active external field cancellation by feedback is made. In the apparatus developed by Park et al. [5.138], schematically shown in Fig. 5.50, the standard solenoid, surrounded by a triaxial Helmholtz coil, is kept with its axis aligned in the East-West direction in a wooden building, far from power lines and other buildings. A series connected auxiliary triaxial Helmholtz coil, located at a distance of about 50 m in order to avoid interference with the solenoid, is used in association with an optically p u m p e d Cs magnetometer placed at its center. Any drift of the environmental field is detected by the atomic resonance magnetometer and converted into a current of proportional strength, which is injected into the main Helmholtz coils. A standard deviation of the resulting compensated field of the order of 0.1 nT is demonstrated against an actual drift of 19 nT over a period of I h. The standard solenoid, whose constant is determined by means of a C s - H e 4 magnetometer [5.132], is endowed with a number of supplementary windings, all laid on the same fused silica former of radius 10.1 cm and length 0.938 m, leading to uniformity better than 0.5 x 10 -6 over a 4 cm long central region [5.139]. A standard coil is characterized by a definite temperature dependence of its constant kH --p, oH/i, where i is the current, which must always be arranged as a function of the actual measuring temperature. An increase of temperature normally leads to a decrease of kH because of the thermal expansion of the former and the winding. Values of (1/kH)(dkH/dT) of the order of 10-5-10 -4 K -1 can be found in standard

266

CHAPTER 5 Measurement of Magnetic Fields Cs-He AMR magnetometer

Auxiliary compensation coil

Current source

l/

urc__ so e source

tL..[L l

,.

ii

~

"

I,,

Standard solenoi J

cs,

.

I

....

I ......

magnetometer ~

'

.

.

.

.

.

\

Main

compensation coil

FIGURE 5.50 Highly accurate standard for low strength magnetic fields, endowed with active nulling of the earth magnetic field. Compensation better than I nT is obtained by keeping the reference field source, a solenoid with axial field uniformity better than 0.5 x 10-6 over a 4 cm wide central region, within a compensating triaxial Helmholtz coil. Drift vs. time of the earth field is actively compensated by detecting it through an optically pumped Cs magnetometer, place d at the center of an auxiliary triaxial Helmholtz coil. The ensuing signal is converted into an adjusting current, which is injected in the main Helmholtz coil. The field generated by the standard solenoid is measured by means of a Cs-He 4 atomic magnetic resonance (AMR) magnetometer (adapted from Ref. [5.138]).

coils [5.140] and are sometimes compensated by use of an extra winding [5.132]. Stable and controlled temperature conditions are therefore required in any calibration procedure, with the actual coil temperature measured, for example, by means of a Pt resistor or determined by means of a separate measurement of its electrical resistance. It m a y happen that a certain coil cannot provide a sufficiently homogeneous field to be calibrated by means of the NMR method. If an NMR calibrated coil having suitable size is available, transfer of its coil constant can be envisaged by coaxially inserting the smaller coil within the bigger one and supplying the two coils with currents generating opposite directed fields [5.94]. By placing a sensitive field sensor (for example, a fluxgate magnetometer) in the central position of the coil axis, the current values are determined, via calibrated resistors,

5.5 MAGNETIC FIELD STANDARDS AND TRACEABILITY

267

which lead to zero field indication. The ratio of these values provides the ratio of the coil constants. The main source of error with this calibration procedure, which m a y be affected by a relative uncertainty of the order of 10 -3 , is related to the imperfect alignment of the coil axes. A possible and generally available alternative m e t h o d for transferring the constant kH from an N M R calibrated coil to a coil of u n k n o w n constant consists in comparing the readings of a Hall

R

1

T

"O

V2

L2

O

D' fM

O

w

Standard coil

1.02-

R = 30.-Q L = 12mH C = 50 pF

O E3

~= 1.01 O <

1.00-

,i

10

.

"

......

i

. . . . . . . .

1 O0

i

1000

9

.

'. . . . . .

!

9

10000

Frequency (Hz) F I G U R E 5.51 Simplified equivalent circuit of a standard coil of inductance L1, resistance R1, and stray capacitance C1 and related measuring setup, made of search coil L2 and digital voltmeter. The curve shows the calculated frequency dependence of the coil constant kH,AC(f), normalized with respect to the DC value kH,DC (Eq. (5.114)).

CHAPTER 5 Measurement of Magnetic Fields

268

magnetometer when switching the same sensing head between the two coils. The currents can be regulated in such a way so as to obtain close readings so that the uncertainty becomes quite independent of the absolute accuracy of the Hall magnetometer. By judicious choice of the type of sensing head (i.e. tradeoff between sensitivity and thermal stability) and tight control of the head direction during the measurement and compensation of the earth field, a relative uncertainty lower than 10-3 can be achieved. The constant kH of a standard coil, measured under AC excitation, is a function of frequency f. In order to determine the kH(f) behavior, a small search coil of turn-area aw can be placed within the standard coil and the correspondingly induced signal can be measured by means of a digital voltmeter. If the search coil is made of a few well separated turns and it is connected by a short low-capacitance coaxial cable to the voltmeter, the problem can be treated through the simplified equivalent circuit of Fig. 5.51, where R1, C1, and L1 represent the resistance, the stray capacitance, and the inductance of the standard coil, respectively. Let V2, i0, and il be the r.m.s, values of the voltage induced in the search coil, of the supply current, and of the active current, respectively. Since 172 = 2"all, Haw, the AC coil constant kH,AC(f ) = /~0/~fi0 is obtained as 172 kH'ACr

1

i 0 2"nfaw"

By solving the primary circuit, we find that depends on frequency as

(5.113)

kH,AC(f)--kH,DCil/iO

1

kH'AC(f) -- kH'DC ~(1 - 41r2f2L1C 1)2 _}_ 4"rr2f2R2C2

(5.114)

Figure 5.51 provides an example of the predicted frequency dependence of the ratio kH,AC(f)/kH,DC calculated on a standard solenoid. A recent international comparison on a coil standard, involving seven different NMIs, has demonstrated good overall agreement on the determined values of kH,AC(f)- The resulting expanded relative uncertainty is U = 1.3 x 10 -3 for frequencies up to 2 kHz and U = 1.9 x 10 -3 in the range 2-20 kHz [5.140].

References

5.1. A.E. Drake, "Traceable magnetic measurements," J. Magn. Magn. Mater., 133 (1994), 371-376. 5.2. http://www.bipm.fr/BIPM-KCDB.

REFERENCES

269

5.3. K. Weyand, "Magnetic field," in Units and Fundamental Constants in Physics and Chemistry, Landolt-B6rnstein New Series (J. Bortfeldt and B. Kramer, eds., Springer Verlag, Berlin, 1991), 268-288. 5.4. http://www.metrolab.com. 5.5. W.V. Hassenzahl, W.S. Gilbert, M.I. Green, and P.J. Barale, "Magnetic field measurements of model SSC dipoles," IEEE Trans. Magn., 23 (1987), 484-487. 5.6. O. Kaul, Y. Holler, K. Sinram, and U. Berghaus, "Field measurements on the HERA e-ring magnets," IEEE Trans. Magn., 24 (1988), 966-969. 5.7. W.F. Brown and J.H. Sweer, "The fluxball," Rev. Sci. Instrum., 16 (1945), 276-279. 5.8. H. Zijlstra, Experimental Methods in Magnetism (Amsterdam: North-Holland, 1967), Vol. 2, p. 7. 5.9. A.E. Drake and A. Hartland, "A vibrating coil magnetometer for the determination of the magnetization coercive force of soft magnetic materials," J. Phys. E: Sci. Instrum., 6 (1973), 901-902. 5.10. W. Hagedorn and H.H. Mende, "A method for inductive measurement of magnetic flux density with high geometrical resolution," J. Phys. E: Sci. Instrum., 9 (1976), 44-46. 5.11. A.P. Chattock, "On a magnetic potentiometer," Philos. Mag., 24 (1887), 94-96. 5.12. H. Zijlstra, Experimental Methods in Magnetism (Amsterdam: North-Holland, 1967), Vol. 2, p. 56. 5.13. H. Pf~itzner and P. Sch6nuber, "On the problem of the field detection for single sheet tester," IEEE Trans. Magn., 27 (1991), 778-785. 5.14. IEC Standard Publication 60404-3, Methods of measurement of the magnetic properties of magnetic sheet and strip by means of single sheet tester (Geneva: IEC Central Office, 1992). 5.15. T. Nakata, Y. Kawase, and M. Nakano, "Improvement of measuring accuracy of magnetic field strength in single sheet testers by using two H-coils," IEEE Trans. Magn., 23 (1987), 2596-2598. 5.16. E. Steingroever, Magnetic Measuring Techniques (K61n: Magnet Physik, 1989). 5.17. B.D. Cullity, Introduction to Magnetic Materials (Reading, MA: Addison & Wesley, 1972), p. 35. 5.18. E.J. Kennedy, Operational Amplifier Circuits (New York: Holt, Rinehart and Winston, 1988), p. 1. 5.19. K.B. Klaassen, Electronic Measurement and Instrumentation (Cambridge: Cambridge University Press, 1996), p. 204. 5.20. BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, Guide to the Expression of Uncertainty in Measurement (Geneva, Switzerland: International Organization for Standardization, 1993).

270

CHAPTER 5 Measurement of Magnetic Fields

5.21. L. Rahf, "A precise versatile calibrator for fluxmeters," J. Magn. Magn. Mater., 83 (1990), 541-542. 5.22. H. Zijlstra, Experimental Methods in Magnetism (Amsterdam: North-Holland, 1967), Vol. 2, p. 49. 5.23. A charge in crossed electric and magnetic fields actually follows a cycloidal trajectory, but the velocity is proportionally increased with respect to a straight path. No effect on the resistivity is consequently expected. See for instance: R.S. Allgaier, "Some general input-output rules governing Hall coefficient behavior," in The Halt Effect and its Applications (C.L. Chien and C.R. Westgate, eds., New York: Plenum Press, 1980), 375-397. 5.24. A.C. Beer, "Hall effect and the beauty and challenges of science," in The Hall Effect and its Applications (C.L. Chien and C.R. Westgate, eds., New York: Plenum Press, 1980), 299-338. 5.25. H.P. Baltes and R.S. Popovic, "Integrated semiconductor magnetic field sensors," Proc. IEEE, 74 (1986), 1107-1132. 5.26. J.M. Ziman, Principles of the Theory of Solids (Cambridge: Cambridge University Press, 1979), p. 250. 5.27. F.Y. Yang, K. Liu, K. Hong, D.H. Reich, P.C. Searson, and L. Chien, "Large magnetoresistance of electrodeposited single-crystal bismuth thin films," Science, 284 (1999), 1335-1337. 5.28. R.M. Bozorth, Ferromagnetism (New York: Van Nostrand, 1951), p. 745. 5.29. I.A. Campbell and A. Fert, "Transport properties of ferromagnets," (E.P. Wohlfarth, ed., Amsterdam: North-Holland, Ferromagnetic Materials, 1982), Vol. 3, 747-804; A. Fert, and D.K. Lottis, "Magnetotransport phenomena," in Concise Encyclopedia of Magnetic and Superconducting Materials (J. Evetts, ed., Oxford: Pergamon, 1982), 287-296. 5.30. M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, "Giant magnetoresistance of (001) Fe/(001) Cr magnetic superlattices," Phys. Rev. Lett., 61 (1988), 2472-2475. 5.31. J.E. Simpkins, "Microminiature Hall probes for use in liquid helium," Rev. Sci. Instrum., 39 (1968), 570-575. 5.32. J. Zhang, P. Sheldon, and R.K. Ahrenkiel, "A Hall probe technique for characterizing high-temperature superconductors," Rev. Sci. Instrum., 63 (1992), 2259-2262. 5.33. C.R. Schott, R.S. Popovic, S. Alberti, and M.Q. Tran, "High accuracy magnetic field measurements with a Hall probe," Rev. Sci. Instrum., 70 (1999), 2703-2707. 5.34. C.D. Lustig, A.W. Baird, W.F. Chaurette, H. Minden, W.T. Maloney, and A.J. Kurtzig, "High-resolution magnetic field measurement system for recording heads and disks," Rev. Sci. Instrum., 50 (1979), 321-325.

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5.53. Schonsted, E. O. Saturable measuring device and magnetic core, US Patent No. 2,916,696. 5.54. O.V. Nielsen, B. Hernando, J.R. Petersen, and F. Primdahl, "Miniaturisation of low-cost metallic glass fux-gate sensors," J. Magn. Magn. Mater., 83 (1990), 405-406. 5.55. D. Son, "A new type of fluxgate magnetometer for low magnetic fields," Phys. Scr., 39 (1989), 535-537. 5.56. S. Takeuchii and K. Harada, "A resonant-type amorphous ribbon magnetometer driven by an operational amplifier," IEEE Trans. Magn., 20 (1984), 1723-1725. 5.57. K. Mohri, K. Kasai, T. Kondo, and H. Fujiwara, "Magnetometers using two amorphous core multivibrator bridge," IEEE Trans. Magn., 19 (1983), 2142-2144. 5.58. S. Kawahito, Y. Sasaki, H. Sato, T. Nakamura, and Y. Tadokoro, "A fluxgate magnetic sensor with microsolenoids and electroplated permalloy cores," Sens. Actuators, 43 (1994), 128-134. 5.59. P. Kejik, L. Chiesi, B. Janossy, and R.S. Popovic, "A new compact 2D planar fluxgate sensor with amorphous metal core," Sens. Actuators, 81 (2000), 180-183. 5.60. P. Ripka, "New directions in fluxgate sensors," J. Magn. Magn. Mater., 2 1 5 216 (2000), 735-739. 5.61. P. Ripka, "Noise and stability of magnetic sensors," J. Magn. Magn. Mater., 1 5 7 - 1 5 8 (1996), 424-427. 5.62. M.D. Mermelstein and A. Dandridge, "Low-frequency magnetic field detection with a magnetostrictive amorphous metal ribbon," Appl. Phys. Left., 51 (1987), 545 - 547. 5.63. F. Bucholtz, D.M. Dagenais, C.A. Villaruel, C.K. Kirkendall, J.A. McVicker, A.R. Davis, S.S. Patrick, K.P. Koo, G. Wang, H. Valo, E.J. Eidem, A. Andersen, T. Lund, R. Gjessing, and T. Knudsen, "Demonstration of a fiber optic array of three-axis magnetometers for undersea application," IEEE Trans. Magn., 31 (1995), 3194-3196. 5.64. F.G. West, W.J. Odom, J.A. Rice, and T.C. Penn, "Detection of low-intensity 'magnetic fields by means of ferromagnetic films," J. Appl. Phys., 34 (1963), 1163-1164. 5.65. C.J. Bader and C.S. DeRenzi, "Recent advances in the thin-film inductance-variation magnetometer," IEEE Trans. Magn., 10 (1974), 524-527. 5.66. M. Takajo, J. Yamasaki, and EB. Humphrey, "Domain observation in Fe and Co based amorphous wires," IEEE Trans. Magn., 29 (1993), 3484-3486.

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5.67. L.V. Panina, K. Mohri, K. Bushida, and M. Noda, "Giant magnetoimpedance and magneto-inductive effects in amorphous alloys," J. Appl. Phys., 76 (1994), 6198-6203. 5.68. K. Mohri, K. Kawashima, T. Kohzawa, and H. Yoshida, "Magneto-inductive element," IEEE Trans. Magn., 29 (1993), 1245-1248. 5.69. L.V. Panina, K. Mohri, T. Uchiyama, and M. Noda, "Giant magnetoimpedance in Co-rich amorphous wires and films," IEEE Trans. Magn., 31 (1995), 1249-1260. 5.70. M. Senda, O. Ishii, Y. Koshimoto, and T. Toshima, "Thin-film magnetic sensor using high frequency magneto-impedance (HFMI) effect," IEEE Trans. Magn., 30 (1994), 4611-4613. 5.71. K. Mohri, T. Uchiyama, L.P. Shen, C.M. Cai, and L.V. Panina, "Sensitive micro magnetic sensor family utilizing magneto-impedance (MI) and stressimpedance (SI) effects for intelligent measurements and controls," Sens. Actuators A, 91 (2001), 85-90. 5.72. A.D. Kerse~ D.A. Jackson, and M. Corke, "Single-mode fibre-optic magnetometer with DC bias stabilization," IEEE J. Lightwave Technot., 3 (1985), 836-840. 5.73. A. Pantinaikis and D.A. Jackson, "High-sensitivity low-frequency magnetometer using magnetostrictive primary sensing and piezoelectric signal recovery," Electron. Lett., 22 (1986), 737-738. 5.74. M.D. Mermelstein, "Magnetoelastic amorphous metal fluxgate magnetometer," Electron. Lett., 22 (1986), 525-526. 5.75. P.T. Squire and M.R.J. Gibbs, "Shear-wave magnetometry," IEEE Trans. Magn., 24 (1988), 1755-1757. 5.76. H. Chiriac, M. Pletea, and E. Hristoforou, "Fe-based amorphous thin films as a magnetoelastic sensor material," Sens. Actuators, 81 (2000), 166-169. 5.77. P. Grivet, "High field magnetometry," in Proc. Int. Conf. on High Magnetic Fields (H. Kolm, B. Lax, F. Bitter, and R. Mills, eds., Cambridge, MA: MIT Press and Wiley, 1962), 54-84. 5.78. M.N. Deeter, A.H. Rose, and G.W. Day, "Sensitivity limits to ferrimagnetic Faraday effect magnetic field sensors," J. Appl. Phys., 70 (1991), 6407-6409. 5.79. M.V. Valeiko, P.M. Vetoshko, R.I. Knonov, A.Ya. Perlov, M.Yu. Sharonov, and A.Yu. Toporov, "Magneto-optical visualizer-magnetometer of high magnetic fields," IEEE Trans. Magn., 31 (1995), 4293-4296. 5.80. M.N. Deeter, G.W. Day, R. Wolfe, and V.J. Fratello, "Magneto-optic magnetic field sensors based on uniaxial iron garnet films in optical waveguide geometry," IEEE Trans. Magn., 29 (1993), 3402-3404. 5.81. O. Kamada, H. Minemoto, and S. Ishizuka, "Mixed rare-earth garnet (TbY)IG for magnetic field sensors," J. Appl. Phys., 61 (1987), 3268-3270.

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5.82. R.P. Cowburn, A.M. Moulin, and M.E. Welland, "High sensitivity measurement of magnetic fields using microcantilevers," Appl. Phys. Lett., 71 (1997), 2202-2204. 5.83. C. Rossel, M. Willemin, A. Gasser, H. Bothuizen, G.I. Mijer, and H. Keller, "Torsion cantilever as magnetic torque sensor," Rev. Sci. Instrum., 69 (1998), 3199-3203. 5.84. B. Eyre, K.S.J. Pister, and W. Kaiser, "Resonant mechanical magnetic sensor in standard CMOS," IEEE Electron. Device Lett., 19 (1998), 496-498. 5.85. K. Weyand, "An NMR marginal oscillator for measuring magnetic fields below 50 mT," IEEE Trans. Instrum. Meas., 38 (1989), 410-414. 5.86. C.P. Slichter, Principles of Magnetic Resonance (Berlin: Springer, 1990), 1. 5.87. E Bloch, "Nuclear induction," Phys. Rev., 70 (1946), 460-474. 5.88. See for instance C.P. Slichter, Principles of Magnetic Resonance (Berlin: Springer, 1990), 33; A.P. Guimar~es, Magnetism and Magnetic Resonance in Solids (New York: Wilej6 1998), p. 189. 5.89. C. Kittel, Introduction to Solid State Physics (New York: Wiley, 1968), p. 501. 5.90. P.J. Mohr and B.N. Taylor, "CODATA recommended values of the fundamental physical constants: 1998," J. Phys. Chem. Ref. Data, 28 (1999), 1713-1852. 5.91. F. Bloch, W.H. Hansen, and M. Packard, "The nuclear induction experiment," Phys. Rev., 70 (1946), 474-485. 5.92. K. Weyand, "Magnetometer calibration setup controlled by nuclear magnetic resonance," IEEE Trans. Instrum. Meas., 48 (1999), 668-671. 5.93. Z. Zhong-Hua and W. Den-An, "An NMR magnetic field meter for measuring high fields at liquid-helium temperature," IEEE Trans. Instrum. Meas., 36 (1987), 280-284. 5.94. K. Weyand, "Maintenance and dissemination of the magnetic field unit at PTB," IEEE Trans. Instrum. Meas., 50 (2001), 470-473. 5.95. G. Boero, J. Frouchi, B. Furrer, P.A. Besse, and R.S. Popovic, "Fully integrated probe for nuclear magnetic resonance magnetometry," Rev. Sci. Instrum., 72 (2001), 2764-2768. 5.96. M. Packard and R. Varian, "Free nuclear induction in the earth's magnetic field," Phys. Rev., 93 (1954), 941. 5.97. F.E. Acker, "Calculation of the signal voltage induced in a toroid proton precession magnetometer sensor," IEEE Trans. Geosci. Electron., 9 (1971), 98-103. 5.98. G. Faini and O. Svelto, "Signal-to-noise considerations in a nuclear magnetometer," Nuovo Cimento, 23S (1962), 55-66. 5.99. N. Bloembergen and R.V. Pound, "Radiation damping in magnetic resonance experiments," Phys. Rev., 95 (1954), 8-12.

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5.100. W.G. Clark T.W. Hijmans, and W.H. Wong, "Multiple coil pulsed NMR method for measuring the multipole moments of particle accelerator bending magnets," J. Appl. Phys., 63 (1988), 4185-4186. 5.101. S. Kato, K. Maheata, K. Ishibashi, Y. Wakuta, and T. Shintomi, "Development of a small-probe nuclear magnetic resonance system for measuring magnetic field uniformity," Rev. Sci. Instrum., 68 (1998), 1469-1473. .

5.102. D.I. Hoult, "Fast recovery, high sensitivity NMR probe and preamplifier for low frequencies," Rev. Sci. Instrum., 50 (1979), 193-200. 5.103. C. Sherman, "High-precision measurement of the average value of a magnetic field over an extended path in space," Rev. Sci. Instrum., 30 (1959), 568-575. 5.104. J.M. Pendlebury, K. Smith, P. Unsworth, G.L. Greene, and W. Mampe, "Precision field averaging NMR magnetometer for low and high fields, using flowing water," Rev. Sci. Instrum., 50 (1979), 535-540. 5.105. C.G. Kim, K.S. Ryu, B.C. Woo, and C.S. Kim, "Low magnetic field measurement by NMR using polarized flowing water," IEEE Trans. Magn., 29 (1993), 3198-3200. 5.106. C.P. Slichter, Principles of Magnetic Resonance (Berlin: Springer, 1990), p. 11. 5.107. N.F. Ramsey, Molecular Beams (London: Oxford University Press, 1965), p. 145. 5.108. I.I. Rabi, "Space quantization in a gyrating magnetic field," Phys. Rev., 51 (1937), 652. 5.109. B.C. Woo, C.G. Kim, P.G. Park, C.S. Kim, and V.Y. Shifrin, "Low magnetic field measurement by separated NMR detector using flowing water," IEEE Trans. Magn., 33 (1997), 4345-4347. 5.110. http://www.fwmagnetometer.com/ 5.111. A. Overhauser, "Polarization of nuclei in metals," Phys. Rev., 89 (1953), 411-415. 5.112. T.R. Carver and C.E Slichter, "Experimental verification of the Overhauser nuclear polarization effect," Phys. Rev., 102 (1956), 975-980. 5.113. I. Solomon, "Relaxation processes in a system of two spins," Phys. Rev., 99 (1955), 559-565. 5.114. H.G. Beljers, L. van der Kint, and J.S. van Wieringen, "Overhauser effect in free radicals," Phys. Rev., 95 (1954), 1683. 5.115. The magnetic dipolar interaction is the basic physical mechanism responsible for resonant line broadening. A spin carrier in motion probes a random interaction with fields of both signs, whose dephasing effect is the lower the faster is the carrier. The ensuing resonant line narrowing effect is called motional narrowing. With electrons, the similar effect of

276

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exchange narrowing can additionally take place. See, for instance, C. Kittel, Introduction to Solid State Physics (New York: Wiley, 1968). 5.116. http://www.gemsys.on.ca/ 5.117. N. Kernevez and H. G16nat, "Description of a high sensitivity CW scalar DNP-NMR magnetometer," IEEE Trans. Magn., 27 (1991), 5402-5404. 5.118. N. Kernevez, D. Duret, M. Moussavi, and J.M. Leger, "Weak field NMR and ESR spectrometers and magnetometers," IEEE Trans. Magn., 28 (1992), 3054-3059. 5.119. F. Primdahl, "Scalar magnetometers for space applications," in Measurement Techniques in Space Plasmas: Fields (R.E Pfaff, J.E. Borovsky, and D.T. Young, eds., Washington, DC: American Geophysics Union, 1998), 85-89. 5.120. E Hartmann, "Resonance magnetometers," IEEE Trans. Magn., 8 (1972), 66-75. 5.121. D. Duret, M. Moussavi, and M. Beranger, "Use of high performance electron spin resonance materials for the design of scalar and vectorial magnetometers," IEEE Trans. Magn., 27 (1991), 5405-5407. 5.122. H. Gebhardt and E. Dormann, "ESR gaussmeter for low-field applications," J. Phys. E: Sci. Instrum., 22 (1989), 321-324. , 5.123. According to the standard spectroscopic notation rules, the symbol 11S0has the following meaning: 1 is the first principal quantum number, S stands for the value L = 0 of the total orbital angular momentum (the symbol P would stand for L -- 1, D for L -- 2, etc.), the subscript number 0 represents the value of the total angular momentum J, the superscript number 1 designates the multiplicity, that is the number of possible J values for a given L. J is given by the combination of L and the spin total momentum S (Russel-Saunders coupling of spin and orbital momenta) and runs from L + S through IL - S[. 5.124. F.D. Colegrove and P.A. Franken, "Optical pumping of helium in the 3S1 metastable state," Phys. Rev., 119 (1960), 680-690. 5.125. D.D. McGregor, "High-sensitivity helium resonance magnetometers," Rev. Sci. Instrum., 58 (1987), 1067-1076. 5.126. Primdhal, F. (2001). "Resonance magnetometers", in Magnetic Sensors and Magnetometers (P. Ripka, ed., Norwood, MA: Artech House, 2001), 267-304. 5.127. A.L. Bloom, "Principles of operation of the rubidium vapor magnetometer," Appl. Opt., 1 (1962), 61-68. 5.128. R.D. Evans, The Atomic Nucleus (New York: McGraw-Hill, 1955), p. 181. 5.129. W.E. Bell and A.L. Bloom, "Optical detection of resonance in alkali metal vapor," Phys. Rev., 107 (1957), 1559-1565.

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5.130. C.G. Kim and H.S. Lee, "Optical pumping magnetic resonance in Cs atoms for use in precise low-field magnetometry," Rev. Sci. Instrum., 69 (1998), 4152-4155. 5.131. W.H. Farthing and W.C. Folz, "Rubidium vapor magnetometer for near Earth orbiting spacecraft," Rev. Sci. Instrum., 38 (1967), 1023-1030. 5.132. V.Ya. Shifrin, E.B. Alexandrov, T.I. Chikvadze, V.N. Kalabin, N.N. Yakobson, V.N. Khorev, and P.G. Park, "Magnetic flux density standards for geomagnetometers," Metrologia, 37 (2000), 219-227. 5.133. F.D. Colegrove, L.D. Schearer, and G.K. Walters, "Polarization of He 3 gas by optical pumping," Phys. Rev., 132 (1963), 2561-2572. 5.134. R.E. Slocum and B.I. Marton, IEEE Trans. Magn., 10 (1974), 528-5531. 5.135. J.L. Flowers, B.W. Petley, and M.G. Richards, "A measurement of the nuclear magnetic moment of the helium-3 atom in terms of that of the proton," Metrologia, 30 (1993), 75-87. 5.136. BIPM, Mutual Recognition of National Measurement Standards and of

Calibration and Measurement Certificates Issued by the National Metrology Institutes (S6vres: Bureau International des Poids et Mesures, 1999). 5.137. The NMR measurement is scalar in nature. Automatic cancellation of the earth field cannot be obtained, as in the vectorial magnetometers (e.g. Hall sensors), by repeating the measurement with the current inverted in the calibration coil. 5.138. P.G. Park, V.Ya. Shifrin, V.N. Khorev, and Y.G. Kim, "Magnetic flux density standard for low magnetic field at KRISS," in Digest of the CPEM 2000 Conference, Sydney, May 2000 (J. Hunter and L. Johnson, eds., 2000), p. 242, Paper TUP4-4. 5.139. V.Ya. Shifrin, C.G. Kim, and P.G. Park, "Atomic magnetic resonance based current source," Rev. Sci. Instrum., 67 (1996), 833-836. 5.140. K. Weyand, E. Simon, L. Henderson, J. Bartholomew, G.M. Teunisse, I. Blanc, G. Crotti, J. Kupec, A. Jeglic, G. Gersak, J. Humar, and L.O. Puranen, "International comparison of magnetic flux density by means of field coil transfer standards," Metrologia, 38 (2001), 187-191.

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PART III

Characterization of Magnetic Materials

The magnetic characterization of materials has two basic aims: (1) the measurement of the intrinsic magnetic parameters, such as the saturation magnetization, the Curie temperature, the exchange constant, the anisotropy and the magnetostriction constants; (2) the determination of the constitutive law of the material, i.e. the dependence of the magnetization on the effective field, which is manifest in the manifold phenomenology of magnetization curves and hysteresis and the related energy aspects. In this part of the book, we shall discuss a number of experimental methods that have been developed accordingly. They can be classified on the basis of the specific effects that are exploited to reveal and measure the magnetic quantities of interest, keeping in mind that different techniques can naturally display intersecting domains of application. A given experimental principle can be applied, for example, either to the determination of fundamental magnetic parameters or, under a different context, to the technical characterization of materials. In summary, a possible classification of the measuring techniques can be attempted according to the following scheme: (1) Techniques based on the measurement of forces and mechanical torques. A non-uniform field generates a translational force on a magnetized sample, while a uniform field can provide a torque. (2) Inductive methods, where the signal induced, according to the Faraday-Maxwell law, on a pickup coil linked with the sample is integrated to provide the material magnetization. The related flux measuring principle has been introduced in Chapter 5. (3) Techniques based on the measurement of the stray fields emitted by magnetized open samples (magnetometric methods). (4) Magnetooptical techniques (e.g. Kerr effect hysteresisgraphs). (5) Magnetostrictive

280

PART III Characterization of Magnetic Materials

techniques, based on the measurement of the magnetization-related dimensional changes of the sample. (6) Magnetic resonance methods. The basic aim of this book is one of providing a guide in the field of magnetic characterization of technical materials, namely soft and hard magnets. Consequently, a good deal of the discussion here will be devoted to inductive measuring techniques and their application to the determination of the magnetization curves. A material-related, rather than technique-related, presentation will be pursued, with some emphasis placed on standardized procedures and the specific constraints imposed by different materials on defined measuring methods. In our presentation, we will systematically refer, when applicable, to the international written measuring standards, specifically the IEC 60404 series standards. A short presentation will also be provided of the base methods employed in the determination of the intrinsic magnetic parameters. Finally, attention will be devoted to metrological issues, with a synthetic discussion on the subject of uncertainty and traceability in magnetic measurements.

CHAPTER 6

Magnetic Circuits and General Measuring Problems

When we talk of magnetic characterization of materials, we refer in most cases to experiments aiming at the determination of the dependence of the macroscopic magnetization M on the effective magnetic field in the material H. The pursued M(H) relationship is obviously assumed to have a meaning at the macroscopic level, which implies that M is a quantity resulting from spatial averaging over the measuring region or the whole test specimen. Provided the conditions of physical homogeneity of the samples are satisfied, this is totally acceptable and it is what we basically need in most applications. Here, the important issues in the evaluation of the performances of magnetic cores are the hysteresis phenomena and the associated energetic features. We could, of course, define local relationships between field and magnetization, for instance, at the scale of the magnetic domains, and carry out the pertaining experiments. It is an approach to the magnetization process which, though of fundamental importance in providing a physically based microscopic background to the presently treated macroscopic material features, cannot be discussed here, because it lies outside the general scope of our treatise. To measure the intrinsic dependence of magnetization on the field in the tested soft or hard materials is an ideal and somewhat elusive goal because the long-range nature of the demagnetizing fields makes the behavior of any test specimen crucially related to its geometrical features. Sometimes this combines with undesired application of uncontrolled stress, when accommodating the sample in the testing fixture. Thus, practical constraints, like those related to the realization of a convenient magnetic circuit or general acceptance of specific measuring methods by the industry, may inevitably lead to approximate realizations of the intrinsic measurements. The chief objective is actually always one of achieving excellent measuring reproducibility, i.e. equivalence of results obtained at different times in different laboratories. Measuring standards 281

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CHAPTER 6 Magnetic Circuits

are purportedly developed to achieve such an objective, the price to pay for it being often represented by a systematic deviation of the determined values from the intrinsic values of the measured quantities. Central to the problem of measurements in magnetic materials is the role of the demagnetizing fields. They unavoidably arise any time the magnetization vector M suffers a discontinuity (V.M # 0) and largely influence the measuring methodology and accuracy. They can, for example, impose additional energetic burden on the field sources, generate macroscopic magnetization inhomogeneifies in the test specimens, and give rise to inaccuracies in the determination of the effective fields. We have previously discussed (Section 1.2) the concept of demagnetizing fields and stressed how only by having ellipsoidal samples one can achieve a definite demagnetizing coefficient and uniform magnetization in the material. As discussed in the following, there are reasons to choose open test samples in many experiments. Ellipsoidal or spherical samples are as far as possible adopted, but practical reasons are often invoked for adopting cylindrical or parallelepipedic shapes. We know that with these shapes the demagnetizing field H d and, consequently, the magnetization M are no more uniform and the determination of the effective field is forcedly approximate.

6.1 C L O S E D

MAGNETIC

CIRCUITS

Closed magnetic circuits are by and large preferred to open samples in the measurement of magnetization curve and hysteresis, provided the nature of the material, the specimen size, and the values of field and magnetization make such a possibility available. The sample itself can either be shaped in order to achieve flux closure by itself or it can be provided with a suitable yoke, made of a high-permeability material. Crucial to the notion of magnetic circuit and the magnetic testing of materials, in general, is the assumption of a macroscopic scalar relationship between the effective field H and M. This is a reasonably achieved condition in most cases of practical interest, dealing, for example, with isotropic, nearly isotropic or near-zero anisotropy energy materials, as well as highly oriented alloys or single crystals excited along an easy axis. Whenever H and M happen to be non-collinear, it might not be totally clear how to define and measure an intrinsic magnetic behavior of the material, since the measured properties eventually depend on how the magnetic circuit is realized in more than one dimension, though, in the limiting case, an infinitely extended body in three dimensions can, in principle, be emulated [6.1] and

6.1 CLOSED MAGNETIC CIRCUITS

283

a complete tensorial M(H) relationship can be provided. If we concentrate, for the time being, on the typical case of scalar magnetic susceptibility materials, we can consider some basic ways to realize open and closed magnetic circuits suitable to meaningful M(H) determination. The intuitive approach to the closed magnetic circuit calls for the realization of a toroidal specimen. Figure 6.1 provides few examples of practical toroids, obtained by cutting and stacking a number of rings (a), spirally winding a ribbon sample (b), and pressing or sintering powders (c). It is clear that, if the primary winding is uniformly laid around the core and the previous scalar condition is satisfied, the flux lines must follow the m i n i m u m energy circumferential paths. Stacked specimens are suitable to isotropic magnetic laminations and, in the case of moderate anisotropy, like the one encountered in conventional nonoriented electrical steels, they provide properties averaged over the lamination plane. With strip-wound cores the magnetic testing is associated with a definite direction in the plane of the sheet, which often coincides with the rolling direction (RD). In most cases RD is a preferred direction, where the anisotropy energy of the material is m i n i m u m both in crystalline and amorphous alloys. We can actually think that a strip-wound core behaves as an infinitely long ribbon sample, although some minor interlamina flux propagation effect can

(a)

(b)

(c) (d)

FIGURE 6.1 Examples of practical toroidal cores, obtained by stacking rings punched out of a lamination (a), winding a ribbon-like sample (b), sintering or bonding a magnetic powder (c). The generation of a high strength circumferential field by means of a current i flowing in an axial conductor of large cross-sectional area is schematically shown in (c). The magnetic path length is calculated as the arithmetic average l m - - ( ~ r / 2 ) ( D i q- D o ) of the inside and outside circumferential length if the condition Do ~ 1.1Di is, as shown in (d), fulfilled.

284

CHAPTER 6 Magnetic Circuits

complicate, to some extent, the detailed analysis of the measured material properties [6.2]. Though excellent from the viewpoint of flux closure, the toroidal geometry suffers from certain testing drawbacks. It is useful to summarize them, because we can have, through their analysis, a glimpse at the difficulties and ambiguities frequently accompanying magnetic measurements. (1) The preparation of the core and of the primary and secondary windings can be extremely tedious. Every new test specimen requires, for instance, the preparation of new windings, though, sometimes, semi-rigid holders can be made with provisions for sample insertion and successive sealing, thereby forming uniformly wound toroids. (2) When need exists for characterizing a lamination along a definite direction (e.g. RD in Fe-(3 wt%)Si and in crystalline and amorphous ribbons), a strip-wound core must be built, which implies the creation of bending stresses (i.e. placing half strip cross-section in compression and the other half in tension) and dependence of the measured properties on the core diameter. This kind of testing, which is generally accompanied by suitable stress relief and, in a significant number of cases, anisotropy-inducing annealing treatments, is very often specifically aimed at determining the features of a given core (or even a final component), rather than pursuing the characterization of the intrinsic material properties. (3) Basically, only very soft magnets can be fully tested as toroids because the field strength available through the primary winding is limited. In some cases, it may be difficult to approach the technical saturation of the material and it is expedient to generate the required field strength by making the primary current to flow in an axial copper rod of large crosssectional area (see Fig. 6.1c). (4) A crucial problem in accurate measurements is represented by the decrease of the applied field on passing from the inside to the outside boundary of the toroid. The circumferential field lines generated by a uniformly laid primary winding are obviously associated with an inverse dependence of the field strength on their diameter D, being the associated magnetic path length I(D) = vrD (see Fig. 6.2a). If the intrinsic material characterization is the chief goal of the measurement, we need to maintain good uniformity of the magnetization over the sample cross-section. The rule is therefore prescribed that the ratio between outside and inside diameters D o / D i ~- 1.1. A mean magnetic path length lm -- 'rtDm, with Dr. - (Di + Do)/2, is defined and with N1 primary turns and a current il, the field is given by H = Nlil/Im [6.3]. Figure 6.1d shows how slender the sample must be to comply with this rule, which is seldom fulfilled in practical inductive components. Here, the measured and the real magnetization curves appear as qualitatively shown in Fig. 6.2b.

6.1 CLOSED MAGNETIC CIRCUITS

285

t

D

LVJ

Hi Ho 0 (a)

q

Do

Diameter

True curve

cO

~-

/m =

N

rcDm

C

(b)

Field

FIGURE 6.2 The circumferential field generated by a uniformly laid primary winding decreases, on passing from the inside to the outside boundaries of a toroidal sample, like the inverse of the diameter D, because of the corresponding variation of the magnetic path length I(D)= frD. By taking the field as H -- Nlil/Im, with N1 and il the number of turns and the current, respectively, and l m the mean path length lrn = frDm~ w i t h D m = ( D i q - D o ) / 2 , the measured curve appears as qualitatively shown by the dashed line in (b).

To cope with the relatively high values of the ratio D o / D i in components, Irn is sometimes defined as the length of the m e a n field line, i.e. Im -f r ( D o - Di)/ln(Do/Di). Numerical interpolation methods, based on the acceptance of local B(H) relationships, defined for each circumferential field line, have actually been proposed, in order to approximately recover the true normal magnetization curve from the m e a s u r e d one [6.4].

286

CHAPTER 6 Magnetic Circuits

It is not clear, however, how to extend these methods in order to take into account hysteresis, as well as to introduce the eddy current effects under AC excitation. As previously stressed, we might better assume that the measurement pertains more to the characterization of a component than that of a material. This is much more so by considering that strip-wound cores may come with shapes well different from the circular one and can even be of the three-limb type. Empirical rules are then defined for the calculation of lm [6.5]. (5) A uniform primary winding generates a field around it, which is equivalent to the field generated by a hypothetical single turn located along the median circumference. A uniformly distributed secondary winding, equally acting as a single turn, correspondingly detects a flux, which spuriously adds to the flux in the material. This effect might become of some importance when the material permeability is very low (as it occurs on the approach to technical saturation) and it might require some obvious compensating arrangement. The strip-wound geometry, with the mentioned limitations regarding stresses, is the usual choice with thin laminations, for example, with rapidly solidified alloys, typically prepared with maximum thickness around 50 ~m, N i - F e tapes, and thinned Fe-Si laminations. On the contrary, it is not a practical option with the conventional low-carbon steels and Fe-Si laminations, generally ranging in thickness between 0.23 and I mm. At the same time, if we are interested in the material properties along a definite direction, as in grain-oriented materials, we cannot adopt the stacked-ring method. This requires, in any case, such a cumbersome sample preparation, which includes mandatory annealing to relieve the stresses introduced by the punching of the rings, that the favored closed magnetic circuit has become, since a number of decades, the one obtained by making a square assembly with suitably wide strips cut along the desired testing direction. This is universally known as the Epstein test frame. It is a standard for measurements in steel sheets from DC to 10 kHz [6.6, 6.7]. The Epstein rig is realized as schematically shown in Fig. 6.3. Four (or a multiple of four) strips of width 30 m m and length variable from 280 to 305 m m are prepared by cutting, possibly annealed for stress relief, and superposed at the comers to form a complete square. Great care must be devoted to geometrical perfection of the strips, which are required to remain flat after cutting and annealing and to have clean burr-free edges. The flux continuity at the square corners is ensured by double-lapped joints, as shown in Fig. 6.3b. A weight of 1 N placed on each corner joint provides good and reproducible flux closure. A decisive argument in favor of the Epstein frame in industrial testing is the possibility it offers of easily assembling and disassembling the magnetic

6.1 CLOSED MAGNETIC CIRCUITS

287

f

S

f

d=d" d

(a)

(b) 30

mm

(c)

FIGURE 6.3 Soft magnetic lamination strips in an Epstein square. (a) Formation of the closed magnetic circuit by superposition of the strips at the comers. (b) Double lapped joints. (c) Schematic view of the final arrangement of the strips, which are inserted within four equal formers, each provided with a secondary winding and, external to it, a primary winding. Both windings have the same number of turns and the four sections are series connected. The magnetic path length adopted in the pertaining standards is 0.94 m, slightly shorter than the median perimeter (dashed line).

circuit, with the strips either slipped into or taken out of a fixed winding arrangement. Each side of the square is provided with a secondary and, external to it, a primary winding, enwrapped on a rigid insulating former having rectangular cross-section (Fig. 6.3c). The solenoids have all the same n u m b e r of turns and each of them covers the same length of 19 cm. There is a total 700 primary and secondary turns in the frame used for DC and power frequency measurements (IEC 60404-2) and 200 turns in the frame recommended for m e d i u m frequency testing (IEC 60404-10). The four solenoids are series connected, somewhat emulating uniformly distributed windings. There is a wealth of results demonstrating the excellent repeatability and reproducibility of the measurements performed by using the Epstein frame method [6.8]. According to the IEC

288

CHAPTER 6 Magnetic Circuits

60404-2 measuring standard, the fixed magnetic path length lm -- 0.94 m is assumed throughout the experiments. In spite of the verified reproducibility, one might legitimately pose the question of how the measured properties are representative of the intrinsic material properties. The realized magnetic circuit, with its double overlapping corners, is not homogeneous and the assumption of a definite lm value, valid for all kinds of laminations and testing inductions, is expectedly the cause of systematic errors. Starting from the early experiments by Dieterly [6.9], such an assumption has been tested in several ways, as summarized by Sievert [6.10]. Figure 6.4a provides an example of the homogeneity of the magnetization in non-oriented (NO, 0.35 mm thick)) and grain-oriented (GO, 0.30 mm thick) Fe-Si strips assembled in a standard Epstein frame. It shows, in particular, that the polarization J(x), measured by means of a localized few-turn coil, decays at most by about 3 and 1.5% in NO and GO laminations, respectively, on going from the center (x -- 0) to the end (x = +9.5 cm) of the strip length covered by the windings. We can therefore reasonably talk of homogeneous magnetization in the measuring region and consequently assign a meaning to the concept of homogeneous effective field, defined via a given magnetic path length. If, for example, we make a power loss measurement, we can safely disregard the measuring uncertainty related to the spatial magnetization fluctuation shown in Fig. 6.4a (i.e. we find (j--~)2 _~ j(x)2). The problem remains that, depending on the kind of material and the magnetization level, a deviation of the actual magnetic path length with respect to the assumed value lm --- 0.94 rn can occur. The true loss value (and, obviously, true hysteresis loop) might then be significantly different from the measured value. To realize the importance of the systematic deviation imposed by the fixed 1m value, we should accurately determine the true loss value in some representative instances. A possible way to do so is by measuring J(x) and the effective field H(x) over the median region (x = 0) of the strip in Fig. 6.4a, by using local B-coils and calibrated flat H-coils (for the concept of H-coil see Section 5.1.1 and Fig. 5.3b), and making the required integrations (see also Section 7.2). The H-coil, laid on the strip surface, must be sufficiently thin (1 mm thick in the present experiments, including the wound wire), in order to provide a signal faithfully proportional to the tangential field. As previously shown, J(x) is quite homogeneous around x = 0 (negligible stray flux) and H is tangential. The ratio of the so-obtained true power loss Ptrue to the conventional power loss figure PEp, determined by measuring the primary current il and calculating the field as H = N~il/lm, with lm = 0.94 m, is given, for two types of NO Fe-(3 wt%)Si laminations, in Fig. 6.4b (curves (a) and (b)). We see that true and conventional loss figures differ in this

6.1 CLOSED MAGNETIC CIRCUITS

289 :'.,t

1.01

\

/C_K"<

~'

1.00

"'

G"13. 0.99 ~'

,, ,,

/

f7

A ; ~

', ,'

#

,.' j~

0.97,

,1NOJEp= 1.5 T,

0.96

.NO 9 fEP. 7"5 31..Fe~(3.wtYo),Si.

Z==f

"i I /==f ' X'

X

J

-8

-4

1.10

,'

Epsteinframe

0

4

x(cm)

(a)

c~. LU

/~

0.98

Epstein frame Fe-(3wt%)Si

8

e~/[

1.05

a.

.

.

.

.

.

.

a

a.+ 1.oo H-coil

/ 7 /

0.95

Eli ~ . l i I ' 0'.5 . . . . . . . .

(b)

/7

B-coil

1'.6 . . . . . . . . J (T)

1'.5 "

FIGURE 6.4 (a) Behavior of the local magnetization J(x), normalized to the value JEp obtained with the secondary windings, measured along one of the arms of an Epstein frame qEp = J(X-)) The local signal is detected at different distances x from the center of the strip (see inset) by means of a few-turn winding. The results refer to non-oriented (NO) and grain-oriented (GO) alloys. One strip is inserted on each side of a standard frame. (b) Ratio of 50 Hz true power loss to the power loss figure obtained with Epstein frame and fixed magnetic path length lm = 0.94 m (IEC 60404-2) in NO and GO laminations. (a)-(c) NO laminations. (d), (e) GO laminations. Curves (c)-(e) are adapted from results reported by Ahlers et al. [6.11]. The true power loss figures are obtained by measuring the effective field with tangential coils.

290

CHAPTER 6 Magnetic Circuits

representative case, where 0.35 and 0.50 m m thick sheets have been tested, at most around 1-2% at 1.5 T. The results of Ahlers et al. [6.11], which obtained the true loss value by means of a single strip tester, show discrepancies up to 5% in NO laminations (curve (c)) and 8% in GO laminations (curves (d) and (e)). From the viewpoint of standardization and reproducibility, this finding may be relatively unimportant, insofar as all laboratories incur in the same systematic deviation. It is of far greater concern when theoretical predictions, which always aim at the intrinsic material properties, have to be made and the correlation between different methods must be assessed. A closed magnetic circuit can be realized, whenever it is not possible or desirable to form it using the test specimen itself, by means of a yoke. It may occur, for example, that samples are shaped as bars or rods or, simply, high fields are needed. The yoke is an as soft as possible magnetic core, by which the ends of the specimen are connected. If it has much larger crosssectional area than the specimen, it provides, according to Eq. (3.22), nearzero reluctance return path for the flux, subjected to the condition of careful assembling of the circuit. To be stressed that, with permanent magnets, there is no reasonable alternative to flux closure by means of soft yokes (no ring samples), examples of which are schematically shown in Fig. 6.5. The circuit in Fig. 6.5a, used for testing of sheet and strip samples, is realized by means of a double strip-wound C-core. The strip is inserted between the pole faces of the yoke, whose upper portion is possibly acted on by a suspension device, in order to counterbalance, if required, part of its weight. The yoke can also be formed by stacking side-by-side individual laminations. If these are made of grain-oriented alloys, it is required that each half-yoke is made of three parts, joined at the comers, in order to offer the highest permeability route to the magnetic flux. With this kind of stacking one can also form a closed frame and place the strip sample along the median axis as a bridge between opposite sides, thereby realizing the so-called horizontal-type single sheet tester [6.12]. The magnetizing coil is generally wound on a former surrounding the sample and encloses both the secondary winding and the sensor of the effective field, which can be either a flat multi-turn H-coil or a Hall sensing device. We will see in the following that, in order to attain the measuring reproducibility suited to the industrial requirements, a fight geometrical arrangement is imposed by the standards to the yoke and the sheet sample [6.13]. A meaningful value for the magnetic path length is consequently obtained, so that the effective field strength can be reliably determined by measuring the magnetizing current. Figure 6.5b schematically illustrates bulk-shaped specimen testing by means of an electromagnet-type yoke. The field is generated, by placing the primary winding on the central limb

6.1 CLOSED MAGNETIC CIRCUITS

291

Primary winding

(a)

/

"x Hall-probe

"x

/ (b)

j

FIGURE 6.5 Schematic examples of magnetic flux closure by means of laminated yokes. These are made of magnetically soft alloys and have a large crosssectional area, in order to provide a near-zero reluctance flux path. Yoke (a) is made as a double-C core and the magnetizing coil surrounds the strip sample. The flux-sensing coil is tightly wound around the specimen and a fiat H-coil is used to determine the effective field strength. In (b) the bulk specimen is gently pressed between the pole faces of an electromagnet-type yoke. The effective field can be determined either by means of an H-coil or a Hall field-sensing probe.

of the yoke and the sample is m a d e to fill exactly the gap. Both hard and soft magnets can be tested using this kind of arrangement. Flux closing yokes are generally built by assembling either N i - F e or GO F e - S i sheets, with the former to be preferred w h e n testing extra-soft materials, such as Nis0Fe20 alloys and a m o r p h o u s ribbons. Yokes m a d e of GO sheets prove, in any case, to be perfectly suited to magnetic testing of electrical steels. Demagnetization prior to use is r e c o m m e n d e d , in order

292

CHAPTER 6 Magnetic Circuits

to avoid biasing of the sample by the residual field. It is carried out, at a conveniently low frequency, by joining the two halves of the yoke, excited by means of supplementary windings, before inserting the sample. The so-built magnetic circuits have obvious frequency limitations, depending on thickness, permeability and resistivity of the yoke laminations. For experiments above the kHz range, either Ni-Fe tape or amorphous ribbon cores are therefore employed. Ahlers et al. have shown that flux closure can be conveniently achieved up to 100 kHz using wound cut cores made of 50 ~m thick Mumetal tape [6.14]. The electromagnet-type yokes are often required to provide very high and uniform field over a wide and variable gap. This is what occurs with permanent magnet testing [6.15]. They are therefore usually built from solid pieces of lowcarbon steel, which associate reasonable cost with acceptable mechanical properties, while affording a high value of the saturation magnetization. Fe-based bulk-core electromagnets can be employed only for quasi-static magnetic characterization because the combination of low resistivity and bulk size is conducive, under time-dependent magnetizing current, to important eddy current shielding and lack of flux penetration. It can be stated, in general, that for a meaningful use of an electromagnet of crosssectional area S and apparent relative permeability ~r the exciting frequency should be lower than about f0 = (Scr/Zr/Z0)-1,

(6.1)

where cr is the material conductivity. This condition amounts to saying that the flux penetration depth 3 = 1/x/~rcr/Zr/Z0f0 should be higher than the radius r0 of the pole pieces. For a low-carbon steel electromagnet with / d , r - - " 100 and r0 -- 0.1 m, a limiting frequency f0 = 3 x 10 -2 Hz is estimated. It is clear, from the foregoing considerations, that the condition of practical closed magnetic circuits is quite never associated with the absence of magnetic charges. Even in toroidal specimens we have free poles, unless the material is perfectly homogeneous, the easy direction is circumferential and the field is applied by means of a uniformly distributed winding. Also in Epstein circuits free charges are inevitable. Actually, the field they generate is indispensable for providing, in combination with the applied field Ha, an eventually homogeneous effective field H over the sample portion covered by the windings and for giving meaning to the notion of integration path running along the whole frame. Figure 6.6, comparing the behaviors of Ha and H along one arm of the Epstein frame, following the results of the local measurements in NO and GO laminations reported in Fig. 6.4, illustrates

6.1 CLOSED MAGNETIC CIRCUITS

293

1.0-

0.9 ~" 0.8"

0.6 0.5

I

....Z _ _ / 7

~,z%,:,~

/

~0.7-

~

I

Epste'm frame -8

-4

0

x(cm)

4

J

8

FIGURE 6.6 The field Ha applied by one of the four magnetizing coils (length 19 cm) of an Epstein frame decreases on going from the midpoint (x = 0) to the coil ends, where it falls off by a factor of about 2 (solid line, see also Fig. 4.3). The effective field H (dashed lines), arising from the combination of Ha and the field generated by the free poles, appears remarkably uniform over the same length. The example reported here refers to the NO and GO laminations and the measuring conditions illustrated in Fig. 6.4.

the homogenizing effect of the free-poles field. Channeling of the effective field lines (and, afortiori, the flux lines) through the yoke limbs in the flux-closing arrangements of Fig. 6.5 is evidently related to the combined effects of the applied field and the magnetostafic field due to the free surface charges on the yoke. A minimum-energy line pattern is eventually achieved, examples of which are sketched, for the electromagnet case, in Fig. 4.23. To be remarked that with very soft and homogeneous materials we always expect the volume charges to be negligible. We can assume in fact B ~-/~0M, so that, being V.B = 0, we conclude that V.M = - p ~ 0. Of course, the magnetization might become non-uniform and produce extra charges in the neighborhood of the pole faces, if good mechanical contact between sample and yoke is not ensured. While with permanent magnet test specimens the near-zero reluctance offered by the soft yokes can be taken for granted, at least far from iron saturation, the overall effectiveness of flux closure with soft magnet testing can be appreciated by simplified analysis of the magnetic circuit. Under the assumption of constant cross-sectional areas Sm and Sy

294

CHAPTER 6 Magnetic Circuits

in the sample and the yoke, having lengths lm and ly, respectively, the continuity equation for the magnetic flux and Amp6re's law for the field provide, in the absence of flux leakage, BInS m = BySy,

Nlil

- - H m l m nt- H y l y ,

(6.2)

where the inductions and the fields are denoted with Bin, By, Hm, and Hy. Since, disregarding hysteresis, Bm = / x 0 H m q-/x0M m =/x0(1 q- Xm)Hm and By = txoHy +/X0My =/x0(1 + Xy)Hy, where Xm and Xy are the susceptibilities, we obtain that the field in the yoke Hy is related to the field in the sample Hm by the equation: lq-Xm S m

Hy= l+xy

S---yHm"

(6.3)

The drop of the magnetomotive force in the yoke neglected if

Hyly can thus be

Xy Sy lm >> 1. Xm Sm ly

The effective field in the sample is then given to a good approximation by H m = Nlil/Im. If we pose Bm ~/.toM m and By ~ g0My, we find Mm/My = Sy/Sm. To remark that, whatever the method used to form it, a closed magnetic circuit is remarkably insensitive to external spurious fields, the larger the permeability, the higher the shielding effect. If we associate to the circuit a demagnetizing factor Nd and an intrinsic relative permeability/Xr, we obtain that the response to an external field is characterized by the apparent permeability: /Xr /~ar-- 1 - } - N d ( ~ r - 1)

9

(6.4)

Applied fields and spurious fields of equal strength produce induction variations in the material that are in the ratio /Xr//Xar--~ 1 + Nd/xr- For Nd = 0.05 and/Xr = 104 we find that gr//Xar ~ 5 X 102. We conclude this section by mentioning the subject of air flux compensation, to be treated in further detail in the following. The problem chiefly consists in the fact that, especially with thin laminations and films, the secondary winding may embrace a far larger cross-sectional area than the one occupied by the material and the detected flux includes then a contribution from a region where the induction is Ba =/x0H, while in the material we have B =/x0H +/-~0M. This extra contribution is, in general, automatically eliminated by connecting in series opposition the secondary winding and another winding with suitable t u r n - a r e a product, which

6.2 OPEN SAMPLES

295

is linked only to the induction Ba. With closed magnetic circuits, this process does not imply special difficulties because the field H is always perfectly defined and identical inside and outside the sample. In the Epstein frame the automatic air-flux compensation is actually carried out by means of a mutual inductor, so that only the polarization J =/z0M is left [6.6]. In other cases, both B and J can be obtained by combination of compensating coils and calculation [6.3].

6.2 OPEN SAMPLES We have previously remarked that the adoption of open magnetic circuits is conducive to homogeneous magnetization only in ellipsoids and spheres, a condition seldom attained in practical samples. More common specimen shapes are cylinders, parallelepipeds, and strips, for which the magnetization is not uniform and either the magnetometric or the fluxmetric demagnetizing factor approximations must be adopted (see Section 1.2). In a number of important cases, however, there are no specific advantages in closing the magnetic circuit, the flux closure is not required, or it is even impossible. A case in point is that regarding the measurement of the magnetic moment of a sample by force methods or by extraction. It turns out, at the end, that measurements with open samples are ubiquitously performed and accepted. We will introduce here some peculiar problem associated with the open sample condition, with the provision that details on techniques and methods will be discussed, within their appropriate context, in other parts of this book. Bulk soft magnets are frequently available as open specimens and are tested, as far as possible, enclosing them in a flux-closing yoke, more or less of the kind shown in Fig. 6.5 (permeameter method, see Section 7.1). One could actually shape the samples as ellipsoids and carry out a full J ( H ) characterization of them by making an exact determination of the demagnetizing field, but it is difficult to envisage easy application of this approach at low and medium induction values. Applied and demagnetizing fields would be in fact so close that the measuring accuracy in the determination of the effective field H = Ha - H d = Ha - N d M - "

Ha - (Nd/P,O)J

(6.5)

would be impaired. In addition, external spurious fields (e.g. the earth's magnetic field) might cause substantial errors in high-permeability materials. Measurements on open soft magnets must then be conducted in a shielded environment, obtained either with active field cancellation

296

CHAPTER 6 Magnetic Circuits

with large Helmholtz coils (see Fig. 4.8) or by enclosing the sample in a suitably large box made of Mumetal or similarly high-permeability alloys. A simple additional measure to reduce interference from the earth s field in samples of elongated shape is to align them with the east-west direction. Shielding by soft magnetic enclosures should obviously avoid relevant coupling of the shield with the sample and the field generating setup because it would introduce unwanted distortions of the field lines. The measuring accuracy is expected to increase on approaching the material saturation, where the applied field largely overcomes the demagnetizing field. Practical considerations suggest, however, to use cylindrical samples instead of ellipsoidal ones, with the secondary coil located around the mid-section and the effective field determined by use of the fluxmetric (also called ballistic) demagnetizing factor N(df) in Eq. (6.5). N(df) is either calculated by standard formulas [6.16] or it is determined by comparison with recorded data concerning precisely machined ellipsoidal samples. With the latter method, illustrated in Fig. 6.7, a measuring uncertainty lower than 1% is claimed, for example,

2.2

2.1

elli

2.0 Iv-

1.9

/

I

1.8 1.7

! t

1.6

'

Soft solid steel

r i

0.0

_ _

.

!

,-

--,

4.0x10 4

H

--

.

!

8.0x10 4

,

.

- r - -

'

1.2x10 s

(A/m)

FIGURE 6.7 Open sample measurements. The true normal magnetization curve of a soft steel specimen, shaped as a prolate ellipsoid (solid curve), is compared with the apparent curve of a cylindrical specimen (length 200 mm, diameter 10 mm) circumscribed to it. The ballistic demagnetizing factor of the cylindrical sample is experimentally determined by this comparison, which puts in evidence, on approaching the material saturation, a demagnetizing field at the sample midsection of the order of 3 X 10 3 A/m (adapted from Ref. [6.17]).

6.2 OPEN SAMPLES

297

in the determination of the high-field normal DC magnetization curve in 1 0 m m diameter and 2 0 0 m m long cylindrical steel samples [6.17]. No special difficulties would additionally be met for measuring the coercive field Hc. For this we only need to make the magnetization to recoil from the saturated state and record the value of the applied field at which the average magnetization in the sample, i.e. the average demagnetizing field, is zero. Such a condition can be verified by detecting the condition of zero stray field around the sample [6.18] (see Chapter 7). Bulk soft magnets have a relatively narrow field of applications, being chiefly used as DC flux multipliers. Most soft magnetic cores, being subjected to AC fields, are obtained by assembling sheets or ribbons, which, under certain circumstances, are tested as open samples. The magnetic anisotropy and the rotational hysteresis of magnetic laminations are, for example, typically measured in disk-shaped samples, for which we can take, as a first approximation, the demagnetizing factor of the oblate ellipsoid. Alternatively, the effective field is directly measured by use of a flat H-coil or a Chattock coil, provided the size of the sample allows for the use of such sensors. With extra-soft strip-like samples and unidirectional field, both flux-closed and open sample configurations bristle with difficulties. If the strip is wound as a toroid or even assembled to form an Epstein circuit [6.19], stresses build-up in the sample [6.20]. On the other hand, yokes may not provide the sought after magnetic short circuit, a faint remanent magnetization of it being detrimental to the meaningful determination of the intrinsic properties of the tested material. One might then resort to measurements on long open strips, but even in samples with high aspect ratio a substantial demagnetizing field correction by means of Eq. (6.5) is required if the material is very soft. Figure 6.8 illustrates the case of a near-zero magnetostrictive amorphous ribbon (length 202 mm, width 8.9 mm, thickness 19.6 ~m) annealed at 320 ~ under a saturating longitudinal magnetic field. This sample is known to have an intrinsic rectangular hysteresis loop with very low coercivity (Hc ~" 0.5 A/m) [6.21]. This loop can be recovered from the experimental one, obtained on an open strip sample (solid line, measurement performed on a east-west oriented strip contained in a large Mumetal shielding box), by calculating the effective field through Eq. (6.5). The value of the resulting fluxmetric demagnetizing factor (N~df~ = 1.25 X 10 -5) is about five times higher than the value predicted by Aharoni for a uniformly magnetized strip (X = 0) of equal aspect ratio [6.22] and closer (within a factor 2) to the prediction for high-permeability cylindrical samples of equal length and cross-sectional area [6.16]. This example illustrates that making accurate correction for the demagnetizing effects can be very difficult in practically shaped soft magnets, the softer

298

CHAPTER 6 Magnetic Circuits 0.8 C071Fe4B15Silo

,', ", i

amorphousstrip

0.4.

'i

!

o.o,

)

"--0.4"

-0.8:

J . i ,,..

-10

i" , ,,

w,

I , , , . ,

-5

k".,.

,.

0

, , , , , . . , .

Ha, H(A/m)

/= 202 mm

w = 8.9 mm

d= 19.6llm i,..,

5

,..r

:,

i.:

10

FIGURE 6.8 A high-permeability near-zero magnetostrictive amorphous strip

(length 202 ram, width 8.9 mm, thickness 19.6 ~m) annealed under a saturating longitudinal field develops a uniaxial magnetic anisotropy and is characterized by an intrinsic rectangular hysteresis loop (dashed line, Hc --" 0.5 A/m). This can be recovered from the sheared experimental hysteresis loop (solid line), which is measured, upon application of a uniform field, using a secondary coil localized on the sample mid-section. The effective field H = H a - N(dOMis calculated using the fluxmetric demagnetizing factor N (f) = 1.25 x 10-5.

the material the more approximate the calculation of the intrinsic magnetization curve. Aharoni's formulations for both fluxmetric and magnetometric demagnetizing factors [6.23] appear better suited to thin film structures, which are generally characterized by much lower permeability values than their bulk counterparts. Numerical methods may, at the end, be required to calculate the stray fields in open sample arrangements. Notice, however, that rough corrections for the demagnetizing field are totally acceptable in a number of cases. Weak magnets, either homogeneous or made of magnetic particles or second phase precipitates dispersed in a non-magnetic matrix, provide an obvious example where correction for the demagnetizing effect is of little or no relevance. We can equally content ourselves with approximate estimates for the demagnetizing field when testing hard and semi-hard magnetic thin films, for their aspect ratio is such as to generally make Hd small with respect to coercivity. It may also happen that these films have uniaxial anisotropy perpendicular to the substrate plane, in which case the demagnetizing effect is tightly knit to the intrinsic material properties and

6.2 OPEN SAMPLES

299

the conventional measuring methods are superseded by specific techniques for the measurement of the perpendicular magnetization (e.g. magneto-optic Kerr or Faraday methods, SQUID detection). It is to be stressed that the air flux compensation in open samples, if automatically accomplished as previously mentioned for closed magnetic circuits, does provide the material polarization J in the limit of low values of the demagnetizing coefficient only. With reference to Fig. 6.9, we assume that the compensating coil and the secondary coil, of turn-area NcSc and N2S2, respectively, are linked with the fluxes ~c = NcScla,oHa and ([')2 ~---N2S2(H,oHa -/z0Hd) q- X2Sml -- N2S2.H,oH q- N2SrnJ~ if Sm is the cross-sectional area of the sample. If N2S2- N~Sc, by connecting the two coils in series opposition we evidently obtain zero signal in the absence of the sample. In the presence of it, we measure, according to Eq. (6.5), the resulting flux:

$2 - 2Sm'(1 mNd )

(6.6)

The compensated flux is then proportional to the polarization in the sample as far as the demagnetizing field is negligible with respect to the magnetization M, a condition normally realized in strips, ribbons, and thin films. The use of a closed magnetic circuit, made of a cylindrical or parallelepipedic sample enclosed between the pole faces of an electromagnet (Fig. 6.5b), is the standard in the measurement of the J(H) and B(H) curves in permanent magnets [6.15]. Open sample measurement methods in hard magnets are, however, very often required or simply preferred for a number of reasons, ranging from speed of measurement to costs or nature of the available specimens. With conveniently large-sized cylindrical specimens, we can perform the conventional measurement of the induction in the material using tightly wound

FIGURE 6.9 Air flux compensation in a strip-like sample. The compensating coil and the secondary coil have equal turn-area products NcSc = N2S2 and are connected in series opposition, so that, in the absence of the sample, the total flux variation with changing applied field Ha is zero. In the presence of the sample, the resulting linked flux is proportional to the material polarization J (i.e. magnetization M), provided the demagnetizing factor Nd KK1.

300

CHAPTER 6 Magnetic Circuits

secondary coils and either conventional or superconducting solenoids as field sources. On the other hand, a host of methods have been developed by which the magnetic m o m e n t of the whole sample, instead of the induction across a given cross-section of the sample, is obtained. They are characterized by high sensitivities, of the order of 10 -7 A m 2 (10 -4 emu) or better, and, consequently, they require small samples (for example, a 10 m m 3 specimen magnetized to 1 T is endowed with a total magnetic m o m e n t of about 8 x 1 0 - 3 A m 2 (8 emu)). Besides the classical force methods, with their extra-sensitive AC version (alternating gradient force magnetometer), inductive techniques have been developed, where the sample is made to play the role of a magnetic dipole, which is either vibrated, rotated, or displaced with respect to one or more sensing coils. Alternatively, one can keep the sample fixed and apply a high intensity pulsated field. When the specimen fulfills the dipole approximation, its magnetic m o m e n t can be determined by measuring the flux linked with a surrounding coil, which is related to it by a definite relationship of proportionality. Let us consider, as in Fig. 6.10, a point-like magnetic m o m e n t I n in the plane of a search coil of radius R and its equivalent current loop im of area a, such that m = a'im. The coordinates (x0, Y0) define the position of the magnetic moment. If we suppose, in a purely fictitious way, that a current is circulates also in the search coil, the two loops can be viewed as coupled circuits, characterized by a mutual inductance M [6.24]. They are linked through the fluxes ~ms = M'is (from the search coil to the small loop)

|

,

search coil

Z

! i i i

x

FIGURE 6.10 A small (ideally point-like) sample of magnetic moment m is represented as a loop of area a with a current im flowing in it (m -- a'im). The flux emitted by the loop which links with the search coil is 9 = k(xo,Yo).m, where k(xo,Yo) is the value of the coil constant associated with the loop coordinates (x0, Y0)- If m has components (rex, my, mz) and z is the axis of the search coil, then c~ = k(xo, yo).mz. A magnetic moment located at the center of a filamentary search coil of radius R provides 9 = #o(mz/2R).

6.2 OPEN SAMPLES

301

and ~sm = M'im (in the opposite direction). The search coil generates the magnetic induction Bz(x0, Y0), directed along the z-axis, in the point occupied by the magnetic dipole and is consequently characterized by a constant k(xo,Yo)= Bz(xo,Yo)/is. We can then also write ~ms = Bz(xo, yo)'a. It follows that M = [Bz(x0, yo)/is].a = k(xo, yo).a and the flux linked with the search coil eventually turns out to be CI)sm - -

k(xo, yo).m.

(6.7)

If the magnetic dipole is in the generic point of coordinates (x, y, z) and has components (rex, my, mz), the general relationship holds (I)sm

--

k(x, y, z).m = kx(x, y, z).mx + ky(x, y, z).my + kz(x, y, z).mz,

(6.8)

with ki(x, y,z) = Bi(x, y,z)/is. For a magnetic dipole located at the center of the coil, where k = ia,o/2R (see Eq. (4.4)), we obtain ~srn = (la,o/2R)mz, while, when a Helmholtz pair is used as search coil, we have from Eq. (4.20) ~sm = 0.7155(ia,oN/R)mz. This is just the flux variation we measure if we extract the sample from the coil. On the other hand, we know from Eq. (4.24) that if we connect the two coils of the Helmholtz pair in series opposition, we obtain a uniform gradient Ok(z)/Oz of the coil constant around the origin. This means that if we make the dipole to oscillate around this position, we can measure, according to Eq. (6.7), an induced voltage again proportional to the magnetic moment m. This is the principle, to be discussed in some detail in Chapter 8, exploited in the vibrating sample magnetometer. It was previously remarked that, if shielding during measurements is required, coupling of the shield with the sample and the useful field source might result in a distortion of the flux lines, eventually influencing the experiments to ill-defined extent. This effect is actually part of a general problem, always to be taken into account each time open samples are tested in the neighborhood of soft magnetic bodies. It can be illustrated by taking the case of a magnetic dipole of moment m brought in proximity of the flat surface of a magnetic body having relative permeability /Zr. It provides a schematic view of what happens, for example, when a permanent magnet sample is placed between the pole faces of an electromagnet. We obviously expect that, in response to the presence of the dipole field, the body will magnetize to an extent depending on the value of/Zr. This brings about a distortion of the field lines emerging from the dipole. The problem can be quantitatively understood, making use, for instance, of the scheme in Fig. 6.11 and

302

CHAPTER 6 Magnetic Circuits

Hm.

rd'y l'41"lm ~

I" ~

~

dx

.....

" .---~

r

,

/lr> 1

(a)

/ ]1r > i

/ x, (b)

/dr < 1

(c)

FIGURE 6.11 Image effect. The field distribution around a magnetic dipole is perturbed in proximity of a magnetic medium. For a dipole of strength m this amounts to the presence of a fictitious dipole of strength m~= [(/~r - 1)/(/~r + 1)]m mirroring the real one. It can be demonstrated that with such an image dipole the continuity conditions for the tangential field and the normal induction component through the air-medium boundary are satisfied. With a ferromagnetic medium (/~r > 1), the image dipole is oriented like in (a) and (b). With a diamagnetic medium, it is oriented like in (c).

considering the continuity conditions on field and induction at the body surface. These regard the tangential component of the magnetic field (Hit(P)--HIt(P1)) and the normal component of the magnetic induction (B• ( P ) = B• (P/)), where we define with P and/Y two points adjacent to the surface, in the air and in the body, respectively. Let us therefore assume that a dipole of moment m is facing the flat surface of an indefinitely extended soft magnetic body. It is common experience that a magnet is attracted to a block of iron, as unlike charges attract themselves. One can therefore reasonably think that the field profile satisfying the previous continuity equations could be achieved by the combination of the dipole in the air and a fictitious dipole in the material, mirroring the real one. This can be demonstrated by calculating HII and B • in both P and P~. We consider then in Fig. 6.11 the field generated in P by the combined

6.2 OPEN SAMPLES

303

contribution of the real dipole of moment m (field Hm) and the notional image dipole of moment m ~(field Hm, ). Using standard formulas (see, for example, Ref. [6.25]) and noting the symmetrical position of m and m ~ with respect to the boundary, we obtain for the tangential field component and the normal induction component: 1 dxdy HII(P) = 4vr r 5 (m - m~),

I~o 3d 2 - r2 B• (P) --- 4vr ~ (m 4- m~),

(6.9)

where the distance r -- ~d 2 + d2 and the positive sign of B • (P), m, and m ! is conventionally associated with the direction of m and m ~ shown in Fig. 6.11a. In order to calculate the same quantities within the body at point P', we refer to a momentarily unknown moment m" (and the related field Hm") in place of m. If the relative permeability of the material is/d,r~ we obtain 1 d,.dy m", HI1(P~)- 4~r r 5

B

(19/) --

l

/d'0/d'r

4rr

3d2 - r2 m" ---7-"

(6.10)

We impose now the previous continuity conditions on HII and B • which provide a couple of equations in the unknown variables m~ and m': m - m~= m',

m + m~=/xr.m'.

(6.11)

By solving them, we find, in particular, that the image dipole is endowed with the magnetic moment:

nil ---- -~-rm- . 1 /J,r 4- 1

(6.12)

We see that in the presence of a high-permeability medium the flux lines emerging from a dipole of moment m are modified as if a mirror dipole of equal strength m/ = m were present within the material. This situation is depicted in Fig. 6.11a and b. It corresponds to the case of an open sample in the air gap of an electromagnet. To be remarked that, in such a case, the permeability of the pole faces tends to rapidly decrease on approaching the saturation of iron, so that, according to Eq. (6.12), the strength of the image dipole is correspondingly decreased. The distribution of the field lines of a sample having defined magnetic moment is consequently affected, together with the flux linkage with any measuring coil. Errors in measurements are consequently introduced. Equation (6.12) also shows that, if the medium is a perfect diamagnet (/d, r "- 0), m~= - m . Under the previous convention on the signs of m and m/, this means that the real and the image dipoles are oriented like in Fig. 6.11c. They have equally

CHAPTER 6 Magnetic Circuits

304

directed tangential components and oppositely directed normal components. The image effect is therefore to be taken into account both w h e n the sample is placed within the pole faces of an electromagnet and inside a superconducting solenoid.

References 6.1. A. Hubert and R. Sch/ifer, Magnetic Domains (Berlin: Springer, 1998), p. 184. 6.2. EC.Y. Ling, A.J. Moses, and W. Grimmond, "Investigation of magnetic flux distribution in wound toroidal cores taking account of geometrical factors," Anal. Fis., B-86 (1990), 99-101. 6.3. IEC Standard Publication 60404-4, Methods of Measurement of the d.c. Magnetic Properties of Magnetically Soft Materials (Geneva: IEC Central Office, 1995). 6.4 T. Nakata, N. Takahashi, K. Fujiwara, M. Nakano, Y. Ogura, and K. Matsubara, "An improved method for determining the DC magnetization curve using a ring specimen," IEEE Trans. Magn., 28 (1992), 2456-2458. 6.5. H. B611,Handbook of Soft Magnetic Materials (London: Heyden, 1978), p. 64. 6.6. IEC Standard Publication 60404-2, Methods of Measurement of the Magnetic

Properties of Electrical Steel Sheet and Strip by Means of an Epstein Frame (Geneva: IEC Central Office, 1996). 6.7. IEC Standard Publication 60404-10, Methods of Measurement of Magnetic Properties of Magnetic Sheet and Strip at Medium Frequencies (Geneva: IEC Central Office, 1988). 6.8. J. Sievert, "Recent advances in the one- and two-dimensional magnetic measurement technique for electrical sheet steel," IEEE Trans. Magn., 26 (1990), 2553- 2558. 6.9. D.C. Dieterly, "DC permeability testing of Epstein samples with double-lap joints," ASTM Spec. Tech. Publ., 85 (1949), 39-62. 6.10. J. Sievert, "Determination of AC magnetic power loss of electrical steel sheet: present status and trends," IEEE Trans. Magn., 20 (1984), 1702-1707. 6.11. H. Ahlers, J.D. Sievert, and Qu.-ch. Qu, "Comparison of a single strip tester and Epstein frame measurements," J. Magn. Magn. Mater., 26 (1982), 176-178. 6.12. T. Nakata, N. Takahashi, K. Fujiwara, and M. Nakano, "Study of horizontaltype single sheet testers," J. Magn. Magn. Mater., 133 (1994), 416-418. 6.13. IEC Standard Publication 60404-3, Methods of Measurement of Magnetic Properties of Magnetic Sheet and Strip by Means of a Single Sheet Tester (Geneva: IEC Central Office, 1992). 6.14. H. Ahlers, A. Nafalski, L. Rahf, S. Siebert, J. Sievert, and D. Son, "The measurement of magnetic properties of amorphous strips at higher frequencies using a yoke system," J. Magn. Magn. Mater., 112 (1992), 88-90.

REFERENCES

305

6.15. IEC Standard Publication 60404-5, Permanent Magnet (Magnetically Hard) Materials. Methods of Measurement of Magnetic Properties (Geneva: IEC Central Office, 1993). 6.16. D.X. Chen, J.A. Brug, and R.B. Goldfarb, "Demagnetizing factors for cylinders," IEEE Trans. Magn., 27 (1991), 3601-3619. 6.17. H.R. Boesch, Accurate measurement of the DC magnetization of steel using simple cylindrical rods, Proc. Second Int. Conf. Soft Magn. Mater. (Cardiff, UK), 1975), 280-283. 6.18. IEC Standard Publication 60404-7, Method of Measurement of the Coercivity of Magnetic Materials in an Open Magnetic Circuit (Geneva: IEC Central Office, 1982). 6.19. A. Kedous-Lebouc and P. Brissonneau, "Magnetoelastic effects on practical properties of amorphous ribbons," IEEE Trans. Magn., 22 (1986), 439-441. 6.20. One is often interested, from the viewpoint of applications, in the final properties of a specific ring sample, once convenient thermal or thermomagnetic treatments have been carried out on it. Little interest would then be attached to the intrinsic magnetic properties of the material at start and their possible determination using open samples. 6.21. C. Beatrice, private communication. 6.22. A. Aharoni, L. Pust, and M. Kief, "Comparing theoretical demagnetizing factors with the observed saturation process in rectangular shields," J. Appl. Phys., 87 (2000), 6564-6566. 6.23. A. Aharoni, "Demagnetizing factors for rectangular ferromagnetic prisms," J. Appl. Phys., 83 (1998), 3432-3434. 6.24. It can be demonstrated that, whatever the coupled circuits, the coefficients of mutual inductance M21 -- ci)21//1 and M12 = ci)12//2, by which we denote the flux delivered in the coil 2 by a unitary current circulating in the circuit 1 and the flux delivered in the coil 1 by a unitary current circulating in the circuit 2, respectively, are equal (M21 -- M12 -- M). The demonstration of this statement (but not only this specific demonstration) is often called "reciprocity theorem". 6.25. D. Craik, Magnetism: Principles and Applications (Chichester: Wiley, 1995), p. 304.

This Page Intentionally Left Blank

CHAPTER 7

Characterization of Soft Magnetic Materials

This chapter will review and discuss current methods in the determination of the DC and AC magnetization curves of soft magnets and the related physical parameters. Basically, this means that we will chiefly present applications of inductive measuring techniques to the characterization of laminations, ribbons, and bulk samples, including sintered powder materials. The specific problems associated with soft magnetic thin films will be briefly dealt with in this and in the next chapter. We have highlighted in the previous chapter some general problems regarding the geometry of the test specimen and the flux-closing magnetic circuit. In this chapter, we shall be more specific on the type of materials subjected to testing and the most frequently employed measuring arrangements. In particular, we shall pay special attention to the solutions endorsed by the technical committees responsible for the creation and revision of the magnetic measurement standards. We shall thus consider the testing arrangements employing toroidal, Epstein, single-sheet, single-strip, and bulk rod-like samples. They apply to the conventional conditions where magnetization and field are properly assumed to be in a scalar relationship. If magnetization and field (defined as average macroscopic quantities) are not collinear, further specifications are required in order to provide meaning to the measurement of the magnetization curves. These will be discussed to some extent in association with the testing of laminations under rotational magnetic field, which, although not yet sanctioned by a measuring standard, is the subject of increasing interest in the domain of computation and design of electrical machine cores. In defining and measuring the M(H) relationship in a magnetic material, we must specify whether we are looking at DC or AC properties. Actually, if we are to determine the magnetization curves, we necessarily have to change the strength of the applied field with time. Strictly speaking, we talk of DC curves when this change is accomplished in such 307

308

CHAPTER 7 Characterization of Soft Magnetic Materials

a way that every recorded M(H) point corresponds to an equilibrium stable microscopic configuration of the system. A rate-independent hysteresis loop is, for example, determined when the applied field is changed so slowly that the evolution of the system through successive metastable equilibrium states, occurring by means of Barkhausen jumps, becomes totally independent of the field rate of change. This is a somewhat ideal measuring condition because relaxation effects, due either to eddy currents or thermally activated processes, can make it difficult in practice to achieve a truly rate-independent M(H) behavior. The DC characterization can be accomplished by measuring the stray field emitted by an open sample as a function of H (magnetometric method), but the inductive method on a flux-closed configuration is largely preferred. Two basic inductive measuring procedures can be adopted. The first one, called ballistic or point-by-point method, consists in changing the field in a step-like fashion and determining each time the corresponding flux variations while the system is allowed to relax to a novel equilibrium state. In the second one, called continuous recording or the hysteresisgraph method, the field is slowly changed according to a suitable continuous law. These two experimental approaches do not always provide the very same results, reflecting the awkward definition of DC magnetization curve and hysteresis. When the previous conditions characterizing the DC behavior no longer apply, we fall into the general domain of AC testing. Basically, this implies that for a given induction rate/~, the applied field strength has to compensate for an additional counterfield related to/~, which is associated with energy dissipation phenomena. For the practically relevant case of soft magnetic laminations and ribbons, the chief source by far of energy losses is represented by eddy currents. Three overlapping domains of investigation are normally considered. The first one is of interest for power frequency applications and typically extends up to 400 Hz (the operating frequency of airborne transformers). A medium-to-high frequency region can then be identified, extending up to around 1 MHz, where the role of stray parameters must be duly accounted for in measurements. We eventually deal with the domain of radiofrequencies, where soft magnets (e.g. ferrites, garnets, thin films, and microwires) are finding increasing applications. However, while in the low and medium frequency range it is mainly the high-induction non-linear magnetization regime that is important, the small induction linear behavior is chiefly considered in theory and experiment at high frequencies.

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

309

7.1 BULK SAMPLES, LAMINATIONS, A N D RIBBONS: TEST SPECIMENS, MAGNETIZERS, STANDARDS

MEASURING

7.1.1 Bulk samples Metallic magnetic materials in bulk form are subjected to DC characterization only, because eddy currents already shield the interior of the core at very low induction rates. If the test specimens are shaped as toroids, their dimensions should conform to the rules given in Section 6.1 and the magnetic path length should be calculated accordingly. With ferrites and sintered or bonded metal particle aggregates, AC characterization can also be performed. The winding arrangement follows the usual rules, where the secondary winding is as close as possible to the specimen surface and the magnetizing winding is external to it, both being evenly laid around the core. Given the relatively large cross-sectional area of the test specimen, usually in the range 50-500 mm2; and the small diameter of the wire used for the secondary winding (0.1-0.2 mm), minor correction for the air flux is generally required, even at high fields. In some special cases (for example, when making measurements at high temperatures), the ring sample must be encased in a rigid container and there can be a substantial area included between the specimen and the windings. A correction must then be made for the extra flux detected by the secondary winding. If the cross-sectional areas of the sample and the secondary winding are A and A2, respectively, and the field is H, as determined by the measurement of the primary current, the sample induction B is obtained by subtracting, per unit turn, the flux contribution / ~ - #0H(A 2 - A ) to the measured flux. This can be easily accomplished by calculation, if A 2 is exactly known. Alternatively, a dummy specimen made of an identical empty container and identical windings can be used. Total air flux compensation is achieved in this case by connecting the primary and the secondary windings of tested and dummy specimens in series and series opposition, respectively. The resulting signal is then proportional to the polarization of the material J -- B -/z0H, from which B is easily retrieved if desired. Notice here that if testing at high field strength (i.e. very low permeability values) is performed, further compensation should be devised for the stray axial flux generated by the equivalent circular turn, with diameter equal to the mean diameter of the ring, formed by the primary winding and collected by the equivalent turn formed by the secondary winding (see also Section 6.1). If the magnetizing winding is made of one layer, compensation can be obtained by winding back one turn along the median circumference. Multiple

310

CHAPTER 7 Characterization of Soft Magnetic Materials

layers should be laid in pairs, with alternate layers wound clockwise and anti-clockwise around the ring. DC magnetic properties of bars, rods and thick-strip specimens, as obtained, for example, by Casting, forging, extrusion, hot rolling, powder compacting or sintering, are generally determined with the use of permeameters, soft magnetic structures of the type shown in Fig. 6.5a realizing a closed magnetic circuit. The yokes in a permeameter are preferably, but not necessarily, of the laminated type. In this case, high permeability Fe-Si or Fe-Ni laminations are employed, which are either U-bent and superposed to form a double-C structure schematically shown in Fig. 7.1 (as shown) or cut, stacked side by side, and assembled with staggered butt joints (Fig. 7.2a). If solid yokes'are used, they should be made of precisely machined soft iron or low-carbon steel. The two basic permeameter arrangements in use today, as recommended by the IEC 60404-4 standard [7.1], are qualitatively illustrated in Fig. 7.1. They both make use of double C-yokes and differ in the way the magnetic field is applied. In the so-called Type-A permeameter (Fig. 7.1a), the magnetizing coil is wound around the specimen, while in the Type-B permeameter (Fig. 7.1b) it is wound around the yoke. The latter solution, which was popular in the version offered by the Fahy permeameter [7.2], is adopted at present by commercial setups [7.3, 7.4]. The minimum recommended specimen length is 250 and 100 mm in Type-A and Type-B permeameters, respectively. Care must be paid to specimen clamping in the yokes in order to minimize the reluctance of the joints. The pole faces must be rectified and coplanar and, for tests on bars and rods, additional pairs of soft iron pole pieces, as shown in Fig. 7.2b, should be employed in order to closely accommodate the test specimen between the yoke pole faces. The flux sensing coil (3), centered on the specimen mid-section, has length between 10 and 50 ram. The experiments show that with typical samples, quite uniform induction is obtained over this length in both permeameter types. Figure 7.3 shows the dependence of B on the distance from the mid-section in a 2 mm thick, 21 mm wide, and 271 mm long soft iron bar (permeability of the order of 103 for H = 103 A/m), as determined by means of a localized few-turn secondary winding made to traverse the length of the specimen [7.5, 7.6]. This figure also shows correspondingly good uniformity of the effective field H, which can then confidently be determined using either a localized probe (i.e. a Hall device) or an H-sensing coil placed on the specimen surface. The Hall method is by far the quickest and simplest although some provision must be made in building the coils for admitting the small sensing head (either transverse or tangential) close to the specimen surface. The earth's magnetic field obviously combines with H in the region occupied by

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

250 mm

311

~1

(a)

350 mm' ,

I

(b) FIGURE 7.1 Permeameters for the characterization of bulk soft magnets, according to the IEC 60404-4 standard. In (a) the field is applied by means of a solenoid (1) wound on a former around the specimen (2), which is clamped between the two halves of the double-C laminated yoke (type A permeameter). The flux-sensing coil (3) has length between 10 and 50 mm and the tangential field is measured either by means of a Hall probe (4) or a fiat H-coil. In the Type-B permeameter (b) the magnetizing windings (1) are wound around the yoke and the sample (2) can be shorter (down to 100 instead of 250mm in Type-A permeameter). A compensated flux-sensing coil (3) can directly provide the sample polarization J. The field can be measured either by using a Hall probe (4) a fiat H-coil, or a Rogowski-Chattock potentiometer (5). It is assumed here that the yokes are of the strip-wound type, obtained by U-bending and stacking either grain-oriented Fe-Si or Ni-Fe laminations.

