Advanced Computer Techniques in Applied Electromagnetics (Studies in Applied Electromagnetics)

ADVANCED COMPUTER TECHNIQUES IN APPLIED ELECTROMAGNETICS

Studies in Applied Electromagnetics and Mechanics Series Editors: K. Miya, A.J. Moses, Y. Uchikawa, A. Bossavit, R. Collins, T. Honma, G.A. Maugin, F.C. Moon, G. Rubinacci, H. Troger and S.-A. Zhou

Volume 30 Previously published in this series: Vol. 29. Vol. 28. Vol. 27. Vol. 26. Vol. 25. Vol. 24. Vol. 23. Vol. 22. Vol. 21. Vol. 20. Vol. 19. Vol. 18. Vol. 17. Vol. 16. Vol. 15. Vol. 14. Vol. 13. Vol. 12.

A. Krawczyk, R. Kubacki, S. Wiak and C. Lemos Antunes (Eds.), Electromagnetic Field, Health and Environment – Proceedings of EHE’07 S. Takahashi and H. Kikuchi (Eds.), Electromagnetic Nondestructive Evaluation (X) A. Krawczyk, S. Wiak and X.M. Lopez-Fernandez (Eds.), Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering G. Dobmann (Ed.), Electromagnetic Nondestructive Evaluation (VII) L. Udpa and N. Bowler (Eds.), Electromagnetic Nondestructive Evaluation (IX) T. Sollier, D. Prémel and D. Lesselier (Eds.), Electromagnetic Nondestructive Evaluation (VIII) F. Kojima, T. Takagi, S.S. Udpa and J. Pávó (Eds.), Electromagnetic Nondestructive Evaluation (VI) A. Krawczyk and S. Wiak (Eds.), Electromagnetic Fields in Electrical Engineering J. Pávó, G. Vértesy, T. Takagi and S.S. Udpa (Eds.), Electromagnetic Nondestructive Evaluation (V) Z. Haznadar and Ž. Štih, Electromagnetic Fields, Waves and Numerical Methods J.S. Yang and G.A. Maugin (Eds.), Mechanics of Electromagnetic Materials and Structures P. Di Barba and A. Savini (Eds.), Non-Linear Electromagnetic Systems S.S. Udpa, T. Takagi, J. Pávó and R. Albanese (Eds.), Electromagnetic Nondestructive Evaluation (IV) H. Tsuboi and I. Vajda (Eds.), Applied Electromagnetics and Computational Technology II D. Lesselier and A. Razek (Eds.), Electromagnetic Nondestructive Evaluation (III) R. Albanese, G. Rubinacci, T. Takagi and S.S. Udpa (Eds.), Electromagnetic Nondestructive Evaluation (II) V. Kose and J. Sievert (Eds.), Non-Linear Electromagnetic Systems T. Takagi, J.R. Bowler and Y. Yoshida (Eds.), Electromagnetic Nondestructive Evaluation

Volumes 1–6 were published by Elsevier Science under the series title “Elsevier Studies in Applied Electromagnetics in Materials”.

ISSN 1383-7281

Advanced Computer Techniques in Applied Electromagnetics

Edited by

Sławomir Wiak Institute of Mechatronics and Information Systems, Technical University of Lodz, Poland

Andrzej Krawczyk Institute for Labour Protection, Warsaw, Poland

and

Ivo Dolezel Technical University of Prague, Czech Republic

Amsterdam • Berlin • Oxford • Tokyo • Washington, DC

© 2008 The authors and IOS Press. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher. ISBN 978-1-58603-895-3 Library of Congress Control Number: 2008931370 Publisher IOS Press Nieuwe Hemweg 6B 1013 BG Amsterdam Netherlands fax: +31 20 687 0019 e-mail: [email protected] Distributor in the UK and Ireland Gazelle Books Services Ltd. White Cross Mills Hightown Lancaster LA1 4XS United Kingdom fax: +44 1524 63232 e-mail: [email protected]

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LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information. PRINTED IN THE NETHERLANDS

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved.