312

CHAPTER 7 Characterization of Soft Magnetic Materials

j

IIiI1~

i-"

~::

i.. ,,j.. 1--1,, I,

-v-

i I I

(b)

(a)

test specim~p,

N2A2 cl

ff=~ (c)

NcAc2 (d)

FIGURE 7.2 (a) Detailed view of a stacked C-yoke comer with staggered butt joints. (b) Pairs of soft polar pieces, housing the specimen (circular or square crosssection) in the region of contact with the yoke pole faces. They ensure a low reluctance path for the magnetic flux. (c) Determination of the tangential field by use of series connected fiat coils placed on opposite sides of the specimen. (d) Coaxial coil arrangement providing automatic air-flux compensation and secondary signal proportional to the sample polarization J. the probe and for this reason the sample is conveniently oriented along the East-West direction. The residual field is automatically subtracted by inverting the applied field polarity and averaging the obtained indications. Use of the Hall method with the Type-B permeameter is totally acceptable, provided there is negligible radial dependence of the field strength over the region occupied by the Hall plate. The two flat coils shown in Fig. 7.2c, placed on opposite sides of the sample and series connected, can be used for the inductive determination of the effective field. Alternatively, two coaxial coils of different diameters connected in series opposition are preferably employed with cylindrical and bar-like specimens. A magnetic potentiometer (Rogowski-Chattock) can also be applied with the Type-B permeameter (see Fig. 7.1b), under the condition of good contact of the coil end faces with the specimen surface and uniform turn density. It should be remarked, in any case, that the objective difficulty of accurately determining low field strengths

313

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

Type-A permeameter 1.04 O

v

:z-

1.02

Ht(x)/Ht(O)

s" :,, "" e -

l

~" 1.00

0.98

. . . .

-30

I

-20

.

.

.

.

I

-10

I_

. . . .

I

0

'

'

'

'

I

10

. . . .

I

20

'

""

'

'

'

:

30

x (mm)

Type-B permeameter 1.04

/

I

' O

v

I I1'

Ht(x) IHt(O)

,',J

I I /9 I I

~- 1.02

:Z-

v

\

0

I

\

.J~ .i~..--~ ~'~"~~

IK..

v

/

r

"~" 1.00

B(x) /B(O) 0.98

. . . .

-30

,

-20

. . . .

,

-10

....

'

'

0 x (mm)

'

'

I

10

'

'

'

'

I

20

'

'

'

'

30

FIGURE 7.3 Magnetic induction B(x) and tangential field Ht(x) measured as a function of the distance x from the mid-section in a soft iron bar (thickness 2.1 mm) tested in the two permeameter types shown in Fig. 7.1. The displayed quantities are normalized to their values at x = 0. Open dots: H ( 0 ) = 800 A/m. Full dots: H(0) = 8000 A/m. The data are taken from Refs. [7.5] (Type-A permeameter) and [7.6] (Type-B permeameter).

314

CHAPTER 7 Characterization of Soft Magnetic Materials

(say around a few A/m) makes the ring method more suitable than the permeameter method in the DC characterization of very soft magnetic bulk samples (for example, large-grained very pure Fe). Conversely, the permeameter method, allowing for the application of high fields, is preferentially employed on approaching the magnetic saturation. Concerning the problem of air-flux compensation, we can, as previously discussed for the ring setup, either obtain it by calculation, once we know the cross-sectional areas of winding and specimen and the tangential field Ht (generally assumed to coincide with the internal field H), or by automatic subtraction via coils connected in series opposition. Figure 7.2d provides a cross-sectional view of a secondary coil directly providing the sample polarization J q-compensated coils). An inner winding of turn-area N2A2 is series connected with two outer compensating windings of turn-areas NcA~I and NcAc2, which are, in turn, connected in series opposition. The flux linked with the outer coils, related to the shaded annulus in Fig. 7.2d, is cI)c = N c ( a c 2 - Ad)p,oH and totally compensates the air-flux linked with the inner winding if N2A2--Nc(Ac2-Acl). The flux globally linked with this triple-coil arrangement then becomes ~c - N2AJ, if A is the cross-sectional area of the specimen. It may happen that Nc and Ac2- Ad cannot perfectly satisfy the previous condition. In such a case, one may try to achieve Nc(Ac2 - Acl ) slightly larger than N2A2 and to fine adjust it connecting a resistance in parallel to the compensating coils. The Type-A permeameter can, at least in principle, be arranged in such way that the value of the effective field H is directly obtained by measuring the current il circulating in the magnetizing winding. This is the concept that has led to the development of the compensated permeameters, notable examples of which are provided by the Burrows [7.7] and the Iliovici [7.8] permeameters. What is compensated in these devices is the drop of the magnetomotive force occurring in the magnetic circuit outside the portion of the test specimen covered by the magnetizing winding. If this has length Im and the number of turns is N1, the effective field is then given by H = Nlil/lm. In order to achieve this condition, auxiliary magnetizing windings are employed, normally placed in proximity the yoke pole faces, and their supply current is suitably adjusted. Classical compensated DC permeameters appear rather obsolete nowadays, given the tedious point-by-point operations involved in the current adjustment and the present availability of quick and precise methods for the measurement of the effective field in the measuring region by means of sensors. An improved AC version of the compensated permeameter for use on single strips and sheets will be discussed below.

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

315

7.1.2 Sheet, strip, and ribbon specimens Soft magnets are applied for the most part in AC devices and for that reason they are generally produced as sheets and ribbons. To characterize them under a closed magnetic circuit configuration, we can, as discussed in Section 6.1, either build ring or Epstein frame samples, or resort to fluxclosure by a means of high-permeability large cross-sectional area yokes. Preparing a test specimen and measuring circuit is a delicate problem because we need to balance the ideal goal of determining the intrinsic magnetic behavior of the material with the practical constraints imposed by the necessity of having a reasonable sample size and geometry, a convenient arrangement of the testing apparatus, and reproducible measurements. Conventional magnetic laminations are usually delivered as 0.5-1.5 m wide sheets, from which testing samples must be cut. Rapidly solidified alloys are instead produced and tested as ribbons of variable width, from 1-2 to around 100 mm, and sometimes as wires. The latter have such a high aspect ratio that demagnetizing field corrections are irrelevant and open sample testing is appropriate, on condition that accurate shielding and East-West alignment of the specimen are provided. Tape-wound ring samples are the usual solution for testing amorphous ribbons. However, tape winding implies the buildup of stresses, half compressive and half tensile, and the magnetic properties of the sample become dependent, through magnetostrictive coupling, on the ring radius R. If Ey is the Young modulus of the material and d is the tape thickness, the maximum strain is ~ m a x - - d / 2 R and the corresponding tensile/compressive stress is O'max =

Eyd/2R.

(7.1)

A 30 ~m thick amorphous ribbon (Ey = 150 GPa) wound on a 2 cm diameter ring is subjected to a maximum stress of the order of 200 MPa. The correspondingly induced average magnetoelastic energy density Erae __ 3 As O.max~ where As is the saturation magnetostriction, can range between some 1 5 J / m 3 in Co-based alloy (As "-" 10 -7) and about 4.5 x 103 J / m 3 in Fe-based alloys (As ---30x 10-6). This brings about a drastic change of domain structure and magnetization curve with respect to the free ribbon, whose intrinsic behavior can possibly be measured only by using an open strip sample and applying the correction for the demagnetizing field (Fig. 6.8). However, we might be interested in the properties of the final ring sample, as they result upon convenient thermal and thermomagnetic treatments. These will generally be different from the properties of the free ribbon, even if the very same treatment sequence is applied in both samples, and, the complete stress-relief being difficult to

316

CHAPTER 7 Characterization of Soft Magnetic Materials

achieve without incurring some incipient crystalline transformation, somewhat dependent on the ring radius. With crystalline laminations, plastic straining can occur below a certain R value. A 0.23 rnm thick Fe(3 wt%)Si lamination (Ey -- 120 Gpa, yield stress cry --- 200 MPa) should be bent, for instance, over a radius larger than Rmin "" 70 mm in order to avoid permanent deformation. Of course, annealing can relieve both elastic and plastic straining, including the work hardening effect associated with strip cutting from the parent lamination but it can also permanently modify the structure of the starting material (for instance, by increasing the grain size) and, again, the measured properties may not be precisely the ones originally aimed at. An important point in tape winding, and common also to stacking, both in ring and Epstein specimens, is represented by the interlaminar insulation. This is provided by a few micrometer thick coating in the industrial Fe-Si laminations or, simply, by surface oxidation in lowcarbon steels and Fe-Ni alloys. Amorphous ribbons are uncoated and very little oxidized and interlaminar currents could potentially arise, in the cores. A number of studies-have actually shown that such currents do not have relevant effects [7.9], even if tension winding is applied in order to increase the packing fraction because of surface roughness. Tensionwound cores can nevertheless exhibit additional losses with respect to loosely wound or ribbon-coated cores after annealing due to the greatly increased number of shorts associated with bonding of adjacent laminations at contact points [7.10]. Note that shorts can also occur between the stacked Epstein strips if lamination cutting does not leave the strip edges completely burr-free. The accuracy of cutting should also be high regarding the geometrical tolerances, a maximum deviation of _ 1~ for example, being allowed, according to the standards, for the direction of cutting with respect to the rolling direction (RD) in grain-oriented alloys [7.11]. The Epstein test method is a widely accepted industry standard, characterized by a high degree of reproducibility, as shown by intercomparisons carried out by National Metrological Institutes and specialized industrial laboratories [7.12, 7.13]. Indeed, the reproducibility of measurements is central to the acceptance and assessment of a method as a standard because it attaches to the economic value of the material being characterized. For all its merits, including many years of solid experience by laboratories worldwide, the Epstein method has certain drawbacks, making its application difficult or not totally appropriate. For one thing, sample preparation, which can include stress-relief annealing after cutting the 3 0 m m wide strips, is time consuming and expensive. With high-permeability grain-oriented

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

317

laminations, stress-relief is mandatory, but in case of laser-scribed materials it is basically inapplicable because it would interfere with the physical mechanisms responsible for domain refining. Its use also appears questionable with ribbon-like samples, in particular amorphous alloys, which can be prepared narrower or wider than 30 mm. With large ribbons, longitudinal cutting cannot be envisaged. In addition, property degradation by magnetoelastic effects upon sample insertion in the frame can occur, which is difficult to control and impairs the measuring reproducibility [7.14]. If we also consider that the Epstein method does not provide absolute results, there are good reasons to look at the single plate test method, as schematically envisaged in Fig. 6.5a, as a practical and flexible alternative, where the absolute determination of the effective field is also possible. Of course, some working rule should be agreed on how to provide general consistency to the results obtained with different methods. Different kinds of single-strip/single-sheet testers (SST), such as those employing horizontal single, double, and symmetrical yokes, or the vertical single and double C-yokes, have been investigated in the literature and have been variously adopted in national and international measuring standards. An example of a double horizontal yoke magnetizer, as proposed by Yamamoto and Ohya [7.15], is shown in Fig. 7.4. The yoke frame is obtained by stacking 100 mm wide grainoriented Fe-Si strips, cut along RD, up to a thickness d > 11 mm [7.16] and the field strength is determined by means of a flat H-coil. Horizontal-type yokes lend themselves to quick operation, with automatic insertion and extraction of the sheet sample, and are therefore attractive from the viewpoint of quality control in the plant. Since strip specimens are in general wide (e.g. 200 mm), negligible effects from cutting stresses are expected and laser-scribed laminations can consequently be easily tested. Horizontal yokes with H-coils have found widespread acceptance in Japan and have been adopted in the JIS standards. The asymmetric structure of the yoke in Fig. 7.4 is conducive, however, to a systematic measuring error, as can appear in the vertical single C-yoke, which becomes relevant when the test plate is longer than the frame side (overhang effect) [7.17]. The overhang error chiefly arises because eddy currents are generated in the plane of the lamination by the magnetic flux leaving the sample and flowing into the yoke limbs and vice versa [7.18]. As schematically shown in Fig. 7.5, these currents give rise to an extra field He, which generates a systematic error in the H-coil signal, as well as extra losses, which are reflected in an overhang-dependent magnetic path length Im, if the field is determined through the measurement of the magnetizing current (MC method).

318

CHAPTER 7 Characterization of Soft Magnetic Materials 100 mm

800

500

(a)

testspec[,~n secondary winding

J

.~,

~

g winding

'

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

, . . . . . . .

L . . . . . . . .

L ~H-coil 9

~nn mm

(b)

FIGURE 7.4 Example of horizontal-type double-yoke single strip tester used for measurements on Fe-Si laminations. With this arrangement, relatively wide strips (e.g. 200 mm) can be tested, thereby making the error introduced by cutting stresses negligible. The air-flux compensation can be obtained by means of a mutual inductor (not shown in the figure). The effective field is measured over the region of maximum homogeneity of the magnetization by means of an H-coil (adapted from Ref. [7.15]). The overhang problem is basically eliminated by use of a symmetric horizontal yoke, where the test plate is sandwiched between two identical frames [7.16], or a double-C yoke. Note, however, that the single vertical C yoke is admitted, in spite of its asymmetry, by the MCbased ASTM standards, both with conventional steel-sheet laminations and the amorphous ribbons [7.19, 7.20]. It is also possible to get rid of asymmetry effects by using the double H-coil, as sketched in Fig. 7.5a. The double H-coil solution in SST is generally applied for the sake of accuracy in the measurement of the tangential field. This varies with the distance x from the sheet surface and the true value H can only be obtained in the ideal condition of an infinitely thin H-coil. Figure 7.5b provides an idea, for two different materials tested in a vertical C-yoke, of the variation of the tangential field H(x) on passing from the sheet surface to the inner surface of the magnetizing solenoid [7.21]. Remarkably, a linear increase of H(x) is found, which permits one to

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

test specimen "~

319

H-coils .~ H

/

He

(a) 15

10

v

!

x v

0

2

x(mm)

4

6

(b) FIGURE 7.5 (a) Eddy currents in the lamination plane associated with the flux flowing into and out of the yoke limbs. The eddy current field H e is the source of additional losses and of a systematic measuring error. This effect, which is equally observed in the horizontal-type yoke of Fig. 7.4, is basically eliminated either by using a synmletric yoke (e.g. vertical double C-yoke) or a double H-coil (adapted from Ref. [7.18]). (b) Increase of the tangential magnetic field with distance x from the surface of the sheet specimen. The effective field at the surface H can be determined by linear extrapolation to x -- 0 of the field strength values measured by two flat coils at different distances xl and x2 (from Ref. [7.21]).

extrapolate to x = 0 the e x p e r i m e n t a l values m e a s u r e d w i t h t w o coils of finite thickness p l a c e d at distances xl a n d x2. We find in p a r t i c u l a r H = H(0) =

x2H(Xl) - -

x 1H(x2)

.

(7.2)

X 2 -- X 1

A l t h o u g h H(Xl) a n d H(x2) can be affected, via the field He, b y the sheet o v e r h a n g , H is not b e c a u s e at the surface H e -- 0. It has b e e n s u g g e s t e d

320

CHAPTER 7 Characterization of Soft Magnetic Materials

that Eq. (7.2) can still be applied using a single H-coil if two measurements are performed, one of them with the coil lifted to a convenient distance from the test plate surface [7.22]. The H-coil method is, in principle, exactly what we need for the measurement of the magnetic properties of soft magnetic laminations because, by providing the value of the effective field directly, it does not require awkward assumptions regarding the magnetic path length. The condition of homogeneous sample magnetization must, of course, be fulfilled over the region covered by the B and H measuring coils. However, the application of this method to the magnetic measurement standards has been so far limited to Japan (JIS Standard H-7152 [7.23]). It is indeed difficult to envisage its general adoption in the industrial environment for a number of reasons: (1) The signal generated in the H-winding is usually small and prone to disturbances by interfering electromagnetic fields. (2) The turn-area calibration must be performed with maximum accuracy. Consequently, it requires a reference magnetic flux density source, traceable to the standards kept by the National Metrological Institutes. (3) The stability with time of the winding t u r n area, crucial to the measuring accuracy, is critical. It calls for a low temperature coefficient and a rigid non-aging structure. Experience shows that given the requirement of low coil thickness (typically around 1-2 mm), the latter is not easily obtained using a non-metallic former. In addition, the installed coil often becomes inaccessible to non-destructive inspection and adjustment. (4) The signal must be integrated in order to achieve the field H(t). The integrating chain can be the source of further instabilities and possible phase errors, especially relevant with analogic integration [7.24]. Consequently, if our key objective is to achieve excellent measurement reproducibility, besides coming reasonably close to the intrinsic material properties, it is acceptable to base the measurement of the magnetic field on the simple and accurate determination of the current circulating in the primary winding. This is just what we do with the Epstein frame, although we know, as discussed in the previous chapter (see Fig. 6.4), that in doing so we can incur in a systematic error. We have previously mentioned the concept of the compensated permeameter, where, in spite of the inhomogeneity of the magnetic circuit, the effective field on the measuring region is directly determined from the measured value of the primary current. Such a possibility arises because we are able to compensate for the effect of air gap and yoke reluctance through supplementary magnetizing coils, located near the pole faces of the yoke. Such compensation can be made automatic by means of a Rogowski-Chattock potentiometer (RCP) and a feedback circuit, as illustrated in Fig. 7.6 [7.25, 7.26]. If we consider in this figure

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS <

.......

300

rnrn

321

-~'

,,,

1 FIGURE 7 . 6 Cross-sectional and side views of the field compensated single strip tester. The principle of this arrangement is one of automatically compensating the drop of the magnetomotive force outside the measuring region of length/rn, SO that for a current il flowing in the magnetizing winding (1), having N1 turns over lm, the effective field on the test specimen (2) is H = Nlil/lm. This condition implies zero signal detected by the RCP (3). The supplementary coils (4) are energized by this signal, which becomes negligibly small when the amplifier gain is very high. The magnetic flux density is detected by the air-flux compensated coil (5) (adapted from Refs. [7.25, 7.26]).

the closed integration path formed by the portion of test plate of length lm, around which N1 turns of the magnetizing winding are wrapped, and the semicircle of length Lp described by the potentiometer winding, we can write, by denoting with il the primary current, ~

H.dx+~ tm

H.dl=Nlil

(7.3)

Lp

A signal proportional to the time derivative of the line integral f Lp H.dl is generated in the potentiometer (see Eq. (5.14)), which is fed, via a highgain amplifier, into the supplementary coils. It is a classical feed-back arrangement, an example of which was given in a different context in Fig. 4.9, which can maintain a condition of near-zero signal at the output of the potentiometer. This is no longer used, as in the previously discussed examples, to measure the tangential field, but it merely acts as a zero state indicator and, as such, it does not need calibration. By keeping, as shown in Fig. 7.6, the length of the magnetizing winding longer than lm and by

322

CHAPTER 7 Characterization of Soft Magnetic Materials

suitably arranging the compensating coils, it is possible to keep the tangential field in the measuring region uniform within a few percent [7.27] so that we can write with a good approximation H -- N1 il/lm

(7.4)

and we can accurately determine the effective field by means of a current measurement. It has been suggested that the compensated single-strip tester is especially appropriate for the measurement of the AC magnetic properties of amorphous ribbons and high permeability laminations, where the value of the field strength determined by the single H-coil method is apparently affected by a substantial error [7.28]. Of course, the structural stability of the potentiometer coil can still be of concern, although it is not so critical as in the case of direct measurement of the field with the H-coil. However, again, if we look at the compensated single-strip/sheet tester from the viewpoint of standardization and general applicability in the industrial laboratories, we might find it rather complicated, both for what concerns the arrangement of the coils and the electronic circuitry. Probably for this reason, it has not been considered in the discussion concerning the development of the measuring standards. The IEC 60404-3 standard [7.29] has adopted, for example, the MC-based SST represented in Fig. 7.7, where the magnetic path length lm is assumed fixed and equal to 0.45 m, that is, coincident with the internal distance between the pole faces of the yoke. The standard prescribes in great detail the structure of the yoke and the winding arrangement. As a result, excellent reproducibility of the measurements is obtained, as demonstrated by a number of intercomparisons. A comparison exercise, involving six different laboratories in the measurement of magnetic power losses at 50 Hz in Fe-Si laminations [7.30, 7.31], is reported and analyzed in Table 10.2 and Fig. 10.2. Another example, concerning the determination, according to IEC 60404-3, of the same quantity in 15 different types of grain-oriented laminations by three different National Metrological Institutes, is given in Fig. 7.8 [7.13]. The histogram shown here provides, for each laboratory, the number of declared values of the power loss P associated with a given relative deviation (Pi - Pref)/Pref of the individual best estimate Pi with respect to the reference value Pref. The latter is obtained, as discussed in detail in Chapter 10 (Eq. (10.34)), as the weighted average of the best estimates provided by the different laboratories together with their combined standard uncertainties. We see that the width of this distribution is smaller than 1%. The reproducibility of the SST method according to the IEC 60404-3 is thus comparable to the reproducibility of the Epstein method according to the IEC 60404-2 [7.12, 7.13]. What both methods are unable to provide, with

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

323

180- 300 mm I

. 2 5 mm

magnetizing winding

"x secondary winding suspension I

,

test specime f

I 500 mm

I

1~

FIGURE 7.7 Single-sheet tester according to the standard IEC 60404-3 [7.29]. The test specimen is 500 mm long and its width can vary between 300 and 500 mm. The double C-yoke, 500 mm wide, is made of either grain-oriented Fe-Si or Ni-Fe laminations. The pole faces are 25 mm thick and are accurately machined and made coplanar within 0.5 mm, so that no gap is formed between opposite pole faces. The upper yoke is movable upwards to permit insertion of the test specimen and is acted on by a suspension, by which its weight is counterbalanced up to a maximum vertical force of 200 N applied on the sheet. The primary and secondary windings are at least 440 mm long. The primary winding can be made of a 400 turn single layer of a I mm diameter wire. The number of turns of the secondary winding, air-flux compensated by means of a mutual inductor, is the one suited to accurate signal acquisition.

their a priori defined value of the magnetic path length, is the true value of the p o w e r loss. We have s h o w n in Fig. 6.4 that with the Epstein m e t h o d we underestimate the p o w e r loss value at inductions of technical interest. It has been equally d e m o n s t r a t e d that by relying on the SST m e t h o d contemplated in the standard IEC 60404-3, one instead tends to overestimate the loss. The globally arising difference, which can be as high as 10%, is s o m e w h a t disturbing, because it will eventually affect the design of magnetic cores. For example, designers accustomed to the Epstein figures, which are often a s s u m e d to be the physically correct values, m a y find unacceptable changes in the building factor of transformers on passing to the SST results [7.32]. There has consequently been general d e m a n d for some procedure connecting the Epstein and the SST results. Two different procedures are suggested as informative annexes to IEC 60404-3. The first one (Annex B) calls for the calibration (that is the determination of the magnetic path length) of the SST tester by means of long Epstein strips. Twelve of them are first tested according to

CHAPTER 7 Characterization of Soft Magnetic Materials

324

GO Fe-Si sheets

10

[~NMI1 NMI2

[~ ~

E ~

Jp=I"7T f= 50 Hz

5 ,,

0

[iT

-1.5 -1.0 -0.5 0.0 0.5 (Pi-Pref) /Pref (%)

1.0

1.5

FIGURE 7.8 Overview of an intercomparison on the magnetic power loss at given frequency and peak polarization value carried out by three different national metrological laboratories (NMIs) on 15 different types of grain-oriented Fe-Si laminations according to the IEC 60404-3 standard. For each lamination, the relative deviation (Pi - Pref)/Pref, where Pi is the best estimate of the ith laboratory and Pref is the reference value, is obtained. The width of the distribution shown here, where N is the number of measurements associated with a given relative deviation, is contained in the 1% range (adapted from Ref. [7.13]).

IEC 60404-2 and are then placed side by side in the SST yoke and tested again with the same apparatus and at the same peak polarization value Jp. An effective magnetic path length of the SST tester is then calculated, which is the one making the Epstein and the SST loss figures coincide. If PSST is the specific power loss measured assuming the conventional magnetic path length lm = 0.45 m, the effective path length then becomes PSST /eft -- lm ~ . PEpst

(7.5)

Note that if non-oriented laminations are tested, two independent determinations of/eft are made, one for the rolling direction and the one for the transverse direction (TD). Of course, such a procedure must be repeated for any different material grade and polarization value and is therefore of somewhat limited value. It may also require stress-relief treatment of the strips after cutting. The experiments show that/eft varies,

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

325

depending on the Jp value, between 0.43 and 0.48 m in non-oriented laminations and between 0.45 and 0.50 rn in grain-oriented laminations [7.31]. Annex C in IEC 60404-3 instead provides a general relationship between Epstein and SST measurements in the case where only one type of measurement has been performed. It is valid only for grain-oriented materials in the polarization range 1 T ~ Jp -< 1.8 T. It is an empirical formula based on the analysis of a very large set of measurements, carried out at 50 Hz on about 250 sample sets (Epstein strips plus SST plates) cut from different heats of differently graded materials provided by a number of manufacturers [7.33]. The experiments were performed by a single laboratory (PTB in Braunschweig) in order to avoid any scatter of the results coming from the use of different setups. The Epstein strips were subjected, as usual, to stress-relief treatment before testing. In summary, the conversion between the values of the Epstein power loss PEpst~ the field HEpst ~ and the apparent power SEpst and the corresponding SST values, obtained as averages from the statistical analysis of the measurements, is calculated according to the following expressions PSST -- PEpst'(1 q- 3p/100),

HSS T --

HEpst.(1 q- 3H/100), (7.6)

SSS w = SEpst'(1 -}- 3H/100),

where the parameters 3p and 3 H a r e related to Jp by the empirical relationships 3I, = 1.7 +

kpJ5,

3H = 6.0 +

kHJ~~

(7.7)

with kp -- 0.2423 T -5 and kH -- 0.1030 T -1~ In conclusion, the final acceptance and application of a given measuring method in soft magnetic strips and sheets eventually depends on our specific objective. The approach to the intrinsic material properties calls for either a slender ring specimen, when applicable, or the H-coil method, by which a reasonably correct value of the effective field H is obtained. In principle, with an open sample we can also arrive at the same fundamental result, provided it has an ellipsoidal shape or it has such a high aspect ratio (e.g. a long wire) to make the demagnetizing field correction inessential. A practical approach, reflecting the needs of producers and users of the materials, points primarily to the reproducibility of results, which has been demonstrated to be generally obtainable by the adoption of the Epstein and the SST methods. The price to pay for the ensuing simplified assumptions on the effective magnetic field is a systematic deviation of the obtained results from the intrinsic ones. Specific aims and circumstances, critical judgment by the researcher,

326

CHAPTER 7 Characterization of Soft Magnetic Materials

and available equipment are instrumental in the adoption of one or both of these measuring approaches.

7.1.3 Anisotropic materials and two-dimensional testing In all our discussions, we have so far tacitly assumed that the relationship between the effective magnetic field and the material magnetization is scalar in nature. This is justified under practical circumstances, as the ones contemplated in the standards, where we deal with strips, sheets, ribbons, bars, and rods, tested along their length. In isotropic or nearly isotropic materials, the macroscopic magnetization vector is always aligned with the applied field, that is, with the sample edges. The only obstacle to the determination of the intrinsic M(H) relationship and the related parameters may derive from our inability to have a perfectly defined value of the effective field because of the previously discussed limitations of the magnetic circuit. In materials provided with one or more easy axes on a macroscopic scale, we need to be specific regarding size and shape of the specimen and the orientation relationships between easy axes and applied field. If our conventional elongated specimens have their axis coincident with an easy direction, as occurs with grain-oriented sheets cut along RD and longitudinally field annealed amorphous ribbons, we can still confidently rely on the scalar approximation. We may wonder, however, what can be a rational approach to testing of a strongly anisotropic material along a direction different from an easy axis. Should we able to flux-close the specimen in order to emulate infinite extension in all directions, we would actually determine what we could call the real intrinsic response of the material to a field applied in whatever direction. Such a response would consist, in general, in a tensorial relationship between M and H. In practice, we may have partial or no flux closure at all and only by applying the field along a few high symmetry directions can we unambiguously define a geometry independent magnetization curve. Let us consider, for example, the case of an amorphous ribbon, fluxclosed at the ends, endowed with an easy axis making an angle 0 with respect to the longitudinal direction, that is, the direction of the applied field H a. If 0 is different from 0 and 90 ~ a demagnetizing field transverse to H a arises during the magnetization process, which, besides hindering the domain wall displacements, imposes a supplementary torque on the magnetic moments. The mechanisms involved with such an effect are rather complex and can be treated with the so-called/z*-method [7.34]. It turns out at the end that the ribbon width acts on the magnetization process in such a way that it becomes, so to speak, one of the intrinsic parameters of the material.

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

327

The classical problem of the magnetization curve in a single iron crystal demonstrates that it is possible to identify specific crystallographic directions along which the intrinsic material behavior can be measured. Looking at Fig. 7.9, we can easily realize that we can form closed magnetic circuits in the cubic Fe crystal by cutting parallelogram-shaped frames having all sides parallel to the (100), (110), and (111) directions, respectively [7.35]. In all these circuits, no surface free charges arise during the magnetization process. By adopting the terminology of N6el's magnetic phase theory [7.36], we can say that in the demagnetized state we expect to have two, four, and six equidistributed phases within each leg of the {100), {110), and {111) frames, respectively. Each phase is formed by domains having the magnetization directed along one of the six possible {100) orientations. If we look at the magnetization process in these three cases, we conclude that: (1) In the {100) frame, the magnetization proceeds from the demagnetized state to saturation by domain wall displacements only (mode I). (2) In the (110) frame mode I, ending at M/Ms-- 1/.r (Kaya's rule) is followed, up to saturation, by mode III, which consists in the rotation towards {110) of the magnetization in the two remaining

[00~]

[110]

[110]

111

I9s

[100]

[010]

9149

[111] m

FIGURE 7.9 The intrinsic magnetization curves along the high symmetry (100), (110), and (111) directions of an Fe single crystal can be determined by cutting parallelogram-shaped frames, which are completely free of demagnetizing fields.

328

CHAPTER

7

Characterization

of Soft Magnetic

2.0-

X-stackedsheets 1.5-

"'< E ;steinstrip

C ~.o~ i'

l

0.5; :

0.0.

" i:

. . . . . .

10

lOO

H (Nm)

Epstein

i

: ;

(a)

: '

. . . . . . . .

lOOO

10ooo

X-stack

1.0-

GO Fe-(3wt%)Si 0=75 ~

0.5-

Epsteinstri~ ~ " ' 1 "..>'~'i,../ o.o

-0.5 -1.0

X-stacked sheets

/!

.~

-41 ,o . . . . . .

(bl '4~o ........

o""

H (Nm)

i " , , , , ,

. . ,

2~o

.

,,

,i

400

Materials

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

329

phases. (3) In the (111) frame, three equi-occupied phases result at the end of mode I, which lasts until M / M s = 1/x/3. The remaining part of the process is covered by rotations (mode II). The same results regarding these three high symmetry directions could have been obtained by experiments on disks, cut parallel to the (100) or (110) planes, but for the additional complication related to the correction for the (isotropic) demagnetizing factor. If the magnetization curve along a generic direction is required, there are no equivalent frame structures to exploit, although we can still conceive, at least in principle, completely flux-closed arrangements in three dimensions [7.37], which can be assumed to provide intrinsic curves. In practice, we test crystals shaped as rods or strips, flux-closed at the ends, or disks. The domain wall processes, the rotations, and the equilibrium between the different phases, guided by the internal field, vector combination of the applied field and the demagnetizing field, now become dependent on the geometry of the specimen and the intrinsic curve cannot be retrieved. For sufficiently elongated rods and strips, the transverse demagnetizing coefficient is high enough to make the transverse magnetization component negligible, that is, M and H m (coincident with Ha in samples flux-closed at the ends) collinear. The magnetization curve along whatever direction can be correspondingly calculated, with mode I assumed to occur under zero coercivity, using N6el's theory [7.36]. An example of interest in applications is the one connected with the characterization of Fe-Si grain-oriented laminations in directions different from RD. Because of their high crystallographic perfection, these materials can be basically treated as single crystals. If tested as Epstein strips, as often done in the literature [7.38], they are expected to behave as N6el's rod [7.36] and the resulting magnetization curve beyond mode I can be predicted accordingly. The prediction along mode I instead requires pre-emptive determination of the intrinsic material properties of the lamination along RD and TD (the (100) and

FIGURE 7.10 Magnetic behavior of high-permeability grain-oriented Fe-Si laminations under application of a magnetic field making an angle 0 with respect to the rolling direction. The measurements have been performed under quasistatic conditions by means of a single-sheet tester [7.39]. In the narrow Epstein strip, the polarization J and the longitudinal field Hm are collinear. In the Xstacked sheet configuration (see inset), emulating an infinitely extended sample, the magnetization can acquire a substantial transversal component. (a) Normal magnetization curves for 0 = 30~ (b) Hysteresis loops for 0 = 75~

330

CHAPTER 7 Characterization of Soft Magnetic Materials

(110) directions in the single-crystal approximation) [7.39]. With larger strips and sheets, like those typically employed in a variety of SST, a certain component of the magnetization transverse to the applied field can arise and the measured M(H) curve becomes dependent on the width of the strip. Note, however, that the curves measured with cutting angles 0 = 0r 0--90~ and, beyond mode I, 0 = 54.7~ preserve their intrinsic character. By cross-stacking the strip specimens, complete flux closure in the plane of the lamination can be obtained [7.40] and the limit of the infinitely extended sample is emulated. We can talk here of intrinsic behavior of the material, which is equally predicted from analysis of the magnetization modes [7.39]. Figure 7.10 shows examples of hysteresis loops and normal magnetization curves measured in highpermeability Fe-(3 wt%)Si laminations, cut at different angles 0, under the Epstein strip and the cross-stacked (X-stacked) sheet configurations. The related measurements are conveniently performed employing one of the previously discussed SST. Recent trends in the development of magnetizers for soft magnetic laminations have favored a comprehensive approach to material testing, where the same setup is employed for measurements under one-(1D) and two-dimensional (2D) fields. These are generated by two- or three-phase supplied yoke magnetizers, which are designed to provide uniform fields, either rotating or alternating along a given direction, within a suitably large gap where the test plate is inserted. The applied field components are generally controlled by means of a feedback system in order to achieve a prescribed time dependence of the two components of the magnetization in the lamination plane (i.e. defined flux loci). These magnetizers, generally known under the name of rotational SST (RSST), have not yet been standardized and have mainly been developed on a laboratory scale. Consequently, in view also of their inherent complexity, there are in practice as many types of RSST magnetizers as the number of laboratories making use of them, as well as no general consensus on their optimal configuration. It is, therefore, not surprising that intercomparison exercises have shown poor reproducibility of the measurements under rotational fields [7.41]. Basically, the RSST magnetizers can be distinguished for the type of supply used (two- or three-phase), the type of yoke (vertical/horizontal, simple/double), and the specimen shape (square/ circular, with or without air gap, cross-shaped, etc.). The whole subject has been discussed quite extensively in the recent literature, especially in a series of lively workshops devoted also to some general problems regarding measurements in soft magnetic materials [7.42, 7.43]. Figure 7.11 provides a schematic view of the horizontal-type RSST magnetizer developed at PTB [7.44]. An alternating/rotating field is applied, by

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

331

Magnetizing windings

B-coils

sample

i

.........

.............

!

~..~/H-cods

..:i::. < .......................... 480 mm

,-

FIGURE 7.11 The PTB horizontal-type magnetizer for the 2D testing of soft magnetic laminations (upside and cross-sectional views) [7.44]. The laminated double yoke can provide either rotational or unidirectional field in the gap, according to the programmed time dependence of the current flowing in the exciting windings. The sample is an 80 x 80 r e _ I n 2 square, placed at the center of the double yoke and separated from it by a narrow (---1 mm) air-gap. The induction and the field are detected upon a 20 mm wide area, where both quantities achieve acceptable homogeneity. The two orthogonal B-coils are threaded through holes of 0.5 mm diameter. The corresponding multi-turn H-coils are wound on a thin former and placed on the lower sample surface. A couple of crossed RogowskiChattock coils can also be employed in place of the fiat H-coils.

suitably p r o g r a m m i n g phase and a m p l i t u d e of the currents circulating in the m a g n e t i z i n g windings, to a square specimen of 80 m m side, which is placed exactly in the m i d - p l a n e of the gap and is separated from the apex of the w e d g e - s h a p e d pole faces by a n a r r o w air-gap (~-1 mm). The two

332

CHAPTER 7 Characterization of Soft Magnetic Materials

components of the induction in the sample are detected by means of two orthogonal few-turn windings, which are threaded through 0.5 mm diameter holes drilled at a distance of 20 mm. This is the relatively small region upon which the uniformity of the magnetization suffices to provide acceptable measuring accuracy. Indeed, the notable distance between the pole faces, the leakage of flux in the orthogonal arm, and the non-ellipsoidal shape of the sample all combine to produce relatively poor homogeneity of the magnetization, but for such a limited area. To avoid hole drilling and the related localized work hardening, pairs of needle probes are sometimes used, by which the average value of the electrical field strength between the points of contact is measured [7.45]. Pole tapering and narrow air-gap both imply minimum power requirement in field generation. The air-gap cannot be too small, however, because it helps in achieving good homogeneity and waveshape control of the induction in the sample. By increasing its width, we also obtain that geometrical imperfections of sample and pole faces have little effect. Tapering is also associated with the rapid decrease in the field strength on leaving the sample surface (see Fig. 4.24). Consequently, it requires precise positioning of specimen and H-coils. The horizontal-type double yoke magnetizer is of simple construction and it has been adopted, with more or less significant variations (e.g. flat H-coil vs. RCP, tapered vs. untapered poles), by a good number of laboratories [7.46-7.49]. The more complex vertical double-yoke 2D magnetizer, sketched in Fig. 7.12, is sometimes preferred to the horizontal-type 2D magnetizer because the two orthogonal applied field components are generated by means of two nearly independent magnetic circuits [7.50, 7.51]. It is generally recognized, however, that with both these types of magnetizers it is difficult to control accurately the rotation of the magnetization in strongly anisotropic materials. This typically applies to grain-oriented Fe-Si laminations, which are very soft along RD, but quite hard along the direction making an angle of 54.7~ to RD (that is quite close to [001] and [111] in the individual grains, respectively). A three-phase 2D field source is expected to provide, at least in principle, better control of the flux loci, besides calling for less exciting power in each channel. An example of a 2D magnetizer with threephase supply and hexagonal test plate is shown in Fig. 7.13 [7.52]. An equivalent system is obtained in a simpler way by generating the field with the statoric core of a three-phase induction motor and placing a circular specimen in the mid-plane of the bore [7.53]. Figure 7.14 illustrates an example of the radial dependence of the magnetization in a non-oriented Fe-Si single disk of diameter 140 mm and thickness 0.50 mm, placed in a bore of diameter 200 mm and height 100 mm and subjected to a rotational field [7.54]. It is fair to say that the measured value of the tangential field

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

333

/ A sample~

(

/

H-coil

;gin FIGURE 7.12 Vertical-type double-yoke magnetizer.

mple A

nd H-sensors

FIGURE 7.13 2D field system with three-phase supplied magnetizer and hexagonal s}anmetry [7.52]. It is basically equivalent to the classical field source obtained by using the statoric core of a three-phase induction motor and a circular sample [7.53].

334

CHAPTER 7 Characterization of Soft Magnetic Materials 140 mm 2.0

.,.,

1.5

.

NO Fe-(3 wt%),a d= 0.50 mm i

oi

B-coils

I

,

~

~- 1.o cr ~

0.s

Disk sample

~ r I

0.0 ~ -60

-40

-20

0 20 x(mm)

40

60

FIGURE 7.14 Radial dependence of the magnetic polarization in a 140mm diameter Fe-(3 wt%) non-oriented Fe-Si disk placed in the bore of a three-phase statoric core of a rotating machine. The characterization of the material under 2D flux loci is performed over the central region of diameter 40 mm.

can, d e p e n d i n g on the actual value of the polarization 1, be s o m e w h a t different from the real one because it varies rapidly with the distance from the lamination surface. It is easily seen, however, that the m e a s u r e d energy loss is, u n d e r the usual condition of thin flat H-coil ( 1 - 2 m m ) or carefully realized RCP, not affected, at least in principle, from the distance of the coil from the sample surface. Let us consider the sheet sample, having demagnetizing coefficient Nd, and i m m e r s e d in a h o m o g e n e o u s applied magnetic field H a . An H-coil placed on the specimen surface will detect the tangential field H = H a - (Nd//Z0). J. The energy loss per cycle and unit volume can be expressed (see also Section 7.3) as W --

H. ~-~ dt,

w h e r e T is the period. Since

f ~ Nd ~ J. ~-~dt dj = /z0

0,

(7.8)

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

335

f

d

FIGURE 7.15 Rotating field and magnetization in the plane of an open test plate. The applied field H a leads the magnetization by the angle ~a. The field FI1 measured by a flat H-coil at a distance h 1 from the sample surface is H1 = H a - klJ and leads by the angle ~1. If the distance is increased to h 2 > h i , the field becomes H 2 - - H a - k2J , w i t h k 2 < k 1 and the leading angle becomes ~2 < ~1. The vector diagram shows that Ha sin ~a = /-/1 Sin ~1 = /-/2 Sin ~ , which implies that the measured loss does not depend, in principle, on the distance of the H-coil from the test plate surface.

we can equivalently write W --

dj

Ha" - ~ dt.

This is obviously u n d e r s t o o d as due to the demagnetizing field being in phase with the magnetization. It also means that whatever the distance h of the H-coil from the test plate surface, the phase shift between the detected field H(h) and J will change in such a w a y that the integral in Eq. (7.8) will not change at all. Figure 7.15 shows this clearly for the 2D case [7.55], where the vector J trails the m e a s u r e d field H(h) by an angle ~, which d e p e n d s on h in such a w a y that the quantity H(h)sin~(h)), proportional to the loss, is constant. The example reported in Fig. 7.16, which compares the hysteresis loops m e a s u r e d on the same lamination, first as a single strip in a closed circuit, then as a disk in a 2D yoke, both using a I m m thick flat H-coil, further illustrates the point. The loop m e a s u r e d on the single strip can be considered as intrinsic because the demagnetizing field is very small as is the increase of the tangential field versus distance from the strip surface (see Fig. 7.5b). Such an increase is m u c h faster with the open disk sample and the resulting m e a s u r e d loop appears sheared with respect to the previous one although, as expected, with the same area and coercive field. In practice, the previous

336

CHAPTER 7 Characterization of Soft Magnetic Materials N.O. Fe-(3 wt%)Si

1.0

t = 0.35 mm

S

f= 50 Hz

, ,..-

- -;-"

//

0.5 t-v r

o

N ._ i,..

0.0

t~ o t2_

I -0.5

-1.0

single strip

Field (A/m)

FIGURE 7.16 The same non-oriented Fe-Si lamination is tested as a single-strip in a closed yoke and as a 140 mm diameter disk in a 2D yoke (Fig. 7.14), in both cases using a I mm thick fiat H-coil. The corresponding hysteresis loops differ as regards to their shape, because of the correspondingly different dependence of the tangential field strength on the distance from the sample surface, but they have closely similar areas and coercivities.

relationships require that the magnetization in the sample is homogeneous, a condition only partially fulfilled. The property of conservation of the loop area is approximated to a more or less good extent, depending on the sample shape and the type of yoke employed.

7.2 M E A S U R E M E N T OF THE DC M A G N E T I Z A T I O N CURVES A N D THE RELATED PARAMETERS Ferromagnetic materials are complex physical systems, whose real defective structure brings about a manifold of metastable states responsible for stochastic microscopic behavior, hysteresis, and nonlocal m e m o r y effects. There is no simple w a y to define and unambiguously measure the DC magnetization curves and hysteresis loops of magnetic materials. It is understood that these represent the ratei n d e p e n d e n t J(H) behavior, the one related only to the sequence of values attained by the applied field and not to its rate of change. However,

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

337

the very same metastability leading to hysteresis can make the magnetization process prone to thermally activated processes. The method by which the applied field is changed with time might then interfere with the path followed by the system in the phase space and make somewhat elusive the concept of rate-independent magnetization process. Fortunately, in many practical soft magnets and at temperatures of interest for applications, thermally activated relaxation processes (after-effects), either due to diffusion of soluted atomic species (for example interstitial C atoms in Fe) or caused by thermal fluctuations, have little or negligible effect and a true rate-independent magnetic behavior of the material can be reasonably approached. Nevertheless, the intrinsic stochastic character of the magnetization process and, in metallic materials, residual eddy-current related relaxation effects can frequently cause, a certain lack of reproducibility of the J(H) curves, especially at low and intermediate polarization values. A conventional distinction between methods for measuring the DC magnetization curves is based, besides the way in which the time variation of the applied field is imposed, on how the sample magnetization is determined. We talk of magnetometric methods when J is induced from the measurement of the stray field generated by the open sample and of inductive methods when it is obtained by integrating the flux variation ensuing either from a variation of the relative position of open sample and sensing coils (vibrating/rotating sample, vibrating/rotating coil, extraction method) or from a variation of the applied field, both with open and closed samples. We have previously stressed, however, that soft magnets are seldom tested as open samples. Applied and demagnetizing fields can have very close values, the latter being also spatially nonuniform in ordinary test specimens, and the precise determination of the effective field is difficult. A notable exception, discussed in Section 7.1.3, is found with the two-dimensional testing of laminations, where methods for applying a rotational field with complete flux closure in two dimensions (for example, using cross-shaped specimens) do not ensure acceptably uniform magnetization in the sample [7.56]. The J(H) behavior of soft magnetic materials is then mostly determined by the inductive method using closed magnetic circuit configurations.

7.2.1 M a g n e t o m e t r i c m e t h o d s Among the open sample methods, the ones based on the measurement of forces and torques are rooted in the early history of magnetic measurements. An emblematic example comes from the experimental

338

CHAPTER 7 Characterization of Soft Magnetic Materials

setup developed by Lord Rayleigh in experiments on the low-field behavior of iron [7.57]. At the time there was debate as to whether the response of iron to feeble fields had a threshold or, as already hinted by Maxwell, it had first linear (elastic) then non-linear character. Lord Rayleigh was able to demonstrate that not only the initial susceptibility had a finite value, but that the magnetization depended quadratically on the field (the Rayleigh law). He was able to do so by means of a highsensitivity torque magnetometer, based on the mechanical action exerted by the stray field emerging from the end of the magnetized sample, a straight piece of iron wire, on a suspended needle. A null reading method was realized, where the needle was kept in its rest position by the compensating action of the field generated by a coil, coaxial with the wire, connected in series with the magnetizing solenoid. A measurement of this kind is characterized by a degree of sensitivity that is still challenging for present-time conventional inductive DC measurements on closed samples. Sensitivity is indeed the landmark property of the force methods that are frequently adopted for measurements on weakly magnetic materials (for example, with m o d e m versions of the Faraday balance) or the determination of very small magnetic moments (for example, by use of the alternating force gradient magnetometer), as we shall discuss below. However, if we are required to characterize the conventional bulk, sheet, or ribbon soft magnetic materials, inductive measurement methods, using the test specimens and magnetizers discussed in Chapter 6 and Section 7.1, are the rule. The magnetometric method can directly provide the total magnetic moment of the sample if this is placed at such a long distance from the field-sensing probe that it behaves as a dipole. At short distances, it is possible to relate the measured stray field with the material polarization by analytical formulation if the sample is ellipsoidally shaped. In all cases, the earth magnetic field and the field generated by the magnetizing winding contribute to the field sensor reading and some way must be found to suppress their interference. Remarkably, the magnetometric measurement techniques can find practical application as zero magnetization detecting methods. If we drive the sample, having regular ellipsoidal or cylindrical shape, to magnetic saturation and bring it back along the recoil curve, we can reveal the passage through the demagnetized state by sensing the zero stray field condition. At that moment, the applied field coincides with the coercive field, which is then measured, more easily and quickly than by determining the whole hysteresis loop and finding the passage of the curve through J = 0. Figure 7.17 illustrates recommended solutions for the magnetometric measurement of the coercive field in open samples. In the arrangement

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES test specimen

~./'///////////////////////////~/////////////~ ~////////,,~

~

/

--t- .............. ,i~*~"~i;~ ................ ............. i ~/'/!!1/1111111/11111111111/I//!!1111/!/11/11/~,

339

field sensor

~,'///////////////////////////~ ~////////////////'~

.............

ti

(a) t East-West Hall plate

field sensors__._~ ...I

~i~";'~ .. ~.1,

i (bl

, (c)

FIGURE 7.17 Coercive field measured by detecting the zero stray field condition in open samples upon recoil from the saturated state. The symmetry of the arrangement makes the sensors immune to the field generated by the magnetizing windings. (a) The field sensor (Hall plate or fluxgate probe) is placed exactly at mid-point between two identical solenoids, one of which contains the test specimen. (b) Stray-field sensing is obtained either with a Hall plate, a s ~ e trically placed near the end face of the sample, or a coil vibrating along the axis of the solenoid. (c) Differentially connected probes are symmetrically placed outside the solenoid and detect the radial field emanated by the sample, which is placed off-center. Shielding against the ambient field is required if low-coercivity materials are to be tested.

s h o w n in (a) [7.58], the test specimen is placed inside one of two identical solenoids, which are designed to provide sufficient field strength to bring the material to saturation. Empirically, one can confidently assume that the material is, to all practical effects, saturated w h e n a 50% increase of the magnetizing field brings about less than 1% increase of the m e a s u r e d coercive field. The field sensor, a Hall plate or a fluxgate probe, is placed exactly at mid-point, where the fields generated by the two solenoids mutually neutralize. Once the specimen saturation is attained, the field is slowly decreased by decreasing the supply current i, which is then increased in the opposite direction till zero field reading is achieved. If this condition, corresponding to the vanishing of the stray field generated by the sample, is reached with a magnetizing current ic, the coercive field

340

CHAPTER 7 Characterization of Soft Magnetic Materials

is obtained as Hc = kHic,

(7.9)

if kH is the constant of the magnetizing winding. Notice that it is not required that the sample has ellipsoidal shape in order to define a demagnetized state, provided it is homogeneous and the applied field is uniform over its volume. Cylindrical test samples are normally used, for which the measured coercive field will correspond to a state of zero volume-averaged polarization. Of course, a coercimeter working on this principle can be employed for testing both soft and hard magnets, the latter case suffering, however, from obvious limitations in the available field strength. For soft magnets, measures must be taken to prevent the effect of the ambient magnetic field, which can affect both the field sensor reading and the sample magnetization. The simplest way to deal with this problem is to align the solenoid along the East-West direction and to make two measurements with inverted currents in the solenoids. For coercivities below 40 A / m , shielding against the environmental magnetic field is prescribed by the pertaining standard [7.59]. Active screening by triaxial Helmholtz coils or an other suitable combination of windings surrounding the measuring apparatus [7.58] is preferred to the use of magnetic shields made of high-permeability alloys because of the distortions introduced by the coupling of the test sample with the shield (image effect, see Chapter 6). The sensor in Fig. 7.17a is pretty far from the specimen and lacks sensitivity. The arrangements shown in Fig. 7.17b and c are then possibly adopted [7.59]. In (b), a Hall plate is placed horizontally at one of the ends of the sample so that it detects only the stray field component normal to the axis of the solenoid. Alternatively, a vibrating axial coil, again insensitive to the applied field, can be used in order to detect the zero stray field condition [7.60]. Finally, two differentially connected, ambient field compensating flux sensing probes are used in (c), symmetrically placed immediately outside the solenoid. The radial component of the stray field emanating from the sample is detected in this way. To achieve good sensitivity, the sample is placed offcenter. In all cases, an elongated sample (say with ratio length/diameter of the order of 5 or higher) is preferred because it is more easily saturated and the uncertainty in the coercivity determination due to shape effects is not significant.

7.2.2 I n d u c t i v e m e t h o d s When talking of DC (or, more appropriately, quasi-static) characterization of soft magnets with inductive methods, we tacitly assume that we are

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

341

exploring the J(H) relationship in such a way that time-dependent effects are irrelevant. We have previously mentioned, however, that conceptual and practical difficulties may arise in certain cases where various relaxation effects take place, having time constants comparable with the measuring times. When looking at the accuracy and reproducibility of the measurement of the DC hysteresis loops and magnetization curves, we should be aware of these difficulties, which often combine with relevant stochastic effects (Barkhausen noise), associated with the discrete character of the magnetization process. There are two typical ways to determine the magnetization curves under quasi-static conditions: (1) The magnetizing field strength is changed in a step-like fashion and the curves are obtained by a point-bypoint procedure. (2) The magnetizing field is changed in a continuous fashion, as slowly as reasonable to avoid eddy current effects (hysteresisgraph method). Ideally, the two methods should lead to the same results, but differences are often found. The point-by-point inductive determination of the magnetization curves was originally introduced to overcome many practical drawbacks in the torque method, which include the difficulty in performing absolute measurements, the need to use open samples, and the sensitivity of sample and sensing devices to the external fields and their fluctuations. What is detected with this technique is the transient voltage induced on a secondary winding by the step-like applied field variation &Ha~ which is integrated over a time interval sufficient to allow for complete decay of the eddy currents in order to determine the associated flux variation &~. Since we deal with flux variations, we always need to define a reference condition. This is normally identified with either the saturated or the demagnetized state. The latter can be attained either by bringing the sample beyond the Curie temperature and letting it to cool down in the absence of external field or by applying an alternating field (with no offset) whose amplitude is progressively decreased to zero, starting from a peak value so high as to attain technical saturation. The hysteretic behavior of the material also requires that the field history following the achievement of the reference state must be perfectly controlled. It is not uncommon that field transients are inadvertently applied (for example, switching off and on power supplies), subverting the prescribed orderly sequence of field values and leading to false results. If, after thermal demagnetization, a monotonically increasing field is applied, the material is brought along the so-called virgin magnetization curve. If this is done after cyclic field demagnetization, the initial magnetization curve is obtained. In some permanent magnets (e.g. nucleation-type magnets), these two curves can be somewhat different, because of different domain wall populations, and only

342

CHAPTER 7 Characterization of Soft Magnetic Materials

after thermal demagnetization can one confidently assume having achieved a reference state. In soft magnets, only minor differences can possibly occur, provided no complications arise from after-effects, and thermal demagnetization is seldom performed. An example of demagnetization process by decrease of cyclic field strength is presented in Fig. 7.18. Note that that the curve actually obtained in most cases and universally exploited in designing the magnetic cores of electrical machines and devices is the normal magnetization curve. This is defined as the locus of the tip points of the symmetric hysteresis loops extending from the demagnetized state to saturation. Again, for all practical purposes, the normal magnetization curve in soft magnets coincides with the initial magnetization curve. Let us now discuss how the normal magnetization curve and the symmetric hysteresis loops can be measured, under quasi-static conditions, by means of the previously introduced methods. Figure 7.19 shows the scheme of principle of the setup currently envisaged, according to the recommendations of the pertaining standards [7.1, 7.61], for the application of the point-by-point method. The primary circuit is designed in order to provide a discrete and defined cyclic sequence of field values through a stabilized DC power supply and a combination of two switches and two rheostats. In particular, the switch S~ is used to

1.5. GO Fe-(3 wt%)Si 1.0'

i

0.5i A

o.o, -0.5 -1.0

! 1' S

-1.5 -40

-20

0 H (A/m)

20

40

FIGURE 7.18 Demagnetization of a grain-oriented Fe-Si lamination by application of an alternating field of progressively decreasing amplitude at a frequency of 10 Hz.

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

,

343

I PG ]

B (T)

,

s

O S'

P

~'.S~'"

(G, p, FIGURE 7.19 Setup for point-by-point determination of normal magnetization curve and symmetric hysteresis loops in a closed sample (ring, Epstein frame, SST). The magnetizing current is provided by a power supply or rechargeable batteries. It can be inverted by the switch $1, changed in a step-like fashion using the switch $2, and regulated by means of the rheostats R1 and R 2. The normal magnetization curve is determined keeping the switch $2 closed, regulating R1 and switching $1 back and forth. The points belonging to the hysteresis loop are found, after having fixed the R1 value (that is the (Hp, Bp) coordinate), by regulating R 2 and acting on the switch $2. A supplementary circuit supplying the winding N3 c a n be used to generate a bias field Hb. By performing a demagnetization procedure at different values of Hb, the anhysteretic curve is obtained. Note that the magnetizing winding is, here as in all testing arrangements discussed in this book, uniformly distributed along the test specimen, in order to ensure full coupling with the secondary windings.

invert the direction of the magnetizing current and the switch S 2 to p r o d u c e step-like changes of its amplitude, according to the values set for R1 and R 2. The p o w e r s u p p l y is conveniently used as a voltagecontrolled current source, which ensures safe behavior u n d e r switching

344

CHAPTER 7 Characterization of Soft Magnetic Materials

in the output circuit. With typical sample arrangements, as recommended by the standards, and ensuing reactances, time constants in the current vs. time curves of the order of a few ms at most are expected. If, as in the example shown in the figure, a closed sample configuration with defined magnetic path length is realized (i.e. ring specimen, Epstein frame, SST), the field and induction values are determined by means of an ammeter in the primary circuit and a fluxmeter connected to the secondary winding, respectively. If the permeameter configuration is adopted (Figs. 7.1 and 7.4), either a Hall device or a second fluxmeter connected to the H-coil is employed for field reading. In the latter case, only field variations are determined and current reading is still required for setting specified field strengths. The fluxmeter calibration is done in the simplest way by using the same current source and a calibrated mutual inductor, as schematically illustrated in Fig. 5.8, and provision must be made for minimization and correction of drift. Crucial to the measuring accuracy of the magnetic induction is the use of a regular specimen, which is normally the case with industrial products, with precisely determined cross-sectional area A. With sheets and ribbons, the direct measurement of thickness and width is not recommended. One should instead calculate the value of A from knowledge of the sample mass and the material density. Preliminary to any measurement is sample demagnetization, which is carried out starting from a suitably high value of the magnetizing current and decreasing it in a continuous and slow fashion, while switching $1 back and forth (that is, inverting each time the current direction). The switch $3 is kept closed during this operation to maintain the flux integrator at zero. To find the normal magnetization curve, the current is initially increased from zero to a low value il, then $1 is inverted several times to achieve a steady cyclic state between the symmetric points (H1, B1) and (-H1, -B1). Once the cyclic stabilization is reached, $3 is thrown open and the flux variation A~I = 2N2AB1, where N2 is the number of turns of the secondary winding and A is the sample cross-sectional area, is measured each time the current is reversed. After the point (H1,B1) is determined by making the average of the two readings obtained from back and forth reversals, the current is increased to a suitable value i2 and the previous operation is repeated to find the novel point (H2,B2) , and so on. Of course, each time a step-like field variation AHa of the field is imposed, the corresponding flux variation A~ will be fully established over the sample cross-section with a certain time delay because of the rise and decay of the eddy-current generated counterfield. A classical calculation permits one to estimate the associated time constant ~. For example, in a lamination of relative

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

345

permeability/Zr, conductivity ~r, and thickness d, we obtain 7 " - /d,0/d,rO'd2/8,

(7.10)

which is found to be sufficiently small for making drift problems negligible with conventional fluxmeters. In the quite limiting case of a 10 m m thick pure Fe slab of relative permeability P'r- 103~ Eq. (7.10) provides ~'---0.15 s. Substantial immunity to drift (under the proper sequence of field steps) is the basic reason for the persisting interest in the point-by-point method (also called the ballistic method because of the earlier use of ballistic galvanometers as integrating devices), in spite of the apparent complication and tediousness of the measuring procedure. Notice that the noise developed because of arcing between switch contacts during the reversal of current could be a source of error in the fluxmeter reading. Mercury switches in place of knife switches are recommended for arc suppression. Once the cyclic state between a defined couple of symmetric points (Hp, Bp) and ( - H p , - B p ) has been stabilized, the point-by-point determination of the associated hysteresis loop can proceed [7.62]. We notice that having fixed the value of al and reached the upper tip point (Hp, Bp), the magnetizing current can be decreased from the peak value ip to zero, first by opening the switch $2 and increasing the value of R 2 and then by opening the switch $1. The remanence point R is reached in this way. If $1 is now closed in the reverse direction and R2 is decreased to zero, the lower tip point ( - H p , - B p ) is attained and we conclude that the whole hysteresis loop can be traversed by acting in the right sequence on $1, $2, and R 2. This is obviously done in a step-like fashion, but it is complicated by the fact that it is not convenient to run across the loop from the upper to the lower tip points by successive field steps AHa, while determining and summing up the corresponding flux variations A~. The uncertainty of the individual readings due to drift would sum up at each step, with the additional problem that if a fine subdivision of steps is required, sensitivity and noise problems can further impair the measuring accuracy. Consequently, the field step sequence is devised in such a way that any point on the loop is determined with reference to the tip points P and V. Two different sequences are followed to determine the points (like Q) included between P and the remanence R and the points (like S) included between remanence and the lower tip point P~. Consider starting, after stabilization of the cyclic state, from point P, which corresponds to the condition: switch $2 closed, switch $1 on the up position. R 2 is regulated to a convenient value, then $2 is opened, thereby generating a sudden decrease AHpQ of the field. The corresponding flux variation &(I)pQ is

346

CHAPTER 7 Characterization of Soft Magnetic Materials

recorded and from the calculated induction variation ABpQ =

A~pQ/N2A

(7.11)

the point of coordinates (Hp - AHpQ, Bp - ABpQ) is obtained. To improve the measuring accuracy and account for minor asymmetries in the measuring system, the operation is repeated starting from the tip point pi and the actually considered variation A(I)pQ is the average of the two readings (ABpQ Jr- ABp,Q,)/2. To find a novel point closer to remanence, the resistance R 2 is increased up to a point where the remanent induction BR at point R is attained by simply opening the switch $1. The points (like S) belonging to the half loop portion going from remanence to P~ are found, again starting from P, in two steps, after having regulated R 2 to a convenient value as usual. First, $1 is opened and the flux variation A(I)pR from P to R is measured. Then we open $2 and close $1 in the reverse direction, so that an additional negative field step is generated and the variation A~RS from R to S is detected and measured. The induction variation will be ABps -~ (~(I)pR q-

A~Rs)/N2A.

(7.12)

Again, ABps will be re-determined starting from the lower tip point P~ and the average of the two readings will be taken. The hysteresis loop measurement with the ballistic method is lengthy and involved, and, even if the previously described sequence can be at least in part made automatic using a programmable bipolar power supply as current source, it is scarcely suited to present-day industrial requirements. On the other hand, the point-by-point determination of the normal magnetization curve is accurate and simple to make, more so than with the continuous recording method where we have to determine a sequence of symmetric hysteresis loops and recover their tip points with the usual problems of control and compensation of the drift of the flux signal. Alternatively, the initial magnetization curve can be measured immediately after demagnetization by applying a field ramp, but drift may introduce uncertainties on the measured induction at high fields, while heat dissipation may pose substantial limits to the maximum achievable magnetizing current. Good accuracy is instead demonstrated in the high field region by the ballistic method due to the virtual absence of drift and minor Joule heating effects in the primary winding thanks to the very short integration time (Eq. (7.10)). Examples of normal magnetization curves detected in this way up to saturation in nonoriented Fe-Si laminations are shown in Fig. 7.20a. It is equally simple to apply the point-by-point method in the determination of the anhysteretic

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

347

GOFe-(3wt% ,,,~~

2.0 !N_OFe-Si!iminati'on__s - 13' ~.'_~

'"',-..., .....

(Hb, Jb)

/,

~lnl1"5 i Fe-(3"[~:/c 1 /,,,~LFe.(1 i~~ ~"~ ........ ........... wt%)Si--t

,/

-1 -1 (a)

;o3

io, ....

H (A/m) (b) 2.0 Anhysteretic

-60 -40 -20

0

20

40

60

H (A/ m)

o"

,.i~

o.5

0.0 ,

(c)

0

20

~

H (A/m)

40

60

FIGURE 7.20 (a) Normal magnetization curves J(H) in low-Si and high-Si non-oriented laminations determined by means of the point-by-point technique (ballistic method). The B(H) curve is also shown for the Fe-(1 wt%)Si alloy (dashed line). It coincides with the J(H) curve up to about 1.9 T. (b) Demagnetization process under bias field in a grain-oriented Fe-Si lamination and resulting anhysteretic curve (full dots). (c) Low-field behavior of normal and anhysterestic magnetization curves in the same lamination, as obtained by means of the ballistic method.

curve (or ideal magnetization curve). A n y point (Hb,Jb) of this curve is obtained by applying a bias field Hb and carrying out the demagnetization process a r o u n d it [7.63]. To apply Hb, a s u p p l e m e n t a r y w i n d i n g can be used, as s h o w n in Fig. 7.19, in association with an adjustable DC source and an inductance connected in series, which has the role of decoupling

348

CHAPTER 7 Characterization of Soft Magnetic Materials

the supplementary circuit from the imposed magnetization transients. From the physical viewpoint, it is assumed that the anhysterefic state (Hb, Jb) associates to a given field Hb the polarization value Jb realizing the condition of absolute minimum of energy, that is, the condition of thermodynamic equilibrium towards which the system would drift by thermally assisted processes if given the time to do so. In other words, it can be stated that the anhysteretic curve would characterize the response of the material to an applied field if only reversible processes could occur. It therefore provides a measure of the internal demagnetizing fields, provided rotations are not significant. It also represents an important piece of information in the physical modeling of magnetic hysteresis [7.64], besides having practical appeal in a number of applications (for example, in analog magnetic recording) [7.65]. An example of biased demagnetization ending in, a point (Hb, Jb) on the anhysterefic curve is shown in Fig. 7.20b. Figure 7.20c illustrates the significant difference existing at low fields between the normal and the anhysteretic curves, which at high fields, however, with the ending of the domain wall processes, become coincident. Incidentally, it also demonstrates that a demagnetizing procedure carried out on a soft open sample with relatively low demagnetizing coefficient in the presence of an unrecognized external field (for example, the earth's magnetic field) can actually magnetize the sample! After the demagnetization under a given bias field Hb, carried out as previously described for the unbiased condition, is completed, $1 is opened. The resistance R1 is then regulated to a low value such that when $1 is closed again in the up position, the current in the primary winding jumps to a high positive value, sufficient to generate, in combination with the bias field, a near-saturating field Hp. The ensuing flux variation, corresponding to a polarization jump of amplitude Jp - Jb, is recorded. Since the point (Hp,Jp) also belongs to the normal magnetization curve, which is independently determined, Jb is immediately obtained. By repeating this procedure for a conveniently large number of HB values, the whole anhysterefic curve is achieved. In its simplest realization, the continuous recording method employs a magnetizing current source, realized with a function generator and a power amplifier, and a flux measuring device connected to the secondary winding. The function generator is typically set to provide a triangular voltage waveform, with frequency as low as reasonable for reliable signal detection and handling. At such a low frequency, the combination of small signal and long integration times impose a tight control on the drift in the secondary circuit. Stable electronic components must be employed so that even if there is some residual drift, it can be corrected after analog-to-digital (ADC) conversion by linear numerical

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

349

compensation over a full period. In addition, the impedance of the primary circuit is mostly determined by the resistance of the winding and the current monitoring resistor so that due to the non-linear behavior of the magnetic core, the induction derivative in the sample has uncontrolled shape. The question on how low the magnetizing frequency should be in order to talk confidently of DC magnetization curves and hysteresis loops is intertwined with the problem of the control of the magnetization rate. If we exclude the case of insulating or near-insulating materials, like ferrites, and do not for the time being, consider the complications arising from diffusion or thermal fluctuation after-effects [7.66, 7.67], what we need is to drive the magnetization in the sample at a sufficiently low speed to avoid eddy current effects. This can be a demanding requirement, especially in bulk specimens. For example, let us make a rough estimation of the eddy current field by a classical calculation in a lamination of thickness d and conductivity cr subjected to the rate of change of the induction/~. We find in the lamination mid-plane

crd2 Heddy --- - ~ / ~ .

(7.13)

For a I m m thick iron sheet, magnetized at a frequency of 0.1 Hz between + 1.5 T, Eq. (7.13) provides Heddy = 0.75 A/m, which can be appreciated in large-grained good purity samples. In a rod sample of diameter D, the calculation provides Heady--crD2/~/16 on the longitudinal axis. For a 1 0 m m diameter iron cylinder cyclically tested between +1 T, it is required that the magnetization period is longer than 250 s for Heddy to become lower than I A / m . Notice that the time required to achieve, in the same sample, 99% decay of Heady when the same flux reversal is obtained by the ballistic method can be estimated (Eq. (7.10)) to be around I s or lower. This would be a good reason for adopting the point-by-point method in DC bulk sample testing, but, as we shall see later, it may occur that the specific nature of the magnetization process (e.g. domain wall nucleation vs. domain wall displacement) calls instead for the application of the continuous recording method. In the continuous recording method it is often required that/~ is held constant. Besides being an obvious reference condition for the investigation of the magnetization process (for example, with some further constraints, in Barkhausen noise experiments [7.68]), a controlled constant magnetization rate permits one unambiguously to define and minimize, according to Eq. (7.13), the role of eddy currents. Figure 7.21 reports the hysteresis loop determined between + 1.7 T in a 0.30 m m thick grainoriented Fe-Si lamination at a frequency f = 0.25 Hz. At such a low

350

CHAPTER 7 Characterization of Soft Magnetic Materials

1.5

GO Fe-(3 wt%)Si f= 0.25 Hz

~"~ f ~- /

1.0

:| /;

0.5 #~,

:://i

o.o

H

-0.5

V\

'I

-1.0

.

.

.

.

.

.

.

.

t k

V"

,

'

i/

I]' ~ t " ~ ( t )

-1.5 -80 '

(a)

'

'

-60 '

|

'

'

'

-40 '

I

'

'

'

-20 '

|

9

.

.

.

.

.

.

15 10

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i

.

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.

.

.

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.

60

.

.

.

80

I

I

jl

(b)

9

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I I

v

0

,',

II I I I

A

I!

.

0 20 H (A/m)

f

0.0

' . . . .

~IL,,

,,,,i I r'.~, ~ . "

0.5

~,

9

. . . .

110 Time (s)

!

1.5

. . . .

2.0

FIGURE 7.21 Quasi-static hysteresis loop in a grain-oriented Fe-Si lamination measured at the frequency f = 0.25Hz under two different conditions: (1) Constant polarization rate of change (J = 1.7 T/s, solid line). (2) Constant field rate of change (H - 80 A / m s, dashed line). By the first condition, we closely approach the rate-independent (DC) hysteresis loop. In the second case, the additional dynamic loss contribution brings about enlargement of the loop, depending on the instantaneous value of the induction rate. The corresponding time dependence of dJ/dt over a half-period is shown in (b).

7.2 MEASUREMENTOF THE DC MAGNETIZATION CURVES

351

frequency, we can approach the rate-independent loop if B is kept constant. In the example reported in the figure we can estimate, using Eq. (7.13), Heddy ~'~ 0.04 A/m (vs. the coercive field value Hc --6.1 A/m) for the constant induction rate B = 1.7 T/s. Lack of control of/~ gives rise to additional rate-dependent loss contribution and the loop shape and area are modified. This is illustrated in Fig. 7.21a by the enlargement of the hysteresis loop occurring when, instead of the polarization, the time dependence of the applied field is controlled (dashed line). The sharp variation of the induction derivative along the loop (Fig. 7.21b), dictated by the strongly non-linear response of the material, gives rise to an additional dynamic loss contribution. Of course, any controlled B(t) waveshape (for example, sinusoidal) can lead to the rate-independent hysteresis loop provided it is always low enough to satisfy the condition that Heddy is negligible with respect to Hc. Another detrimental effect of uncontrolled B(t) behavior is apparent in Fig. 7.21b, where the peaked shape of the secondary signal strains the dynamic range of amplifiers and A / D converters, resulting in distortions and poor reproducibility of results. This problem is typically met in the characterization of circumferentially field-treated cores made of extra-soft materials (for example, amorphous, nanocrystalline, or permalloy ribbons), which exhibit near-rectangular hysteresis loops. Hysteresis loop tracers endowed with control of the induction rate are nowadays chiefly based on digital methods. They typically work on the principle of generating the magnetic field waveform by digital means ensuring the desired B(t) (i.e. J(t)) time dependence, besides handling the measuring procedure and recording of data by means of a computing unit. Systems employing analog feedback are still in use in some cases and sometimes preferred, under the condition that the employed electronic components have excellent thermal stability, where real-time control of the magnetization is important. Analog negative feedback in DC and lowfrequency hysteresis loop tracers is actually made difficult, as previously stressed, by the small value of the induced signal and the resistive nature of the primary circuit, which call for high amplification of the feedback chain and make the accurate control of the drift signal difficult. A number of apparatuses have been proposed in the literature, based either on the control of the flux derivative [7.69, 7.70] or the flux itself [7.71, 7.72]. Figure 7.22 provides a schematic description of the high-sensitivity, highstability analog electronic loop tracer holding constant induction derivative (i.e. triangular induction waveshape) developed by Mazzetti and Soardo [7.69]. The control of the magnetization rate in this device is accomplished by comparison of the flux derivative in the sample, detected by means of a separate feedback winding N3 and amplified by

352

CHAPTER 7 Characterization of Soft Magnetic Materials

~H(0 9

V'-"

b

1

---

Oomara,orp FIGURE 7.22 Block diagram of very low frequency hysteresis loop tracer with analog feedback imposing constant rate of change of induction dB/dt. Feedback is accomplished by comparison of the dB/dt signal with a reference rectangular waveform signal provided by a voltage comparator circuit and integration of their difference. Since the gain on the integrator I2 is very high, such a difference can be kept vanishingly small. Use of a high-performance low-noise low-drift DC amplifier A2 in the B chain is mandatory for good measuring accuracy over integration times of several minutes (adapted from Ref. [7.69]).

the low-noise DC-coupled amplifier A1, with a defined rectangular waveform generated by a voltage comparator circuit. The amplifier bandwidth is at least of the order of a few kHz to ensure stability in the closed loop feedback operation. The transitions between the two states of the comparator are driven by the passage of the output of the integrator I2 through two fixed values, which, for a symmetric loop, are equal and of opposite sign. I2 has a very large gain and the difference between the inputs a and b tends to vanish correspondingly. This implies that the voltage at the output of the DC amplifier A~ becomes constant and so does dB/dt, whose actual value can be changed by varying the resistance Re. The use of the extremely low-noise low-drift amplifier A2 in the secondary circuit is the key to good accuracy of the B(t) measurement over long periods. A drift value lower than 10 -8 W b / s is obtained at the integrator output, which amounts to an uncertainty lower than 1% with a flux rate of 10-6 Wb/s. A peculiar property of the devices accomplishing the control of the magnetization rate via negative feedback is that they can interfere with the microscopic mechanisms of the domain processes, to an extent

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

353

depending on the nature of the material, the b a n d w i d t h of the feedback chain, and its gain. It is observed in such cases that the Barkhausen noise is partially suppressed in the secondary winding to appear in the magnetizing current [7.71]. A classical example is represented by the so-called re-entrant loops, as typically observed in picture-frame Fe-Si single crystals, brought back and forth under tightly constant magnetization rate between the two saturated states [7.73], and in extra-pure, largegrained iron rings, as illustrated in Fig. 7.23. That the loop must be of re-entrant type follows from the fact that the nucleation of the domain walls requires a higher field H~ than the field Hc needed to drive the walls through the pinning centers in the material. It is clear that a different loop shape w o u l d be obtained instead using the ballistic method, by which

Purified iron

1.0

f

T = 1080 s

0.5

"0.... 4 I

0.0

I

-

I I I I I

-0.5

J

-1.0 , , , , l , , , l l , , , , l , , l ,

-20

-15

-10

-5

,

i

,

0

I

I

I

5

10

15

20

H (Nm)

FIGURE 7.23 Initial magnetization curve and hysteresis loop determined on hydrogen-purified Fe ring specimen 7 mm thick at constant rate of change of polarization by means of the loop tracer exploiting analog feedback shown in Fig. 7.22. The loop is traced in a time T -- 1080 s, in order to avoid eddy current effects, and displays a re-entrant shape, thereby revealing the existence of a threshold field Hn for the reversal of the magnetization, which is higher than the coercive field He. The feedback control lowers the magnetic field before reversal takes place and permits one to determine the value of Hc. In the absence of feedback, the hysteresis loop is expected to follow the trajectory described by the dashed lines (taken from Ref. [7.69]).

354

CHAPTER 7 Characterization of Soft Magnetic Materials

we would be unable to define the central portion of the loop, or applying a triangular field waveform. In the latter case, switching occurs once Hn is attained and from that point the magnetization reversal proceeds at a rate J(t) oc (Hn - Hc(t))~ where Hc(t) is the instantaneous value of the pinning field. The loop area is now increased (dashed lines in Fig. 7.23) because additional energy dissipation takes place during this transition, the instantaneous extra power loss term being Pdyn(t) -- (Hn - Hc(t))~l(t). Analog-feedback DC loop tracers are delicate setups, traditionally developed and applied in basic research, which have nowadays given way to fully computer controlled systems, where both field generation and signal treatment are digitally handled. There are two basic ways of implementing digital feedback. One consists in trying to emulate by computation the real time control of the sample magnetization realized by means of analog feedback, the other in programming the suitable time dependence of the magnetizing current by iterative augmentation of the input using an inverse approach. Computing requirements impose the basic limitation to the feedback chain bandwidth in real time control. The operations involved basically consist in: (1) Acquisition at given instants of time, separated by conveniently small intervals, and A / D conversion of a reference signal, describing the desired dB/dt waveform, and of the actual measured waveform. (2) Comparison of these two signals and computation, by means of a regulation algorithm, of the correct value of the magnetizing current, taking into account the composition of the primary circuit. (3) Digital-to-analog conversion and generation of the calculated current. It is expected that with the application of increasingly fast digital signal processor (DSP) cards, real-time digital control will gain general acceptance, both in commercial and laboratory setups. At present, the iterative method is most commonly applied to achieve the desired Jm(t) trajectory [7.74-7.76]. It can be realized by adopting the scheme shown in Fig. 7.24, which describes in summary the general structure of a computer-controlled digital hysteresis loop tracer. The operation of recursive digital control of J(t) starts with the generation of a function e(t), normally similar to the desired induction derivative dB/dt. The magnetizing winding is then supplied by a current ill(t) via a power amplifier used as a voltage amplifier with resistiveinductive load. Safe operation of the power amplifier in face of possible overvoltages at the output (which could turn up when working in current mode) is ensured in this way. The calibrated resistor RH provides a voltage drop uH(t) proportional to the magnetic field strength, which is detected and A / D converted, at least over one full period, together with the signal u2(t) -- -N2A dB/dt on the secondary winding. A two-channel digital oscilloscope or an acquisition card performing synchronous

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

Arb. function generator

I

T

=

ii I

'H ili

Ir--~l

Digital

N2

1

osci,,oscoe

PC ,,,i

355

I

dB/dt czdB

L~~Ul2(t)

(a) 0.06 ~

NO Fe-Si lamination Epstein frame f = 0.5 Hz; Jp = 1.32 T

0.04 1 >~ ~.. 0.02 v~-..4

o.oo

|

,.. . . . . .

[

UH(t)

i',, /

UL! t)

.",-f'--"'~,"

'~'1 ,i/

/i s

\/,,,. ........"q

I

-~ -0.02 -0.04

uG (t)

,., .0 '~ !

-0.06

'~ o.o . . . .

o15 . . . .

1:o . . . .

Time (s) (b)

l'.s . . . .

2.o

FIGURE 7.24 Basic scheme of a hysteresis loop measuring setup imposing a prescribed time dependence of the material polarization J (that is of the secondary voltage u2(t)) by means of a digitally controlled recursive technique. Air-flux compensation is automatically achieved by means of the mutual inductance Ma. After a first run with a sample e(t) waveform, from which an approximate J(H) relationship is obtained, the appropriate ill(t) function is computed and a novel e(t) function is generated. The process is iterated until the defined criterion for acceptance of the generated J(t) (e.g. the form factor) is met. A couple of identical DC-coupled variable-gain low-noise amplifiers is conveniently interposed between the H(t) and dJ/dt signal sources and the acquisition device. The diagram in (b), taken from experiments on non-oriented Fe-Si laminations, illustrates the behavior of the voltage signals in the primary circuit over a period once the iteration process leading to sinusoidal J(t) function is concluded.

356

CHAPTER 7 Characterization of Soft Magnetic Materials

acquisition can be used for this purpose. These devices are characterized by input impedance typically around 1 Mf~ or higher and do not load the secondary circuit. Multiplexing is not recommended as it introduces a time delay between channels. At the very low magnetizing frequencies involved with quasi-static hysteresis loop measurements, the sampling rate can be relatively low and high-resolution ADC converters can be employed. Typically, a few hundred kHz sampling rate in commercial digital signal analyzers is associated with 16-bit resolution and synchronous triggering over the two channels (interchannel delay and trigger jitter both lower than 10 -1~ s). A large number of sampled points per period (normally more than 103) makes negligible the error made in the measurement of the loop area, when the elementary time intervals have duration not commensurable with the magnetization period. It is also important, depending on the degree of control of the induction rate, when rectangular loops have to be measured. For maximum accuracy of the numerical integration, the trapezoidal rule or the Simpson's rule are typically applied. Normally, the secondary signal is small and needs to be amplified by a DC-coupled, variable gain, very stable low-noise amplifier. Commercial high-quality devices are usually endowed with RC filtering and may introduce small phase shifts. Consequently, both UH(t) and u2(t) should be passed through the same amplifier types (although usually with very different gains). The digitized and recorded signals are then stored into a PC (for example, via an IEEE 488 interface card), where integration is performed, the residual offsets and drifts are numerically eliminated and, in the absence of automatic compensation, correction for the air-flux is made. Thermal stability of the amplifiers is mandatory for meaningful linear drift compensation of the induction signal. Both H(t) and J(t) are thus calculated and the suitable adjustments on the magnetizing current strength are made in order to attain the desired Jp value. The time functions H(t) and J(t) can now be regarded as parametric representations of a hysteresis loop bearing a substantial similarity with the final loop. By changing H(t) we then expect that J(t) will be modified as dictated by the behavior of the function H(B). It is an easy matter to compute such a function, that is, the field H(t) depending on time in such a way that J(t) turns out to be identical to the desired one (for example, the one with constant J(t) value). A novel e(t) function can be calculated accordingly, programmed into the arbitrary function generator and delivered to the power amplifier. To this end, the equation of the primary circuit is considered, which can be written according to the scheme in Fig. 7.24 as

Ge(t) = uG(t) = UR(t) if- UL(t) -- (Rs + RH)iH(t) + N1A dJ

dt'

(7.14)

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

357

where G is the gain of the power amplifier, (Rs + RH) is the resistance of the whole primary circuit, including the winding resistance, A is the cross-sectional area of the sample, and the air-flux enclosed by the primary winding is disregarded. If the magnetic path length Im is defined, the current i(t) in this equation is related to the programmed field according to the usual equation ill(t)= H(t)N1/Im. In those cases where the air-flux contribution cannot be neglected, the voltage UL in Eq. (7.14) is better expressed as UL(t)= NI(A dJ/dt + At/t 0 dH/dt), with At the total area enclosed by magnetizing winding. An example of measured relationship between uG(t), UR(t), and uc(t) under controlled sinusoidal induction is reported in Fig. 7.24b for the case of a nonoriented Fe-Si lamination tested in an Epstein frame at a frequency of 0.5 Hz and peak polarization value Jp = 1.32 T. After the thus calculated e(t) function is generated, signal acquisition and calculation of H(t) and J(t) are repeated. Normally, the process cannot be concluded in a single step and e(t) is re-calculated and applied again. Iteration will proceed until the desired ](t) behavior is achieved, as objectively judged, for example, from the deviation of the actual value of the form factor from the theoretical one. Note that this feedback procedure, being based on analytical considerations regarding the loop shape, is in principle independent of the specific magnetizing frequency and is free of autooscillatory behavior. To improve the accuracy and speed of the feedback process, the reactive and the resistive terms in Eq. (7.14) should possibly be of comparable values. At very low magnetizing frequencies, this calls for increased mass of the specimen and reasonably low values of Rs and R H. It may occur in experiments that the hysteresis loop is of the reentrant type (Fig. 7.23) or that in order to emulate specific working conditions of magnetic cores in applications, complex time histories of J(t), including local minima, have to be considered. The previous feedback procedure becomes difficult to apply in such cases and it might be convenient to pose the whole problem under more general terms, where the input function e(t) is recursively calculated by introducing error terms proportional to the difference between actual and desired output values. A relationship between the values taken by e(t) upon successive iterations of the type

ek+l(t) = ek(t) + a(Bo(t) - Bk(t)) + ]3(/30(t) -/3k(t)),

(7.15)

where Bo(t) and /30(t) are the desired induction function and its time derivative, respectively, a and /3 are suitable constants, and k is the iteration order, can be envisaged in particular [7.77]. Mathematically,

358

CHAPTER 7 Characterization of Soft Magnetic Materials

Eq. (7.15) implements the search for the fixed point of the functional F(e(t)) -- e(t) + a(Bo(t) - B(e(t)) +/~(/~0(t) -/~(t)),

(7.16)

which, for suitable values of the constants c~ and/3, exists and is unique. An example of the application of Eq. (7.15) is illustrated in Fig. 7.25, showing the evolution of the polarization J(t) along the iteration process in the specific case of prescribed constant J(t) with local minima in a nonoriented Fe-Si lamination. The corresponding quasi-static hysteresis loop is endowed with minor loops. The foregoing considerations make clear that the measurement of the DC magnetization curves and hysteresis loops may have a somewhat elusive character. For example, even if we make negligible the role of eddy currents by magnetizing at extremely low speeds, time-phenomena may equally play a role because of the thermal, diffusive, or structural relaxation processes (after-effects). Specific anomalies, globally labeled as magnetic viscosity phenomena, can be found, for example, in pure Fe with faint concentrations of C or N at temperatures below 0 ~ Fe-Ni, Fe-Si and Fe-A1 alloys at the temperatures where atom diffusion occurs in times comparable to the measuring times [7.78]. In addition, however, when we are basically free from magnetic viscosity effects, there is no certainty that the point-by-point and the continuous recording methods should provide nearly identical results. The previously discussed reentrant loops is a somewhat extreme case, where large differences are found with the two methods and we can only speak of quasi-static behavior appropriately when J(t) is controlled. From a practical viewpoint, it is interesting to see to what extent the two methods can agree in applicative materials. The experiments show that in crystalline and amorphous laminations differences can be demonstrated, although comprehensive results on this point are not available. The reason for such differences is not clear, but it is understood that the way in which the field variation is imposed (continuous vs. discontinuous) can be expected to bring the system through different trajectories in the phase space and to slightly different end-points. Figure 7.26 provides two examples of comparison of the DC loop areas (i.e. energy loss per cycle) found with the continuous (J(t) = const.) and the ballistic methods in grain-oriented and non-oriented Fe-Si laminations, respectively. To accurately determine the DC energy loss W under continuous waveform magnetization, the hysteresis loop and losses are determined, under controlled sinusoidal induction waveshape, as a function of frequency and the extrapolation procedure to f = 0 normally adopted for achieving separation of the loss components is applied. There is a solid theoretical background to this

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

.5

,

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,

i

,

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.

,

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,

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A #-,~ .~-~; i',.~,,,

1.0 0.5 v

,

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359

,

NO Fe-(3wt% )Si = 1.4 T f = 0.5 Hz

j~ ''

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(b)

.

.

.

.

I

-100

|

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,

. . . . . . . . . 100 H (A/m) i

0

200

FIGURE 7.25 (a) The prescribed time dependence with local minima of the polarization in a non-oriented Fe-(3 wt%)Si lamination (solid line) is attained upon a convenient number of iterations performed according to Eq. (7.15). The measurement is performed on an Epstein test frame by means of a digital hysteresis loop tracer like the one shown in the previous figure. The evolution of the J(t) waveshape vs. the number of iterations (1, 2, 5, and 12) is described by the dashed lines. At the end of the iteration process the quasi-static hysteresis loop with minor loops shown in (b) is obtained (courtesy of E. Barbisio and C. Ragusa).

360

CHAPTER 7 Characterization of Soft Magnetic Materials

16

GO Fe-(3wt%)Si d = 0.29 mm

14 O3

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~o 10

./.

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.

'

2

.

.

.

.

I

.

.

.

.

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4 6 fl/2 (Hzl/2)

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9

9

9

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,,

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FIGURE 7.26 Comparison of the DC hysteresis loss values determined with both a hysteresis loop tracer-wattmeter (Wh) and a ballistic apparatus (Wh.ball) in two types of Fe-Si laminations. Using the continuous method, Wh is found by extrapolating t o f -- 0 the quantity W - Wd, where W and Wd are the total and the classical loss component, respectively, plotted as a function of fl/2. The fitting function is provided by the statistical theory of losses [7.79].

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

361

simple and effective approach, provided by the statistical theory of magnetic losses [7.79]. In practice, the method consists in subtracting, at each frequency, the classical energy loss component Wd, an exactly calculated quantity (Eq. (2.3)), to the total loss W and in plotting the quantity W - Wd as a function of fl/2. The best fitting function of this quantity provided by the theory extrapolates to Wh for f = 0 as shown in Fig. 7.26. We see here that in one case (GO lamination) the thus found Wh value coincides, within the measuring uncertainty, with the value Wh,ball obtained by the point-point method. In the other case (NO lamination), W h > Wh,ball by a few percent. The method of loss separation can be applied, at least in principle, for the determination of Wh under a rotational field. It is difficult, however, to provide accurate energy loss figures at low frequencies because the field signal, determined either with a flat H-coil or a RCP, is very low. An alternative method consists in measuring the parasitic torque that is generated by the dissipation mechanisms during the rotation of the field. It is a very old method employing a sensitive torque magnetometer, providing the torque L(O) per unit volume as a function of the angle 0 made by the magnetization (or the field) with respect to a reference direction [7.80]. If we wish to make the measurement up to magnetic saturation, the classical setup with disc sample and rotating electromagnet can appropriately be used. The experiment consists in rotating the magnet by 360 ~ and recording the related L(O) behavior. L(0) is made of a reversible oscillating part, generated by the magnetic anisotropy, and a frictional irreversible contribution. The reversible torque averages out to zero upon integration over the whole period. The hysteresis rotational loss is then obtained as WRh = --

L( O)d O.

(7.17)

To improve the measuring accuracy, the torque is averaged upon both clockwise (c.w.) and counterclockwise (c.c.w.) rotation. Actually, the parasitic torque Lw also oscillates with 0 in an anisotropic material. To retrieve the Lw(0) behavior, we can make the difference for each 0 value between the c.w. and c.c.w, torques, thereby eliminating the reversible part, and dividing the result by 2 [7.81]. Laboratories engaged in the DC characterization of soft magnets can make their measurements traceable to the relevant base and derived SI units through accredited laboratories or by direct comparison with the National Metrological Institutes (NMIs). The NMIs provide a list of measurement capabilities with stated uncertainties as they result from

362

CHAPTER 7 Characterization of Soft Magnetic Materials

intercomparison exercises [7.82]. Few illustrative examples regarding the measurement of DC magnetic parameters in bulk, powder, and sheet soft magnetic materials are provided in Table 7.1. Achieving absolute calibration of the measuring setups and direct traceability to the SI units requires considerable effort and specialized equipment. For the sake of routine calibrations, reference samples can be used, with crosssectional area and number of turns appropriate to the ranges to be covered. Very pure Fe or fully decarburized and stabilized Fe-Si alloy samples can be employed for reference purposes. The material will be annealed for stress relief and stabilized against aging by applying, for example, prolonged thermal treatment at a temperature around 200 ~ Ni is not a truly soft material and is better used for the calibration of systems employed in the characterization of hard magnets. It should be stressed, in any case, that the intrinsic stochastic nature of the domain wall processes is responsible for relatively poor repeatability of the measurements in soft magnets at low polarization values, that is, below about J =Js/2, where the use of reference samples might not be totally satisfactory.

7.3 A C M E A S U R E M E N T S The normal operating conditions of soft magnetic cores in devices call for time-varying fields. Soft magnets then have commercial value when they are categorized according to a minimum set of magnetic properties determined under defined AC excitation. Technical difficulties and costs often limit the amount of information provided by manufacturers, while designers, who need to compare different materials in order to optimize their devices at reasonable costs, benefit from as large as possible an ensemble of significant material parameters, obtained by characterization of the material under both AC and DC fields. The need for measurements not necessarily limited to the base figures provided in the data sheets is therefore widespread and shared by research and industrial laboratories. It justifies efforts to present and discuss comprehensive measuring methods. As stressed in the previous sections, the determination of the intrinsic material behavior sometimes appears as an elusive goal. We can approach it to a reasonable extent, while preserving good measuring reproducibility, not only by applying tightly controlled measuring conditions, including the consensus rules dictated by the written standards, but also by understanding the physical problems lying behind the measured material properties. One can indeed observe a complex evolution of the magnetic phenomenology with the magnetizing

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7.3 AC MEASUREMENTS

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364

CHAPTER 7 Characterization of Soft Magnetic Materials

frequency, which relates to a corresponding evolution of the relaxation processes, ranging from eddy current phenomena to various resonance effects occurring at radiofrequencies. In this section we shall treat the problems involved with the magnetic measurements under AC excitation in the light of the physical mechanisms of the magnetization process. A broad distinction will be made between measurements at low frequencies, where stray parameters have little or no influence (roughly speaking, up to a few hundred Hz), and measurements up to the MHz range (mediumto-high frequencies). Characterization methods at radiofrequencies will be summarized in the last part of this section.

7.3.1 Low and power frequencies: basic measurements Soft magnets employed in static and dynamic electrical machines are classified and marketed for their properties at 50 or 60 Hz and at such frequencies they are usually tested. This makes sense from a commercial viewpoint but it is rather unsatisfactory from the perspective of physical investigation or design of devices. Basically, one should aim at characterizing the material as a function of the exciting frequency, starting from quasi-static testing, under controlled (typically sinusoidal) induction waveshape, possibly assessing the whole phenomenology in the frame of physical theories [7.83]. The measuring setup schematically described in Fig. 7.24, exploiting the digital feedback procedure discussed in Section 7.2.2 (Eqs. (7.14)-(7.16)), is totally appropriate to this aim. With it one can perform precise and reproducible measurements of hysteresis loops and losses as a function of magnetizing frequency in soft magnetic laminations, powder cores, and ferrites. No special restrictions exist as to the induction waveform, thanks to digital feedback and use of a programmable arbitrary function generator, but modifications of the magnetizer configuration and hardware of the measuring setup are required on approaching the kHz range. We have described in Chapter 6 and in Section 7.1 the basic specimen configurations that can be devised for AC testing of soft magnetic materials: ring, Epstein frame, single-strip/single-sheet tester, and open samples. The permeameter arrangement with bulk specimens applies to DC characterization only. In all cases, we consider regular samples only, having defined cross-sectional area A, and a magnetic path length lm is identified. In the measuring setup illustrated in Fig. 7.24, the field is determined by measuring the current in the magnetizing winding. We can therefore make use of this circuit when testing rings, Epstein frames, and single-sheet assemblies. It basically fulfills the requirements of

7.3 AC MEASUREMENTS

365

the standards IEC 60404-2 [7.11], IEC 60404-3 [7.29], ASTM A804 [7.19], ASTM A343 [7.84], ASTM A912 [7.85], and ASTM A932 [7.20]. If flat H-coils or a RCP, placed on the sheet surface, are employed for the determination of the tangential field H(t) (Figs. 7.4-7.6), provision should be made for an additional acquisition channel in order to deal with the related signal, proportional to the time derivative of H(t). We will discuss this point to some extent in Section 7.3.3, while presenting measuring methods under rotational fields. The standard IEC 60404-2 deals with the measurements on steel sheets using the Epstein magnetizer from DC to 400 Hz. The standard IEC 60404-3 is devoted to the testing of single sheets (SST) at power frequencies. We have introduced them in Sections 6.1 (Figs. 6.3, 6.4, and 6.6) and 7.1.2 (Figs. 7.7 and 7.8). We have discussed to some extent there how the assumption of an a priori fixed value of the magnetic path length lm was reflected into a somewhat conventional determination of the magnetic parameters, namely the power loss, with ensuing discrepancies between results obtained on the same material by the two different methods. Procedures to reconcile such results in some specific cases have been considered (Section 7.1.2, Eqs. (7.5)-(7.7)). The basic provisions of the IEC 60404-2 and IEC 60404-3 standards and the specific features of magnetizer and specimens are summarized in Table 7.2. Whatever the specimen configuration, any digital hysteresisgraphwattmeter built according to the scheme of Fig. 7.24 can deliver complete information on the magnetic properties of the material over the appropriate range of magnetizing frequencies and defined induction waveforms: major and minor hysteresis loops, normal magnetization curve, permeability, apparent power, and power losses. All desired quantities are obtained in it by numerical elaboration after A / D conversion. Using high-resolution high sampling rate acquisition devices with synchronous sampling over the different channels, we can achieve excellent reproducibility of results [7.13]. It is of course possible, as envisaged in the standards, to employ physically different devices for the determination of the required quantities. The peak value Jp of the magnetic polarization can be obtained from the mean rectified value 0 2 o f the secondary, as provided by an average value voltmeter, according to the equation 02 =

4fN2AJp.

(7.18)

The standards prescribe that u2(t ) is sinusoidal (form factor, equal to the ratio between r.m.s, value and average rectified value, FF -- 1.1107 + 1%), but Eq. (7.18) is valid, in the absence of minor loops, whatever the secondary voltage waveform. The peak value of the magnetic field

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C h a r a c t e r i z a t i o n of Soft M a g n e t i c Materials

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CHAPTER 7

"~ Vl

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U3

7.3 AC MEASUREMENTS

36/

strength Hp is obtained by reading with a peak voltmeter the peak value/~H of the voltage drop across the calibrated resistor RH Hp-

N1 /hH Im RH"

(7.19)

The r.m.s, value H of the field strength can similarly be obtained by measuring /~H = RHi'H by means of an r.m.s, voltmeter (with accurate response in the presence of high crest factors). An r.m.s, voltmeter connected across the secondary winding, in parallel with the average value voltmeter, will equally provide u2, that is, the r.m.s, value of dB/dt. The maximum allowed uncertainties are + 0.2% for the average type and r.m.s, voltmeter readings and + 0.5% for the peak voltmeter reading. The specific apparent power, defined as N1 1 S -- ZHU2N2 ma

(7.20)

is immediately obtained. In this equation, the quantity ma -- 3lmA , with 3 the material density, is defined as the active mass of the specimen. It appears in place of the total mass to account for the fact that the magnetic path length lm can be different (as in the Epstein and SST testing) from the actual average length of the specimen. More complete information is retrieved, however, using the previously explained method of synchronous acquisition of the signals uH(t ) and u2(t), A / D conversion, and numerical computation of the desired quantities. We can thus obtain, at a given peak polarization value Jp, the hysteresis loop and its area (Eq. (1.29)) W=

H d J = f~ H(t) dJ(t) dt dt,

(7.21)

that is, energy loss per cycle and unit volume. The latter quantity or, equivalently, the specific average power loss per unit mass P=

ff~

dJ(t)

H(t) dt dt

(7.22)

is the base technical parameter used in the designation of the different material grades. Equation (7.22) can be derived under very general terms from Poynting theorem [7.83]. With a closed magnetic configuration, H(t) coincides with the applied field and under AC excitation it is the field existing at the specimen surface, where the eddy-current counterfield is zero. H(t) is then equal to the sum of the field required by the DC constitutive equation of the hysteresis loop and the additional field that must be applied at any instant of time in order to antagonize the eddy

368

CHAPTER 7 Characterization of Soft Magnetic Materials

current counterfield and preserve the same value J(t) of the polarization, averaged across the specimen cross-section. Let us consider the equivalent circuit in Fig. 7.27, where we have assumed, for the time being, that there are no leakage inductances and stray capacitances and that the resistance of the secondary winding is negligible with respect to the input resistance R 2 of the measuring instrument (either voltmeter, pre-amplifier, or acquisition device). We also assume that R 2 is so high that i2 ~ 0. The average power delivered by the generator into the magnetizing winding, purged of the ohmic losses in the winding resistance Rwl, is given, per unit of effective sample mass, by

Pw-- 11f~UL(t)iH(t)dt"

(7.23)

ma T

Since the available voltage on the primary circuit is ul(t) and not uL(t), the magnetic loss determination by direct application of Eq. (7.23) is possible only if the power dissipated in the winding resistance Rwl is calculated and subtracted from the loss measured on the primary circuit. This procedure is not desirable in general because the ohmic losses in Rwl are comparable to, or even higher than, the magnetic losses and they tend to change during the measurement because the winding temperature can change. The secondary voltage is therefore considered (virtual open circuit) and, since N2 u2(t ) --- _ ~ UL(t)'

by introducing it in Eq. (7.23), we obtain

Pw =

1 N1 N2 T11~u2(t)iH(t)dt"

(7.24)

ma

Equations (7.22) and (7.24) are equivalent because u2(t)= -N2A(dB/dt) and ill(t) = H(t)(Im/N1),the latter relationship implying that the magnetic circuit is closed and lm has a defined value. In fact, by substituting u2(t) and ill(t) in Eq. (7.24) we obtain Pwand, since

f I~ H(t) dB(t)dt dt-

B(t) = I~H(t) + J(t), we can also write T dH(t) d/(t) Pw=f lo[IZ~ +H(t) di dl(t) = f-~ I~ H(t) -~-dt=P,

(7.25)

]dt

(7.26)

7.3 AC MEASUREMENTS

369

Rs.

.Rwl JH

/2

iH [ RH m m

UH.---.~

(a)

U L

H

(b) FIGURE 7.27 (a) Equivalent AC circuit of the hysteresisgraph-wattmeter shown in Fig. 7.24. The magnetizing frequency is assumed to be sufficiently low as to permit one to neglect the effects of stray capacitances and leakage inductances of the primary and secondary windings. Rwl is the resistance of the magnetizing winding. The resistance of the secondary winding is negligible with respect to the input resistance R 2 of the secondary instrument. The e.m.f, u2(t) appearing across the secondary winding is related to the voltage uc(t) by the equation u2(t)= -(N2/N 1)uc(t). (b) Corresponding vector diagram (linear approximation) for nonnegligible load current i2(t) and imposed flux. The total magnetizing current im(t) (i.e. the field H(t)), phase shifted with respect to the flux ~(t) because of iron loss, is imposed and, consequently an extra-current i2~(t)= -(N2/N1)i2(t) must flow in the primary winding to counter the effect of i2(t) in the secondary winding. The total current ill(t) = ira(t) + i21(t) to be supplied in the primary winding accounts then for the additional power consumption in the load. The vector diagram is drawn here for N2/N1 = I and the proportions of i2 are somewhat exaggerated for the sake of clarity.

CHAPTER 7 Characterization of Soft Magnetic Materials

370

where the first term within square brackets integrates to zero over a period. We can therefore carry out the measurement of the power loss either by time averaging the product of primary current and secondary voltage or by calculating the hysteresis loop area. Equation (7.24) rests on the condition of negligible value of the secondary current i2(t), which is satisfied under normal measuring operations, where high input impedance signal conditioning devices (preamplifiers, acquisition cards, electronic wattmeters, etc.) are used. However, when old-fashioned electrodynamic wattmeters, especially if connected in parallel with an average value voltmeter and an r.m.s, value voltmeter, are employed, we might have to account for the additional power consumption in the instruments brought about by the circulatha.g current i2(t). Loading by the secondary circuit results, under rated flux, in an additional current i2/(t)--(N2/N1)i2(t) in the primary winding, as defined by the condition that the total magnetizing current ira(t)ill(t)- i2/(t) is imposed (see the vector diagram in Fig. 7.27b). The generated field, resulting from the currents circulating in the primary and secondary circuits, is then

N1 N2 H(t) - ~m ill(t) + -~m i2(t)"

(7.27)

By introducing it, together with the expression for the induction derivative dB u2(t) dt

N2A '

in Eq. (7.25), one obtains for the specific power loss P=

= -

H(t)- dt

,, ma

dt

u2(t)iH(t) d t -~2

0

(7.28)

u2(t)ia(t) d t "

The second term within square brackets is the power consumed in the secondary circuit, which must then be subtracted from the indication of the wattmeter in order to obtain the magnetic power loss. It may happen that AC testing is to be done on open sheet or strip samples. We know that in this case the effective field H is better measured, for instance by means of tangential H-coils, than calculated using the demagnetizing coefficient. If we are unable to determine

7.3 AC MEASUREMENTS

371

directly H, we might equally use the applied field Ha instead of the true Ha(t)in Eq. field involved in the problem. By posing (7.25), we find

H(t)=

P=

f f i"Ha(t) dJ(t) dt dt

(Nd/lzO)J

(7.29)

This equation is noteworthy, since it contains the applied field, which is the quantity directly under our control, and the polarization, which describes the magnetic state of the material under test. The magnetostatic energy is stored and released in a reversible fashion and averages out to zero over a complete hysteresis cycle. Four equivalent expressions for the specific power loss can then be implemented, according to Eqs. (7.22), (7.24), (7.25), and (7.29). To measure the specific power loss P, as well as the apparent power S, digital methods are nowadays the rule, both to set the magnetizing conditions (frequency, peak polarization value, and control of the induction (polarization) waveform) and to carry out all signal handling, computation, data storing and retrieval. Several examples are discussed in the literature [7.75, 7.76, 7.86-7.88] and commercial solutions are offered by instrument manufacturers [7.3, 7.89]. Regarding the digital control of magnetic field generation and induction waveform, this is in principle little dependent on the magnetizing frequency and the discussion given in Section 7.2.2 for the quasi-static case also applies to the AC regime. Significantly, the devised digital feedback methods, while being totally appropriate to satisfy the requirement of sinusoidal voltage u2(t) , are the ideal solution for emulating the non-sinusoidal induction waveforms expected in many applications, whether they are associated with distortions or are deliberately generated for specific needs. This is the case, for instance, for inductor cores used in switched mode power supplies or stator cores in variable speed motors supplied by means of pulse width modulated (PWM) voltages. Whatever the method of field control, the key device in the primary circuit is the power amplifier, which is required to handle signals having a large dynamic range, to be injected into strongly inductive loads. For measurements on Epstein frames up to 400 Hz, peak current values and voltages higher than 10 A and 100 V, respectively, might be required. As previously remarked, safe operation of the power amplifier can be obtained by using it in voltage mode, with a suitable power resistor in series with the magnetizing winding. Concerning data acquisition and the computation of P and S with the associated uncertainties (see also the related discussion in Section 10.4), a few basic points will be discussed here. Reference is made to the scheme of

372

CHAPTER 7 Characterization of Soft Magnetic Materials

Fig. 7.24a and the equivalent circuit of Fig. 7.27, and to a number of related studies dealing with this problem in the more or less recent literature [7.13, 7.24, 7.90-7.94]. (1) Field signal. The signal UH(t)----RHiH(t), proportional to the field H(t) in closed samples, is detected on a stable resistor calibrated against a standard. At low and power frequencies, anti-inductively wound manganin alloy wire resistors are appropriate, thanks to their near-zero temperature coefficient. Different kinds of power resistors (metal foil, metal film, and molded) can equally be employed, but their temperature coefficient must be checked and a heat sink made available. Calibration must be performed frequently and is indispensable to repeat it whenever thermal shocks due to uncontrolled current surges have occurred. Different contacts for current and voltage leads should also be adopted. The relative uncertainty on the resistance value can be kept around some 10 -4 , that is, negligible to all practical effects. If an H-coil is to be used, the related t u r n - a r e a product must be determined with the aid of a calibrated flux density source. It is typicall~ achieved with relative uncertainty in the range 1 x10 - 3 5 x 10 - o (Table 10.3), but its stability with time is critical and calibration must be frequently repeated. The range of the UH(t) signal is determined by the maximum J value required by the experiments and by the frequency. Exciting the material beyond the knee of the magnetization curve can soon strain both the power amplifier and the dynamic range of the acquisition device in the H-channel. If the measurement is made under sinusoidal polarization, as required by the measuring standards, only the fundamental harmonic UHI(t) will contribute to the power loss, since the products of the higher harmonics with the sinusoidal function u2(t ) will average to zero in the loss integrals (Eq. (7.24)). This brings about a reduced effective dynamic range of the signal conditioning devices because UHl(t) is only a fraction, increasingly smaller with increasing Jp, of the total signal to be handled in the H-channel. (2) Induction signal. The secondary signal is detected by means of 700 turns in the Epstein test frame and a convenient number of turns when using the SST, ring specimens, or open strips. Even by limiting the upper frequency to 400 Hz, a large dynamic range can result if the sample cross-sectional area remains unchanged. Figure 7.26 shows that the accurate determination of the quasi-static energy loss Wh requires that measurements are performed down to a frequency as low as 0.5-0.25 Hz, which implies more than 60 dB range in the induced voltage amplitude. To cope with it, it is convenient to interpose a DC-coupled low-noise amplifier with calibrated gain, variable in

7.3 AC MEASUREMENTS

373

a step-like fashion, between the secondary winding and the input of the acquisition device. An identical amplifier, with identically set upper cut-off frequency, is introduced in the H-channel in order to avoid any possible spurious phase shift between primary current and secondary voltage signals. Under normal conditions, the relative uncertainty of the uH(t) and u2(t) values related to gain and distortion of the pre-amplifiers is lower than 2 x 10 -3, while the imperfect compensation of the air-flux can provide a contribution to the uncertainty on u2(t) around 10 -3. A trivial and dangerous source of error comes from imperfect determination of the cross-sectional area A of the specimen. For example, a 1% uncertainty in the value of A, proportionally reflected in the value of Jp, can propagate into around 2-2.5% uncertainty in the power loss P and even 15-20% uncertainty in the apparent power S in a non-oriented material at 1.6 T and 50 Hz. The direct measurement of the strip/sheet gauge does not guarantee a sufficiently accurate determination of A. Amorphous ribbons, for example, which are inconveniently thin in this respect (typically 10-50 ~m), show a decrease of thickness going from the strip axis to the edges and their cross-section has more or less an elliptical profile. It is then recommended that mass m and length I of the test specimen are measured and that the cross-sectional area is calculated as A = m/81, with the density 8 known from composition or obtained by measurement (for example, with the immersion method). Mass and length of the specimen can be measured with very good accuracy (e.g. relative uncertainties lower than 10 -3 in Epstein strips) so that the major contribution to the uncertainty in the determination of the value of A comes from the measurement of the density. A relative uncertainty u ( A ) / A ~ 2 x 10 -3 can be achieved in Epstein specimens (see also Section 10.4). The case of non-oriented sheet steels, while having obvious industrial relevance, is made somewhat complicated by the variety of compositions associated with the different grades, which would often suggest direct measurement of the density. Such a measurement, however, may appear complicated and expensive in routine magnetic quality testing. We know that the electrical resistivity p and the density 8 are, within the usual compositional limits of the nonoriented alloys (random solutions), both monotonically dependent on the concentration of Si, A1, and Mn [7.95]. It has been accordingly verified by experiments that within the concentration limits for Si and AI: c(Si) <-- 4 wt%, c(A1) = 0.17c(Si) + 0.28 wt% (with c(A1) ~ 0), respectively, and concentration of all other additive elements globally lower than 0.4 wt%, the following linear relationship between 8 and

374

CHAPTER 7

Characterization of Soft Magnetic Materials

the product p8 exists

8-- 80 - kspS,

(7.30)

where 30 = 7973 k g / m 3 and ks = 8.92 x 104 ~-~-1 m - 1 as obtained by linear regression analysis of experimental data [7.96]. Equation (7.30) is graphically represented in Fig. 7.28a. Note that the product p8 can be experimentally obtained in strips (e.g. IEC 60404-2) and square specimens (e.g. IEC 60404-3) by means of mass, resistance, and length measurements. Figure 7.28b shows the scheme for the measurement of the resistance R -- ple/A between points separated by the distance le in a strip of length l and mass m = 31A. The product p8 is given by

i)8 -

am

(7.31)

lel

and the material density is immediately obtained by means of Eq. (7.30). For a 0.30 m m thick Epstein strip of Fe-Si 3 wt% with le -- 250 mm, R is of the order of 1.5 x 10 -2 ~, which is accurately measured with a test current of a few amperes. If a square sample is used, as in the SST method (IEC 60404-3), the resistance measurement is carried out by the Van der Pauw method [7.97]. Following the scheme of Fig. 7.28c, four electrical contacts are symmetrically arranged at the median points A, B, C, D. If the test current is made to circulate between the points A and B, the voltage difference between the points C and D is measured and the ratio RAB,CD -- VCD/iAB is calculated. In a similar fashion, the resistance RBC,DA -- VDA/iBC, expected to be very close to RAB,CD, is obtained. The material resistivity is obtained applying Van der Pauw's theory and the product p8 correspondingly becomes ,/rm

P(~ -- l2 In 2

RAB,CD q- RBC,DA

2

"

(7.32)

The density is again obtained from Eq. (7.30). The specific details of the measuring procedure can be found in the standard IEC 60404-13 [7.98], while a study on the repeatability and reproducibility of the method has been provided by Sievert [7.99]. Since the relative uncertainty of the density value u(8)/8 is, by virtue of the previous equations, connected to the relative uncertainty of the resistance value u(R)/R by a sensitivity coefficient lower than 0.1 (for a definition of these quantities see Section 10.2), it is concluded that an accuracy of the order of 1% in the measurement of resistance is sufficient for all purposes, since it brings about a m a x i m u m 0.1% uncertainty of the density value.

7.3 AC MEASUREMENTS

375

7900

7800 E 7700

7600

7500

Silicon sheet steel 0.0

.

.

.

.

w

.

.

.

.

i

.

.

.

.

i

.

.

.

.

|

1.0x10 -3 2.0x10 -3 3.0x10 -3 4.0x10 -3 p 5 (~ kg / m 2)

(a)

B

---~-Z"

A /~ / (b)

,z""

IsSS _

(3

~./i (c)

FIGURE 7.28 (a) Experiments show that the density 3 of silicon steel is linearly related to the product pS, where p is the electrical resistivity [7.96]. The product p3 can be obtained by independent mass and resistance measurements. (b) The resistance of strips over a length Ie is obtained by making a suitably high current i to flow between separate contacts traversing the whole strip width and detecting the potential difference between two inner contact points. (c) With a square sheet of side l with syTrunetrically placed contacts A, B, C, and D, the Van der Pauw method is applied [7.97]. Here the resistance RAB,CD (RBC,DA), obtained as the ratio between the voltage appearing between the contacts C - D (D-A) when the test current is made to flow between the contacts A - B (B-C), is used to obtain the product p3 according to Eq. (7.32). (3) Signal acquisition, A/D conversion, computation. The signals u2(t ) a n d uH(t ) d e r i v e d from the s e c o n d a r y w i n d i n g a n d the calibrated resistor R H are amplified a n d fed into the acquisition stage, w h e r e they are s y n c h r o n o u s l y s a m p l e d at a c o n v e n i e n t l y fast rate a n d A / D converted.

376

CHAPTER 7 Characterization of Soft Magnetic Materials

An interfaced computing unit will then be used to store the data and perform all necessary calculations, besides the general control of the operations of the whole hysteresisgraph-wattmeter. The converted voltage u2(t), now available as a time sequence of values u~k), with the index running over k = 1,..., n over a period, is numerically integrated, possible drifts are compensated and, after multiplication by the appropriate factors, the corresponding sequence of values j(k) is obtained. A sequence of points i(kH)(H(k)), synchronous with the u(2k)(J(k)) points, is similarly available (obviously without performing integration) on the second channel. If an H-coil is employed instead of the current method for the measurement of the field, the H (k) values are computed exactly as for the j(k) values. In those cases where compensation for the air flux is not automatically accomplished (e.g. by means of a mutual inductor or a compensated secondary winding), the H (k) values are calculated first and the suitable correction on the signal, proportional to la,oH(k), is performed afterwards. The specific power loss is calculated according to Eq. (7.24), from the arrays u(2k) and i~) as

p =

1 N1 1 ~ , (k);(k) - ma N2 n

k=l~2 "H,

(7.33)

or, equivalently, from the area of the hysteresis loop (Eq. (7.21)), using the arrays H (k) and j(k). The numerical integration is performed according to the conventional trapezoidal or Simpson's rules. The specific apparent power S is similarly obtained by separately computing the r.m.s, values

=

(1 ~()2) 1/2 n k=l

and

~ZH-" (l~(/(Hk) 1/2-n k=l /2) and multiplying them according to Eq. (7.20). All relevant parameters of the hysteresis loop (permeability, coercivity, remanence, etc.) are immediately calculated from the stored H (k) and j(k) arrays. Figure 7.29 provides an example of determined hysteresis loops in grain-oriented Fe-Si laminations at different frequencies and the corresponding behavior of their area, the energy loss per cycle and unit volume. Notice the loss-separation procedure by which the hysteresis (Wh), classical (Wd), and excess We• loss components are identified at each frequency

7.3 AC MEASUREMENTS

377

7

G rai n-orientecJ,,~7--~f~ [ d = 0.30 mmt / Jp= 1.7T

1.5 Fe-(3wt%)Si I 1.0 0.5

0.2

o.o ,%

-0.5

~ 1 5 0 Hz -1.0

"•400

-1.5

Hz

.

50

.

.

.

I

.

-100

.

.

.

i

-50

.

.

.

. . . .

.

!

0

. . . .

i

50

. . . .

1O0

15O

H m (A/m) 600

Grain-oriented Fe-(3wt%)Si d = 0.30 mm

soo E

"--

.'1

J~=I.7T

400

,, "

Wexc

/

o

300

s

..e-

r

-"

/Ir

~oo

r

.. " " "

Wcl

.-"

,,

0

I

,,-

~

s

. ,'J"

,,

LU 200

,, ,- "

,, ,"

"

.

.

.

.

I

100

.

.

.

.

I

200

.

.

.

.

I

300

.

.

.

.

I

400

Frequency (Hz)

FIGURE 7.29 Hysteresis loops and energy losses in a grain-oriented Fe-Si lamination (thickness 0.30 mm), m e a s u r e d as a function of frequency by means of a digital h y s t e r e s i s g r a p h - w a t t m e t e r (Fig. 7.24). The induction w a v e f o r m is controlled sinusoidal, with peak polarization value ]p = 1.7 T. The acquisition is m a d e by sampling more than 2000 points per period, with a m i n i m u m resolution of 12 bits. At the lowest frequencies, use is m a d e of low-noise large b a n d w i d t h identical pre-amplifiers for both H and B channels.

378

CHAPTER 7 Characterization of Soft Magnetic Materials

[7.79]. The frequency-dependent part W d ~ = Wd + Wexc = W - Wh, called dynamic loss, is frequently considered. The sample is, as a rule, demagnetized before taking any measuring point, but the demagnetizing frequency is not specified in the measuring standards. It is implicitly assumed that demagnetization and testing are made at the same frequency, but it is also expected that it plays a negligible role under ordinary test conditions, provided it is not so high that flux penetration is incomplete. Under certain conditions, however, where samples with wide-spaced domains are tested at low Jp values, a certain dependence of the measured losses on the demagnetizing frequency can be demonstrated (Fig. 7.30) [7.100] and it should be duly considered when specifying the measuring conditions. The physical reason for this effect lies in the role of the demagnetization frequency fd on the domain wall spacing finally attained, which is the larger the lower fd. This implies a correspondingly larger excess loss component if the testing is made at a sufficiently low Jp value and

6-

"I1 ,

GO Fe-(3wt%)Si

1 1 ",a

=o.TT

I

'I ,I

1/tt

i .o t

o .,."i

fd = 50 HZ

1 .,,,,

LU

i,I,

~

f11~

fd = 1 Hz

1

0

.." 1 1

O/

o

0

20

40

60

80

100

Frequency (Hz)

FIGURE 7.30 Energy loss per cycle vs. magnetizing frequency measured at Jp 0.7 T in a grain-oriented Fe-Si sheet following demagnetization at two different frequencies: fd = 1 and 50 Hz. The observed differences are due to the variation of the excess loss component with the domain spacing, which increases with decreasing fd (adapted from Ref. [7.100]). -

-

7.3 AC MEASUREMENTS

379

extensive rearrangements of the domain structure during the magnetization process do not take place. (4) Sources of measuring uncertainty. The chief measuring uncertainty sources in the AC characterization of soft magnetic materials can be broadly classified as related to [7.90, 7.101]: (a) magnetic state of the material, which is affected by intrinsic randomness of the magnetization process, complex time histories, and after-effects; (b) temperature, stress, and physical parameters of the material (e.g. density and sample crosssectional area), geometry of the magnetic circuit (e.g. method of flux closure and magnetic path length); (c) stability, offset, and noise of the magnetizing current; (d) signal treatment: gains, phase shifts, distortions, form-factor of the secondary voltage, sampling, A / D conversion, and computation. In Section 10.4, we present a simplified approach to the estimation of the uncertainty in the measurement of the power loss in soft magnetic laminations, which takes into account the most significant of the above contributions. Some of them have obvious effects and require obvious steps. More complex is the role of signal treatment, whose optimization in terms of measuring uncertainty is not immune from considerations of operating costs, simplicity of use, and speed of measurement. Recent intercomparison exercises on Epstein and SST samples have shown that digital signal-conditioning systems, endowed with defined and suitable technical specifications and traceable calibrations, can provide, under standard testing conditions, minor contributions to the measuring uncertainty [7.13]. All necessary operations of integration, offset and drift correction, and signal multiplication, being numerically performed, intrinsically satisfy the requirements of stability and temperature independence. The performance of these systems can be summarized by two basic parameters: sampling frequency and amplitude resolution. They are not independent because the resolution of A / D converters decreases with increase of the sampling frequency. In ordinary experiments with Epstein and SST samples at peak inductions of technical interest, 12-bit resolution with Nyquist sampling (i.e. with the sampling frequency an integral multiple of the magnetizing frequency) over 128 points per period appears to provide adequate measuring accuracy [7.90]. The theoretical calculations and computer simulations reported in Fig. 7.31 show that in this case the relative uncertainty u(P)/P contributed by the A / D converters on the measured loss at power frequencies can be deemed low or negligible with respect to the other uncertainty sources. For example, in the worst case reported in Fig. 7.31, that of a non-oriented Fe-(3 wt%)Si lamination excited at 1.7 T peak polarization at 50 Hz, a value u(P)/P--- 2 x 10 -3 is estimated for 12-bit resolution. In practice, with present-day commercially available

380

CHAPTER 7

Characterization of Soft Magnetic Materials

0.1 Fe-Si sheet steel "O...,.

0.01

"-.4,.

1E-3

1

2

--.....o.,.,...Q....,...

K" v 1 E-4

3

"

1E-5

1E-6

'

1'0

"

1'2

.

1'4

'

1'6

Resolution (bit)

FIGURE 7.31 Relative uncertainty in the measurement of the power loss in non-oriented and grain-oriented Fe-(3 wt%)Si laminations as a function of the amplitude resolution of the A / D converters. Theoretical calculations and computer simulations, made under the assumption of Nyquist sampling at 50 Hz on 128 points per period, provide the lines and the points, respectively. (1) NO alloy, Jp = 1 79 T", (2) NO alloy, Jp -- 1.0 T; (3) high-permeability GO alloy, Jp = 1.8 T; (4) high-permeability GO alloy, Jp = 1.5 T (adapted from Ref. [7.90]).

digital acquisition setups, this resolution can be attained with sampling rates around 100 MHz and higher, exploiting the large redundancy of sampled data. With points per period easily in the order of a few thousand, there is no need for Nyquist sampling, and the small error associated with imperfect duration of the integration time can always be numerically corrected. Digital treatment of the signal does not give rise to significant spurious phase shift between channels. This prevalently derives, in fact, from inevitable differences between the analog devices in the two chains, stray parameters, and noise. To reduce the effect of the latter, a convenient number N of single shot determinations of either P or S is performed and the results are averaged. The example reported in Table 10.6 shows that for typical loss measurements (e.g. GO Fe-Si, Jp = 1.7 T, f - 50 Hz), the related uncertainty contribution (Type A, going as 1/x/N 1) becomes

7.3 AC MEASUREMENTS

381

lower than I x 10 -3 for N > 7. Note that the method of averaging the/H(t) and u2(t) signals upon successive acquisitions and integrating their product over the period to obtain P [7.102] is methodologically incorrect and should be avoided. Spurious phase shifts can actually be the source of a large and dominant contribution to the uncertainty on P when Jp goes far beyond the knee of the magnetization curve. The fundamental component of the magnetizing current/Hi(t) tends to go in quadrature with respect to u2(t ) at high inductions (i.e. the power factor P/S becomes very low). Consequently, even very small fluctuations of their phase shift can engender large fluctuations of the measured power loss value. In fact, if we write, by developing Eq. (7.24) for sinusoidal u2(t), 1 NI~ ~ P . . . . ZH1u2 cos ~ ma N2

(7.34)

and we disregard the other uncertainty sources, we obtain that the uncertainty of P is related to the uncertainty of the phase shift according to the equation

u(P) p

u(qo)

= ~ tan ~ ~

q0

(7.35)

and tends to diverge in the limit ~--* 90 ~ The calculations reported in Fig. 7.32 regarding Fe-Si laminations [7.24] show that a phase uncertainty as low as u(~)---0.1 ~ can generate unacceptable uncertainty u(P)/P above 1.6-1.7T in non-oriented alloys and 1.8-1.9 T in grainoriented alloys. A further unavoidable source of error at high measuring inductions comes from the distortion of the secondary voltage waveform, control of which becomes extremely difficult to achieve. Correction for such an error on the power loss measurement can be envisaged, but a sound procedure requires that a loss separation procedure be carried out at the very same high Jp value, which can be obtained to a rough extent only [7.103, 7.104]. Correction for distortion can easily be made with the apparent power S (Eq. (7.20)), whose value is, however, markedly affected at high inductions by the uncertainty of the setting of the peak polarization value. This occurs because the slope of the magnetization curve rapidly falls on approaching the magnetic saturation and a fluctuation, AJp is reflected in a large field fluctuation, AHp. If we assume that flux compensation is made and J(t) is sinusoidal, Eq. (7.20) can be written as O9

S = -~M(j)j,

(7.36)

382

CHAPTER 7 Characterization of Soft Magnetic Materials

1 u(@) = O. 1 ~

3.0x10 "2

a. 2.0x10 .2

1.0xlO "2

o.o

9

'

o I

.

.

.

1.0

.

~o"--I

.

1.2

.

.

5---~"~>~-o

.

'~

.

.

.

.

1.4

Jp(T)

I

1.6

'

'

----~ '

'

I

'

1.8

FIGURE 7.32 A faint spurious phase shift fluctuation between primary current and secondary voltage can be the cause of a large uncertainty in the power loss value when the peak polarization value is beyond the knee of the magnetization curve (Eq. (7.35)). The examples reported here refer to calculations made in four different soft magnetic laminations: (1) non-oriented Fe-(3 wt%)Si; (2) low-carbon steel; (3) conventional grain-oriented Fe-(3 wt%)Si; (4) high-permeability grainoriented Fe-(3 wt%)Si (adapted from Ref. [7.24]).

where we have made explicit the dependence of/z/on j (i.e. By taking the derivative of Eq. (7.36), we obtain that uncertainty u(j)/J of the polarization amplifies into uncertainty u(S)/S of the apparent power, according to the

u(S) s

j

1

fi ollo

u(j) i

of Hp on Jp). the relative the relative equation (7.37)

In practice, it is expected that measurements of the apparent power, performed according to recommendations of the IEC Standards at Jp values of practical interest, are affected by relative uncertainties varying between 1 and 5%. A summary of the present capabilities on AC measurements in soft magnetic materials declared by NMIs is provided in Table 7.3. All these measurements are traceable to the appropriate SI units.

r~

o

,4..a

8

~d

I

X tr3

?

X

7.3 AC MEASUREMENTS

oo

o 0

Z

o~

~8

0

L~

.

~>

~6 ~a O

7

X tr~ I

I

X

? X u3

~d

X

7

? X

I

oo

r

O

v

383

384

o

~9

r~

0

~3

c~

G;

4

r

G)

kO

G~

kO

~6

Characterization of Soft Magnetic Materials

?

[/1

? x

~,.~

',D t',.

x

~

b,o

K

~"

0

<

7

x

?

CHAPTER 7

r~

0

'.n

o.~

7.3 AC MEASUREMENTS

385

7.3.2 L o w and p o w e r frequencies: special m e a s u r e m e n t s The measuring problems and methods we have discussed in the previous section relate more or less explicitly to existing standards and cover the basic issues of AC testing of soft magnets. However, both research in materials and specific industrial needs may require a certain degree of flexibility in the measuring approach. Non-conventional methodologies are therefore in frequent use, ranging from on-line testing of magnetic laminations to characterization under two-dimensional fields. For the variety of adopted solutions, accuracy and traceability remain central issues of measurements and should as far as possible be considered and declared along with the reported results. Most of the interest attached to AC testing regards the power loss figure in view of its role as a fundamental technical parameter of the material. We will largely devote our discussion to its measurement.

7.3.2.1 On-line testing of magnetic sheet steel. The single-sheet testing method is widely accepted in the industrial laboratories and as far as possible adopted in lieu of the Epstein strip testing method because it is quicker and easier to operate. It may be desirable, however, to make a quality assessment of the whole lamination in the plant during the final phase of production, that is, to make magnetic testing on the continuously running steel strip before coiling. This must obviously be non-contact measurement and special solutions must be devised to reliably detect B and H, which is all we need to characterize the material. Figure 7.33a provides a schematic view of an arrangement for on-line lamination testing along the rolling direction (that is, the direction of motion of the strip) [7.105]. The steel strip slips through an enwrapping magnetizing solenoid and a B-sensing coil, which are enclosed by a double-C yoke, placed at the smallest possible distance from the moving strip allowing for safe operations. A fiat H-sensing coil, placed as near the surface of the lamination as practicable, can additionally be employed. The comparatively approximate nature of the resulting measurement, which is valuable chiefly in terms of comparative testing and continuous monitoring of the lamination properties along the coil, is the inevitable consequence of the poor quality of the magnetic circuit and the tension on the strip. The effect of the latter must be accounted for by known dependence on stress of the magnetic properties of the investigated material. Figure 7.33b illustrates the realization of an on-line testing method where, in analogy with the principle of the fieldcompensated single-strip tester described in Fig. 7.6, a RCP is employed to automatically detect and compensate, via the set of supplementary

386

CHAPTER 7 Characterization of Soft Magnetic Materials Direction of motion

~

~

solenoid

(a)

RCP

C

c /

J'fllllllillllllll

IIIIIIIIIII!1I!!1 C

IIIIilllllll .u,.a.es

I,,,m,m~ c

moving strip

r

c l,mm,l~ C

/

\

d~/dt

C

magnetizing winding

(b) FIGURE 7.33 On-line testing can provide quality control on the whole strip length in the rolling mill. The strip is passed through a magnetizing tunnel where a solenoid generates the field and a non-contact flux-closing double-C yoke ensures a certain degree of homogeneity of the magnetization in the material (a) [7.105]. The induction is measured by means of an enwrapping coil and the field by means of a tangential H-coil. In the arrangement shown in (b), however, a Rogowski-Chattock coil is used in association with compensation windings (C) and copper plates to maintain uniform field in the measuring region, so that the magnetizing current provides a measure of the field (see also Section 7.1.2 and Eqs. (7.3) and (7.4))[7.106].

7.3 AC MEASUREMENTS

387

coils (C), the drop of magnetomotive force occurring outside the measuring region [7.106, 7.107]. In this way, the value of the magnetizing current can be taken as a measure of the field in the material (see also the discussion in Section 7.1.1). Due to the distance between the ends of the RCP and the moving sheet surface, the compensation is inevitably affected by an error (though reputedly small) and some form of calibration is likely required. Two Cu plates, 1-2 m m thick, are inserted between the magnetizing winding and the strip in order to reinforce field homogeneity. This occurs because the eddy current fields generated in the plates restrain the AC flux component normal to the sheet plane. With non-oriented laminations, the problem arises of testing the moving strip in the TD. This cannot be done with enwrapping magnetizing solenoids. A solution can been devised, which consists in generating the transverse magnetizing field by means of current sheets, obtained by laterally removable arrays of parallel conductors placed immediately above and beneath the moving strip (see Fig. 7.34) [7.108]. The induction signal can be obtained by a couple of enwrapping coils, crossing the strip at 45 ~ and connected in such a way that the detected variations of the flux component along the rolling direction subtract and those along the TD add one to the other. Non-enwrapping secondary windings can also be devised, as discussed in detail by Beckley [7.108]. To cope with the strong demagnetizing fields ensuing from the finite width of the strip, a transversally placed flux-closure yoke should be used. If this is not convenient, as may happen when strips of different

Ha

"

.s.s

s-""

"

,,. '"

secondary

TD ~

magnetizingwmndmng winding

FIGURE 7.34 Example of field generation by means of non-enwrapping windings for on-line testing of a strip along the TD. The two removable arrays of parallel conductors, placed both above and beneath the strip, realize current sheets and the ensuing flux along TD is flux-closed by means of a double-C yoke (not shown in the figure) with end faces placed as close as possible to the strip edges [7.108].

388

CHAPTER 7 Characterization of Soft Magnetic Materials

widths are tested, the density of the conductors in the magnetizing arrays can be increased in the region where the strip edge is expected to lie, thereby compensating to some extent for the flux leaving the strip before it can reach the edges. Note that the precise measurement of the sample magnetization depends on accurate knowledge of the crosssectional area A of the strip. This can be a challenging feat with moving laminations. It has been suggested that A can be obtained by making a measurement of the polarization J under a very high field (H a > 50 kA/m) [7.105]. The correct value of A is the one making such a value to coincide with the saturation polarization Js, which is known from the material composition. In this measurement, the induction waveform does not play any role and the required power can be taken directly from the mains.

7.3.2.2 Power loss with thermal methods. Local measurements. We have remarked how the measurement of the power loss at very high induction values is affected b y a large uncertainty and cannot be reasonably pursued by means of the wattmeter-hysteresisgraph method. Phase shift fluctuations are the chief reason for this limitation, as quantitatively illustrated by Eq. (7.35). There are devices, however, where saturation in the core or some parts of it is approached (e.g. the teeth in the statoric core of motors) and interest has been raised in the determination of the related loss figure. The natural approach to the measurement of energy loss in this case is based on calorimetric techniques. The temperature of a ferromagnetic sample kept under adiabatic conditions actually evolves along a whole hysteresis loop, partly in a reversible fashion because the entropy term associated with the spin configuration does correspondingly change, and partly irreversibly because of the various energy, dissipation mechanisms. In metallic samples, these mechanisms are almost exclusively due to eddy currents. If we are interested in the measurement of the power loss at a given Jp value, we can apply the AC field to the thermally insulated specimen for a convenient time interval and look either at the correspondingly generated global heat or at the rate of heating. The measurement of the total heat by means of conventional calorimeters can be envisaged in small samples (a few m m lateral size), not amenable to conventional magnetic measurements [7.109]. There are certain complications and sources of measuring uncertainty with this method (including, for example, the fitting of the exciting coil in the calorimeter and non-uniformity of the magnetization in the open sample) and little application of it has been reported. In the classical approach to the thermal measurement of power losses, which can be performed on the very same specimens prepared for

7.3 AC MEASUREMENTS

389

conventional measurements according to the standards, the rate of change of the temperature dT/dt is determined by placing either thermocouples or thermistors on the sample surface. For an adiabatic process, this quantity is in fact proportional to the dissipated power dT dV 1 P = Cp--~ - Cp dt (dV/dT)'

(7.38)

where c, is the specific heat per unit mass, d V / d t is the rate of increase of the generated thermoelectromotive force (t.e.m.f.), and d V / d T is the Seebeck coefficient of the adopted junction. Consequently, the AC power loss at a defined Jp level can be simply obtained by the slope of the straight line representing the increase of temperature with time in a perfectly insulated sample. It is actually expected that the previously mentioned reversible process of cyclic heat exchange between the spin system and the lattice (magnetocaloric effect) would affect the T(t) signal. However, the ensuing fluctuation of dT/dt is filtered out at the usual test frequencies and does not interfere with the observed steady increase of T(t). The simplest approach to the measurement of P with the thermal method is that of placing a thermocouple, possibly obtained by joining small diameter (0.1 mm or less) wires, in good contact with the surface of the sample. An identical and closely located cold junction, physically separated from the sample, is connected in series opposition to it and the resulting differential signal is amplified via a calibrated nanovoltmeter (Fig. 7.35). Provision should be made for excellent thermal insulation of the test specimen and generation of the suitable AC field through a feedback procedure. It is fair to stress again that the induction (or polarization) waveform control on approaching the saturated state becomes possible only to a limited extent and correction for distortion, though approximated, should be made. The rate of rise of temperature method was applied for the first time to the measurement of the high-induction losses by Brailsford [7.110], who actually used the compensating thermocouple attached to a guard strip of known specific heat, which was heated by a current flowing in it. This current could be adjusted in such a way that the same rate of heating occurred in the guard strip and the specimen, and the power loss in the latter could then be determined under resulting zero signal. Localized loss measurements were demonstrated by placing micro-thermocouples in different grains of grain-oriented Fe-Si sheets by Overshott et al., unambiguously showing the connection between domain wall spacing and excess losses [7.111]. The problem with thermocouples is that they generate a very low signal. This requires careful protection against

390

CHAPTER 7 Characterization of Soft Magnetic Materials insulating block

TC2 /

oltm.

~ Field supply

I

Digital

oscilloscope

~

FIGURE 7.35 Schematic view of a loss measuring setup based on the rate of rise of temperature method. The strip test specimen either makes part of an Epstein frame or it is endowed with a flux-closing yoke, altogether provided with suitable thermal insulation (not shown here). Alternatively, it may be cut into elongated elliptical form and placed in a vacuum bell, in turn placed within a magnetizing solenoid. The average temperature over a representative sample region is detected by means of the extended junction TC1, connected in series opposition with the identical junction TC2, insulated from the sample, by which the ambient temperature and its fluctuations are compensated. noise and electromagnetic interference, and avoidance of any spurious electromotive force. The use of a precise nanovoltmeter for amplification of the dV/dt signal is highly recommended. To make an example, we find from Eq. (7.38) that a power loss of I W / k g (the typical loss figure at 50 Hz and 1.7T of a high- permeability grain-oriented Fe-Si lamination) generates in a copper-constantan junction a rate of increase of the t.e.m.f, dV/dt = 87.7 x 10 -9 V/s, corresponding to a temperature rate of change dT/dt = 2.17 x 10 -3 ~ This is a perfectly manageable signal, as illustrated by the example of the measured dT/dt behavior in a nonoriented Fe-Si specimen (thickness 0.35 mm) reported in Fig. 7.36. It refers to exl~eriments performed on a 140 m m disk, placed in a vacuum bell (p < 10 -~ Pa) and subjected to an alternating field provided by a magnetizer of the type shown in Fig. 7.14 and directed along RD. The induction waveform, measured over a central region of the disk by threading a few-turn search coil through 0.7 m m diameter holes drilled at a distance of 40 mm, is kept sinusoidal by means of digital feedback. An extended copper-constantan junction, glued with silver paint over

7.3 AC MEASUREMENTS

391

0.20 Field off

0.15 0 o

&.

0.10

v

0.05 NO Fe-(3wt%)Si Jp= 1.60T f= 50 Hz

0.00 I

0

(a)

0.12 0.10 0

'"

'

~

'

'I

20

'

"'

'

''

I

'

'

''

'

I

40 60 Time (s)

NO Fe-(3wt%)Si I. 60T f= 50 Hz

'

'

"'

''

I

80

'

I

,

,

100

,~,/ ..~ z

/

r

0.08

o

0.06 v

1

0.04 0.02 0.00 (b)

' ' O' . . . .

5' . . . . 10' . . . . Time (s)

15' . . . .

2'0

FIGURE 7.36 Temperature increase vs. time in a non-oriented Fe-Si lamination upon application of a 50 Hz alternating field along the rolling direction. The polarization is sinusoidal and the peak value is Jp - 1.6 T. The sample is diskshaped and is held in a vacuum bell (p < 10 -2 Pa), fitted in a magnetizer like the one shown in Fig. 7.14. Heat loss after field switch-off is apparent in (a). The associated decay of temperature is basically described by Eq. (7.41). It is observed in (b) that the experimental increase in temperature (open dots) is accurately described by Eq. (7.40) (dashed line). The adiabatic approximation (dash-dotted straight line, Eq. (7.42) is verified within 2% up to t --- 10 s).

392

CHAPTER 7 Characterization of Soft Magnetic Materials

the same distance, is employed to detect the rise of temperature following the switching-on of the field and is connected, as shown in Fig. 7.35, in series opposition with an identical junction, thermally insulated from the steel plate. Figure 7.36 provides a good idea of the practical extent to which adiabatic conditions are emulated. To assess this problem, we can model it by considering a real sample of unit volume where the energy P dissipated in unit time goes partly to increase the sample temperature and is partly exchanged with the surrounding reservoir. If at switch-on of the applied field Ha (t - 0) the sample and the reservoir are at the same temperature To, we can write, by denoting with d Q i n / d t and d Q o u t / d t the rates at which heat is stored in the sample and lost to the environment, respectively, the dynamic energy balance equation at time t > 0 p_

dQout

dQ~ dt

~

dt

d ( T - To) - Cp

dt

4- K~ (T - To) ,

(7.39)

where K1 is the heat transmission coefficient. The increase of the specimen temperature T with time is then obtained by solving this equation, which provides ( T - T o ) - - ~1 1 - e x p

-~pt

.

(7.40)

In the limit K1 ---, 0, this equation reduces to the linear relationship (7.38). The coefficient K1 is easily found by fitting the exponential decay of temperature observed after field switch-off. If at this instant of time the temperature is T1, we obtain, by posing P = 0 in Eq. (7.39), that the temperature decreases as

(K,)

( T - To)= (T1 - To)exp --~-pt .

(7.41)

For the case shown here, it is found kl -- 1.35 J/(kg ~ s). The value of P is then determined, as shown in Fig. 7.36b, by complete fitting of the experimental ( T - To) vs. t behavior upon application of Ha (dashed line). The same result could equally be obtained, and more simply, by drawing the tangent P (T - To) -- - - t Cp

(7.42)

7.3 AC MEASUREMENTS

393

(dash-dotted line in Fig. 7.36b) to the initial portion of the experimental curve. In fact, a straight line is a good approximation to it up to a time t such that K2 t<
394

CHAPTER 7 Characterization of Soft Magnetic Materials

high inductions in non-oriented steel sheets by use of thermistors and a bridge circuit [7.113], while Derebasi et al. claim accurate measurements at 50 Hz as a function of Jp in amorphous ribbons in spite of the very low values of the loss figures involved [7.114]. The obvious disadvantage of thermistors is that they are active devices, which must be supplied by a stable current source. They consume some amount of power (typically a few per cent of the measured power) and a correction for the related error on the loss figure must be applied. The quest for local power loss measurements is motivated by the desire to investigate the behavior of the material to a scale suggested by the geometrical features of the sample (for example, the teeth in the statoric core of a motor), the structure of the material (e.g. individual crystal in grain-oriented laminations), or even the domain structure. To this end, a number of techniques, besides the thermal one, have been implemented, differing in the methods used to detect local field and induction. In the scheme reported in Fig. 7.37a, the local tangential field is measured by means of a miniature magnetoresistive sensor made of permalloy, biased by a small permanent magnet (for a discussion on magnetoresistive magnetometers, see Section 5.2.2), and the corresponding magnetization is induced from the signal detected by a coil wound around the yoke limb [7.115]. This flux-sensing method requires a calibration procedure on a test sheet of known properties. The local induction value can actually be determined in a most simple way by making holes and threading a few-turn winding through them. A non-destructive approach is, however, preferred for practical and quick material testing and, whenever sensitivity is not a big problem, needle probes can be advantageously employed. Figure 7.37b schematically shows two possible arrangements for local induction measurement with needle probes, and Fig. 7.37c illustrates the underlying working principle. If the contacts are placed at a distance l much greater than the lamination thickness and the flux variation occurs homogeneously upon such a scale, it is obtained from FaradayMaxwell's law that

1 ~ dBy Via = 734 = - 2

A,, - ~

1 d a n = 2 Vc~

(7.43)

where V12 is the voltage measured between the contact points of the needles, A n is the portion of the cross-sectional area of the sheet lying beneath the contacts, By is the y-directed induction, and Vc the voltage induced in the one-turn search coil wound around the same area. In deriving this equation, it is implicitly assumed that the lamination edges

7.3 AC MEASUREMENTS

395

magnetizing

,.,

coil

-

!]2 o (a)

.'c d 1Or.,

(b)

/ly>

-

(c)

FIGURE 7.37 Methods for the local measurement of hysteresis loop and power losses in magnetic sheets. (a) The applied field is provided by a yoke and the local field is detected by means of a magnetoresistive sensor (sensing area 1.5 mm • 10 ~m). The flux in the measuring region is obtained, upon suitable calibration, by measuring the flux in the yoke (adapted from Ref. [7.115]). (b) Local measurement of the induction by needle probes in one dimension [7.117] and in two dimensions [7.118]. The local field is determined either by means of a Hall sensor or an H-coil. (c) The voltage V12 = V34 measured by using needle probes over a distance l much greater than the lamination thickness (exaggerated for clarity here) and the domain wall spacing is V12 = 89 Vc, where Vc is the voltage induced in a one-turn search coil wound around the same portion A,, of the lamination cross-sectional area. The horizontal lines schematically represent the eddy current trajectories. are far from the m e a s u r i n g region and the e d d y currents are flowing only in the x-direction. Detailed theoretical analysis shows that the presence of the d o m a i n walls does not introduce i m p o r t a n t differences b e t w e e n the time evolution of the fluxes detected by the single turn and the needles. Discrepancies can be of the order of a few percent w h e n the d o m a i n wall spacing D becomes a substantial fraction of the m e a s u r i n g

396

CHAPTER 7 Characterization of Soft Magnetic Materials

distance (D >---0.21) [7.116]. Reasonable figures are therefore expected using needle probes placed at a distance around 10 m m in grain-oriented laminations, as this arrangement, illustrated in Fig. 7.37b, provides a reasonable D/I value, while allowing one to put in evidence the grain-tograin loss fluctuation. Notice that the needle leads must always be arranged in such a way that as shown in the figure, they do not embrace a significant area. In the examples here reported, the local field is detected either by means of a Hall plate [7.117] or by a small H-coil [7.118]. A combination of double flat H-coil and double B-sensing needle pairs in a hand-held device has also been developed as a practical means for the analysis of the local distribution of fields and losses in laminated soft magnetic cores [7.119].

7.3.2.3 Measurements under two-dimensional fields. Rotational losses. There are regions in the magnetic cores of rotating machines and three-phase transformers where the field rotates and the flux consequently describes a two-dimensional pattern. There is no simple way to connect the conventional scalar characterization of the soft magnetic laminations with the properties displayed under a two-dimensional (2D) field. The standard AC measurements should then be supplemented by additional experiments performed under controlled circular/elliptical flux loci. With these idealized patterns, one can, for example, make reasonable predictions of the energy losses associated with real 2D patterns [7.120] and employ them in the numerical calculation of the losses in the cores. The 2D characterization of magnetic sheet samples can be accomplished using one of the two- or three-phase magnetizers discussed in Section 7.1.3 and shown in Figs. 7.11-7.14. As of now, there are no decisive arguments in favor of a specific type of magnetizer. It has been shown that the simpler two-phase magnetizer, with either horizontal or vertical double yoke, provides more homogeneous magnetization in square samples than circular ones and it can be conveniently employed for measurements in non-oriented materials [7.55, 7.121]. Three-phase magnetizers instead appear more suited to the characterization of highly anisotropic alloys (e.g. high permeability GO Fe-Si laminations), provided a circular or hexagonal specimen is used because one can better cope with the presence of a hard direction in trying to achieve optimum control of the 2D flux loci [7.52]. Three-phase systems have the additional advantage of requiring less exciting power to the single supply channels for the same value of the applied field. Closed samples are seldom tested under rotational fields. Examples are provided by the classical cross-shaped specimen [7.122], where induction is homogeneous in a very small central

7.3 AC MEASUREMENTS

397

region of the sample only, and Epstein strips with diagonal secondary coils (similar to Fig. 7.34), where flux loci control is very demanding because of the strong asymmetry of the sample [7.123]. Whatever the magnetizer, some form of 2D adaptive control of the modulus IJI and the angular velocity ~ of the magnetic polarization vector must be accomplished. With circular loci, both these quantities are to be kept constant. This is automatically obtained under a field vector H a of constant modulus which rotates at constant speed if the sample is circular and perfectly isotropic or it is cut as such a small disk that the demagnetizing field alone is sufficient to impose the required feedback conditions [7.124]. Under typical measuring conditions, with real anisotropic materials, active feedback is generally required. In this case, the simplest approach to the generation of rotational or elliptical flux loci with two-phase systems consists in duplicating the single-phase digital feedback circuit and procedure illustrated in Section 7.2.2 for the conventional measurements under alternating fields. The usual iteration process is carried out independently for both polarization components Jy and J~ (IJI = ~/J~ 4- ].2) in order to achieve them as sinusoidal functions of time. Typically, the y and x reference axes in the lamination plane are taken to coincide with RD and TD, respectively. A circular polarization locus of radius Jp is obtained by imposing 90 ~ phase shifted longitudinal and transverse sinusoidal polarizations Jy(t)=Jp cos(2zrft) and Jx(t)-Jp sin(2-rrft). It will transform into an elliptical locus for peak polarization values Jpy va Jpx, with either a vertical or horizontal major axis according to whether lpy > Jpx or Jpy (Jpx. Sinusoidal voltages uy(t) and Ux(t) of proportionally related peak amplitudes (under the usual condition B ~- J) will be detected across the two orthogonally placed search coils (or needle probes). This procedure, which would be justified only in the limit where the macroscopic two-dimensional magnetization process can be approximated as a linear combination of independent microscopic unidirectional processes, is experimentally found to be appropriate only for low anisotropy materials. Under normal conditions, coupling between Jy(t) and Jx(t) should be taken into account. One way to do so is by introducing in the iterative algorithms used for the generation of the suitable supply currents ix(t) and iy(t), the experimental relationships Jy(t)--f(iy(t), ix(t)) and Jx(t)=f(iy(t), ix(t)), which can be approximately obtained by an initial testing essay with sinusoidal 90 ~ phase shifted voltages impressed at the input of the magnetizing circuit [7.125]. A comprehensive approach to 2D feedback in anisotropic samples can actually be carried out in the following way. Let us consider, as in Fig. 7.38a,

398

CHAPTER 7 Characterization of Soft Magnetic Materials

Y[t . . . . . . . -.

'Gy

/

Ad'"-

I'/1

J I

(a)

--....

Ha• r

(b)

FIGURE 7.38 (a) The magnetic polarization v e c t o r Jp can be made to describe a circular locus at constant angular velocity dOi/dt in a non-isotropic lamination by applying a suitable rotating field H0(t) whose modulus and angular velocity can be determined as a function of time by means of 2D digital feedback. (b) Twophase and three-phase exciting fields provide the same rotational fields when they are related by Eq. (7.44). the polarization vector Jp, which is to be set in rotation at constant angular velocity dOI/dt, describing a circular locus of radius Jp. If at the start a uniformly rotating field of modulus Ha is applied by generating the sinusoidal field components Hay(t) = Ha cos 2vrft and Hax(t) = Ha sin 2"rrft, a vector Ja(t) will be obtained. This will not describe, as eventually required, a circular locus at constant speed. The deviation from the ideal behavior will depend on the degree of magnetic anisotropy and the value of the demagnetizing coefficient of the employed specimen. At a given instant of time, the relationship between field and polarization can be represented as shown in Fig. 7.38a, where the actual polarization Ja differs from the desired polarization Jp by the error vector &J. In order to remove this error, the field H0 = Ha q- AH should be applied. We can arrive at it by iteration, where the correction term &H is progressively updated through increasingly better information on the Jp(Ha) relationship. After a first essay with the circular field Ha, the values taken by Jy(0j), Jx(0j), Hay(0j), and Hax(~) in correspondence with a reasonably high number N of equispaced angles 0j, distributed over the whole period, are calculated and stored. The angle ~ is normally taken with respect to a reference direction, related to the geometrical or crystallographic properties of the parent sheet. In the present context, we take RD as such a direction, which is made to coincide with the y-axis. Iteration is started, applying first, for example,

7.3 AC MEASUREMENTS

399

the previously mentioned conventional feedback procedure, where the y and x channels are independently treated. This operation leads not only to a new set of N values for each of the previous quantities that are expected to be closer to the final ones, but it also provides, for each angle 01, the desired information on the differential 2D response of the material. That is, we find a sort of permeability/d, d y ( 0 j ) - - AJy/AHay and / d ' d x ( ~ ) = AJx(Oj)/AHax(O J) in the neighborhood of the final state (H0(0j), Jp(0j)), by which the N values Ha,j(0j) and Hax(0j) can be updated. This means, in practice, updating the corresponding current values ix(t) and iy(t), the correspondence between times and angles being imposed by the constraint d Oj/dt = const. The experiments show that convergence of this method is fast and little iteration is required to achieve the desired circular flux locus. In addition, no special difficulties arise on passing from circular to elliptical loci. The foregoing discussion has considered a two-phase system, but it has been noted that three-phase setups might sometimes be preferred for 2D testing of highly anisotropic laminations. This only implies a transformation of the computed field components Hay(t) and Hax(t) in the equivalent triplet

Hi(t) = Hay(t)~1.5;

H2(t) = I-I~x(t)/x/3- Hay(t)~3;

(7.44)

H3(t) = -Hax(t)/~/3- Hay(t)/3 as schematically shown in Fig. 7.38b, and in the corresponding threephase currents i 1(t),/2(t), and/3(t). It should be stressed again that we do not directly drive these currents, but we supply the voltages UG:(t), UG2(t), and UG3(t) (Fig. 7.39). We can relate currents and voltages by writing, for each supply channel, the primary circuit equation. We take into account in this equation that the inductance of the magnetizing yoke Ly is normally very large and very little influenced by the presence of the test specimen. It is also constant because the yoke is always kept far from saturation and is affected by a large demagnetizing coefficient, and we can reasonably assume that supply current and field in the gap are proportional. We therefore obtain, according to the scheme of Fig. 7.39, that the voltage UGk(t) to be programmed into the arbitrary function generator of the kth channel (k = 1, 2, 3) is related to the computed current ik(t) as

dik(t) UGk(t) = R~kik(t) + L,/ dt "

(7.45)

Examples of circular polarization loci obtained with this feedback procedure in non-oriented and grain-oriented Fe-Si alloys are provided

400

CHAPTER 7 Characterization of Soft Magnetic Materials

B-coils

I R~3 i3/~u~s~

Y#

Arb. function generator --)~x UG1

R~2

iI

TC Nanovoltm.

I v

~DT(t) PC

ocdHy/dt I

Digital oscilloscope

b~ v

ocddy~t FIGURE 7.39 Setup for testing soft magnetic laminations under controlled 2D flux loci. The yoke is energized by a three-phase supply system and generates a rotating field in the gap. The test specimen is disk-shaped and is endowed with pairs of orthogonal B-coils (or needles) and H-coils. The desired 2D pattern (circular/elliptical) of the polarization vector J is obtained by means of an iterative digital feedback procedure, by which the three-phase function generator is programmed to provide, through the power amplifiers (PA), the suitable voltages ucl(t), UG2(t), and UG3(t) to the primary circuit (see text). Once the circular/elliptical polarization is achieved with prescribed degree of accuracy, the signals detected by search coils are amplified, fed into the acquisition device, A/D converted and elaborated by software. The 2D energy loss is obtained by summing up the areas of the determined (Hy,Jy) and (H~,Jx) hysteresis loops. At high induction values, the power loss is preferably obtained by measuring the rate of rise of the sample temperature, detected, for example, by means of microthermocouples stuck on the lamination surface (Eq. (7.38)). in Fig. 7.40. Certain difficulties exist in achieving perfectly circular loci in the strongly anisotropic GO lamination; the higher Jp the more difficult the process. This is not surprising; because the applied field amplitude must change by more than an order of magnitude on passing from easy to hard

7.3 AC MEASUREMENTS

401

GO

ff

iI.i

....

, ....

-10

Fe-Si

, ....

';Jx (T)

-o5

-0.5

NO

Fe-Si

400

o~ 300 "o i.>.,

Jp= 1.5T

_go 200 O

Jp= 1.7T

(1)

>

1._

O3 c"

<

100 f = 50 Hz 0

.

.

.

.

!

2

. . . .

I

4

.

.

.

.

i

6

.

.

.

.

!

8

.

.

.

.

10

Time (ms) FIGURE 7.40 (a) Polarization loci obtained in non-oriented and grain-oriented Fe-Si laminations after 2D feedback digital procedure, carried out at 50 Hz in a three-phase setup like the one shown in Fig. 7.39. The sample is disk shaped (diameter 140 mm) and the signal is detected over a 40 mm diameter region by means of orthogonal few-turn coils threaded through 0.7 mm holes. The dashed circles show the desired polarization loci. (b) Corresponding behavior of the angular velocity, fluctuating around the prescribed average value, along a semiperiod.

402

CHAPTER 7 Characterization of Soft Magnetic Materials

axes if Jp is to be kept constant [7.125]. Of course, the relatively poor control of the circularity of the polarization locus is associated with substantial fluctuation of the angular velocity d0j/dt around the prescribed average value, as illustrated in Fig. 7.40b. Here we see, by contrast, the tight control one can achieve in the non-oriented alloys. Notably, however, the error introduced by the fluctuation of dOj/dt on the energy loss is relatively small and it can be corrected to a good extent. To this end, we separate the loss contributions at the test frequency and we evaluate for each of them the correction to be made. It has been actually verified by experiments that the energy losses under rotational fields follow the same power law dependence on frequency observed with alternating fields; i.e. the hysteresis loss power loss PRh is proportional to f, the classical power loss PRcl goes like f2, and the excess power loss PRexc follows f3/2 [7.126]. The physical analysis of the magnetization process shows that the concept of loss separation can be extended down to instantaneous quantities [7.83]. The instantaneous equations for the loss components will follow a power law dependence on the angular velocity and, if we write, in particular PR(t) -

PRh(t) 4- PRcl(t) 4- PRexc(t)

d0j d0j) 2 ----kh--~ 4-kcl ( - ~ 4-kexc(d~) 3/2,--~

(7.46)

with kh, kcl, and kexc proportionality constants, we conclude that the average values PRh, PRcl, and PRexc will be proportionally related to (dOj/dt), ((d0j/dt)2), and ((dOj/dt)3/2), respectively. This means, in particular, that PRh depends on the average value of the angular velocity and is immune from the fluctuations of dOi/dt, which affect the other two terms in a predictable manner. Applied to the results reported in Fig. 7.40, these relationships provide for the grain-oriented lamination at Jp --- 1.7 T, an increase of PRcI around 6% caused by the fluctuating profile of the angular velocity. The increase of PRe• is similarly estimated to be of the order of 4%. The same calculations provide an estimated increase of PRd and PRexc around 0.5% only, if applied to the non-oriented lamination and its weakly fluctuating dOj/dt profile in Fig. 7.40. The directly measured value P of the total loss can then be suitably corrected, possibly also taking into account the related fluctuations of the amplitude Jp. The latter, however, are expected to provide a negligible contribution with respect to the fluctuations of dOj/dt with typical flux loci control. The determination of the power loss PR (or the energy loss in a period WR) is actually the basic objective of most measurements done and attempted under rotational fields. Several methods have been devised in

7.3 AC MEASUREMENTS

403

the literature, which can be classified along three basic categories: mechanical, thermal, and fieldmetric methods [7.56]. We have already described in summary in Section 7.2.2 how the energy loss WRh in a sheet sample subjected to quasi-static rotational field can be determined by measuring the average parasitic torque over a 360 ~ rotation of the applied field (Eq. (7.17)). Such a torque is also measurable under dynamic conditions. As shown by Brix [7.127], it is sufficient to increase the inertia of the torque-sensing device to the point where the fluctuating torque associated with the anisotropy effects is completely smoothed out. This is most effectively obtained at frequencies much higher than the eigenfrequencies of the system (f > 25 Hz in Brix's device). The measurement of the average parasitic torque provides the energy loss under pure rotational flux. Under general 2D magnetization process and AC exciting conditions, the magnetic field at the sheet surface H = Hy~ + Hx~ and the polarization J - Jy~ + Jx~ averaged over the sheet cross-section, are conveniently determined. A pair of flat H-coils, orthogonally placed on the sample surface across the region of homogeneous magnetization (Figs. 7.11-7.14 and 7.39) can be employed to detect the components Hy and Hx. Alternatively, a pair of RCPs covering the same region can be used [7.128]. The use of Hall sensors can also be envisaged. In this case, however, the measurement is somewhat localized and the active character of the device is a disadvantage. The polarization components Jy and Jx can be obtained either with B-coils or needle probes, as illustrated in Fig. 7.37. The measuring region, however, must be sufficiently large to encompass the structural inhomogeneities of the tested material. With grain-oriented laminations this is not easily achieved and averaging of the results obtained on a number of samples might be required. The directly detected signals are obviously proportional to the time derivative of the above quantities, which are then obtained by numerical integration (Fig. 7.39). In this way, we have all we need for the measurement of the 2D power losses. Under very general terms, it can be stated that the electromagnetic energy flowing in unit time into a given region of the lamination, bounded by the surface S~ is given by the integral - ~ s ( E X I-I).n dS~ where the product (E x I-I)~ with E and H the electric and magnetic field at the lamination surface, is known as the Poynting vector. By integrating the instantaneous value of the surface integral of the Poynting vector over a full period we obtain the energy loss W. Under normal measuring conditions, the edge effects are irrelevant and the energy streams only through the top and bottom surfaces of the lamination and we can conclude that the energy loss throughout the whole sample volume can be determined by means of a surface measurement of the electric and magnetic fields. These are exactly the quantities we obtain by means of

404

CHAPTER 7 Characterization of Soft Magnetic Materials

our needle probes (Fig. 7.37), which provide the voltages Vy = Eyl and Vx = Exl, and by the H-coils, from which the field components Hy and Hx are obtained after integration. With the symmetry of our problem, where the field is applied in the lamination plane and E and H do not have components along the z direction and are constant upon the measuring region of area l2, we obtain that the instantaneous power dissipated in the volume 12d, where d is the lamination thickness, is

P(t) = -21(E x H)I.I2 -- 212(EyHx - ExHy) = 2l(VyHx - VxHy),

(7.47)

from which the average power loss per unit mass is obtained as

PR-- 3dl21 TII~ P(t)dt" Thus, by needle contacts, we satisfy both the Poynting vector formulation of the loss and, under the condition l>> d, the need for precise determination of the induction value. However, the low signal level and the difficulty of avoiding linkage with stray flux often make it preferable to resort to the alternative method of detecting the induction derivative by windings threaded through tiny holes drilled at the distance l. The ofteninvoked detrimental effects related to local hardening by drilling can be safely avoided if this operation is carried out with care. In this way, the signals dBy/dt and dBx/dt become available. They are related to Ex and Ey, respectively, by the Faraday-Maxwell law and, by substitution in Eq. (7.47) with the appropriate sign convention, we obtain

PR-P+PRy =

1

dBy, 1 ~ x dt +Hy---~-~-)dt= --3Tf~(Hx~t+Hy~t)dt" (7.48)

The energy loss under 2D excitation can therefore be measured by summing the areas of the hysteresis loops taken along two orthogonal directions. In a perfectly isotropic material, subjected to a field Hp of constant modulus rotating with constant speed, the vector Jp has equally constant modulus and lags behind Hp by a fixed angle ~H. The resulting (Hx,Jx) and (Hy,Jy) loops are ellipses and the resulting specific rotational power loss is

PR-- 2-~~3HpJpsin ~H.

(7.49)

7.3 AC MEASUREMENTS

405

This condition is approximated at low inductions in ordinary nonoriented alloys (inset in Fig. 7.41). In particular, in the region of validity of the Rayleigh law, it is possible to predict theoretically the rotational hysteresis loss from measurements performed under alternating fields and suitable hypotheses on the statistical distribution of the grain orientations [7.129]. On increasing Jp, the actual anisotropic behavior of the material comes largely into play and both the amplitude Hp of the rotating field and the phase shift qOjHundergo fluctuations while trying to proceed, with the help of feedback, according to the desired time dependence (e.g. independent of time for circular polarization locus) of Jp and dOj/dt. The measured hysteresis loops then take the characteristic re-entrant shape shown in Fig. 7.41. Remarkably, there are instances where anisotropy is

1.5~ NO Fe'(3 wt%)Si f= 10Hz

l

1.0 0.5 '~ "~

0.0

0.5

-0.5 -1.0 -1.5-1000

-500

0

Hy (A/m)

500

1000

FIGURE 7.41 Hysteresis loops in non-oriented laminations measured under controlled circular polarization qp = const., dOj/dt = const.) at different Jp values by taking the components Hy and Jy of rotating field and polarization aIong the rolling direction over a period. The sample is a disk placed within a three-phase yoke system (Fig. 7.39). Hy and Jy are detected by means of a fiat H-coil and a few-turn B-winding, respectively, over a region of uniform magnetization in the sample. The pseudo-elliptic hysteresis loops obtained at low induction values are shown in the inset. The actual anisotropic behavior of the material leads to oscillations of the amplitude Hp of the field and of the phase shift {pbetween Hp and Jp, which are apparent from the shape of the loops.

406

CHAPTER 7 Characterization of Soft Magnetic Materials

sufficiently strong to bring Jp ahead of Hp during the rotation, which leads to typical buttonholes in the loops, signaling net energy release by the sample to the external world. This can add to the uncertainty of the determination of the area of the loops. Figure 7.42 illustrates the measurement of energy loss with circular polarization Jp=l.5T in 0.35 m m thick Fe-(3 wt%)Si non-oriented laminations via determination of the (Hx,Jx) and (Hy,Jy) hysteresis loops. These are actually the result of loop averaging upon clockwise and counterclockwise rotation of the applied field, by which spurious phase shifts, responsible for substantial errors in the measured loop area, are for the most part eliminated. Possible asymmetries of the measuring apparatus, especially misaligrunents of H-coils and B-coils, can detrimentally affect the loss-measuring accuracy if they are not suitably compensated. We see in the typical case reported in Fig. 7.42b (non-oriented alloy at Jp = 1.5T) that Jp lags behind Hp by an average angle around ~H "" 3~ This means that an all too comrnon 0.5-1 ~ misalignment of the windings can result in an intolerable 10-30% error in the measured loss. To recognize qualitatively the effect of compensation by c.w. and c.c.w, field rotation, we can try to evaluate, for example, the effect of a spurious misalignment between the Hx and Jx windings [7.130, 7.131]. ! If the error angle is A~x and the measured loss figures are PRcwxqp~Aqox)and P~Rccwx(Jp,A~px)for c.w. and c.c.w, rotation, the actual power loss PI~ = ~

Hx

dt

is related to the result of averaging by the equation i +P~ccw~PI~ = PRcwx ".

2cos(A~x)

(7.50)

Compensation is therefore effective under typical measuring arrangements, where A~px can be kept within 1~ and it can be safely assumed cos(A~x)= 1. It is generally acknowledged that it is difficult to perform acceptably accurate rotational loss measurements with the fieldmetric method discussed here beyond about 1.5 T in non-oriented Fe-Si laminations [7.41]. The chief limitation arises, as for the alternating case, from the corresponding rapid decrease of the average value of q0jH~which becomes dwarfed by fluctuations. In grain-oriented alloys already at low inductions, the problem is exacerbated, by the wild fluctuations undergone by both Hp and q0jH~ as demanded by the control of the polarization loci. Under such difficult conditions, a reasonable alternative solution is offered by the rate of rise of temperature method, as previously discussed

7.3 AC MEASUREMENTS

407

NO Fe-(3 wt%)Si f= 10Hz

1.5 1.o 0.5 0.0 --0.5

-1.0 Jp= 1.5T

-1.5 '

-

'

I

. . . .

I

. . . .

I

. . . .

. . . .

1500 -1000 -500

I

. . . .

500

0

I

. . . .

I

'

'

1000 1500

Hy, Hx (A/m)

(a) 12

NO Fe-(3 wt%)Si

/~

Jp=1.5T 10Hz

f=

o

8 o v

o

9CO

-(b

(/)

g 4

. . . . . . . . . . . . . . .

"~o~176

t13.

o

0

'

(b)

0.00

'

'

'

I

'

0.01

'

'

'

I

'

'

'

'

I

.

0.02 0.03 Time (s)

.

.

.

I

'

0.04

'

'

9

0.05

FIGURE 7.42 (a) Example of energy loss measurement under circular polarization locus in non-oriented Fe-Si, laminations. The areas of the hysteresis loops associated with the components of effective field and polarization along the two orthogonal axes x and y, made to coincide with RD and TD, respectively, are summed up to provide, according to Eq. (7.48), the energy dissipated in a magnetization period V W R q p , f ) = W R y q p , f ) + WRx(Jp,f) = P ( J p , f ) / f . (b) Because of anisotropy, the vector Jp is not always lagging behind H.. during rotation and @H Can be negative. The small value of the average phase sh~ft (horizontal dashed line) can make the accurate measurement of the loss difficult and a calorimetric approach can therefore be preferred if testing at higher Jp values is required.

408

CHAPTER 7 Characterization of Soft Magnetic Materials

to some extent and as schematically shown in Fig. 7.39. The signal provided by the thermal junctions or the thermistors in a given time interval, as defined by the requirement of operating close to the adiabatic regime, is directly proportional to the dissipated power. It is then proportional to the product J'~f"', where n can range between 1.5 and 2 and m is roughly around 1.5. Because of this power law dependence, it is difficult to apply the thermal method at low frequencies and low inductions. A reasonable approach to the whole problem would then call for application of the simpler and more informative fieldmetric method as far as the uncertainty of ~H is acceptably low and to make use of the thermal method in the high induction range only, provided a suitable overlap region is identified. Agreement of the measurements performed upon such a region would provide a stringent check of the accuracy of both methods. Of course, the expectedly imperfect control of the 2D magnetization pattern on approaching the magnetic saturation undoubtedly affects the accuracy of the thermal method, but, as already stressed, some kind of correction of the raw result (Eq. (7.46)) can be attempted. It should finally be remarked that, if the objective of the measurement is exclusively the determination of the rotational losses, sufficiently large DC fields are available, and accuracy is not a very stringent requisite, we could refrain from setting up complicated controls of the polarization loci and difficult signal handling by preparing a suitably small disk-shaped or cylinder-shaped sample, placing it within the polar faces of an electromagnet, setting it into spinning motion, and finally looking at the way damping is affected by the presence of the steady field. Basically, this means that as in the previously mentioned dynamic torque method [7.127], a measure is made of the average parasitic torque, now by rotating the sample instead of the magnetic field. With small diameter disks (e.g. 10-20 mm) the demagnetizing effect is sufficiently high to impose a nearcircular polarization locus even in highly anisotropic materials [7.132]. After setting the applied field to the desired value and bringing the sample to a spinning rate of 50-200 Hz, the drive is removed and the sample is allowed to spin freely under the restraining action of the mechanical and the eddy-current induced frictional torques. At the same time, the rate of change d~/dt of the decaying angular velocity is measured [7.133]. Such a measurement, usually performed by optical means, amounts to a determination of the parasitic torque Lw-- I dco/dt, where I is the moment of inertia of the sample, as a function of the spinning frequency f, ideally down to f = 0. The average torque at any given frequency provides then the rotational power loss PR -- 2Vr~w once the spurious mechanical parasitic torques are eliminated by making two identical measurements with and without the applied field.

7.3 AC MEASUREMENTS

409

The spinning sample technique appears attractive in those cases where the size of the sample is so small or its nature is so specific that the fieldmetric technique cannot be applied and it is desirable to measure the losses up to saturation. Contrary to the case where a rotating field is produced, saturation is not difficult to achieve by means of a DC source. Examples of rotational loss separations obtained by the spinning sample method have been reported by Cecchetti et al. in grain-oriented Fe-Si, non-oriented Fe-Si and amorphous ribbons [7.134, 7.135]. It is expected that this method can be applied to soft magnetic composites, where the necessarily bulk samples would offer too large a demagnetizing coefficient to any applied rotating field or, in general, to powder aggregates. Experiments on hard and semi-hard powders, where the theory of rotational hysteresis plays a major role in the physical modeling of the magnetization process, have been reported [7.136].

7.3.3 M e d i u m - t o - h i g h f r e q u e n c y m e a s u r e m e n t s There is an increasing trend towards the use of electrical machines and various types of devices over a wide range of frequencies and with a variety of supply methods, which call for the precise characterization of soft magnetic materials beyond the assessed DC and power frequency domain. There are indeed many kinds of excitation methods, which impose, within broad frequency limits, not only sinusoidal flux conditions, but also different types of rated voltage waveforms, with and without bias field, or pulsed magnetization (either with determined current or voltage pulses) [7.137]. While there are no special additional difficulties in setting up a system by which one can drive a particular type of excitation, many problems arise with the increase of the frequency. They can be faced by means of a rigorous approach to the measuring principles and their implementation in the testing operations. However, to adapt the conventional measuring setups to the characterization of soft magnets up to the MHz range might become a relatively complex task. We shall discuss here the main potential difficulties and problems arising with the increase of the magnetizing frequency, in the limit where electromagnetic propagation phenomena are still irrelevant, that is, up to frequencies where wavelengths are much larger than the size of the region occupied by the specimen. (1) The flux penetration in the test sample can be incomplete (skin effect) This effect can be evaluated by calculating the skin depth = ~/2//~0/~rCr~o~ that is, the depth in the sample where the induction value falls by a factor 1/e with respect to the induction value at the surface. Under these conditions, thickness-dependent instead of intrinsic

410

CHAPTER 7 Characterization of Soft Magnetic Materials

properties are measured. Figure 7.43 provides illustrative examples of induction profiles vs. thickness in non-oriented Fe(3 wt%) laminations (at 400 and 1000 Hz) and in amorphous and nanocrystalline tapes (at 1 MHz). The latter samples, endowed with very close resistivity values and the same thickness, are tested as strip-wound toroids, after having been optimized for high-frequency applications by means of annealing treatment under a transverse saturating field. They exhibit remarkably different profiles, which are due to their different permeability values (4.5 x 103 in the amorphous ribbon and 32 x 103 in the nanocrystalline sample). The skin effect can equally occur in the conductors and increase their AC resistance, often prompting the use of windings made of copper strips or multiconductor Lietz wire. It has troublesome consequences when it affects the calibrated resistor RH employed in the primary circuit as current probe. The frequency response of RH should therefore be verified before starting the measurements. Wirewound resistors are not recommended in general, while carbon resistors can display a flat response up to 10-20 MHz. Proximity effects, that is the interaction existing between closely placed conductors carrying an AC current, can also add to the skin effect in giving rise to an increase of the resistance of windings and leads. (2) The temperature of the sample can appreciably rise during the measurement. A 0.050 mm thick grain-oriented sheet, suitably developed for high-frequency applications, can display, for example, a power loss of about 500W/kg at 1.0T and 10 kHz. The sample temperature is correspondingly expected to rise at a rate around 1 ~ In a M n - Z n ferrite tested at 1 MHz and peak polarization Jp = 0.1 T, sample heating can proceed at a rate higher than 2 ~ Excessive heating is obviously detrimental to the measuring accuracy because the physical properties of the material can rapidly change with the temperature. This should be measured by placing a micro-thermocouple in contact with the specimen, which, in turn, should be suitably cooled, for example, by keeping it into an oil bath. If the specimen is encapsulated, the junction will be put in contact with the core material by making a small hole in the container. The method of single-shot acquisition and digital control of the measurement is the most appropriate when looking for minimum temperature increase because the time interval where the sample is excited is at a minimum. (3) The increase of the required exciting power P(t) = uc(t)iH(t ) with the frequency poses serious limitations on the achievable peak polarization value. Under most circumstances of high-frequency testing, the characterization of the material for Jp not far from the knee of the magnetization curve can only be done under pulse excitation [7.138]. To make an example, a M n - Z n ring of average diameter 50 mm and

7.3 AC MEASUREMENTS

1.0

"-

411

- I

.