v

Preface This book contains papers presented at the International Symposium on Electromagnetic Fields in Electrical Engineering ISEF’07 which was held in Prague, the Czech Republic, on September 13–15, 2007. ISEF conferences have been organized since 1985 as a common initiative of Polish and European researchers who deal with electromagnetic field applied to electrical engineering. Until the present the conferences have been held every two years either in Poland or in one of European academic centres renowned for electromagnetic research. Technical University of Prague and the Chech Academy of Sciences make Prague be such a centre. Additionally, Prague is well-known in the world for its beauty and charm and it is called “Golden Prague”. The city of Prague is one of the six most frequently visited cities in Europe. Indeed, it is indisputable that Prague can attract every has the opportunity to visit it. The long, more then 20-year-old, tradition of ISEF meetings is that they try to tangle quite a vast area of computational and applied electromagnetics. Moreover, ISEF symposia aim at joining theory and practice, thus the majority of papers are deeply rooted in engineering problems and simultaneously present high theoretical level. Bearing this tradition, we attempt to touch the core of electromagnetic phenomena. After the selection process 237 papers were accepted for the presentation at the Symposium and almost all of them were presented at the conference, both orally and in the poster sessions. The papers have been divided into the following groups: • • • • • • •

Micro and Special Devices Electromagnetic Engineering Computational Electromagnetics Coupled Problems and Special Applications Measurement Monitoring and Testing Techniques Bioelectromagnetics Magnetic Material Modelling

The papers which were presented at the symposium had been reviewed and assessed by the sessions’ chairmen and the Editorial Board assembled for the postconference issue of ISEF’07. All the papers accepted for further publication were divided into three groups: 1) of more computational aspect, 2) of information technology aspect and 3) of more applicable nature. The latter ones are published in this volume while the first ones went to COMPEL journal (COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 27, No. 3/2008) and the second group to Springer Verlag (series on Studies in Computational Intelligence, vol. 119, 2008). The papers selected for this volume have been grouped in three chapters and seven sub-chapters. The division introduces some order in the pile of papers and the titles of chapters mirror the content of the papers to some extent. Names of chapters and subchapters are as follows: Chapter A Fundamental Problems and Methods Fundamental Problems Methods

vi

Chapter B Computer Methods in Applied Electromagnetism Computational Methods Numerical Modelling of Devices Chapter C Applications Electrical Machines and Transformers Actuators and Special Devices Special Applications The papers gathered in Chapter A are mainly devoted to physics of electromagnetic materials and mathematical approaches to electromagnetic problems. In the first sub-chapter papers concern physical phenomena, like magnetostriction, vibrations, anisotropy, occuring in the various electromagnetic materials from ferromagnetics to dielectromagnetics. And the second sub-chapter consists of papers concerning methods of analysis of electromagnetic phenomena in their methodological aspects. Chapter B contains papers dealing with numerical (or computer) analysis of electromagnetic devices and phenomena. The first sub-chapter shows how mathematical methods are realised numerically, i.e. how to make real calculation, based on numbers. And the papers gathered in the second sub-chapter deal with numerical modelling of some groups of devices. Chapter C, in turn, reveals the world of engineering problems, showing how theoretical and methodological considerations can be transferred to real engineering problems. Indeed, the chapter gives the image of real applied electromagnetics. The first sub-chapter is devoted to the very classical electrical devices, namely transformers and electrical machines. In spite of avery long tradition of numerical analysis of electromagnetic phenomena in such devices, the papers bring some new ideas and approaches. The second sub-chapter shows newer applications like sensors and actuators, and thus the area of engineering called mechatronics. Special approaches are needed inthe analysis of these devices as their size and operation features are quite different fromthe previous ones. And the last sub-chapter gathers a few papers dealing with very special applications based, for example, on superconductivity or ferroresonance. Needless to add that the electromagnetic analysis in such cases requires again new techniques and methods. The division of the papers is far from clear distinction of the papers’ topics and content. It is a very rough distinction which gives prospective readers some suggestion on how to find a paper of their personal interest. Summarising this introductory remarks we, the Editors of the book, would like to express our hope that the book you have in your hands will help the world-wide electromagnetic community, both academic and engineering, in better understanding electromagnetism itself and its application to technical problems. At the end of these remarks let us be allowed to express our thanks to our colleagues who have contributed to the book by submitting their papers or/and by peerreviewing the papers at the conference as well as in the publishing process. We also convey our thanks to IOS Press Publisher for their effective collaboration in giving this very attractive shape of the book and its promoting. Let us also express our strong belief that ISEF conference will maintain strong links with IOS Press in the future. Ivo Dolezel Chairman of the Organising Committee

Andrzej Krawczyk Scientific Secretary

Sławomir Wiak Chairman of the ISEF Symposium

vii

Contents Preface Ivo Dolezel, Andrzej Krawczyk and Sławomir Wiak

v

Chapter A. Fundamental Problems and Methods A1. Fundamental Problems Power Effect in Magnetic Lamination Taking into Account Elliptical Hysteresis Approach Kazimierz Zakrzewski

3

Study of Electromagnetic Field Properties in the Neighbourhood of the Metallic Corners Stanisław Apanasewicz and Stanisław Pawłowski

8

Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming Jan Albert, Wolfgang Hafla, André Buchau and Wolfgang M. Rucker Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in Metals Dagmar Faktorová

16

21

Electromagnetic Field Energy in Ferromagnetic Barriers Ryszard Niedbała, Daniel Kucharski and Marcin Wesołowski

26

The Influence of Temperature on Mechanical Properties of Dielectromagnetics Barbara Slusarek, Piotr Gawrys and Marek Przybylski

34

Influence of the Magnetic Anisotropy on Electrical Machines M. Herranz Gracia and K. Hameyer

39

Analysis of Structural Deformation and Vibration of Electrical Steel Sheet by Using Magnetic Property of Magnetostriction Wataru Kitagawa, Koji Fujiwara, Yoshiyuki Ishihara and Toshiyuki Todaka

47

A2. Methods Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction J. Turowski, Xose M. Lopez-Fernandez, A. Soto Rodriguez and D. Souto Revenga

53

Application of Logarithmic Potential to Electromagnetic Field Calculation in Convex Bars Stanisław Apanasewicz

58

Multi-Frequency Sensitivity Analysis of 3D Models Utilizing Impedance Boundary Condition with Scalar Magnetic Potential Konstanty Marek Gawrylczyk and Piotr Putek

64

viii

Very Fast and Easy to Compute Analytical Model of the Magnetic Field in Induction Machines with Distributed Windings Manuel Pineda, Jose Roger Folch, Juan Perez and Ruben Puche Coupling Thermal Radiation to an Inductive Heating Computation Christian Scheiblich, Karsten Frenner, Wolfgang Hafla and Wolfgang M. Rucker Consideration of Coupling Between Electromagnetic and Thermal Fields in Electrodynamic Computation of Heavy-Current Electric Equipment Karol Bednarek Force Computation with the Integral Equation Method Wolfgang Hafla, André Buchau and Wolfgang M. Rucker

72 80

85 93

Chapter B. Computer Methods in Applied Electromagnetism B1. Computation Methods Numerical Simulation of Non-Linear Electromagnet Coupled with Circuit to Rise up the Coil Current Slawomir Stepien, Grzegorz Szymanski and Kay Hameyer

101

Numerical Calculation of Power Losses and Short-Circuit Forces in Isolated-Phase Generator Busbar Dalibor Gorenc and Ivica Marusic

108

Numerical Methods for Calculation of Eddy Current Losses in Permanent Magnets of Synchronous Machines Lj. Petrovic, A. Binder, Cs. Deak, D. Irimie, K. Reichert and C. Purcarea

116

3-D Finite Element Analysis of Interior Permanent Magnet Motors with Stepwise Skewed Rotor Yoshihiro Kawase, Tadashi Yamaguchi, Hidetomo Shiota, Kazuo Ida and Akio Yamagiwa Advance Computer Techniques in Modelling of High-Speed Induction Motor Maria Dems and Krzysztof Komęza

124

130

Computation of the Equivalent Characteristics of Anisotropic Laminated Magnetic Cores E. Napieralska-Juszczak, D. Roger, S. Duchesne and J.-Ph. Lecointe

137

Improving Solution Time in Obtaining 3D Electric Fields Emanated from High Voltage Power Lines Carlos Lemos Antunes, José Cecílio and Hugo Valente

144

Thermal Distribution Evaluation Directly from the Electromagnetic Field Finite Elements Analysis A. di Napoli, A. Lidozzi, V. Serrao and L. Solero

151

Coordination of Surge Protective Devices Using “Spice” Student Version Carlos Antonio França Sartori, Otávio Luís de Oliveira and José Roberto Cardoso

158

ix

B2. Numerical Models of Devices Nonlinear Electromagnetic Transient Analysis of Special Transformers Marija Cundeva-Blajer, Snezana Cundeva and Ljupco Arsov

167

Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine Y. Tamto, A. Foggia, J.-C. Mipo and L. Kobylanski