.

.

.

-

. . . .

v

f= 1000 Hz

0.5

NO Fe-(3 wt%)Si d = 0.34 mm Jp=lT 0.0

'

'

I

'

'

'

'

I

'

'

'

'

I

'

'

'

'

I

'

-0.15 -0.10 -0.05 0.00 x(mm)

'

'

'

I

'

'

0.05

5

'

'

I

'

'

0.10

'

'

I

'

0.15

'

Jp = 0.005 T f= 1 MHz /

4

~o.

3

~'Q

2

1

Nanocrystalline =20 gm

/ -

/

Amorphous

-

,

I

-10

'

'

'

'

I

-5

'

'

'

'

I

0

~

_

'

'

'

'

I

5

'

'

'

'

I

10

x (l~m)

FIGURE 7.43 Examples of eddy current-induced profiles of the reduced peak polarization J (x)/Jp vs. distance x from mid-plane of 0.34 mm thick non-oriented Fe-(3 wt%)SiP laminations and 20 ~m thick amorphous and nanocrystalline ribbons. The profiles are calculated by an FEM technique taking into account the experimental DC B(H) hysteresis loop curve of the material. The amorphous and nanocrystalline ribbons have the same thickness and very close resistivity values, but the latter are endowed with a one order of magnitude larger permeability (courtesy of O. Bottauscio).

412

CHAPTER 7 Characterization of Soft Magnetic Materials

cross-sectional area A = 50 mm 2 requires about 100 V A peak power to be excited at Jp -- 0.1 T at the frequency of I MHz. The general trend is one of using small cores with few primary turns to limit the primary voltage. The number of turns of the secondary winding is equally small, but full coupling is to be ensured with the magnetizing winding, which must then be uniformly wound along the core [7.139]. As a rule, testing is made on ring-shaped specimens. If single strips are to be characterized, a fluxclosing yoke should be devised whose soft magnetic behavior associates with minimum skin effect. This condition may be satisfied by a double-C core made by assembling mumetal or amorphous tapes of as low a thickness as practical. It has been shown that with such an arrangement, where the field strength is determined either by means of an H-coil or by measuring the magnetizing current and adopting a magnetic path length equal to the internal yoke diameter, measurements on amorphous strips can be reliably carried out up to 100 kHz [7.140]. Note that it would be very difficult to overcome this frequency limit with H-coils, as their behavior is inevitably affected by self-capacitances. Measurements on open samples are obviously possible but they are not recommended in general. Besides the obvious correction for the demagnetizing field, they require certain precautions regarding the generation of the rapidly varying stray field in the surrounding milieu because of unwanted electromotive forces generated in metallic parts and conductors forming closed loops, which can eventually interact, through the generation of spurious fields, with the magnetization process. In the following, we will refer to the most common condition of closed samples (e.g. Epstein or ring specimen) with the field determined via the measurement of the magnetizing current. (4) Fast A / D converters are required to satisfy our requirement of single-shot signal acquisition and real-time analysis. This implies a certain limitation in signal-amplitude resolution and if we are to make a complete characterization starting from DC properties, an acquisition device (for example, a digital oscilloscope or a VXI system) with high resolution and relatively low sampling rate can be employed in the lower frequency range, to be substituted by a faster one with lower resolution at high frequencies. This makes sense because the contribution to the measuring uncertainty coming from the uncertainty in the phase shift ~p between the fundamental component of the magnetizing current and the secondary voltage (i.e. on the phase shift qOjH--('/r/2)- ~ between the fundamental component of the field and the polarization), whose minimization calls for a high number of sampling points and high amplitude resolution, is dominant only at low frequencies. The example shown in Fig. 7.44 of the evolution with frequency of the hysteresis loops

7.3 AC MEASUREMENTS ,

~

I

413 ~

~

i

I

,

,

,

I

,

,

,

nanocrystalline ribbon

Jp

0.004

I

~

~

,

I

,

,

~

I

~

~

,

I

,

,

14

0.002 o.ooo

-0.002 Hz

'

-0.004

-0.6-0.4-0.2

0.0 0.2 H (A/m)

0.4

0.6

70 nanocrystalline ribbon 60 50 "~ o v

40

"1-

30 20 10

I

1

10

100

Frequency (kHz)

1000

FIGURE 7.44 Evolution from quasi-static conditions to I MHz of hysteresis loops taken at Jp = 5 mT in nanocrystalline ribbons of composition Fe73.sCutNb3B9Si13.5 and corresponding behavior of the phase shift ~ I between the fundamental component of the field waveform and the polarization. The loops shown in the figure have been taken at the following frequencies: 100 Hz, 10, 50, 100, 300, 500 kHz, and 1 MHz.

414

CHAPTER 7 Characterization of Soft Magnetic Materials

and the phase shift ~H in an extra-low loss nanocrystalline ribbon demonstrates that the resolution features required for the A / D converters in order to appreciate the loop area are the more demanding the lower the frequency. These results, in particular, have been obtained by making signal acquisition by means of a 150 M sample/s digital oscilloscope having amplitude resolution decreasing, together with the sampled time window, from 14 to 8 bits. The record length is correspondingly decreased from 5000 samples to 150 samples minimum at 1 MHz. (5) With the increase of the magnetizing frequency above the kHz range, it becomes important to consider the role of stray inductances and capacitances. This is a most basic issue, the one making the real difference between low-frequency and high-frequency measurements, at least up to the radiofrequency domain, where the wavelength of the electromagnetic field becomes comparable with the dimensions of the test specimen. We need to reduce the effect of the stray parameters to the largest possible extent if we wish to determine the J(H) behavior as we do in the lowfrequency regime, without specific constraints regarding the non-linear behavior of the material. Under many practical circumstances, however, we might be specifically interested in the weak field response of the test sample, or simply in its behavior as an inductor, and correction for the effect of the stray parameters of windings and connecting cables can be attempted. While the effect of leakage inductances, that is, spurious flux linkages with windings and winding leads, can be generally kept negligible with respect to the flux generated by the magnetization in the material by closely fitting the windings on the magnetic core and by using short leads, the capacitance related effects require special consideration. Any winding is endowed with self-capacitance, which increasingly tends to drain current with increase in the frequency. This effect is reinforced by interwinding capacitance and capacitance in the connecting cables and at the input of the acquisition device. For example, a coaxial cable has a self-capacitance of the order of 50-100 pF/m, while the input impedance of a digital voltmeter or oscilloscope is typically around 1 Mf~ with a capacitance around 10-50 pF in parallel. The self-capacitances C1 and C2 of primary and secondary windings of the standard 700-turn Epstein frame can be of the order of 50-200 pF, much lower than the interwinding capacitance Co, which can be as high as 1000-3000 pF. Interposing an electrostatic screen between the windings and connecting it to the ground can eliminate the effect of Co, resulting, however, in a correspondingly enhanced value of C1 and C2. For magnetic sheet- and strip-testing beyond 400 Hz, this frame should be substituted, according to the Standard IEC 60404-10, by a 200-turn frame, which is deemed appropriate up to 10 kHz [7.141]. Because of the reduced number of turns, the effect of

7.3 AC MEASUREMENTS

415

interwinding capacitance can be minimized by laying out primary and secondary windings, which have the same number of turns, as a bifilar single layer with suitably spaced conductors. In this way, neighboring conductors are at the same potential and current leakage at high frequencies in the interposed dielectric is largely prevented. It is estimated that Co "~ 300 pF. To reduce both self and mutual capacitances of the windings and the related dielectric losses, not only must some space be allowed between successive turns, but also the dielectric material lying beneath the conductors must have low permittivity. Polysterene could be such a material. With ferrite cores, it is the test material itself that happens to be endowed with a high value of the dielectric constant, thereby favoring capacitive coupling between the neighboring winding turns. To limit this effect, a low permittivity dielectric tape should be wound beforehand on the core. Figure 7.45 provides an idea of the effect on the hysteresis loop, observed at 1 MHz in an M n - Z n ferrite ring, of a capacitance of 50 pF, equivalent to about a I m long connecting cable, inserted in parallel with the magnetizing winding (N1 = 5, N2 = 5, average ring diameter 30 mm). One can notice the tilting of the loop towards the second quadrant due to the fact that the measured primary current is the sum of the current leaking through the stray capacitance and the active magnetizing current and is consequently associated with an abnormal phase relationship with the magnetic induction. Figure 7.46a provides a qualitative description of the arrangement of connections and windings in a setup for the characterization of soft magnetic cores at medium and high frequencies. Bifilar single-layer windings are used and the connections are made by means of shielded cables. These cables should be short, as a rule, that is, of the order of a few centimeters at most in the MHz range to minimize the associated capacitances. In addition, the resistor R H should be connected to the magnetizing winding by a very short lead. Since the sample is flux-closed, there are no stray fields generated by it and the value of a H is not perturbed. Because N1 and N2 can be very low at high frequencies, the effect of coupling between the fictitious primary and secondary single turns located along the median circumference of the ring specimen could be appreciated. This effect should therefore be checked and possibly compensated. The setup in Fig. 7.46a is given a complete description in terms of lumped and stray parameters by the equivalent circuit shown in Fig. 7.47. Here, in particular, we have considered the self-capacitances (C1, C2) , the leakage inductances (Lwl, Lw2)~ and the resistances (Rwl ~Rw2) (primary and secondary windings), the interwinding capacitance Co, and the capacitances CH and Cj, which include the contribution of the cables and the input channels of the acquisition device. The value of RH~

416

CHAPTER 7 Characterization of Soft Magnetic Materials

0.006 " Mn-Zn ferrite ring sample 0.004 9 '

'

i

.

0.002

.

.

i

.

.

.

.

I

.

.

.

.

i

.

.

.

.

I

.

.

.

.

I

.

.

f

.

.

i

.

,

,

I-

.

o.ooo

-0.002

,.,/ C1- 50 pF

-0.004

t ,P

Jp=5 -0.006 f= 1 MHz ,

,

9 I

,

9 i

-1.5

,

I

. . . .

-1.0

I

,

-0.5

.

,

9 I

,

0.0

,

,

,

H (A/m)

i

,

0.5

,

,

.

,

1'.0

,

,

,

i

,

1.5

j

L

FIGURE 7.45 Hysteresis loop measured in a M n - Z n ferrite ring at 1 MHz with Jp = 5 mT before (solid line) and after (dashed line) insertion of a 50 pF capacitor in parallel with the primary winding. This capacitor emulates the effect of the stray capacitance introduced by a coaxial connecting cable about I m long. The primary and secondary windings are each made of five well-separated turns, wound in a bifilar layer. The tilting of the loop observed after insertion of the capacitor derives from the additional contribution of the current leaking through the capacitance to the measured primary current ill(t). Notice that the area of the loops (i.e. the energy loss) remains unchanged upon insertion of the capacitor.

wavfm, generatori ( ) power amplifier ~

1

b11~

"",~i

t

digital acquisition device

I' FIGURE 7.46 Schematic description of the setup for the characterization of soft magnets at medium and high frequencies. Primary and secondary windings are laid down as a single bifilar layer, with well-separated turns. A digital voltmeter or a digital oscilloscope can be employed for synchronous two-channel signal acquisition. The calibrated resistor RH is physically located close to the leads of the magnetizing winding and the shielded cables, which connect RH and the secondary winding with the acquisition device, are as short as possible. Notice that the lowpotential lead of the primary winding is separated from the ground by RH.

7.3 AC MEASUREMENTS

417 Co

G

i ~ R,-

T

=.L~ 1

Lwl

i

l:

UL1

lllI ILW2

c.I FIGURE 7.47 Equivalent circuit of the measuring setup in Fig. 7.46 taking into account the stray parameters: ul = voltage across the magnetizing winding; UL1= voltage balancing the primary e.m.f.; UL2 = -(N2/N1)UL1 secondary e.m.f.; u2=voltage across the secondary winding; C1,C2,Lwl,Lw2,Rwl,Rw2=self capacitances, leakage inductances, and resistances of primary and secondary windings; Cj, CH = capacitances of the connecting cables; Co = interwinding capacitance; R2 -- input resistance of the acquisition device.

typically ranging between I and 10 f~, is actually so small with respect to XcH = 1/~CH that we can safely disregard the role of CH and assume that the related voltage drop is always uH(t ) -- RHiH(t ). In fact, using a 20 cm long connecting cable and taking into account that with a digital oscilloscope we have typically R2 - 1 Mf~ with 10 pF capacity in parallel, we obtain XcH ~" 8 kf~ at 1 MHz. The basic question we pose here is how we can estimate the error introduced by the distributed parameters on the measured values of the power loss P and apparent power S. Both these quantities are experimentally determined, according to the base equations (7.20) and (7.24), starting from the current ill(t) supplied to the primary winding and the voltage u2(t ) appearing at the input of the acquisition device. In the presence of stray parameters, the quantities to be considered will be, instead, the magnetizing current im(t), resulting from the composition of ill(t) with the current i2~(t) (see Fig. 7.27b), and the voltage Uc2(t ) (see Fig. 7.47). The current ira(t) = (Im/N1)H(t) is then the current that flowing in the magnetizing winding in the absence of any secondary load, would

418

CHAPTER 7 Characterization of Soft Magnetic Materials

provide the field H(t) ensuring the rated induction B(t). If as is often the case, we disregard, the effect of the capacitance Co, we write for the current im(t) N2 im(t) - i l l ( t ) - icl(t)+ ~-~i2(t),

(7.51)

where icl(t) is the current leaking through the self-capacitance C1. To simplify the matter, we treat the material as a linear system, with the time dependence of field and induction described by phase-shifted sinusoidal functions. It is an acceptable approach because, on the one hand, we are often only interested in order of magnitude estimates of the errors deriving from the interference of the distributed parameters with field application and signal detection. On the other hand, a linear-like response of the material at increasing test frequencies is observed, due both to the prevalence of the classical eddy current loss contribution with respect to the domain-wall dependent loss contributions and the natural limitation on the achievable peak induction values. We have already introduced in Section 7.3.1 (Eqs. (7.27) and (7.28)) and illustrated with the equivalent circuit and the vector diagram in Fig. 7.27b the case where current is drained in the secondary circuit by the measuring instrument of input resistance R2. The correspondingly dissipated power AP must be subtracted from the measured loss Pmeasin order to obtain the actual power loss P in the material. With reference to Fig. 7.27a and b, no stray parameters being considered, we re-formulate Eq. (7.28) for the specific power loss under the assumption of sinusoidal time-dependent quantities.

P=

H(t) dt

is thus calculated by posing

H(t)-- Nlim(t)/lm and dB(t) dt

u2(t) N2A '

thereby obtaining p

~

~

~

1 N1

~

ma N2 imU2 cos qo--

~_~3Hp

Bp sin ~H

1 Xl~ ~ ma N2 IHU2 COS q012-- ma R2 -- Pmeas - hP,

(7.52)

7.3 AC MEASUREMENTS

419

where ~12 is the phase shift between ill(t) and u2(t)~ the measured power loss 1 Xl~ ~ ma N2 ZHU2 COS q~12

Pmeas - -

and the power dissipated in the load is &p-

1 522 ma R 2 "

Also, the measured and actual values of the specific apparent power (Smeas and S, respectively) differ because of the current circulating in the resistive load. We define them, based on Eq. (7.20), as ~ ~ N1 1 Smeas - - I H U 2 ~ ~ ~

S --

N1 1 l'mR2 ~ ~

N2ma

(7.53)

N2ma

and, being in this case

N2

im(t) -- ill(t) + ~ / 2 ( t ) , we obtain, under the general assumption i 2
(7.54)

where ~ is the phase shift between/m(t) and u2(t ) (see Fig. 7.27b). Notice that ~o12 and ~ are both larger than vr/2, so that the terms cos ~012 and cos ~ in Eqs. (7.52) and (7.54), respectively, are always negative. The terms AP and AS are relevant only when using relatively low impedance r.m.s, voltmeters for the reading of 52 or somewhat obsolete electrodynamic wattmeters for the measurement of P. With R 2 --1 Mf~, no significant corrections are expected. For example, a sample made of 12 strips of 0.10 m m thick grain-oriented strips tested by means of an Epstein frame, modified according to the Standard IEC 60404-10, will display, at 10 kHz and Jp = 1 T, a power loss figure of the order of 1 x 10 ~ W / k g , a secondary r.m.s, voltage 52 = 80 V~ and the specific apparent power Smeas ~ 1.2 X 103 V A/kg. The relative deviations AP/p and AS/S turn out to be, according to Eqs. (7.52) and (7.54), lower than 10 - 3. The equivalent circuit in Fig. 7.27a is of limited validity at m e d i u m and high frequencies and we should try to make our evaluation of AP/P and AS/S having in mind the complete circuit shown in Fig. 7.47. We start by assuming that either an electrostatic screen is used or the windings are

420

CHAPTER 7

Characterization of Soft M a g n e t i c Materials

laid upon the specimen in such a way that the interwinding capacitance Co is negligible. We equally neglect the leakage inductances and the winding resistances, so that the equivalent circuit and the associated vector diagram become as shown in Fig. 7.48. Here, we have posed Cs -C2 + Cj. The magnetizing current im(t) is related to the supply current ill(t), the secondary current i2(t), and the leakage current icl(t) by Eq. (7.51), now written in vector notation as im --" iH + jtoC1 N1

N2

(7.55)

~22U2 -{- ~11i2 9

The measured power loss is given by the usual expression Pmeas---- f --

N1 f ~ u2(t)iH(t) dt =

m---a N22

1 NI ~'HU2 ,

- m--~ ~

cos r

and is related to the actual power loss P via Eq. (7.55). We obtain, for a given value Jp of the peak polarization, 1

NI_

~

Pmeas -- - ~m a N---2IHU2 COS q::~12=

1

NI~

~

1

2

ma N2 lmU 2 COS q0 q- ~ ~ ma R2

= P + AP.

(7.56)

As was to be expected, the sole presence of the self-capacitances in the primary and secondary circuit does not affect the measured power loss, having the extra-current to be provided to the primary winding a purely reactive character. The example reported in Fig. 7.45 is one such case where the change of the hysteresis loop shape after addition of a small capacitor in parallel with the magnetizing winding leaves the loop area unchanged. The measured apparent power is instead affected by the presence of C1 and Cs. Starting from the definition of Smeasand S provided by Eq. (7.53), we find again the relationship between Zm and iH through Eq. (7.55), based on the analysis of the equivalent circuit and the vector diagram shown in Fig. 7.48. It turns out, provided the conditions l~C1 <
'

N2

N2

N1 iCs << G,

N1 iR2 << Zm,

are satisfied, that Smeas -- S + AS ~ S --

1 ~2 COS q0 ma R2

_ 1 ((N1/N2)2C ma

1 nu

Cs)tO~22 s i n qo.

(7.57)

7.3 AC MEASUREMENTS Rs

421

iC1

ul

G ~-

~

O

ULI

RH ~VV~

iH

i2

iI ~r

iH

,

iR2

!l T ,cs t f u. ==cs J I, [ o (a)

iC1 U 1 - UL1

~

9

il

., 12

p-

Ics U2 -- UL2

(b) FIGURE 7.48 (a) The medium-to-high frequency equivalent circuit of test sample and measuring setup arrangement shown in the previous figure is simplified under the assumption that leakage inductances, winding resistances and interwinding capacitances play a negligible role. This circuit is somewhat idealized and is useful to single out the effect of the self-capacitances of windings and connecting cables. It can be emulated by eliminating the interwinding capacitance by means of an electrostatic screen inserted between primary and secondary windings. (b) Associated vector diagram. Under rated flux conditions the measured power loss is not affected by the presence of C1 and Cs = C2 + Cj and it differs from the actual loss in the material only by the power Ap consumed in the load, exactly as obtained with the circuit of Fig. 7.27 (Eqs. (7.52) and (7.56)). The measured apparent power is instead changed by a quantity depending on the leakage currents ic1 and ics (Eq. (7.57)).

422

CHAPTER 7 Characterization of Soft Magnetic Materials

The example reported in Fig. 7.45 does not satisfy the condition ic1
jcoC1~N1 U

2 -}- ~N2 11i2-

jtoCl(Rwl +jwLwl)il,

(7.58)

where the current il = im + i2/ -- im -- (N1/N2)i2. If we substitute il in Eq. (7.58), we obtain the desired relationship between the quantities appearing in the definition of Pmeas and P. The calculations and the due approximations i2 KKi r a ,

co2LwlC1<< 1,

~oRwlC1<< 1

(7.59)

done, we eventually obtain that measured and actual specific power loss are related by the equation Pmeas -- P + AP ~

P(1 - co2C1nwl) -}- ~

1 NI_ _ + ~oC1Rwl~ma N22zmu2 sin ~.

ma R2 (7.60)

7.3 AC MEASUREMENTS

423

fRw1

.~t7--- r

icl

/]]

IJ

IR2

u 2 ~ Uk2

FIGURE 7.49 If the equivalent circuit in Fig. 7.48a is modified with the addition of the leakage inductance Lwl and the resistance Rwl of the magnetizing winding, the vector diagram in Fig. 7.48b is accordingly changed, by adding the corresponding contributions URw 1 = Rwlil and ULw I - - j r a L w l i l . Including the additional conditions posed for the derivation of Eq. (7.57), we also calculate the relationship between measured and actual specific apparent power as Smeas = S -~-/kS ~ S(1 - a~2CiLwl) -

- L((N1/N2)2C1

ma

1 ~ ma R2

+ C s ) a ~ 2 s i n qo.

cos q0 (7.61)

If we consider the previous example of Fe-Si sheet testing by means of an Epstein frame, accurately built according to the IEC Standard 60404-10 in order to have negligible interwinding capacitance, we still find that the only significant contribution to Ap comes from the power fft2/R2 dissipated in the measuring instrument. The role of stray capacitance appears more important with the apparent power. With C1 = Cs = 1000 pF and U2 = 8 0 V, we estimate from Eq. (7.61) AS/S---10-2. To provide another example, we take an M n - Z n ferrite ring sample

424

CHAPTER 7 Characterization of Soft Magnetic Materials

(Ira -- 0.1 m, A = 50 x 10 -6 m2), tested at 1 MHz and Jp = 0.05 T, where the measured loss is Pmeas-~ 250 W/kg. The material appears to behave almost linearly under these test conditions. The desired sinusoidal induction is obtained applying a sinusoidal field of peak amplitude Hp = 75 A / m by means of a uniformly distributed primary winding with N1 = 5 and a current Zm = 1.06 A is used. The voltage detected upon the secondary winding (N2 --5) has r.m.s, value 2 2 = 55 V. With C1 = 10 pF, Cs = 20 pF, Lwl = Lw2 = 1 ~H, R 2 -- 1 Mf~, we calculate with Eq. (7.60) A P / P m e a s "--" 2 X 1 0 -3 and with Eq. (9.61) A S / S m e a s "" 5 X 10 -2. Notice that with relative permeability of the ferrite material ~ r - - 5 X 103, the self-resonant frequency is expected to be of the order of 5 MHz. To complete our discussion on the effect of the stray parameters on the loss measurements, we make an estimate of the contribution of the interwinding capacitance Co. We limit our analysis to the simple case where NI = N2. It is a common arrangement, which is mandatory in Epstein frames [7.141, 7.142]. Following Brugel et al. [7.143], we proceed to the simplification of the equivalent circuit illustrated in Fig. 7.47, first reducing it as shown in Fig. 7.50a, then, by applying the delta-star transformation, into the circuit in Fig. 7.50b. Here, we have posed R1 R H 4- awl and advantage has been taken of the fact that the input impedance of the instrument is always high with respect to leakage reactance and resistance of the secondary winding, IRw2 4- J~

R2 << 1 + jwCsR2 "

It is also assumed in the delta-star transformation that the following conditions are fulfilled: (R1 + Rw2)/(1/wCo) << 1, (coLwl + roLw2)/(1/coCo) ~< 1. The impedance Z0 is then calculated as

Zo = Ro + jXo = -coCo(wLw2R1 4- 0aLwlRw2) 4- jwCo(Ri Rw2 4- ro2LwlLw2).

(7.62)

To find out the relationship between Pmeas- (1/ma)IHU 2 COS q012 and P = - ( 1 / m a ) i m ~ m cos q0, we start with the balance equation i H --ic1 4im + i2 for the currents in the circuit of Fig. 7.50b and we look for the equation relating iH.u 2 and im.Um. We write then u2.i H -- u2.ic1 4- u2.i m 4- u2.i 2

(7.63)

and we pose ic1 =jwClUl, with ul = u2 +(R1 +floL1)i2 +(R1 +floL1)im, and u2.i m -- Um'im 4- R0im.i m. By substituting u2.ic1 and u2.i m in Eq. (7.63),

7.3 AC MEASUREMENTS

425 Co

II II

TT,o i

i2

Lw2

Rw2

uL

R2

L

I

(a)

iH

R1

L wl

i __L_ C~

ul ,,IF-

i~,

~-

i2

;o l

|

2

R2

(b) FIGURE 7.50 (a) The equivalent circuit in Fig. 7.47 is simplified for the representative case where primary and secondary windings have the same number of turns (N1 - N2). It is assumed that the lower ends of the windings are at the same potential and the resistance R H is lumped with the winding resistance, so that R1 -- RH q-Rwl. (b) The same circuit after delta-star transformation and simplification based on the following approximations: (R1 q-Rw2)/(1/~Co)<< 1, (roLwl + oaLw2)/(1/OJCo) << 1, IRw2 q- jcoLw21 << IR2/(1 q- jcoCsR2 I. w e eventually obtain, u n d e r the a p p r o x i m a t i o n s given b y Eq. (7.59), 1 ~2 Pmeas -- P + AP --~ P(1 - co2C1Lwl) -t

+

1 ma

(R0(1 - co2C1Lwl) -

ma R2

1 ~- c o C i R w l - Z'rnUm sin ma

XocoRiC1)'z2.

(7.64)

426

CHAPTER 7 Characterization of Soft Magnetic Materials

The effect of the interwinding capacitance on the power loss is therefore lumped in the last term of this equation. Applied to the previous example of ferrite ring testing at 1 MHz, it provides a correction AP/P of the order of some 10 -3 for Co = 50 pF and a H - 2 fL There is of course a practical way of looking for the effect of stray capacitances, which consists in making measurements with and without condensers having suitably small capacity values, placed across the windings in order to emulate the effect of C1, C2, and Co. One should pay attention to the quality of the dielectric used because the parasitic currents in the self-capacitances of the windings might give rise to additional dielectric losses. They can be roughly estimated, for a dielectric loss angle i5, by considering the equivalent parallel circuit of the condensers and the extra-currents Aicl = ul roC1 tan 3 and A/C2 -- u2roC 2 tan 3 flowing into the equivalent parallel resistances. The related loss contribution AP~ ~ ~1- u22ro(C1+ C2 ) tan 3

is often not negligible. For example, an Epstein frame with coils impregnated with a common dielectric can display a value tan 3---0.1 and AP~/P might amount to few percent [7.143]. We have previously remarked that an acceptable simplification in the analytical treatment of power loss and apparent power at high frequencies is obtained by treating the material as a linear medium. It is possible in this case to generalize the basic concept of permeability, defined as the ratio / z - - B / H between induction and field strength when the material is taken along the normal curve or the anhysteretic curve, in order to account for the AC hysteresis. Because of energy dissipation, induction lags in time behind the field and with H(t)= Hp cos rot we can assume B(t) = Bp cos(rot - 3) under combined low peak induction values and high frequencies [7.144]. We can then describe B(t) = Bp cos 3 cos ~ot + Bp sin 3 sin rot as the sum of two 90~phase-shifted sinusoids of peak amplitude B1 --- Bp cos 3 and B2 - Bp sin 3. Equivalently, we can write, using the complex notation, H(t)= Hp d~t and B(t)= Bp ~(oJt-8). The 90~ component of the induction is evidently connected with the dissipation of energy. From the definition of power loss per unit volume, we obtain

P =f

~~

dB(t)

H(t) dt dt =frrHpBp sin 3.

(7.65)

7.3 AC MEASUREMENTS

427

On the other hand, we can apply the definition of permeability to both in-phase and 90 ~ out-of-phase components of B(t) /j/:

Bp ~pp cos 3,

,

Bp = ~pp sin 3.

(7.66)

/z~ and tz" can be viewed as the components of a complex quantity /z -

Bp ej(~t-a) p/ jtz" = p e i~

(7.67)

and accordingly take the name of real and imaginary permeability, respectively. It is immediately obtained that power loss per unit volume and imaginary permeability are related by the equation

P =fzrHpl~'.

(7.68)

Thus, the sole hypothesis of linearity permits us to describe the dynamic magnetic behavior of the material by means of the complex permeability. On very general grounds, one can actually state that tz~ and tz" are not independent, but are related by a defined analytical relationship (Kramers-Kronig relations). An example of correlated behavior of tz~ and k~"as a function of frequency has been provided for a number of soft ferrites in Fig. 2.23. The standard characterization of soft ring samples with the methods described in this section provides us with the magnetic parameters of the material we are interested in: take P, Bp~ Hp~ 3~ tz/~ p,'. We can use them to define the electrical properties of the inductor under testing. For example, the ratio between the imaginary and real permeabilities can serve as a measure of the departure of an inductor with ferromagnetic core from a pure reactance. According to Eq. (7.66), such a ratio is /z' tan 3. (7.69) The quantity tan 3, called "loss factor", coincides with the inverse of the quality factor Q of the inductor. The latter is defined, in fact, as the ratio Q = 2r

EL R,

BpHp cos ~ and E R = "rrBpHpsin 8 are the m a x i m u m stored where EL = 89 energy and the energy dissipated during one cycle per unit volume of the material, respectively. A real inductor can be described by means of the R-L series circuit shown in Fig. 7.51a, where the resistor Rs lumps the effect of iron loss and

428

CHAPTER 7 Characterization of Soft Magnetic Materials

i~

T

..... 7 I

R~

G

jcoLsis

Rp

I

Lp

Ls

]!

,.~c~ !

Ls

', I

Rds

~'

1

(a)

(b)

(c)

FIGURE 7.51 (a) Equivalent R-L series circuit of the real inductor, where the resistance Rs lumps the effect of iron loss. The quality factor of the inductor is given by the ratio Q = ~Ls/Rs = 1/tan 3. (b) Equivalent R-L parallel circuit. Rp and a~Lp are related to Rs and a~Lsby Eq. (7.71) and Q = Rp/~Lp. (c) If the stray capacitance Cp is taken into account, the measurements provide values R~sand oJL~s which, for sufficiently small values of Cp are given by Eq. (7.72).

the winding resistance is neglected. For a ring core of average diameter D and cross-sectional area A, provided with a winding of N turns, the equivalent resistance Rs and reactance ~oLs turn out to be related to the imaginary and real permeabilities according to the equations Rs -- coN2 - A ~ /z' ,

~oLs - coN2 - A ~ tzI

(7.70)

and the quality factor is Q-

1 tan 3

~oLs Rs

A parallel R - L equivalent circuit, as shown in Fig. 9.51b, can equally be adopted, where the parallel resistance Rp and the parallel reactance oJLp are related to their series counterpart by the equations R 2 -}- a~2Ls2

Rp =-

Rs

'

coLp =

R 2 + ~o2Ls2 oJLs

(7.71)

and the quality factor is correspondingly expressed as Q = Rp/~Lp. The complete equivalent circuit should actually also contain the stray capacitance and the modified R - L series circuit shown in Fig. 7.51c would in this case be more appropriately adopted. The values R~s and ~0L~s provided by the measurement should therefore be corrected in order to

7.3 AC MEASUREMENTS

429

obtain the true values Rs and roLs. For sufficiently small values of the stray capacitance Cp, the following relationships hold: R's

as , 1 - 2 ro2Ls Cp

,OL's

roLs 1 - (.02Ls Cp

,

(7.72) tan g

tan 1 - ro2LsCp

If an open sample is characterized and the demagnetizing coefficient Nd is known, we can derive the effective field H(t) from the applied one Ha(t) = Hap cos rot according to the expression

H(t)

~-

Hap cos rot -

(Nd/~o)Bp

cos(rot - ~a)~

(7.73)

where we have posed J(t)~ B(t). Applied and effective fields have the same out-of-phase component amplitude, while the in-phase component peak values differ by the quantity

Ha= Nd /p. /-to

The effective values of the magnetic parameters can then be numerically retrieved and the parameters of the real inductor are calculated. A very important point, always to be recalled when making weak field measurements, regards the polarization bias, which is the natural consequence of hysteresis and must always be eliminated or duly taken into account. Thus, specimen demagnetization is required before measurements, unless the bias is explicitly desired. The trouble is that even if the correct procedure is followed, unrecognized transients originating in the supply system (for instance, in the digital waveform synthesizer at the outset of voltage waveform generation) may bring the material along an unspecified trajectory in the q,H) plane, ending in an unknown remanence point. The weak field loop obtained by testing around this point is likely to display different properties with respect to the loop centered on the demagnetized state and totally meaningless results might be obtained. A further common complication in weak field experiments in alloys has to do with after-effects, that is, the time decay of permeability deriving from microscopic relaxation phenomena that interfere with the small-amplitude oscillations of the domain walls. To achieve reproducible results, the time interval upon which the measurement is performed and the measuring temperature must be specified.

430

CHAPTER 7 Characterization of Soft Magnetic Materials

The foregoing discussion demonstrates that the characterization of the magnetic component, rather than the search for the material properties, might actually be our prime objective. An unsuitable shape of the test specimen, its possible heterogeneous nature, or the mere necessity of testing for quality control are possible reasons for loose interest in the material behavior. Commercial ring cores most often come with a ratio between outside and inside diameter that is far higher than the recommended value Do/Di <- 1.1, resulting in measured properties which depend on sample shape. To characterize inductors, AC bridge methods are universally employed. They are classical methods, belonging to the realm of electrical measurements, which have been assessed over many decades and are treated by abundant literature. They are of minor interest for the sake of material characterization, for which the previously discussed approach based on the multiplication of the voltages proportional to field and induction derivative is the rule. ACbridge measurements are useful and possible only for very low magnetic flux densities, where the material behaves linearly and a limited supply power is required. They may in any case need a correction for the dissipation of energy in the winding. Digital impedance analyzers (LCR meters) are nowadays available, which directly provide the values of the equivalent parallel resistance Rp and inductance Lp of the core. From such values, the magnetic power loss and the permeability of the test specimen can be retrieved. The use of LCR meters is contemplated, under the above-mentioned limitations, in the measuring standards IEC 60404-6, which deals with the properties of soft magnetic cores and powder composites between 20 Hz and 200 kHz [7.145], and IEC 620442, specifically devoted to soft ferrite cores [7.146]. A basic measuring arrangement for inductor testing with an LCR meter is shown in Fig. 9.52a [7.147]. A four-terminal shielded configuration is adopted here, where the effect of lead impedances is minimized by using independent current and voltage sensing cables, with the outer conductors all connected to a guard terminal. The measured impedance Zm can still receive a contribution from the test fixture, which must be suitably compensated. The most popular compensation technique is based on the open/short procedure, schematically shown in Fig. 7.52b. It is assumed that the series of resistance Rsh and inductance Lsh and parallel of capacity Cop and conductance Gop represent the test fixture, where Rsh q-jcOLsh ~ (Gop q-jcoCop)-1. The impedance Zsh -- Rsh q-jcOLsh is measured when the terminals are short-circuited, the impedance Zop = (Yop)-1-- (Gop q-j~op) -1 is measured when the terminals are open. By connecting the terminals to the inductor under test, the impedance Zm is measured, which can be related to the impedance of the inductor

7.3 AC MEASUREMENTS

431

LCR meter . . . . .

)

I

) )

! (a) Rsh

Zsh

Lsh

Ti

v Cop

A

z~

-~r.E >

/ Yop

z,

/ \\ (b)

FIGURE 7.52 Measurement of the inductor impedance with LCR meter and four-terminal shielded configuration. The residual impedance of the test fixture is compensated using the equivalent circuit shown in (b), where Zsh = ash qjmLsh is the impedance measured with short-circuited terminals and Yop OJCop q-Gop is the admittance measured with open terminals (open/short compensation method). Zm is the measured impedance and ZL is the impedance of the inductor under test. =

by the equation Zm - Zsh ZL -- 1 - Y o p ( Z m - Zsh ) "

(7.74)

At sufficiently high frequencies, the contribution of the w i n d i n g resistance to loss dissipation is negligible (provided the skin effect in the conductors is not important) and the p o w e r loss can therefore be obtained from the m e a s u r e d value of Rp as 1 ~2 P -

m a Rp'

(7.75)

432

CHAPTER 7 Characterization of Soft Magnetic Materials

where U2 is the r.m.s, value of the voltage detected by a flux sensing secondary winding having the same number of turns as the magnetizing winding connected to the bridge.

7.3.4 Measurements at radiofrequencies The methods for the characterization of magnetic materials at radiofrequencies were promoted in the 1940s and 1950s by the discovery and the industrial development of the spinel ferrites (Section 2.6), whose combination of soft magnetic properties and semi-insulating character made them ideal for applications in communication engineering and as computer memory cores. More recently, trends towards miniaturization of inductive components and increasing areal density in magnetic recording, both associated with increasing operating frequencies, have been instrumental in the development of soft magnetic thin films and the related characterization methods well beyond the MHz range. The measuring approaches described so far in this chapter can be applied with due precautions (see the previous discussion in Section 7.3.3) to soft ferrites and thin films up to a few MHz. For thin films, in particular, problems related to their feeble cross-sectional area can be solved by accurate compensation of the inevitable large air-flux contribution to the secondary signal. Since thin films are usually tested as open samples, correction for the demagnetizing effect should additionally be considered. This may eventually result in minor adjustment because thin-film samples have a favorable aspect ratio and seldom attain very high permeability values. On approaching and going beyond the 10 MHz region a somewhat radical departure from the previously described classical fluxmetric measuring methods making use of primary and secondary windings is required. For one thing, the role of the magnetizing winding is greatly affected by self-capacitance and interwinding capacitance. With ferrite samples, the test material itself, being endowed with appreciable dielectric constant, may provide further undesirable current drain and phase shift. Reducing the number of primary turns, as imposed also by obvious limits of power supplies, can be done only to the point where a uniform magnetizing field, as ideally generated by a continuous sample-enveloping current sheet, is preserved and the effect of leakage inductance by the leads is restrained. Alternative approaches to homogeneous sample excitation at high frequencies come naturally into play by looking at the basic properties of radiofrequencies, which reside in the wavelike nature of the electromagnetic field.

7.3 AC MEASUREMENTS

433

7.3.4.1 General properties of transmission lines. The notion of simultaneity between cause and effect, as implied by ordinary electric circuit theory (closing a switch at some point does make ensuing currents and voltages appear at once at all points in the circuit), must give w a y at high frequencies to the concept of waves traveling through connecting cables and circuits in times non-negligible with respect to the oscillation period. The theory of transmission lines must then be applied to analyze the behavior of circuits and, in particular, the role of the magnetic components and materials found in them. A signal of frequency f = 50 MHz propagating at the velocity of light c = 3.108 m / s has wavelength A = c/f = 6 m. If we take as significant a time delay of 1/30 of a period, we conclude that traveling wave techniques should be used at such a frequency to deal with phenomena taking place over distances larger than about 20 cm. When transmission line structures (e.g. the diameter of the connecting cables) are significantly smaller than the signal wavelength, the analysis can be performed satisfactorily in terms of line voltages and currents. If this is not the case, a full electromagnetic field solution should be considered. We can assume as reasonable, with the structures ordinarily employed in magnetic characterization up to a few GHz, to deal on a voltage and current basis. It should be stressed that at radiofrequencies the characterization is identified in most cases with the determination of the real and imaginary permeability c o m p o n e n t s / ~ and/~" (see Eqs. (7.65)-(7.68)), which amounts to assuming that under the AC fields normally available for testing, the material behaves linearly. To see how the limitations of the low-frequency testing approach can be overcome in the framework of the transmission line theory, we need to summarize some basic conclusions of this theory, which is useful to achieve the specific aims of magnetic characterization. For a complete theoretical discussion, the reader should refer to one of many textbooks on the subject [7.148, 7.149]. Let us therefore consider, as in Fig. 7.53, the general case of a transmission line of length I connecting a generator with internal impedance Z s with a load of impedance ZL. For the range of frequencies and the cases of interest here, the transmission line can be thought to be made of a coaxial cable (circular cross-section) or a stripline (planar configuration) [7.150]. Coaxial systems are used in most lowpower laboratory applications at hyperfrequencies (wavelengths ranging from few centimeters to few millimeters) also and little use is nowadays made of waveguides. Whatever the case, we can treat the wave propagation along the line by considering this as the seat of distributed parameters. To simplify the matter, we assume that the line is lossless,

434

CHAPTER 7 Characterization of Soft Magnetic Materials

IL

z0 T

T i ZL

0

0

i

Source~ i plane

f

|

i~

~

'

Load

plane

(a) H

NI, l ~ 2 r ~ .

.. 1

i L,~" .."~.-'"

2

3

",4 (b) FIGURE 7.53 (a) Scheme of transmission line connecting the load impedance Z L with a generator of internal impedance Z_. A lossless line is considered here, having characteristic impedance Z0 =~x/~C, where L and C are the inductance and capacitance per unit length, and phase constant /3 = ~ox/~. Analysis of the circuit by the transmission line theory is required when the time taken for the electromagnetic field to propagate from source to load cannot be neglected with respect to the period. Notice in this scheme that the origin of coordinates is taken at the load plane. (b) Common transmission lines: (1) coaxial cable; (2) stripline; (3) microstrip line.

a condition satisfactorily emulated in m a n y practical situations. Thus, disregarding the series ohmic resistance and parallel conductance, we can consider any infinitesimal piece of line of length dx associated with a series inductance L dx and a parallel capacitance C dx, where L and C are the inductance and capacitance per unit length of the line. By considering the relationship between input and o u t p u t currents and voltages in the infinitesimal piece of line and considering sinusoidal time dependence for both (u(t)= V eJ~~ i(t)= I ~o~t), we obtain the two linear differential equations d2V - -]32V~ dx 2

d2/ - -/32/, dx 2

(7.76)

7.3 AC MEASUREMENTS

435

having the solutions

V(x) = V+eO13x+ V - e -j~x,

V+ V- -j~x I(x) = -~o eO~x- Z----~e ,

(7.77)

where we have assumed the origin of the coordinate x coincident with the load plane (see Fig. 7.53). 13 = ~ o ~ is the propagation constant (phase variation per unit length in a lossless line) and Zo - v/-L/C is defined as characteristic impedance of the line. In a lossless line, Z0 is a real quantity. We see in Eq. (7.77) that/3 -- 27r/h. It is understood that the equations for the instantaneous voltages and currents u(x, t) and i(x, t) are obtained multiplying the terms in Eq. (7.77) by the factor ej~t. It is thus immediately evident that the two components of u(x, t) = V(x) ej~~ and i(x, t) -- I(x)e j~ represent two waves traveling in opposite directions. Since there is no loss, traveling is not associated with attenuation and only phase relationships are involved in Eq. (7.77). The phase velocity of the wave is immediately obtained as v~-

~o 1 /3 - Lye-C"

(7.78)

If o0 is sufficiently small and the line sufficiently short, the phase/3x--, 0 and the simultaneity of the signal in different parts of the circuit is recovered. We see that the velocity v~ is independent of frequency, that is, propagation occurs in a non-dispersive fashion, as a consequence of the assumed lossless nature of the line. For a line made of a coaxial cable having the radii of the inner and outer conductors r and R0, respectively (Fig. 7.53b), filled with a dielectric of constants 8 r and /t/,r, we find, with good approximation, L -- /Xr/X~In R~ 27r r and C = 27rSrSO(ln--~-) -1 The characteristic impedance and the propagation constant then turn out to be 1 dP'r/XOlnRO Zo = ~ V ereO r

(7.79)

and -- r

P,r/-/,0.

(7.80)

436

CHAPTER 7 Characterization of Soft Magnetic Materials

Equation (7.77) written at the load plane (x = 0) becomes V + _ VV(0) = V + q- V - = VL,

1(0) --

Z0

Z0 = lc,

(7.81)

where V + and V- are the amplitudes of the incident and reflected voltage waves at the load. At the source plane it becomes V+ V - -j131 I(l) = -~-0 ejt31 - Z--~-e = Is. (7.82)

V(l) = V+eJ~l + V-e-J/31 = Vs,

We define the reflection coefficient at the load as g

FL-

~

V+.

(7.83)

FL is a complex quantity, whose modulus varies between 1 (complete reflection) and 0 (no reflection). Since it is also V(0)= ZcI(0), we obtain from Eq. (7.81) FL -- ZL -- Z0

ZL + Z0 "

(7.84)

We come across the well-known property of the lines terminated with the characteristic impedance (ZL = Z0) of totally absorbing the power associated with the incident electromagnetic waves at the load (FL = 0, properly terminated lines). In contrast, short-circuited (ZL = 0) and openended (ZL = oo) lines generate total reflection. In particular, at shortcircuited terminations the voltage VL must be zero, which implies FL = -- 1, while with open ends IL = 0 and FL = 1. Of course, a purely reactive termination ZL = +--jXL does always impose total reflection (IFLI = 1), the phase relationship between incident and reflected waves varying in the range ~r u 2 tan-~ IXLI. Z0 Power can be absorbed only by a resistive load RL, which is therefore associated with partial reflection, or no reflection at all when RL = Z0. Whenever partial or total reflection takes place, the simultaneous existence of incident and reflected waves gives rise to a standing wave. In correspondence with a short-circuited termination, the voltage standing wave has a node and the current wave has an antinode. This has the important consequence that high-frequency magnetic testing might be performed advantageously by placing the sample closely against such a termination. In this region, not only the magnetic field is maximum, but the electrical field is negligibly small and one can disregard dielectric effects.

7.3 AC MEASUREMENTS

437

A basic quantity to be determined with the circuit of Fig. 7.53 is the input impedance Zin = Vs/Is at the source plane. It is clear that when the transit of the signal from source to load cannot be considered as instantaneous, the impedance one is looking into at different points of the line can be considerably different from the load impedance. Making the ratio Z(x)= V(x)/I(x) in Eq. (7.77), we obtain, in fact, after elementary manipulation of the exponential terms,

ZL "q'-jZo tan(C/x) Z(x) = Zo Zo T,_~CC tan(C/x)"

(7.85)

This expression provides, as required, Z ( 0 ) = ZL and for x = l the input impedance

ZL q-jZo tan(]31) Zin = Z0 Z0 + ~ L tan(]31) "

(7.86)

There are cases of special interest that can be highlighted by means of Eqs. (7.85) and (7.86). With ZL = Z0, we obviously obtain Zin -- Z0, as Z(x) = Zo at any point of the line, and also Zin = Z0 for a line of infinite length (there can be no reflection from the load if the line is infinite). By posing ZL = oo and ZL = 0 in Eq. (7.86), the input impedance of a line of length l with either open or short-circuited terminations is obtained as Zin,op =

-jZo cot(/3l),

Zin,sh =

jZo tan(C3/).

(7.87)

Zin,op and Zin,s h a r e pure reactances, changing from inductive to capacitive (or vice versa) with the length of the line according to the behavior of ~l = 2,rd/A. Such a change occurs for any r value integral multiple of Ir/2, that is, any time the length of the line passes through a multiple value of A/4. According to Eq. (7.87), a shorted line of length l = A/4 has input impedance Zin,s h = OO. It is inductive for I < A/4, capacitive for A/4 < l < A/2, and so on. For l = A/2, we have Zin,s h = 0. The behavior of the shorted line around I -- A/4 is thus closely similar to that of a circuit with parallel lumped inductance and capacitance, resonating at ~o = v~13. Conversely, around I - A/2, it fits into a description of a series resonating lumped LC circuit. Symmetrical behavior of Zin,op with respect to Zin,s h is predicted by Eq. (7.87). In any case, it is notably obtained from such equation

Zin,opZin,sh -- Z 2.

(7.88)

Equation (7.84) demonstrates the relationship between wave reflection and load impedance. It can be written as I+FL ZL = Z0 1 - FL"

(7.89)

438

CHAPTER 7 Characterization of Soft Magnetic Materials

We can, therefore, conveniently express the input impedance at the source plane in terms of the reflection coefficient at the load 1?L. Introducing Eq. (7.89) in Eq. (7.86), we obtain 1 + i-'L e -j2/31 Zi~ = Z0 1 - FL e -j2~t "

(7.90)

It is actually possible to associate a reflection coefficient with any plane of coordinate x, as provided by the corresponding ratio F(x) between the amplitudes of the incident and reflected voltages. At the source plane, F(I)= FLe-j2~l and Eq. (7.90) can be rewritten, in analogy with Eq. (7.89), as 1 + F(I) Zin = Z0 1 - F(I)"

(7.91)

Most experiments involving transmission lines are nowadays performed using vector network analyzers. These devices have built-in signal sources and can measure the complex reflection and transmission coefficients of two-port and one-port networks over a broad range of frequencies. Figure 7.54a illustrates the general case of a two-port network with incident, reflected, and transmitted signals. The performance of such a network, that is, the quantities relating incoming and outgoing signals at the two ports, can be defined by means of the scattering matrix IS]. With reference to this figure, we define with al and a 2 the voltage signals entering port 1 and port 2, respectively, and with bl and b2 the voltage signals, resulting from the composition of both transmitted and reflected signals, leaving these ports. The scattering matrix relates these signals according to the equation = b2

[sl Sl2][a] .

$21 $22

(7.92)

a2

The scattering parameters, in general complex quantities, are then given by the following signal ratios: (1) $11 = (bl/al)ia2=o, reflection coefficient at port 1, and $21 --(b2/al)la2=O, transmission coefficient from port 1 to port 2 (forward gain) when the incoming signal at port 2 is equal to zero; (2) $22--(b2/a2)la~=O, reflection coefficient at port 2, and $12 = bl/a2la2=O, transmission coefficient from port 1 to port 2 (reverse gain) when the incoming signal at port I is equal to zero. A network analyzer test system as shown in Fig. 7.54b can implement these conditions by exciting the network under test at one port and loading the other port with the characteristic impedance (no reflection from it), and permits one to determine the scattering parameters. It essentially consists of

7.3 AC MEASUREMENTS

a

1

) Port 1 S_, ,'~" bl

439

i

21

"

I' ]~

1

Two-port network ~

Port 1

,

~

I OUT ]l

s~2 (a)

b

2 ~,

;'1~ ~ / L $22 ~_~

Port 2

a2

Port2

F .~ceiver

A/D

. _ .

Source ,,

Phase lock (b)

FIGURE 7.54 (a) The performance of a two-port network inserted into a transmission line can be defined by means of the scattering matrix [S]. It relates the signals a l , a 2 , b l , b 2 entering and leaving the device through the reflection coefficients $11,822 and the transmission coefficients $21, $12 (Eq. (7.92)). (b) Basic scheme of a network analyzer. The swept radiofrequency signal (typically from some 100 kHz to a few GHz) generated by a high-resolution synthesized source can be delivered to either port 1 or port 2 of the device under test (DUT). Directional couplers separate incident and reflected signals. The signals leaving ports I and 2 are routed to the inputs A and B of the receiver, respectively, while the incident signal is sent to the reference input R. Transmission and reflection measurements can be done in both forward and reverse direction. An attenuator permits one to adjust the power incident on the test device without changing the power in the reference path. Internal and external loads are not shown in the figure.

the following parts: (1) A s y n t h e s i z e d signal source, p r o d u c i n g a s w e p t r a d i o f r e q u e n c y signal, typically in the range of a few h u n d r e d k H z to a few GHz. The delivered p o w e r is typically of the order of 10-100 mW. The fraction of it incident on the t w o - p o r t n e t w o r k u n d e r test (DUT) can be adjusted b y m e a n s of an attenuator. (2) A test set circuit, s e p a r a t i n g the incident signal from the t r a n s m i t t e d a n d reflected signals

440

CHAPTER 7 Characterization of Soft Magnetic Materials

(e.g. via directional couplers) and routing them to the receiver ports A and B (reflected and transmitted signals) and the reference port R (incident signal). A switch makes it possible to pass from direct (port 1 to port 2) to reverse (port 2 to port 1) measurements without changing the connections to the test device. (3) A receiver with the adapted ports A, B, and R (e.g. 50f~ characteristic impedance). The signals are sampled and mixed and directed to the A / D converter. A processor performs all required mathematical operations involved with measurement and setup calibration. Notice that forward (reverse) gain in a lossless DUT is IS211= 1 (IS121= 1). DUT and termination, whether adapted or not, together play the role of a one-port network. The measured input reflection coefficient $11 = bl/al ($22 = b2/a2) does then always provide the total impedance connected to port 1 (port 2). In strict analogy with Eq. (7.89), we can then write the input impedance at the port 1 plane as 1 +$11 Zin = Z0 1 _ $11"

(7.93)

Of course, the planes of the DUT ports do not physically coincide with the planes of the analyzer ports as they are separated by the length of connectors and cables. The cables introduce frequency-dependent phase shifts (r = fll for a cable of length l), which must be accounted for and compensated. One might appropriately choose the length of the cables so as to create equal phase delays on the different connecting lines (e.g. the reference (R) and the unknown (A, B) signal paths). However, one can actually impose, exploiting the mathematical capabilities of the network analyzer, an electrical delay by software, that is, numerically add or subtract equivalent lengths of a lossless transmission line and effectively move the measurement plane. A phase shift ~ (radians) at a given frequency f will be compensated by emulating an air-filled piece of cable of length l = 0.4771 x 10s(~/f). Operation with a network analyzer may be affected by appreciable systematic contributions to the measuring uncertainty due to non-ideal behavior of the apparatus. They should as far as possible be removed by means of a calibration procedure [7.151]. The main sources of systematic errors are: (1) Imperfect separation of incident and reflected signals in the directional couplers. A small amount of the incident signal appears in the coupled arm and spurious reflections from the internal termination and the output connector may also occur. (2) Imperfect matching at the source and the load. Some of the signal reflected by the DUT towards the source is reflected back to the DUT, which re-reflects it again. Part of the signal transmitted through port 2 of the DUT is reflected back from the receiver

7.3 AC MEASUREMENTS

441

port and can be transmitted in reverse from port 2 to port I and the source. (3) Cross-talk between different analyzer signal paths. (4) Frequency response. This error is due to different variations in magnitude and phase versus frequency of the signals taken along the reference (R) and test (A, B) paths. To correct the systematic errors due to the above-mentioned effects, modeling is required and the measured scattering parameters are mathematically related to the actual ones. The model can be implemented by performing suitable reflection and transmission tests at all frequencies of interest using reference terminations and a transmission standard. In the calibration tests, the analyzer automatically provides for the corrections and stores the results for further retrieval. For example, three reference impedances with accurately known reflection coefficient $11 A a r e tested as one-port devices in the calibration of the reflection measurement: a short-circuit, an open termination, and a perfect load. The fact that the open termination actually exhibits a certain fringe capacitance and a related small phase shift is automatically taken into account by the analyzer. Three separate measurements, SllM~ are made with the three terminations connected at the measurement plane (either port 1 or port 2) and comparison with $11 A provides the three major error terms (directivity, source match, and frequency), which are removed by software. The correction applies also to the measured reflection coefficient of two-port devices, provided they are terminated with the system characteristic impedance. A full two-port error model is implemented for calibration of the four S-parameters of a two-port network [7.151, 7.152]. It first requires the execution of the previous calibration procedure for reflection measurement on both ports 1 and 2. The ports are then connected using the transmission standard. The reflection coefficient and the transmission path frequency response are consequently measured in the forward and reverse direction. The 12 main error terms envisaged by the two-port model are determined in this way and mathematically removed. For microstrip lines, where mismatches at the transition from the coaxial ports of the analyzer to the microstrip circuits are produced, a specific calibration procedure has been developed (through-reflect-line calibration technique) [7.152]. We have previously remarked that radiofrequency magnetic characterization of materials amounts in most cases to the determination of the scalar differential permeability ~ -- ~ + j~'. Linear material response is assumed and the previous standard concepts regarding the theory of the transmission lines can be applied. This has a practical effect, since weak field excitation is the rule in applications. High levels of excitation would generally imply prohibitively high values of losses. In some

442

CHAPTER 7 Characterization of Soft Magnetic Materials

instances, the anisotropic properties of the material may make it necessary to look at the permeability as a tensor. Because of energy dissipation, each tensor component consists of a real and an imaginary part. When the oscillating field frequency approaches the natural precession frequency of the moments in a constant magnetic field, the permeability of the material, following from the gyroscopic equation for the angular m o m e n t u m (see Section 5.4), is always described by a tensor (Polder matrix). It is transverse permeability, describing the fact that it is the component of the AC field in the plane perpendicular to the direction of the constant field (either applied or due to anisotropy; thereby providing a restoring torque on the magnetic moments), which is associated with an oscillating magnetization. In the following, we shall provide a synthetic view of the current methods applied in the weak field radiofrequency testing of ferrites and thin films.

7.3.4.2 Measurements in ferrites. A popular approach to the measurement of the complex permeability in ferrite toroids is based on the use of coaxial lines. They are the seat of a well-defined field configuration and provide a convenient fixture for sample testing. In particular, by placing the sample at the bottom of a shorted coaxial line, where the electric field (voltage) has a node and the magnetic field (current), azimuthally directed, is maximum, we can ignore dielectric effects, provided the sample thickness h is small with respect to the quarter wavelength A/4 of the electromagnetic field. This is easily verified by considering the input impedance Zin,sh of a piece of line of length h (K A/4 with shorted termination (Fig. 7.55). From Eq. (7.87), we obtain that the input impedance at the sample plane is, jZo tan(flh) -- jZoflh. and, since fl -- ~oLv~-Cand Z0 = ~/r/C, it turns out Zin,sh =

Zin,s h =

jtoLh,

(7.94)

(7.95)

L being the inductance of the line per unit length. For an air-filled lossless coaxial line of radii R and r, where the field lines are concentric with the inner conductor, we have Zin,s h --

jo)~--~_In --Rh. 7"

(7.96)

If the piece of line is filled with a ferrite toroid, a dissipation term will appear, which is taken into account introducing the complex permeability

7.3 A C M E A S U R E M E N T S

443

Zin,M ..

r

I :'

~ --~--" rm

'

,

m,< i I-h

i

"-' h ' ~

Shorted termination

~" '

Sample plane

FIGURE 7.55 The high-frequency permeability in a toroidal test sample can be determined by measuring the input impedance Zin,s h of a piece of coaxial line holding the sample against the shorted termination. If the sample thickness h is much smaller than a quarter wavelength A/4 of the electromagnetic field, dielectric effects can be ignored. The sample does not need to fit into the line. As usual with practical toroids, the azimuthal magnetic field strength can appreciably vary across the sample section and the characterization may pertain more to the sample than to the material properties. /z --/z0/Zr -j/z0pt'. The input impedance becomes Zin,s h -- jr

h(/Zr - j/z')ln rR"

(7.97)

The toroid does not need to fit the line. The total impedance of the piece of line holding the sample of thickness h, which, due to magnetic loss, must contain a resistive part, is always obtained as the sum of the impedances provided by the region occupied by the material and that of the remaining air-filled part. In fact, inductances and resistances in a line add in series. It is essential in this case that the sample be centered in the coaxial line, that is, concentric with the field lines, so as to avoid demagnetizing effects [7.153]. The impedance is then measured with and without the sample inserted in the line. In the latter c a s e , Zin,s h is purely reactive while, with the sample inserted, it is complex. The reactive and resistive parts are separated. The two impedance measurements thus provide two values Lm and La of the inductance, whose difference is AL = Lm - La

--

(/d, tr --

/z0 h In Rm 1)~---~ ~rm ,

(7.98)

where Rm ~ R0 and r m ~ r are the outer and inner radii of the toroidal sample, respectively. From this equation, the real component of

444

CHAPTER 7 Characterization of Soft Magnetic Materials

the permeability is obtained as /z~

~ (h Rm) -1 = ~r~0 = ~O + aL ~--~ln .

(7.99)

Ym

The measured resistive term, Rs = Re[Zin,sh], which is zero in the absence of the sample, relates to the imaginary component of the permeability (see Eq. (7.70)) according to Eq. (7.97) (see also the discussion in Section 7.3.3). We obtain from this equation ~" = R s . ( f h lnRm) -1 rm

(7.100)

and the related quantifies tan 3 = / z ' / # and Q = 1/tan 3. Goldfarb and Bussey [7.153], using a commercial coaxial connector and an impedance analyzer, have shown excellent repeatability of the permeability measurements made according to the previous scheme on a number of M n - Z n and N i - Z n ferrite toroids up to 40 MHz. No theoretical limits are envisaged below about 1 GHz. At such frequencies, resonance modes in the connector may be excited and the condition on the sample thickness consistent with hypothesis of negligible dielectric effects (h << A/4) may not be fulfilled. This depends on the actual values of the relative permittivity and permeability, being the electrical length of the toroid he - - h ~ r / Z!r . ! The value of Zin,s h w e have discussed so far makes reference to the sample plane and we must relate it to the impedance Zin,M actually looked into at the analyzer port plane. If I > h is the length of the lossless transmission line, whose characteristic impedance Z0 and propagation constant 13 are known, containing the sample at the shorted bottom, the related input impedance is from Eq. (7.86) Zin,s h -}-jZ o tan fl(l - h) Zin,M -- Z0 jZin,sh tan fl(l - h) + Zo

and Zin,sh can be determined by calculation. With network analyzers, we can actually move the calibration plane to the desired location using internal software, which simulates a variable length transmission line and the related phase shift. We can, in particular, compensate for the electrical length of the piece of coaxial line separating the sample plane from the analyzer port plane. In this case, we shall directly apply Eq. (7.93) 1 +$11

Zin'sh -- Z0 1 - $11 '

7.3 AC MEASUREMENTS

445

300

i

L

P"r

~9

~

rr

2oo-

lOO

101

10 2

10 3

10 4

10 5

10 6

10 7

10 8

Frequency (Hz) FIGURE 7.56 Real and imaginary components of the initial permeability in a YIG toroidal sample measured with a coaxial line method making use of single-loop pickup coils for signal detection. The lower frequency range (below 100 kHz) is covered with a conventional type measuring setup (adapted from Ref. [7.154]).

with Z0 the characteristic impedance of the analyzer port and $11 the one-port network reflection coefficient, at the sample plane. Lacking the network analyzer, it is possible, for instance, still to exploit the coaxial line as a source of azimuthal magnetic field at high frequencies, but, instead of making an impedance measurement, to detect field and induction in the sample separately by means of two one-turn loops, one around the sample cross-section and the other close to it. The signals can be delivered to an analyzer, which provides for the separation between real and imaginary parts. Guyot and Cagan have applied this approach to the determination of initial permeability in YIG toroids from 100 kHz to 1 GHz and combined it with a conventional measurement at low frequencies [7.154]. The result shown in Fig. 7.56 puts in evidence diffusion relaxation (after-effect) and domain wall relaxation around I kHz and a few MHz, respectively. A weak ferromagnetic resonance effect is observed beyond 100 MHz. A shorted coaxial line passes through resonant conditions every time its length is an integral multiple of a quarter wavelength A/4. Thus, if a ferrite toroid (or indeed any toroidal magnetic specimen made of a material with insulating or near-insulating character) is placed within

446

CHAPTER 7 Characterization of Soft Magnetic Materials

a ,~/4 line, it brings about, together with an increase AL of the inductance L of the line, a decrease of its resonance frequency f0 by a quantity zXf directly related to the material permeability. As usual, the sample is placed in contact with the shorted termination in order to avoid dielectric effects. We know that for high-Q resonant lines and circuits AL/L = - 2 k f / f o . A toroidal sample of relative permeability/Z'r (real component) introduced in the line adds a contribution AL given by Eq. (7.98). If a reference specimen with known v a l u e /tl]r,ref of the relative permeability having identical size as the test specimen were available, we would immediately obtain from the previous equation AL = ALre f

/Z~r-1

_

Af

/ f -- 1 - /d,r,re

Afref

,

(7.101)

with Afref the variation of the resonant frequency upon introduction of the reference sample. We notice that the permeability/z~r of the test sample is here obtained by determination of the variation of the resonance frequency of the line on134 and no further information on the parameters of the line or the sample is needed. Lindenhovius and van der Breggen suggested the use of a solid copper ring as reference sample [7.155]. Because of the complete shielding of the ring interior by eddy currents, the region it occupies becomes field-free and /d,r,re ' f ---~ 0. Equation (7.101) then provides for the permeability of the test sample af /Zr, -- 1 -- &fref "

(7.102)

Again, a regular sample shape and centring are required to avoid the disturbance of demagnetizing fields. To determine the loss-related imaginary component of the permeability/z', we may look at the x/2-bandwidth of the resonance curve (i.e. the Q-factor) of the line and its variation upon insertion of the sample. It is shown that/z" can be obtained by determining the bandwidths (A~o)c and (AoJ)tot before and after insertion of the sample, and the associated amount of detuning A~o [7.155]. In particular it is found, tan 3 -

/d,~

/x~

- -

(Ao))tot -- (Ar

2Aro

"

(7.103)

The use of the ,~/4 resonating line might be found too cumbersome below a few hundred MHz (,t ~ I m at 300 MHz) and re-entrant resonant cavities have therefore been variously applied in the lower frequency range [7.155-7.157]. A re-entrant coaxial resonator is essentially a section

7.3 AC MEASUREMENTS

447

Coupler~ ~

.................................. I]' D Toroidal sample

/

...........

/

. ...........................

Capacitorplates

/

=,,

FIGURE 7.57 Schematic of toroid testing with re-entrant resonant coaxial cavity. Changing the distance between the condenser plates regulates cavity tuning.

of coaxial transmission line that is short-circuited at one end and capacitively coupled at the other end (Fig. 7.57). Resonance occurs when the reactance of the capacitor Xc = 1/joJC is equal to and opposite in sign to the input impedance Zin,s h = jZo tan/31. The resonance frequency, obtained by equating Zin,s h and Xc, is 1

1

f0 = ~ 27rZ0 tan(27rl/A) " As the capacitance becomes larger, the length of the resonating re-entrant cavity becomes shorter. In conclusion it is possible, to achieve resonant conditions over distances much lower than the free-space A/4 at relatively low frequencies (say down to 30-50 MHz) with reasonably sized cavities (say, less than 300 m m long). Variable tuning can be achieved by changing the distance between the capacitor plates by means of a fine micrometer drive [7.156]. For all its sensitivity, the resonant cavity method, which requires a high and reproducible Q value, is little used today because of its strict mechanical requirements, relatively small tunable range, and complexity of setup. Instead, the determination of the scattering parameters in a coaxial line with network analyzers is largely applied, both with one-port and two-port devices, as a fast and reliable method for the determination of permeability of ferrites up to the microwave range. Used in the swept frequency mode, network analyzers can automatically provide a broadband characterization of the device under test, ensuring wide dynamic range (typically larger than 100 dB) and a fully calibrated environment. Such a characterization can be more general than so far discussed. In particular, we can extend it beyond the rather narrow case of permeability

448

CHAPTER 7 Characterization of Soft Magnetic Materials

measurements in toroidal specimens of thickness h << A/4. On approaching the GHz range, A/4 might become comparable with the sample size, making it difficult to separate magnetic and dielectric effects. On the other hand, we might be interested in the simultaneous measurement of both permeability and permittivity. This poses certain measuring difficulties, which can be treated to a good extent relying on the concept of scattering matrix in signal analysis and the use of network analyzers in the determination of the reflection and transmission coefficients [7.158-7.160]. Let us therefore consider the quite general case of a twoport network made of a transmission line of length 2l + h loaded in the center with a sample of thickness h completely filling the line crosssection, as schematically shown in Fig. 7.58. In the same figure, we show a stripline measuring cell developed by Barry [7.160], implementing this model scheme on ferrite test samples. In this specific case, 5 m m thick ferrite plates sandwiching the center conductor (a 0.48 m m thick copperberyllium strip) are housed in a 10 m m thick aluminum frame. The magnetic field lines, enwrapping the central conducting strip, are conducive to small demagnetizing effects if the magnetic plates are in good contact. This may require grooving of the test plates for better fit with the center strip. The cell is accurately designed for excellent matching with coaxial cables and analyzer ports in order to avoid undesired reflections at connections. To determine the sample complex permeability and permittivity as a function of frequency, the measurement of a pair of independent complex parameters is required. With a two-port network, the accessible parameters are the scattering coefficients $11 and $21, which define the signal reflection and transmission properties of the device. By virtue of their definition, $11 and $21 depend on both characteristic impedance Zm and propagation constant J~m of the test sample, which, in turn, are related to the values of permeability ~ = /~0(/~r- j/~') and permittivity ~ - - ~ 0 ( ~ r - j~') (Eqs. (7.79) and (7.80)). To work out the relationship between scattering parameters, permeability, and permittivity, we start by writing Zm and J~m as a function of the characteristic impedance Z0 and propagation constant fl of the air-filled stripline. From Eqs. (7.79) and (7.80) we obtain

Zm = Z0ff/d'--r ,Sr

~m -- ~ ~ r S r 9

(7.104)

Wave reflection at the boundaries between medium and line depends on the associated variation of the characteristic impedance. The theory of transverse electromagnetic (TEM) wave propagation provides for the reflection coefficients at the line-medium and medium-line boundaries

7.3 AC MEASUREMENTS

449

Port 2

Port 1

I I I

s21 .

.

.

.

.

.

.

.

.

.

.

.

Zo, # ...... . . . . . . . . .

., "

.

#

.

.

I

1-"..... Zo,# ....... .

.

.

.

,.L, "T'

,.I., "T"

h

I

:

II

(a)

.

~: I :

#

/

Ground plane ~

~ n t e r

d[

i

conductor

.\

/

.........

E'~/// / / / / /

r./ ,s,ec,.n

I,,

i

/ , / /

j

i

.......... _ _

Ground plane (b) FIGURE 7.58 (a) Transmission line loaded with test sample of characteristic impedance Zm and propagation constant ]~m at its center. The signal is generated at port 1 and the line is matched. Complex permeability and permittivity of the material can be determined by measuring the scattering parameters $11 and $21 of the whole network (measured at analyzer port 1 and port 2). The scattering parameters are related to the value of the reflection coefficient F = ( Z m Z0)/(Zm + Z0) at the sample boundaries I and II, the propagation constants in the line and the sample, the lengths I and h. The measuring principle is implemented on ferrite plates with the stripline cell shown in (b), where the sample fills completely the cell cross-section. The cell has typical size of few centimeters (adapted from Ref. [7.160]).

450

CHAPTER 7

C h a r a c t e r i z a t i o n of Soft M a g n e t i c M a t e r i a l s

(lines I and II in Fig. 7.58a) the quantities F and - F , respectively, with F = Zm - Z0 Zm + Z0 9

(7.105)

Combining Eqs. (7.104) and (7.105), we obtain /

. ,

]d'r = ]d'r - - ] ~ r --

/3m I + F ]3 1 - F '

I

9//

8r = 8r -- ]Sr --

/3m 1 - F ]~ 1 + F"

(7.106)

The propagation constant/3m and the reflection coefficient F are connected by a definite relationship with the scattering parameters $11 and $21 measured at the reflection port 1 and transmission port 2, respectively, and the parameters of the line. One can find such a relationship by expressing the transmission matrix [T] connecting the voltage signals al (incident) and bl (reflected) at port I with the signals a 2 and b2 at port 2, both in terms of/3m and F and of $11 and $21 (see, for example, the detailed derivation by Barry [7.160]). The calculations provide the following expressions relating ]3m and F to the experimental quantities $11, $21 ~mh -- cos- 1

(

)

e-J4~l + S~1 S~1 2e -j2~IS21

F=

e

-j2~l

- $2~e

-j~mh"

(7.107)

~r and Sr can then be obtained from the measured scattering parameters through Eq. (7.106). The first of Eqs. (7.107) has periodic solutions for 27r ~m h -

~m h

(if ~-m is the wavelength within the material). Only the zero-order solution is acceptable in these experiments because the specimen thickness must be kept in the range 0 - h-< I/2. Results in thicker materials may be affected by dimensional resonances [7.161] and do not comply any more with the previous equations. The base procedure followed in the measurement of the scattering parameters can be summarized as follows: (1) The systematic errors of the analyzer are corrected by performing the previously sketched calibration procedure. Since we are to test a two-port network, the full 12-term error procedure must be carried out, using the calibration kit and the software facilities provided by the manufacturer of the network analyzer (see, for example, Ref. [7.151]). (2) The parameter $21 is determined for the empty stripline, thereby obtaining the total electrical length of the two-port network. By subtracting from it the actual specimen thickness h, we obtain the length parameter l to be used in the previous equations, which is different from the physical length only if the line is not air-filled

7.3 AC MEASUREMENTS

451

(we have here defined/3 = r (3) The line is loaded in the center with the specimen and all four parameters $11, $21, $22~$12 are measured. Forward and direct measurements are averaged in order to compensate for slight asymmetries in the loaded cell (e.g. errors in sample centering). Should cavity modes be excited in the GHz range, a way should be found to suppress them, for example by means of radiofrequency absorbers. A typical experimental result on permeability and permittivity in a ferrite sample obtained in the frequency interval 500 MHz-5.5 GHz by the stripline method is shown in Fig. 7.59 [7.160]. Note that with the employed test geometry the measured permeability value is expected to be little affected by demagnetizing effects. In the case where the gap between the two parts of the specimen cannot be neglected and a demagnetizing coefficient Nd can be estimated, a correction may be made according to the standard equation/Z r "-- /d, a r ( 1 - - Nd)/(1 - - N d / - t , a r ) , relating effective and apparent relative permeabilities /Zr and /Zar. When the sample does not fill entirely the line cross-section, the stripline method can still be applied provided the assumption of quasi-TEM mode

20

Ferrite sample

15~o

{

lO

~'r

\

-

\

5

0

~ ~'~"~'"~" ~.~

'

'

I

'

.

'

.

'

I

'

'

4

2

Frequency (GHz)

FIGURE 7.59 Example of frequency behavior of the real and imaginary components of relative permeability and permittivity in ferrite test plates, as obtained by the measurement of the scattering parameters $11 and Sn in a stripline cell of the type shown in Fig. 7.58 (from Ref. [7.160]).

452

CHAPTER 7 Characterization of Soft Magnetic Materials

propagation, where longitudinal electric and magnetic field components are negligible with respect to the transverse ones, still holds. Fessant et al. [7.162] assume that the piece of cell inhomogeneously filled by the sample plus air can be treated as if it were occupied by a fictitious homogeneous material, whose complex permeability and permittivity values are determined with a measurement of the scattering parameters of the device. They are calculated at the same time by a self-consistent method, based on a quasi-TEM electromagnetic model of the real structure of the cell cross-section at the sample position. Comparison of calculation and experiments eventually provides the effective values of the parameters/z and s in the specimen [7.163].

7.3.4.3 Magnetoimpedance measurements. In recent times, there has been a good deal of interest in the phenomenon of giant magnetoimpedance in amorphous wires and its measurement at radiofrequencies. The impedance that a wire offers to a radiofrequency current flowing in it can suffer a large variation when the intensity of an externally applied DC field is changed [7.164]. This occurs because the circumferential permeability depends on such a field via the domain structure and the penetration depth 3 ~/p/ld,ola,rq~f (p, resistivity; /d,r6, circumferential permeability) of the radiofkequency circumferential field associated with the current is consequently affected. Magnetoimpedance experiments can be performed with conventional impedance analyzers and four-probe methods up to around 10MHz, while at higher frequencies, the methods employing transmission lines and network analyzers are to be applied. The basic approach to the determination of the wire impedance at such frequencies consists in using the wire under test as the center conductor of a suitably terminated coaxial cell, playing the role of one-port network, which is characterized in reflection by means of a network analyzer [7.165, 7.166]. The wire is subjected to the usual azimuthal magnetic field, whose intensity is kept low enough to ensure linear magnetoimpedance effect. The size of the cell should be small enough (length around 10-20 ram) to avoid the establishment of unwanted cavity modes at the investigated frequencies, that is, up to a few GHz. Figure 7.60a provides a schematic view of the measuring arrangement with the terminated cell and the network analyzer [7.166]. The impedance of the wire can be determined as a function of frequency for different values of the applied DC field HDc in a relatively simple way using a shorted termination. This should be very close to an ideal short circuit, that is, associated with negligibly small electrical length. At the same time, the electrical length of the input connector of the cell should be precisely determined and the reference plane correspondingly =

7.3 AC MEASUREMENTS

453

Amorphouswire Network analyzer

s 1 ~ oi i ....

O' i

/

Termination

gc

Outer conductor

(a)

2.0 1.5

HDc = 10 kA/~,;

II

!

1.o N

N 0.5 <~

-

0.0 !

-0.5,

"-

' .

-1.0

0.1

.

.

.

....................................... 1

10

100

Frequency(MHz) (b)

1000

FIGURE 7.60 (a) Schematics of measurement of giant magnetoimpedance in amorphous wires. The wire specimen, subjected to a longitudinal DC field H D C ~ is held as central conductor in a coaxial cell with termination. The network analyzer provides the radiofrequency signal and measures the reflection parameter $11 of the one-port cell at the reference plane 0 0 !. (From Brunetti et al. [7.166]). (b) Relative variation of the impedance of a Ni-Co-based amorphous wire with frequency of the supply current for two values of the field H D C . The peaked behavior is attributed to ferromagnetic resonance (adapted from Ref. [7.165]).

m o v e d to OO/, where the wire starts. The input i m p e d a n c e is determined, according to Eq. (7.93), with the m e a s u r e m e n t of the p a r a m e t e r $11 ~ a s Zin,s h = Z0(1 + $11)/(1 -- 811)~ Z0 being the characteristic i m p e d a n c e of the analyzer port. To calculate Zin,s h f r o m the theory of transmission lines, we need to consider the resistance of the magnetic wire. The previous transmission line equations (7.76) lead to the expression for characteristic impedance of the cell Zc -- ~/(R + j~L)/jo~C and the propagation constant 3' = x/( R -4-j~oL)jroC, with R the resistance per unit length of the wire (the resistance of the outer conductor being negligible).

454

CHAPTER 7 Characterization of Soft Magnetic Materials

We have assumed in these expressions that having to deal with an air-fflled cell, we can always disregard the effect of parallel conductance G between central and outer conductors. With a perfectly shorted termination and a wire of length lw, we obtain, in close analogy with the previous equations (7.86) and (7.87), valid for the lossless line, the expression Zin,s h = Z c tanh(),lw).

(7.108)

Zc'), = (R 4-floL) = (R + floLi + floL0),

(7.109)

We find

where the inductance per unit line is L = Li 4- L0, the sum of the intrinsic inductance of the piece of wire Li, and the inductance in air of the line L0. The latter is L0-- ~

h R0 rw'

where rw is the radius of the wire and R0 is the inner radius of the outer conductor. We wish to find the impedance of the wire Z = Zwlw-(R + floLi)lw from the measurement of the input impedance Zin,s h. The impedance per unit length of the wire is from Eq. (7.109)

Zw = Z~3'- j~Lo.

(7.110)

Since Zc = ,),/jo~, with the line capacity per unit length C = 2rrs0(ln (Ro/ rw)) -1, the measured impedance Zin,s h from Eq. (7.108) is Zin,s h -" j - ~

tanh(7lw).

(7.111)

The propagation constant 3, is numerically found from this equation as a function of Zin,s h u s i n g the known quantity C. Substitution in Eq. (7.110) eventually provides Zw --

['}/(Zin,sh, C)] 2 ja~7, - floLo

(7.112)

and the total impedance of the wire Z = Zwlw. Where the condition lw << A/4 holds (typically, at frequencies f << 1 GHz for usual 5-20 m m long wires), tanh(ylw) ~ 71w and Zin,s h ~ Z c , ' Y l w - " (~/ja,C)lw. We find that the wire impedance is directly obtained by measuring the input impedance, according to the equation Z ~ Zin,s h - j c o L o l w .

(7.113)

7.3 AC MEASUREMENTS

455

Figure 7.60b provides an example of behavior Of the thus measured quantity AZ Z

Z(HDc) - Z(0) Z(0) '

the relative variation of the impedance of a Ni-Co-based amorphous wire upon application of the field HDC, in the frequency interval 100 k H z 1 GHz (from [7.165]). The results obtained in the lower frequency range (below 13 MHz) by means of a conventional impedance analyzer are observed in these experiments to agree with those achieved with the coaxial cell and the network analyzer (above 5 MHz) in the overlapping region of frequencies. Once the wire impedance Z = Rlw + jXi is known, the associated high-frequency circumferential permeability /z~ = / z ~ - j / z ~ can be obtained using the standard relationship connecting Z and /z4 at high frequencies, as discussed in the literature [7.167]. Notice that we could equally measure the high-frequency wire impedance by making a double determination of the input impedance Zin, using either shorted or open termination. In this way, the characteristic impedance of the coaxial cell with the wire would be obtained, according to our previous Eq. (7.88), as Z 2 Zin,shZin,o p. However, truly open terminations, providing infinite load impedance to the line, do not exist and we have at least to consider the role of fringing capacitances. This results in a somewhat more involved expression for Zc and it would require separate characterization of the coaxial cell with the magnetic wire substituted by a near-zero resistance center conductor [7.166]. =

7.3.4.4 Measurements in thin films. Magnetic thin films are natural candidates for high-frequency applications. Low thickness is synonymous with low eddy current losses, good flux penetration, and low demagnetizing effects, and the thin-film geometry conveniently integrates in miniature devices for a multitude of applications, including magnetic recording heads, sensors, and telecommunication systems. Given the usual planar structure of thin films, the microstripline or parallel-plate geometries are conveniently employed in their characterization (Fig. 7.1b). This requires carefully arranged transitions between the stripline fixture holding the test sample and the coaxial system providing the connection with the measuring equipment in order to minimize reflections, phase shifts, and signal attenuation. Thin films are often characterized in terms of permeance, sometimes called effective permeability, which is given by the product of permeability and film thickness. The related measuring setups are commonly called permeance

456

CHAPTER 7 Characterization of Soft Magnetic Materials Uref

0.55

mm

U2

,

glass

Sample

FIGURE 7.61 The thin film permeance measuring jig developed by Calcagno and Thompson [7.168]. The figure-eight coil gathers near-zero signal in the absence of the sample, therefore generating to a good approximation a voltage u 2 ( t ) - Sm(dJ/dt) when the sample of cross-sectional area Sm is introduced in the jig. The reference voltage Uref(t)generated in the reference coil, placed close to the sample surface, is assumed to be proportional at a given frequency to the effective field. meters or permeameters, a term that also applies in a very different context in the characterization of bulk soft magnets (Section 7.1.1). An earlier permeance tester for characterization of permalloy films in the 0.5-5 lxm thickness range up to 100 MHz was developed by Calcagno and Thompson [7.168]. It is in essence a classical fluxmetric device, where a drive strap provides a radiofrequency field in the plane of the film sample, which is held in a measuring jig as schematically shown in Fig. 7.61. The sample is slipped between two thin glass plates having precisely known dimensions. External Helmholtz coils permit one, as usual in thin-film characterization setups, to impose both a defined magnetization history and a reference magnetic state, around which the weak-field high-frequency characterization is performed. The signal u2(t) induced by the time-varying magnetization is collected by one loop of a figure-eight coil, made of interconnected evaporated lines, ideally providing a near-zero signal in the absence of the sample. Thus, with the sample of cross-sectional area Sm inserted in the measuring jig, it is with good approximation u2(t) = Sm(dJ/dt), if J is the material polarization. With typical specimen size and thickness, demagnetizing effects may not be negligible. To obtain the true value of the permeability, a reference single-loop H-coil placed close to the specimen surface is employed. The detected signals are analyzed by means of a conventional digital voltmeter. Self-resonant effects in the detecting coils provide an upper test frequency limit of the order of 100 MHz. The conventional fluxmetric approach to high-frequency permeability measurements in thin films can be advantageously adapted to

7.3 AC MEASUREMENTS

457

oft 2

~~--f

~

Thinfilmsample

FIGURE 7.62 Schematics of a two-port permeance meter based on shorted parallel-plate transmission line. The driving plates are connected to the drive port (Port 1) of a network analyzer. The center conductor generates a uniform magnetic field at the sample position, while, for typical operating frequencies (up to few hundred MHz, depending on the size of the fixture), electrical field is negligible. The pickup coils are connected in electrical opposition to the receiver port (Port 2) of the analyzer. Since source and receiver have the same characteristic impedance, the scattering parameter $21 of the two-port network is equal to the ratio of transmitted to incident voltage and relates to the sample permeability (adapted from Ref. [7.169]).

the broadband swept-frequency testing methods using network analyzers. Here, the presence of the film sample can be treated as a perturbation in the transmission properties of a two-port network. Figure 7.62 schematically illustrates the permeance meter fixture developed on this principle by Grimes et al. [7.169]. The test fixture is made of a parallelplate transmission line with the usual shorted termination. The measuring jig has a typical size in the range of a few centimeters (e.g. length 6 cm, width 3.2 cm) and is supplied via the drive port of a network analyzer. The current in the center sheet generates a homogeneous radiofrequency field in the upper and lower cavities, which contain identical single-turn loops made of copper strips connected in electrical opposition to the receiver port of the analyzer. The sample is placed, with its plane parallel to the sheet (that is, the field direction), inside one of the loops. Under the usual assumption of line short with respect to quarter wavelength (I << ~), we obtain e-+J~l~ 1 +j~l. Since the reflection

458

CHAPTER 7 Characterization of Soft Magnetic Materials

coefficient at the shorted termination is FL = V - / V + = - 1 , Eq. (7.82) provides for the current flowing in central conductor i ~ 2V+/Zo everywhere. By denoting with Vin = V(1) the voltage at the input of the line, we obtain that the current in the center conductor of width w is 1 Zo jill

Fin

Fin

Gin'

where Gin is the input impedance of the shorted line. The field strength at the sample position is H = i/2w, that is H -~

Fin

2wZin

- kVin.

We suppose, for the time being, that the air-flux is completely compensated and the demagnetizing field is negligible. We can thus express the flux linked to the pickup coil setup as due to the sample only, 9 = AsJ = AsP,0(/zr - 1 ) H , if As is the cross-sectional area of the film of relative permeability IZr. With a sinusoidal field, the output voltage Vout = - d a p / d t is Vout = -jookAslzO(IZr - 1)Vin. Since both source and receiver have 50 12 impedance and reflection at the input port is neglected, the ratio of input to transmitted wave voltage is equal to the scattering parameter $21 of the two-port network Vout/Vin

--

821 --- -joJkAsl~O(IZr - 1),

(7.114)

that is, related to the sample permeability. A background signal is always present and it adds to the previously calculated voltage Vout. It can be subtracted by making two separate measurements of 821 ~ first saturating the sample ($211sat)~ which amounts to eliminate any coupling due to magnetism of the thin film, then bringing it into the desired magnetic state (e.g. at remanence, $2110). If we define the difference AS21 -- $211sa t - 82110~ we re-write Eq. (7.114) as AS21 : -j0akAs/z0(/zr - 1),

(7.115)

an operation which can be numerically performed by the analyzer. The proportionality factor k can be calculated in principle, but the associated uncertainty may be large. It is therefore expedient to perform the very same measurement on a reference sample, whose permeability, in its real and imaginary parts, and cross-sectional area are precisely known. From the quantity (AS21)ref and Eq. (7.115), the permeance of the test sample is thus obtained. To this end, Grimes et al. make use of a 100 nm thick permalloy film, having negligible value of the imaginary permeability up to a few hundred MHz [7.169]. Lack of suitable reference samples limits the high-frequency operations of this device, together with the dimensional

7.3 AC MEASUREMENTS

459

resonances intrinsically associated with the fixture used. Notice that if the sensing coils are successively aligned along three orthogonal directions, while obviously keeping the field in the direction dictated by the drive sheet, we can additionally measure the off-diagonal components of the permeability tensor. These can be important in anisotropic samples, depending on the direction of the applied field with respect to the easy axes, as illustrated by Grimes and Prodan in amorphous Co55Zr45 thin films [7.170]. The need to achieve the GHz range in the characterization of thin magnetic films, instigated by their increasing applications in UHF devices (for example, mobile phone handsets), has prompted both further development of the just described pickup coil methods and adaptation to the thin-film geometry of the TEM transmission line techniques solidly assessed with ferrites. Good progress towards extended frequency permeance meters of the pickup coil type has been achieved in the last decade at the Tohoku University laboratory in Sendai [7.171-7.173]. An example of a permeance measuring jig with 1 MHz-3.5 GHz capabilities developed in this laboratory is shown in Fig. 7.63. The measuring jig is made of a pair of current-carrying driving plates, shorted at the end and connected via an SMA connector to the reflection port of a network analyzer. Its dimensions are such as to provide 50 12 matching. In addition, it is sufficiently short to avoid higher-order resonating modes in the GHz range (•/2 = 43 mm at 3.5 GHz). The pickup coil, which is connected to the receiver port of the network analyzer, is especially designed to be insensitive to the electric fields. As discussed in detail in Ref. [7.172], the pickup loop, placed along the jig axis (that is, with surface perpendicular to the magnetic field) is realized by means of a central conductor and outer conductors. The latter provide shielding against the electric field and are connected to the ground plane of the driving coil near the termination. The sample is placed inside the loop (typical opening size 6-7 x 1.5 mm 2) in-plane with the field, the orientation in the plane typically dictated by the existence of uniaxial anisotropy. A reference measurement is required for calibration. This may be accomplished by saturating the sample along the easy axis by means of a strong DC field HDC. With the radiofrequency field H perpendicular to the DC field, the magnetization process can only occur by rotation of the magnetic moments and the associated permeability is easily obtained by equating the torque provided by H with the resisting torque due to HDC and the anisotropy field Hk. The low-frequency permeability is calculated as/z = Js/Hk + HDC, with H k = 2Ku/Js, if Ku is the anisotropy constant. By increasing the value of HDC, the permeability value is decreased and the ferromagnetic resonance frequency oJ0 = ~/et-~(Hk + HDC) is increased

460

CHAPTER 7 Characterization of Soft Magnetic Materials Pickup coil 1.4 mm x 7 mm

Sample.

Port 2

H

~'" Port 1 x

(a) Shorted termination f = 3 GHz

< X

"

/ -

[

I

P i c k u p coil

,

,

I

(b)

FIGURE 7.63 (a) Pickup coil type permeance meter developed by Yamaguchi et al. [7.173] for magnetic film testing in the frequency range 1-3.5 MHz. The drive parallel plates are terminated with a short circuit and are 50 f~ matched to the reflecting port of a network analyzer. The standing-wave generated magnetic field amplitude varies appreciably with distance from the termination only on approaching the GHz range, as shown in (b) for f -- 3 GHz (A/4 = 25 mm). The pickup coil, which is matched to the receiver port of the network analyzer, is realized by means of a central conductor and outer conductors. The latter provide shielding against the electric field and are connected to the ground plane of the driving coil near the termination.

in a predictable way. These behaviors can be used to validate the calibration procedure [7.173]. The application of the conventional one-port reflection m e t h o d to the characterization of magnetic films, m a k i n g use of Eq. (7.93) on a shorted transmission line loaded by the sample u n d e r test has been d e m o n s t r a t e d up to a few GHz by a n u m b e r of authors [7.174, 7.175]. The procedure in essence is no different from the previously discussed m e t h o d of characterizing a toroidal ferrite specimen placed at the bottom of a shorted line, but for the sensitivities required and the obvious use of a microstrip line instead of a coaxial line. The termination is in the shape of

7.3 AC MEASUREMENTS

461

a loop, similar to but smaller than the one m a d e with the drive plates in Fig. 7.63 (width w ~ 5 - 1 0 ram, length l --- 10 mm). At a frequency f = 2 GHz, these dimensions are still safely lower than ~/4. With a radiofrequency current i circulating in the loop, the generated field, directed along the loop axis, is H = k(i/w), with k -< 1 and the flux additionally linked with the loop after introduction of a film sample of cross-sectional area As is z ~ = AsJ = As/z0(/Zr - 1)H. This brings about a variation of the coil impedance AZ -- joJ/z0(/zr - 1)k(As/w), with complex relative permeability/zr =/Zr - j/z", which is then determined by difference of a double m e a s u r e m e n t with e m p t y and loaded fixtures. Again, calibration with a reference sample amenable to good theoretical prediction is recomm e n d e d . Korenivski et al. [7.174] take a YIG thin film as a model reference sample because of its near-zero internal anisotropy (/z = Js/HDc), lack of e d d y current effects, and n a r r o w ferromagnetic resonance line.

Co-Nb-Zr thickness = O.137 #m

i \ I

1000

= ::t.

500

"r '

t

-'YA', f

~

I

I

\ -500

0

;';0'' 1'0'00'" d00'' dO0'" 2S'O0"";0'00 Frequency (MHz)

F I G U R E 7.64 Real and imaginary part of the initial permeability in a C o - N b - Z r

amorphous thin film sample (thickness 0.137 ~m, width 5.5 mm) measured upon the frequency range 1 MHz-3 GHz in two different laboratories with two different methods. Solid lines: measured at Tohoku University Laboratory with the two-port pickup coil type permeance meter shown in Fig. 7.63 [7.173]. Dashed lines: measured at CEA, using a one-port method with microstripline terminating in a loop loaded with the sample [7.175]. The permeability is measured with the radiofrequency field applied along a direction orthogonal to the easy axis (anisotropy field Hk = 4537 A/m) (adapted from Ref. [7.176].

462

CHAPTER 7 Characterization of Soft Magnetic Materials

Yamaguchi et al. have carried out a comparison of permeability measurements up to 3 GHz in a number of amorphous C o - N b - Z r and F e ~ - C o - M o - S i - B thin films by means of two independent methods: the two-port setup method illustrated in Fig. 7.63 and the approach based on the loading of a shorted microstripline [7.176]. The values found differ, on average, by + 15%, the major reason for the found discrepancies being attributed to the calibration procedure. Figure 7.64 illustrates a case where good agreement was found. It relates to a 0.14 ~m thick C o - N b - Z r film with anisotropy field Hk = 4537 A / m , characterized along the direction orthogonal to the easy axis.

References 7.1. IEC Standard Publication 60404-4, Methods of Measurement of the d.c. Magnetic Properties of Magnetically Soft Materials (Geneva: IEC Central Office, 1995). 7.2. R.L. Sanford and I.L. Cooter, "Basic magnetic quantities and the measurement of the magnetic properties of the materials," Monogr. NBS, 47 (1962), 1-36. 7.3. http: //www.magnet-physik.de 7.4. http://www.laboratorio.elettrofisico.com 7.5. A.E. Drake and C. Ager, "Investigation of the uniformity of the magnetic field and flux density in the NPL permeameter," NPL Report DES, 101 (1990), 1-21. 7.6. E. Genova, private communication. 7.7. C.G. Svala, "An improved, practical Burrows permeameter," IEEE Trans. Magn., 12 (1976), 816-818. 7.8. J.P. Barranger, "Very high temperature permeameter," J. Appl. Phys., 42 (1971), 1796-1797. 7.9. H.J. Price, M.H. Price, and K.J. Overshott, "The effect of coating on the power loss of amorphous ribbon toroids," IEEE Trans. Magn., 19 (1983), 1943-1945. 7.10. D.M. Natasingh, C.H. Smith, and A. Datta, "Effects of coatings on the soft magnetic properties of an iron-based amorphous alloy," IEEE Trans. Magn., 20 (1984), 1332-1334. 7.11. IEC Standard Publication 60404-2, Methods of Measurement of the Magnetic Properties of Electrical Steel Sheet and Strip by Means of the Epstein Frame (Geneva: IEC Central Office, 1996). 7.12. A.E. Drake, "Precise magnetic measurements on electrical sheet steel," EU Report No. EUR 10233, Brussels (1985). 7.13. J. Sievert, H. Ahlers, F. Fiorillo, L. Rocchino, M. Hall, and L. Henderson, "Magnetic measurements on electrical steels using Epstein and SST

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PTB-Bericht, E-74 (2001), 1-28. 7.14. A. Kedous-Lebouc and P. Brissonneau, "Magnetoelastic effects on practical properties of amorphous ribbons," IEEE Trans. Magn., 22 (1986), 439-441. 7.15. T. Yamamoto and Y. Ohya, "Single sheet tester for measuring core losses and permeability in a silicon steel sheet," IEEE Trans. Magn., 10 (1974), 157-159. 7.16. T. Nakata, N. Takahashi, K. Fujiwara, and M. Nakano, "Study of horizontaltype single sheet tester," J. Magn. Magn. Mater., 133 (1994), 416-418. 7.17. K. Matsubara, T. Nakata, N. Takaghashi, K. Fujiwara, and M. Nakano, "Effects of the overhang of a specimen on the accuracy of a single sheet tester," Phys. Scripta, 40 (1989), 529-531. 7.18. J. Sievert, M. Enokizono, and B.Ch. Woo, "Experimental studies on single sheet testers," Anales de Fisica, B-86 (1990), 102-104. 7.19. ASTM Publication A804/A804M-99, Standard Test Methods for Alternating-

Magnetic Properties of Materials at Power Frequencies Using Sheet-Type Test Specimens (West Conshohocken, PA: ASTM International, 1999). 7.20. ASTM Publication A932/A932M-01, Standard Test Methods for AlternatingCurrent Magnetic Properties of Amorphous Materials at Power Frequencies Using Wattmeter-Ammeter-Voltmeter Method with Sheet Specimens (West Conshohocken, PA: ASTM International, 2001). 7.21. T. Nakata, Y. Kawase, and M. Nakano, "Improvement of measuring accuracy of magnetic field strength in single sheet testers by using two H-coils," IEEE Trans. Magn., 23 (1987), 2596-2598. 7.22. H. Pfiitzner and P. Sch6nhuber, "On the problem of field detection for single sheet testers," IEEE Trans. Magn., 27 (1991), 778-785. 7.23. JIS Standard H 7152, (1996) "Methods of test for magnetic properties of amorphous metals using single sheet specimen". 7.24. J. Sievert and H. Ahlers, "Is the Epstein frame replaceable?," Anales de Fisica, B-86 (1990), 102-104. 7.25. M. Mikulec, V. Havli~ek, V. Wiglasz, and D. Cech, "Comparison of loss measurements on sheets and strips," J. Magn. Magn. Mater., 41 (1984), 223-226. 7.26. A. Nafalski, A.J. Moses, T. Meydan, and M.M. Abousetta, "Loss measurements on amorphous materials using a field-compensated single-strip tester," IEEE Trans. Magn., 25 (1989), 4287-4291. 7.27. A. Nafalski and A.J. Moses, "Loss measurements of amorphous materials using single-strip testers," Phys. Scripta, 40 (1989), 532-535. 7.28. T. Nakata, N. Takahashi, and Y. Kawase, "Factors affecting the accuracy of a single sheet tester using an H-coil," Proc. Seventh Int. Conf. Soft Magn. Mater., Blackpool, U.K., (1985), p. 49-51.

464

CHAPTER 7 Characterization of Soft Magnetic Materials

7.29. IEC Standard Publication 60404-3, Methods of Measurement of the Magnetic Properties of Magnetic Sheet and Strip by Means of a Single Sheet Tester (Geneva: IEC Central Office, 1992). 7.30. A.E. Drake and C. Ager, "Intercomparison of AC magnetic measurements on electrical sheet steels using a single sheet tester," EU Report No. EUR 12377 (Bruxelles: BCR, 1989). 7.31. J. Sievert, M. Binder, and L. Rahf, "On the reproducibility of single sheet testers: comparison of different measuring procedures and SST designs," Anales de Fisica, B, 86 (1990), 76-78. 7.32. R.S. Girgis, K. Gramm, J. Sievert, and M.G. Wickramasekara, "The single sheet tester. Its acceptance, reproducibility, and application issues on grainoriented steel," J. Phys. IV (France), 8-Pr2 (1998), 729-732. 7.33. J. Sievert, "The measurement of magnetic properties of electrical sheet steel: survey on methods and situation of standards," J. Magn. Magn. Mater., 2 1 5 216 (2000), 647-651. 7.34. A. Hubert and R. Sch/ifer, Magnetic Domains (Berlin: Springer, 1998), p. 184. 7.35. H.J. Williams, "Magnetic properties of single crystals of silicon iron," Phys. Rev., 52 (1937), 747-751. 7.36. L. N6el, "Les lois de l'aimantation et de la subdivision en domaines 616mentaires d'un monocristal de fer," J. Phys. Radium, 5 (1944), 241-251. 7.37. A. Hubert and R. Sch/ifer, Magnetic Domains (Berlin: Springer, 1998), p. 184. 7.38. S. Taguchi, T. Yamamoto, and A. Sakakura, "New grain-oriented silicon steel with high permeability ORIENTCORE HI-B," IEEE Trans. Magn., 10 (1974), 123-127. 7.39. E Fiorillo, L.R. Dupr6, C. Appino, and A.M. Rietto, "Comprehensive model of magnetization curve, hysteresis loops and losses in any direction in grainoriented Fe-Si," IEEE Trans. Magn., 38 (2002), 1467-1475. 7.40. T. Nakata, N. Takahashi, Y. Kawase, and M. Nakano, "Influence of lamination orientation and stacking on magnetic characteristics of grainoriented silicon steel laminations," IEEE Trans. Magn., 20 (1984), 1774-1776. 7.41. J. Sievert, H. Ahlers, M. Birkfeld, B. Cornut, F. Fiorillo, K.A. Hempel, T. Kochmann, A. Lebouc, T. Meydan, A. Moses, and A.M. Rietto, "European intercomparison of measurements of rotational power loss in electrical sheet steel," J. Magn. Magn. Mater., 160 (1996), 115-118. 7.42. Proc. Fifth Int. Workshop on Two-Dimensional Magnetization Problems (A. Kedous-Lebouc, ed., Les Ulis, France: EDP Sciences, 1998). 7.43. Proc. Sixth Int. Workshop on 1 and 2-Dimensional Magnetic Measurement and Testing (H. Pfiitzner, ed., Wien: Vienna University of Technology, 2001). 7.44. M. Enokizono, T. Suzuki, J. Sievert, and J. Xu, "Rotational power loss of silicon steel sheet," IEEE Trans. Magn., 26 (1990), 2562-2564.

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7.45. G. Loisos and A.J. Moses, "Critical evaluation and limitations of localized flux density measurements in electrical steels," IEEE Trans. Magn., 37 (2001), 2755-2757. 7.46. Y. Tamura, Y. Ishihara, and T. Todaka, "Measurement of magnetic characteristics of silicon steel in any direction by RSST and SST: method and relationship," J. Magn. Magn. Mater., 133 (1994), 382-385. 7.47. W. Salz, "A two-dimensional measuring equipment for electrical steel," IEEE Trans. Magn., 30 (1994), 1253-1257. 7.48. J.G. Zhu and V.S. Ramsden, "Two dimensional measurement of magnetic field and core loss using a square specimen tester," IEEE Trans. Magn., 29 (1993), 2995-2997. 7.49. D. Makaveev, M. von Rauch, M. De Wulf, and J. Melkebeek, "Accurate field strength measurement in rotational single sheet tester," J. Magn. Magn. Mater., 215-216 (2000), 673-676. 7.50. T. Nakata, N. Takahashi, K. Fujiwara, and M. Nakano, "Measurement of magnetic characteristics along arbitrary directions of grain-oriented silicon steel up to high flux densities," IEEE Trans. Magn., 29 (1993), 3544-3546. 7.51. S. Zouzou, A. Kedous-Lebouc, and P. Brissonneau, "Magnetic properties under unidirectional and rotational field," J. Magn. Magn. Mater., 112 (1992), 106-108. 7.52. A. Hasenzagl, B. Weiser, and H. Pftitzner, "Novel 3-phase excited single sheet tester for rotational magnetization," J. Magn. Magn. Mater., 160 (1996), 180-182. 7.53. F. Fiorillo and A.M. Rietto, "Extended induction range analysis of rotational losses in soft magnetic materials," IEEE Trans. Magn., 24 (1988), 1960-1962. 7.54. A.M. Rietto, private communication. 7.55. J. Xu and J. Sievert, "On the reproducibility, standardization aspects and error sources of the fieldmetric method for the determination of 2D magnetic properties of electrical sheet steel," in Proc. Fifth Int. Workshop on Two-Dimensional Magnetization Problems (A. Kedous-Lebouc, ed., Les Ulis, France: EDP Sciences, 1998), 43-54. 7.56. J. Sievert, "On measuring the magnetic properties of electrical sheet steel under rotational magnetization," J. Magn. Magn. Mater., 112 (1992), 50-57. 7.57. J.W. Strutt (Lord Rayleigh), "On the behaviour of iron and steel under the operation of feeble magnetic forces," Phil. Mag., 23 (1887), 225-245. 7.58. H. Ahlers, J. Lfidke, and J. Sievert, "Coercivity meter with earth's magnetic field compensation," J. Magn. Magn. Mater., 160 (1996), 187-188. 7.59. IEC Standard Publication 60404-7, Method of Measurement of the Coercivity of Magnetic Materials in an OpenMagnetic Circuit (Geneva: IEC Central Office, 1982).

466

CHAPTER 7 Characterization of Soft Magnetic Materials

7.60. A.E. Drake and A. Hartland, "A vibrating coil magnetometer for the determination of the magnetization coercive force of soft magnetic materials," J. Phys. E: Sci. Instr., 6 (1973), 901-902. 7.61. ASTM Publication A341/A341M-00, Standard Test Method for Direct Current

Magnetic Properties of Materials Using DC Permeameters and the Ballistic Test Methods (West Conshohocken, PA: ASTM International, 2000). 7.62. In the present context, as is usual the case with soft magnets tested at technical inductions, the terms "magnetic polarization" and "magnetic induction" and the corresponding time derivatives are interchangeable. There is no appreciable difference between B and J under practical circumstances. Distinction should be made, however, on the approach to saturation. For example, an Fe-Si non-oriented lamination is considered technically saturated for applied fields of the order of 5 x 104 A / m (see Fig. 7.20a). This provides a term/~0H = 0.063 T and under this rather extreme experimental condition the difference between B and J is around 3%. 7.63. P. Allia, M. Celasco, A. Ferro, A. Masoero, and A. Stepanescu, "Transverse closure domains and the behavior of the magnetization in grain-oriented polycrystalline magnetic sheets," J. Appl. Phys., 52 (1981), 1439-1447. 7.64. D.C. Jiles and D.L. Atherton, "Theory of ferromagnetic hysteresis," J. Magn. Magn. Mater., 61 (1986), 48-60. 7.65. A steel ship is magnetized from stem to stern under the action of the earth magnetic field. Its magnetization tends to increase with time, because the fluctuating mechanical stresses due to sea waves favor the progressive evolution of the system towards lower energy states, that is the anhysteretic curve. 7.66. A. Ferro and G.P. Soardo, "Directional ordering as a source of coercive force and asymmetry of the loop," IEEE Trans. Magn., 6 (1970), 110-115. 7.67. G. Ferrari, J.L. Porteseil, and R. Vergne, "Thermally activated motion of a 180~ Bloch wall in an Fe-Si single crystal," IEEE Trans. Magn., 14 (1978), 764-766. 7.68. G. Bertotti, F. Fiorillo, and M.P. Sassi, "Barkhausen noise and domain structure dynamics in Si-Fe at different points of the magnetization curve," J. Magn. Magn. Mater., 23 (1981), 136-148. 7.69. P. Mazzetti and P. Soardo, "Electronic hysteresigraph holds dB/dt constant," Rev. Sci. Instr., 37 (1966), 548-552. 7.70. W. Grosse-Nobis and W. Schoenfelder, "Stabilization field of a 180~wall in a frame type silicon-iron single crystal with solute carbon measured with controlled wall velocity," Physica B, 80 (1975), 407-420. 7.71. J.A. Baldwin, Jr., "A controlled-flux hysteresis loop tracer," Rev. Sci. Instr., 41 (1970), 468. 7.72. T.H. O'DeU, "A low frequency (M, B) loop tracer for amorphous ribbons," IEEE Trans. Magn., 17 (1981), 3364-3366.

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7.73. G. Hellmiss and L. Storm, "Movement of an individual Bloch wall in a single-crystal picture frame of silicon iron at very low velocities," IEEE Trans. Magn., 10 (1974), 36-39. 7.74. G. Birkelbach, K.A. Hempel, and F.J. Schulte, "Very low frequency magnetic hysteresis measurements with well-defined time dependence of the flux density," IEEE Trans. Magn., 22 (1986), 505-507. 7.75. G. Bertotti, E. Ferrara, F. Fiorillo, and M. Pasquale, "Loss measurements on amorphous alloys under sinusoidal and distorted induction waveform using a digital feedback technique," J. Appl. Phys., 73 (1993), 5375-5377. 7.76. K. Matsubara, N. Takahashi, K. Fujiwara, T. Nakata, M. Nakano, and H. Aoki, "Acceleration technique of waveform control for single sheet tester," IEEE Trans. Magn., 31 (1995), 3400-3402. 7.77. F. Fiorillo, C. Ragusa, and E. Barbisio, "Measurement and analysis of magnetic power losses with non-sinusoidal induction and minor loops," PTB-Bericht, E-81 (2003), 53-66. 7.78. A. Ferro, P. Mazzetti, and G. Montalenti, "Anomalous Barkhausen effect in ferromagnetic alloys," Appl. Phys. Lett., 7 (1965), 118-119. 7.79. G. Bertotti, "Physical interpretation of eddy current losses in ferromagnetic materials," J. Appl. Phys., 57 (1985), 2110-2126. 7.80. F. Brailsford, "Rotational hysteresis loss in electrical sheet steels," J. IEE, 83 (1938), 566-575. 7.81. J. Zbroszcyk, J. Drabecki, and B. Wyslocki, "Angular distribution of rotational hysteresis losses in Fe-3.25% Si single crystals with orientations (001) and (011)," IEEE Trans. Magn., 17 (1981), 1275-1282. 7.82. http://www.bipm.fr/BIPM-KCDB 7.83. G. Bertotti, Hysteresis in Magnetism (San Diego: Academic Press, 1998). 7.84. ASTM Publication A343-97, Standard Test Method for Alternating-Current

Magnetic Properties of Materials at Power Frequencies Using WattmeterAmmeter-Voltmeter Method and 25-cm Epstein Test Frame (West Conshohocken, PA: ASTM International, 1997). 7.85. ASTM Publication A912-93, Standard Test Method for Alternating-Current

Magnetic Properties of Amorphous Materials at Power Frequencies Using Wattmeter-Ammeter-Voltmeter Method with Toroidal Specimens (West Conshohocken, PA: ASTM International, 1993). 7.86. L. Rahf and J.D. Sievert, "Microprocessor-controlled test assembly for electrical steel sheets," IEEE Trans. Instr. Meas., 36 (1987), 847-850. 7.87. G. Birkelbach, J.V. Freeden, and K.A. Hempel, "Device for the automatic determination of the magnetic properties of electrical steel sheet," IEEE Trans. Magn., 20 (1984), 1711-1713.

468

CHAPTER 7 Characterization of Soft Magnetic Materials

7.88. M. De Wulf, L. Dupr6, and J. Melkebeek, "Real-time controlled arbitrary excitation for identification of electromagnetic properties of non-oriented steel," J. Phys. IV (France), 8-Pr2 (1998), 705-708. 7.89. http://www.brockhaus.net 7.90. H. Ahlers and J.D. Sievert, "Uncertainties of magnetic loss measurements, particularly in digital procedures," PTB-Mitteilungen, 94 (1984), 99-107. 7.91. H.J. Stanbury, "An industry-based standards system for the measurement of power loss and apparent power of electrical sheet steel," Proc. IEE, 132A (1985), 129-132. 7.92. H. Ahlers, L. Rahf, H. Hartwig, and J.D. Sievert, "Programmable digital two-channel function generator for testing power meters," IEEE Trans. Instr. Meas., 34 (1985), 231-234. 7.93. M. De Wulf and J. Melkebeek, "On the advantages and drawbacks of using digital acquisition systems for the determination of magnetic properties of electrical steel sheet and strip," J. Magn. Magn. Mater., 196-197 (1999), 940-942. 7.94. A.E. Drake, "Traceable magnetic measurements," J. Magn. Magn. Mater., 133 (1994), 371-376. 7.95. G. Bertotti, and F. Fiorillo, "Crystalline Fe-Si, Fe-A1, Fe-Si-A1 alloys," in Landolt-B6rnstein New Series, vol. 19 "Magnetic properties of metals," Sub-vol. il, (Berlin Heidelberg: Springer Verlag, 1994), p. 33. 7.96. K.H. Schmidt and H. Huneus, "Determination of the density of electrical steel made from iron-silicon alloys with small aluminium content," Techn. Messen, 48 (1981), 375-379. 7.97. L.J. Van der Pauw, "A Method of Measuring Specific Resistivity and Hall Effect of Discs of Arbitrary Shape," Philips Res. Rep., 13 (1958), 1-9. 7.98. IEC Standard Publication 60404-13, Methods of measurement of density, resistivity and stacking factor of electrical steel sheet and strip (Geneva: IEC Central Office, 1995). 7.99. J. Sievert, "Determination of the density of magnetic sheet steel using strip and sheet specimens," J. Magn. Magn. Mater., 133 (1994), 390-392. 7.100. G. Bertotti, "General properties of power losses in soft ferromagnetic materials," IEEE Trans. Magn., 24 (1988), 621-630. 7.101. H. Ahlers and J. Liidke, "The uncertainties of magnetic properties measurements of electrical sheet steel," J. Magn. Magn. Mater., 215-216 (2000), 711-713. 7.102. Y. Ishihara, A. Isozumi, T. Todaka, and T. Nakata, "Comparison of two averaging methods for improving the measurement accuracy of power loss," J. Magn. Magn. Mater., 215-216 (2000), 696-699.

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7.103. F. Fiorillo and A. Novikov, "An improved approach to power losses in magnetic laminations under nonsinusoidal induction waveform," IEEE Trans. Magn., 26 (1990), 2904-2910. 7.104. The IEC Standards do not consider any longer the correction for the effect of distortion of the secondary voltage, which is recommended to have a form factor FF = 1.111 +_ 1%. The ASTM Standards A932/A932M01 (amorphous strips) [7.20] and A912-93 (amorphous ribbons wound as toroids) [7.85] suggest a procedure for the approximate correction of the P value measured with distorted u2(t). By denoting as Us = x/2vrN2AJpf the r.m.s, value of the desired voltage, the dynamic component of the measured power loss is multiplied by a factor K = (Us/U2)2~ in order to recover the corresponding quantity under sinusoidal induction waveform. This correction is based, however, on the assumption that the energy loss per cycle W = P/f linearly depends on frequency, which is not verified by the experiments (see Fig. 7.29), and it should be critically applied. 7.105. P. Beckley, "Industrial magnetic measurements," J. Magn. Magn. Mater., 215-216 (2000), 664-668. 7.106. H. Iranmanesh, B. Tahouri, A.J. Moses, and P. Beckley, "A computerised Rogowski-Chattock potentiometer (RCP) compensated on-line power loss measuring system for use on grain-oriented electrical steel production lines," J. Magn. Magn. Mater., 112 (1992), 99-102. 7.107. V. Havlicek and M. Mikulec, "On-line testing device using the compensation method," Phys. Scripta, 39 (1989), 513-515. 7.108. P. Beckley, "Continuous power loss measurement with and against the rolling direction of electrical steel strip using nonenwrapping magnetizers," Proc. IEE, 130A (1983), 313-321. 7.109. G.S. Canright and R.L. Anderson, "Calorimeter for measuring ac magnetic losses in small samples," J. Appl. Phys., 53 (1982), 8269-8271. 7.110. F. Brailsford and C.G. Bradshaw, "Iron losses at high magnetic flux densities in electrical sheet steels," IEE Proc., 102 (1955), 463475. 7.111. K.J. Overshott, I. Preece, and J.E. Thompson, "Magnetic properties of grain oriented iron losses at high magnetic flux densities in electrical sheet steels," Proc. IEE, 115 (1968), 1840-1845. 7.112. R.S. Albir and A.J. Moses, "Improved DC bridge method employed to measure local power loss in electrical steels and amorphous materials," J. Magn. Magn. Mater., 83 (1990), 553-554. 7.113. H.R. Boesch, "Accurate measurement of the DC magnetization of steel using simple cylindrical rods," Proc. Second Int. Conf. Soft Magn. Mater. (U.K. Cardiff, 1975), 280-283.

470

CHAPTER 7 Characterization of Soft Magnetic Materials

7.114. N. Derebasi, T. Meydan, M. Goktepe, and M.H. So, "Computerised DC bridge method of thermistor measurement of localised power loss in magnetic materials," IEEE Trans. Magn., 28 (1992), 2467-2469. 7.115. A. Moghaddam and A.J. Moses, "Localised power loss measurements using remote sensors," IEEE Trans. Magn., 29 (1993), 2998-3000. 7.116. T. Yamaguchi, K. Senda, M. Ishida, K. Sato, A. Honda, and T. Yamamoto, "Theoretical analysis of localized magnetic flux measurement by needle probe," J. Phys. IV (France), 8-Pr2 (1998), 717-720. 7.117. K. Senda, M. Kurosawa, M. Ishida, M. Komatsubara, and T. Yamaguchi, "Local magnetic properties in grain-oriented electrical steel measured by modified needle probe method," J. Magn. Magn. Mater., 215-216 (2000), 136-139. 7.118. M. Enokinozono, I. Tanabe, and T. Kubota, "Local distribution on magnetic properties in grain-oriented silicon steel sheet," J. Appl. Phys., 83 (1998), 6486-6488. 7.119. G. Krismanic, H. Pfiitzner, and N. Baumgartinger, "A hand-held sensor for analyses of local distributions of magnetic fields and losses," J. Magn. Magn. Mater., 215-216 (2000), 720-722. 7.120. C. Appino, F. Fiorillo, and A.M. Rietto, "The energy loss components under alternating, elliptical, and circular flux in non-oriented alloys,"

in Proc. Fifth Int. Workshop on Two-Dimensional Magnetization Problems (A. Kedous-Lebouc, ed., Les Ulis, France: EDP Sciences, 1998), 55-61. 7.121. N. Nencib, A. Kedous-Lebouc, and B. Cornut, "2D analysis of rotational loss tester," IEEE Trans. Magn., 31 (1995), 3388-3390. 7.122. A.J. Moses and B. Thomas, "Measurement of rotating flux in silicon iron laminations," IEEE Trans. Magn., 9 (1973), 651-654. 7.123. T. Sasaki, M. Imamura, S. Takada, and Y. Suzuki, "Measurement of rotational power losses in silicon-iron sheets using wattmeter method," IEEE Trans. Magn., 21 (1985), 1918-1920. 7.124. T. Yamaguchi and K. Narita, "Rotational power loss in commercial siliconiron laminations," Electr. Eng. Jpn., 96 (1976), 15-21. 7.125. D. Makaveev, J. Maes, and J. Melkebeek, "Controlled circular magnetization of electrical steel in rotational single sheet testers," IEEE Trans. Magn., 37 (2001), 2740-2742. 7.126. L.R. Dupr6, F. Fiorillo, C. Appino, A.M. Rietto, and J. Melkebeek, "Rotational loss separation in grain-oriented Fe-Si," J. Appl. Phys., 87 (2000), 6511-6513. 7.127. W. Brix, "Measurements of the rotational power loss in 3% silicon-iron at various frequencies using a torque magnetometer," J. Magn. Magn. Mater., 26 (1982), 193-195.

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7.128. W. Salz, "A two-dimensional measuring equipment for electrical steel," IEEE Trans. Magn., 30 (1994), 1253-1257. 7.129. F. Fiorillo and A.M. Rietto, "Rotational versus alternating hysteresis losses in nonoriented soft magnetic laminations," J. Appl. Phys., 73 (1993), 6615-6617. 7.130. J. Sievert, J. Xu, L. Rahf, M. Enokizono, and H. Ahlers, "Studies on the rotational power loss measurement problem," Anales de Fisica, B-86 (1990), 35-37. 7.131. W. Salz and K.H. Hempel, "Which field sensors are suitable for a rotating flux apparatus?," PTB-Bericht, E-43 (1992), 117-126. 7.132. K. Narita and T. Yamaguchi, "Rotational hysteresis loss in silicon-iron single crystal with (001) surfaces," IEEE Trans. Magn., 10 (1974), 165-167. 7.133. J.M. Kelly, "New technique for measuring rotational hysteresis in ferromagnetic materials," Rev. Sci. Instr., 28 (1957), 1038-1040. 7.134. A. Cecchetti, G. Ferrari, F. Masoli, and G.P. Soardo, "Rotational power loss in 3% Si-Fe as a function of frequency," IEEE Trans. Magn., 14 (1978), 356-358. 7.135. G. Bottoni, D. Candolfo, A. Cecchetti, and F. Masoli, "Rotational and alternating loss in amorphous ferromagnet," J. Magn. Magn. Mater., 26 (1982), 130-132. 7.136. G. Bottoni, D. Candolfo, A. Cecchetti, and F. Masoli, "Ratio of the rotational loss to hysteresis loss in ferrimagnetic powders," IEEE Trans. Magn., 10 (1974), 317-320. 7.137. H. B611,Handbook of Soft Magnetic Materials (London: Heyden, 1978), p. 85. 7.138. H. Seki, S. Takemori, and T. Sato, "Measurement of losses and saturated permeability of saturable reactor cores under submicrosecond saturation," IEEE Trans. Magn., 32 (1996), 5251-5255. 7.139. M. Birkfeld and K.A. Hempel, "A device for measuring the magnetic properties of ring specimens at high frequencies," J. Magn. Magn. Mater., 133 (1994), 393-395. 7.140. H. Ahlers, A. Nafalski, L. Rahf, S. Siebert, J. Sievert, and D. Son, "The measurement of magnetic properties of amorphous strips at higher frequencies using a yoke system," J. Magn. Magn. Mater., 112 (1992), 88-90. 7.141. IEC Standard Publication 60404-10, Methods of Measurement of Magnetic Properties of Magnetic Steel Sheet and Strip at Medium Frequencies (Geneva: IEC Central Office, 1988). 7.142. ASTM Publication A348/A348M-00, Standard Test Methods for Alternating-

Current Magnetic Properties of Materials Using the Wattmeter-AmmeterVoltmeter Method, 100 to 10 O00Hz and 25cm Epstein Frame (West Conshohocken, PA: ASTM International, 2000).

472

CHAPTER 7 Characterization of Soft Magnetic Materials

7.143. L. Brugel, P. Brissonneau, A. Kedous, and J.C. Perrier, "Effects of the Epstein frame imperfections on the accuracy of power measurements at medium frequencies," J. Magn. Magn. Mater., 41 (1984), 230-232. 7.144. The phase shift between B(t) (i.e. l(t)) and H(t) was previously defined as ~H. In the specific context of this discussion it is called ~ according to the conventional notation adopted in the literature. 7.145. IEC Standard Publication 60404-6, "Methods of measurement of the magnetic properties of magnetically soft metallic and powder materials at frequencies in the range 20 Hz to 200 kHz by the use of ring specimens," (Geneva: IEC Central Office, 2003). 7.146. IEC Standard Publication 62044-2, Cores Made of Magnetically Soft Ferrites. Measuring Methods. Part2: Magnetic Properties at Low Excitation Level, (Geneva: IEC Central Office, 2004). 7.147. M. Honda, The hnpedance Measurement Handbook, (Yokogawa-HewlettPackard, 1989). 7.148. R.K. Moore, Traveling-Wave Engineering (New York: McGraw-Hill, 1960). 7.149. S. Ramo, J.R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (New York: Wiley, 1965). 7.150. K.C. Gupta, R. Garg, and I.J. Bahl, Microstrip Lines and Slotlines (Norwood, MA: Artech House, 1979). 7.151. HP 8753E RF Vector Analyzer, User's Guide (Hewlett-Packard Company; 1998). 7.152. F. Gardiol, Microstrip Circuits (New York: Wiley, 1994). 7.153. R.B. Goldfarb and H.E. Bussey, "Method for measuring complex permeability at radio frequencies," Rev. Sci. Instr., 58 (1987), 624-627. 7.154. V. Cagan and M. Guyot, "Fast and convenient technique for broadband measurement of the complex initial permeability of ferrimagnets," IEEE Trans. Magn., 20 (1984), 1372-1374. 7.155. H.J. Lindenhovius and J.C. van der Breggen, "The measurement of permeability and magnetic losses of non-conducting ferromagnetic materials at high frequencies," Philips Res. Rep., 3 (1948), 37-45. 7.156. R.D. Harrington, R.C. Powell, and P.H. Haas, "A re-entrant cavity for measurement of complex permeability in the very high frequency region," I. Res. NBS, 56 (1956), 129-134. 7.157. C.A. Hoer and R.D. Harrington, "Parallel reversible permeability measurement techniques from 50 kc/s to 3 Gc/s," J. Res. NBS, 67C, (1963), 259-265. 7.158. A.M. Nicolson and G.F. Ross, "Measurement of the intrinsic properties of materials by time-domain techniques," IEEE Trans. Instr. Meas., 19 (1970), 377-382. 7.159. W.B. Weir, "Automatic measurement of complex dielectric constant and permeability at microwave frequencies," Proc. IEEE., 20 (1974), 33-36.

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7.160. W. Barry, "A broad-band, automated, stripline technique for the simultaneous measurement of complex permittivity and permeability," IEEE Trans. Microwave Th. Tech., 34 (1986), 80-84. 7.161. F.G. Brockman, P.H. Dowling, and W.G. Steneck, "Dimensional effects resulting from a high dielectric constant found in a ferromagnetic ferrite," Phys. Rev., 77 (1950), 85-93. 7.162. A. Fessant, J. Gieraltowski, J. Loa6c, and H. LeGall, "Strip-line method for measuring the complex permeability of magnetic materials," J. Magn. Magn. Mater., 83 (1990), 557-558. 7.163. A. Fessant, J. Gieraltowski, J. Loa6c, H. LeGall, and A. Rakii, "Influence of in-plane anisotropy and eddy current on the frequency spectra of the complex permeability of amorphous CoZr thin films," IEEE Trans. Magn., 29 (1993), 82-87. 7.164. M. Knobel and K.R. Pirota, "Giant magnetoimpedance: concepts and recent progress," J. Magn. Magn. Mater., 242-245 (2002), 33-40. 7.165. D. M6nard, M. Britel, P. Ciureanu, A. Yelon, V.P. Paramonov, A.S. Antonov, P. Rudkowski, and J.O. Str6m-Olsen, "High frequency impedance spectra of soft amorphous fibers," J. Appl. Phys., 81 (1997), 4032-4034. 7.166. L. Brunetti, M. Coisson, P. Tiberto, and F. Vinai, "Magneto-impedance measurements in amorphous Co-based magnetic wires at high frequency," J. Magn. Magn. Mater., 249 (2002), 310-314. 7.167. L.G.C. Melo, P. Ciureanu, and A. Yelon, "Permeability deduced from impedance measurements at microwave frequencies," J. Magn. Magn. Mater., 249 (2002), 337-341. 7.168. P.A. Calcagno and D.A. Thompson, "Semiautomatic permeance tester for thick magnetic films," Rev. Sci. h~str., 46 (1975), 904-908. 7.169. C.G. Grimes, P.L. Trouilloud, and R.M. Walser, "A new swept-frequency permeameter for measuring the complex permeability og thin magnetic films," IEEE Trans. Magn., 24 (1988), 603-610. 7.170. C.G. Grimes and J.V. Prodan, "Swept frequency permeameters for measuring the complex, off-diagonal permeability tensor components of anisotropic, thin magnetic films," J. Appl. Phys., 73 (1993), 6989-6991. 7.171. T. Kawazu, M. Yamaguchi, and K.I. Arai, "A new microstrip pickup coil for thin-film peremeance meters," IEEE Trans. Magn., 30 (1994), 4641-4643. 7.172. M. Yamaguchi, S. Yabukami, and K.I. Arai, "A new 1 MHz-2 GHz permeance meter for metallic thin films," IEEE Trans. Magn., 33 (1997), 3619-3621. 7.173. M. Yamaguchi, S. Yabukami, and K.I. Arai, "Development of multilayer planar flux sensing coil and its application to 1 MHz-3.5 GHz thin film permeance meter," Sensors and Actuators, 81 (2000), 212-215.

474

CHAPTER 7 Characterization of Soft Magnetic Materials

7.174. V. Korenivski, R.B. van Dover, P.M. Mankievich, Z.X. Ma, A.J. Becker, P.A. Polakos, and V.J. Fratello, "A method to measure the complex permeability of thin films at ultra-high frequencies," IEEE Trans. Magn., 32 (1996), 4905-4907. 7.175. D. Pain, M. Ledieu, O. Acher, A.L. Adenot, and F. Duverger, "An improved permeameter for thin film measurements up to 6 GHz," J. Appl. Phys., 85 (1999), 5151-5153. 7.176. M. Yamaguchi, O. Acher, Y. Miyazawa, K.I. Arai, and M. Ledieu, "Cross measurements of thin-film permeability up to the UHF range," J. Magn. Magn. Mater., 242-245 (2002), 970-972.

CHAPTER 8

Characterization of Hard Magnets

Through the forces they exert or they are subjected to, permanent magnets offer the most intuitive demonstration of magnetic phenomena. The history of magnetic measurements is naturally linked to the various methods by which such forces have been exploited, either to sense magnetic fields or to characteriz:e the material itself. The old magnetometer method, where a suspended or pivoted needle-shaped permanent magnet of total magnetic moment m, immersed in a uniform and precisely known magnetic field H, takes an orientation resulting from the equilibrium of the torque 9 = m x/~0H applied by the field and a counteracting torque applied by a spring or a torsion wire, can provide an accurate measurement of the volume-averaged magnetization of the specimen. By letting such a specimen oscillate about its equilibrium position, one obtains a torsion pendulum, whose natural resonance frequency is directly related to the value m and the moment of inertia of the pendulum [8.1]. If the test sample, assimilated to a dipole, is instead placed in a non-uniform field, it will be subjected to a translatory force, F -- V(m./~0H)~ whose measurement in a precisely defined field gradient will equally provide the value m of the magnetic moment. The popular Faraday and Gouy methods are based on this principle. The measurement of the force is accomplished in them with the use of an electronic balance. Using forces and torques, we can in principle perform absolute measurements, but the difficulty of accurately calibrating the field gradient often calls for the use of reference samples. For all their sensitivity and accuracy, the force methods require rather cumbersome apparatus and are not frequently used to characterize permanent magnets. Torque methods remain important for the determination of the magnetic anisotropy, while the Faraday and Gouy methods are mostly applied to susceptibility measurements in weakly magnetic and paramagnetic materials. Here, we shall touch upon the subject of force 475

476

CHAPTER 8 Characterization of Hard Magnets

magnetometers only for the special case of alternating gradient force magnetometers (also called vibrating reed magnetometers). These instruments work on the principle of applying an alternating force to the sample by means of an AC field gradient. The sample is cemented to some kind of support, whose oscillations can be brought to resonance with appropriate choice of the AC field frequency and consequently amplified by the Q factor of the resonating device. In this chapter, we shall chiefly discuss the inductive methods of characterization of permanent magnets, including particulate recording media. We will therefore talk about general characterization of the materials, excluding, in particular, specific test methods devoted to the measurement of recording system performances, which are the subject of specialized literature [8.2]. We shall focus, as previously with soft magnets, on the hysteresis loop and its parameters. We shall distinguish between the classical fluxmetric measurements, prevalently employing closed magnetic circuits and exploiting the methods for detection and conditioning of the field and flux derivative signals already described in the previous chapter, and the open sample methods based on the determination of the magnetic moment of the sample. The latter will still be prevalently of inductive type. We will consider, in particular, the vibrating sample and the extraction method magnetometers, where a signal is induced in one or more search coils by means of fast sample motion, and the pulsed field magnetometer, where the sample is fixed and the field is applied in a burst-like fashion. The methods based on closed magnetic circuit measurement are absolute in nature, while reference samples are generally employed for calibration purposes when dealing with open circuits. These samples are, in general, extra-pure Ni spheres of diameter 2-3 mm with certified value of the mass and the magnetization value in a definite field-strength interval in the region of the approach to saturation. Ni standards, which are preferred to Fe standards because of their chemical stability, are available at national metrological laboratories suitably equipped for the absolute determination of their magnetic moment. Secondary standards can be obtained using a magnetometer calibrated against a primary standard (for example the 99.999% purity NIST 772a Ni sphere [8.3]) and are often provided by manufacturers of magnetic instrumentation. Reference Ni disk samples (diameter 6 mm) have also recently been made available at NIST. The disk geometry is closer to actual samples used for testing of recording media (disks and tapes). Figure 8.1 shows examples of high field polarization behavior of reference Ni spheres, as measured at PTB [8.4] and NIST [8.5] at room temperature. The PTB result is obtained by means of a vibrating sample magnetometer (VSM) calibrated against

CHARACTERIZATION OF HARD MAGNETS

41'/

0.624 0.622 1

0.620

2

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0.618

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0.616 0.614

Pure Ni sphere 0.612

.

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.

.

.

i

2000

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i

4000

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6000

Ha (kNm) F I G U R E 8.1 Polarization in extra-pure annealed Ni spherical samples as a function of the applied field in the region of approach to saturation. (1) Eckert and Sievert [8.4] at 23 ~ by means of a vibrating sample magnetometer calibrated against absolute fluxmetric measurements [8.6]. Sphere diameter 3.1 mm. (2) Shull et al. by means of a Faraday method [8.5]. Sphere diameter 2.382 mm. The dashed line is a plot of the empirical fitting function J = 0.61574(1 + 0.0026 In(Ha/398)), where Ha is expressed in kA/m. (3) Adapted from the measurements of Crangle and Goodman on 4 mm x 2 mm ellipsoidal samples [8.7].

absolute m e a s u r e m e n t s m a d e on c o m p a n i o n cylindrical samples with a fluxmetric setup using an electromagnet [8.6]. In the NIST measurement, the absolute value of the magnetic m o m e n t m of a Ni sphere of diameter 2.382 m m and mass 63.16 m g is obtained by means of a Faraday method, where the magnetic field gradient is d e t e r m i n e d using three i n d e p e n d e n t techniques. The e x p a n d e d relative m e a s u r i n g uncertainty is U = 3 x 10 -3 and the t e m p e r a t u r e d e p e n d e n c e is ( l / m ) x (dm/dT) = 5 x 10 -4 K -1. Older absolute m e a s u r e m e n t s by Crangle and G o o d m a n , m a d e on ellipsoidal samples by means of a F a r a d a y m e t h o d (4 m m x 2 mm), are also reported [8.7]. Notice that the conventional saturation polarization value Js is a zero-field extrapolated quantity and it is not the most appropriate value to be taken for the calibration of the m a g n e t o m e t e r s because this operation is m a d e at high fields. Notice also that a p e r m a n e n t m a g n e t secondary standard, calibrated

478

CHAPTER 8 Characterization of Hard Magnets

against the Ni standard, could possibly be used to calibrate search coils, but long-term stability and effects of temperature must be carefully checked and frequent re-calibration might be required. When we talk of hard magnet testing, we implicitly assume that it is accomplished under DC conditions. In practice, time effects are often important, even if we look for quasi-static measuring conditions, and their role should consequently be evaluated. We find again, somewhat amplified, the same ambiguity in the practical recognition of rateindependent hysteresis as seen in soft magnets. Materials like Alnicos and sintered N d - F e - B based magnets, which have metallic conductivity, can exhibit significant time effects due to eddy currents. Since very high field values must be applied to the latter material if the saturated state is to be approached before determining the hysteresis loop, the pulsed field method can be envisaged for their characterization. In this case, however, complete loop traversing is accomplished in a few milliseconds only and specific corrections of the raw data are required in order to retrieve the static loop. On the other hand, a very low magnetization rate might not represent a definite solution to time effects because of magnetic viscosity, i.e. the spontaneous variation of the magnetization induced by thermally activated microscopic magnetization processes (thermal fluctuation aftereffect). This variation, which depends on the actual magnetic state of the material (it is highest on approaching the coercive field), shows logarithmic time dependence and adds to the magnetization change exerted by the external field. This amounts to measuring a coercive field Hc that decreases by increasing the time scale of the experiment. The gamma ferric oxide ~-Fe203 particulate recording media, for example, while being immune to eddy currents, display relevant magnetic viscosity effects. In this case, one might usefully resort to predictive formulations for the time dependence of Hc [8.8]. Also, the measurements of the N d Fe-B magnets suffer from interference from magnetic viscosity [8.9]. One finds, in particular, that with the hysteresisgraph and VSM methods the variation of the measured Hc value over the range of practical measuring times may be of the order of 1-2% in N d - F e - B magnets [8.10]. Full AC measurements are obviously not considered with hard magnets (the continuous-wave and pulsated techniques for read-write efficiency testing of recording media being merely oriented to determine the device performances), but reversible susceptibility can indeed be usefully investigated in permanent magnets. Apart from the scale factor on the amplitude of the AC field and of the possible bias DC field, susceptibility measurements can convey as useful information on the magnetization process as in soft magnets: real and imaginary parts, domain wall displacements and rotations, magnetic phase transitions, etc.

CHARACTERIZATION OF HARD MAGNETS

479

An interesting example of information on the microscopic magnetization processes given by AC susceptibility measurements has been given by Pareti and Turilli with their determination of the transverse susceptibility x(H) as a function of the longitudinal effective field H in Ba ferrites [8.11]. They verify that x(H) exhibits the peaked profile predicted to occur from the Stoner-Wohlfarth theory of single domain particles, the peaks occurring in correspondence with the anisotropy field and the coercive field values, which are then obtained with this measurement. Permanent magnets can have significant temperature-dependent properties and measurements should be performed under tight control of temperature. For example, the coercive field in Ba and Sr ferrites has positive temperature coefficient ac = (dHc/dT)/Hc, of the order of 0.30.5%/~ and saturation magnetization coefficient C~M---(dMs/dT)/Ms around -0.2%/~ The same quantities in N d - F e - B compounds are of the order of - 0 . 6 and -0.1%/~ respectively. It turns out that the customarily allowed fluctuation of + 5 ~ around the room temperature (23 ~ in technical measurements [8.12] may involve too high a measuring uncertainty or may require awkward corrections. In the characterization of permanent magnets, our basic technical objective is to achieve the demagnetization curve B(H) and possibly the associated J(H) curve in the second quadrant. From them, we obtain the base parameters: remanent induction Br = Jr, coercive fields Hcl and HoB, and energy product (BH) (Fig. 3.2). Full loop properties are in any case desired if we go beyond routine material testing. With recording media, we are also interested in achieving parameters specifically connected with the recording process. Squareness is one such parameter. It is defined as S* -- 1 - (Mr/XoHc), where X0 = (dM/dH)H=Hc is the differential susceptibility measured at the M -- 0. S * varies from I for a perfectly square loop (X0 = oo) to 0 for X0 = Mr/Hc. This definition is appropriate for recording media, where a value S * close to 1 indicates spatial sharpness of the magnetization transition, and it differs from the usual definition of sharpness S*--Mr~Ms. The remanence magnetization curves shown in Fig. 8.2 are also conveniently employed in the characterization of recording media, where, for instance, the initial curve is paralleled by a remanence curve (Fig. 8.2a). Any point of it is obtained by bringing the material from the demagnetized state to a point (M, H) of the initial curve, releasing the field and recording the resulting remanent magnetization value Mr(H). This curve, which saturates at the remanence value Mr(Hp) , provides the magnetization change irreversibly accomplished by means of particle switching. The points of the reverse remanence curve are similarly determined by releasing the field after having saturated

480

CHAPTER 8 Characterization of Hard Magnets M (/-/)

/

Mr(H.) M r (P)

\

(a)

H-

.....

M(H)

M

!

,'/ H

\

\,'

I1' / '/

,'j/

'~

II !

/i I/"

/,' /' /I /,

Ho

\,

__

H~\

/,'

/!'

~ \i

ii !

,-,

\

(b) FIGURE 8.2 (a) Initial magnetization curve M(H) and remanence curve Mr(H ) Each point p/ of the latter is obtained by recoiling with the field from the corresponding point P on the initial curve to the remanent magnetization value Mr(P) and taking the coordinate (H(P), Mr (P)). (b) The ensemble of remanence points associated with the hysteresis loop M(H) generate the remanence loop Mr(H) (dashed line). The material can be brought to the demagnetized state by proceeding along the reverse curve and releasing the field after the appropriate value Hcr (remanence coercivity) is attained (adapted from Ref. [8.2]).

8.1 CLOSED MAGNETIC CIRCUIT MEASUREMENTS

481

the material and having brought the material to the desired (Me H) points on the second or third quadrant. The remanence loop (Mr~ H) is eventually obtained, as shown by the dashed curve in Fig. 8.2b. Of interest for analog recording is the modified anhysteretic magnetization curve. It differs from the ideal anhysteretic curve discussed in Section 7.2.2 in that both the alternating field and the bias DC field are simultaneously reduced to zero at the same rate, in analogy with the process involved with the socalled AC bias recording.

8.1 CLOSED MAGNETIC CIRCUIT MEASUREMENTS The discussion in Chapter 7 on measurements in plates, strips, ribbons, and bulk soft magnets has demonstrated that fluxmetric methods applied to samples forming closed magnetic circuits, either by themselves or with the aid of yokes, do provide the most accurate and practical solution to magnetic testing of low coercivity materials. It is natural to assume that the same principle could apply, if practical, to the characterization of hard magnets, now using electromagnets both for providing the exciting field and constituting a soft return path for the flux. However, there are reasons in many cases for resorting to open sample measurements. For one thing, coercivities in permanent magnets can be very high (for example around 1.5 X 1 0 6 A / m and higher in rare-earth based magnets) and fields in the range of some 5 x 106 A / m and higher may be required in order to approach the saturated state (an empirical rule being that a field strength from three to five times the intrinsic coercive field Hcl is required). Electromagnets have a natural limitation in the saturation of iron, although most commercial apparatus now come equipped with interchangeable Fe-Co tronco-conical pole faces (Js = 2.35 T~ see Section 2.5). If the iron core is brought far beyond the knee of the magnetization curve, the spatial homogeneity of the generated field is impaired and incomplete magnetic testing can result. We see in the results reported in Fig. 8.3 an example of the very high fields required to attain the limiting demagnetization curve on some N d - F e - B magnets [8.13]. These are outside the capabilities of electromagnets. In addition, many practical samples may come in sizes and shapes unsuitable for any kind of characterization with closed magnetic circuits. Nevertheless, the hysteresisgraph method with closed magnetic circuit is in general and prevalent use for industrial testing of permanent magnets. If the right conditions regarding the sample geometry and the field magnitude requirements are met, one can confidently rely on the measuring accuracy and reproducibility of this method. On the one hand, it employs practically

CHAPTER 8 Characterization of Hard Magnets

482

1.0 Melt spun Nd'Fe'B I

~

/

0.8

p.

0.6 0.4

0.2

0.0

.,

-1000

-500

0

H (kA/m)

500

1000

1500

FIGURE 8.3 Initial magnetization curve and demagnetization curves in melt spun Nd13.5Fe81.3B5.2ribbons. These curves differ by the value of the maximum effective field Hp attained along the initial curve before starting the reversal. These values are shown in kA/m on each curve (adapted from Ref. [8.13]). the same setups for dB/dt and H signal detection and conditioning applied and solidly validated with soft magnets. It is little burden for a laboratory engaged in the testing of soft magnetic materials to expand their measuring capabilities in the field of hard magnets in this way. On the other hand, it is officially backed by specific IEC and ASTM measuring standards [8.12, 8.14], which define in detail test conditions and field of application, parameters to be measured, and measuring procedure. As previously stressed, it is an absolute method of measurement, traceable to the base SI measuring units. However, given the considerable effort required for absolute calibration, the use of reference Ni samples can also be envisaged [8.6]. Figure 8.4 provides the basic scheme of a hysteresisgraph for permanent magnet testing. There are no fundamental differences with the permeameter arrangement used in bulk soft magnet testing (see Fig. 7.1b), but for the constraints now imposed by the requirement of high field strengths and specific sample size, and no special control of the field waveform is normally required. The differential permeability of permanent magnets in the steepest part of the loop is

8.1 CLOSED MAGNETIC CIRCUIT MEASUREMENTS

483

orders of m a g n i t u d e lower than in soft magnets and e d d y currents in the sample are always low or absent w h e n traversing the loop with the typical rates allowed by e d d y current shielding in the bulk structure of the electromagnet (say a r o u n d 50-200 s for full loop determination). With reference to Fig. 8.4 and the m e a s u r i n g standards, we s u m m a r i z e here the features of the hysteresisgraph m e t h o d specifically connected to the characterization of p e r m a n e n t magnets. (1) Electromagnet. A uniform field of high strength can be generated in the gap of a soft iron electromagnet. The iron core has the role of

Bipolar supply

Function generator

.a,,

=i iiiiiii~ii' i

'

AT

gaussmeter Digital oscilloscope

PC

FIGURE 8.4 Testing of permanent magnets using an electromagnet with variable gap and the hysteresisgraph method. The sample is a cylinder with circular or rectangular cross-section and regular end faces. D 2 is the distance over which the radial decrease of the field strength is less than 1%, le is the gap length, and D O is the diameter of the pole faces. To ensure axial anci transverse un"iformity of the field, the conditions given by Eqs. (8.1) and (8.2) on D2, Ig, and Do are recommended by the measuring standards [8.12, 8.14]. Compensated coaxial search coils of the type illustrated in Fig. 7.2, directly providing the polarization J in the material, can be used in order to accommodate different sample diameters. The field strength can be determined either by means of a Hall plate or a multiturn H-coil, embedded with the search coil (perm-card). The magnetizing frequency is typically around some 10 -2 Hz or less and is largely determined by the condition of full flux penetration in the core of the electromagnet. Alternatively, a point-bypoint procedure as described in Section 7.2.2 can be applied.

484

CHAPTER 8 Characterization of Hard Magnets

channeling and amplifying the field produced by means of a pair of windings located close to the gap, according to the scheme of principle shown in Fig. 4.23. Normally, one of the poles is made movable to exactly accommodate the sample in the gap, and an H-frame is used to reduce the reluctance of the soft magnetic path. The general appearance of a variable gap electromagnet is like the one sketched in Fig. 4.22. The basic physical principles associated with field generation have been discussed in Section 4.5. As far as the permeability of the iron core is high, its role is one of bringing the whole magnetomotive force across the gap. By tapering the pole faces (with optimum taper angle /3 = 54.74~ the available field in the gap can notably increase for a given magnetization value in the back core (see Figs. 4.23 and 4.27), as demonstrated by Eqs. (4.61) and (4.77). This is appreciated whenever the induction in the gap Bg--/z0Hg is high enough to bring the magnetization in the iron core towards and beyond the knee of the magnetization curve (Figs. 4.24 and 4.25). Pole tapering, possibly using Fe-Co caps, is actually indispensable for testing of rare-earth based magnets in a closed magnetic circuit. The price one has to pay for the increased field strength available with tapered poles is reduced field homogeneity in the gap. Figure 8.5, based on the known magnetization curve of iron up to saturation, shows the result of FEM calculations on the model C-core electromagnet shown in Fig. 4.23. The calculations concern, in particular, the behavior of the reduced axial component of the flux density la,oHg(x,O)/ia, oHg(O,O ) along the axial coordinate x and/z0Hg(z, 0)//z0Hg(0, 0) along the radial coordinate z, as predicted to exist for a given core diameter Do and gap length lg, and a defined value of the magnetomotive force Ni. In this specific case, we pose Do -- 180 mm, lg = 25 m m and we consider two supply conditions: Ni = 25 x 103 A and Ni -- 125 x 103 A. We see that, with untapered pole faces, axial field homogeneity in the gap is ensured within a few parts in 103 under both supply conditions. The axial field is equally homogeneous on moving along the radial direction over distances of the order of lg. With 54.74 ~tapering and diameter of the pole cap Dg -- 21g -- 50 mm, ~ H g falls by 1-2% along the x-axis on passing from the pole surface to the center of the gap. The field variation along the radial direction z turns out to be equally relevant, although it is contained within 1% for a distance z0 < 10 m m from the axis. There is no advantage in using tapered poles in the low-medium induction range. There is an obvious advantage when we wish to approach and overcome the 2 T range in the gap (see in the example shown in Figs. 4.24 and 4.27 how, for the same core diameter and gap length, the flux density in the gap passes from 1.8 to 2.5 T after tapering), but, in order to accommodate practical samples, one should adopt as large diameter Do of the core as permitted by available supply

8.1 CLOSED MAGNETIC CIRCUIT MEASUREMENTS

485

1.02

,

/oHg

z

=2-5T

A 0

~ oC 1.01 0.9\ ", "

o

.,.

% 9 ,,,

:g

, ' sS

1.8 T

\

,,

.

.

.

.

.

.

.

## %%

"."

/

,, -

"',

_

1.00

.

Dg

... "~,~,.~,_

,. ,~ , , ~ "

_

/ 0.9 T Soft Fe C-core Ig = 25 mm Do= 180 mm Dg= 50 mm

0.99

0.98

'

9 i

. . . .

-10

I

. . . .

-5

I

. . . .

5

. , " "" /

/

I I

0.96

,,

~"

I

' " '/,

.

.

.

/zo H o = 2.5 T

", ~

,

9

PoHg

I

', ,'

I

0.9 T ->'x

" ',

',,

I I

l

II

o...

9

...........~ 0.9 T

0.98

,~o

I

10

_

.8 T

~,

. . . .

0 x (mm)

(a) 1.00

I

--:~x Oo

,, ~ I, I I X , , .

Soft Fe C-core DO= 180 mm

oo omm

0.92 . . . . . . . . . , . . . . . . . . . . . . . . . . . . . -20 -10 0 (b) z (mm)

Z

l I Il

10

",1' -)---)!t---I ~:::r ~ ' J V ,uoHg

-)" x

20

FIGURE 8.5 Behavior of the reduced value of the axial flux density on the gap along the axial coordinate x (a) and the radial coordinate z (b) predicted by FEM calculation on the model C-core electromagnet shown in Fig. 4.23. The iron core is assumed without hysteresis and its magnetization curve up to saturation is known. Tapered (/3 = 54.74 ~ and untapered pole faces are considered, with gap length lg, core diameter Do, and pole cap diameter Dg as shown in the figure. The comparison is made for two values of the magnetomotive force. For N i = 25 x 103A we obtain /~0Hg(0, 0)---0.9 T, with both fiat and tapered pole faces. With N i = 125 x 103, tapering brings about 40% increase of/z0Hg(0, 0) (from 1.8 to 2.5 T) at the cost of reduced uniformity of the field in the gap.

486

CHAPTER 8 Characterization of Hard Magnets

power and cost of the apparatus, roughly scaling as D~. Notice that, the permeability of permanent magnets being very low, there is no shaping of the field by the sample, as is always the case with soft magnets (see Section 6.1), and the field generated by the electromagnet must necessarily be highly uniform. Let us consider the permanent magnet specimen inserted in the gap of the electromagnet and let us assume that the iron core is far from saturation. We can reasonably state, in view of the large differences between cross-sectional areas and permeabilities of sample and soft core, that the latter is a short circuit for the magnetomotive force generated by the specimen and that the magnetomotive force Ni exerted by the windings appears all across the specimen. An ideal magnet placed between the pole pieces of the electromagnet is perfectly flux closed and, as predicted by FEM calculations in the example provided in Fig. 8.6 (where the magnet saturation polarization Js -- 1 T has been assumed), the effective field on the sample H - - H g is zero (i.e. the demagnetizing field disappears) and the remanent state is attained (Br--Jr = Js). If the electromagnet is excited, the field appearing across the specimen is uniform and equal to H - - H g ~ Ni/lg as far as the permeability of the iron core is high. The superposition principle applies and the measured induction in the sample is B = Js-/z0Hg. Once the magnet polarization becomes balanced by the induction generated by the electromagnet, the coercive point Hcs is attained. The distribution of the flux lines in the neighborhood of this condition is described in Fig. 8.6b. A real magnet will actually suffer a decrease of its polarization, which implies that Hcs will be lower, but the homogeneity of field and polarization will be preserved. This will continue to be the case even when the field H is increased beyond the intrinsic coercive field Hcl, up to the point where the knee of the magnetization curve of the iron core is left behind and the saturated state is approached, at least, for wedgeshaped poles, in the region immediately behind the apex of the pole. Under these conditions, not only the field generated by the electromagnet will be increasingly non-uniform, but the soft core will progressively repel the flux lines generated by the magnet under test. The field distribution around the sample will then approach that of an open magnet immersed in the field generated by the electromagnet. In a real magnet, both polarization and effective field will be nonhomogeneous and the measuring accuracy will be impaired. Figure 8.7 provides a view of this effect through FEM calculations on the model system discussed in the previous figure. (2) Sample arrangement and signal detection. The test specimen is normally shaped, according to the recommendation of the IEC and ASTM

8.1 CLOSED MAGNETIC CIRCUIT MEASUREMENTS

\

.i"" B - -rT,c

/"

..... ~ ....

. .... j l ....

(a)

!!i .........

.,..,..-- - . . . . . .

I,

,,,... . . . .

.

-

487

.............. .......-................

i
\'

",it-, "

>

~---,~

:.: .....iiii .....

,..

41[ ~

....

..........

Hg=O

(b)

Hg ,-, HcB

FIGURE 8.6 FEM modeling of flux distribution in ideal permanent magnet (saturation polarization Js = 1 T) and soft iron pole pieces (Armco type iron). (a) The permanent magnet, inserted between the pole faces of an electromagnet with demagnetized core and zero supply current, is perfectly flux closed by the nearzero reluctance soft core. The field in the gap is zero and the magnet is at remanence ( B r - Jr - - J s ) . ( b ) The electromagnet is supplied and the magnetomotive force Ni appears almost totally across the magnet (Ni Hglg). The soft core is in the high permeability region and the field in the gap is uniform. In the condition shown, the induction in the sample B - Js - ~0Hg is nearly zero and the HcB point is being approached. The calculation is made for the model electromagnet of Fig. 4.23, but for use of 45~ tapered pole pieces (courtesy of O. Bottauscio). =

Standards [8.12, 8.14], as a cylinder, having either rectangular or circular cross-section, with smooth and as parallel as possible end faces and uniform cross-sectional area (within 1%). Their m i n i m u m height should be 5 mm. If not, it is expedient to stack the specimens, p r o v i d e d they are identical. Whatever the case, it is suggested, in accordance with the Standards, that, in order to guarantee axial and radial field uniformity (within 1%) in the region occupied by the specimen, the final sample height lm -- Ig~ the diameter Do of the pole faces, and the diameter D 2 of the cylindrical region that contains the specimen and the field m e a s u r i n g sensor (see Fig. 8.4), are related by the conditions Do - 21m,

Do --> D2 q- 1.2lm,

(8.1)

488

CHAPTER 8 Characterization of Hard Magnets 1.82 T

2.92 T

1.06 T

2.06 T

0.20 T

1.20 T

FIGURE 8.7 FEM calculation of magnetic flux distribution in ideal permanent magnet test sample (saturation polarization Js = 1 T) and surrounding area for two largely different levels of the applied magnetomotive force. Electromagnet and sample are arranged as in the previous figure. (a) The induction in the sample is B = 1.82 T and the induction in the back core is lower than 1 T. Consequently~ the effective field in the sample is homogeneous. (b) With induction in the sample overcoming 2.9 T, the induction in the back core is larger than 1.9 T, flux closure by the iron core is poor, and both effective field and induction in the sample are inhomogeneous (courtesy of O. Bottauscio). where Dg will replace Do with tapered poles. The first condition automatically defines the maximum sample length for a given electromagnet. The second condition depends on the magnetic state of the iron core and poses an upper limit to its exploitable polarization value. The example reported in Fig. 8.5, regarding the numerical prediction of axial and radial field uniformity in the gap of the model C-core electromagnet shown in Fig. 4.23, satisfies these conditions with untapered polar faces (/g -- 25 mm, Do = 180 mm) for D 2 ~ 100 m m ([~0Hg - 1.8 T) and D 2 140 m m (/~0Hg = 0.9 T). We see that excellent uniformity of the field in the gap is obtained even when the polarization value in the core is of the order of 1.8 T (see also Figs. 4.24 and 4.27), quite beyond the upper limit J = 1.0 T recommended in the IEC and ASTM Standards. With 54.74 ~ pole tapering and high flux density in the gap (/~0Hg -- 2.5 T, corresponding to a polarization J --- 2 T in the region of the core surrounding the pole faces), we fulfill the condition on radial uniformity of/~Hg for D2 < 18 mm, but the predicted axial variation is around 2%. From what we have previously said and shown in Fig. 8.7b, inhomogeneity of the generated field in the gap is only part of the problem because, with the iron core offering poor

8.1 CLOSED MAGNETIC CIRCUIT MEASUREMENTS

489

closure to the flux emanating from the permanent magnet, non-uniform demagnetizing field arises and it compounds with the applied field. The measurement of the effective field, made with an H-coil or a Hall plate placed on the sample surface (see Fig. 8.4), then becomes affected by a substantial error. The demagnetizing field of the sample is small around the intrinsic coercive field Hc! and the transition through this point may occur in a relatively smooth fashion. It turns out that a good measurement of Hcl (say, up to around 1500 k A / m ) can often also be made in standard rare-earth magnets by means of an electromagnet-based setup and F e - C o tapered caps. We see in Fig. 8.8 a typical consequence of field inhomogeneity in the shape of the measured hysteresis loop (J,H), where the value of J, either calculated by subtracting the term/~0H from the measured induction value or automatically obtained by means of a compensated coil of the type shown in Fig. 7.2, becomes incorrect on approaching the magnet saturation.

J

s

0.8

0.4 f t

0.0 Nd30Fe63.sDYsAIo.4B1 -0.4

-0.8 '

-2000

' ' " '

'

| '

' '

-

I

.

.

1000

.

.

.

.

.

.

.

.

.

S .

.

.

0 H (kA/m)

'

'"'

'

I

'

1000

'

'

'

'

'

'

'

'

2000

FIGURE 8.8 Initial magnetization curve and hysteresis loop measured on a rareearth compound using the hysteresisgraph method and electromagnet with iron poles. The somewhat anomalous behavior of the upper portion of the loop is attributed to inhomogeneity of the effective field H, as predicted by calculation in Fig. 8.7b, and ensuing incorrect determination of the material polarization upon subtraction of the field contribution/~0H to the measured magnetic induction. The initial magnetization curve reveals a coercivity mechanism of nucleation type and the discontinuity at remanence indicates the presence of a decoupled softer phase.

490

CHAPTER 8 Characterization of Hard Magnets

Poor contact between the test specimen and the pole faces is a further source of measuring error. For this reason, good finishing and parallelism of the sample end faces should be ensured. Air gaps are, however, unavoidable, especially with brittle materials like the ferrites, which can only withstand slight pressure without crushing. Because of the air gaps, the free charges on the ends of the magnet are no more perfectly neutralized by the corresponding images in the iron poles (as it happens in the ideal flux-closing condition depicted in Fig. 8.6a), and a nonhomogeneous demagnetizing field will appear. According to the measuring Standards [8.12, 8.14], an estimate of the maximum relative error on the measured field strength in correspondence with a defined point (Bm, Hm) on the demagnetization curve can be given by the equation AHm

2d

Hm

lm ~ H I

Bm

(8.2)

for a specimen separated from the pole faces by two gaps of thickness d. The determination of the B(H) and J(H) relationships in hard magnets with a closed circuit arrangement does not show particular differences with respect to the DC testing of soft magnets using the permeameter method. As sketched in Fig. 8.4, a secondary winding with a conveniently high number of turns, as imposed by the relatively low value of the magnetization rate, is used to detect the flux derivative. Air flux compensation or subtraction is carried out according to the simple rules described in Section 7.1.1. For example, one can automatically obtain the polarization in the sample by means of the triple-compensated J-coil illustrated in Fig. 7.2. The tangential field at the sample surface is again detected by means of a Hall plate or a coaxial H-coil. In the simplest case, we make use of a secondary winding of known number of turns and calibrated turn area N2A 2 and we simultaneously detect the flux derivative dcI)2/dt- N2(A21-~odH/dt + AdJ/dt),

(8.3)

where A is the cross-sectional area of the specimen, and the effective field H by means of a tangential sensor. The signal associated with the variation of the polarization in the sample, with reasonable values for dJ/dt, N2, and A, is in the mV range at most and it is conveniently passed through a low-noise calibrated amplifier. The H signal is normally retrieved as analog output of the Hall gaussmeter. If obtained by means of an H-coil, it will follow the same destiny of the d ~ 2 / d t signal. Signal acquisition by

8.1 CLOSED MAGNETIC CIRCUIT MEASUREMENTS

491

means of a two-channel digital oscilloscope or voltmeter, A / D conversion, and digital treatment of the signal are then performed so that B(t), H(t), and J(t) are determined, together with all the parameters related to the hysteresis loop (at least over the second quadrant), as already discussed for measurements on soft magnetic materials. Hysteresis loop and initial magnetization curve can be determined by means of the pointby-point technique, as described in Section 7.2.2. It has the distinctive advantage of low drift, but it can only be applied where the saturated state can be cyclically and easily achieved and it is therefore reserved for the characterization of Alnico and ferrite materials. The continuous field hysteresisgraph method is the rule nowadays. The problems associated with the long integration times, going hand-in-hand with the small amplitude of the signal, are overcome with the use of low-noise and thermally stable amplifiers and digital correction of the drift. A practical difficulty associated with measurements on permanent magnets with closed magnetic circuits derives from the shape of the components to be tested, which often does not fit the requirements of the Standards. Machining the specimens is difficult and costly because the material is normally brittle, and open sample experiments can ultimately be preferred, when applicable. Very simple devices can be made for grinding small spheres suitable for testing with a host of open sample methods [8.15]. A ferrite sphere of diameter 2 - 3 m m can be prepared in a few minutes starting from an irregular shaped fragment of material, which is let free inside a box internally lined with abrasive paper. A blow of air under pressure channeled into the box sets the sample into a spinning motion, making it randomly hit the abrasive walls. Since the particle in flight has spinning axis perpendicular to its maximum diameter, it will be preferentially abraded along its largest size. Because of their mechanical hardness, the sintered rare-earth magnets may require several hours of grinding, besides the use of an inert gas in place of air. In order to test components without machining a sample out of them, it is possible to devise special soft iron adapters (polar shoes), placed in intimate contact with both the hard magnet and the polar faces of the electromagnet. If the samples have irregular or difficult-to-measure crosssectional area, we might content ourselves with the determination of the material properties close to the surface, adopting Steingroever's concept of interchangeable polar pieces equipped with pole coils, as sketched in Fig. 8.9 [8.16]. The working principle of the pole coils is that the axial component of the induction is preserved at the interface between the pole and the sample. Thus, with pole coil 1, we can measure the induction in the sample or, at least, in the region close to the surface in contact with the polar face. In a similar way, pole coil 2 can provide a measure of

492

CHAPTER 8 Characterization of Hard Magnets

I 4iiiiii!i!!![il/ (a)

(b)

FIGURE 8.9 (a) Electromagnet with interchangeable pole pieces endowed with embedded coils. The coil (1) placed immediately below the end face of the test specimen provides a measure of the induction in the magnet, at least in the region close to the pole face. The other coil (2) can be used either to determine the field in the gap, which is the same as the effective field on the sample if homogeneity conditions are fulfilled, or to provide, by connection in series opposition with the other coil, a measure of the material polarization J. (b) Upper view of the interchangeable pole piece with inserted coils (adapted from Ref. [8.16]).

the induction in the gap ~0Hg -~/~0H, where H is the effective field on the sample. With an arrangement of this kind, which can be supplemented by a Hall plate for measuring the field at the surface of the sample, we are able to determine the B(H) and J(H) curves, even if the cross-sectional area of the specimen is unknown or ill-defined. A further adaptation of the electromagnet pole pieces must be devised whenever measurements above room temperature have to be performed. As previously remarked, ferrites and rare-earth magnets have an appreciably high temperature coefficient of both saturation magnetization and coercive field. This not only implies that a strict control of the temperature during the measurement must be applied, but it also frequently calls for material testing at elevated temperatures. However, should the conventional Standards be applied for testing at elevated temperatures, the complicated and expensive solution of immersing the whole magnetic circuit in an oil-filled tank should be adopted. The arrangement of sample and polar pieces sketched in Fig. 8.10, as

8.1 CLOSED MAGNETIC CIRCUIT MEASUREMENTS ! !

thermal insulator~ heati ng plat e N,.~

'

'-

pole piece (cold)

v_/~/////,///////_/~///////////////,,///~_...Z__---~measuring pole (warm)

~////~,d,c/YW//d/,;E;Y~,,.,,

'

/

permcard FIGURE 8.10

493

i

~'N

thermocouple

test specimen

Testing of permanent magnets up to 200~

(adapted from

Ref. [8.17]).

proposed by the IEC Technical Report 61807 [8.17], affords a reasonably accurate and economical solution to the characterization of permanent magnets up to 200~ that can be applied within the industrial environment. In this setup, the final portion of the polar pieces and the specimen are warmed up by a pair of heating plates, which are thermally insulated from the remaining part of the magnetic circuit. The sample temperature is controlled with the aid of a thermocouple, inserted deeply within one of the warm poles, and making use of pre-emptive determination of the relationship existing between the temperature of the sample and that of the pole. A compensated J-coil and an H-coil, inserted in a temperature-resistant holder (permcard), are conveniently employed for signal detection. (3) Measurement procedure, calibration, uncertainty. The characterization of permanent magnets using electromagnet and closed magnetic circuits can be performed in a complete fashion on materials that can be saturated in the maximum uniform field available in the gap. In commercial apparatus, such a field is of the order of 2000 k A / m (flux density around 2.5 T, see Fig. 4.25), which is perfectly adequate for full testing of ferrite, alnico, and ordered 3d alloy products. As we shall see, we can also determine the demagnetization curve along the second quadrant in most rare-earth high-energy product magnets, provided the saturation of the materials is separately accomplished either in a pulse magnetizer or in a superconducting solenoid. Magnetic tapes, which have coercivities ranging between 30 and 100 k A / m , can be tested using permeametertype apparatus, like the ones described in Section 7.1 and Fig. 7.1, or even as open strips in thick cooled solenoids. In the latter case, a compensated J-coil can be realized, as shown in Fig. 6.9. It is obtained by connecting in series opposition the search coil enwrapping the tape and an identical winding linked only to the induction Ba --/~0Ha, where Ha is the applied field.

494

CHAPTER 8 Characterization of Hard Magnets

Taking into account that the demagnetizing coefficient of the tape specimen can be of the order of 10 -5 or less, it can be assumed, according to Eq. (6.6), that this compensated coil always provides a good measure of the polarization of the material. For a measurement directed only at determining the value of the coercive field, one of the magnetometric methods described in Section 7.2.1 and Fig. 7.17 can be applied. To ensure that the material under test is saturated as fully as practical, the old rule stated that the maximum field Hp was to be from three to five times the intrinsic coercive field Hcl. According to the pertaining ASTM Standard [8.14], the specimen is considered to be saturated if an increase of the magnetizing field strength by 50% changes the values of Hcl, Br, and the anisotropy field Hk by less than 1%. The rule suggested by the IEC Standard [8.12] is based on the determination of the maximum energy product (BH)max at increasing fields. It states, in particular, that saturation is considered to be attained when the maximum energy products (BH)ma• and (BH)max2 determined in correspondence with two values of the field strength Hi and H2, with H 2 -> 1.2H1 are in the relationship (BH)max2 < ( H 2 ) 0"02454 (BH)maxl- H1 "

(8.4)

Alternatively, the same condition is applied to the coercivity ratio After having accurately centered the cylindrical sample surrounded by the secondary coil between the pole faces of the electromagnet, displaced the moveable pole and gently pressed it against the sample end faces, the magnetic field is brought to the maximum value Hp, known to satisfy Eq. (8.4), by applying a voltage ramp to the bipolar power supply via the programmable function generator. This can be followed by a triangular voltage function lasting one period. At the same time, the signals from search coil and Hall sensor (or H-coil) are detected, sampled, and A / D converted and the major hysteresis loop with all the relevant parameters is calculated. The supply can be used in either current mode or voltage mode. The former is more efficient, but the latter is to be preferred because it ensures protection against possible faults in the output circuit. Good information on the nature of the magnetization process (e.g. domain nucleation vs. domain wall pinning) comes from the initial magnetization curve. Its measurement requires that not only the sample, but also the iron core be fully demagnetized. By switching off the supply of the electromagnet at the end of a measurement run and taking off the specimen, a residual field in the gap is detected, the higher its value the narrower the gap. For a 10-20 mm gap, useful for typical samples, this

HcB2/HcB 1.

8.1 CLOSED MAGNETIC CIRCUIT MEASUREMENTS

495

field can be of the order of 10-20 k A / m , sufficient to miss or perturb a portion of the initial magnetization curve. This occurs even in rare-earth magnets when a decoupled soft phase is present. The best way to cope with this problem is to demagnetize the iron core by putting the pole faces in contact and applying the usual procedure. Since the magnetization rate must in any case be very low, in order to always guarantee full penetration of the field in the core, and the whole process can last more than 30 min, one could alternatively adjust the pole pieces to the right distance and then inject a slowly increasing demagnetizing supply current. This is eventually regulated to the value corresponding to zero indication by a Hall sensor placed in the gap. If the test magnet can be saturated within the electromagnet, the demagnetization procedure can be done with the specimen in place. Many alternations are required to complete the process reliably. To demagnetize the specimen outside the electromagnet, a pulsed field source of the type described in Section 4.3 can be employed. For rare-earth magnets, a difficulty could arise because of eddy currents. The source should then be designed in order to provide a high initial peak field value and conveniently low oscillation frequency. In order to refine the process, a number of repetitions can be made, each time decreasing the charging voltage by a convenient factor. In this case, there can be considerable heat production in the solenoid and the sample temperature must be checked. Thermal demagnetization can be safely applied on ferrites, while it requires caution in all other cases. Rare-earth magnets, in particular, may suffer from uncontrolled structural modifications, not least oxidation, on temperature rising above the Curie point. Figure 8.11 provides an example of initial magnetization curve and hysteresis loop obtained in a 10 mm high anisotropic Ba ferrite cylinder of diameter 20 m m using an electromagnet with untapered pole faces of diameter Do = 250 mm. The sample was demagnetized via an oscillating discharge in a pulsed field solenoid with the maximum value of the peak field 2400 k A / m (flux density ---3 T), oscillating period around 5 ms, and time constant ~"= 18 ms. A triangular voltage waveform was generated, starting from zero value, in a burst-like fashion over 5/4 of a period T = 150 s and supplied to a 5 kVA DC-coupled power amplifier. The dJ/dt and dH/dt signals, achieved by means of a permcard, were amplified by means of two identical DC-coupled low-noise 100 Mf~ input impedance amplifiers and A / D converted via a 14-bit digital oscilloscope. The full magnetic characterization of the extra-hard rare-earth magnets cannot be carried out by means of closed circuit methods. There is little room for these methods in basic research on extra-hard materials. However, the determination of the parameters defining quality and domain of application of industrial products is generally possible,

496

CHAPTER 8 Characterization of Hard Magnets 1.5

Barium ferrite in electrom T= 150 s

I

1.0

0.5 .

.

.

j"-

.

0.0

i

Z

I-" v

..~ -0.5

-1.0

-1.5 I

'

I

'

I

'

I

-800 - 6 0 0 - 4 0 0 - 2 0 0

'

'

I

0 200 H (kA/m)

'

I

"

400

'

'1

600

'

I

800

FIGURE 8.11 Initial magnetization curve and J(H) and B(H) hysteresis loops measured in an anisotropic Ba ferrite permanent magnet using a closed magnetic circuit and the hysteresisgraph method. The test specimen was demagnetized by oscillating discharge in a pulsed field source (maximum field 2400 kA/m). The full hysteresis loop was traced in a time interval T = 150 s. according to the recommendation of the measuring Standards, because the material can be driven along the demagnetization curve in the second quadrant by means of an electromagnet generated field. Of course, other field sources must be employed to bring the material into the saturated state before starting the travel along the demagnetization curve. Magnetic flux densities up to 9 T (~-6400 k A / m ) are usually produced using NbTi superconducting solenoids at 4.2 K. Pulsed field solenoids capable of producing transient fields up to about 8000 k A / m over a useful volume by capacitor discharge are commercially available. Used in the LR mode (see Section 4.3), they can saturate most practical rare-earth magnets. Because of possible lack of flux penetration in the test sample during application of the transient field, it is good empirical practice to ensure that the field strength remains higher than about 3000 k A / m for a time interval longer than 10 ms. Once saturated, the sample is brought into the gap of the electromagnet, where the iron core is demagnetized, and it is inserted in the search coil, which is kept axially centered by means of a suitable holder. It is understood that the electromagnet is of the tapered

8.1 CLOSED MAGNETIC CIRCUIT MEASUREMENTS

49V

type, possibly with Fe-Co pole caps, and the supply system is adequately sized. Immediately before insertion of the sample in the coil, acquisition of the flux derivative starts. It continues over the successive steps of the process, which consist in the closure of the pole pieces over the sample and the application of a voltage ramp to the power supply in order to provide as high as possible a field in the same direction of the magnetization in the sample (the "forward direction"). When the peak field value Hp is reached, the corresponding magnetization value, calculated by integrating the signal collected through the search coil since start and accurately eliminating the drift, is recorded. All these operations are carried out by digital methods and computer control. We can reasonably assume that, since the materials to which this procedure applies have all very high Hcj value, only a small fraction of the magnetization is irreversibly lost under the influence of the demagnetizing field and that most of it is recovered by remagnetizing the sample to the maximum field Hp. This conclusion is expected to apply both to conventional magnets and nanostructured spring magnets. One can prove it empirically by repeating the experiment with samples having different aspect ratios. It is in any case recommended that the height-todiameter ratio of the test specimen be greater than 1. In conclusion, it is expected that, once Hp has been reached, the magnetizing current is decreased to zero, and finally it is reversed down to the symmetric - H p field, that the whole second quadrant along the limiting curves B(H) and J(H) is covered. The absolute determination of the B(H) and J(H) curves via closed magnetic circuit and the hysteresisgraph or ballistic method can be made traceable to the base SI units by means of a suitable calibration procedure which involves: (1) Measurement of the cross-sectional area A of the cylindrical specimen via the measurement of the diameter with a calibrated micrometer. Alternatively, A is obtained by a measurement of mass, density, and length. Traceability is made to the standard of length in the first case and to the standards of length and mass in the second case. (2) Calibration of the turn-area product of compensated coils and H-coil. It is made using a standard field source (solenoid or Helmholtz coil), supplied by a precisely known AC current. Traceability is made to voltage and resistance standards (for the supply current) and the frequency standard (for the generated field, via NMR measurement). (3) Calibration of the Hall setup for the measurement of the magnetic field. To this end, field sources (solenoids and electromagnets) calibrated through NMR measurements are used. Again, traceability is made against the frequency standard. (4) Calibration of amplifiers and acquisition setup by means of reference sources and voltmeters, ensuring traceability to the voltage

498

CHAPTER 8 Characterization of Hard Magnets

standard. In order to avoid the burden of absolute calibration, reference Ni samples, if available, can be conveniently exploited [8.6]. On the whole, the measurement of the technical parameters of the permanent magnets, which are associated with the behavior of the B(H) and ](H) curves in the second quadrant, suffers from a certain lack of reproducibility if compared with measurements made on typical soft magnetic materials. This is confirmed by the results of intercomparison exercises, such as those on Fe-Si laminations mentioned in Chapter 7 (Fig. 7.8) and Chapter 10 (Figs. 10.2 and 10.4 and Table 10.2) and those on N d - F e - B specimens reported in Fig. 8.12 [8.10]. From the statistical analysis of the 1700 1600

E 1500 1400 1300

Nd-Fe-B '

I

'

I

2

'

I

4

6

'

I

'

I

'

I

8 10 Laboratories

'

I

12

14

'

I

16

240 co

E

230

~ 220 E

~ 210 200

Nd-Fe-B '

I

'

I

.

'

I

6

'

;

'

Laboratories

'

,'2

'

1'.'1'6

FIGURE 8.12 Intercomparison exercise on the measurement with the closed magnetic circuit method of the intrinsic coercive field Hq and the maximum energy product (BH)max on Nd-Fe-B magnet specimens (15 mm diameter, 5 mm height cylinders). Each point represents the best estimate of the measured quantity made by each laboratory. The calculated relative standard deviation of the results around the reference value (the unweighted average), calculated after having excluded the outliers, turns out to be 4.7% for Hcl and 3.6% for (BH)max (adapted from Ref. [8.10]).

8.2 OPEN SAMPLE MEASUREMENTS

499

latter results, one finds, for example, that the best estimates on the intrinsic coercive field Hc! reported by the different laboratories fluctuate around the reference value (the unweighted average) with a relative standard deviation cr = 4.7%. For the maximum energy product (BH)max cr = 3.6% is obtained. It is not surprising that materials like rare-earth magnets, having extreme and somewhat unstable magnetic behavior that is strongly temperature dependent and influenced by magnetic viscosity effects, exhibit appreciable scattering of the measured properties from laboratory to laboratory. According to the reported ensemble of results, this conclusion can be drawn for both the closed magnetic circuit and the open magnetic circuit methods.