175

Reduction of Cogging Torque in Permanent Magnet Motors Combining Rotor Design Techniques Andrej Černigoj, Lovrenc Gašparin and Rastko Fišer

179

Optimum Design of Linear Motor for Weight Reduction Using Response Surface Methodology Do-Kwan Hong, Byung-Chul Woo, and Do-Hyun Kang

184

Analytical Evaluation of Flux-Linkages and Electromotive Forces in Synchronous Machines Considering Slotting, Saliency and Saturation Effects 192 Antonino di Gerlando, Gianmaria Foglia and Roberto Perini Radiation in Modeling of Induction Heating Systems Jerzy Barglik, Michał Czerwiński, Mieczysław Hering and Marcin Wesołowski Time-Domain Analysis of Self-Complementary and Interleaved Log-Periodic Antennas A.X. Lalas, N.V. Kantartzis and T.D. Tsiboukis New Spherical Resonant Actuator Y. Hasegawa, T. Yamamoto, K. Hirata, Y. Mitsutake and T. Ota

202

212 220

Chapter C. Applications C1. Electrical Machines and Transformers Influence of the Correlated Location of Cores of TPZ Class Protective Current Transformers on Their Transient State Parameters Elzbieta Lesniewska and Wieslaw Jalmuzny Machine with a Rotor Structure Supported Only by Buried Magnets Jere Kolehmainen

231 240

FEM Study of the Rotor Slot Design Influences on the Induction Machine Characteristics Joya Kappatou, Kostas Gyftakis and Athanasios Safacas

247

Concentrated Wound Permanent Magnet Motors with Different Pole Pair Numbers Pia Salminen, Hanne Jussila, Markku Niemelä and Juha Pyrhönen

253

Air-Gap Magnetic Field of the Unsaturated Slotted Electric Machines Ioan-Adrian Viorel, Larisa Strete, Vasile Iancu and Cosmina Nicula

259

x

Dynamic Simulation of the Transverse Flux Reluctance Linear Motor for Drive Systems Ioan-Adrian Viorel, Larisa Strete and Do-Hyun Kang

268

Influence of Air Gap Diameter to the Performance of Concentrated Wound Permanent Magnet Motors Pia Salminen, Asko Parviainen, Markku Niemelä and Juha Pyrhönen

276

Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries V. Fireţeanu

284

Analysis and Performance of a Hybrid Excitation Single-Phase Synchronous Generator Nobuyuki Naoe, Akiyuki Minamide and Kazuya Takemata

294

Numerical Calculation of Eddy Current Losses in Permanent Magnets of BLDC Machine Damijan Miljavec and Bogomir Zidarič

299

Analysis of High Frequency Power Transformer Windings for Leakage Inductance Calculation Mauricio Valencia Ferreira da Luz and Patrick Dular

307

Influence of the Stator Slot Opening Configuration on the Performance of an Axial-Flux Induction Motor Asko Parviainen and Mikko Valtonen

313

Characteristics of Special Linear Induction Motor for LRV Nobuo Fujii, Kentaro Sakata and Takeshi Mizuma

318

Electromagnetic Computations in the End Zone of Power Turbogenerator M. Roytgarts, Yu. Varlamov and А. Smirnov

324

C2. Actuators and Special Devices The Impact of Magnetic Circuit Saturation on Properties of Specially Designed Induction Motor for Polymerization Reactor Andrzej Popenda and Andrzej Rusek

335

Electromagnetic Design of Variable-Reluctance Transducer for Linear Position Sensing J. Corda and S.M. Jamil

343

The Influence of the Matrix Movement in a High Gradient Magnetic Filter on the Critical Temperature Distribution in the Superconducting Coil Antoni Cieśla and Bartłomiej Garda

350

Macro- and Microscopic Approach to the Problem of Distribution of Magnetic Field in the Working Space of the Separator Antoni Cieśla

356

Electric Field Exposure Near the Poles of a MV Line D. Desideri, A. Maschio and E. Poli

363

xi

Study on High Efficiency Swithced Reluctance Drive for Centrifugal Pumping System Jian Li, Junho Cha and Yunhyun Cho