8.2 O P E N SAMPLE M E A S U R E M E N T S Hard magnets have their properties influenced by the demagnetizing fields, but they are only slightly sensitive to the environmental fields, which can disrupt or falsify measurements made on open soft magnetic strips and sheets. Consequently, testing hard magnets as open samples is not only methodologically correct, but it opens a good, wide scenario in terms of novel measuring techniques and flexibility as to the type of materials to be characterized and the size and shape of the test specimens. Open samples reveal their presence through the stray field they emit and their state of magnetization can be determined through, besides the previously mentioned torque and force methods, the measurement of such fields. Homogeneous and ellipsoidally shaped samples have uniform magnetization, which can be determined by measuring the field in a point close to the sample, for example in a point belonging to one axis of the ellipsoid. In this case, in fact, stray field and magnetization are related by closed analytical expression. Search for the zero stray field condition upon the application of an increasing reverse field on a previously saturated sample leads to the determination of the intrinsic coercive field Hcl. For a non-ellipsoidal sample, the condition of zero average sample polarization is detected. The measuring arrangements shown in Fig. 7.17 and the procedure described in detail in the Standard IEC 60404-7 [8.18] can be applied to both hard and soft magnets, as discussed in Section 7.2.1, with the limitations imposed by the relatively low field strengths which can be generated. With the field-sensing probe placed at a sufficiently large distance from the sample, only the dipolar field is detected and the magnetic moment can be determined, again providing the volume-averaged polarization. In all cases, if the measurement is made in the presence of an applied field, methods must be devised

500

CHAPTER 8 Characterization of Hard Magnets

for compensating its effect. In permanent magnets, a field can easily result much larger than the field generated by the test specimen at the probe location. For example, the field generated on its axis by a 5 m m x 5 m m cylindrical permanent magnet with polarization 1 T at a distance of 20 m m is of the order of 1.5 k A / m , orders of magnitude lower than the field to be applied to bring it into such a magnetic state.

8.2.1 Vibrating sample magnetometer The flux linked with a sensing coil placed at a certain distance from an open sample subjected to an intense magnetizing field can be seen as the sum of a main contribution due to such a field plus a perturbation originating from the sample. We are interested in measuring such a perturbation. An effective and simple way of separating it from the background is a sort of AC magnetometric method, where linkage of the sensing coil with the signal generated by the sample is made to vary rapidly with time, all the rest remaining unaltered. This can be obtained by imparting a vibrating motion to the sample, so as to produce an AC signal while making a DC characterisation. Any background constant flux is automatically filtered out and signal optimization can be pursued if some degree of flexibility exists in the amplitude and frequency of the oscillation and the arrangement of the sensing coils. The popular vibrating sample magnetometer (VSM) is based on this principle [8.19]. It is a general purpose, high-sensitivity magnetic moment measuring device, not only perfectly suited to the characterization of permanent magnets and recording media, both in advanced research and routine industrial testing, but also applicable to weakly magnetic and paramagnetic materials. Its sensitivity can be made very high and a lower limit for measurable magnetic moment is typically around some 10 -9 A m 2 in commercial setups. This limit can be extended, by optimization of the coupling between sample and sensing coils, down to a noise floor of the order of 10 -12 A m 2 (10 -9 emu) [8.20]. On the other hand, the upper limit for the measurable moment can be of the order of 0.1 A m 2 and higher (for example, the magnetic moment of a 7 m m diameter iron sphere is around 0.3 A m2). To see how the variable magnetic field generated by the oscillating sample links to the pickup coils, one may start from the equivalent dipole of moment m and the known formulation for its field, from which the flux threading the coils and its time variation upon the dipole vibration are calculated. A more elegant and simpler way of treating this problem can be pursued by invoking the reciprocity principle [8.21]. We have discussed it in Section 6.2. In essence, this principle amounts to the well-known theorem stating that the flux mutually linking

8.2 OPEN SAMPLE MEASUREMENTS

501

two coils is independent of which one carries the current, i.e. a unique mutual inductance coefficient M -- M12 -- M21 can be identified. By virtue of the equivalence between magnetic dipole and current loop, it can be generally stated that the magnetic flux 9 originating from a dipole of moment m located in a point of coordinates (x, y, z) and threading a coil (or a system of coils) characterized by the constant k(x, y, z) = B(x, y, z)/is, the ratio between the flux density B(x, y, z) generated by the coil in this point when a current is circulates in it, is given by the equation = k(x, y, z).m -- kx(x, y, z).mx + ky(x, y, z).my 4- kz(x, y, z).mz

(8.5)

(already presented as Eq. (6.8)). A dipole moving with velocity t will therefore induce in these coils the instantaneous voltage

u(t)-

d(I) dt - (1/is).grad(B.m).r.

(8.6)

For a voltage to be generated upon vibration of the dipole, it is required that the gradient of the flux component along the direction of m that would be generated by the fictitious current is circulating in the coil be different from zero. A simple coil arrangement, ensuring defined value of this gradient together with symmetry properties, is obtained with the series opposing pair shown in Fig. 8.13a, already introduced in Section 4.1.3. They generate an axial field Bx(x) always passing through the zero value at the origin, where the gradient is maximum and attains a value depending on the ratio between the radius of the coils a and their distance d. When d - - a , they form the so-called inverse Helmholtz pair and the induction derivative at the origin is

Nis dBx(0) dx - 0"8587/~~ a 2 if N is the number of turns in each coil. Because of the antisymmetrical configuration of the coils, the field derivative contains only even terms in the development around the origin. By placing the coils at the distance d --- x/3a, the third order term in the field disappears and the linearity is improved. With the magnetic moment m directed along the axis of the coil pair, as shown in the figure, and moving with velocity • Eq. (8.6) takes the form d u(x, t) = ~x (mkx(x)).• = m.gx(X).•

(8.7)

502

CHAPTER 8 Characterization of Hard Magnets z

T.a

I

(a)

w

m w-

I d

<

~,

>

1.01

r

gx (x)

1.00

d = .v/3a

-4

0.99

-2

i

...... ""

i

~

k (b)

-0.4

-0.2

0.0

x/a

0.2

0.4

2

.,, //J"

4

)

Cc)

FIGURE 8.13 Relative sensitivity function vs. displacement of a small sample of magnetic moment m along the x axis of a thin-coil pair for different values of the ratio between intercoil distance d and coil radius a. The coils are connected in series opposition. Curve 1: d = a (inverse Helmholtz pair). Curve 2: d - x/3a (maximum homogeneity of the sensitivity function) (Eq. (4.24)). Curve 3: d = 1.848a. (Graphics in b), (adapted from Ref. [8.22]). (c) The sensitivity function averages out to zero over a region of the order of 4a.

w h e r e k x ( x ) - Bx(x)/is. The function

gx(X) is called reduced radius a different

dkx(x) dx

(8.8)

sensitivity function. Figure 8.13b s h o w s the b e h a v i o r of the sensitivity function gx(x)/gx(O) for the o p p o s i n g coil pair of as a function of the r e d u c e d distance from the origin for three values of the coil interdistance d. As p r e v i o u s l y r e m a r k e d ,

8.2 OPEN SAMPLE MEASUREMENTS

503

the maximum homoj~eneity around the origin of the sensitivity function is obtained with d -- ~/3a. Figure 8.13c shows that this comes at the expense of a certain reduction in the maximum value of the function attained at the origin. The vanishing of the derivative Ogx(X)/Ox at the origin is evidently due to the symmetry of the coil arrangement. In general, one looks for sensing coil arrangements making the point at the origin a saddle point for the sensitivity function because, in the neighborhood of such a point, the signal is insensitive to the first order to the sample position. For a small-amplitude vibration of the magnetic moment around the origin, we can safely assume gx(X)-~ gx(O). If, for example, the time-dependent amplitude for an oscillation in the x direction is x ( t ) - - X o sin ~ot, the sinusoidal voltage u(t) = mgx(O).k = mgx(O).XooJ cos o~t

(8.9)

is induced in the coils. The symmetry conditions on the coil arrangement leading to a saddle point can have a large number of solutions. If the field is applied by means of a superconducting solenoid, the vibration is necessarily exerted along the axial direction (x-axis) and the series opposition coil pair we have just discussed is commonly employed. In practice, however, the filamentary coil approximation may not apply and one should make use of numerical methods in coil design. Recall that general rules for the realization of the Helmholtz condition with thick coils have been discussed in Section 4.2. When the field is applied by conventional electromagnets or by permanent magnet sources, transverse sample vibration (z-axis) is generally adopted. In such a case, the previous Eq. (8.7) becomes d u(z, t) = -~z (mkx(z)).• = m.gx(Z).•

(8.10)

with kx(z) = Bx(0, 0, z)/is and the sensitivity function gx(Z) dkx(O,O,z)/dz. Some of the principal coil arrangements employed in VSM with transverse vibration are sketched in Fig. 8.14 [8.22]. They employ two, four, and eight identical coils, respectively, and are endowed with the symmetry properties making the point at the origin a saddle point. Notice that all these arrangements, like the previously discussed inverse coil pair, are of the compensating type, making the measured signal insensitive to variable external fields. The marked arrows indicate the way in which the signals from the coils have to be added. The four-coil configuration is known as the Mallinson set [8.21], while with the eight-coil arrangement one can intercept the components of the magnetic moment along x, y, and z. In particular, with

504

CHAPTER 8

Characterization of Hard Magnets

X

Ha

@ |

1

h

i

! ~

(a)

(b)

1.2

2-coil

x

1.0

@ ...."-i ..... (c)

d

~5 o

0.8

1

Y" Z" "4-coil ~

x

.__. 0.6 (D

rr 0.4 x

0.2 0.0 -0.3 -0.2 -0.1 0.0 0.1 x/a, y/a, z/a (d)

Y 0.2

0.3

FIGURE 8.14 Examples of saddle point coil arrangements in a VSM for vibration perpendicular (z-axis) to the direction of the magnetic moment (x-axis). The sample is attached to a non-magnetic vibrating rod. The arrows marked on the coils identify the way in which the signals from the coils have to be added. The four-coil system is the so-called Mallinson's set [8.21], while with the eight-coil configuration and a different set of connections, the component of the magnetic moment along the y axis can also be detected [8.23]. The diagram (d) shows the behavior of the relative magnetometer output (i.e. the sensitivity function gx(Z)) for transverse vibration when the position of the vibrating sample is displaced along one of the reference axes x, y, z. The calculations are made assuming small coil size with respect to the intercoil distances (small coil approximation) and the relative output is normalized to the unit turn-area of the coil system. The upper and lower families of curves for the two-coil arrangement are obtained for flat coils (d = 1, h - - 0 ) and long coils (d -- 1, h -- x/~), respectively. Four-coil arrangement: the dashed lines are obtained with dx = 1, dz = 0.389, while the continuous lines correspond to d.,. = 1, dz = 0.678. Eight-coil system: d.,. = dy = dz = 1/~/2. This system exhibits, by virtue of its cubic symmetry, identical response along the three axes (adapted from Ref. [8.22]).

8.2 OPEN SAMPLE MEASUREMENTS

505

the connections made as indicated in the figure, it detects the mx component, while with a different set of connections it can be made to detect one of the other two components [8.23]. The diagram in Fig. 8.14 provides an overview of the response of these different coil arrangements, with m directed along the x-axis, calculated under the assumption of a coil size small with respect to the intercoil distance [8.22]. In particular, the behavior of the relative magnetometer output, that is, the behavior of the associated sensitivity function, is calculated as a function of the position along the three reference axes of the transversally vibrating sample in the neighborhood of the saddle point. It is generally observed that increased flatness of the sensitivity function comes at the expense of its maximum value, that is, of the voltage output of the pickup coils. To increase the output value, one could increase the number of turns as far as this implies a gain with respect to the associated increase of the Johnson noise, which goes proportionally with the total coil resistance. The discussion so far on the prediction of the sensitivity function for the various sensing coil arrangements has been based on the assumption that the test specimen can be assimilated to a point-like dipole. If this is not the case, we shall have to consider the variation of the sensitivity function and, in case of non-ellipsoidal samples, of the magnetization over the sample volume. If we take the simple case of the inverse coil pair in Fig. 8.13a and a sample of volume V oscillating along the x-axis, the instantaneous induced voltage will be obtained by generalization of Eq. (8.7) by integrating it over the sample volume. If we assume that the center of gravity of the homogeneous sample has coordinate x, we obtain

Ux(X, t) -- ~v Mx(r' x)gx(r' x)dr3"•

(8.11)

where r identifies a generic point within the sample. For ellipsoidal samples, Mx is uniform and Eq. (8.11) reduces to

mf

Ux(X, t) = -~

v gx(r, x)dr3.x,

(8.12)

where, in place of the sensitivity function at point x, we have an average over the sample volume, shifted by a distance x from the origin. It can be induced from Fig. 8.13c, showing the extended behavior of the sensitivity function gx(X), that a long sample in the x direction generates a low signal and practically no signal at all when its length is a few times the coil radius. An interesting consequence of this fact is that,

506

CHAPTER 8 Characterization of Hard Magnets

since ~o gx(x)dx = 0, the vibrating rod is not expected to contribute to the measured signal. To apply the vibrating sample principle in a permanent magnet measuring setup we basically need a stable and rugged vibrating assembly, a digitally driven magnetic field source (an electromagnet or superconducting solenoid), a lock-in amplifier for the voltage induced in the pickup coils, an auxiliary signal source synchronous with the frequency of oscillation of the sample, to be exploited for precisely driving the vibration amplitude, a field sensing system and, if required, a temperature controller. A computer is used for general control of the measuring procedure and for analysis of the results. A measuring arrangement implementing these requirements is schematically represented in Fig. 8.15. We summarize here the basic operations performed with a system like this one and a few general problems associated with the VSM method. (1) Field generation and control. The measurement of the magnetic moment of the test specimen as a function of the field strength can be performed by continuous variation of the field with time, as conventionally carried out with the hysteresisgraph method (sweeping mode). The voltage simultaneously induced in the pickup coils is simultaneously detected and processed to determine the magnetic moment. However, in order to improve the signal-to-noise ratio, averaging of the signal must be performed with a reasonable integration time of the order of I s or longer. The lower the signal, the longer the integration time and the larger the lagging of the magnetic moment with respect to the field. This problem can be addressed by generating the field in steps. After each step, the measurement of the magnetic moment takes place. Because of the non-linear behavior of the material, it might be desired to regulate the step amplitude along the hysteresis loop in order to make homogeneous the vertical resolution. This can be done by means of a real-time feedback procedure, based on the reading of the field in the gap by the Hall sensor, comparison with the target field value, and generation of the appropriate magnetizing current via a suitable algorithm. An interfaced DC source, driving a bipolar power supply connected to the magnetizing winding, is used to this purpose. A complete hysteresis loop can be traversed in several minutes, a far longer time than with the conventional hysteresisgraph method. In addition, the maximum available field in the gap of the electromagnet is lower when employed in the VSM mode because space must be allowed for the vibrating rod and the pickup coils. The latter, in particular, should be placed close to the sample for obvious sensitivity reasons [8.20] but, at the same time, keeping them at some distance from the pole

8.2 OPEN SAMPLE MEASUREMENTS

507

P C - Vibration control, field control, processing.

I

DC

Vibrating

Ref. magnet

head

to the

oven / cryostat

source

/

Bipolarpower supply

_

Ref. coils Sample

Pickup coils'[

i

p

.

1

I

.

g rod

/ .

.

.

/

Pickup coils

Hall plate

FIGURE 8.15 Scheme of vibrating sample magnetometer using an electromagnet as field source. The law of variation of the magnetizing current with time is defined by software, implementing real-time control of the field strength by means of feedback, driven by continuous reading by an interfaced Hall unit. The current is generated by means of a bipolar power supply driven by an interfaced DC source. The voltage induced in the pickup coils is amplified by means of a lock-in amplifier, whose internal reference signal, driven via a computer-controlled procedure relying on the signal generated by the vibrating reference magnet, is used to feed the power amplifier supplying the vibrator. Tracing a complete hysteresis loop can take several minutes.

508

CHAPTER 8 Characterization of Hard Magnets

faces helps in reducing the image effect (Fig. 6.11). In this way, the field lines emerging from the sample and intercepted by the coils are the least distorted by such an effect. (2) Test sample arrangement and vibrating system. A permanent magnet test sample most commonly comes shaped as a 2 - 3 m m diameter sphere. With the usual size and arrangement of practical pickup coils, during the measurement it lies in a region of uniform sensitivity function. The spherical geometry is obviously the ideal one because it guarantees uniform magnetization in the sample and accurate retrieval of the effective magnetization curve after correction for the demagnetizing field. It is also very easy to prepare small spheres using the method of random grinding mentioned in Section 8.1. Other shapes, even irregular ones, can be tested as well. For cylinders and parallelepipeds, the measurement will provide the average polarization in the sample and the magnetometric demagnetization coefficient will be used for field correction [8.24]. With magnetic tapes and thin films, the disk geometry can be conveniently adopted and the associated demagnetization factor will approximately correspond to that of the oblate ellipsoid with the same axes length. For prismatic thin films, the magnetometric demagnetizing coefficients calculated by Aharoni could be adopted [8.25]. Notice, in any case, that the combination of small thickness-todiameter ratio and hard magnetic properties make even an approximate correction for the demagnetizing effect acceptable in many cases. The test sample is firmly held in a small container, which is screwed into the end of the vibrating rod. If the material is anisotropic, grinding a small sphere out of the bulk wipes out any visible information on the direction of the macroscopic anisotropy axis. We can recover it by letting the spherical sample freely orient itself in a sufficiently high field. For a precise determination, we can search for the orientation of the sphere in a VSM setup associated with the maximum value of the remanence. Figure 8.16a provides an example of hysteresis loops measured in a 3 m m diameter sphere of anisotropic Ba ferrite along different directions in a defined plane. Successive directions are tested and the preferential axis is determined by making the sample rotate first around a generic axis, to identify a plane containing the anisotropy axis, then around an axis perpendicular to such a plane. We see here how one can easily and advantageously exploit the open sample geometry of the VSM method and the spherical shape of the test specimen for obtaining physical information on the properties of the material, which are difficult to obtain with the conventional closed circuit method. Preliminary to the measurement, the sample must be centered to find the saddle point of the sensitivity

8.2 OPEN SAMPLE MEASUREMENTS

0.4

509

Anisotropic bariu'm f ~

Isotropic strontium ferrite

0.4

,/f

0.2

0.2

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/

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t !

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.

500

.

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.

1000

FIGURE 8.16 (a) Hysteresis loops measured with a VSM on an anisotropic sintered BaFe12019 sample, tested as a 3 mm diameter sphere. The preferential direction (0~ and the distribution of the easy axes around it on a plane, induced from the value of the remanence, are determined by making the sample to rotate first around a generic axis, to identify a plane containing the anisotropy axis, then around an axis perpendicular to such a plane. (b) The same experiment made on nominally isotropic SrFe12019 spherical sample provides nearly coincident hysteresis loops. (c) The hysteresis loop taken along the preferred direction is compared with the loop obtained on the parent bulk sample with the closed magnetic circuit and hysteresisgraph method.

function gx(Z) (transverse vibration), to avoid a n y d i s t u r b i n g a s y m m e t r y r e g a r d i n g the mechanical action of the charged pole faces on the sample, a n d to ensure reproducible m e a s u r e m e n t s . It is a s s u m e d that the coils h a v e a l r e a d y b e e n set in place before centering a n d they have

510

CHAPTER 8 Characterization of Hard Magnets

been tightly locked to the pole caps. The centering operation consists in starting the vibration, applying a sufficiently high DC field, and making mechanical regulations on the position of the sample along the x-, y-, z-axes (as defined in Fig. 8.14), in order to eventually leave the sample in the saddle point. This amounts to finding the position where the signal induced in the coils is minimum for displacements along the x-axis and is maximum for displacements along the y- and z-axes (e.g. Mallinsons's coils). Notice that at the end of this operation the sample is left at some remanence point. It might therefore be necessary to demagnetize it before starting the measurements. Notice also in Fig. 8.16c the comparison between the hysteresis loops obtained by the VSM and the closed circuit method in the previous anisotropic Ba ferrite. The coercive field appears to be slightly lower in the VSM determined loop. Since the time taken to traverse the whole loop is much longer in this case (about 20 min vs. about 100 s), it is plausible to attribute such a difference to the thermal fluctuation aftereffect. A fundamental requisite of the vibrating system regards the stability of frequency and amplitude of the oscillation imparted to the sample. To this end, the vibrating head is supplied by a reference signal generated by the lock-in amplifier and suitably amplified. Mechanical and electrical effects may, however, cause a drift in the performance of the vibrator. For this reason, a reference signal is generated in a pair of coils by a permanent magnet attached to the vibrating rod at a distant position from the measuring pickup coils, it is amplified and compared by software to the target signal. Any difference is numerically compensated and the driving signal of the lock-in amplifier is modified in order to recover the programmed vibration frequency and amplitude. There are no special restrictions as to the frequency of vibration, provided it is far from any mechanical resonance frequency of the apparatus. It is also useful to keep it incommensurate with the line frequency. Since frequency and amplitude of the vibration go hand-in-hand, it is required that their product does guarantee useful signal amplitude. Typically, f ranges between few Hz and some 100 Hz, with the sample oscillating from around 0.1 mm to a few mm. (3) Calibration, sensitivity, and noise. The calculation of the sensitivity function gx(Z)~ either in closed form or by means of a numerical procedure, does in principle provide the means for an absolute measurement of the magnetic moment. In practice, the burden of an absolute approach is not worth the expected resulting low accuracy. Consequently, calibration by comparison with the measured magnetization of a standard nickel sphere at a defined high field value is the rule (Fig. 8.1). However, this procedure also has certain limitations: (1) The measurement may be more or less

8.2 OPEN SAMPLE MEASUREMENTS

511

affected by the image effect, which is automatically taken into account by means of the comparative calibration procedure. However, the image effect depends on the permeability of the pole faces, while the calibration factor for the measured magnetic moment is determined for the defined value of the field applied to the Ni sphere (i.e. of the permeability). Such a factor is therefore expected to change slightly with the strength of the field in the gap, for example along the hysteresis loop. (2) If the test sample and reference Ni sphere are very different in size, calibration may be affected by non-flatness of the sensitivity function. (3) The comparative calibration with the Ni sphere is based on the dipole approximation. If non-spherical samples are to be tested (for example, particulate media and thin films) and their size is larger than the size of the Ni sphere, calibration can introduce appreciable systematic errors [8.26]. The most direct way to cope with this problem would consist in the absolute calibration of reference Ni samples as far as possible similar to the actual non-spherical test specimens. The maximum sensitivity of commercial VSM setups is around 10 - 9 A m 2 (10 - 6 emu). This is totally appropriate for the majority of testing requirements in bulk permanent magnets and recording media. Weak magnets and paramagnets can equally be tested, while the high value of the demagnetizing coefficient of the test specimen makes the VSM method unsuitable for the characterization of soft magnets. Foner has actually shown that, since the signal intercepted by the coils is expected to increase approximately inversely as the cube of the distance between the detection coils and the sample, while the noise decreases by decreasing the size of the coils, a very large sensitivity gain can be obtained by simultaneously decreasing the coil size and the coil distance from the sample [8.20]. Ultimately, a change in magnetic moment around 10-12A m 2 (10 -9 emu) should be detectable. In practical setups, small intercoil distances introduce certain complications. For example, all operations regarding sample insertion and centering become more difficult, the sensitivity function gx(Z) becomes sharper, and calibration uncertainties due to different size of sample and Ni standard are enhanced. In any case, maximum control and reduction of the background signal and noise must be sought, an increasingly difficult task with increasing sensitivity. In general, the following sources of background signal and noise are expected to play a role: (1) Signal from the paramagnetic or diamagnetic sample holder and, if any, from the sample substrate. In case of thin films or hard disks, the contribution from the substrate can be even larger than the contribution from the magnetic material. It must be subtracted by making measurements with and without the magnetic layer. (2) Interference from sources of

512

CHAPTER 8 Characterization of Hard Magnets

electromagnetic fields in the neighborhood of the apparatus and within the apparatus itself. The coils are intrinsically insensitive to external variable fields, but compensation is never perfect. (3) Vibrations of the pickup coils. Spurious signals can be induced in the coils if they vibrate in the applied field because this, besides being very high, can also be slightly inhomogeneous. Vibrations are principally caused by mechanical coupling between the vibrating assembly and the coils. They represent a troublesome problem because the ensuing signal, having the same frequency as the signal induced by the vibrating sample, cannot be filtered out by the lock-in amplifier. The simplest way to minimize the effect of vibrations is by locking the coils to the magnet. An additional countermeasure consists in interposing vibration dampers or even active antivibration elements between the vibrating head and the rest of the apparatus [8.27]. (4) Johnson noise generated in the coils. A few thousand turns of small diameter copper wire (0.1 m m or less) are normally used in each coil, which implies a resistance R around 100 12 and higher. Johnson noise has a white spectrum of density ~(f) = 4kTR, with k the Boltzmann's constant and T the absolute temperature. If the measurement bandwidth is hf, the associated r.m.s, voltage is uj = 4 k T R A f .

The degree of reproducibility of the measurements made on permanent magnets using the VSM method is comparable with that expected for measurements made with the closed magnetic circuit method. This has been demonstrated by means of an interlaboratory comparison, where both methods have been applied [8.10]. For the VSM-based comparison, Ni spheres of diameter 2.79-2.99 m m have been circulated among the laboratories. We have mentioned the results concerning the closed magnetic circuit in Fig. 8.12. A similar analysis is presented for the VSM tested N d - F e - B magnets in Fig. 8.17. If outliers are excluded from the analysis it is found that the best estimates made by the different laboratories of the intrinsic coercive field Hc! fluctuate around the unweighted average value with a relative standard deviation cr = 3.7%. For the maximum energy product (BH)max, it is found cr-- 2.3%. For investigation on anisotropic materials, information on the magnetization value and hysteresis behavior only in the direction of the applied field may provide too limited information. The effective field can, in fact, be very different not in only in magnitude, but also in direction with respect to the applied field, and the simultaneous measurement of the components of the magnetization along the field direction and orthogonal to it (the x- and y-axes in Fig. 8.14) is generally required. The simplest way to achieve vector measurement is by adding a set of pickup coils in the direction orthogonal to the field direction. It is also possible to

8.2 OPEN SAMPLE MEASUREMENTS

513

2500

Nd-Fe-B

--, 2000

E

:1::~ 1500

1000

9

'

I

'

2

I

6

'

I

Laboratories

9

I

8

10

'

9

I

12

240 E

220 200

E

~-

18o 160 140

Nd-Fe-B '

I

2

'

I

4

'

I

6

'

Laboratories

I

8

'

I

10

'

I

12

FIGURE 8.17 (a) Intercomparison exercise on the measurement with the VSM method of the intrinsic coercive field Hcj and the maximum energy product (BH)max on N d - F e - B magnets. The measurements have been performed on spheres of diameter around 3 mm circulated among the laboratories. Each point represents the best estimate of the measured quantity made by each laboratory. The calculated relative standard deviation of the results around the reference value (the unweighted average), calculated after having excluded the outliers, turns out to be 3.6% for Hcl and 2.3% for (BH)max (adapted from Ref. [8.10]).

detect either mx or my with the very same coil assembly, p r o v i d e d the connections between the individual coils are modified in a suitable way. Four-, eight-, and twelve-coil configurations have been devised on purpose, as discussed in detail by Bernards [8.23]. Vector m e a s u r e m e n t s are frequently used in the investigation and characterization of recording media. If a small disk-shaped thin film or particulate media sample is placed with its plane coincident with the (x, y) plane and its hysteresis loop is d e t e r m i n e d as usual along the direction of the applied field (x-axis), we

514

CHAPTER 8 Characterization of Hard Magnets

have an indirect method for measuring the anisotropy field by taking loops at different angles of rotation of the sample around the z-axis (Fig. 8.18a). If the magnetization component My is simultaneously determined together with the component Mx, we obtain the instantaneous

Sample /

/~

g Easy axis

i',, Pickup coil (a)

Easy ax"

/.//"

(b) FIGURE 8.18 (a) Schematic view of a vector VSM setup employing two orthogonal sets of pickup coils and circular test sample (not in scale). The disk-shaped sample is vibrated along the z-axis. Under sufficiently high field I-I,, the anisotropy constant of the material can be determined by measuring the component My of the magnetization as a function of the angle 8. (b) In perpendicular/oblique-anisotropy recording media (uniaxial anisotropy), there is interest in determining the magnetic properties as a function of the angle made by ~ with respect to the normal to the sample plane. To this end, the test plate is placed at the center of the gap, with its plane perpendicular to the (x, y) plane, and it is rotated by an angle cr about the z-axis. The demagnetizing field Ha is normal to the sample plane.

8.2 OPEN SAMPLE MEASUREMENTS

515

value of the torque per unit volume exerted by the field "r =/z0M • Ha -- la,oMyHa ~..

(8.13)

At equilibrium, this torque is balanced by the torque due to the magnetic anisotropy. In fact, for sufficiently high applied field strength, the residual parasitic torque due to domain wall processes is negligible and iMI = Ms. In the case of uniaxial anisotropy, the energy per unit volume arising from the rotation of the magnetization vector by an angle 0 with respect to the easy axis is EA = K1 sin20 (disregarding higher order constants) and the associated torque is "rA = (dEA/dO)~.. At equilibrium, ~'A = ~'H, that is la,oMyHa = 2K1 sin 0 cos 0, and the orthogonal component My oscillates when the sample is made to rotate around the z-axis. The measured maximum value (My)max, attained when 2 sin 0 cos 0 = 1, then provides

K1 -= tzo(My)maxHa.

(8.14)

There are cases where the easy axis does not lie in the plane of the film or tape test sample. This occurs, for example, in thin films for perpendicular recording (e.g. C o - C r layers), or in obliquely evaporated metal tapes. In these materials, one is often interested in investigating the dependence of the magnetic properties on the angle made by the applied field with the normal to the sample plane [8.28-8.30]. To this end, the sample plane, placed perpendicular to the (x,y) plane, is rotated about the z-axis (Fig. 8.18b). For each angle c~made by the normal to the sample plane with respect to the x-axis (i.e. the direction of the applied field Ha), the magnetization components Mx and My are measured and the components M• normal to the sample plane, and MII , lying within the plane, are obtained as M• -- Mx cos a - My sin a and MII = Mx sin a 4- My cos cr The problem with this measurement is the existence of a large demagnetizing coefficient Nd associated with the M• component, which is close to 1. The demagnetizing field Hd -- - N d M z combines with the applied field to provide the effective field Heff -- H a 4- H d. If the angle c~is kept fixed and the applied field amplitude is cyclically varied, the hysteresis loops (Mx, Ha) and (My, Ha) are measured. They do not evidently reflect the intrinsic magnetic properties of the material and a correction for the demagnetizing field must be applied. However, for a fixed value of the angle c~,the magnetization M changes in magnitude and direction with the value of the applied field and so does the effective field Heft . One can estimate the intrinsic magnetic behavior of the tape along the direction of the applied field by calculating the effective field Heff,x = H a - - H d cos o~ and taking the loops (Mx,Heff~x) and (My, Heft,x) [8.29]. Bernards and Cramer have developed a method where the angle made by

516

CHAPTER 8 Characterization of Hard Magnets

Heft with the sample normal is maintained constant along the measurement [8.31]. This is achieved by simultaneous variation and adaptation in small steps of the value of the applied field and the sample orientation. In this way, the magnetization components Mx and My are measured as a function of Heff and the related hysteresis loops (Mx, Heff) and (My, Heff) are obtained. Most VSM apparatuses in industry and research make use of an electromagnet as a variable field source. The limitations on the maximum available field in the gap are somewhat stronger with respect to the closed circuit arrangement because of the space required for the coils. The image effect is also a disturbing factor with narrow gaps. It turns out that full investigation of the high-coercivity high saturation field magnets, like SmCo5 and the N d - F e - B based compounds, is out of reach of conventional electromagnet-based VSM apparatus. The obvious solution to the requirement of very high fields is provided by the superconducting solenoids. It is a relatively complex solution, characterized by high running costs, because it requires continuous refrigeration at the liquid helium temperature. It is normally unsuitable to industry needs, but it is fundamental to many physical investigations. A superconducting solenoid is, in principle, the same device using copper wire at room temperature. In practice, things are rather more complex, both regarding the manufacturing of the solenoid and the operating procedures for the generation of fixed or slowly variable fields. A superconducting solenoid is realized by winding a superconducting cable on a former (made, for instance, of aluminum or stainless steel), upon which it is firmly clamped with the help of special impregnating resins, contrasting the strong Lorentz forces tending to expand radially and compress axially the solenoid (see Fig. 4.12). The superconducting cable is made of a very large number of superconducting filaments immersed in a resistive matrix (normally copper). The filaments are made of either NbTi, by which practical maximum fields around 7000 k A / m (~-9 T) are reached at the boiling point of He (4.2 K), or the more expensive and brittle Nb3Sn, normally providing maximum fields of the order of 16,000 kA/rn (---20 T). These compounds are Type II superconductors. In them, the Meissner effect (the disappearance of the superconducting state engendered by a magnetic field) takes place gradually above a certain critical field through the creation of an increasing number of vortexes, i.e. flux tubes enclosing a flux quantum. The vortexes are subjected to a force by the flowing supercurrent and, by rearranging themselves in the cross-section of the conductor, they create electrical fields, i.e. resistive phenomena and energy dissipation. With the use of a large number of micrometer-sized superconducting filaments, the motion of vortexes is restrained. Safe

8.2 OPEN SAMPLE MEASUREMENTS

517

operations are carried out by keeping the superconducting solenoid at 4.2 K, far from the critical temperatures of NbTi (Tc = 9.5 K) and Nb3Sn (To = 18 K). The solenoid is housed in a cryostat, a vessel shielding it from heat transfer by conduction, convection, and radiation, where it is immersed in the liquid He bath. The sample is contained in a variable temperature insert (VTI) placed in the bore of the solenoid. Figure 8.19 provides a simplified view of a VSM magnetometer using a superconducting solenoid field source [8.32]. This setup, developed at the Laboratoire Louis N6el, is specifically designed for high-sensitivity measurements (resolution 2 x 1 0 - 1 ~ 2) in the temperature range 1.4-300 K, with the variable field supplied either in sweeping or stepping mode. The pickup coils are of the axial type, as imposed by the geometry of the field source. In order to compensate the spurious flux variations due to vibrations, two sets of coils connected in series opposition are used. They have the same area-turn product, but obviously different sensitivity functions, so that the strong reduction of the background signal comes at the cost of a partial reduction only of the signal generated by the vibrating sample and the signal-to-noise ratio is improved. The vibrating rod, connected by C u - B e springs at the top and bottom of the VTI, is acted on by the oscillating force generated on a permanent magnet affixed to the rod by two AC-fed superconducting coils connected in series opposition, which generate an AC gradient at the position of the magnet. In order to minimize the AC field possibly generated in the measuring area, a second pair of coaxial coils of larger diameter and same area-turn product is employed and the whole vibrator is shielded. The sample is held at the center of a Perspex slab, which, by extending far from the pickup coils, does not generate any additional signal (see Fig. 8.13c). The background signal, measured without the test sample on the rod, is reproducible and corresponds to an equivalent magnetic moment of 1.5 x 10 -8 A m 2. It is precisely subtracted taking into account its weak dependence on field strength and temperature. A superconducting solenoid used at 4.2 K is a perfect or nearly perfect diamagnet. Any magnetized sample placed inside it will have its field lines distorted, according to the notional idea of the image dipole (Fig. 6.11). The main factors affecting the image effect in a superconducting solenoid are the ratio 2a/D of the pickup coil diameter to the solenoid bore diameter and the ratio A of the superconducting volume to the total volume of the windings. In a Type II superconductor, A decreases with increasing the field strength and so does the image effect. Figure 8.20 shows the dependence of the normalized VSM output on the field strength measured by Zieba and Foner in a N b - T i 9 T magnet working at 4.2 K [8.33]. The experiment was made using a field-independent

518

CHAPTER 8 Characterization of Hard Magnets Magnet

Spring

Shield

S

Vibrating coils

Vibrating rod j

Pickup coils

Sample

\ Superconducting solenoid ~.fJ-.f]

Spring

He bath

He exchanger

FIGURE 8.19 Example of VSM setup using a superconducting solenoid field source. The axial pickup coils are compensated by concentric coils connected in series opposition having same area-turn product and far lower sensitivity function. The vibration is generated by a couple of AC-supplied superconducting coils connected in series opposition, which create an AC force on a magnet affixed to the vibrating rod. The vibration frequency is 14 Hz and the peak-to-peak oscillation amplitude is 4 mm (adapted from Ref. [8.32]).

8.2 OPEN SAMPLE MEASUREMENTS

519

magnetic moment, obtained by a small solenoid carrying a precisely known constant current, and pickup coils with maximum-homogeneity interdistance d - - 4~a and diameter 2a = 0.66D. With zero applied field and the magnet in the superconducting state, the measured magnetic m o m e n t is nearly 8% lower than the actual value. It tends then to increase by increasing the superconducting current because of the decrease of the factor K The change speeds up on approaching the critical field (around 9000 k A / m (11 T) at 4.2 K). From a practical viewpoint, we do not perform absolute measurements and the measuring uncertainty is only due to the field dependence of the image effect, if calibration is made with reference Ni sample. A complete correction as a function of the applied field can be made by testing a calibrated magnetic moment, obtained by means of a current-carrying small coil, which is field independent.

0.94t Nb-Tisolenoid

o

"o

.N_ 0.93o Z

0.92 '

I

2000

'

'

'

'

I

4000

'

Field(kA/m)

'

'

'

I

'

'

'

6000

FIGURE 8.20 Normalized output of VSM using Nb-Ti superconducting solenoid as a function of the applied field. A precisely known magnetic moment, realized with a small solenoid carrying a constant current is measured with axial pickup coils of diameter 2a and interdistance d = x/~a. The diameter of the solenoid bore is D--3.03a. After magnet cool down, the image effect makes the measured magnetic moment nearly 8% smaller than the actual one. On increasing the field, the image effect decreases because magnetic flux creeps in the superconducting material and the output signal increases (adapted from Ref. [8.33]).

520

CHAPTER 8 Characterization of Hard Magnets

Coey and co-workers have realized an interesting development in VSM setups through an apparatus exploiting a permanent magnet-based variable field source [8.34]. As schematically shown in Fig. 8.21, the basic elements of this source are a pair of nested Halbach's cylinders made of N d - F e - B . We have illustrated this structure and the working principle of the Halbach's cylinder in Section 4.4 (Figs. 4.19-4.21). Thanks to the exceptionally high value of the anisotropy field (Hk -- 8 T in Nd2Fe14B), which make these magnets close to ideal magnets, any building block of the cylinder is transparent to the field generated by itself and the other blocks, and a stable high field normal to the cylinder axis is obtained in the

Vibrator" Y

Hallbachcylinders

Hallplate

~.

I //J //

Sample

Pickup coils FIGURE 8.21 Permanent magnet based vibrating sample magnetometer developed by Coey and co-workers [8.34]. A field Ha of fixed direction and variable amplitude is generated in the cavity by making two nested Halbach's cylinders made of Nd-Fe-B to rotate in opposite directions. Each cylinder generates a field of strength H0 in the bore and the resulting field varies with the angle of rotation a as H(a)= 2H0 cos a, making a full period for a 360~ rotation of the cylinders (adapted from Ref. [8.34]).

8.2 OPEN SAMPLE MEASUREMENTS

521

shell cavity. Since the generated field strength is H - - Brln r~ /~0 rg with r0 and rg the radius of the cylinder and the radius of the cavity, respectively, two nested cylinders endowed with the same ratio ro/rg generate the same field strength on their axis. In the setup shown in Fig. 8.21, these two cylinders, realized in practice by means of an octagonal structure, are rotated in opposite directions in small successive steps by means of a pair of motors, with a belt and pulley mechanism. The corresponding fields, of equal modulus H0, rotate with them. It is easily seen, looking at Figs. 4.20a and 4.21a, that the resulting field in the cavity has a fixed direction and strength depending on the angle a covered by the counter-rotating cylinders, according to the equation H ( a ) = 2H0 x cos a. This field then oscillates between +2H0 through a 360 ~ rotation of the nested cylinders. In the setup realized by Coey et al., the outer and inner cylinders have outside diameter 108 and 52 ram, inside diameter 52 and 26 mm, and height 115 and 65 mm, respectively. The resulting device turns out to be extremely compact, with a mass around 20 kg. A conventional electromagnet, providing approximately the same field strength with the same degree of homogeneity, has typically a mass around a few hundred kilograms. Each cylinder generates a field H0 = 470 k A / m and the peak value of the total field is therefore 940 k A / m , less than half the maximum value achievable in electromagnet-based VSMs. Considering that the field strength follows a logarithmic law on the ratio ro/rg and that the diameter of the cavity cannot shrink below obvious values, there is little room for increasing the maximum available field strength by increasing the volume of the cylinders.

8.2.2 Alternating gradient force magnetometer Increasing trends towards the miniaturization of devices and the development of artificial structures with faint magnetic moments can impose demanding requirements in terms of measuring sensitivity. The VSM is in general the preferred solution for the determination of low magnetic moments, down to the some 10-8-10 -9 A m 2, since it combines ruggedness and a relatively simple measuring procedure with good accuracy, solid experience in many laboratories, and availability of reliable commercial setups. It may happen, however, that magnetic moments lower than the typical VSM noise floor have to be determined. For example, one might be interested in following the basic magnetization process in isolated particles. A BaFe12019 single particle of about 5 ~m has

522

CHAPTER 8 Characterization of Hard Magnets

a moment of the order of 5 x 10 -11 A m 2, far below the VSM sensitivity. Besides the SQUID magnetometers, with their well-known problems of high running costs and impractically long measuring times, a solution to very low magnetic moment measurements is provided by the alternating gradient force magnetometer (AGFM), also called the vibrating reed magnetometer (VRM). It is realized as a sort of inverted VSM. We can imagine, in fact, energizing the pickup coils in a VSM by means of an AC current. These generate a non-uniform alternating field in the gap, so that a moment-bearing sample placed between them becomes subjected to an oscillating force. The sample displacement could then be revealed, at least in principle, across the vibrator. Under general terms, we can express the force acting on a magnetic dipole of moment m subjected to an inhomogeneous field H as F = V(m./.~0H).

(8.15)

The component of the force along the x-axis is therefore given by

Fx = ~

[

0Hx 0Hy 0/-/z ] mx--~x + m y ~ + m~~

(8.16)

and totally analogous expressions are obtained for the Fy and F~ components. With the typical configuration of the VSM pickup coils shown in Figs. 8.13 and 8.14, the field generated in the gap has defined symmetry properties. The force it exerts on the dipole depends, according to Eq. (8.15), on the value taken by the gradient of its components Hx, H v, and H~ along the reference directions. This force is then proportional to the sensitivity function of the coils. If we assume, in particular, that the field is generated by a pair of identical coils connected in series opposition, like those described in Fig. 8.13, and that the dipole of moment m is placed at the center and directed along the x-axis, we find that the force acting on it is obtained from Eq. (8.15) as

F = Fx - ~ m ~.OHx Ox

(8.17)

For symmetry reasons, Fy = i~om(OHx/Oy) and Fz = I.~om(OHx/OZ)are equal to zero on the (y,z) midplane. With a sinusoidal current i ( t ) - io sin ~ot supplying the coils, a sinusoidally varying force Fx(t)- F0 sin ~ot is applied to the sample. For small oscillations around the center, we obtain, according to Eq. (8.8), F0 = mgx(O)io, with gx(O) the value of the sensitivity function at the origin. A VRM was originally developed by H. Zijlstra in order to investigate the magnetic behavior of single micrometer-sized hard magnetic particles [8.35]. This device was based on the idea of placing the sample on the tip of a micro-cantilever beam, in turn located at

8.2 OPEN SAMPLE MEASUREMENTS

523

the center of a pair of AC supplied coils, connected in series opposition and 2 m m spaced. The typical maximum field gradient value obtained at the sample location is of the order 4000 A / m m m -1. Under these conditions, an oscillating force, as given by Eq. (8.17), is applied to the beam tip, which is made to vibrate. The ensuing deflection amplitude can be largely magnified by bringing the system made by cantilever and the sample to resonance. For sufficiently small oscillation amplitudes, the resonating system, characterized by a quality factor Q, behaves linearly and the peak vibration amplitude is x0 = XDcQ, if XDC is the deflection suffered by the beam tip when subjected to the same maximum force F0 under static conditions. A 20 m m long golden wire with diameter 38 p~m, cemented at one end to a brass bar and endowed with a Q factor around 100, was used as a resonating beam in Zijlstra's setup. The oscillation amplitude of the beam tip was measured by observing with a microscope the stationary image of the deflected reed, as obtained by stroboscopic illumination. A modified version of Zijlstra's magnetometer, developed by Roos et al. [8.36], is schematically illustrated in Fig. 8.22a. Again, a golden wire reed (diameter 18 ~m, length 10mm) is used, but the mechanical vibration is converted into an electrical signal by means of a bimorph piezoelectric plate (20 m m x 1.5 m m x 0.5 mm), connected to the reed by means of glass fiber (length 70 mm, diameter 150 ~m). The sample is stuck at the tip of the reed using wax and it can be oriented under a DC field by heating the wax. The signal provided by the bimorph (resonance frequency around 62Hz), proportional to the magnetic moment of the sample, and the signal proportional to the slowly varying magnetizing field, measured by means of a Hall device, are then used to trace the hysteresis loop behavior. Given the microscopic size of the sample, a substantial negative contribution to the measured magnetization comes from the diamagnetic golden wire, which must then be subtracted from the total signal. A large improvement in the quality factor of the resonating system, which depends on the viscosity of the surrounding gas, can be achieved by enclosing the reed assembly in an evacuated holder. This is demonstrated by the behavior of the output signal vs. the vibration frequency for different air pressures shown in Fig. 8.22b. Under vacuum conditions (P -- 1 Pa), a quality factor as high as Q = 550 is obtained and a magnetic moment measuring sensitivity of the order of 10-13A m 2 is ultimately achieved. However, the use of stiffer reeds can be envisaged, so that the effect of air damping can be strongly decreased and vacuum can be dispensed with [8.37]. General use of an AGFM setup in the laboratory requires a certain degree of ruggedness, relative ease of operation in mounting and substituting the sample and elements of the resonating-detecting

524

CHAPTER 8 Characterization of Hard Magnets

Lock-in

Oscillat~ Magnet

(31ass fiber"',,, Piezoceramic

Sample

Hall probe

(a)

1500 >:::L -~ 1000 t-

v

Q.

WW-d

0

40 Pa

Pa

500

,

(b)

//1

,

,

i

.

61.5

.

.

.

i

,

62.0 62.5 Frequency (Hz)

FIGURE 8.22 Scheme of the alternating gradient force magnetometer using vibrating reed, developed by Roos et al. [8.36]. The reed vibration is transmitted to a piezoelectric element by means of a glass fiber. The non-homogeneous AC magnetic field is created in the gap of the electromagnet by a couple of coils connected in series opposition (diameter 3 mm, distance 2 mm). They are supplied at the resonant frequency of the reed, whose oscillation amplitude is amplified by the quality factor of the resonating system. The hysteresis behavior of the material is obtained by plotting the signal proportional to the DC field, provided by a Hall sensor, and the signal proportional to the sample magnetization, provided by the piezoelectric bimorph and amplified by the lock-in amplifier. The measuring sensitivity is strongly affected by the viscosity of the surrounding gas and can be increased by decreasing the gas pressure. This is demonstrated in (b) by

8.2 OPEN SAMPLE MEASUREMENTS

525

assembly, possible application over a range of temperatures, and some kind of calibration procedure. Flanders has developed two kinds of AGFMs which appear basically to satisfy these requirements. Figure 8.23a shows the arrangement of sample and support system, which incorporates the piezoelectric element, in the horizontal gradient setup [8.38]. The vertical gradient AGFM is shown in Fig. 8.23b [8.39]. Let us consider the first instrument. Here, the horizontal AC field gradient is generated along either the x- or the y-axis, depending on the specific coil arrangements. With the magnetic moment m directed along x, the force Fx ensuing from the x-directed gradient is given by Eq. (8.17), while with the y-directed gradient it is Fy = Ixom(OHx/Oy). For a cantilever rod of thickness d, length l, density 3, and Young modulus Y, the fundamental resonance frequency is

foc-- 27r -g l

"

If a mass ms is fastened at the end of the cantilever of mass mc, the resonance frequency decreases to the value fos ---focx/[mc/(mc + 4.2ms)] = focx/-R. The complete cantilever system (bimorph plus extension with fastened sample) then resonates at the frequency fo =

fo~

,

(8.18)

~oc/fob)2 'q- 1/R where fob is the fundamental resonance frequency of the free bimorph. Extensions made of either glass, plastic, or copper have been used in this device, with lengths varying between 10 and 70 m m and thicknesses between 0.12 and 0.6mm, depending on the material employed. Experiments have been carried out at frequencies in the range 1 0 0 H z - l k H z . The output voltage V0 generated by the bimorph, detected by metallic contacts at the clamped end of the bimorph, is supplied to a high impedance pre-amplifier, feeding the lock-in amplifier. This voltage can be calculated and is approximately given,

the behavior of the voltage measured at the output of the lock-in amplifier around the resonance frequency. The quality factor of the resonating system passes from Q = 70 (P - 105 Pa, atmospheric pressure) to Q = 190 (P = 40 Pa) and Q = 550 (P -- 1 Pa) (adapted from Ref. [8.36]).

526

CHAPTER 8 Characterization of Hard Magnets

Clamp

zT

Contacts

Ceramic mount Rubber~

Piezo bimorph

ro~

~'~ i / kx\\\\\\\\\\\\\\\x~

..I] II

~1211Jklh2/,/,2/2,1d~ I I /

Glass/quartz Glass

.._ 11D e r

- - - ' l l =] J~

Extension

1

<~1 i ~ S[~~mample

!1).

Bimorph

Thermal shields

k

x"

Glass mount ~

i

I Fz, mple mocouple

I Side.view l (a)

(b)

FIGURE 8.23 Horizontal gradient (a) and vertical gradient (b) AFGM setups developed by Flanders. The conventional coil pair shown in (a), provides a gradient of the AC field along the x-axis. It can be replaced by a different coil arrangement, in order to achieve the gradient (i.e. the alternating force) along the y-axis. The piezo bimorph, the extension rod, and the sample are tuned to mechanical resonance as a whole. In the other setup, a vertical AC field gradient is generated by suitable coil arrangement (see, for example, Fig. 8.14). The vibration of the sample (having moment directed along the x-axis) along the z-direction is transmitted to the cantilevered piezoelectric bimorph by means of a glass or quartz fiber, fastened to the tip of the bimorph. This geometry is useful for measurements upon a wide range of temperatures, because the bimorph plate is far from the sample and thermally shielded from it, so that its temperature is always constant. Both types of magnetometers can operate at different frequencies, ranging between about 100 Hz and I kHz (adapted from Refs. [8.38, 8.39]). for x-directed AC gradient, by the expression V0 = 3FxG(Ib +

le)g31/2Wbdb,

(8.19)

w h e r e G is the mechanical gain, equal to the Q factor at resonance, Ib and Ie are the lengths of b i m o r p h and extension, respectively, g31 is

8.2 OPEN SAMPLE MEASUREMENTS

527

the transverse piezoelectric modulus, wb and db are width and thickness of the bimorph. Similar expressions for fo and V0 are found in the vertical gradient magnetometer, where the sample is subjected to the z-directed alternating force F ~ - ~m(OHx/Oz). Here, the resonating object is the bimorph cantilever only, with the additional masses of the fastened fiber and the sample. The value of the resonance frequency fo is obtained by applying to the resonance frequency of the free bimorph fob the correction for the presence of such masses. We therefore write

fo = fob~fmb/(mb + 4.2mrs),

(8.20)

where mb is the mass of the bimorph and mrs is the combined mass of fiber and sample. In a similar way, the output voltage V0 will be provided by Eq. (8.19), posing the length le --0. The vertical gradient AGFM appears to display a somewhat superior performance with respect to similar existing devices, being endowed, in particular, with the following properties: (1) Magnetic moments can be measured over a wide range of values by changing the combination of bimorph and extension rod. It can be adapted, in particular, to measure relatively massive specimens. Accurate measurements from m = 10 -3 A m 2 (1 emu) to m = 6 x 10 -9 A m 2 (6 x 10 -6 emu) have been demonstrated [8.40]. (2) The sample temperature can be varied without influencing the resonance parameters because the bimorph is located in a constant temperature environment. (3) The sample orientation can be changed without remounting the specimen. (4) Specimen mounting is simplified. (5) Magnetic impurities in the piezoelectric bimorph do not contribute to the signal. A critical aspect of the measurements with the AGFM lies in the calibration procedure. The standard method based on the use of a reference sample with known magnetic moment is not always applicable. In some cases, for example, the test specimen cannot be removed from the reed and a separate reed housing the reference sample may not duplicate the behavior of the reed in use. The compensation method, based on the use of a small coil having calibrated turn-area NA and carrying a precisely known direct current i is frequently used. This coil, having its plane normal to the field gradient, is made to surround the sample. The current in it can be regulated up to the point where the total measured moment is zero (i.e. the force on the ensemble made by the coil and the test specimen becomes small to the point of vanishing). The sample moment is correspondingly determined as m = NAi. Its measurement is therefore reduced to the measurement of the current providing the null moment

528

CHAPTER 8 Characterization of Hard Magnets

condition [8.41]. Of course, the compensation extends also to that part of the sample holder that extends into the region of the non-zero field gradient, whose moment contribution should be separately evaluated. In addition, the coil mass adds to the mass of the specimen and the copper wire might provide a spurious contribution if it contains magnetic impurities. Direct use of the theoretical prediction, such as the one provided by Eqs. (8.17) and (8.19), could be envisaged. The associated relative uncertainty is predicted, for the horizontal gradient AGFM in Fig. 8.23a, to be of the order of 10% [8.38]. We have so far assumed that under common measuring conditions the magnetic moment is always collinear with the DC field. The vibrating assembly centered on the use of the cantilevered bimorph sensor is perfectly appropriate to reveal the unidirectional forces arising when m only changes in amplitude and not in direction. If it happens that the vector behavior of the magnetization is important or it needs to be revealed, a different magnetization sensing method should be applied. Let us consider the basic inverted coil pair arrangement and coordinate system of Figs. 8.22 and 8.23a, with the magnetic moment m = m x ~ + m ~ placed at the center and the vibrating reed directed along the z-axis. Since the coils generate a field gradient at the sample position both along the x-axis and in the radial direction, forces will arise along the three axes. The force along z is ineffective because it is compensated by the reed, and the bending forces acting on the reed tip along the x- and y-axes are, according to Eq. (8.15) and by virtue of the symmetry of the system, OHx OHy F x = t-~om x - - ~ - , F y = la,om y Oy

(8.21)

In order to reveal the mx and my components, Zimmermann et al. [8.42] have developed a vectorial magnetometer, where the reed is formed with the final portion of a single-mode optical fiber, supplied at a wavelength of 780nm by a laser diode. The light beam is focused onto a semiconductor sensor (position-sensitive detector), from which the position of the spot is retrieved. In order to calibrate the mechanical response of the reed with the sample mounted, an acoustical excitation of precisely known intensity at the reed position is applied. Figure 8.24 shows an example of vector measurement on a BaFe12019 single particle, with diameter around 2 ~m, oriented with the easy axis at 90 ~with respect to the applied DC field. The evolution of the magnetization in the direction perpendicular to the field, related to the nucleation process, is demonstrated.

8.2 OPEN SAMPLE MEASUREMENTS

529

1.50x10 -11

I o~r i o/.~

%

<.%

/i

t(l)

E E

.

0.00

O

O .m

O r O~

-i!~ I-v" \

\o ~/

oo.oO08~ -1.50xl 0-11 -600

m ,

,

,

,

x

,

,

0

,

,

,

,

600

Field (kA/m)

FIGURE 8.24 Vector AGFM measurement on a BaFe120~9 single particle, with diameter around 2 ~m and easy axis (y direction) oriented at 90~ with respect to the applied DC field (x direction). The observed variation of my vs. applied field provides information on the irreversible magnetization processes (adapted from Ref. [8.42]).

An original application of the alternating field gradient principle has been realized by Asti and Solzi, with their vibrating wire susceptometer (VWS) [8.43, 8.44]. It is in essence a remanent moment and differential susceptibility measuring device, which dispenses with the DC field and is specifically suited to high-sensitivity measurements in a wide temperature range. Instead of a reed, a wire is made to vibrate by the AC gradient, which is not generated by coils, but by a tiny soft ferromagnetic platelet. The basic scheme of the VWS setup is shown in Fig. 8.25. The sample is fixed at the midpoint of a 300 mm long 0.04 m m diameter tungsten wire, held in place by means of a ceramic bar and a spring. It faces at a distance of about I m m the local gradient generator (LGG), an 8 m m long, 2 m m wide Fe-(3 wt%)Si platelet cut from a single crystal sheet. Its major side is parallel to the [001] easy axis and is aligned with the direction of the wire (x-axis). An AC field is applied along the x-direction by means of an external coil, by which the LGG is periodically magnetized close to saturation. The sample is eventually subjected to the field He, the sum of the applied field and the oppositely directed stray field emanating from the LGG, which is characterized by a transverse gradient. With the

530

CHAPTER 8 Characterization of Hard Magnets

Oo,.z

/ tube

Thermo- I < ..... ]NNN. H?er~[~~.~~Cerami~ regulator

|/=..

Lock-in il I

r~

Sample / /N Fe-Si single crystal !

FIGURE 8.25 Vibrating wire magnetometer. The sample is stuck onto a tungsten wire, which vibrates under the transverse AC field gradient generated by the local gradient generator (LGG). This is an Fe-Si elongated platelet cut out of an Fe(3 wt%)Si single crystal, with the longer side parallel to the [001] direction. The LGG is magnetized at a convenient frequency (typically few hundred Hz), as dictated by wire resonance, and peak field value (few hundred A/m) by means of an external coil. The wire vibration is sensed by means of a piezoelectric device. The length of the wire, which is held in place by means of a ceramic bar and a spring, is 300 mm. The magnetometer can typically operate up to a temperature close to the Curie point of the Fe-Si LGG (---750 ~ Minimum detectable magnetic moment is estimated to less than 10 - u A m 2 (taken from Ref. [8.43]). magnetic m o m e n t m of the sample aligned with the x-axis and the field gradient directed along the z-axis, we obtain from Eq. (8.15) that the sample is subjected to the z-directed alternating force F z = liom(OHe/OZ). In the general case, the sample has remanent m o m e n t mr and is characterized by the apparent differential susceptibility Xa-X/(1 + NdX), where Nd is the demagnetizing coefficient and X is the effective susceptibility. If X is different from zero, there is a time-varying

8.2 OPEN SAMPLE MEASUREMENTS

531

contribution to the magnetic moment and the total force can be written as 0He X H e ~0He Fz =/J,0mr-~z +/J'0V 1 + Nd----------~ Oz

(8.22)

where V is the sample volume. In response to an oscillating effective field He(t) = Heo sin ~ot, the wire will vibrate under the transverse force

0Go

vx

OH2o

G(t) =/~0mr 0z sin cot +/~0 4(1 + NdX) (1 -- cos 2cot) 0---Z"

(8.23)

The response at the fundamental frequency then provides a measure of mr, while the field-dependent part of the magnetization is associated with the double frequency. To reveal the vibration, a piezoelectric sensor placed in contact with the wire is employed.

8.2.3 Extraction m e t h o d In practical applications of permanent magnets, it is often preferable or even required to magnetize the individual elements before assembling them in the final device. It is a particular possibility offered by the rareearth based magnets of behaving in quasi-ideal fashion so that they can resist their own demagnetizing field and the field created by neighboring magnets, and thus complex assemblies can be built with the individual pieces already charged. Quick and simple testing of these pieces before assembly is desirable. The extraction method does provide a practical approach to the measurement of the magnetic moment of the permanent magnets, which can combine good accuracy with relatively inexpensive apparatus and ease of use. In its classical realization, a magnetized bar specimen is removed from a rest position, where it is fully inserted within a short search coil, to a distant place, where flux linkage with the coil is zero. The ensuing flux variation, measured by a fluxmeter connected with the coil, is A~ = BA = i~o(M- Hd)A --/~0MA(1 - Nd), if A is the crosssectional area of the bar sample and the search coil is assumed to be closely fitted at the start upon the bar mid-section. The magnetization M is then determined introducing the known value of the ballistic demagnetizing factor Nd. This method applies basically to cylindrical and parallelepipedic specimens and becomes increasingly inconvenient with sample shortening because the coil fixture becomes difficult to accomplish and unreliable. However, high coercive field materials often come as short samples because the induction behaves nearly linearly in the second quadrant and the maximum energy product is obtained with a high value of the demagnetizing coefficient (Nd ~ 1/2). An elegant solution to the measurement of the magnetic moment of a short sample by the extraction

532

CHAPTER 8 Characterization of Hard Magnets

method is offered, as in the VSM and AGFM methods, by the application of the reciprocity principle. The fundamental Eq. (8.5) shows that any coil brought in proximity to a small sample having magnetic moment m is linked with a certain amount of flux ~ = k(x, y, z).m, where k(x, y, z) is the coil constant at the sample position. If we apply the previous sample withdrawal operation starting from a rest position where the coil constant has a convenient value, we obtain the sample magnetization from the measurement of the ensuing flux variation and knowledge of the geometrical properties of the coil. The sample does not actually need to be considered point-like, but it is certainly desirable that the coil constant is uniform over the sample volume V. If we generalize the reciprocity relationship through the expression = f v k(x, y, z).M(x, y, z)dV

(8.24)

and we assume that the coil has axial symmetry, besides being endowed with uniform coefficient kx over the region occupied by the sample, we obtain

= kx f v Mx(x, y, z)dV = kxmx.

(8.25)

A solenoid coil excellently satisfies the condition of uniform kx, but it is not convenient for practical use, while Helmholtz coils allow easy and accurate positioning of the sample. For a filamentary Helmholtz pair of radius a, with N-turn coils, we have kx(x)= Bx(x)/i s = 0.71551a,o(N/a). Figure 8.26a provides a schematic view of sample arrangement at the center of a Helmholtz pair and of the determination of its magnetic moment by withdrawal and measurement of the flux cI). It is assumed that thick coils are used, with sides w and h of the winding area such as to ensure maximum uniformity (see Section 4.2). The shaded area about the center approximately delimits the region where the relative variation of kx is lower than 1%. It is apparent that a good, wide range of specimen sizes can be accommodated with appropriately designed Helmholtz pairs. If a conventional electronic fluxmeter is used to integrate the flux variation, possible correction of the instrument reading for the coil resistance should be considered. It should be stressed that sample rotation by 180~ around the y-axis can equally provide the variation of linked flux required for the determination of the magnetic moment. Both withdrawal and rotation are contemplated in the IEC Standard 60404-14 [8.45]. The same standard suggests an arrangement for the measurement of the saturation polarization of the sample, which makes use of an assembly of high coercivity permanent magnets to provide the magnetizing field

8.2 OPEN SAMPLE MEASUREMENTS

533

Z~

JI

t h

t

Fluxmeter

X

~

Sample

(a) Permanent

magnets

. . . . . . .

Helmholtz pair

(I

...........

....... .....0 ~

I . . . . . . .

~

,,

'."...... .' ".'..'~........

.

.

Slide

I I

'i1~

....

.

~ I

.

2>

Specimen

i

(b) FIGURE 8.26 Measurement of the magnetic moment of permanent magnets by sample extraction. The flux variation Aq~detected upon withdrawal of the sample from the center of a Helmholtz pair up to a distant position is proportionally related to the component of the magnetic moment along the coil axis. The saturation magnetization of hard materials can also be determined using an assembly of high-coercivity permanent magnets to generate a conveniently high field at the center of the Helmholtz pair. (Fig. 8.26b) [8.46]. The recoil permeability of these magnets is close to zero and no interference by the image effect is expected. There is an obvious limitation to the available maximum field values and saturation can be achieved only in moderately hard magnets.

534

CHAPTER 8 Characterization of Hard Magnets

Helmholtz coils are sensitive to the axial component mx of the sample moment. If the magnetization is not aligned with the x-axis, but it has a defined direction, the measurement will be repeated after successive 90 ~ rotations of the sample in order to detect the my and mz components. The arc magnets used in motors, which are radially magnetized, represent a case of special interest [8.47]. Let us assume that the arc of angle 2a0 and volume V, sufficiently thin to be considered uniformly magnetized, is placed at the center of the coils, where it is bisected by the x-axis. If M is the value of the radial magnetization, an infinitesimal arc element making an angle a with the x-axis has moment component d m x - - M V ( c o s a / 2 a o ) d a , which, integrated between - a 0 and +a0, provides for the measured magnetic moment, mx = MV(sin ao/ao). If it is assumed that the arc sample covers a region where kx is uniform, mx is directly measured upon sample withdrawal, according to Eq. (8.25). The radial magnetization is therefore obtained as M-

mx ao V sina0

(8.26)

Interference by external variable fields is a disturbing factor and it should be suppressed as far as possible. To this end, we can connect in series opposition to the Helmholtz pair a compensating coil, totally uncoupled to the sample, having equal turn-area product. Alternatively, the whole assembly can be shielded, providing an aperture for sample pulling. In the latter case, however, the flux lines emerging from the sample are perturbed and an absolute measurement becomes impossible. Calibration with a reference permanent magnet, the same size as the test specimen, is required. If the permanent magnet assembly in Fig. 8.26b is used, a pure Ni reference sample is the best choice. An interesting variant of the extraction method involves the movement of the sample between the centers of two identical search coils connected in series opposition and placed at a convenient distance along their common axis x. At the start, the sample is located at the center point of coil I and the linked flux is ~1- No flux is linked with coil 2. The sample is then pulled at the center of coil 2. Since this coil is oppositely connected and no residual flux is linked with coil 1, the globally linked flux is now -~1. The global flux variation is A~ = 2~1. If kx is the coefficient of the two coils and the magnetic moment is rex, a fluxmeter integrating the generated voltage pulse will provide, according to Eq. (8.25), the quantity A ~ = 2k~mx.

(8.27)

8.2 O P E N SAMPLE MEASUREMENTS

535

By supplying an axial field H a by means of a solenoid, one can make a determination of the polarization J = I~o(mx/V) as a function of it. The effective field H can be calculated by subtracting the demagnetizing field and the magnetization curves can be obtained. A modified coil arrangement, allowing one to apply the extraction method using an electromagnet as field source, can also be envisaged. However, the image effect may markedly affect the results and some form of compensation or calibration with reference samples of identical size should be implemented. For a complete characterization of high coercivity materials, the field should be applied by means of superconducting solenoids [8.48, 8.49]. Figure 8.27a provides an example of pickup coil assembly, where the magnetic moment is determined by displacement of the sample over a length of about 4:2 m m (travel time ---0.5 s) [8.4:8]. This system is designed for use within a superconducting solenoid. Figure 8.27b

; .,"-;~FdT"d]~3

Pickup

coils

A N

A'N

i

_

"

%%

I

............ i ................. , ..... 2 . " :

/

:

v 0.0

.........

I

:

-I

............ :.............. ~........................... i! I: I l

_

-0.5

l i

....... 1 .... } ......... ","" } ............. " ..............

:: !

", ~

B~

I

(a)

I

:

42 mm

_

i

i

I--

l

....

Sample!

I I

Pickup coils

,, ! i

,

.................

-1.0 9

-40

.'"~'

9

|

-20

.

.

.

.

I

0

'.

9

9

9

,

20

.

.

.

.

40

x(mm)

(b)

FIGURE 8.27 (a) Vertical axis pickup coil assembly in the vector extraction m a g n e t o m e t e r of Dufeu et al. [8.48]. It provides the vertical c o m p o n e n t mx of the magnetic m o m e n t of the sample w h e n this is m o v e d b e t w e e n point I a n d point 2. These points are at the center of two couples of concentric coils. The behavior of the coil coefficient kx(x) is s h o w n in (b). The external field is supplied by means of a s u p e r c o n d u c t i n g solenoid.

536

CHAPTER 8 Characterization of Hard Magnets

demonstrates the fact that the kx coefficient of the sensing coil system exhibits a flat maximum in correspondence to the sample rest points 1 and 2. This means that the measuring accuracy is very little dependent on the precise location of such points and that the sample size can be varied within relatively wide limits. It should be stressed that the inverted character of coils 1 and 2 is instrumental to the cancellation of the interferences by external fields (which instead disturb the conventional extraction method using Helmholtz coils). Actually, this may not be sufficient for compensating the effect of mechanical vibrations, which can be a relevant source of background noise because they may occur under a very high applied field (up to 8 T). Each coil is then made of inner and outer layers (A-A / and B-B ! in Fig. 8.27a), having the same turn-area value and connected in series opposition and altogether they efficiently reject the external disturbances. The outer windings A / and B~ have their number of turns reduced with respect to the inner windings A and B roughly in the ratio of their diameter squared. They correspondingly have quite a low relative value of the coefficient kx, thereby introducing only a minor compensation of the flux variation generated by the sample displacement. It is fair to remark that the double-coil extraction magnetometer can have, as already remarked for the VSM and AGFM methods, general application in the field of small magnetic moment measurements due to its excellent sensitivity (around 1 0 - ' A m2). Consequently, besides permanent magnets, it can be applied, to the characterization of thin films, recording media, artificial structures, and weak magnetic materials.

8.2.4 P u l s e d field m e t h o d Applied magnetism is fond of extreme properties. The search for extrasoft and extra-hard magnets has produced impressively good materials but, inevitably, substantial problems in their characterization. The rareearth permanent magnets belong to the category of difficult materials, challenging the measuring capabilities of existing commercial setups because the same reason that makes them so attractive from the viewpoint of applications, the very high values of coercive field and energy product, strains conventional field sources, even for incomplete characterization. In addition, these materials tend to be somewhat unstable and prone to aftereffects. Sintered N d - F e - B , SmC05, and Sm2C017 magnets exhibit coercivities typically ranging between 1000 and 2000kA/m, but a high-performance electromagnet with Fe-Co pole caps can barely demagnetize a magnet with coercive field beyond about 1500 kA/m. The demagnetization curve can in general be obtained, but it requires

8.2 OPEN SAMPLE MEASUREMENTS

537

the relatively complex procedure described in Section 8.1, calling for previous specimen magnetization to saturation using either a pulsed field magnetizer or a superconducting source. One can actually perform complete material characterization using open samples and a VSM with a superconducting solenoid, but this is neither an economic nor a flexible approach to testing and there is little interest in it from the industrial viewpoint. For this reason, considerable interest has been attached in recent times to the development of hysteresisgraphs using pulsed field sources, which combine the generation of high peak field values with quick testing procedures, at relatively low capital costs and minimal running costs. Not all problems associated with this method have been solved or fully understood, but there is good industrial interest in it and progress is being made, to the point that pulsed field magnetometers (PFM) for general use in production and quality testing are nowadays available [8.50]. Pulsed field sources, such as those discussed in Section 4.3, capable of generating both oscillating and non-oscillating field transients, have been traditionally applied to routine magnetizationdemagnetization operations on industrial products. They can generate high enough field strengths to saturate rare-earth based magnets, as well as to fully demagnetize them. With the addition of a suitable sensing coil system and signal analysis setup, and sound recognition of the physical phenomena occurring when imposing fast magnetization rates, they can be employed in the realization of practical magnetometers for permanent magnets. A pulsed field source is a thick solenoid with bore diameter and length suited to sample housing in a region of uniform field, assembled in such a way as to display good mechanical strength and good thermal dissipation. We have discussed in Section 4.3 (see Figs. 4.10-4.12) these sources, which are supplied by discharging a bank of capacitors, and the way they can generate either a damped oscillatory field or a monotonically decaying field after attaining a maximum value. Typical parameters are listed in Table 4.1. Normally, the capacitor bank (C--103 ~F) is charged at a voltage of a few kV. Peak fields of the order of a few M A / m can be generated in the bore of the coil (having typical inductance L in the m H range and resistance R around 0.1 f~) upon the condenser discharge, by releasing a relatively modest energy amount of a few kJ. The oscillation period T - 2~r/I/1/LC- R2/4L2 is around a few ms and the time constant ~-= 2L/R of the damped oscillation is of the order of 1030 ms. Ideally, the measurement of the hysteresis loop in permanent magnets should be carried out under quasi-static conditions. The larger the sample, the longer should be T and T, and thereby the higher L and C, in order to minimize the effect of eddy currents. However, the increase of

538

CHAPTER 8 Characterization of Hard Magnets

energy and the cost of the magnetizer for significantly increasing T and ~would make it economically unattractive. The general arrangement of sample and pickup coils for measurement of the hysteresis loop in a PFM can be as illustrated in Fig. 8.28. In these assemblies, special care must be devoted to the precise adjustment of the compensation for the signal induced in the sensing coils by the time-varying applied field. It is easily realized that the signal generated by the variation of the magnetization in the sample can be only a small fraction of the signal induced by the variation of the field. We see here all the advantage in sensitivity and accuracy of the previously discussed open sample methods, where the applied field is kept constant during the measurement. Two possible compensation schemes are shown in Fig. 8.28. The first one adopts the two concentric coils arrangement [8.51] (see also Figs. 8.19 and 8.27). The coils, connected in series opposition, have the same turn-area value NiAi = NoAo and, with the additional help of a series connected adjustable small coil, they cancel out the contribution of the applied field. At the same time,

PC

~

' H-coil ~

dH/dt

Acquisition

_'e'u.__ xj ITI

/

I/i

' I \

', !/

i:

--Sample Solenoid

(a)

J-coil"

ii

Air-flow

i i

/

1 i |

!

-

1

"

i

(b)

FIGURE 8.28 Cross-sectional view of typical coil assemblies in the pulsed field magnetometer. They are fitted in the bore of the magnetizer. The signal collected by the J-coils is compensated for the linkage with the applied field. The arrangement in (a) makes use of two concentric coils connected in series opposition, which have the same turn-area and different coefficient kx (adapted from Ref. [8.51]). The compensation coils in (b) are symmetrically placed with respect to the sensing coil surrounding the sample and are not linked with the flux lines generated by it. A small additional coil is added in series for fine adjustment of the compensation.

8.2 OPEN SAMPLE MEASUREMENTS

539

they link with the flux -- (kix - kox)mx~

(8.28)

where the coefficient of the inner winding is much larger than the coefficient of the outer winding kix ~ kox. It is assumed here that in the region occupied by the specimen the coefficient of the coils is uniform. If this is not so (for example, with short coils and large specimens), the general equation (8.24) will be used. With the arrangement shown in Fig. 8.28b, the compensating coils are symmetrically placed with respect to the moment-sensing coil and are not linked with the flux lines generated by the sample. The measured flux is then related to the magnetization as 9 = k x m x = k x M x V ~ i.e. Jx = ~ o ( ~ / k x V ) 9Of course, given the large background contribution from the applied field, compensation can never be perfect and the residual signal (zero signal) detected in the absence of the test specimen must be accounted for. Any measurement must then be repeated with and without the sample and the zero signal must be mathematically subtracted. The maximum geometrical accuracy and mechanical stability of the sample holder and coil assembly must be ensured in order to achieve the required measuring repeatability. Solenoid heating is a primary cause of poor repeatability and measures should be taken to avoid it, either by avoiding high repetition rates or by means of forced cooling. Notice that N d - F e - B magnets have extreme sensitivity to temperature variations, which also call for strict control of the room temperature. If the solenoid holder contains metallic parts (e.g. screws and bolts), it may become subjected to electrodynamic forces and ensuing vibrations during the field burst. They are not normally tolerated and these parts should be eliminated or reduced. There are two possible procedures in the determination of the hysteresis behavior of a permanent magnet using a PFM, according to whether a damped oscillatory field or an exponentially decaying transient field are employed. In the first case, there is an intrinsic asymmetry in the sequence of the applied field peak values and a complete major hysteresis loop can only be determined if the second positive peak field is still sufficiently high to bring the material into saturation. Figure 8.29 provides an example where this does not occur. Notice, however, that the chief pieces of information on the material behavior, the demagnetization curve in the second quadrant and the value of the coercive field Hc]~ are obtained to a good approximation. If the exponentially decaying transient field pulse is applied, the measurement of the hysteresis loop is made in more than one step. In particular, starting from the demagnetized state, a first pulse is applied, which brings the material into the working point in the second quadrant, passing through the tip point of the loop, possibly

~40

CHAPTER 8 Characterization of Hard Magnets

10

, ( 7 VSM

0.5

J

0.0

-0.5

-1.0

~ ''

'

'

'

I

.~

.

Nda0Fe63.6DY5AI0.4B1 .

.

.

.

.

-3000 -2000 -1000

.

.

.

.

0

.

I

'

1000

'

'

'

I

. . . .

2 0 0 0 3000

H (kA/m) FIGURE 8.29 Hysteresis behavior in a N d - F e - D y - A 1 - B magnet obtained under oscillatory damped field burst in a pulsed field magnetizer starting from the demagnetized state. The first peak field value is Hp1 = 2700 k A / m and time constant is ~'---20 ms. This measurement provides an example of incomplete material characterization because of insufficient peak field amplitude. The demagnetization curve described on returning from the first point of reversal at 2700 k A / m and the related coercive field should be considered as approximate. Loop centering is possible in spite of asymmetry, because the whole trajectory starts from and ends into the demagnetized state. Comparison with a hysteresis loop determined, after sample demagnetization, in a VSM setup using an electromagnet with maximum available field Hmax =

1600 kA/m is also shown.

lying in the saturated region. A second identical pulse with inverted polarities makes the sample attain the symmetrical working point in the fourth quadrant. A third pulse, with restored polarities, will close the loop in the second quadrant. Loop centering is ensured by the symmetry of the signal because the rest points are symmetrically located in the second and fourth quadrants. The field is determined by integration of the signal detected with sensing coils located in the bore of the solenoid, with known initial conditions (Ha -" 0). The measured applied field must be corrected for the demagnetizing field, which, using the J-coil arrangements shown in Fig. 8.28, will be associated with the magnetometric demagnetizing factor.

8.2 OPEN SAMPLE MEASUREMENTS

541

The PFM measurement must be associated with a defined calibration procedure because the absolute measurement based on the calculated J-coil coefficient can be affected by a large uncertainty. The calibration is based on the experimental determination of the coil coefficient in response to the presence of a known magnetic moment located in the position occupied by the test specimen. A calibrated magnetic moment can be formed with a small coil having turn-area NA, determined using a reference field source. By letting a precisely known AC current of r.m.s, value is flow in this small coil placed at the center of the J-coil and measuring the correspondingly induced voltage, the associated mutual inductance M12 is determined. The effective coil coefficient is correspondingly obtained as kx = M12/NA. Alternatively, a Ba-ferrite or soft ferrite specimen, pre-emptively characterized up to saturation in a VSM setup in association with a pure Ni sample, is tested. The coil coefficient is determined by equating the magnetic moments at a specified high applied field value (e.g. 400kA/m) obtained with both methods. A final point of fundamental importance for general acceptance of the PFM method concerns the role of eddy currents and the methods to account for it in order to recover a true quasi-static hysteresis behavior. The problem chiefly arises in sintered N d - F e - B and S m - C o magnets, which have a metallic character, their resistivities being around 150 x 10 -8 and 60-90 x 10-8 f~ m, respectively. On a laboratory scale, it is expedient to use small samples (for example, 2-3 m m diameter spheres) and as high as practical T and T values. In this way, the correction to be applied to the hysteresis loop, enlarged with respect to the quasi-static one, in order to purge it of the additional area brought about by the long-range eddy currents, is minimal. On an industrial scale, relatively large samples must be used and the correction may be substantial. This can be performed either by experiment or by numerical modeling, both approaches in any case being based on certain approximations and simplifications. The experimental correction is based on the assumption that the contribution to loop enlargement, that is, the additional field contribution required to sustain a certain induction rate, is proportional to the magnetizing frequency [8.52]. In the conventional language of magnetic loss theory, this is called classical approximation. Given the present poor knowledge on the subject of losses in hard magnets, we tentatively accept this approximation and we assume that the field which is associated with a certain value of the magnetic polarization J on the loop at a certain magnetization frequency fl is H q , f l ) = Hh(J)q-Tfl, where y is a proportionality constant. H is the effective field, the difference between the applied

542

CHAPTER 8 Characterization of Hard Magnets

field and the demagnetizing field. At the different frequency f2, it will be measurements made under the same conditions at these two frequencies will then provide the constant = (H2 - H1)/(f2 - f l) and the "true" static field Hh(J) will be recovered. This correction is easily implemented by software for any point on the loop. A finite element simulation can alternatively be pursued in order to achieve loop correction [8.53, 8.54]. Certain simplifications, like imposing an axi-symmetric geometry and a defined permeability value, are also introduced in this case. This method can deal in principle with the condition of incomplete flux penetration in the sample, but from the viewpoint of material characterization such a condition, somewhat undefined, is not desirable. An important conclusion of these simulations is that, with a typical 5 m m diameter N d - F e - B sample, the correction to be provided becomes lower than 5% when the period T becomes longer than 5 ms. The question to be posed, in any case, regards the real meaning of the reconstructed hysteresis loop. It is known that the very nature of the magnetization process makes the role

Hq,f2)- Hhq)+ 3'f2. Two

0.4

0.2

0.0

-0.2 1.76 mT/s -0.4

4 mT/s -1140

-1120

-1100

H (kA/m)

FIGURE 8.30 A portion of the demagnetization curve in a Nd-Fe-B commercial magnet is shown for three different rates of change of the polarization. The measurements have been performed using a VSM setup. With dJ/dt decreasing by an order of magnitude, from 17.6 to 1.76 roT/s, the coercive field decreases by about 1.5%, because of magnetic viscosity (adapted from Ref. [8.56]).

REFERENCES

543

of measuring time relevant even in the absence of eddy current effects. This is due to the phenomenon of magnetic viscosity (aftereffect), which can be distinguished from time effects related to eddy currents because of its independence from material resistivity and sample size. It can be phenomenologically defined through a magnetic viscosity coefficient Sv, which describes the narrowing of the hysteresis loop on increasing the measuring time [8.9]. This coefficient can be given a microscopic physical interpretation in terms of thermally activated crossing of energy barriers by the local magnetization reversal processes, reflecting into a change of coercivity M-/c when two different field rates/:/ and /:/~ are considered. We can write, in particular, AHc = Sv ln(/:/~//~ [8.55]. On passing from the pulsed field measurement to a standard measurement made with an electromagnet or a superconducting solenoid, the magnetization rate does change by about five or six orders of magnitude and one may possibly incur a non-negligible additional contribution to the measuring uncertainty by magnetic viscosity. The experiments show that the hysteresis loop measurements on barium and strontium ferrites are moderately affected by the magnetic viscosity. In contrast, a large effect is observed in S m - C o and N d - F e - B magnets. Figure 8.30 provides an idea of the change of the demagnetization curve around the coercive field due to magnetic viscosity in a commercial sintered N d - F e - B magnet when the magnetization rate is changed by an order of magnitude [8.56].

References 8.1. H. Zijlstra, Experimental Methods in Magnetism, (Amsterdam: North-Holland, 1967), Vol. 2, 100. 8.2. J.E. Monson, "Recording measurements," in Magnetic Recording Handbook (C.D. Mee and E.D. Daniel, eds., New York: McGraw Hill, 1989), p. 396. 8.3. http://www.metallurgy.nist.gov. 8.4. D. Eckert and J. Sievert, "On the calibration of vibrating sample magnetometers with the help of nickel reference samples," IEEE Trans. Magn., 29 (1993), 3001-3003. 8.5. R.D. Shull, R.D. McMichael, L.J. Swartzendruber, and S.D. Leigh, "Absolute magnetic moment measurements of nickel spheres," J. Appl. Phys., 87 (2000), 5992-5994. 8.6. J. Sievert, H. Ahlers, S. Siebert, and M. Enokizono, "On the calibration of magnetometers having electromagnets with the help of cylindrical nickel reference samples," IEEE Trans. Magn., 26 (1990), 2052-2054.

544

CHAPTER 8 Characterization of Hard Magnets

8.7. J. Crangle and G.M. Goodman, "The magnetization of pure iron and nickel," Proc. Roy. Soc. Lond., A-321 (1971), 477-491. 8.8. M.P. Sharrock, "Particle-size effect on the switching behavior of uniaxial and multiaxial magnetic recording materials," IEEE Trans. Magn., 20 (1984), 754-756. 8.9. D. Givord and M.F. Rossignol, "Coercivity," in Rare Earth Iron Permanent Magnets (J.M.D. Coey, ed., Oxford: Clarendon Press, 1996), p. 218. 8.10. J. Sievert, H. Ahlers, J. Liidke, S. Siebert, L. Pareti, and M. Solzi, "European intercomparison of measurements on permanent magnets," IEEE Trans. Magn., 29 (1993), 2887-2889. 8.11. L. Pareti and G. Turilli, "Detection of singularities in the reversible transverse susceptibility of a uniaxial ferromagnet," J. Appl. Phys., 61 (1987), 5098-5101. 8.12. IEC Standard Publication 60404-5, Permanent Magnet (Magnetically Hard) Materials--Methods of Measurement of Magnetic Properties (Geneva: IEC Central Office, 1993). 8.13. F.E. Pinkerton and D.J. Van Wigerden, "Magnetization process in rapidly solidified neodymium-iron-boron permanent magnet materials," J. Appl. Phys., 60 (1986), 3685-3690. 8.14. ASTM Publication A977/A977M-02, Standard Test Methods for Magnetic

Properties of High-coercivity Permanent Magnet Materials Using Hysteresigraphs (West Conshohocken, PA: ASTM International, 2002). 8.15. H. Zijlstra, Experimental Methods in Magnetism (Amsterdam: NorthHolland, 1967), Vol. 2, p. 85. 8.16. E. Steingroever, Magnetic Measuring Techniques (K61n: Magnet Physik, 1989). 8.17. IEC Technical Report 61807, Magnetic Properties of Magnetically Hard Materials at Elevated Temperatures. Methods of Measurement (Geneva: IEC Central Office, 1999). 8.18. IEC Standard Publication 60404-7, Method of Measurement of the Coercivity of Magnetic Materials in an Open Magnetic Circuit (Geneva: IEC Central Office, 1982). 8.19. S. Foner, "Versatile and sensitive vibrating sample magnetometer," Rev. Sci. Instrum., 30 (1959), 548-557. 8.20. S. Foner, "Further improvements in vibrating sample magnetometer sensitivity," Rev. Sci. Instrum., 46 (1975), 1425-1426. 8.21. J. Mallinson, "Magnetometer coils and reciprocity," J. Appl. Phys., 37 (1966), 2514-2515. 8.22. A. Zieba and S. Foner, "Detection coil, sensitivity function, and sample geometry effects for vibrating sample magnetometers," Rev. Sci. Instrum., 53 (1982), 1344-1354.