370

C3. Special Applications Power Quality Effects on Ferroresonance Luca Barbieri, Sonia Leva, Vincenzo Maugeri and Adriano P. Morando FEM Computation of Flashover Condition for a Sphere Spark Gap and for a Special Three-Electrode Spark Gap Design Matjaž Gaber and Mladen Trlep Recent Developments in Magnetic Sensing Barbaros Yaman, Sadık Sehit and Ozge Sahin Modelling of Open Magnetic Shields’ Operation to Limit Magnetic Field of High-Current Lines R. Goleman, A. Wac-Włodarczyk, T. Giżewski and D. Czerwiński Selected Problems of the Flux Pinning in HTc Superconductors J. Sosnowski

381

388 396

403 410

The Effect of the Direction of Incident Light on the Frequency Response of p-i-n Photodiodes Jorge Manuel Torres Pereira

417

3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis Algorithm S. Coco, A. Laudani, F. Riganti Fulginei and A. Salvini

425

Electro-Quasistatic High Voltage Field Simulation of Large Scale 3D Insulator Structures Including 2D Models for Conductive Pollution Layers Daniel Weida, Thorsten Steinmetz, Markus Clemens, Jens Seifert and Volker Hinrichsen Electromagnetic Aspects of Data Transmission Liliana Byczkowska-Lipińska and Sławomir Wiak

431

438

Application of the Magnetic Field Distribution in Diagnostic Method of Special Construction Wheel Traction Motors Zygmunt Szymański

449

Author Index

457

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Chapter A. Fundamental Problems and Methods A1. Fundamental Problems

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-3

3

Power Effect in Magnetic Lamination Taking into Account Elliptical Hysteresis Approach Kazimierz ZAKRZEWSKI Institute of Mechatronics and Information Systems, Technical University of Lodz Stefanowskiego 18/22 str., 90-924 Lodz, Poland [email protected] Abstract. On the grounds the author’s early works the analytical formulae for unit active power (losses) and reactive power in magnetic lamination were presented. The elliptical hysteresis approach of magnetic loops was assumed. In particular, these powers have been refereed to magnetization frequency. The universal functions F1, F2 for the case of magnetic flux forcing and F3, F4 for magnetic strength application on the booth side of lamination have been deduced in the work.

Introduction Despite many efforts, the full satisfactory analytical formulae for active and reactive power in magnetic laminations used in electromagnetic devices and electromechanical converters are not elaborated. The microscopic phenomena which we transferred for macroscopic effects are not exact described analytically. The numerous approximations which model static or dynamic hysteresis loop [1–3] not enable to calculate the power losses or reactive power with exactness acceptable in electrical engineering. Author, for dozen years prefers the use of equivalent elliptical hysteresis loops by approximation of real magnetization characteristics which enable to take account the hysteresis and first kind of eddy current losses. Thanks to introduction so called anomaly coefficient A n the analytical results may be adapt to total losses obtained experimentally [5,6].

Forced Magnetic Flux in Lamination The introduction of elliptic hysteresis loops is connected with assumption that all function of electromagnetic field are sinusoidal in time. The authors formulae deduced in [4] will need for the next discussion. In the paper [4] the active and reactive power were refereed to lamination segment with side surface area equal to 1 m2, in this work will be recalculated for 1 kG of the lamination weight. The measured active power may be expressed Pφ meas = An ( Bav, f )

k3 φm2 ξφ 2 σ μ m2 γ d

(1)

4

K. Zakrzewski / Power Effect in Magnetic Lamination

Table 1. Average values of

An

for ET3 lamination

Bav

T

0,2

0,5

1,0

1,1

1,3

1,5

1,7

An

–

2,0

2,0

1,92

1,76

1,45

1,29

1,29

where: Pφ meas – measured active power, An – anomaly coefficient as function of amplitude Bav (average induction in a cross section of lamination) and frequency f k = π f μ mσ

(2)

μ m – magnetic permeability for Bav, σ – conductivity of lamination material, γ – spe-

cific weight of lamination material, d – thicknees of lamination, φm – forced magnetic flux in lamination cross-section 1 ⋅ d m 2 φm = Bav ⋅ d [Wb/m] ξφ =

a sinh ( akd ) − b sin ( bkd ) cosh ( akd ) − cos ( bkd )

(3)

(4)

a = cos

δ δ + sin 2 2

(5)

b = cos

δ δ − sin 2 2

(6)

δ – angle of elliptical and symmetric hysteresis loop with amplitude Bav. The value An was called anomaly coefficient because takes into account the additional losses in relation to calculated ones, as a result of elliptical and continuous alternating magnetization inside the lamination material. In the work [6], the coefficient An for the transformer lamination ET3 0,35 mm was investigated and described. With a same approximation it is possible to assume the average values of An coefficients in dependence on frequency in a range (5 ÷ 300) Hz as a constant values. The function An ( Bav) is presented in Table 1. The forced magnetic flux appears in laminated core of transformer by voltage excitation. Essential dependence in the transformer praxis is the total losses reference to the frequency as a measurement method for hysteresis losses extrapolation in the case f →0. It will be indicated that frequently used extrapolation by application a direct line is not correct. Using (1)

5

K. Zakrzewski / Power Effect in Magnetic Lamination

Pφ meas

= An ( Bav )

f

Pφ

(7)

f

where Pφ f

=

π φm2 ( kd ) ξφ . 2μ m d 2γ

(8)

From (8) may be introduce the universal function (not dimensional) F1 = ( kd ) ξφ .

(9)

This formula may be presented in dependence not dimensional value ( kd ) . The 2

relation between f and ( kd ) is 2

f =

( kd )

2

π μm σ d 2

(10)

In the case of reactive power it is possible to introduce the function F2 = ( kd )ψ φ

(11)

and Qφ

=

f

π φm2 ( kd )ψ φ 2 μ m d 2γ

(12)

where ψφ =

b sinh ( akd ) + a sin ( bkd ) cosh ( akd ) − cos ( bkd )

(13)

The reactive power was less investigated than active power. From author’s experience results that an adaptive coefficient for measured power is equal to HI/Hm relation, where HI – first harmonic amplitude, Hm – amplitude of magnetic strength during sinusoidal change of induction with amplitude Bav in conditions of the real hysteresis loop magnetization. Qφ meas f

=

H I Qφ Hm f

The functions F1, F2 are illustrated in Fig. 1.

(14)

6

K. Zakrzewski / Power Effect in Magnetic Lamination

Figure 1. Diagrams of functions F1, F2.

Forced Magnetic Field Strenght on Lateral Surface of Lamination Another means of practical generation of electromagnetic field in lamination is an excitation by electrical current with forced magnetic field strength on lateral surfaces of lamination. The adequate formulae on the base [4] are for active power (losses) PH π μ m 2 ζ H = Hm f γ ( kd )

(15)

where: ζH =

a sinh ( akd ) − b sin ( bkd ) cosh ( akd ) + cos ( bkd )

(16)

for reactive power QH π μ m 2 ψ H = Hm f γ ( kd )

(17)

where: ψH =

b sinh ( akd ) + a sin ( bkd ) cosh ( akd ) + cos ( bkd )

(18)

Unfortunately, the formulae (15), (17) with adaptation to results of measurements was not yet investigated (open problem).

K. Zakrzewski / Power Effect in Magnetic Lamination

7

Figure 2. Diagrams of functions F3, F4.

Analogically to F1, F2 it is possible to introduce the universal functions F3 = ζH /(kd), F4 = ψH /(kd) which are illustrated in Fig. 2. Conclusion The universal functions F1 and F2 may be used for investigation of losses and reactive power in a wide range of magnetic flux frequency in lamination (voltage excitation). Adequately, the functions F3 and F4 are interesting for this investigation in a case of forced magnetic field strength on lateral surfaces of lamination (current excitation). The analytical formulae may be helpful in the praxis for design of different electromagnetic devices. References [1] D.C. Jiles, D.L. Atherton: “Theory of Ferromagnetic Hysteresis”, Journal of Magnetism and Magnetic Materials 61 (1986), North-Holland, Amsterdam, pp. 48-60. [2] D. Mayergoyz: “Mathematical Models of Hysteresis”, Springer-Verlag, New York 1991. [3] J.K. Sykulski (editor): “Computational Magnetics” Chapman and Hall, 1995 London, Glasgow, Weinheim, New York, Tokyo, Melbourne. [4] K. Zakrzewski: “Berechnung der Wirk und Blindleistung in einem ferromagnetischen Blach unter Berücksichtigung der Komplexen magnetischen Permeabilität”, Wiss. Z.TH Ilmenau (1970), H.5, s. 101-105. [5] K. Zakrzewski: “Method of calculations of unit power losses and unit reactive power in magnetic laminations in a wide range change of induction and frequency”, Proceedings of ISEF’99 – 12th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering, ISEF’99, Pavia, Italy, September 23-25 1999, s. 208-211. [6] K. Zakrzewski, W. Kubiak, J. Szulakowski: „Wyznaczanie współczynnika anomalii strat w blachach magnetycznych anizotropowych, Prace Naukowe IMNPE Politechniki Wrocławskiej, Studia i Materiały Nr 20, Wrocław 2000, s. 299-305.