REFERENCES

545

8.23. J.P.C. Bernards, "Design of a detection coil system for a biaxial vibrating sample magnetometer and some applications," Rev. Sci. Instrum., 64 (1993), 1918-1930. 8.24. H. Zijlstra, Experimental Methods in Magnetism (Amsterdam: North-Holland, 1967), Vol. 2, p. 20. 8.25. A. Aharoni, "Demagnetizing factors for rectangular ferromagnetic prisms," J. Appl. Phys., 83 (1998), 3432-3434. 8.26. J. Lindemuth, J. Krause, and B. Dodrill, "Finite sample size effects on the calibration of vibrating sample magnetometer," IEEE Trans. Magn., 37 (2001), 2752-2754. 8.27. E.O. Samwel, T. Bolhuis, and J.C. Lodder, "An alternative approach to vector vibrating sample magnetometer detection coil setup," Rev. Sci. Instrum., 69 (1998), 3205-3209. 8.28. J.P.C. Bernards, G.J.P. van Engelen, C.P.G. Schrauwen, H.A.J. Cramer, and S.B. Luitjensl, "Simulation of the recording process on Co-Cr layers with a VSM," IEEE Trans. Magn., 26 (1990), 216-218. 8.29. K. Ouchi and S. Iwasaki, "Analysis of perpendicular recording media using a biaxial vibrating sample magnetometer," IEEE Trans. Magn., 24 (1988), 3009-3011. 8.30. D.E. Speliotis and J.P. Judge, "Magnetic and thermomagnetic analysis of metal evaporated tape," J. Appl. Phys., 69 (1991), 5157-5159. 8.31. J.P.C. Bernards and H.A.J. Cramer, "Vector magnetisation of recording media," IEEE Trans. Magn., 27 (1991), 4873-4875. 8.32. D. Dufeu and P. Lethuillier, "High sensitivity 2T vibrating sample magnetometer," Rev. Sci. Instrum., 70 (1999), 3035-3039. 8.33. A. Zieba and S. Foner, "Superconducting magnet image effects observed with a vibrating sample magnetometer," Rev. Sci. Instrum., 54 (1983), 137-145. 8.34. O. Cugat, R. Byrne, J. McCaulay, and J.M.D. Coey, "A compact vibrating sample magnetometer with variable permanent magnet flux source," Rev. Sci. Instrum., 65 (1994), 3570-3573. 8.35. H. Zijlstra, "A vibrating reed magnetometer for microscopic particles," Rev. Sci. Instrum., 41 (1970), 1241-1243. 8.36. W. Roos, K.A. Hempel, C. Voigt, H. Dederichs, and R. Schippan, "High sensitivity vibrating reed magnetometer," Rev. Sci. Instrum., 51 (1980), 612-613. 8.37. H.J. Richter, K.A. Hempel, and J. Pfeiffer, "Improvement of sensitivity of the vibrating reed magnetometer," Rev. Sci. Instrum., 59 (1988), 1388-1393. 8.38. P.J. Flanders, "An alternating-gradient magnetometer," J. Appl. Phys., 63 (1988), 3940-3945.

546

CHAPTER 8 Characterization of Hard Magnets

8.39. P.J. Flanders, "A vertical force alternating-gradient magnetometer," Rev. Sci. Instrum., 61 (1990), 839-847. 8.40. P.J. Flanders and C.D. Graham, Jr., "DC and low frequency magnetic measuring techniques," Rep. Prog. Phys., 56 (1993), 431-492. 8.41. Th. Frey, W. Janz, and R. Stibal, "Compensating vibrating reed magnetometer," J. Appl. Phys., 64 (1988), 6002-6007. 8.42. G. Zimmermann, K.A. Hempel, J. Dodel, and M. Schmitz, "A vectorial vibrating reed magnetometer with high sensitivity," IEEE Trans. Magn., 32 (1996), 416-420. 8.43. G. Asti and M. Solzi, "A wide temperature range susceptometer," IEEE Trans. Magn., 32 (1996), 4893-4898. 8.44. G. Asti and M. Solzi, "Vibrating wire magnetic susceptometer," Rev. Sci. Instrum., 67 (1996), 3543-3552. 8.45. IEC Standard Publication 60404-14, Methods of Measurement of the Magnetic Dipole Moment of a Ferromagnetic Material Specimen by the Withdrawal or Rotation Method (Geneva: IEC Central Office, 2002). 8.46. E. Steingroever, "A magnetic saturation measuring coil system," IEEE Trans. Magn., 14 (1978), 572-573. 8.47. S.R. Trout, "Use of Helmholtz coils for magnetic measurements," IEEE Trans. Magn., 24 (1988), 2108-2111. 8.48. D. Dufeu, T. Eyraud, and P. Lethuillier, "An efficient 8 T extraction magnetometer with sample rotation for routine operation," Rev. Sci. Instrum., 71 (2000), 458-461. 8.49. J.K. Krause, "Extraction magnetometry in an ac susceptometer," IEEE Trans. Magn., 28 (1992), 3066-3071. 8.50. R. Cornelius, J. Dudding, P. Knell, R. Gr/Sssinger, B. Enzberg-Mahlke, W. Fernengel, M. K/ipferling, M. Taraba, J.C. Toussaint, A. Wimmer, and D. Edwards, "Pulsed field magnetometer for industrial use," IEEE Trans. Magn., 38 (2002), 2462-2464. 8.51. K. Seiichi and K. Giyuu, "Pulsed field magnetometer for low temperature study of high-performance permanent magnets," IEEE Trans. Magn., 36 (2000), 3634-3636. 8.52. R. Gr6ssinger, G.W. Jewell, J. Dudding, and D. Howe, "Pulsed field magnetometry," IEEE Trans. Magn., 29 (1993), 2980-2982. 8.53. G.W. Jewell, D. Howe, C. Schotzko, and R. Gr6ssinger, "A method for assessing eddy current effects in pulsed magnetometry," IEEE Trans. Magn., 28 (1992), 3114-3116. 8.54. C. Golovanov, G. Reyne, G. Meunier, R. Gr6ssinger, and J. Dudding, "Finite element modeling of permanent magnets under pulsed fields," IEEE Trans. Magn., 36 (2000), 1222-1225.

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547

8.55. J.C. T611ez Blanco, R. Sato Turtelli, R. Gr6ssinger, E. Est6vez-Rams, and J. Fidler, "Giant magnetic viscosity in SmCo5-xCux alloys," J. Appl. Phys., 86 (1999), 5157-5163. 8.56. R. Gr6ssinger, R. Sato Turtelli, and J.C. T611ez Blanco, "The influence of magnetic viscosity on pulsed field measurements", J. Optoelectron. Adv. Mater., 6 (2004), 557-564.

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CHAPTER 9

Measurement of Intrinsic Magnetic Properties of Ferromagnets Magnetic parameters are defined as intrinsic when they depend only on composition and not on the specific structural properties of the materials. Spontaneous magnetization cr~ Curie temperature Tc~ anisotropy and magnetostriction constants in single crystals comply with this definition. In several instances, however, there is no clear borderline between intrinsic and structure-dependent properties and a pragmatic approach to the previous quantities is pursued. The rapidly quenched alloys provide a case in point, where profound structural modifications may ensue from annealing treatments and induce dramatic evolution in the intrinsic magnetic parameters, the composition remaining the same. In this chapter, we shall thus briefly describe some commonly adopted methods in the measurement of cr and Tc~ and of the anisotropic properties of ferromagnets, their often loose meaning as intrinsic quantities being understood. These measurements touch upon fundamental problems in the magnetism of materials, but they have clear relevance also under applicative instances and their execution should possibly be within reach of any laboratory involved in both research and product testing.

9.1 S P O N T A N E O U S TEMPERATURE

MAGNETIZATION

AND CURIE

Ferromagnets have the stunning property of being spontaneously magnetized in a zero applied field. The spontaneous magnetization Ms existing within any domain is a decreasing function of temperature T, as accurately described by the law Ms(T) = Ms(0)(1 - AT3/2)~ and vanishes at the ferromagnetic Curie temperature TCF. In order to determine Ms(T) (for which the term "magnetic saturation" is equivalently used in the present context) in the material under investigation, be it a single 549

550

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

crystal, a polycrystal, or an amorphous alloy, we need to apply a sufficiently high field to wipe out the domain walls and complete the magnetization rotation towards the direction of the effective field against anisotropy. It is classically held that at high fields the magnetization curve can be described by means of the following expression:

(a

M ( H ) = Ms 1 - H

H2

+ kH,

(9.1)

where a, b, and k are suitable temperature-dependent constants and H is the effective field (applied field minus the demagnetizing field). The term kH is assumed to describe the forced increase of the spontaneous magnetization and to be generally negligible at temperatures well below TCF and fields lower than about 106 A / m . The constants a and b are related to the presence of structural inhomogeneities (voids, dislocations, etc.) and to the anisotropy, respectively. Under normal conditions available in the laboratory, with fields provided by sources like electromagnets, we may determine Ms(T) in soft and semi-hard magnets by fitting the highfield portion of the experimental curve through the first term in the righthand side of Eq. (9.1). One way to do so, depending on the tested material, is by plotting M ( H ) as a function of either 1/H 2 or 1/H. In polycrystalline Fe and Ni, for example, the term a / H is valid only within some finite range of strength of the field [9.1] and M~(T) can be obtained by extrapolating the linear portion of the M ( H ) vs. 1 / H 2 curve to the limit 1 / H 2 = 0 (see, e.g. [9.2]). If the applied field is so high that the term kH can be observed, one may instead linearly extrapolate the final portion of the M ( H ) curve to H = 0. Equation (9.1) has been assumed to hold for single crystals and polycrystals, but there is no consensus on its general validity. Pauthenet has demonstrated by means of extremely accurate measurements up to about 14 M A / m that the magnetization curve at high fields in Fe, Co, Ni and YIG single crystals can be described by the equation: M ( H ) = M s + A H 1/2 + BH,

(9.2)

with A and B temperature-dependent constants [9.3]. The term proportional to H 1/2 is associated with the Holstein-Primakoff spin-wave amplitude reduction by the applied field [9.4], while the additional susceptibility B is attributed to the Pauli paramagnetism. Figure 9.1a shows Pauthenet's experimental magnetization curve on a single pure Fe crystal taken at T = 286.4 K and its fitting by Eq. (9.2). The value of spontaneous magnetization per unit mass or= 2 1 8 . 2 A m 2 / k g is obtained according to such a fitting. The interpretation of the highfield magnetization behavior in amorphous alloys, where there may be a special situation of large moment canting at room temperature, has brought about a number of theoretical formulations [9.5, 9.6].

9.1 SPONTANEOUS MAGNETIZATION AND CURIE TEMPERATURE

551

221 Fe single crystal

[ 100]-o riented 220

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219

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218

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.

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.

,,

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H(kA/m) 180 Amorphous ribbon

Fe-B-Si

175

O3 o,]

E <

170

T = 295 K

165 . . . . ,

0 (b)

....

000

, .... 2000

, . . . ~ 3000

000

H (kA / m)

FIGURE 9.1 Isothermal high-field magnetization curve along the [100] easy axis in a pure Fe single crystal (a) and an Fe-based amorphous ribbon (b). The experimental curve in Fe, due to Pauthenet [9.3], is accurately described by Eq. (9.2), while Eq. (9.3) is shown to apply to the high-field portion of the curve found in Fe-based amorphous alloys [9.7].

552

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

Szymczak et al. have verified in Fe-based amorphous ribbons that Eq. (9.2) may fit the experimental data, but with a certain difficulty in the interpretation of the coefficient B [9.7]. Good agreement has been found with the equation proposed by Chudnovsky [9.6]:

M(H) = Ms + A1H -1/2 + B1H.

(9.3)

For the determination of the coefficients of all these equations, it is expedient, as previously mentioned, to resort to a representation of either M(H) or dM(H)/dH as a function of the appropriate power of either H or 1/H, to highlight the existence of a power law through the slope of the thus found straight lines. Small errors in the measurement of the absolute value of M(H) have then little influence in the determination of the coefficients. In Fig. 9.2, we show an example of 1/H 2 representation of a few isothermal curves J(H)= ~oM(H) in an amorphous alloy of composition Fe40Ni40P14B6 [9.5]. The coefficient b in Eq. (9.1) is determined from the slope of the obtained straight lines and the 1.24

Amorphous ribbon

1.22

3

Fe4~176176

~

1.20

........

T= 80 K

"'~

-----.o

,_,~ 1.18 ~ 1 16

T= 180 11:10K

1.14 1.12

T=260 K o

. . . .

. . . .

260 . . . .

36o

400

(1 /#o H) 2 (T-2)

FIGURE 9.2 1/H2 representation of magnetic polarization curves measured in an Fe40Ni40B20amorphous alloy. In the field interval -~40 kA/m -< H ~ --- 160 kA/m the magnetization follows a 1/H2 law and the slope of the experimental straight lines provides the coefficient b in Eq. (9.1). These lines extrapolate to the value of the spontaneous magnetization Ms(T) = Js(T)/la,o,given by their intercept with the ordinate axis (see inset). The steep increase of J(H) towards 1/H2 ~ 0 is associated with the forced paramagnetic effect (adapted from Ref. [9.5]).

9.1 SPONTANEOUS MAGNETIZATION AND CURIE TEMPERATURE

553

spontaneous magnetization Ms(T) = Js(T)/la, o is given by the intercept of these straight lines with the ordinate axis. The measurement of the spontaneous magnetization runs into substantial complications on approaching the Curie temperature because magnetic ordering created by a saturating magnetic field adds to the weakened intrinsic ordering due to the exchange interaction, leading to an increase of the magnetization value with respect to the spontaneous one. It is observed, in particular, that the experimental Ms(T) vs. T curve displays a characteristic transition region from the ferromagnetic to the paramagnetic state, whose extension depends on the strength of the applied field, making it difficult to identify unambiguously the Curie temperature TCF. Neutron diffraction experiments actually show that long-range order gives way to short-range order at T = ZcF and that the individual spins become independent of neighbors only far from TCF [9.8]. Figure 9.3

-(3 wt%)Si 2.0

E

Js

1.5

-3

0

K1

1.0

0.5

0.0

0

200

400 T(~

600

/

800

/OF

FIGURE 9.3 Saturation polarization Js and anisotropy constant K1 determined as

a function of temperature in an Fe-(3 wt%)Si alloy. K1 has been measured on a single crystal disk by means of a torque magnetometer. The ferromagnetic Curie temperature TCF is empirically obtained by extrapolating the Js(T) curve from the inflexion point and taking the intercept with the horizontal axis. The anisotropy constant is seen to vanish before attaining the Curie temperature. This is conducive to strong magnetic softening under weak fields and a corresponding peak of the initial susceptibility value (Hopkinson's effect), which drops to a negligible value for T = TCF (adapted from Ref. [9.9]).

554

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

provides an experimental example of temperature dependence of the saturation magnetization in an Fe-(3 wt%)Si alloy, measured with a field of the order of 106 A/m, displaying a smooth transition at the Curie temperature [9.9]. The zero-field final portion of the Js(T) curve and TCF can be empirically obtained by extrapolating the experimental Js(T) curve at the inflexion point and taking the temperature value corresponding to the intercept with the horizontal axis. In a similarly approximate way, one can exploit Hopkinson's effect, where the weak field susceptibility exhibits a rapid increase on approaching TCF and drops sharply at T = TCF. This effect can be explained by considering the decrease of the magnetic anisotropy with increasing temperature, which leads, as shown for the constant K1 in Fig. 9.3, to the practical disappearance of the related energy term and ensuing magnetic softening well before reaching TCF ~ followed by a precipitous drop of susceptibility when long-range ordering disappears. A well-known quantitative method for the precise determination of the Curie temperature, by Arrott [9.10], is based on the analysis of the Weiss-Brillouin function for the spontaneous magnetization Ms:

Ms

= Mso tanh( I~~

+ ~/Ms))

kT

(9.4)

in the neighborhood of TCF. The quantity Ms0 in this equation is the spontaneous magnetization at zero absolute temperature, m is the elementary magnetic moment, and y is the mean field constant. It is shown that, under the condition Ms KKMs0, Eq. (9.4) provides the following relationship between internal (effective) field H and magnetization Ms [9.10]:

Ms)3 ( T _ T c F ) M s

1( g ~

+

T

Mso

i~omH. kT

(9.5)

Equation (9.5) predicts that an isothermal magnetization curve taken at the Curie temperature (T = TCF) obeys the equation: 1 ( Ms )3__ I~omH

-3 ~

kT

and is therefore represented by a straight line passing through the origin in a M 3 vs. H representation. It predicts instead either downward or upward bending of the curve according to whether the temperature is slightly higher or lower than TCF, as illustrated in the example reported in Fig. 9.4 [9.11].

9.1 SPONTANEOUS MAGNETIZATION AND CURIE TEMPERATURE

1.5

555

66 K ~ EuO

/

70K

%

72 K 0.5

IIII 0.0

,","

0

.

.

.

.

.

.

.

.

.

500

.

.

.

H (kA/m)

.

.

.

.

.

.

','

1000

FIGURE 9.4 Arrott plots in the ferromagnet EuO. Isothermal high-field magnetization curves taken in the neighborhood of the Curie temperature TCF are plotted in terms of the cube of magnetization vs. the effective field. As predicted by Eq. (9.5), this results in a linear behavior of o~ vs. H at T = TCF.In the present case, it is obtained TCF ~ 69.5 K (adapted from Ref. [9.11]). The Curie temperature and the paramagnetic susceptibility above the Curie point can be determined in soft magnets with a single experiment by measuring, upon the suitable range of temperatures, the magnetic moment of a spherical sample under relatively weak applied magnetic fields [9.12]. Because of the magnetic softness of the material, the internal field H in a spherical sample is always much lower than the demagnetizing field H d - - N d M - - M / 3 , which can then be assumed practically to coincide with (i.e. to balance) the applied field Ha = H + Hd. It then turns out that magnetization and applied field are proportional M ~-3Ha, as far as the Curie temperature is attained. This is demonstrated by the experimental dependence on temperature of the magnetization in iron under a constant Ha value shown in Fig. 9.5a [9.13]. It is observed here that M is independent of temperature up to T = TCF ~where the M vs. T curve suffers a sharp bending and starts to decrease. The internal field correspondingly increases, eventually tending to approach the value Ha. The so-measured paramagnetic susceptibility, obtained calculating the internal field as the difference between

556

CHAPTER 9 Measurement of Intrinsic Magnetic Properties 0.15 ' ~ - H a = 3.6 104 A/m I Fe sphere} 0.10

~ Ha= 2.9 104 ~m

~~

yb-

-Ha= 2.16 104 Nm

0.05

i ~ ~

0.00 i.

Ha= 1.4 104 A/m

o'4o

lo'6o ....

Tc F

(a)

lo7o

T(K)

10

Ni

O3 I.-

Fe

6 4

~ ~-~

0 (b)

,

~ f

,

a,

~

\

,

i

10

.

Tp -TcF

.

.

.

.

I

.

.

20 T- TCF (~

.

.

I

30

.

.

.

.

I

40

FIGURE 9.5 Ferromagnetic Curie temperature TCF determined in pure Fe by measurement under constant applied field of the magnetization as a function of temperature. Use of a spherical sample ensures high demagnetizing factor and internal field much lower than the demagnetizing field, so that the magnetization remains constant up to T = TCF. The susceptibility in the region immediately following TCF, shown in (b) for Fe and Ni samples, follows a law of the type X = a / ( T - TCF)~, with a a constant and a = 1.2-1.3. Tp is the paramagnetic Curie temperature (adapted from Refs. [9.13, 9.14]).

9.1 SPONTANEOUS MAGNETIZATION AND CURIE TEMPERATURE

557

the applied and the demagnetizing field, is observed to follow, in the region close to TCF ~ a law of the type X - a/(T - TCF)~, with a a constant, where typically c~ = 1.2-1.3. In the conventional 1/Xp vs. T representation, this appears as a deviation from the linear Curie-Weiss law behavior (Fig. 9.5b, for Fe and Ni [9.14]), following the existence of short-range magnetic order. When we are chiefly interested in the determination of the Curie temperature of a ferromagnet, we may resort to some indirect method where the sharp variation of a physical quantity affected by magnetic ordering is recorded. Like all second-order phase transitions, the passage from the ferromagnetic to the paramagnetic state is accompanied by a discontinuity in heat capacity and coefficient of thermal expansion. The thermal coefficient of resistivity d p / d T also shows a discontinuity at T - TcF (see Fig. 5.11). The specific heat of a magnetic material contains, besides the dominant lattice vibration term, electronic, thermal expansion, and magnetic contributions, all increasing with the temperature. With zero applied field, the magnetic contribution can be written %, = d U m / d T , where dUm =-/~0He dMs and H e = ~/Ms is the Weiss mean field. Cm is thus expected to increase with temperature as Cm=-

~0 ~/dM2 2 dT

and to drop to zero once TCF is attained. Again, the decrease of Cm will occur over a certain temperature interval beyond TCF. Differential thermal analysis (DTA) and differential scanning calorimetry (DSC) are commonly employed techniques in the investigation of the thermal properties of materials and can be used to reveal the magnetic phase transition. They both analyze the response of the test sample to a defined heating schedule by comparison with a suitable reference sample, which has similar heat capacity and is immune from transformations in the temperature range of interest. With DTA, a furnace transfers the same heat per unit time to both samples and any transformation occurring in the test specimen appears as a temperature difference AT between the samples. With DSC, the heat rates supplied by two independent furnaces to the samples are regulated in such a way that always AT = 0. The quantity of heat released or absorbed by the test specimen during a transformation is then determined from a corresponding variation of the energy supplied by the furnaces. Figure 9.6 provides an example of the determination of TCF in an Fe-based amorphous alloy using DSC. The sample is brought at constant heating rate through the ferromagnetic phase transition and beyond, until the crystallization process is completed. The DSC trace displays an endothermic trend before attaining the Curie temperature, consistent

558

CHAPTER 9 Measurement of Intrinsic Magnetic Properties Fe78B13Si9 amorphous alloy

as

a r

6 t--

~,, sample

v 4 o m o x

7-

uJ

o

/

"0 c-

uJ

TCF ,

200

,

,

'

'

360

'

'

'

dT / dt = 0.33 oC/s I

,

,

400 Temperature (~

,

,

,

s60

,

,

,

600

FIGURE 9.6 DSC trace obtained upon heating an amorphous alloy of composiIt tion Fe78B13Si9 at a constant rate of temperature change dT/dt = 0.33 ~ shows the increase of the heat capacity of the test sample and the drop occurring at the Curie temperature, together with the large exothermic peak associated with the crystallization process. DSC works on the principle of supplying the test sample and a reference sample with heat flows Qs and Qr in such a ratio that zero temperature difference is maintained between them (courtesy of E. Ferrara). with the increase of Cm with T, followed by a sharp turn to exothermic behavior and large heat release associated with the amorphous to crystalline transformation. The measurement of spontaneous magnetization and Curie temperature and the analysis of ferromagnetic to paramagnetic transition can be carried out with the closed circuit or open sample methods described in Chapters 7 and 8. Great accuracy and tight control of the parameters involved is required in the determination of the magnetization curve on the approach to saturation, especially if absolute calibration is required. Closed-circuit methods are not in frequent use because the only practical way of making precise high-field measurements is by use of an electromagnet as a field source. This must be large enough to provide maximum uniformity of the field in the gap, at least in the region occupied by the sample. Due to saturation of the pole faces, the maximum field strength

9.1 SPONTANEOUS MAGNETIZATION AND CURIE TEMPERATURE

559

with the required homogeneity is limited to around 1.5-2 x 106 A / m . Absolute determination of the saturation polarization in pure Ni samples tested in a closed magnetic circuit with an electromagnet has been performed by Sievert et al. [9.15]. These authors, using an arrangement similar to the conventional one shown in Fig. 8.4, have performed the measurement of the magnetization curve up to about 106A/m in 10 m m x 10 mm cylinders obtained from zone-melted Ni ingots. The cross-sectional area of the specimen is known with an uncertainty of the order of 10 -3. The air-flux compensated sensing coils are obtained with a minimum residual signal in the absence of the sample. This is shown to contribute to a final uncertainty on the measured polarization value of the order of 0.05%. The turn-area product of the sensing coils is obtainedusing a standard solenoid source calibrated with an NMR magnetometer. The measurement of the effective field is performed by means of a Hall plate, placed on the sample surface orthogonal to its axis. Again, calibration of this probe relies on the use of an NMR calibrated source. The electronic apparatus used for signal amplification and treatment has sufficient resolution and stability for contributing negligibly to the final measurement uncertainty. In contrast, a possible detrimental effect may arise from: (1) Imperfect flux closure at the contact between the pole faces and the sample ends. (2) The effect of temperature on the Hall probe coefficient (typically 0.1-0.2%/K) and on the value of the spontaneous magnetization [(1/Ms)(dMs/dt) ~ 5 x 10 -4 K -1 in Ni]. (3) Clamping pressure of the pole pieces on the sample and its repeatability. Sievert et al. estimate that keeping all these factors under tight control is conducive to a final measuring uncertainty (lo') of the found room temperature value of the saturation magnetization in Ni of 0.15%. Open sample methods make general use of spherical samples, where a certain ease of preparation is associated with the fundamental requirement of a homogeneous demagnetizing field. Studies on ellipsoidal samples have also been reported [9.16]. The VSM and the extraction methods, as discussed in Chapter 8, are among the favorite measuring approaches for the determination of the magnetic moment of the spherical sample up to the very high fields obtainable with superconducting solenoids or Bitter coils. These methods are not prone to absolute measurements and reference samples are generally employed. The highly accurate measurement by Pauthenet beyond 10 000 k A / m applied field strength reported in Fig. 9.1a [9.3] has been carried out with an extraction magnetometer of the type shown in Fig. 8.27 (see Section 8.2.3). The single crystal spherical sample is placed between the centers of two coils connected in series opposition, which are immersed in a homogeneous magnetic field provided by a Bitter magnet. For typical samples, with

560

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

a magnetic moment of the order of 0.05 A m 2 (50 emu), the relative measuring accuracy is between 10 -4 and 10 -5 . The calibration of the absolute moment value is obtained using a reference Ni sample. A similar experimental arrangement has been adopted by Foner et al. in their magnetic moment measurements in Fe and Ni up to 12 000 k A / m [9.17]. A VSM setup with field supplied by a superconducting coil (see Fig. 8.19) has been used by Eckert and Sievert in their measurements of Ni spheres up to about 6500 k A / m (shown in Fig. 8.1). SQUID magnetometers have also been employed for the determination of the saturation magnetization of magnetic glasses [9.7, 9.18]. Accurate measurements of the magnetic moment in a range of temperatures, of the Curie temperature, and the susceptibility beyond the Curie point can be performed by means of the Faraday force method (Faraday balance) [9.13, 9.16, 9.19]. This ancient technique has resisted the general evolution of methods and materials and is still appreciated because of its intrinsically high sensitivity and accuracy, and is frequently applied in the measurement of the susceptibility of paramagnets and weak magnets [9.20]. The magnetic moment measuring capability of the Faraday balance stems from the creation of a translatory force F = X7(m.~0H) acting upon a magnetic dipole of moment m immersed in a non-uniform field H. If certain symmetry conditions regarding the field, its gradient and the direction of the magnetic moment are satisfied, the modulus of the moment can be obtained by means of a scalar-type measurement of the force. We are already familiar with gradient coils and forces on magnetic moments, discussed in Chapter 8 in association with the VSM and AGFM magnetometers. The basic difference between AGFM and Faraday magnetometers resides in the character of the forces operating in them. In the AGFM, the gradient coils are AC supplied and the magnetic sample is subjected to an oscillatory force, which makes it appropriate to measure the oscillating sample displacement. In the Faraday balance, the field gradient is static and the force, rather than the displacement, is conveniently measured. It is useful to express such a force in terms of its Cartesian components. They are obtained from the previous vector relationship as

Onx

oG

]

OHm.

OH,./

OHz ]

Onx

oH,,

oG ]

Fx =/-to mx--~x + my---~x + mz--~x ,

=

.o

mx- y + my- V + m,- V ,

Fz - P'o m x - - ~z- + m y - - ~z- + mz---~z

,

(9.6)

9.1 SPONTANEOUS MAGNETIZATION AND CURIE TEMPERATURE

561

where mx~ my~ and mz are the moment components. Either vertical- or horizontal-field geometries are adopted in Faraday magnetometers. We take the center of symmetry of the arrangements shown in Fig. 9.7 as the origin of the coordinate system. In the vertical-field setup schematically represented in Fig. 9.7a, the DC main field generated by a solenoid imposes a certain state of magnetization, i.e. a magnetic moment mz~ along the z-axis in a spherical sample. At the same time, two coaxial identical coils connected in series opposition generate a uniform field gradient at the sample position. The behavior of the field gradient OHz/OZ along the symmetry axis, which depends on the ratio a/d between coil diameter and distance between the coils, is as illustrated in Fig. 8.13 for a few representative values of such a ratio (the field gradient there appears as the sensitivity function of the axial VSM). Notice that the maximum homogeneity of OHz/OZ is obtained, with filamentary coils, for d = x/3a. From Eq. (4.24), we obtain in such a case ~ OHz _ 0 . 6 4 1 3Ni Oz

(9.7)

a2

where N and i are the number of turns and the current flowing in each coil, respectively. We assume that mx = my = 0~ i.e. mz = m. Equation (9.6) in this case provides

0H~ F x - / ~ 0 m 0x ~

0G Fy =/~0m 0y ~

0G Fz -/~0m 0--~--"

(9.8)

The sample is kept at the center of the coil assembly by means of a suspension connecting it to the arm of an electrobalance, by which the pulling force Fz is measured. The balance works on the principle of null deflection, where the force is measured through the value of the current signal required to keep the balance pan at its rest position. Once the field gradient OHz/OZ is known, the force Fz directly provides the value of the sample moment m. To make a numerical example, we take the case of two coils for which Eq. (9.7) applies. For coil diameter 2 a - - 5 cm, N = 100~ i = 5 A, we get the gradient OHz/OZ = 5.1 x 105 A / m 2. We assume that the microbalance can reliably measure force changes of 10 ~g (--- 10 -7 N). This amounts, according to Eq. (9.8), to a measurable magnetic moment around 1.5 x 10 -7 A m 2 (1.5 x 10 -4 emu). Lateral instability may occur with relatively short solenoids. In fact, while there is no radial gradient at the sample position in the field generated by the coils (Fx--Fy = 0 in Eq. (9.8)), lateral force on the sample may come from the gradient of the applied main field. In fact, in a non-infinite solenoid, this field tends to increase on going from the axis to the inner walls (see Eq. (4.15). The associated gradient OHz/Or--[OHz/OXl = IOHz/Oyl > 0, where r is

562

CHAPTER 9

Measurement of Intrinsic Magnetic lJroperties

r////.4

IN.

I !

NF-a-

j'l|

]|

_..L_

Gradient coils ~

!

d

'

~

Sample

~

Magnetizing coil

~

Z

i

i !

(a)

Y

Gradient coils

x~

iiiiiii

O

Hx r

(b)

. . . . . . .

i

,

,i

FIGURE 9.7 Schematics of vertical-field (a) and horizontal-field (b) Faraday magnetometers. One can realize with them the measurement of the magnetic moment m of a sample through the measurement of the pulling force Fz exerted on it by a magnetic field gradient. In the first case, the vertical magnetizing field Hz and the gradient OHz/OZ are generated by a solenoid and by identical windings connected in series opposition, respectively. The resulting force on the sample is Fz = la,omz(OHz/Oz). In the second case, the field Hx is generated by the electromagnet and the gradient OHx/OZ is provided by the coils placed on the pole faces. The downward pulling force is Fz = p,omx(OHx/OZ). The set of gradient coils used with the electromagnet is shown in some detail, and the circulating currents are indicated. The scheme shown in (b) is adapted from Ref. [9.26].

9.1 SPONTANEOUS MAGNETIZATION AND CURIE TEMPERATURE

563

the radial coordinate, is proportional to r and results, according to Eq. (9.8), in a force tending to pull the sample away from the solenoid axis. In order to achieve very high fields and gradients, superconducting solenoid and coils have been employed [9.21, 9.22]. Bitter solenoids have also been used. Interestingly, it has been verified in the latter case that it is possible to dispense with the gradient coils. In fact, the generated magnetizing field can be so large that its own axial gradient may suffice to achieve the desired moment measuring sensitivity [9.23, 9.24]. In this case, the sample is placed near or in correspondence to the end plane of the solenoid, where the maximum value of OHz/OZ is attained. If maximum sensitivity is not required, there is convenience in placing the sample immediately beyond the end of the solenoid [9.23]. In fact, on leaving the interior region of the solenoid, the radial dependence of the field intensity is reversed and so is the resulting lateral force on the sample, which, instead of being pulled away from the solenoid axis, is automatically centered. In order to gain easier access to the measuring region, the horizontalfield type Faraday magnetometer with electromagnet field source can be employed. In the classical device of this type, the electromagnet is provided with polar caps having a specific profile, by which the generated field becomes endowed with a suitable gradient OHx/OZ along the vertical axis [9.25]. Such gradient is dependent, however, on the field strength, and measurements at low fields necessarily imply low gradients, an undesirable limitation when dealing with ferromagnetic samples, for instance. On the other hand, very high gradients are not required with ferromagnets, where independent adjustment of main field and gradient are appropriate instead. This objective can be reached by adopting an arrangement as shown in Fig. 9.7b, where coils placed in the gap of the electromagnet make possible the desired independent regulations [9.26]. The set of coils adopted in this specific case realize a scheme similar to Mallison's set, for which the behavior of the gradient OHx/Oz is given in Fig. 8.14d. If we pose my = mz = 0, Eq. (9.6) becomes

0Hx

Fx = I~omx 0---~'

0HI

G = ~omx 0---y-'

0HK

F~ = ~omx O---z-"

(9.9)

Again, the symmetrical arrangement of the coils makes the gradients OHx/OX --- OHx/Oy - 0 at the rest position of the sample. Only the vertical force F~, pulling the sample downward when the currents circulate in the coils as shown in Fig. 9.7b, is different from zero. On the other hand, if the sample is slightly displaced from the center of symmetry along the x-axis, a lateral force due to the magnetic image will arise, which is roughly proportional to the displacement [9.27].

564

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

To simplify the generation of the field gradient, the use of a pair of identical strips in the gap has been suggested [9.28]. The strips are parallel to the pole faces and are oriented along the z-direction. They bear the same current, thereby producing zero field and non-zero field gradient at the center of the gap, i.e. at the sample position. Shull et al. have used such a field gradient generating setup in their Faraday magnetometer, by which they have performed absolute measurements of the magnetic moment of pure Ni spheres (m = 3.47x 10 -3 A m 2) [9.19]. Absolute determination of m with stated expanded relative uncertainty of 3 x 10 -3 has been obtained by making a direct measurement of the generated field gradient.

9.2 M A G N E T I C

ANISOTROPY

Magnetic materials seldom behave isotropically. The classical Heisenberg exchange interaction -JqSi.Sj is isotropic, but an anisotropic exchange term, bonding the spins more tightly when they point along certain directions, can exist. In addition, the spins individually interact with the crystalline field, which, being endowed with the symmetry properties of the host lattice, provides preferential orientations for the exchangecoupled spins. This property can be phenomenologically described by expressing the associated energy term by means of a function provided with suitable symmetry properties. Thus, the crystalline magnetic anisotropy energy in a cubic crystal is appropriately defined through a polynomial series in even powers of the direction cosines 6r 6r and OZ3 made by the direction of the magnetization with the cube edges. The relevant equation is Ea -- K1(cr 2 cr2 q- c~2c~2 q- cr cr

q- K2cr 2 ~2202

(9.10)

higher order terms being normally of little or no relevance. K1 and K2 are constants characterizing a given material and are expressed in J / m 3 (we consider here and in the following only quantities related to the unit volume). Their amplitude and sign determine the directions along which the anisotropy energy is minimum. In Fe crystals, it is K1 > 0 and K1 >> IK21, which makes the (100) axes the m i n i m u m energy directions for the magnetization (easy axes). In hexagonal crystals (e.g.h.c.p. Co), the energy is expressed as Ea = K1 sin 20 + K2 sin 40,

(9.11)

where 0 is the angle made by the magnetization with the c-axis. With K1 > 0 and K2 ~ - K 1, the c-axis is an easy axis, a condition met at room

9.2 MAGNETIC ANISOTROPY

565

temperature in Co and Ba ferrite single crystals. Polycrystalline materials combine the crystallographic orientations of the single grains in a variety of ways. The resulting texture translates into more or less pronounced magnetically anisotropic behavior, which, however, is not univocally related to the distribution of the orientations. Besides crystallographic texture, demagnetizing fields, stresses, and various effects of directional atomic ordering can induce anisotropic effects. In these cases, we have to deal in general with uniaxial anisotropies. For most purposes, the dependence of the uniaxial anisotropy energy on the angle 0 between magnetization and easy axis is described by the equation: Ea -- Ku sin20.

(9.12)

Demagnetizing fields are the source of shape anisotropy, which inevitably affects all non-spherical samples. With ellipsoidal samples, the demagnetizing field Hd = - N d M is homogeneous and the constant Ku can be exactly defined in terms of the difference between demagnetizing coefficients pertaining to the minor and major axes, respectively. If Xda and Ndc are such coefficients and 0 is the angle between the uniform magnetization M and the major axis, we obtain the magnetostafic energy as Ems--(~/2)M2(Nda cos2Oq-NdcSin20). We thus define the shape anisotropy energy as Ea--(IJ, o/2)M2(Nda- Ndc)Sin20 = Ku sin20. If a tensile/compressive stress is applied to a sample and the ensuing magnetoelastic energy can be described by means of an isotropic magnetostriction coefficient As, the stress anisotropy energy turns out as Ea = -~Aso"sin20-- Ku sin20. The stress axis is an easy axis if the product AsO" is positive (e.g. tensile stress and positive magnetostriction) and a hard axis if this product is negative (e.g. compressive stress and positive magnetostriction). In the latter case, the plane normal to the stress axis is an easy plane. A straightforward measurement of magnetic anisotropy can be performed by determining the magnetization curve up to saturation along different directions. This method implements the definition of anisotropy energy as the difference in energy required to saturate the material along different axes. If the easy axis is known or it has been identified and the magnetization curves have been obtained along both such an axis and the direction under investigation, one makes the difference of the areas included between these magnetization curves and the J-axis (Fig. 1.6b). It is understood that all other energy terms have changed negligibly on passing from one direction to the other. This method is made somewhat complicated and often unreliable by a number of factors: (1) Measurements on closed magnetic circuits might require

566

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

complex flux-closing arrangements in two or even three dimensions to cope with the presence of the demagnetizing field pointing along a direction different from the direction of the applied field [9.29]. Measurements on open samples should preferably be made on spheres, but the huge shearing of the curves can make the determination of the difference of areas imprecise. (2) There is an hysteresis effect, which should be eliminated by considering the anhysteretic curve or, at least, the median curve between ascending and descending branches of the hysteresis loop. (3) The approach to saturation may be affected by the presence of various defects, inclusions and localized stresses to an extent depending on the specific investigated crystallographic direction. Anisotropies can be measured in a direct fashion using a torque magnetometer. Spherical or disk-shaped samples are subjected in it to a slowly rotating field, whose strength must be sufficient to eliminate the domain walls and drag the magnetization without discontinuities along a whole 360 ~ turn. Let us consider the scheme shown in Fig. 9.8a, where the applied rotating field H a makes, at a given instant of time, an angle ~0 with the easy axis (taken as the reference direction) in a sample with uniaxial anisotropy. The magnetization Ms makes the angle 0 with the easy axis when the torque per unit volume: ZH =/zoMsHa sin(~o- 0),

(9.13)

due to the field is balanced by the intrinsic torque ZK = -OEa/O0 due to the anisotropy. This follows from the minimization condition OE/O0 = 0 imposed onto the total energy: E = -/zoMs'Ha q- Ea,

(9.14)

sum of the anisotropy and field interaction energies. With Ea given by Eq. (9.12), it is rK = Ku sin 20.

(9.15)

At equilibrium, TH = rK = Z and, according to Eq. (9.13), the angle 0 is related to ~ by the equation: 0 = ~0- sin -1

~(~o) . /z0MsHa

(9.16)

The applied field appears in Eq. (9.13) instead of the effective field H - - H a - N d M s . The demagnetizing field, being co-linear with the magnetization, cannot in fact contribute to the torque. The experiment requires that H a is larger than a threshold value, beyond which totally reversible rotation of the magnetization can occur. Such a value is

9.2 MAGNETIC ANISOTROPY

567

Easy axis

,

Ha

(a) 1000

/

r(~)

500 04

E

I

z

~1 ~:/2

(1) '-I 0 I--

-500

K u = 1000 J / m 3 Ha = H k

(b)

~ ~

/ i II / iIII ~ /11

-1000

FIGURE 9.8 Torque in a sample with uniaxial anisotropy. (a) A slowly rotating applied field Ha exerts a torque on the saturation magnetization Ms in a spherical or disk-shaped sample, dragging it along a complete 360 ~ turn. The field torque per unit volume of the sample TH = p,oMsHa sin(~o - 0) is balanced at any time by the intrinsic torque ZK = -OEa/OO due to magnetic anisotropy. ~0 and O are the angles made by Ha and Ms with the reference axis, respectively. (b) From the measured oscillatory torque ~(~) (solid line), the desired behavior of the torque 9(O) (dashed line) is obtained. The example reported here refers to uniaxial anisotropy. Both curves attain the same peak value, which coincides with the value of the anisotropy constant Ku.

568

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

obtained imposing the additional condition of stability on the second derivative O2:Ea/O02 and it turns out to be the value of the so-called anisotropy field Hk = 2Ku/la,oMs. Figure 9.8b illustrates the behavior of the torque curve ~(~) measured with a field slightly larger than Hk and its transformation into the "r(0) curve, the one achievable with saturating fields, using Eq. (9.16). The value of the anisotropy constant Ku coincides, according to Eq. (9.15), with the maximum of the torque curve. This transformation procedure is general, but requires careful application because it might easily generate errors in the following harmonic expansion of the torque curve, if this is required to reveal higher order anisotropy terms. When possible, it is appropriate to apply a sufficiently large field H a in order to have a small difference ~ - 0 and consequently linearize the transformation given by Eq. (9.16). In many experimental circumstances, the applied field strength is indeed so high that 0 = ~, no transformation is required, and the anisotropy is directly provided by the measured curve. Notice, however, that the measured anisotropy may not be observed to saturate with the applied field. Kouvel and Graham, making experiments on Fe-Si disk-shaped single crystal samples (thickness 0.3 mm), ascribed this effect to the presence of residual edge domains, depending in a complicated way on the direction of the field [9.30]. These are expected to introduce small variations of the magnetization amplitude as the saturating field rotates in the plane of the disk, resulting in an apparent small extra torque. There also appears to be a more fundamental problem of anisotropy of the saturation magnetization, as demonstrated by Aubert in Ni single crystals [9.31]. Figure 9.9 provides an experimental example of field dependence of the anisotropy constant K1 in the saturation region in Fe-Si single crystals, determined in spherical samples [9.32]. This effect is attributed to the orientation dependence of the saturation magnetization. Notice the steep increase of the measured K1 value in the field region going up to Ha ~ Ms~3. It is related to the progressive disappearance of the Bloch walls and their contribution to the magnetization process. A classical realization of the torque magnetometer is shown in Fig. 9.10. Here, the field is obtained by means of an electromagnet, installed on a rotating platform. The sample, located at the center of the gap, is either a disk or a sphere. It is held solidly in place in the interior of a quartz tube, which is connected by means of a taut suspension (e.g. a tungsten wire) to a rigid non-vibrating structure. A weight attached at the bottom of the sample provides stabilization against lateral movement. The whole assembly is kept in vacuum making it possible to perform measurements as a function of temperature whilst avoiding the disturbing action of air convection. On top of the quartz tube and integral

9.2 MAGNETIC ANISOTROPY

569

_

40

_

~

Fe-(3.4 %)Si T = 373 K

30 co

iI !

20

o

T-v

v

Fe-(6.1%)Si T = 293 K

iI

E

v--

10 Hk I ~ 9

'

I--4

i I;

500

'

'

'

'

'

I0'00

'

'

'

....

i ....

1500

H a (kA I m)

FIGURE 9.9 Magnetic anisotropy constant K1 in two Fe-Si single crystals of different Si content as a function of the applied field H a. The measurements have been performed by means of a torque magnetometer on 5 mm diameter spheres. The steep increase of K1 with the field for H a < Ms~3, where applied field and demagnetizing field have nearly same strength, is due to the gradual disappearance of the Bloch walls. Once magnetic saturation is attained, K 1 shows a feeble increase with Ha, which is attributed to the anisotropy of magnetization (adapted from Ref. [9.32]).

with it, there are a mirror and a multiturn coil, which are part of a servo system keeping the sample firmly in place against the torque applied by the rotating field (automatic force-balancing method). When the field starts rotating, the sample tends to follow because of the action of the torque ~H (Eq. (9.13)). The collimated light beam, striking the mirror and evenly reflecting at rest position into a dual photocell, generates unbalanced photocurrents, which are amplified by the high-gain DC amplifier and injected into the multiturn coil attached to the tube. This coil, being immersed in the gap of a permanent magnet, imparts a restoring torque to the assembly, the higher the gain of the amplifier, the tighter the balancing of TH. One can thus take the value of the current circulating in the coil as a measure of the magnetic torque and record it together with a signal proportional to the angle ~ in order to recover the anisotropy curve shown in Fig. 9.8b.

570

CHAPTER 9

Measurement of Intrinsic Magnetic Properties

V//////Z Tungsten wire

/

Permanent magnet

~",, S js

s

,~

",

Photocells

DC a

Acquisition setup

Sample

J

,

,

i

)

1

s/SS"

FIGURE 9.10 Schematics of a torque magnetometer for magnetic anisotropy measurement. An electromagnet on a rotating platform provides large and uniform field over 360 ~ to a disk-shaped or spherical sample, placed in the center of the gap. The sample is solidly held within a quartz tube, which is suspended at the upper end by means of a taut wire and is provided with a suitable stabilizing weight at the bottom. A servo system keeps the sample in a fixed position during rotation of the field, by injecting a suitable current in a multi-turn coil integral with the tube. This is immersed in the field of a permanent magnet and provides a balancing torque to the tube. The current circulating in the coil provides a measure of the torque and is recorded together with the angle r correspondingly made by the field with the reference direction.

9.2 MAGNETIC ANISOTROPY

571

Torque magnetometers of this type are somewhat cumbersome to use, but they are rugged devices, displaying long-standing performances (the same unit has been in use for 35 years at IEN), in association with good sensitivity and wide dynamic range. With the development of Halbach's cylinders, made of rare-earth-based permanent magnets (Section 4.4), it has become possible to dispense with large electromagnets in the generation of the rotating field (though only below Ha---800 kA/m), resulting in compact devices [9.33]. In general, one can measure with the same apparatus either the large torque offered by Fe single crystals or the faint torques associated with induced anisotropies in soft amorFhous alloys. Typical measuring ranges are between 10 -8 and 1 0 - ~ N m . Especially sensitive devices have been built for measurements on thin films, where torque measurement capabilities may range between some 10 -5 and 10 -12 N m [9.34, 9.35]. Limits to the sensitivity may come from defective centering of the sample, unwanted shape anisotropy effects due to imperfect sample preparation, parasitic torques deriving from the possible presence of metallic parts in the region invested by the rotating field, and noise and instabilities in the servo system. The background signal is determined by means of a measurement made without the sample. Calibration can be performed by means of a reference sample, but absolute calibration is also possible. One way to do this is by measuring the torque exerted by the field Ha on an artificial dipole created by means of a current-carrying loop of known t u r n - a r e a product [9.36]. With this procedure, measurement of the field at the sample position is required. Another method consists in measuring the torsional coefficient of the taut suspension by means of a separate experiment and in determining the current generated by the servo system when, in the absence of field and sample, the assembly itself is rotated by an angle, imposing a known torsional moment to the suspension. The measurement of the magnetic torque can provide unambiguous quantitative information on the magnetocrystalline anisotropy of single crystals and on uniaxial induced anisotropies. A general approach to the analysis of the torque curves -r(q~) is based on a Fourier development, where one can express their periodic dependence on the angle qJ made with respect to a suitable direction as 00

9(~) = Y. An sin(nq~),

(9.17)

n=l

where one can take q~-= 0 in the case of uniaxial anisotropy. In cubic crystals, 4~is connected to the direction cosines appearing in Eq. (9.10). In this way, the coefficients A~ can be related to the anisotropy constants K1,

572

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

K2, and Ku. The application of the Fourier analysis requires unskewed torque curves or, better, the curves obtained under saturating fields, where the magnetization is practically aligned with the field. Let us consider the simple example of uniaxial anisotropy. We have already shown that when the anisotropy energy can be expressed as Ea -- Ku sin 20, the anisotropy constant Ku is provided by the peak value of the torque curve (Fig. 9.8b). If the energy Ea is instead expressed as in Eq. (9.11), with K2 non-negligible with respect to K1, we need to analyze the whole curve. By making the derivative OE/O0and developing it according to Eq. (9.17), we get

K2

"r(0) = (K1 + K2) sin 20 - -~-sin 40 = A 2 sin 20 + A4 sin 40.

(9.18)

Analysis of the curve provides the coefficients A 2 and A 4 and the anisotropy constants are therefore obtained as K1 =

A2 q- 2A4,

K2 = - 2A4.

(9.19)

When applied to polycrystalline materials, the torque analysis can reveal the presence of dominant textures, but it cannot unambiguously provide quantitative information on them without associated X-ray diffraction investigation. Figure 9.11 illustrates the case of a torque curve measured on a non-oriented Fe-(3 wt%)Si lamination. The sample is made of a disk of diameter 15 m m and thickness 0.35 m m and the applied field is of the order of 106 A / m . The curve fits into a (110) [001] texture, seemingly occupying a volume fraction of the sample around 20%. The X-ray ODF analysis does demonstrate the existence of such a texture, though in lower proportion than suggested by the magnetic experiment. This can be understood by considering that the torque results from the combined effects of several crystallographic components, whose volume fractions are not proportionally reflected on the torque curve. It is known, for example, that the contribution of the (111) planes, being related to the small second order constant K2 only, is totally masked by the contributions of other textures. The measurement of the magnetic anisotropy made according to the scheme shown in Fig. 9.8a, where the rotating magnetic field is sufficiently large to drag the magnetization reversibly along a 360 ~ period, but not so large to keep 0----~0, can be made without resorting to the direct determination of the torque with a magnetic balance. We see in Fig. 9.8a that the phase shift ~o- 0 between H a and Ms can be viewed in terms of component of the magnetization M l perpendicular to Ha. Since

9.2 MAGNETIC ANISOTROPY

573

'

I

'

I

Non-oriented Fe-(3 wt%)Si

4000

@

O\

9

o~ :

@

• ood •

2000 eo

E E z

OI

@

i

~ : Ol"

(~ I

0

p9

I

b I 9

O" 0

q~ ,

I

0

~

9

,

aj

P

I

l- -2000

......

b

I

t ,? -4000 0

,

I

45

i

I

90

J

I

135

,

180

e(~ RD FIGURE 9.11 Magnetic anisotropy torque curve in a non-oriented Fe-(3 wt%)Si lamination. Points represent the experiments. The fitting line results from Eq. (9.10) with dominant (110) [001] texture. X-ray diffraction ODF analysis shows that this is only partially the case and puts in evidence the semi-quantitative information on the texture of polycrystalline materials conveyed by magnetic torque experiments.

M• = Ms sin(~0- 0), we write Eq. (9.13) as 7(~o) =/~0M• Ha

(9.20)

and we conclude that, once the applied field is known, the torque is indirectly obtained by the m e a s u r e m e n t of the normal c o m p o n e n t of the magnetization M• This m a y represent an excellent m e a s u r i n g solution in those cases where the material is very hard, like the rare-earth-based magnets, m a k i n g ~0 - 0 seldom negligible d u r i n g the rotation [9.37]. Since we p u t ourselves in a condition where the magnetization components can be directly measured, which is not generally the case with torque balances, we can exploit the simultaneous determination of M• and Mli to obtain 0 = ~o- t a n - l ( M •

that is, the -r(O) torque curve.

(9.21)

574

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

We have shown in Fig. 8.18a an arrangement for the measurement of the torque through the measurement of M~ using a vector VSM and we have briefly discussed its operation in Section 8.2.1. With this method, two pairs of sensing coils are mounted along two orthogonal axes x and y, the field is directed along x, and the disk-shaped or spherical sample is rotated stepwise around the vibration axis z. The orthogonally placed coil pairs gather the signals proportional to M • and MII = Mx, respectively, and the torque -r(0) is eventually obtained via Eq. (9.20), after measurement of the field strength Ha. For uniaxial anisotropy, as described by Eq. (9.12), the constant Ku is provided by the peak value (M• x as Ku = p~(M• If the field, instead of the sample, is rotated, adopting, for example, the compact solution offered by Halbach's cylinders [9.38], we need to detect both the magnetization components Mx and My and the field components Hax and Hay. With reference to Fig. 9.8a, we obtain that M• and MII are related to the measured quantities Mx and My by the equations:

M• = Mx cos ~ - My sin ~,

Mll = My cos ~p+ Mx sin ~p.

(9.22)

Substitution of M• in Eq. (9.20) provides the expression for the torque: 9( ~ ) = a 0 ( M x H a y

-

MyHax),

(9.23)

the relationship between 0 and ~ being given by Eq. (9.21). Use of the vector VSM in torque measurements does not require special modifications of the conventional setup, but for the addition of an orthogonal set of coils and some extra electronics and software. Alternative methods may nevertheless be considered. For example, an extraction magnetometer with suitably modified sets of orthogonal coils has been employed [9.37] and a great deal of activity was carried out in the past using the rotating sample magnetometer [9.39, 9.40]. In the latter case, the voltage generated in the pickup coil positioned to sense M I is proportional to the derivative of the torque curve. The measurement of the anisotropy is often identified with that of the anisotropy field Hk. We have previously stated, that in a uniaxial system with energy given by Eq. (9.12), Hk -- 2Ku/la,oMs happens to be the field at which the magnetization starts to follow the rotating field in a fully reversible fashion. On the other hand, Hk also represents, according to the Stoner-Wohlfartti model, the nucleation field for coherent reversal of the magnetization in single domain particles, with Ku taken to derive from both intrinsic and shape effects. If we subject an ensemble of independent Stoner-Wohlfarth particles to a rotating field Ha, we thus expect that the nucleation field will be just the Ha value corresponding to the vanishing of

9.2 MAGNETIC ANISOTROPY

575

the irreversible processes, that is, of the energy loss. Note that, this continues to be true even for randomly distributed non-ideal StonerWohlfarth particles and the rotational loss measurement (that is, of the field at which it disappears) can therefore be taken as a good measure of the particle anisotropy [9.41]. The experiments on the rotational loss show that single-particle features can be retrieved to some extent from experiments on particle aggregates. It has actually been demonstrated in theory and experiment that investigation of the magnetization curve of a polycrystalline ferromagnet can lead to the determination of the anisotropy field of the individual crystals. The idea is from Asti and Rinaldi [9.42], who took at face value the fact that a single crystallite brought to saturation along a hard direction exhibits a magnetizing curve with a discontinuity at the saturation point. This can easily be demonstrated if, for example, we calculate the magnetization curve in a uniaxial crystal along a direction orthogonal to the easy axis. It is sufficient to take the expression for the

1.0

0.8 t'-

=

0.6

~ 0.4 % BaFe

0.2

0.0

J

0.0

015

.

.

.

.

.

.

1.'0

.

.

I

1.5

H/H k

FIGURE 9.12 Experimental behavior of the second derivative d2M/dH 2 along the magnetization curve in a polycrystalline Ba ferrite sample. H is the effective field H = H a - NckM, where Ha is the applied field and Nd is the demagnetizing coefficient, d ' M / d H 2 exhibits a cusp exactly at the position of the anisotropy field Hk = 2(K1 + 2K2)/M s (adapted from Ref. [9.42]).

576

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

anisotropy energy Ea given by Eq. (9.11) and minimize the total energy E - -/~0Ms.H + Ea to obtain the curve M(H). Having a definite slope, this attains the saturated state at a finite field value, specifically the anisotropy field value Hk = 2(K1 + 2K2)/M s. The derivative d M / d H thus presents a step-like discontinuity at H = Hk, which transforms into a cusp upon making the second derivative d2M/dH 2. Asti and Rinaldi showed that this singularity is not lost when such a crystal is immersed in a sea of randomly oriented grains. They demonstrated this for non-interacting grains, but the experiments revealed that also in dense aggregates a cusp showed up in correspondence with the anisotropy field upon successive derivations of the magnetization curve. Indeed, it is difficult to observe such a singularity in soft magnets, where magnetostatic interactions appear to be very effective in the face of the anisotropy energies. The determination of the anisotropy field in polycrystalline hard magnets using the singular point detection (SPD) technique is best accomplished by analyzing the curve obtained in transient fashion by means of a pulsed field magnetizer (Section 8.2.4) [9.43]. Figure 9.12 provides an example of experimental behavior of d2M/dH 2 along the magnetization curve in a polycrystalline BaFel2019 sample [9.42]. The cusp in the second derivative occurs, as predicted, in correspondence with the anisotropy field. It is verified that this always occurs, independent of the specific distribution of orientation and size of the grains, even when the size is larger than required for achieving the single domain state.

aefeyences 9.1. S. Chikazumi, Physics of Ferromagnetism (Oxford: Oxford University Press, 1997), p. 274. 9.2. H. Zijlstra, (Amsterdam: North-Holland, Experimental Methods in Magnetism, 1967), vol. 2, 182. 9.3. R. Pauthenet, "Experimental verification of spin-wave theory in high fields," I. Appl. Phys., 53 (1982), 8187-8192. 9.4. T. Holstein and H. Primakoff, "Field dependence of the intrinsic domain magnetization of a ferromagnet," Phys. Rev., 58 (1940), 1098-1113. 9.5. H. Kronmfiller, "Micromagnefism in amorphous alloys," IEEE Trans. Magn., 15 (1979), 1218-1225. 9.6. E.M. Chudnovsky, "Magnetic properties of amorphous ferromagnets," J. Appl. Phys., 64 (1988), 5770-5775. 9.7. P. Szymczak, C.D. Graham, Jr., and M.R.J. Gibbs, "High-field magnetization measurements on a ferromagnetic amorphous alloy from 295 to 5 K," IEEE Trans. Magn., 30 (1994), 4788-4790.

REFERENCES

577

9.8. M.K. Wilkinson and C.G. Shull, "Neutron diffraction studies on iron at high temperatures," Phys. Rev., 103 (1956), 516-524. 9.9. A. Ferro, G. Montalenti, and G.P. Soardo, "Temperature dependence of power loss anomalies in directional Fe-Si 3%," IEEE Trans. Magn., 12 (1976), 870-873. 9.10. A. Arrott, "Criterion for ferromagnetism from observations of magnetic isotherms," Phys. Rev., 108 (1957), 1394-1396. 9.11. T.R. McGuire and P.J. Flanders, "Direct current magnetic measurements," in Magnetism and Metallurgy (A.E. Berkowitz and E. Kneller, eds., New York: Academic Press, 1969), p. 123. 9.12. J.E. Noakes and A. Arrott, "Initial susceptibility of ferromagnetic iron and iron-vanadium alloys just above their Curie temperature," J. Appl. Phys., 35 (1964), 931-932. 9.13. S. Arajs and R.V. Colvin, "Ferromagnetic-paramagnetic transition in iron," J. Appl. Phys., 35 (1964), 2424-2426. 9.14. S. Arajs, "Paramagnetic behavior of nickel just above the ferromagnetic Curie temperature," J. Appl. Phys., 36 (1965), 1136-1137. 9.15. J. Sievert, H. Ahlers, S. Siebert, and M. Enokizono, "On the calibration of magnetometers having electromagnets with the help of cylindrical nickel reference samples," IEEE Trans. Magn., 26 (1990), 2052-2054. 9.16. J. Crangle and G.M. Goodman, "The magnetization of pure iron and nickel," Proc. Roy. Soc. Lond., A321 (1971), 477-491. 9.17. S. Foner, A.J. Freeman, N.A. Blum, R.B. Frankel, E.J. McNiff, Jr., and H.C. Praddaude, "High-field studies of band ferromagnetism in Fe and Ni by M6ssbauer and magnetic moment measurements," Phys. Rev., 181 (1969), 863-882. 9.18. M. Liniers, J. Flores, F.J. Bermejo, J.M. Gonzalez, J.L. Vicent, and J. Tejada, "Systematic study of the temperature dependence of the saturation magnetization in Fe, Fe-Ni, and Co-based amorphous alloys," IEEE Trans. Magn., 25 (1989), 3363- 3365. 9.19. R.D. Shull, R.D. McMichael, L.J. Swartzendruber, and S.D. Leigh, "Absolute magnetic moment measurements of nickel spheres," J. Appl. Phys., 87 (2000), 5992-5994. 9.20. L. Petersson and A. Ehrenberg, "Highly sensitive Faraday balance for magnetic susceptibility studies of dilute protein solutions," Rev. Sci. Instrum., 56 (1985), 575-580. 9.21. A.M. Stewart, "The superconducting Faraday magnetometer: error forces and lateral stability," J. Phys. E: Sci. Instrum., 8 (1975), 55-59. 9.22. D. Zhang, Ch. Probst, and K. Andres, "A novel and sensitive Faraday-type magnetometer for the field range from 0 to 12 T," Rev. Sci. Instrum., 68 (1997), 3755-3760.

578

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

9.23. P.J. Flanders and C.D. Graham, "Magnetization measurements using the field gradient of a high-field solenoid," Rev. Sci. Instrum., 50 (1979), 1564-1566. 9.24. G. Felten and Ch. Schwink, "Design of a Faraday magnetometer in Bitter coils," J. Phys. E: Sci. Instrum., 13 (1980), 487-488. 9.25. R.D. Heyding, J.B. Taylor, and M.L. Hair, "Four-inch shaped pole caps for susceptibility measurements by the Curie method," Rev. Sci. Instrum., 32 (1960), 161-163. 9.26. R.T. Lewis, "A Faraday type magnetometer with an adjustable field independent gradient," Rev. Sci. Instrum., 42 (1971), 31-34. 9.27. A.M. Stewart, "Prediction of lateral instabilities in the Faraday magnetometer," J. Phys. E: Sci. Instrum., 2 (1969), 851-854. 9.28. R.D. Spal, "Production of uniform field gradients for magnetometers by means of current-carrying strips," J. Appl. Phys., 48 (1977), 1338-1341. 9.29. A. Hubert and R. Sch/ifer, Magnetic Domains (Berlin: Springer, 1998), 184. 9.30. J.S. Kouvel and C.D. Graham, Jr., "On the determination of magnetocrystalline anisotropy constants from torque measurements," J. Appl. Phys., 28 (1957), 340-343. 9.31. G. Aubert, "Torque measurements of the anisotropy energy and magnetization of nickel," J. Appl. Phys., 39 (1968), 504-510. 9.32. J.D. Sievert, "Anisotropy of energy and magnetization of iron-rich Si-Fe alloys," J. Magn. Magn. Mater., 2 (1976), 162-166. 9.33. B. Comut, S. Catellani, J.C. Perrier, A. Kedous-Lebouc, T. Waeckerl6, and H. Fraisse, "New compact and precise magnetometer," J. Magn. Magn. Mater., 254-255 (2003), 97-99. 9.34. EB. Humprey and A.R. Johnston, "Sensitive automatic torque balance for thin magnetic films," Rev. Sci. Instrum., 34 (1963), 348-358. 9.35. M. Tejedor, A. Fernandez, B. Hemando, and J. Carrizo, "Very simple torque magnetometer for measuring magnetic thin films," Rev. Sci. Instrum., 56 (1985), 2160-2161. 9.36. M.J. Pechan, A. Runge, and M.E. Bait, "Variable temperature ultralow compliance torque magnetometer," Rev. Sci. Instrum., 64 (1993), 802-805. 9.37. E. Joven, A. del Moral, and J.I. Arnaudas, "Magnetometer for anisotropy measurement using perpendicular magnetization," J. Magn. Magn. Mater., 83 (1990), 548- 550. 9.38. O. Cugat, R. Byme, J. McCaulay, and J.M.D. Coey, "A compact vibrating sample magnetometer with variable permanent magnet flux source," Rev. Sci. Instrum., 65 (1994), 3570-3573. 9.39. P.J. Flanders, "Magnetic measurements with the rotating sample magnetometer," IEEE Trans. Magn., 9 (1973), 94-109.

REFERENCES

579

9.40. P.J. Flanders, "A Hall sensing magnetometer for measuring magnetization, anisotropy, rotational loss and time effects," IEEE Trans. Magn., 21 (1985), 1584-1589. 9.41. G. Bottoni, D. Candolfo, A. Cecchetti, and F. Masoli, "Ratio of the rotational loss to hysteresis loss in ferromagnetic powders," IEEE Trans. Magn., 10 (1974), 317-320. 9.42. G. Asti and S. Rinaldi, "Singular points in the magnetization curve of a polycrystalline ferromagnet," J. Appl. Phys., 45 (1974), 3600-3610. 9.43. R. Gr6ssinger, Ch. Gigler, A. Keresztes, and H. Fillunger, "A pulsed field magnetometer for the characterization of hard magnetic materials," IEEE Trans. Magn., 24 (1988), 970-973.

This Page Intentionally Left Blank

CHAPTER 10

Uncertainty and Confidence in Measurements

The ideal objective of any measurement is the determination of the true value of a measurand. The real objective is to make the most accurate estimate of this true ~/alue because no measuring operation can exist without an error. Consequently, a measurement has a meaning if, having defined a measuring method and a measuring procedure, it provides the best estimate of the value of the measurand and a related uncertainty, the latter representing the degree of dispersion of the results around such an estimate. At the core of the concept of measurement lies the principle of reproducibility, which implies the possibility to compare results obtained at different times and in different laboratories. It is not only a vital requirement of any scientific investigation, but it also responds to practical needs in various fields, such as industrial production and quality control, commerce, law, health, and environment. In order to make meaningful comparisons, it is necessary that measuring uncertainties be treated through a consistent approach. Although the subject is very old, general consensus on the procedures to be followed for expressing the uncertainty has only been reached in recent times, under the initiative of the Comitf International des Poids et Mesures (CIPM), the highest authority in the field of metrology. Through the active cooperation of the National Metrological Institutes and various international organizations, the International Standard Organization (ISO) undertook the task of preparing a Guide to the Expression of Uncertainty of Measurement, which was eventually published in 1993 [10.1]. We will refer to this Guide in the following.

10.1 E S T I M A T E O F A M E A S U R A N D MEASURING UNCERTAINTY

VALUE AND

Measuring a physical quantity is a very common activity in everyday life and the concepts of measurement accuracy and repeatability do not 581

582

CHAPTER 10 Uncertainty and Confidence in Measurements

require special competence to be appreciated. It is intuitive to recognize that some kind of stochastic behavior inevitably affects any measuring operation, be it some gross evaluation performed through our senses or some sophisticated measure made by specialists in the laboratory. If we go somewhat deeper into the problem, we can easily verify that, by repeating the very same measurement m a n y times under identical conditions, scattered values of the measurand are found. Once ordered according to the customary histogram or frequency representations, these do provide the idea of an underlying probability distribution function [10.2]. Such an idea was made quantitative a long time ago. By denoting with x the generic value of the measurand subjected to direct determination, it can be shown that, for example, if the condition of stationarity is satisfied, the probability of finding it within a prescribed interval (x, x + dx) is given by the normal distribution function:

(X- ~)2) exp 2~2 dr(x) dx -~r2 ~

"

(10.1)

f(x) is a symmetric function, peaked at the mean value x = ~, and satisfies the normalization condition y~-oof(x) dx --- 1./~ is also the most probable value of x and is identified with the true value of the measurand, with the meaning that this term has in a statistical sense. It would be the result of a perfect measurement and cannot be known. If the outcome of a measurement is x, the difference 8 = x - ~ is defined as the measurement error, again an unknowable quantity, cr2 is the second-order m o m e n t about the mean:

~

oo

o .'2 - -

(x -

~)2f(x)

dx

(10.2)

and is called variance. Its square root cr provides a measure of the dispersion of the measured values around the true value and is called

standard deviation. Historically, the normal function was proposed by Gauss in order to represent the error distribution in the astronomical observations. It is the idealized distribution function associated with a truly stochastic variable. Any reading or measurement of this variable can be thought of as affected by m a n y small contributions of r a n d o m sign and amplitude, which are generated by a large n u m b e r of sources of influence. The central limit theorem [10.2] ensures that, in a case like this, the values taken by the variable closely follow a normal law, whatever the distribution function of the contributing variables. It is therefore understood that fix) can be

10.1 ESTIMATE OF A MEASURAND VALUE

583

assumed of normal type [10.3]. Any practical measuring operation, carefully performed and corrected for any possible bias, can only approximate the generation of a truly Gaussian process and what one achieves, in general, is an estimate of the true value of the measurand. If n independent observations of the measurable quantity x are performed, providing the values X(1),X(2)~...~X(n)~ the best estimate is given by the arithmetic mean:

YC--

~.k

=1 F/

x(k) ,

(10.3)

where the individual outcomes X (i) differ because of random effects. In the limit n--* oo~ it is expected that ~ =/~. In reality, it is difficult in most instances to fulfill the condition of stationarity for a sufficiently long time and a convenient number of repetitions is chosen according to specific conditions imposed by the problem under testing. From a sample of measurements, one can make an estimate of the variance cr 2 of the whole population of the possible values of the measurand by defining an experimental variance s2(x(k)). This characterizes the dispersion of the measured values around ~:

11 82(x(k)) = Yk=l

(x(k) --

~)2

n- 1

'

(10.4)

together with its square root, the experimental standard deviation s(x(k)). Notice that the number of degrees of freedom v = n - 1 is used in the definition of the experimental variance in Eq. (10.4). In fact, of the n terms (x (k) -Yc)~ only n - 1 are independent. Since the experiment provides the value ~ as a best estimate of the true value of the measurand, we wish to know how good such an estimation is. ~ is itself a random quantity and, according to Eq. (10.3), its variance and standard deviation are cr2(Yc) = cr2/n and cr(~)= cr/x/~ , respectively (the law of large numbers). The best estimates of 02(~) and cr(~) are

$2(~) __

S2(x(k))

__

n

ylkZ=l (x(k) __ ~)2 n(n - 1) (10.5)

S(x(k)) s(~)

-

-- .. I ~k=l (x(k) --

~

n ( n - 1)

~)2

584

CHAPTER 10 Uncertainty and Confidence in Measurements

the experimental variance and the standard deviation of the mean, respectively, s(~) is also called the standard uncertainty u(yc) of the best estimate of the measurable quantity x u(~) = s(~)

(10.6)

and the corresponding variance is u2(x)~-s2(x). According to this definition, the standard uncertainty u(~) is a parameter providing a quantitative evaluation of the dispersion of values that can be reasonably attributed to the measurand. By making repeated measurements of the same quantity, stochastic effects are thus revealed and can be quantified through the standard uncertainty. There are, however, further sources of uncertainty, whose contribution remains constant while the measurements are repeated. They can derive from the environmental conditions (e.g. temperature, humidity, and electromagnetic interference), calibration and resolution of the equipment, peculiarities of the electrical circuit (e.g. thermoelectromotive forces), drifts and distortions, inaccurate assumptions about constants and parameters to be used in the data treatment procedure, and personal errors. The related uncertainty is traditionally classified as systematic, in contrast with the random uncertainty, associated with repeated measurements. It is recognized, however, that such a classification applies to a specific measurement only, because what is random in a measurement can become systematic in a further measurement at a different level. For example, an instrument calibration made in a standard laboratory will report the combination of random and systematic components as a single value of the total uncertainty. A laboratory at a lower hierarchical level, making use of this calibrated instrument, will introduce this value as a systematic effect in the derivation of its uncertainty budget. Systematic effects are expected to produce a bias on the random distribution of the x (k) values obtained upon repeated measurand determinations. This bias can be quantified and corrected for a good proportion, as schematically illustrated in Fig. 10.1, but a residual contribution to the uncertainty of the measurement, having a systematic origin, is nevertheless expected to remain. This can be evaluated by judicious appraisal of all available information on the physical quantity being measured, the measuring procedure and the measuring setups, previous knowledge on the subject, etc. Notice that, in some instances, the correction for the bias can be estimated to be zero, without implying that the associated uncertainty contribution is also zero. The method by which an uncertainty contribution deriving from systematic effects is obtained is defined as a Type B evaluation method. For repeated measurements, we speak of a Type A evaluation method of the uncertainty. These definitions

10.1 ESTIMATE OF A MEASURAND VALUE

0.201

585

-",,,,

0.10

o 0.00. (a)

i\ ~

95

1O0

105

x

m,L

,

bias

i--

12

i _ e

A

x

v

i

Z

(b)

85

90

....

i ....

95 x

i ......

100

105

FIGURE 10.1 (a) Normal distribution function f(x) for the probability of finding the value of a measurand in the interval (x,x + dx) (Eq. (10.1)). x is assumed to be a truly stochastic quantity. The mean value /~ is defined as the true value of the measurand, an ideal and unknowable quantity, cr is the standard deviation. (b) Independent repeated measurements generate numbers that, arranged in a histogram, emulate the normal distribution. The raw data are shown on the left, with their mean ~ and the related standard deviation s(2) (not in scale). The standard deviation of the mean is equivalently called standard uncertainty u(~) (see Eqs. (10.5) and (10.6)). After correction for the systematic effects (somewhat exaggerated here for clarity), the best estimate of the true measurand value is characterized by a combined uncertainty uc(x), including the uncertainty on the correction.

586

CHAPTER 10 Uncertainty and Confidence in Measurements

are recommended by the ISO Guide [10.1]. In any case, all components of the uncertainty, be they evaluated with A or B methods, are described by the same statistical methods, characterized by probability densities with variances and standard deviations, and are treated and combined in the same way. For the Type A uncertainties, the probability densities are obtained from observed frequency distributions, while for the Type B uncertainties one makes use of "a priori" probabilities. When applying the B method, an assumption is made regarding the distribution function of the measurand values. If derived from a calibration certificate, this distribution is conveniently assumed as being of the normal type. In other cases, it is only possible to estimate upper and lower bounds x0- and x0+ for the values that x can take in a specific measurement. With no further knowledge on how these values are distributed, one can reasonably assume that they are equally likely to belong to any point of the interval (Xo-,Xo+). The variance and the uncertainty associated with the expectation value ~ = ( x 0 - + x0+)/2 of this rectangular distribution are expressed, posing x 0 - + x0+ = 2a, by the equations: U2(~ ) __ (X0+ -- X 0 _ ) 2

12

10.2 C O M B I N E D

a2

= -~-,

a

u(~) = - ~ .

(10.7)

UNCERTAINTY

In the usual case one does not make a direct determination of the measurand, but a certain number of input quantities are sampled or considered, to which the measurand is related by a functional relationship. Let the functional relationship between the output quantity y and the input quantities Xl,X2, ...,XN be y = g(xl~x2, ..., XN).

(10.8)

If the best estimates of the input quantities a r e Xl~ x2~-..~ XN~ we write for the best estimate of the output quantity:

9 = g(x~,X2,

"", IN).

(10.9)

The identificationof the input quantities is a crucial step of the whole process of uncertainty determination. They can include, besides the quantities subjected to direct measurement, the bias corrections to suggested by the specifically considered measuring procedure. The problem then becomes one of determining variance and uncertainty of from knowledge of the same quantifies for ~i,x2,...,xN, taking into

10.2 COMBINED UNCERTAINTY

587

account possible correlation effects. To this end, we assume that the function g and its derivatives are continuous around the expectation value Y- A Taylor series development, truncated to the first order, provides y-- y = ~ ~ i=1

(10.10)

(Xi - YCi)

for small intervals (X i -- YCi). The square of Eq. (10.10) is

(y_~)2=~.

N (0g)2 N-1 N OR OR ~ ( XOXj ~ (Xi_YCi)2q_2 y . y . ~OXi i=1 i=1 j=i+l

i -- YCi)(Xj -- YCj). (10.11)

By interpreting the differences appearing in Eq. (10.11) as experimental samples and taking the averages, we can express the variance of the output estimate as a combination of the variances u2(xi) and covariances U(YCi,YCj) of the input estimates, according to the law of propagation of

uncertainty: N (Og)2

uc(y)= ~22

i=1

~

N-1N OgOg /,/2(~i)+ 2 y. y. OXi ~u(Yci,Yq). OXi i=1 j=i+l

(10.12)

-

uc(y) is called combined variance and its square root Uc(y) is the combined standard uncertainty. The partial derivatives in Eq. (10.12) are called sensitivity coefficients and by posing ci -~ Og/Oxi we can rewrite Eq. (10.12) as N N-1 N 2Uc(y) = y c2ua(xi)-}-2 y~ y . CiCjU(Xi~YCj). i=1 i=1 j-i+1

(10.13)

Equation (10.13) is of general character and combines variances and covariances of the input quantities irrespective of the type of evaluation method (either A or B) employed in their derivation. An input estimate Xi can be associated with both Type A and Type B uncertainties and the related variance in Eq. (10.13) is written as U2(Xi) = U2A(YCi)-~- U~(YCi).

(10.14)

Notice that the output quantity y is associated in many cases with an approximately normal distribution function, although the distribution functions of the input quantities can be far from normal. One can, in fact, linearize the functional relationship (10.8) around the best estimates of the input quantities by means of a Taylor development truncated to

~8

CHAPTER 10 Uncertainty and Confidence in Measurements

the first order (Eq. (10.10)). The output distribution is then provided by the convolution of the input distributions and, according to the central limit theorem, it can be approximated by a normal distribution, the higher the number of repetitions and the input quantities, the better the approximation. The case where a dominant Type B component of uncertainty exists, with distribution different from normal, is an exception to this rule. 2- -2 In many cases, it is useful to resort to the relative variance Uc(y)/y and the relative standard uncertainty uc(Y)/9. Remarkably enough, if it occurs that the functional relationship relating the output quantity and the input independent quantities has the general form y = m.xPl.x p2, ...,XPNN, with m a constant coefficient and Pl,P2, ...,Pn either positive or negative exponents, we can express the relative variance as

2Uc(y)

N U2(Xi) ~2 __~p2 L-~ 9 i=l Xi

(10.15)

The sensitivity coefficients are sometimes evaluated by experiments by determining the variation of the output quantity y upon a small variation of an input quantity xi, the other input quantities being kept fixed (see Eq. (10.10)). If the input quantities are uncorrelated, the covariance u(Yci,ycj) is zero and the combined uncertainty reduces to

Uc(Y) = i ~'i=1r

(10.16)

A measure of the degree of correlation is provided by the value of the coefficient: u(~i, ~j) r(Yci,Ycj) = u(Yci)u(Ycj),

(10.17)

which varies from 0 to 1 on going from uncorrelated to completely correlated input variables. In the latter case, the combined standard uncertainty becomes

N UcQ~) --- ~ CiU(YCi). i=1

(10.18)

For a Type A evaluation of the uncertainty, the covariance of two correlated input estimates (2i, x]) can be experimentally evaluated by forming the cross-products (xl k)- Yci).(x~k)- 2]) at each repetition and

10.3 EXPANDED UNCERTAINTY AND CONFIDENCE LEVEL

589

averaging them according to the equation: i! u(2i,Yq) = s(2i, 2j)= n(n - 1) y (xlk) -- Xi)'(X~k) -- ~j)"

(10.19)

k=l

For a Type B evaluation, critical analysis of the available information on the reciprocal influence of the input estimates should be carried out. If, for example, it is known that a variation Ai of xi produces a variation &j of ~j, the correlation coefficient can be roughly estimated as r(Yci, Ycj) ~ u(Ycj)a i "

(10.20)

Two input quantities (Xi,Xj) can have their correlation originating from a common set of independent and uncorrelated quantities z,,,. Let gi(Zl, ...,ZM) and gj(zl, ...,zM) be the functional relationships associated with xi and xj, respectively. We can write for the covariance of the best estimates (2i, ~j): M Ogi Ogj U2(~,n). U(YCi'YCJ) = Y" OZ,,, OZ,n llZ--1

(10.21)

10.3 E X P A N D E D U N C E R T A I N T Y A N D C O N F I D E N C E LEVEL. WEIGHTED UNCERTAINTY The discussion of the derivation of the uncertainty budget carried out in the previous sections illustrates the great merit of the procedure recommended by the ISO Guide, which permits one to combine in a consistent way all the contributions to the uncertainty, derived either from repeated measurements or through "a priori" probabilities. The information conveyed by the measurement can then be confidently collected in two parameters: the best estimate (experimental mean) and the combined uncertainty. Except for special cases, the probability distribution of the output quantity, being the convolution of the input distributions, is approximately of normal type (central limit theorem). If z is a quantity described by a normal distribution function, characterized by an expectation value/~ and a standard deviation or, and we define a confidence interval + kcr around/~, by integrating the distribution function over it we will achieve a corresponding confidence level p (the included portion of the area of the distribution). With coverage factors k = 1, 2, and 3, the confidence levels are p = 68, 95.5, and 99.7%, respectively. Let y be a quantity, defined as in Eq. (10.8), subjected to measurement and characterized by an experimental best estimate 9 and

590

CHAPTER 10 Uncertainty and Confidence in Measurements

a combined standard uncertainty Uc(y). We wish to determine the coverage factor k identifying an expanded uncertainty U = kuc(~) , that is, an interval 9 - U --< y - 9 + U, to which the true value of the measurand is expected to belong with a given high confidence level p (e.g. 95%). With knowledge of U, the result of the measurement can be declared in the form: Y = 9 + U.

(10.22)

We do not actually know the expectation value /~ and the standard deviation or of the output distribution, but only the best estimate 9 and the standard uncertainty uc(~). We then consider the quantity t-

y- y

(10.23)

Uc(9)

and its probability distribution function ~ t ) . The integration of q~(t) over a certain interval ( - t p , +tp): f+t~ q~(t) dt

(10.24)

P = d-tp

provides the confidence level for the expanded interval U = t~Uc(~)= kuc(9). In fact, the condition ( - t p ~ t ~ tp) is equivalent, according to Eq. (10.23), to the condition (9 - tpUc(9) ~ y ~ 9 + tpUc(9)). When y is a single quantity subjected to direct measurement and its best estimate 9 = ~ is obtained by means of a series of n independent repeated measurements, q~(t) is described by the Student distribution function (t-distribution): ~b(t) -

~

1

F((v + 1)/2) ~+1)/2, F(u/2) (1 + ta/v) -(

(10.25)

where the properties of the F function are known and v = n - 1 is the n u m b e r of degrees of freedom, q~(t) reduces to the normal distribution in the limit v---+ oo, a condition already well approximated for v --- 50. For the general case where y is a function of two or more input quantities, Eq. (10.25) can be used only as an approximation by introducing an effective number of degrees of freedom /jeff in place of v. /jeff can be calculated in terms of the degrees of freedom/ji of the input uncertainty contributions u2(9) c2ua(xi)~ under the assumption of independent input estimates. It is provided, in particular, by the Welch-Satterthwaite formula: =

4-

/Jeff =

uc(y) 4-

ui (y) /ji

,

(10.26)

where u~(9) = (y../N_au 2i (y)) - ~_. While for the Type A evaluation, z,i = n - 1 , the degrees of freedom in the Type B evaluation can be estimated on

10.3 EXPANDED UNCERTAINTY AND CONFIDENCE LEVEL

591

the principle that the more reliable the standard uncertainty ui(y) the higher ~'i. For the usual case where an "a priori" probability is taken, the uncertainty is completely defined and z,i--* oo. Tabulations are available where, for given ~'eff values, the coverage factors k = tp associated with a confidence level p are provided [10.4]. In most cases, a confidence level p---95% is deemed adequate. The corresponding coverage factors are provided, for different values of ~'eff, in Table 10.1. An application of the concepts discussed in this section is given in Appendix C (Example 2). The procedure for providing the result of a measurement can therefore be summarized as follows: (1) The measurement process is modeled and the mathematical relationship y = g(xl,x2, ...,x N) between the measurand y and the input quantities Xl,X2, ...,XN is expressed. These quantities also include possible bias corrections. (2) The best estimates xl, x2,..., XN of the input quantities are made. (3) The standard uncertainties u(xi) of the input estimates are found. Type A evaluation is applied for input estimates obtained by means of repeated measurements and Type B evaluation for all other kinds of estimates. If there are correlations between input quantities, covariances are considered. (4) Using the functional relationship y = g ( x l , x 2 , . . . , X N ) , the best estimate 9 of the measurand is made. (5) The combined standard uncertainty Uc(y) is calculated by combination, with the appropriate sensitivity coefficients, of the variances and covariances of the input estimates. (6) The expanded uncertainty U = kuc(9) is determined, with the coverage factor k typically ranging from 2 to 3 for a confidence level of 95%. The actual value of k depends on the effective number of degrees of freedom Veff, calculated by means of the Welch-Satterthwaite formula, and are found in generally available tabulations. The result of the measurement is eventually declared a s y = ~/+ U.

TABLE 10.1 Coverage factor k as a function of the effective degrees of freedom Veff

for a confidence level p = 95.45%. Ueff k

1 13.97

2 4.53

3 3.31

4 2.87

5 2.65

7 2.43

10 2.28

20 2.13

50 2.05

oo 2

k is provided by the Student distribution function and coincides, in the limit veff~ o% with the value provided by the normal distribution.

592

CHAPTER 10 Uncertainty and Confidence in Measurements

It may happen that the same quantity is measured by means of different methods or in different laboratories and the problem arises of combining the results in order to obtain the most reliable estimate of the measurand value. It is usually held that, being the different estimates normally associated with different uncertainties, a comprehensive estimate based on the data generated by the whole ensemble of experiments is best obtained by means of weighted averaging. Let us assume that M independent experiments, made of a convenient number of repeated measurements, have produced the best estimates Yl~ Y2~'" "~ YM and the related uncertainties uc(yl),Uc(92),...,Uc(gM). We look for a weighted estimate of the type M

= ZgilJi

(10.27)

i=1

having minimum variance. The weight factors gi must satisfy the condition: M

Z gi = 1.

(10.28)

i=1

We then write the variance of ~ in terms of the variances of the estimates Yl,Y2, ...,YM

M U2(~) = Z gi2 Uc(Yi) 2 i=1

(10.29)

and find the set of factors gl,g2, ...,gM minimizing u2(~) [10.5]. With the use of the Lagrange multiplier k, the variance can be written as u 2 ( Y ) = Z gi2 Uc(Yi) 2 - + ,~ 1 - Z g i i=1 i=1

(lO.3O)

and the minimization conditions

0u2(#) Ogi

= 2giu2(gi)- X = 0

(10.31)

provide the weight factors k

gi = ," cl,y "

(10.32)

as a function of the multiplier JL This is eliminated through the normalization constraint (10.28) and the factors gi are thus obtained as

10.3 EXPANDED UNCERTAINTY AND CONFIDENCE LEVEL

593

a function of the input variances:

U2(~ti)

g~ =

~.iM1

.

(10.33)

1 U2(~/i)

The weighted mean and the associated weighted variance follow from Eqs. (10.27) and (10.29): y.iM1

Yi

u2(yi)

~t ~-

1

,

(10.34)

2 Uc(Yi)

1

u2(~)

M =

1 u2(yi) "

(10.35)

According to these equations, the smaller the uncertainty associated with a result, the higher its role in determining the reference value ~ and the uncertainty u(,0). A confidence interval can be identified with ~ + U r e f = ~t + ku(~t), where the coverage factor k is taken from the t-distribution for v = M - 1 (see Table 10.1). An example of intercomparison of magnetic measurements is shown in Fig. 10.2. Laboratories find the most stringent test of their measuring capabilities in the comparison exercise. At the highest level, the national metrological laboratories organize key comparisons as a technical basis for establishing the equivalence of measurement standards and the mutual recognition of calibration and measurement certificates. The degree of equivalence of each national measurement standard is expressed quantitatively by two terms: its deviation from the key comparison reference value and the uncertainty of this deviation at 95% level of confidence [10.6]. The assumption of the weighted mean (10.34) as the reference value is considered appropriate if the collective measurements are consistent and they can be treated as part of a homogeneous population. Discrepant results often arise in intercomparisons and special approaches have therefore been proposed to deal with the problems, including politically sensitive issues, raised by the presence of inconsistent data [10.7, 10.8]. It is clear that meaningful comparisons can be pursued only where all laboratories follow a common approach to the evaluation of the measuring uncertainty, such as the one provided by the ISO Guide [10.9].

594

CHAPTER 10 Uncertainty and Confidence in Measurements

3.2

t_ - Uref

3.1

cL 3.0

2.9

Laboratories

FIGURE 10.2 Six different laboratories perform the measurement of magnetic power losses on the same set of non-oriented Fe-Si laminations with the SST method [10.10]. They report their best estimates and the related extended uncertainties as shown in this figure. Analysis shows that the result provided by laboratory 6 is to be excluded, because it largely fails the consistency test provided by the calculation of the normalized error (Eq. (10.36)). The reference value Pref solid line) and the expanded uncertainty Uref (delimited by the dashed lines) are then obtained by re-calculating Eqs. (10.34) and (10.35) with the results of laboratories 1-5.