8

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-8

Study of Electromagnetic Field Properties in the Neighbourhood of the Metallic Corners Stanisław APANASEWICZ and Stanisław PAWŁOWSKI Technical University in Rzeszów, Chair of Electrodynamics and Electrical Machinery Systems, Poland

Study of electromagnetic field in the neighbourhood of the metallic corners is an aim of the present paper. Concave and convex corners are considered.

Introduction It is known that in the case of depth of electromagnetic field penetration in the metal that is small in comparison with radius of curvature of metal surface, calculation of the field distribution can be simplified. Impedance boundary conditions are applied. Aim of that is simplification of Helmholtz equation in the metallic area: term with second derivative in the direction tangential to the boundary is rejected from this equation and the second derivative in the transverse direction is left. Such a simplified equation is solved in the open way and it causes impedance boundary conditions in the air area. Normal derivative of z component of vector potential A is proportional to A ie ∂A α A = . In the neighbourhood of the corners such simplifications are not possible ∂n μ r and in such cases calculation of electromagnetic field distribution is more complicated. Primary aim established by authors of the present paper is to find adequate simplifications if there is metallic area of large curvature, radius of which is small in comparison to the depth of field penetration. It turned out that this problem looks differently for the concave and convex corners. Therefore, our first task is to study the essential differences occurring in these both cases. At first, we analyze the corners creating angles π/2 lub π/3; however we omit the analysis of rounded corners. We consider three types of field excitation: a) electrostatic, b) incidence of straight flat wave, c) excitation of eddy currents in the metal by fields generated by sinusoidal currents flowing in wires in air area. In the mentioned case, solution of adequate Helmholtz and Laplace equations can be presented in the form of Fourier integrals and determination of integrand comes down to solve an equation of Fredholm integral equations of the second kind. In this paper we are restricted to two first variants.

9

S. Apanasewicz and S. Pawłowski / Study of Electromagnetic Field Properties y

y

−Q

b

γ =

Q

a

a

γ =∞

b

−Q

b

Q

a

x

a

γ =∞

a) Concave corner

γ =0

b

x

b) Convex corner Figure 1. Sketch of studied corners.

Study of Electrostatic Field We assume that electrostatic field is generated by charges Q and –Q placed for the simplicity symmetrically in relation to the metallic walls in points (a,a) and (b,b) (Fig. 1a and 1b). Electrostatic field is described by scalar potential ϕ = ϕ ( x, y ) that fulfils Laplace’s equation. This potential can be presented in the form of two components: ϕ = ϕ 0 + ϕi

(1)

First of them presents the field generated by charges Q, –Q and the second presents influence of metallic walls. Term ϕ0 is known and it is presented in the following way: ϕ0 =

∞ g cosτ x + g 2 sin τ y Q ( x − a)2 + ( y − a)2 Q ln = eτ x 1 dτ , 4πε 0 ( x − b) 2 + ( y − b) 2 2πε 0 ∫0 τ

x < 0 (2)

where: g1 = e − sb cos sb − e − sa cos sa g 2 = e − sb sin sb − e − sa sin sa

However component ϕi can be presented in the following way: ∞

1 ϕi = ∫ D(τ ) ⎡⎣ e−τ x sin τ y + e −τ y sin τ x ⎤⎦ dτ , τ 0

x > 0,

y>0

(3)

10

S. Apanasewicz and S. Pawłowski / Study of Electromagnetic Field Properties

In the case of concave corner, D function is the only function that should be calculated. Potential ϕ must be equal zero on the surface of metal: ϕ ( x, 0) = ϕ (0, y ) = 0 what is tantamount to the zeroing of tangential components of electrostatic field  E = ( E1 , E2 , 0) = (−ϕ x , −ϕ y , 0) , so the following condition should be fulfilled ∞

⎤ Q ⎡ x−b x−a − = g ( x) ⎢ 2 2 2 2 ⎥ ( x b ) b ( x a ) a − + − + ⎦ 0 ⎣

∫ D(τ ) cosτ xdτ = 2πε 0

(4)

hence D=

∞

2 g ( x) cosτ xdx π ∫0

(5)