Let us analyze, as an example regarding magnetic measurements, some results derived from an intercomparison of magnetic power losses in non-oriented Fe-Si laminations [10.10]. Six laboratories (i = 1, ..., 6) provide, as shown in Fig. 10.2, their best estimates Pi of 50 Hz losses at 1.5 T (full dots) and the associated expanded uncertainties (at 95% confidence level) Ui = ku~(Pi). These data are all included in a preliminary determination of the weighted mean (reference value Pref) and the expanded weighted uncertainty Uref~ according to Eqs. (10.34) and (10.35). The consistency of the reported uncertainties with the observed deviations of the best estimates Pi from Pref is then verified. To this end, the normalized error

Eni

=

IPi - Prefl ~/U2..}_Ur2;

(10.36)

10.4 TRACEABILITYAND UNCERTAINTY

595

is considered [10.11]. When the dispersion of the individual estimates is in a correct relationship with the correspondingly provided uncertainties, it is expected that Eni < 1 [10.12]. In the present case, the reported (P6, U6) values largely fail to satisfy this condition (En6--3.1), due to both unrecognized bias and unrealistic uncertainty estimate. They are consequently excluded from the analysis. Pref and Ure f are then re-calculated by means of Eqs. (10.34) and (10.35), providing the results reported in Table 10.2. It should be stressed that the estimated expanded relative uncertainty of the reference value Uref~re1 is of the order of 1%, typical of this kind of measurement. It is also noticed that the result of laboratory 5 is not completely satisfactory because IP i - Prefl is higher than the related expanded uncertainty: (10.37)

U(Pi - Pref) -- k~/u2(Pi) if- u2(Pref) 9

10.4 T R A C E A B I L I T Y MAGNETIC

AND UNCERTAINTY MEASUREMENTS

IN

Measurements are indispensable for the manufacturing and trade of products and for any conceivably related research activity. They need to be traceable to the relevant base and derived SI units, that is, related to the corresponding standards through an "unbroken chain of comparisons, all having stated uncertainties" [10.13]. Industrial and research laboratories can achieve traceability for a specific kind of measurement through accredited laboratories or directly to National Metrological Laboratories (NMIs). The mission of NMIs is to ensure that the standards are the most accurate realization of the units, so that these can be disseminated to the national measurement network. To ensure this calibration flow, national calibration and accreditation systems have been developed. The NMIs engage in extensive intercomparisons of standards, organized either by the regional metrological organizations (e.g. EUROMET and NORAMET) or the Consultative Committees of the International Committee for Weights and Measures (CIPM). Supervision of the intercomparison activity is carried out by the Bureau International des Poids et Mesures (BIPM), which has the task of ensuring worldwide uniformity of measurements and their traceability to the SI units (Fig. 10.3). Physical standards for magnetic units, traceable with stated uncertainties to the base SI units, are maintained in several NMIs and used for dissemination to measurement and testing laboratories [10.14, 10.15]. Illustrative examples of magnetic standards and calibration capabilities

596

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Uncertainty and Confidence in Measurements

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CHAPTER 10

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10.4 TRACEABILITYAND UNCERTAINTY

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Calibration& TestingLab. FIGURE 10.3 Traceability chain in measurements. The calibration and testing laboratories can relate their measurements to the SI units through a flow of calibrations starting from the National Metrological Laboratories (NMIs). The NMIs perform mutual comparisons of their standards, under supervision by the Bureau International Poids et Mesures (BIPM). developed by NMIs are given in Table 10.3 [10.6, 10.16]. Examples of recent NMI intercomparisons of magnetic measurements, regarding DC and AC flux density and apparent p o w e r / p o w e r loss in electrical steel sheets, are reported in Refs. [10.17, 10.18]. The importance for industrial customers of measurement traceability and calibrations, as ensured today by the NMIs, is easily appreciated for magnetic measurements. For example, a magnetic steel producer can ensure the quality of the grainoriented laminations delivered by one of its plants only through periodic calibrations of its magnetic equipment, traceable to an NMI laboratory. Since a large plant can produce 105 ton/year of this high-quality material, worth around s 10s, the economic impact of traceable magnetic measurements is apparent. It should be stressed that when ferromagnetic (or ferrimagnetic) materials are characterized, several factors can detrimentally affect the reproducibility of the measurements. For one thing, the magnetization process is stochastic in character and strongly affected by the geometrical properties of the sample and magnetic circuit. In addition, time effects are ubiquitous, either due to aging or various types of relaxation effects, and many alloys (e.g. rapidly quenched magnetic ribbons) display an intrinsically metastable behavior. Measurements must then be carried out under tightly prescribed conditions, such as those defined by written standards, and by means of accurately calibrated setups in order to achieve

598

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?

L~

y

?

X

? X Y

I

m 0

I 0

? 0

I ~

, 0

4-

o -,~ ,-..,

~

r~

x

b,,,

x

~

~

~

x

~

x

L~

8

r~

",,0

Y

Z

~

4-1

CHAPTER 10 Uncertainty and Confidence in Measurements

g~ •

D

~

z

r,,,2

10.4 T R A C E A B I L I T Y

r

? x

~

AND

I

O r

o

cf] .i.a

UNCERTAINTY

~2

?

x

?

I

?

e:a3

I

u3

<

•

I

•

I

x

O

Z ~r~ O

4.a

•

I

o

x

r,3

~3 t~

or]

t~

Z

O

t~

o o

.4..a

v t/3

c~

,4..a or]

O

9

O

4..a 9 ,4..a

9

~

599

600

CHAPTER 10 Uncertainty and Confidence in Measurements

good reproducibility. To illustrate this point, we can take some of the results obtained with the previously mentioned intercomparison of grain-oriented steel sheets [10.18], which involved three NMIs and 15 different types of laminations. The statistical analysis of these results (see the discussion in Section 10.3) provides the relative deviations (Pi- Pref)/Pref, where Pi is, for each lamination, the power loss of the ith NMI (J = 1.7 T, f = 50 Hz) and Pref is the associated reference value (Eq. (10.34)) (see the histogram shown in Fig. 10.4a). The relative deviations of the apparent power (Si- Sref)/Sref are represented in Fig. 10.4b. It is observed that, under the excellent measuring conditions attained by these metrological laboratories, (Pi- Pref)/Pref is in the 0.5% range, always in agreement (within the 95% confidence level) with the consistency condition U(Pi- Pref) > [Pi- PrefI. The same condition is satisfied by ( S i - S r e f ) / S r e f , which is found to be in the 1% range. It has been previously emphasized that intercomparisons have meaning if the different laboratories follow a uniform approach to the measuring uncertainty. The ISO Guide provides such an approach. It therefore appears appropriate to discuss a few illustrative cases of correspondingly obtained uncertainty determination in magnetic measurements.

10.4.1 Calibration of a magnetic flux density standard A Helmholtz pair is prepared to serve as a magnetic flux density standard in the range 2 x 10-4T ~ B-----4 x 10-2T. Each winding is made of 2210 turns, with average radius r - 0.125 m, and is provided with a supplementary 200-turn coil, improving the uniformity of the generated axial field. The Helmholtz coil constant kH - - B / i is obtained by the simultaneous measurement of the magnetic flux density in the center of the pair and the circulating current. B is detected by means of a lowfield NMR probe operating in the range 0.034-0.121 T. By placing the Helmholtz pair within a triaxial Helmholtz coil system, the earth's magnetic field is compensated to the level of 0.02 ~T. The calculations show that inside the active volume of the probe (11 m m x 6.5 mm), the produced field is homogeneous enough to permit safe establishment of the resonance conditions. It is observed, in particular, that the maximum relative variation of the field amplitude in this volume is --- 1.5 x 10 -5. The current i (of the order of 1.6 A for B --- 0.036 T) is determined by detecting the voltage drop Vacross a standard resistor R = I f~ (four point contacts). Twenty repeated acquisitions are made over a time span of 10 s, short enough to keep the temperature increase of the windings due to Joule heating within z~T---0.4 ~ The coil temperature is controlled immediately before and after the determination of B by measuring the resistance

10.4 TRACEABILITY AND UNCERTAINTY

601

E ~ NMI1

15.

Jp= 1.7T f = 50 Hz

rrrrrl NMI 2 NMI 3

10.

52

.5

-1.0

-d.5

9 o~o

"

!

0~5 ' 1.0

9

1.5

( ~'mref)/mref (%)

(a) 15.

Epstein test frame, Jp = 1.7T f= 50 Hz

~NMI, r~lT[ NMI2 NMI3

10. ,.,., .,_.

- 3. . . . - 2

(b)

"1

'

6

m

"

i

"

~

"

~

( S i- Sre f) / Sre f ( % )

FIGURE 10.4 An intercomparison of power loss P and apparent power S carried out by three NMIs in 15 different types of grain-oriented Fe-Si laminations tested by the Epstein method identifies, for each lamination, the reference values Pref and Sref. The relative deviations (Pi - P r e f ) / P r e f and (Si - S r e f ) / S r e f of the best estimates from the reference values are in the 0.5 and 1.0% range, respectively (adapted from Ref. [10.18]).

CHAPTER 10 Uncertainty and Confidence in Measurements

602

of the winding. B and V are s i m u l t a n e o u s l y detected and the ratio k H -is calculated. The best estimate is written as

(B/V)R

w

kH = (B/V)R + (~(kH)B -4- (~(kH)V q- ~(kH)R,

(10.38)

w h e r e the first term on the right-hand side is the experimental mean, as p r o v i d e d b y Eq. (10.3) w i t h n = 20 a n d (~(kH)B, (~(kH)v, and (~(kH)R are the bias c o m p e n s a t i o n terms associated w i t h field reading, voltage reading, a n d s t a n d a r d resistance value. The bias terms are d e e m e d negligible a n d w e find, from averaging, kH --0.01952636 T/A. The relative c o m b i n e d u n c e r t a i n t y is provided, according to Eq. (10.16), as a function of the Type A a n d Type B i n p u t uncertainties b y the expression: w

uc(kH) kH

i

u2(kH) +

U2B(BB)A U2B(V)} u2B(R)

~2

e 2

R2

(10.39)

,

as s u m m a r i z e d in Table 10.4. The Type A uncertainty, evaluated for n - - 2 0 , is uA(kH)/kH = 1.5X10 -6. The Type B uncertainty

UB(B)/B

10.4 Uncertainty budget in the calibration of a standard magnetic flux density source

TABLE

Source of uncertainty

Distribution Divisor Relative Sensitivity Degrees of function uncertainty coefficient freedom

Magnetic field reading Voltage reading Standard resistor (calibration) Repeatability Combined relative standard uncertainty Expanded uncertainty (95% confidence level)

Rectangular

x/3

2.1 x 10 -5

1

oo

Rectangular Normal

~/3 2

1.2 x 10 -5 2.5 x 10 -5

1 1

oo oo

Normal Normal

1

1.5 X 10 -6

1

-

3.4 x 10 -5

-

19 137

6.8 x 10 -5

Coverage factor k=2

The source is realized by means of a Helmholtz pair (average winding radius r = 0.125 m) energized by a maximum current of 1.6 A. The calibration is performed via a low field NMR probe at B = 0.036 T. The Helmholtz pair is placed in a field-immune region (residual B--- 20 nT), at the center of a triaxial Helmholtz coil setup providing active compensation of the earth's magnetic field. The maximum field gradient in the active NMR probed central volume 11 mm x 6.5 mm of the standard coil is 3 x 10 -7 T / m m at B = 0.036 T. The result is kH = kH + U = 0.01952636 + 1.33 x 10 -6 T/A.

10.4 TRACEABILITY AND UNCERTAINTY

603

associated with the B reading is the sum of two main independent contributions: (1) spatial inhomogeneity of the generated field and uncertainty in the position of the NMR probe; (2) temperature variation and uncertainty on its value during the measuring time. Regarding contribution (1), we assume a rectangular distribution with upper and lower bounds differing by 2a(1)/B--4x10 -5 and we estimate from Eq. (10.7) u~)(B)/B -- 1.2 x 10 -5. Contribution (2) is estimated on the basis of a series of measurements around room temperature (e.g. between 19 and 30 ~ With measuring temperature fluctuation limits +AT = 0.2 ~ the rectangular distribution of half-width a (2)/B = 3 x 10 -5 is evaluated, leading to U(B2)(B)/B-- 1.7 • 10 -5. Consequently, UB(B)/B= 2.1 x 10 -5. The voltmeter specifications provided by the manufacturer give, in the employed 10 V scale, a 1 year accuracy of 12 p p m of reading + 2 p p m of range. For a 2.3 V read-out on this scale, this corresponds to a semi-amplitude of the distribution a = 4.8 x 10 -5 V. The corresponding relative uncertainty is UB(V)/V = (a/x/3)(1/V)--1.2• 10 -5. The calibration certificate of the standard I f~ resistor provides a 2or uncertainty of + 5 0 p p m . It is then UB(R)/R=2.5xIO -5. The relative combined standard uncertainty is obtained by Eq. (10.39) as uc(kH)/kH = 3.4 X 10 -5 and the expanded uncertainty at 95% confidence level is, with coverage factor k---2, U/fcH = 6.8• 10 -5. We eventually write the Helmholtz coil constant at the temperature T -- 23 ~ as kH = kH + U -- 0.01952636 + 1.33 • 10 -6 T/A.

10.4.2 D e t e r m i n a t i o n of the D C polarization in a ferromagnetic alloy We wish to determine the normal magnetization curve of a non-oriented Fe-(3 wt%)Si lamination with the ballistic method. We want to know, in particular, the uncertainty associated with the determination of the polarization value J at a given applied field. Testing is made, according to standards, on half longitudinal and half transverse Epstein strips on a rig provided with compensation of the air flux. Each point of the curve is obtained, after demagnetization, by switching the field several times between symmetric positive and negative values and eventually recording the flux swing 2 h ~ = 2NSJ, where N is the number of turns of the secondary winding (N = 700) and S is the cross-sectional area of the sample, by means of a calibrated fluxmeter. The measurement is repeated five times, always using the same procedure, and the arithmetic mean h ~ is obtained. The best estimate of the polarization value for

604

CHAPTER 10 Uncertainty and Confidence in Measurements

a given applied field H is A~ J= ~ 4- 3(J)d q- ~)(J)a if- 3(1)s q- 3(J)T,

(10.40)

where 3(J)d, 8(/)a, 3(/)T, and 3(J) s are the bias corrections for fluxmeter reading, residual air flux, temperature, and sample cross-sectional area, respectively. The relative combined standard uncertainty of the polarization value is then expressed through the Type A and Type B contributions (see Eq. (10.15)) as Uc(j[) / U2(A(I )) U2(~-~)d u2(A(I))a U2(~-~)T u2(S) . (10.41) i -- V ~-~2 if- A(I)2 -}- A ~ 2 if- A ~ 2 -} ~ $2

Eight strips (two on each leg of the frame), nominally 305 m m long and 30 m m wide, are tested. The sample cross-sectional area S is determined by the precise measurement of the total mass 8m and the strip length l as S = m/431, assuming the nominal material density 8 = 7650 k g / m 3. It is found S = 2.9529 x 10 -5 m 2. The variance u2(S) is obtained by combination of the variances associated with m, 1 and 3. The uncertainty of the value of 8 is by far the largest and we write

S2

~

(~2

(10.42)

Based on the data provided by the steel producer, it is assumed for 8 a rectangular distribution of semi-amplitude a - 25 k g / m 3. Since UB(3)a/x/-3, we obtain u2(S)/S 2= 3.6• -6. Again, we consider the bias compensation terms equal to zero and from the five repeated measurements, it is obtained, for H - 80 A / m , A ~ - 0.02013 V s, i.e. J = 0 . 9 7 3 8 T . The associated Type A uncertainty is found to be UA(A~)/A(I) = 2 x 10 -3. The fluxmeter, calibrated by means of a standard mutual inductor, is assigned a relative uncertainty (1or) on the employed scale (r/A(I) - 4 x 10 -3, from which UB(A(I))d/A(I) = 4 x 10 -3. The uncertainty for the uncompensated air flux is estimated UB(A(I))a/A(I ) = 5 X 10 -4, while any contribution to the uncertainty of the measured polarization value related to temperature is deemed negligible in these alloys and UB(A(I))T/A(I ) ~" 0. It should be stressed, however, that this last term could become very important in some hard magnets (e.g. N d - F e - B alloys) to which the present discussion clearly applies. The combined and expanded uncertainties are thus obtained by means of Eq. (10.41), as summarized in Table 10.5. This specific result is expressed as J = j + u = 0.9738 + 1.00 x 10 -2 T.

10.4 TRACEABILITY AND UNCERTAINTY

605

T A B L E 10.5 Uncertainty budget in the measurement with the ballistic method of the magnetic polarization in non-oriented Fe-(3 wt%)Si laminations

Source of uncertainty

Distribution Divisor Relative Sensitivity Degrees of function uncertainty coefficient freedom

Normal 1 Fluxmeter reading and drift Rectangular x/3 Air-flux compensation Rectangular x/3 Cross-sectional area of the sample Rectangular x/3 Sample temperature 1 Repeatability Normal Combined relative Normal standard uncertainty Expanded uncertainty (95% confidence level)

4 x 10 -3

1

co

5X

10 - 4

1

co

1.9 X

10 - 3

1

co

1

co

1

4 20

---0 2X

10 - 3

4.9 X 10 -3

1.03 x

10 - 2

-

Coverage factor k=2.1

The uncertainty components are specified in Eq. (10.41). The measurement is performed on a point of the normal magnetization curve, under an applied field H -- 80 A/m, with eight strips inserted in an Epstein rig. The result is J = j + U = 0.9738 ___1.00 x 10 - 2 T.

10.4.3 Measurement of power losses in soft magnetic laminations Soft magnetic materials are p r o d u c e d and sold for use p r e d o m i n a n t l y in energy applications and have their quality classified according to their p o w e r loss figure. The precise and reproducible m e a s u r e m e n t of the p o w e r losses in these materials is industrially significant and is required in m a n y application-oriented research investigations. Specific measurem e n t standards have therefore been developed and u p g r a d e d over the years [10.19-10.21]. Inter laboratory comparisons have been carried out to validate these standards, settling to a broad extent the m e a s u r e m e n t capabilities of metrological and industrial laboratories. Critical to the appraisal of the reproducibility and degree of equivalence of the measurements p e r f o r m e d by different laboratories is the correct determination of

606

CHAPTER 10 Uncertainty and Confidence in Measurements

the measurement uncertainty. This is quite a complex task because many possible contributions have to be taken into account, as thoroughly discussed in Ref. [10.22]. Unduly optimistic or pessimistic evaluations are not infrequent, as revealed by the analysis of intercomparisons (see Fig. 10.2). We shall discuss here a largely simplified approach, focused on the testing of soft magnetic laminations at power frequencies, by considering only the most relevant contributions to the uncertainty and assuming that the signal treatment is performed by digital methods (see also Section 7.3). Let us therefore express the average magnetic power loss per unit mass at the frequency f as

Ps = ~ H dB = ~

/fH(t)---d--~dt ,

(10.43)

where H and B are applied field and induction in the sample, respectively, and 3 is the material density. Equation (10.43) can equivalently be written in terms of the current iH in the primary circuit and the secondary voltage VB as

f NH fl/d VB(t)iH(t)dt (10.44) 3 NB~,*S Jo having posed H(t) = (NH/~,*)iH(t)and VB(t) -- NBS(dB(t)/dt), NH and NB Ps

_

being the number of turns of the primary and the secondary windings, respectively, s the magnetic path length and S the sample cross-sectional area. We assume that the measurement is performed under sinusoidal induction waveform (i.e. sinusoidal secondary voltage VB(t)) and we consequently write 1

Ps-- 3r

N H

NB

1

~

~

RH VBVH1cos qG

(10.45)

where RH is the resistance value of a calibrated shunt in the primary circuit. 17B and 17m are the rms values of the secondary voltage and of the fundamental harmonic of the voltage drop on the shunt, respectively, which are phase shifted by the angle ~. Let us thus consider a possible approach to the evaluation of the uncertainty in the specific practical case of grain-oriented Fe-(3 wt%)Si laminations, tested by means of an Epstein frame and a digital wattmeter. It is assumed that testing is made at peak polarization Jp--1.7 T and frequency f = 50 Hz, with automatic air-flux compensation. We can therefore assume VB(t)-NBS(dJ(t)/dt) = 2~fNBSJp sin tot. A number of repeated determinations of P are made, each time disassembling and assembling the strips in the same order. Sixteen strips, 305 mm long and 30 mm wide, are used

10.4 TRACEABILITYAND UNCERTAINTY

607

(four in each leg of the frame) and the cross-sectional area of the resulting sample is determined measuring the total mass m (8 = 7650 k g / m 3) and the length of the strips. It is obtained that S--3.3569 x 10 -5 m 2. The voltages VB(t) and Vm (t) are amplified and fed by synchronous sampling (e.g. 2000 points per period, interchannel delay lower than 1 x 10 -9 s, trigger jitter ~- 10-1~ -11 s) into a two-channel acquisition setup and A / D converter and Eq. (10.44) is computed by means of suitable software. By denoting with Pmeas the result of such a calculation, we express the best estimate of the power loss as P -- PmeasF(AJp)F(AFF)F(AT)"4- 8(P)VBa q- 8(P)VBg q- 3(P)vH1

(10.46)

+ 8(P)a + 8(P)s + 8(P)~,

where the first term on the right-hand side is the mean, made over the repeated measures, of the values of Pmeas times the correction factors F(AJp), F(AFF), and F(AT). These factors account for the differences AJp, AFF, and AT between the actual and the prescribed values of peak polarization, form factor of VB(t) (FF = 1.1107), and temperature T (23 ~ respectively. They are recorded and automatically multiplied by Pmeas with each measurement repetition. Experiments [10.23-10.25] suggest the following approximate relationships:

( ,p )18 jp+Ajp'

F(T) =

(1-5 x 10-4AT),

F(AFF)=

( ) 1.8 1 + fl 1.1107 + AFF 1.1107 (10.47)

where fl is the ratio between the dynamic and hysteresis loss components at the measuring frequency. The bias correction terms ~(P)VBa, (~(P)VBg, 8(P)vH1 , 8(P)a, 8(P)s, 8(P), are assumed to have zero value and non-zero uncertainty and are associated with air-flux compensation, gain and offset of the secondary voltage channel, gain of the field channel, material density, cross-sectional area, phase shift between VB(t) and VHl(t), respectively. With the employed measuring setup, the contributions to the uncertainty deriving from frequency setting, synchronization error during signal acquisition, standard shunt resistor in the primary circuit, and quantization of the signal by the A / D converters are deemed negligible. The latter might become important at very high inductions, where the peak amplitude of the fundamental harmonic of the field (i.e. VH1) reduces to a small fraction of the peak field amplitude, with ensuing reduction of the effective dynamic range of the field channel.

608

CHAPTER 10 Uncertainty and Confidence in Measurements

Based on the foregoing discussion a n d Eq. (10.16), w e express the relative c o m b i n e d s t a n d a r d u n c e r t a i n t y of the p o w e r loss m e a s u r e m e n t as

u~(P) P = ~ u2(p)~2 + u2(VB)a ~ +

u2(VB)g V-----~ +

U~_II(VH1) V21

U~(~)T +2

32

U~(q0) +qo2tan2cp qo2 (10.48)

TABLE 10.6 Uncertainty budget in the measurement of the magnetic power losses at 50 Hz and 1.7 T peak polarization in a grain-oriented Fe-(3 wt%)Si lamination (Eq. (10.48)) Source of uncertainty

Distribution function

Divisor Relative uncertainty

Sensitivity coefficient

Degrees of freedom

Air-flux compensation Gain and offset of B channel Gain of H channel Sample cross-sectional area Material density Phase shift between VB and VH1 Repeatability Combined relative standard uncertainty Expanded uncertainty (95% confidence level)

Rectangular

x/3

2 x 10-3

1

oo

Rectangular

~/3

2 x 10-3

1

OO

Rectangular

x/3

1 x 10 -3

1

oo

Rectangular

x/3

1.5 x 10 -3

1

oo

Rectangular Rectangular

x/3 ~/3

1.5 x 10 -3 4 x 10 -3

1 co qotan~ - 0.7 oo

Normal Normal

1 -

I x 10 -3 5.5 x 10 -3

1 -

-

-

11 x 10 -3

6 1536

Coverage factor k=2

The measurement is performed by a digital wattmeter, using an Epstein test frame. For this specific case, the phase shift ~ between secondary voltage and fundamental component of the primary current is 43~. The result is expressed as P = P + U = 1.176 + 13 x 10 -3 W/kg.

REFERENCES

609

2.5 2.0

Ni/

!o/o

1.5

:2)

1.0 0.5

.0

O~o_o,=0~o~o=g=Q=0=Q,,0=8~8~/~fo/O/GO

0,0

.

.

.

.

i

0.5

.

.

.

.

i

.

1.0 Jp (T)

.

.

.

I

1.5

.

.

.

.

2.0

FIGURE 10.5 Expanded relative uncertainty and its dependence on peak polarization in the measurement of the magnetic power losses at 50 Hz in grainoriented and non-oriented Fe-Si laminations (PTB laboratory [10.18, 10.22]). where it is assumed, according to Eq. (10.42), that the uncertainty of S is chiefly due to the uncertainty of 8. Notice also that we have treated as independent all the input quantities appearing in Eq. (10.48). This is a crude approximation. In particular, for a given Jp value, VB and ~ are expected to correlate and the associated covariance should be considered, via Eqs. (10.17) and (10.20), in Eq. (10.48). Table 10.6 provides, u n d e r s u c h an approximation, the details of the calculation of uc(P)/P and the related expanded uncertainty U. For the specific measurement reported in this example, we find P = P + U - 1 . 1 7 6 + 11x10-BW/kg. Notice that uc(P)/P is observed to increase rapidly on increasing Jp towards saturation, as illustrated for non-oriented and grain-oriented laminations in Fig. 10.5 [10.18, 10.22]. This is chiefly ascribed to the previously stressed detrimental effect of reduced dynamic range in the field channel and the phase error. In fact, for high J~ values, 17B and VH1 are nearly in quadrature and the term ~2 tan2q0(u2(qo)/~2) in Eq. (10.48) tends to diverge.

References 10.1. BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, Guide to the Expression of Uncertainty in Measurement (Geneva, Switzerland: International Organization for Standardization, 1993).

610

CHAPTER 10 Uncertainty and Confidence in Measurements

10.2. In repeated measurements one may find from time to time seemingly inconsistent outcomes (outliers). A good appraisal of the kind of measurement being performed and expertise should guide the person making the experiment in accepting or rejecting these outcomes. 10.3. A. Papoulis, Probability, Random Variables and Stochastic Processes (Tokyo: McGraw-Hill Kogakusha, 1965), p. 266. 10.4. C.F. Dietrich, Uncertainty, Calibration and Probability (Bristol: Adam Hilger, 1991), p. 10. 10.5. S. Rabinovich, Measurement Errors (New York: AIP, 1995), p. 195. 10.6. BIPM, Mutual Recognition of National Measurement Standards and of

10.7. 10.8. 10.9.

10.10.

10.11. 10.12. 10.13. 10.14. 10.15.

10.16.

10.17.

Calibration and Measurement Certificates Issued by the National Metrology Institutes (S6vres: Bureau International des Poids et Mesures, 1999), http:// www.bipm.fr / BIPM-KCDB. M.G. Cox, "A discussion of approaches for determining a reference value in the analysis of key-comparison data," NPL Report CISE 42 (1999). J.W. M(iller, "Possible advantages of a robust evaluation of comparisons," BIPM Report 95/2 (1995). S. D'Emilio and F. Galliana, "Application of the GUM to measurement situations in metrology," in Proc. 16th IMEKO World Congress (A. AfjehiSadat, M.N. Durakbasa, and P.H. Osanna, eds., Wien: ASMA, 2000), 253-258. J. Sievert, M. Binder, and L. Rahf, "On the reproducibility of single sheet testers: comparison of different measuring procedures and SST designs," Anal. Fis. B, 86 (1990), 76-78. EUROMET, "Guidelines for the organisation of comparisons," (EUROMET Guidance Document No. 3). R. Thalmann, "EUROMET key comparison: cylindrical diameter standards," Metrologia, 37 (2000), 253-260. ISO, International Vocabulary of Basic and General Terms in Metrology (Geneva, Switzerland: International Organization for Standardization, 2000). A.E. Drake, "Traceable magnetic measurements," J. Magn. Magn. Mater., 133 (1994), 371-376. M.J. Hall, A.E. Drake, and L.C.A. Henderson, "Traceable measurement of soft magnetic materials at high frequencies," J. Magn. Magn. Mater., 215216 (2000), 717-719. R.D. ShuU, R.D. McMichael, L.J. Swartzendruber, and S. Leigh, "Absolute magnetic moment measurements of nickel spheres," J. Appl. Phys., 87 (2000), 5992-5994. "Intercomparison of magneticflux density by means ofj~'eld coil transfer standard," (EUROMET Project No. 446, Final Report, 2001).

REFERENCES

611

10.18. J. Sievert, H. Ahlers, F. Fiorillo, L. Rocchino, M. Hall, and L. Henderson, "Magnetic measurements on electrical steels using Epstein and SST methods. Summary report of the EUROMET comparison project no. 489," PTB-Bericht, E-74 (2001), 1-28. 10.19. IEC Standard Publication 60404-2, Methods of Measurement of the Magnetic

Properties of Electrical Steel Sheet and Strip by Means of an Epstein Frame (Geneva: IEC Central Office, 1996). 10.20. IEC Standard Publication 60404-3, Methods of Measurement of the Magnetic Properties of Magnetic Sheet and Strip by Means of Single Sheet Tester (Geneva: IEC Central Office, 1992). 10.21. IEC Standard Publication 60404-10, Methods of Measurement of Magnetic Properties of Magnetic Sheet and Strip at Medium Frequencies (Geneva: IEC Central Office, 1988). 10.22. H. Ahlers and J. Liidke, "The uncertainties of magnetic properties measurements of electrical sheet steel," J. Magn. Magn. Mater., 215-216 (2000), 711-713. 10.23. G. Bertotti, F. Fiorillo, and G.P. Soardo, "Dependence of power losses on peak induction and magnetization frequency in grain-oriented and nonoriented 3% Si-Fe," IEEE Trans. Magn., 23 (1987), 3520-3522. 10.24. F. Fiorillo and A. Novikov, "An improved approach to power losses in magnetic laminations under nonsinusoidal waveform," IEEE Trans. Magn., 26 (1990), 2904-2910. 10.25. A. Ferro, G. Montalenti, and G.P. Soardo, "Temperature dependence of power loss anomalies in directional Fe-Si 3%," IEEE Trans. Magn., 12 (1976), 870-872.

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APPENDIX A

The SI and the CGS Unit Systems in Magnetism

Magnetic measurements are generally traceable to the SI unit system. This means that reference can be made, for any measured quantity, to the standards of the base units maintained by the National Metrological Institutes, a feat accomplished by connecting the primary laboratory with the end user through a chain of comparisons, having stated uncertainties. SI (Sist6me International d'Unit6s) has been adopted and updated through successive CGPM (Conf6rence G6n6rale des Poids et Mesures) resolutions, starting with the task assigned by the 9th CGPM in 1948 to the CIPM (Comit6 International des Poids et Mesures) "to study the establishment of a complete set of rules for units of measurement and to make recommendations on the establishment of a practical system of units of measurement suitable for adoption by all signatories of the Convention du M6tre" [A.1]. SI is an absolute system, which means that the corresponding base units are invariant with time and space and all other units can be derived from them through definite mathematical relationships. In particular, a unit {y} of a quantity y is given in this system in terms of the base units {Xl},{x2}, ..., {x,,}, by an expression of the type: {y} =

(A.1)

with c~,/3, ..., v either positive or negative integers and no numerical factors other than 1. The SI units are then said to form a coherent set of units. SI is based on seven base units: meter (m) for the length, second (s) for the time, kilogram (kg) for the mass, ampere (A) for the electric current, kelvin (K) for the temperature, candela (cd) for the luminous intensity, and the mole (mol) for the amount of substance. The base units relevant for electromagnetic phenomena are m, s, kg, and A. The "vexata quaestio" of unit systems in electromagnetism is rooted in the development of the CGS system in the 19th century, which accompanied over several decades the development of the electrical sciences and their broadening impact on industry and society [A.2]. 613

614

APPENDIX A The SI and the CGS Unit Systems in Magnetism

With CGS all electrical and magnetic units are expressed in terms of three base mechanical units: cm, g, and s. Two different unit systems, however, exist, according to whether electrostatic or magnetostatic base equations are used. The electrostatic units (e.s.u.) are defined starting from Coulomb's law, where the interaction force F between two electrical charges (Q1, Q2) at a distance r is expressed by the equation: F = k I Q1Q2 r2 .

(A.2)

By taking the adimensional proportionality constant kS = 1, the dimensions of the electric charge are [M1/2L3/2T-1] and the corresponding unit is 1 statcoulomb = 1 dyn cm. With another choice of the primary equation, where electrodynamic forces are described, the electromagnetic units (e.m.u.) are defined. The force per unit length F/L mutually exerted by two infinitely long straight wires carrying the currents il and i2 is 9

1 / 2

9

F _ k'2 il i2 -7,

.

--

7

(A.3)

where r is the distance between the wires. With the constant ks/= 1, the dimensions of the current are [M1/2L1/2T -1] and those of the electric charge, which is the time integral of the current, are [M1/2L1/2]. It is therefore apparent that the electric charge has different dimensions in the e.s.u, and the e.m.u, systems, the ratio between them having the dimension of a velocity. This duality is a drawback and, to compound it, the mixed CGS (or Gaussian) system has been introduced, where the e.s.u, units are used for the electric quantities and the e.m.u, units are used for the magnetic quantities. The Gaussian system has been instrumental in the development of the electrical sciences; it has been largely applied in the past literature, and is still applied to some extent in papers and textbooks on magnetism and magnetic materials. But most of its units have inconvenient size with respect to practical engineering needs and objective complications arise from the multiplicity of notations. Consequently, since their official adoption by the British Association for the Advancement of Science in 1873, the absolute Gaussian units have been used in association with practical electrical units, such as the volt, the ampere, and the ohm. In 1901, G. Giorgi, by remarking that the watt could equally be the absolute unit of mechanical and electric power, the latter being the product of voltage and current, proposed a system where a fourth independent electrical quantity was associated with the base mechanical units m, kg, and s [A.3]. By abandoning a purely mechanical system, dimensional factors had evidently to be introduced in the laws of force between charges and between currents, but now all the practical

APPENDIX A The SI and the CGS Unit Systems in Magnetism

615

electrical units slipped coherently into this system. Giorgi's proposal won immediate recognition, but it took several decades for general and official acceptance to be achieved, following thorough discussions by International Union of Applied and Pure Physics (IUPAP), International Electrotechnical Commission (IEC), and the Consultative Committee for Electricity (CCE) of CIPM. In 1946, CIPM approved the adoption of the meter, kilogram, second, ampere system (MKSA) [A.4]. The ampere was thereby defined as "that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular crosssection, and placed I m apart in vacuum, would produce between these conductors a force equal to 2 x 1 0 - 7 N of force per meter of length". MKSA was eventually sanctioned, with the addition of the K and the cd as base units, by the 10th CGPM in 1954 as practical system of units of measurement [A.5]. According to this definition of the ampere, the proportionality constant in Eq. (A.3) has definite dimensions and the value 2 x 10-7 F _ /z0 2 ili2. L 4~r r

(A.4)

The magnetic constant ~0 (formerly called "magnetic permeability of free space") has therefore an exact value by definition/z 0 -- 4rrx 10 -7 N / A 2. The unit of electric charge is now 1 C - - 1 A s and the constant kJ in Coulomb's law (A.2) is no more adimensional F-

1 Q1Q2 47r80 r 2 "

(A.5)

The electric constant 80 is related, according to the solution of Maxwell's equations for the propagation of the electromagnetic field in vacuum, to /z0 and the speed of light c: c = (80/z0)-1/2.

(A.6)

In 1983, the 17th CGPM re-defined the meter, assigning the speed of light the precise value c -- 299 792 458 m / s [A.6]. In force of this decision, the electric constant 8o takes, according to Eq. (A.6), the exact value 80 = 8.854187817 x 10 -12 F/m. The SI unit system is strongly recommended by the international metrological and standardization organizations and is increasingly applied in magnetism. However, a massive body of literature exists which makes use of CGS and a substantial resilience to SI, dictated either by attachment to routinary use of old units or by sheer distaste of the awkward redundancy of fields in free space generated by SI, is observed in some areas (for example, in the field of permanent magnets).

616

t"",l

,4

~x .~+

X

II

+

II

-F

~

x II

o~

"~

II 1~

v

<

i~i~

~

~I~~ ~ ~

~

~

~

~

-,..

+ il x

.,~ ~

~~

~

~

,.~'~" ~

II

~

~ e~l~

.~.~

~

~

x

~

~

II

+

~

~

~

II

~

~

"v~ ~"

X

ii

x

D

x

" I~

II

~

~

o~

o

<

~

II

~

II

~I~ ~~

--

,~

~I

~

,~

ro t~

v

II

II

~.

The SI and the CGS Unit Systems in Magnetism

""

~"

x

r

o~

~

"~

-,~

~

~

x

II

~

9~

+

APPENDIX A

~ " ' I r~

z

~

APPENDIX A

r~

.<

8 <

II

~[o ~

~

~

I=

ai

II

II

~9 1 ~

~

tr

N

p.

II

~+ .~.~

II

as

"0

II

a

g

t~

II

~1~

v

II

a~

~1~

.~..~

The SI and the CGS Unit Systems in Magnetism

~ ,-~

r~

c;

0~

~8

o~.

-~.~ ,-..i

~

m

Q

"~-o

~

-b'.~

.~-~

0

ol--t

617

618

APPENDIX

II II

A

X

X

II II II

9

r~

?

o

~

?

I

I

.~.~.~

~ ~~ .~ ~ .~.~ ~ ~ "~'~'~'~ ~ ~ ~

~

T h e SI a n d t h e C G S U n i t S y s t e m s in M a g n e t i s m

ol

II ~ I

~

APPENDIX A The SI and the CGS Unit Systems in Magnetism

619

It is therefore required in many instances to pass from one system to another. Tables permitting to do so are found in textbooks, but it seems nevertheless appropriate to summarize here the chief magnetic equations under the Gaussian and SI systems, to suggest some simple translation rules and provide a conversion table for the units of interest in magnetism (Tables A. 1 and A.2). Equations in SI can be translated into the corresponding equations in the Gaussian system (and vice versa) by means of a few simple rules [A.7], employing relationships like the following ones:

jG BG

BsI

PG

EsI

HsI

DsI

~0

_

MG

~ tt0'

(A.7)

where the meaning of the symbols is understood from Table A.1. To make, for example, the conversion of an SI equation containing the variables Xsz,YsI, ..., Zsi, we solve Eq. (A.7) for them, which become then expressed in terms of the corresponding Gaussian variables XG,YC, ..., ZG and the magnetic or electric constants. By substituting and making additional use of the relationship e0tt0 = 1/c2~ the equivalent Gaussian equation is obtained. An identical procedure in reverse leads to the translation of the Gaussian equations into the SI ones.

aefr A.1. 9th CGPM, "Resolution 6: proposal for establishing a practical system of units of measurement," Comptes Rendus des S~ances de ta Conf&ence G~n&ale des Poids et Mesures, (1948), 64. A.2. J.J. Roche, The Mathematics of Measurement (London: The Athlone Press, 1998), 163. A.3. G. Giorgi, "Rational units of electromagnetism (in Italian)," Atti dell'Associazione Elettrotecnica Italiana, 5 (1901), 402-418. A.4. CIPM, "Resolution 2: definitions of electric units," Proc~s-Verbaux des S~ances du Comit~ International des Poids et Mesures, 20 (1946), 129-137. A.5. 10th CGPM, "Resolution 6: practical system of units," Comptes Rendus des S~ances de la Conf&ence G~n&ale des Poids et Mesures, (1954), 80. A.6. 17th CGPM, "Resolution 2: recommended value for the speed of light," Comptes Rendus des Sfances de la Conf&ence Gfn&ale des Poids et Mesures, (1983), 103. A.7. A.S.Arrott, "Magnetism in SI units and Gaussian units," in Ultrathin Magnetic Films (B. Heinrich and J.A.C. Bland, eds., Berlin: Springer, 1994), 7-19.

This Page Intentionally Left Blank

APPENDIX B

Physical Constants

The v a l u e s of the p h y s i c a l constants p r e s e n t e d h e r e are those r e c o m m e n d e d b y C o m m i t t e e o n Data for Science a n d T e c h n o l o g y (CODATA), a n o r g a n i z a t i o n of the I n t e r n a t i o n a l C o u n c i l of Scientific U n i o n s , w h i c h has the d u t y of e v a l u a t i n g , storing a n d r e t r i e v i n g d a t a p r o d u c e d in science a n d t e c h n o l o g y in a c o o r d i n a t e d a n d critical fashion. The f u n d a m e n t a l p h y s i c a l c o n s t a n t s p l a y a f u n d a m e n t a l role in science a n d m e a s u r e m e n t s today, since they are increasingly u s e d in d e f i n i n g a n d m a i n t a i n i n g the SI units. The c o n s t a n t s p r e s e n t e d here h a v e r e l e v a n c e in m a g n e t i s m . For a c o m p l e t e set of data, see Ref. [B.1].

Quantity

Symbol

Value

Speed of light in vacuum Magnetic constant Electric constant Elementary charge Avogadro constant Boltzmann constant Planck constant Magnetic flux quantum, h/2e Bohr magneton, eh/2m e Nuclear magneton, eh/2mp Electron magnetic moment Electron g-factor Electron gyromagnetic ratio, 2/~e//t Proton magnetic moment Proton gyromagnetic ratio, 2p,p/~ Proton g-factor Neutron magnetic moment Neutron gyromagnetic ratio, 2/Zn/h

c po e0 e NA k h r P,B P,N /ze ge ~,e /Zp

2.99792458 x 4~r x 8.8541878 x 1.6021765 x 9.274009 x 1.3806503 x 6.6261 x 2.0678336 x 9.274009 x 5.0507832 x - 9.2847636 x - 2.0023193 1.760859794 • 1.4106066 x 2.6752221 • 5.585694675 -0.9662364 x 1.8324719 x

~/p gp p,~ Tn

Unit 10s 10 -7 10 -12 10 -19 10 -24 10- 23 10 -34

10 -15 10- 24 10 - 2 7

10-24 1011

m s -1 N A -2 F m -1 C mo1-1 J K- 1 Js Wb J T- 1 J T -1 J T- 1

108

T -1 s -1 J T -1 T -1 s -1

10 -26 108

J T -1 T -1 s -1

10 -26

e, electron charge; me, electron mass; rap, proton mass.

621

622

APPENDIX B Physical Constants

Reference B.1. P.J. Mohr and B.N. Taylor, "CODATA recommended values of the fundamental physical constants: 1998," J. Phys. Chem. Re]:. Data, 28 (1999), 1713-1852.

APPENDIX C

Evaluation of Measuring Uncertainty C.1 TYPE B M E T H O D OF EVALUATION OF THE UNCERTAINTY 1. A standard resistor in a laboratory is associated with a calibration certificate declaring a value R = 100.0039 f~ + 0.0011 f~, with the expanded uncertainty (see Section 10.3) U(R) = ku(R) = 0.0011 f~ defined with a confidence level of 95%. The coverage factor k is, for the given confidence level, k = 1.96. The resistor can then be assigned the absolute and relative standard uncertainties u(R) = 0.00056 f~ and u(R)/R = 5.6 x 10 -6. 2. A digital voltmeter is used in the 10V range, where the specifications of the manufacturer provide, at a distance of 1 year after the last calibration, an accuracy of 3 5 p p m of r e a d i n g + 7 p p m of range at a temperature of 23 + 5~ following a I h warm-up. By making a series of repeated readin_gs u n d e r the same conditions, we determine an average value V = 7.558244 V. It is a s s u m e d that the accuracy declaration by the instrument maker identifies an interval + a of equally likely values centered on 17. It can alternatively be stated that the correction for the bias AlP-- 0, being it equally likely to lie in the interval + a. We obtain a = 7.558244 x 35 x 10 -6 q- 10 x 7 x 10 -6 -335 x 10 -6 V. From Eq. (10.7), the Type B uncertainty turns out to be u(AlT) -- a/.4~ = 193 x 10 -6 V and u(A~/)/V = 2.55 x 10 -5.

C.2 EVALUATION OF COMBINED UNCERTAINTY C.2.1 Example 1 The resistance R = V/I of a c o m p o n e n t is determined by means of simultaneous voltage and current measurements. Five repeated and simultaneous readings of V and I are made, as s u m m a r i z e d in Table C.1. 623

624

APPENDIX C

Evaluation of Measuring Uncertainty

We focus our attention here on the Type A contributions, a s s u m e d in this specific case d o m i n a n t with respect to the Type B contributions. Two a p p r o a c h e s are possible. In the first one (Mode 1), the best estimate of the o u t p u t quantity is defined as the ratio of the best estimates of the i n p u t q u a n t i t i e s / ~ = I7/i. In the second one (Mode 2), it is defined as P, = (V/I), the average of the ratios of the ordered couples (V,I). Following M o d e 1, the expression for the c o m b i n e d variance of /~ is, according to Eq. (10.12), u2(/~) = (1/h2u2(17) +

() ~

2u2(h - 2( ~- )

~- u(V, I),

(C.1)

w h e r e the covariance u(V,I) is obtained b y application of Eq. (10.19). Following M o d e 2, the same quantity is obtained in the o r d i n a r y way, according to Eqs. (10.4) a n d (10.5)" U2(/~) = u2(V~//) =

((V/I) (i) -~//)2 i=1 n ( n - 1) "

(C.2)

TABLE C.1

i

v (v)

i (ma)

v f l (~)

1 2 3 4 5 Best estimate (arithmetic

4.15558 4.14524 4.15415 4.14001 4.14918 17 = 4.14888

30.8742 30.8332 30.8348 30.9132 30.8944 7 -- 30.8700

134.605 134.441 134.723 133.924 134.302 (V/I) = 134.399

Experimental variance

u2(17)= 8.40

u2(h = 2.54

u2(V/I) -- 0.0192 f~2

Standard uncertainty Covariance correlation coefficient Mode 1

u ( ~ = 2.90 x 10 -3 V

rnean)

Mode 2

x 10 - 6 V 2

x 10 -1~ A 2

u([) = 1.59 x 10 -5 A

u(V/I) = 0.139 f~

u(17, F) = -1.99 x 10 -8 V A, r(17, I) = -0.43

/~ = 134.399, u2(/~) = 0.0192 f~2 Uc(/~) = 0.139 f~, Uc(/~)//~ = 1.03 x 10 -3 iR -- 134.399, u2(/~) = 0.0192 f12, uc(/~) = 0.139 f~, Uc(/~)//~ -- 1.03 • 10 -3

C.2 EVALUATION OF COMBINED UNCERTAINTY

625

It is observed that the two approaches generate the same values for the best estimate of the output quantity and its uncertainty, provided the correlation of the input quantities is considered in Mode 1.

C.2.2 Example 2 A resistor is calibrated by making a stable DC current to circulate in a series formed by the resistor under test and a standard resistor and by measuring the voltages across these resistors by means of two high precision calibrated digital voltmeters. Let us indicate with V and Vs the voltages developed across the unknown resistor and the standard resistor, respectively. R and Rs denote their resistance values. R is determined by comparing the two voltages and is given by the expression: V R = ~ssRs .

(C.3)

For the employed voltmeter scales, the 1-year manufacturer's specifications provide at a temperature of 23 + 2 ~ an accuracy of 12 p p m of reading + 2 p p m of range. The calibration certificate of the standard resistor provides a value Rs--100.0145 f~, an uncertainty of 3 0 p p m (2o-level) at 23 ~ and a thermal drift of 5 p p m / K . V and Vs are subjected to 10 repeated simultaneous readings. The arithmetic mean of the ratio V/Vs and the associated experimental variance (Type A evaluation) are calculated. For a specific measurement, we obtain, in accordance with Eqs. (10.3)-(10.6), ( V / G ) = 1.499939 and UA(V/Vs)= 21 X 10 -6 (n -----10). We consequently obtain / ~ - 150.0156 f~ and the uncertainty UA(/~)-RsUA(V/Vs) -- 2.1 x 10 -3 ~. In order to obtain the best estimate/~ of the unknown resistance value, we express it in terms of the significant input quantities: = g((V/Vs), 8R1, ~)R2~~R3~ ~R4) = (V/Vs)Rs + ~R1 q- ~R2 q- ~R 3 q- 8R 4.

(C.4)

In this equation, the corrections 8R1, 8R2~ 8R 3 and ~R4 for bias compensation have been added to the estimate generated by the repeated readings (Type A evaluation). 8R1 and 8 R 2 a r e the corrections accounting for the bias in V and Vs readings, respectively. 8R 3 is the bias correction associated with the calibration of the standard resistor, whose thermal drift is compensated by ~R4. Although the measurement might be prepared in such a way that these corrections are zero, the associated uncertainties are not and they must be considered in the calculation (Type B evaluation) of the combined uncertainty Uc(/~). This can be estimated

APPENDIX C Evaluation of Measuring Uncertainty

626

by means of Eq. (10.16), since it is fair to assume that all the input quantities are uncorrelated. We therefore write the expression: - - 2 2 UB2(V 2 - s) J- (~r/~rs)2 U2(/~) -- u2(a)J-(Rs/[ds)2U~l([0 -}-(-RsV/Vs) X (u23(Rs) q- u24(Rs)),

(C.5)

w h e r e the Type B variances UB1, ... , UB4 are associated with the corrections BR1, ..., ~R 4. In Eq. (C.5) we have assumed, according to Example 1, (V/Vs) ~-17/17s. The relative combined uncertainty is consequently given by

u~(R) -

=

TABLE

~ u2(a)~2 I U2(/~) U21(~7) U22 (T~rs) u23(as) u24(as) /~2 -] ~'2 J ~s qR2 qR2

(C.6)

C.2

Source of uncertainty

Distribution function

Relative uncertainty

Sensitivity coefficient

Digital voltmeter (V)

Rectangular

UBI(~r) -- 15 X 10 -6 17

1

Digital voltmeter (Vs)

Rectangular

UB2(~r) -- 8 X 10 -6 17

1

Standard resistor (calibration) (Rs) Standard resistor (temperature drift) (Rs) Repeatability

Normal

UB3(as) -- 15 X 10 -6 Rs

1

Rectangular

UB4(Rs) -- 1.5 X 10 -6 Rs

1

UA(/{) _ 14 X 10 -6

1

Normal

(R) Combined relative uncertainty Expanded uncertainty

Normal

R u~(P~) U(/~)

Degrees of freedom

oo oo

oo

Peff--" 124 - 2 7 x 10 - 6

- k u~(___ )

R

= 54x

10 -6

C.2 EVALUATION OF COMBINED UNCERTAINTY

627

The previous specifications are used to estimate the terms deriving from a Type B evaluation. The declared accuracy interval of the e m p l o y e d voltmeter is taken as the semi-amplitude a of a rectangular distribution. We obtain a = 38 x 1 0 - 6 V for the voltage reading on the u n k n o w n resistor (1.5 V read on the 10 V range) and a = 14 x 10 -6 V for the voltage reading on the standard resistor (1 V read on the 1 V range). The corresponding uncertainties ( u ( ~ ) = a / v ~ ) are UBI(~ = 22X 1 0 - 6 V and UB2(T~rs) = 8 X 10 -6 V. The uncertainty p r o v i d e d with the standard resistor calibration certificate is associated with a normal distribution, at a 20 level, and is UB3(Rs) ~ 15 X 10 -6 X 100 -- 15 X 10 -4 ~. With an allowed fluctuation of + 0.5 ~ of the standard resistor temperature, the distribution semi-amplitude is a ~ 2.5 x 10 -6 x 100 -- 2.5 x 10 -4 ~ and u4(Rs) 1.5 x 10 -4 ~. One can n o w apply Eq. (C.6) and calculate the relative combined uncertainty. Table C.2 summarizes the whole procedure. The measuring test report will provide the e x p a n d e d uncertainty U = kuc, with the value of the constant k set according to a defined confidence level. The result of the m e a s u r e m e n t is therefore declared here as R - - / ~ + U = 150.0156 + 5 . 4 x 10 -3 ~. In the present case, a 95% confidence level is assumed, for which k---2. For a discussion on the e x p a n d e d uncertainty and the degrees of freedom, see Section 10.3.

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APPENDIX D

Specifications of Magnetometers

We s u m m a r i z e here the m a i n specifications of c o m m e r c i a l l y available m a g n e t i c field m e a s u r i n g devices. These devices are b r o a d l y classified according to their w o r k i n g principle a n d are a s s i g n e d typical m e a s u r i n g field r a n g e s a n d uncertainties. The p r o v i d e d figures refer to the sensing s y s t e m a n d can be affected to s o m e extent b y electronic circuitry a n d software. Magnetomer t y p e ~ working principle

Field range

Frequency range

Relative uncertainty

Rotating/vibrating coil-fluxmetric method Search coilmfluxmetric method Hall effect in semiconductors AMR and GMR Fluxgate magnetometers Thin-film inductive magnetometers Magnetostriction Continuous wave NMR magnetometers Free proton precession Flowing-water NMR magnetometes Optical pumping SQUID

1 nT-10 T

DC

10-3-10 -2

0.1 nT-10 -2 T

1 Hz-0.5 MHz

10-3-10 -2

10 ~T-10 T

DC--10 kHz

1 ~T- 100 mT I nT-1 mT 100 nT-100 ~T

DC--1 MHz DC--1 kHz DC~10 kHz

10-3-10 -2 (DC) 10-2-5 x 10 -2 (AC) 10- 2 10-3-10 -2 10 -2

I nT-100 ~T 40 mT-20 T

DC--100 Hz DC

10 -2 2 x 10-6-10 -5

20 ~T-100 ~T 5 ~T-2 T

DC DC

2 x 10 -5 10 -5

10 ~T-100 ~T 10 pT-1 mT

DC 5 x 10 -5 T/s

10 -4 10 -3

629

This Page Intentionally Left Blank

List of Symbols

A A B,B Br C Cp

C e

E E EH Eme Ey

f

F,F h

Ha, I~ Hc Hd, Hd

Hk H,H, Hm, Hm i

j,j J, J, Jm, Jm

ls k k k K, K~ L

exchange stiffness coefficient magnetic potential vector magnetic induction remanent induction capacity specific heat capacitance per unit length in transmission lines electron charge electric field energy Hall field magnetoelastic energy density Young's modulus frequency force Planck's constant applied field coercive field demagnetizing field anisotropy field effective field electrical current current density magnetic polarization peak magnetic polarization saturation magnetic polarization Boltzmann's constant coil coefficient coverage factor magnetic anisotropy constant angular momentum 631

632

List of Symbols

L L me

m, m M,M

Mr M~ F/o

Nd N1 N2 P Q RH R

S Sq T

TCF U /,/At /,/Br Uc

U U V~

V

VH W ^

^

^

x~ y, z

Zo

inductance inductance per unit length in transmission lines electron mass dipole magnetic moment magnetization remanent magnetization saturation magnetization number of turns per unit length demagnetizing coefficient number of turns magnetizing winding number of turns secondary winding specific power loss quality factor magnetic reluctance Hall coefficient electrical resistance specific apparent power scattering parameter temperature, period ferromagnetic Curie temperature voltage standard uncertainties energy expanded uncertainty phase velocity sample volume Hall voltage energy product, specific energy loss unit vectors characteristic impedance

Greek symbols OLll OL21 OL3

# # "T F 3 3 8

direction cosines flux leakage factor propagation constant in transmission lines gyromagnetic ratio Weiss constant reflection coefficient mass density penetration depth strain

List of Symbols ~0 8r )t ,~s /Z ~0 /ZB jU,e, jU,h /d,r P PH PM O" Or O"M 0"y T r r r X 60

633 electric constant relative permittivity wavelength saturation magnetostriction magnetic permeability shear modulus magnetic constant Bohr magneton electron mobility, hole mobility relative magnetic permeability electrical resistivity Hall resistivity volume density of magnetic charge electrical conductivity, stress magnetic moment per unit mass surface density of magnetic charge yield stress torque magnetic flux scalar magnetic potential magnetostatic potential magnetic susceptibility angular velocity

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Subject Index

Page ranges in boldface in the index indicate where a subject is discussed in detail. Page numbers followed by (f) or (t) indicate that the subject is to be found in a figure or table, respectively.

A AC field sources, see Magnetic field generation AC field measurement, 159, 190, 205 AC bridge methods in magnet testing, 430-432

AC characterization of soft magnets low and power frequencies, 364-409

medium-to-high frequencies, 409-432

radiofrequencies, 4 3 2 - 4 6 2 Active mass of specimen, 367 Aftereffect (magnetic viscosity), 337, 358, 429, 478, 510, 543 Aging, magnetic, 35, 36(f), 362 Airflux compensation, 299, 309, 294, 312(f), 314, 373, 490 --, y transition, 35, 38(f), 61 A1N precipitates 35, 40, 42, 46 Alternating Gradient Force Magnetometer (AGFM), 5 2 1 - 5 2 9 Amorphous alloys, 51-58, 28, 297, 315-316, 558 approach to saturation, 550-552 losses, 58(f), 60(f) physical parameters, 53(t), 57(t) preparation 50(f), 51

Amorphous thin films, 79, 82-83, 212, 462 Amorphous wires, 55, 452-455 Amorphous-wire field sensor, 205-208 Amorphous-to-crystalline transition, 557, 558(f) Amperian currents, 3 Amplifiers power, 354, 371 operational, 169-170 lock-in, 1 7 1 - 1 7 2 Angular momentum, 3, 71,253, 257, 259 Anhysteretic magnetization curve, see Magnetization curve Anions in spinel ferrites, 67 Anisotropy, see Magnetic anisotropy Anisotropy field, 191, 204, 211, 494, 514, 520 Anisotropy of magnetoresistance (AMR), see Magnetoresistance Antiferromagnetic coupling, 66-67, 76-78 Apparent permeability/susceptibility, 13, 198, 294, 451, 530 Apparent power, 325, 376, 419-423, 600, 601(f) definition, 367 measuring uncertainty, 373, 381-382 635

636 Approach to magnetic saturation, s e e Magnetization curve Arrot's plots, 554 Austenitic transformation, 46

Barber-pole sensor, 192(0, 194(0 Barkhausen noise, 193, 199, 341, 349, 353 Bending stress, 284 Best estimate/experimental mean of measurand value, 581, 583, 586, 589-595 Bias terms/corrections in measurements, 584, 585(0, 602, 604, 607, 625 Bifilar winding, 415 Biot-Savart's law, 4, 105 BIPM, 595, 597 Bloch equations, 221 Bohr magneton, 255 Breit-Rabi equation, 257 Bridge circuit/method, 193-196, 430-432, 201, 249(f)

Calibration and measuring capabilities of NMIs, 363(t), 383(t), 598(t) Calibration of, AGFM, 527, 528 extraction magnetometer, 534 field sources, 262-268 fluxmeters, 1 7 2 - 1 7 5 flux density standard, 600-603 hysteresisgraphs/wattmeters, 497-499, 323, 362, 393, 482, Hall magnetometers, 190 network analyzers, 440-441, 450 permeance meters, 459 PFM, 541 search coils, 163 VSM, 510-512, 519 Carbides, 28

Subject Index Cementite precipitates, 35 Central limit theorem, 582 CGPM, 613, 615 CGS units, s e e Unit systems in magnetism Characteristic impedance, s e e Transmission lines Chemical shift, 236 Chemical Vapour Deposition, 51 CIPM, 595, 581, 613, 615 CODATA, 250, 621 Coercive field definition, 20(f), 21 in amorphous alloys, 57(t) in Fe-Ni and Fe-Co alloys, 63(t) in nanocrystalline alloys, 57 in permanent magnets, 20(f), 93, 479, 498, 499, 542(f) in permendur, 65 in soft magnets, 27, 33(t) measurement, 338-340, 479, 486, 498(f), 513(f) temperature coefficient in permanent magnets, 479 Coercimeter, 3 3 8 - 3 4 0 Coherent units, 613 Cold/hot rolling of magnetic steels, 35(t), 40(t), 45(t) Compensated permeameters, 314, 321(f) Confidence interval, definition, 589 Confidence level, 589-591, 593 Conductivity, s e e Electrical conductivity Constitutive law in a magnetic medium, 16 Conversion of units (SI vs. CGS), 616-619 Core loss, s e e Energy loss Correction for the demagnetizing field, 295, 298(f), 515, 540 Covariance in combined uncertainty, 587-589, 624 Coverage factor, 589, 591(t), 593, 623, 627

Subject Index Cube-on-edge (Goss) texture, 43 Cube-on-face texture, 49, 50 Curie temperature, in amorphous alloys, 52, 53(t), 57(t), in Fe-Ni and Fe-Co alloys, 62, 63(t) in Fe-Si, 53(t) in spinel ferrites, 69(t) measurement, 553-558, 560 Cylindrical test sample, 13-16, 296(f), 310, 339(f), 340, 486-487, 494, 496(f), 531 D

DC characterization of soft magnets,

637

Diffusion, atomic, 337, 358 Dimensional resonance, 450 Dipole line, 142 Disk samples, 329, 334(f), 336(f), 400(f), 401(f), 476, 514(f), 567(f), 572 180~ domain walls, 30 Drift of integrated signal, 170, 345-346, 348, 352 DSC/DTA analysis, 557, 558(f) Dynamic recoil in permanent magnets, 95-98

Dynamic loss, s e e Energy loss Dynamic Nuclear Polarization (DNP), s e e Overhauser effect

336-362

point-by-point method, 3 4 2 - 3 4 8 continuous recording (hysteresisgraph) method, 3 4 8 - 3 5 8 rotational hysteresis (torque magnetometer), 361 Decarburization annealing, 35 Degree of equivalence of national standards, 593 Degrees of freedom in repeated experiments, 583, 590, 591(t), 602(t), 605(t), 608(t), 626(t) Demagnetization AC, 341, 342(f), 344, 378, 495 biased, 347(f) thermal, 341, 495 Demagnetizing field, 8-16, 91, 295-299, 335(f), 387, 429, 488-490, 515, 555, 565, 566 Demagnetizing coefficient, 9-16, 91-94, 214-216, 295-299, 371, 451, 555, 565 fluxmetric/magnetometric, 13 in ellipsoids, 9-11 in cylinders, 11, 14(f), 15(f) in prisms, 12, 16 Demagnetizing tensor, 9 Dielectric losses, 415, 426 Dielectric permittivity, 451 Digital treatment of signal, 171, 354-356, 375-376, 412

Earth magnetic field active cancellation, 123(f), 189, 265, 266(f) measurement, 234, 236, 249, 252 Easy axis, 43, 61, 70, 79-82, 191, 326-327, 564-565 Easy plane, 565 Eddy currents, 30-32, 154, 292, 317, 344, 349, 367, 387, 394, 446, 483, 541 Effective anisotropy, 28 Effective field, 10, 152, 167, 211, 224, 241, 282, 288, 290, 293(f), 294-297, 298(f), 310-314, 322, 325-326, 337, 370, 429, 486 Effective permeability/susceptibility, 13, 198, 451 Electrical analogy of magnetic circuits, 98-100

Electrical resistivity / conductivity, 177 Hall, 178 in amorphous alloys, 52, 57(t) in Fe-Ni and Fe-Co alloys, 63(t) in Fe-Si, 39(f), 49, in semiconductors, 178-181 in spinel ferrites, 69(t) measurement, 375(f) resistivity-density product in Fe-Si, 375(f)

638 and spin-polarized electrons, 76-78, 184-185 vs. angle with M, 183 vs. temperature, 183(f) tensor, 183 Electromagnet, 1 4 5 - 1 5 5 field in the gap, 147, 150(f), 151(f), 152(f), 153(f) flux lines, 149(f), 153(f) H-type, 146(f), 151(f) heat dissipation, 155 testing with, 481-499, 291(f), 292, 408, 507(f), 524(f), 526(f), 570(f) Electromotive force, 161 Electron gyromagnetic ratio, 245 Electron mobility, 177 Electron spin resonance (ESR), 251 Electronic energy levels, 247, 249(f), 253(f), 254(f) Ellipsoidal sample, 9, 10(f), 296(f) Energy/power loss in magnetic materials, 21-22, 37(f), 42(f), 44(t), 47, 48(f), 58(f), 288, 289(f), 354 definition, 367-371, 21 measurement Epstein vs. SST, 322-325 low and power frequencies, 364-396, 4 0 2 - 4 0 9 medium-to-high frequencies, 416-427 on-line, 385-388 rate of rise of temperature method, 388-394 under 2D fields, 334-336, 361, 3 9 6 - 4 0 9 using needle probes, 3 9 4 - 3 9 6 vs. frequency, 30-33, 60(f), 377(f), 378(f), 541 Energy product B H , 92-94, 494, 498(f) Environmental electromagnetic field (ELF / VLF), 163 Epstein test frame, 286-295, 316-317, 112, 371, 373, 379 medium-to-high frequencies, 423-426, 414, 419.

Subject Index Epstein-SST relationship, 322-325. Equivalent circuit of inductor, 427-429

Equivalent circuit of AC testing setup, 417(f), 421(f), 425(f) Errors in measurements, see Measuring uncertainty EUROMET, 595 Exchange anisotropy, 78 Exchange coupling, 564 in amorphous/nanocrystalline alloys, 54, 59 in multilayers, 75-78 in ferrites, 67 Exchange narrowing, 247, 275 Expectation value, 586, 589

Faraday's balance, 5 6 0 - 5 6 4 Faraday rotation, 2 1 2 - 2 1 4 Faraday-Maxwell's law, 5, 235, 394, 404 Fe, pure, 33, 34(t), 353(f), 551(f), 556(f) Fe-A1, 49(f), 51 Fe-Co alloys, 65 Feebly magnetic materials, 475, 500 Feedback, see Induction waveform control Fe-Ni alloys/permalloys, 61-64, 184(f), 191, 192(f), 203, 456 Fermi temperature, 246 Ferrimagnetism, 66-68 Ferrites, see Soft spinel ferrites Ferromagnetic Curie temperature, 549, 553-558 Ferromagnetic resonance, 83, 445(f), 459 Fe-Si, 38-51 intrinsic properties 38-39, 49(f), 553(f), 569(f) phase diagram, 38(f) resistivity and density determination, 3 7 3 - 3 7 5 single crystal, 43(f), 48(f), 327-329

Subject Index Fe-Si, grain-oriented, 4 3 - 4 9 losses, 44(t), 48(f), 58(f), 289(f), 324-325, 360(f), 377(f), 378(f), 382(f), 601(f) normal curve/hysteresis loop, 64(f), 328-330, 342(f), 350(f), 377(f) processing, 4 4 - 4 7 secondary re-crystallization, 46 specifications, 44(t) Fe-Si, non-oriented, 39-43, 573(f) losses, 41(f), 42(f), 58(f), 360(f), 382(f), 391(f), 594(f), 596(t) normal curve/hysteresis loop, 20(f), 336(f), 347(f), 359(f) processing, 40 Fe-(6.5 wt%)Si, 49-51 Fe-Si-A1, 51 Figure-eight coil, 456(f), Finite Difference Method (FDM), 101 Finite Element Method (FEM), 101 applicative examples, 149(f), 152(f), 153(f), 487(f), 488(f) Flux, s e e Magnetic flux Flux-closure, 292-295, 286, 330, 486 -488 Fluxball, 163 Fluxmeter, 169-175 Flux vortexes, 516 Force Lorentz's, 4, 130, 177, 213(f), 217 in a lifting device, 97 / torque on a coil/magnetic dipole, 175, 217, 338, 475, 522, 524(f), 526(f), 528, 530, 560-564 Form factor, of secondary voltage, 125, 365, 607 Free poles, s e e Magnetic charges G g factor, 219, 247 Gapped ring magnet, 90(f) demagnetizing coefficient, 92 field in the gap, 91-92 Garnets, 213-214

639

Gaussian units, s e e Unit systems in magnetism Giant magnetoimpedance (GMI), s e e Magnetoimpedance Giant magnetoresistance effect (GMR), s e e Magnetoresistance Goss texture, 43 Grinding of spheres, 491 Gyromagnetic ratio, bare proton, 219 deuteron, 225 electron, 245, 258 shielded proton, 224, 250 Gyroscopic equation, 220 H

H-coil, 166-168, 311(f), 318(f), 319(f), 331(f), 333(f), 412 and loss measurement, 334-336, 403 -407

double, 318-320, Halbach's cylinder, 142-143 Hall coefficient, 177-181 Hall effect, 175-181 Hall magnetometer, 185-190 Hall probes, 187(f) Hall resistivity/voltage, 178 Hard ferrites, 479, 496(f), 509(f) Helmholtz pair axial/radial field uniformity, 122(f) as standard field source, 123(f), 600 filamentary; 113-117 in extraction method, 533(f) inverse 116, 501, 502(f) square, 117 thick, 121-123

triaxial, 123(f), 266(f) Holes in semiconductors, 179 Hyperfine interaction/splitting, 246, 247, 249(f), 254(f), 257-259 Hysteresis loop area, 21 biased, 78, 347(f)

Subject Index

640

in permanent magnets, 20(f), 482(f), 489(f), 496(f), 509(f), 529(f), 540(f) in soft magnets, 20(f), 22(f), 23(f), 26(f), 29(f), 31(f), 64(f), 298(f), 328(f), 336(f), 342(f), 377(f), 413(f), 416(f) measurement methods in hard magnets, 481-513, 520-543 measurement methods in soft magnets, 340-359 remanent, 479-481, re-entrant, 353 under 2D fields, 405(f), 407(f) vs. magnetizing frequency, 23(f), 31(f), 377(f), 413(f) with minor loops, 359 Hysteresisgraph (continuous recording) method permanent magnet testing, 4 8 1 - 4 9 9 soft magnet testing, 348-358, 365, 377, 405, 407

by digital feedback, 3 5 4 - 3 5 9 under 2D fields, 3 9 6 - 4 0 2 Inductor equivalent circuit, 4 2 7 - 4 2 9 loss factor, 427 Q-factor, 427-428 Integration analog, 169-170, 352(f) digital, 171, 356, 376 Intercomparisons, 593-597, 324(f), 363(t), 601(f) Interlaminar insulation, 316

J J-compensated coils, s e e Pickup coils Joule effect, 393 K

Kerr effect, 280, 299 Kohler's rule, 181 Kopp-Neumann law, 393

I

Ideal permanent magnet, 94(f) Image effect, 301-304, 490, 535 in VSM, 517-519, 506, 511, 516 Impedance analyzer, testing with, 430-432

Impedance magneto-, s e e Magnetoimpedance characteristic, s e e Transmission lines Impurities, effect on losses in NO Fe-Si, 41 in Fe, 34(t) Inhibitors in GO Fe-Si processing, 45 Initial/normal magnetization curve, 17, 18(f), 20(f), 21, 342, 346, 347(f), 480(f), 494, 496(f), 540(f) Input impedance in acquisition devices, 356, 414, 422 in transmission lines, 437-444, 454-455 Induction waveform control by analog feedback, 351-354

Lagrange multipliers, 592 Land6 atomic factor, 257 Larmor frequency, 218, 221, 225, 243, 255, 261, 262 LCR meter, s e e Impedance analyzer Leakage inductance, 422-425 Leakage reluctance, 99, 147 Load line, 92, 93(f), 94(f), 96(f) Loss angle, 72, 426-429 Loss separation, 30-33, 377(f) Low-carbon steels, 3 3 - 3 8 aging, 35-36 magnetic losses, 37(f), 382(f) magnetization curve, 296(f) processing, 35(t) M

Magnet cladding, 139-141 Magnetic anisotropy

Subject Index crystalline, 564, 27, 38, 49, 51, 61-64, 69-71, 569(f), 573(f), 575(f) exchange, 78 in amorphous alloys, 54 in Fe-Si alloys, 39(f) in Fe-Ni allloys, 62(f) in nanocrystalline alloys, 59 in spinel ferrites, 69(t) induced, 29(f), 55, 59, 62(f), 79, 191, 201, 203, 209 measurement, 514-515, 565-576, shape, 216, 565 stress-induced, 28, 54-55, 210, 565 vanishing, 28, 54, 59, 63 Magnetic charges (free poles), 6-8, 90(f), 91, 132-133, 136, 292-293 Magnetic circuit, 98-100, 282-295, 481-492, 140, 147, 327(f), 386(f) Magnetic dipole, 107, 142-143, 152-154, 300, 338, 475, 500-506, 511, 522, 560, 571 Magnetic domains bar-like 43(f), flux-closing 47 Magnetic field (H-field), definition, 4 generation, 1 0 5 - 1 5 5 AC fields, 123-125 DC fields with filamentary coils, 106-117 circular/rectangular loop, 108 solenoid, 108-112, 118-121 Helmholtz coils, 113-117, 121-123, 189, 265, 600 DC fields with thick coils, 117-123 electromagnets, 1 4 5 - 1 5 5 permanent magnet sources, 132-145 pulsed fields, 125-131 superconducting solenoids, 516-517 variable fields with permanent magnets, 1 4 4 - 1 4 5

641

mapping, 163-164, 188, 238 measurement, 1 5 9 - 2 6 8 ESR methods, 251-252 ferromagnetic sensor methods, 196-209 fluxmetric methods, 161-175, 288, 299, 314, 317-322, 335(f), 395(f), 400(f), 403, 538(f) Hall effect methods, 185-191, 291(f), 310-311, 339(f), 395(f), 483(f), 490, 497, 507(f), magneto-optical methods, 212-215 magnetoresistance methods, 191-196, 395(f) magnetostriction methods, 209-212 microtorque-microlever methods, 215-217 NMR methods, 227-251 optical pumping methods, 252-262, 266(f) potentiometer, 168(f) Magnetic flux, definition, 161 detection, 161-164 leakage, 94, 99, 147, 286, 309, 488(f), Magnetic heads, 78-82 Magnetic induction (B-field), definition, 4, 5 Magnetic moment, 3, 218, 262 free precession, 233-235 in magnetic glasses, 52 in spinel ferrites, 68(t), 69(t) of whole sample, measurement, 475-476, 500-543 per unit volume, 4 Magnetic multilayers, 76-79, 185,

186(f)

Magnetic path length Epstein vs. SST, 322-325 in Epstein test frame, 288-290 in ring cores, 284-286, 91 in SST, 317, 322

642 Magnetic permeability/susceptibility apparent/effective, s e e Apparent permeability definition, 5 imaginary/real, absorption/dispersion, 223-224 and loss, 427-428 at radiofrequencies, 443-452, 455-462 in soft ferrites, 72(f), 443, 451(f) in thin films, 461(f) initial/reversible, 55, 71, 72(f), 198, 427 in amorphous/nanocryst, alloys, 60(f) in Ba ferrites, 479 in Fe-Co alloys, 63(t) in Fe-Ni alloys, 63(t) in garnets, 445 in permalloys, 60(f) in soft ferrites, 60(f), 71, 72(f), 451(f) in thin films, 455-462, 81(f) in wires, 455 recoil, 95 soft magnets, 33(t), 53(t), 198 Magnetic polarization, definition, 5 Magnetic potential (scalar), 6-9, 98 Magnetic potential (vector), 100-101 Magnetic viscosity, s e e Aftereffect Magnetization, definition, 4 Magnetization curve anhysterestic, 346-348, 481, 566 approach to saturation, 549-553, 476, 566 initial/normal, 341-345, 18(f), 21, 347(f), 480(f), 494, 496(f), 551(f) in Fe/Fe-Si crystals, 327-330 recoil, 96(f) remanent, 479 second quadrant, 93(f), 94(f), 96(f), 496 virgin, 17, 341 Magnetization modes and phases in single crystals, 327

Subject Index Magnetization process, 16-24, 326-330 domain wall displacements, 27, 31, 204, 206, 327, rotation of magnetization 55, 63,199, 203(f), 329 Magnetoelastic energy, 315, 565 Magnetoimpedance in wires, 205-208, 55, in thin films, 82-83, 208 measurement, 452-455 field sensors, 208 Magnetometers AGFM, 521-529 AMR, 191-194 ESR, 251-252 extraction method, 531-536 Faraday's, 560-564 fluxgate, 197-202 GMR, 194-196 Hall effect, 185-190 inductive, 202-209 magneto-optical, 212-215 magnetostrictive, 209-212 micromechanical, 215-217 NMR, 227-251 continuous wave, 227-233 free-precession, 233- 237 pulsed, 237-239 flowing-water, 239-245 Overhauser (DNP), 247-250 optically pumped, 252-262 pulsed field (PFM), 536-543 rotating coil, 163 rotating sample, 574 SQUID, 202, 218 torque, 566-571 vibrating coil, 164-165 vibrating sample (VSM), 500-521 vibrating wire, 529-531 measuring uncertainty, 629 specifications, 629 Magnetomotive force, definition, 98 in electromagnets, 147, 149 of permanent magnets, 99, 140

Subject Index Magnetoresistance, 181-185 anisotropic (AMR), 182-184, 191-193, 176 giant (GMR), 76-79, 185, 186(f) in Bi, 190 ordinary / spontaneous, 182 Magnetoresistive heads, 80(f) Magnetostatic energy, 8, 92, 96, 371, 565 Magnetostriction in amorphous alloys, 53, 57(t) in Fe-Ni and Fe-Co alloys, 62(f), 65 in Fe-Si and Fe-A1 alloys, 49(f), 51, in spinel ferrites, 69(t) magnetometers, 209-212 Marginal oscillator, 229-230 MBE technique, 75 Measuring standards (IEC/ASTM), 366(t), 33, 34, 40, 44, 287, 310, 320, 322, 365, 374, 414, 430, 499, 532 Measuring uncertainty, budget, 602, 605, 608, 624, 626 combined, 586-589, 172-175, 602, 604-605, 608, 624, 626 expanded definition, 590 determination, 591, 594, 596, 602, 604, 608, 626 in magnetic field measurements fluxmetric method, 163, 172-175, Hall magnetometers, 189-190 NMR magnetometers, 225, 237, 250 specifications of magnetometers, 629 standard field sources, 264(f) in magnetic materials testing, 379-384, 603-609, 594-600, 324(f), 363(t), 367, 372, 373, 412 law of propagation, 587 relative, definition, 588 sensitivity coefficients, 587, 588, 602(t), 605(t), 608(t), 626(t) standard, definition, 584

643

type A evaluation method, 584 type B evaluation method, 585 Meissner effect, 516 MKSA, 615 Molecular beam epitaxy (MBE), 75 Multilayers, s e e Magnetic multilayers Mutual inductor, 173-174, 355(f) N

Nanocrystalline alloys, 59 losses, 58(f), 60(f), 413(f) initial permeability, 60(f) physical properties, 57(t) skin depth, 411(f) N6el temperature, 66(f) Network analyser, 438-441, 444, 447, 450, 453(f), Noise signal, 171, 202(f), 209, 228, 239,500 Normal distribution function, 582, 585(f) Normalized error in comparisons, 594 Nuclear magneton, 218 Nuclear Magnetic Resonance (NMR), 218-227

absorption line, 223, 226, 227(f), 233(f), 244(f) free induction decay, 234-236 linewidth, 232 longitudinal relaxation, 219, 220, 234 resonance frequency, 220, 224, 225, 230, 232 resonator tuning, 230-232, 228, rotating frame, 222, 237, 238(f), 241(f) saturation, 225 transverse relaxation, 221- 222, 232, 236 Nuclear susceptibility, 219, 223-226, 247 Null chamber, 189 Numerical methods in magnetostatics, 100-101

Subject Index

644

O Open sample testing soft magnets, 295-304, 330-336, hard magnets, 499-543 Operational amplifiers, see Amplifiers On-line testing of magnetic sheets, 385-388

Optical pumping, 252-259 Cs 133, 259, 254(f) He 3, 261 He 4, 252-256 Rb 87, 257-258, 254(f) Optically pumped magnetometers He 3, 262 He 4, 256, 253(f) alkali vapours, 259-261 heading error, 259 linewidth, 256, 259 sensitivity, 256, 259, 262 Overhauser effect, 245-248

Paramagnetic Curie temperature, 556 Parasitic torque, 361, 403, 408, 515 Paschen-Back effect, 257 Pauli paramagnetic susceptibility, 246 Peaking-strip technique, 201 Permalloy, 63-64, 80, 191-193, 456 Permanent magnet field sources, 132-145

axial field, 134, 135(f), 137 field confinement, 139-142 fiat vs. tapered polar faces, 138(f), 139(f) variable field sources, 144-145, 520 Permanent magnets characterization, 475- 543 operation, 89-101 Permeability, s e e Magnetic permeability Permeameters, 310-314 type-A/type-B, 311(f), 313(f) compensated types, 314 Permeance, 455

Permeance coefficient, 92 Permeance meters, 456 (f), 457(f), 460(f) Permendur, 65 Phase lag, 335(f), 404-406, Physical constants, 621 Pickup / search coils, 161 - 164 compensated, 299, 309, 312(f), 314, 490, 538(f) concentric, 535(f), 536, 538(f), 539 embedded, 492 figure-eight, 456(f) Helmholtz, 533(f) inverse Helrnholtz 501, 502(f) in VSM 5OO-5O5 Mallinson's, 503 non-enwrapping, 387(f) Rogowski-Chattock, 166-167, 311(f), 321(f), 386(f) tangential, 166, 167, 289(f), 291(f), 318(f), 319(f), 331(f), 333(f), 386(f), 395(f) vibrating, 164-165, 339(f) Picture frame crystals, 327 Poisson's equation, 6 Polar shoes, 491 Pole faces in magnets and electromagnets, fiat vs. tapered, 138(f), 139(f), 149(f), 150(f), 151(f), 152(f), 153(f) Powder cores (soft), 37 Power amplifiers, s e e Amplifiers Power loss, s e e Energy loss Power loss in thick coils, 119 Poynting theorem, 367, 403 Precession of magnetic moments, 219-223, 234, 237-238, 240-242

Precipitates in Fe and Fe-Si, 28, 35, 41, 42, 46 Pulsed field generation, 125-131 energetic efficiency, 128 thermal effects, 129 stresses in coils, 130(f) setup parameters, 131(t) Pulsed Field Magnetometer(PFM), s e e Magnetometers

Subject Index

Q Q-factor, 427, 447, 523, Quantum magnetometers, 227-262 Quantum of flux, 160, 218 R

Radiation damping, 236 Radiofrequency measurements,

64~ Rotational field, s e e Rotational single sheet tester Rotational loss measurement, using H-coils, 334-336, 402-408 rate of rise of temperature method, 388-394 parasitic torque method, 361, 408 Rotational Single Sheet Tester (RSST), 330-334

432-462

Rare-earth based magnets, 18(f), 20(f), 478-479, 482(f), 489(f), 540(f), 542(f) Rayleigh law, 21, 22(f), 338 Real inductor, 427-429 Reciprocity principle, 300-301 Recoil line, 95-97 Rectangular distribution function, 586 Re-entrant cavity, 447 Reference magnets, 172 Reference samples, 476-478, 559, 362 Reluctance, definition, 98 Remanence, definition, 20(f) Resistivity, s e e Electrical resistivity Resonance, electron spin, s e e Electron Spin Resonance ferromagnetic, 83, 445(f), 459 nuclear magnetic, s e e Nuclear Magnetic Resonance piezoelectric, 210 in optically excited atoms, s e e Optical pumping in a coaxial line, 445-447, 231 in AGFM, 523-527 RHEED technique, 75 Ring-core specimens, 283-286, 309, 428, 430, 443-444 Ring-core type fluxgate magnetometer, 200 RKKY coupling, 76 Rogowski-Chattock potentiometer, 166-167, 386(f) in permeameters, 311(f), 320-322

S

Saturation polarization/magnetization in amorphous alloys, 53(f), 57(t), 551(f), 552(f) in Fe, 551(f) in Fe-Ni alloys, 62(f), 63(t) in Fe-Co alloys, 63(t) in Fe-Si, 39(f), 553(f) in soft magnets, 33(t) in spinel ferrites, 69(t), 70(f) measurement, 549-564 Scattering coefficients (network analyzer), 438 Search coils, s e e Pickup coils Seebeck coefficient, 389, 393 Selection rules, 258 Sensitivity of magnetometers, 202(f), 251 continuous-wave NMR, 232 flowing-water NMR, 243 fluxgate, 202 free proton precession, 237 Hall, 190 magneto-optical, 213-214 magnetoresistance, 193, 194 magnetostriction, 212 micromechanical, 216 optically pumped, 256, 259, 261, 262 Overhauser, 249 SQUID, 160 thin-film inductance, 205 Ship's magnetism, 8, 466 Silicon-iron, s e e Fe-Si SI units, s e e Unit systems in magnetism

646

Subject Index

Signal acquisition, 375-376, 354, 365, 367, 379 Single Sheet Tester (SST), 317-326, 364, 374, 379, 383(t), 594(f) Single Strip Tester, 318(f), 321(f) Singular Point Detection (SPD) technique, 575 Skin depth, 409, 411(f), Skin effect in conducting cables, 410 Soft magnetic materials, properties, 25-83 Fe and low C steels, 33-38 NO Fe-Si alloys, 39-43 GO Fe-Si alloys, 4 3 - 4 9 high-Si, Fe-A1, Fe-Si-A1 alloys, 49-51

amorphous and nanocrystalline alloys, 5 1 - 6 0 Fe-Ni and Fe-Co alloys, 61-65 soft ferrites, 65-73 thin films, 73-83 Soft magnetic materials, characterization, 3 0 7 - 4 6 2 DC, 336-362 low and power frequencies, 364-409

medium-to-high frequencies, 409-432

radiofrequencies, 432-462 Soft spinel ferrites, 65-73 cation occupancy, 68(t) general properties, 69(t) processing, 73 energy loss/hysteresis, 72, 416(f), 423, 432 measurements at radiofrequencies, 442-452

Solenoid as a standard field source, 265 filamentary, 1 0 8 - 1 1 2 radial field, 111 rectangular cross-section, 112 with overwound ends, 109 thick, 1 1 7 - 1 2 1 Bitter, 118(f), 120

cooling, 119, 155 dissipated energy; 119 Gaume, 121 superconducting, 5 1 6 - 5 1 7 Specific heat, 389, 393, 557 Spectroscopic notations, 276 Spherical harmonics, 110 Spin-polarized conduction electrons, 77, 184 Spin valve, 77, 81 Spontaneous magnetization, s e e Saturation polarization Sputtering, 74 Squareness, 479 Standard deviation, definition, 582 Standard field sources, 262-268 Stray capacitances, 415-432 Stray field, 7, 164, 338-340, 499, Stresses in amorphous alloys, 28, 54 in Fe-Ni alloys, 184(f) in Fe-Si alloys, 42, 47, 48(f), 316 in strip-wound cores, 284, 315 in thick coils, 130 Student distribution, 590 Superconducting solenoids, 5 1 6 - 5 1 7 Superexchange interaction, 67, 76 Susceptibility, s e e Magnetic susceptibility T Temper rolling, 35 Thermoelectromotive force, 389, 391(f) Thin-film inductive field sensors, 203-205 Thin magnetic films characterization at radiofrequencies, 455-462

DC characterization, 432, 508, 514(f), 515, 571 permeance, 455 preparation, 7 3 - 7 5 Toroidal cores, see Ring-core specimens

Subject Index Torque magnetometer, 566-571 Traceability of measurements, 262-268, 595-600, 162-163, 250, 361, 363(t), 383-384(t), 497 Transmission lines, 4 3 3 - 4 4 2 characteristic impedance, 435-436, 453, 455 coaxial, 434(f), 435, 442, 443(f), 445 input impedance, 437-438, 440, 442-444, 454-455, matched, 436 short-circuited, 436-437, 442-447, 454, 457(f), 460(f) phase velocity, 435 propagation constant, 435, 448-454 reflection coefficient, 436, 438, 450, 458 stripline, 434(f), 441,448-452, 457(f) True value of a measurand, 582 Two-dimensional magnetizer, s e e Rotational single sheet tester Two-carrier conduction, 180-181 U Uncertainty, s e e Measuring uncertainty Unit cell in spinel ferrites, 67 Unit systems in magnetism, 613-619 V Vacquier-type fluxgate magnetometers, 199 Van der Pauw method, 374, 375(f) Variance, combined, 587 definition, 582 experimental, 583 weighted, 593

647

Verdet constant, 212 Virgin magnetization curve, 17 Vibrating coil method, 164-165, 339(f) Vibrating reed magnetometer, s e e AGFM magnetometer Vibrating Sample Magnetometer (VSM), 500-521 image effect, 517-519, 506, 511,516 intercomparisons, 513(f) measurement of anisotropy, 515 permanent magnet field source, 520-521 saddle-point coil arrangements, 504(f) sensitivity function, 501-503 setup, 507(f) superconducting field source, 517-519 vector measurements, 513-515 W

Wattmeter, 355, 370, 400(f) Weighted averaging, 592-593 Welch-Satterthwaite formula, 590

Yield stress, 39(f), 53(t) Yokes, 290-292 Z Zeeman splitting proton, 219 electron, 249(f), in He 3, 261(f) in He 4, 253(f) in Rb87, 254(f)

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