D function and ϕi potential can be calculated in the overt form. Ultimately, omitting laborious computational transformations, we can present solution in the form of mirror images: ϕi ( x, y ) = ϕ∗ (−b, −b) + ϕ∗ (−a, −a ) + ϕ∗ (b, −b) + ϕ∗ (−b, b) − ϕ∗ (a, − a ) − ϕ∗ (− a, a) (6) Q ln ⎡ ( x − a )2 + ( y − b)2 ⎤⎦ . 4πε 0 ⎣ Fictitious Q charges are located in points (a,–a), (b,–b), (–a,–a), (–b,–b), (–a,a), (–b,b). In the event of convex corner in the area x > 0, y > 0 we accept ϕi in the form of Formula (3) and in the area x < 0, y >0:

where ϕ∗ (a, b) =

∞

1 ϕi = ∫ ⎡⎣ R(τ )eτ x sin τ y + R∗ (τ )e −τ y sin τ x ⎤⎦ dτ τ 0

(7)

We assume that ϕi and ϕix are continuous on the line x = 0, y > 0 in order to calculate functions D, R, R*. Additionally, on the line y = 0, x < 0 the condition ϕi + ϕ0 = 0 must be fulfilled. On the basis of these conditions, we obtain: R = D, ∞

∫R

∗

0

hence:

D+R =

cosτ xdτ =

2τ π

Q 2πε 0

∞

∫ [ D( s) − R ( s ) ] s ∗

0

2

ds +τ 2

⎡ ⎤ x −b x−a − = g ( x) , ⎢ 2 2 2 2 ⎥ ( x b ) b ( x a ) a − + − + ⎣ ⎦

(8) x<0

11

S. Apanasewicz and S. Pawłowski / Study of Electromagnetic Field Properties

5000 4000

for cocave corner for convex corner

E1 (V/m)

3000 2000 1000 0 -1000 -2000

0

1

2

3

4

x=y (m)

5



Figure 2. The E1 component of electrical field distribution on y = x line.

D(τ ) =

R∗ (τ ) =

τ ds [ D(s) − R∗ ( s)] 2 2 π s +τ ∞ ⎤ Q ⎡ s+b s+a cos τ sds − ⎢ 2 2 2 2 2 ⎥ ∫ π ε 0 0 ⎣ ( s + b) + b ( s + a ) + a ⎦

(9)

(10)

By elimination of R* from Eqs (9), (10) we obtain integral equation for determination of D function: D(τ ) =

∞

τ ds D( s) 2 − D0 (τ ) ∫ π 0 s +τ 2 ∞

τ ds D0 (τ ) = ∫ R∗ ( s ) 2 2 π 0 τ +s

Equation (11) is solved numerically. Enclosed Figs 2–4 illustrate results of calculations.

(11)

12

S. Apanasewicz and S. Pawłowski / Study of Electromagnetic Field Properties

4000 3000 2000

E 2 (V/m)

1000 0

-1000 5

-2000

4 2 1

2

3

x (m)

1 4

y (m )

3

-3000

5



Figure 3. The E2 component distribution in the case of concave corner.

4000

3000

1000

0

5 4 -4

3 -2

2 0

x (m )

2

1

)

-1000

y( m

E 2 (V/m)

2000

4

Figure 4. The E2 component distribution in the case of convex corner.



S. Apanasewicz and S. Pawłowski / Study of Electromagnetic Field Properties

Figure 5. Concave corner – no shadow; elementary solution.

13

Figure 6. Convex corner – there is a shadow zone, field is described by integral solution.

Study of Diffraction of Straight Waves on the Concave and Convex Corners We consider the following wave structure of the electromagnetic field:  E = (0, 0, E ( x, y )) ,

ΔE = − k 2 E ,

H1 =

 H = ( H1 , H 2 , 0)

jE y ωμ0

,

H2 = −

(12) jEx , ωμ0

k=

ω c

In the event of concave corner we have also elementary solution of the problem. If sinusoidal straight wave with amplitude E* falls at an angle of ϕ (in relation to x axis) on the wall of metallic area (what can be verified easily) complete E field can be determined by one elementary formula: E = 2 E∗ [ 2 cos k ( x cos ϕ + y sin ϕ ) + cos k ( y sin ϕ − x cos ϕ )]

(13)

Course of wave is illustrated on Figs 5 and 6. However in the event of convex corner, situation is completely different. Solution of Helmholtz equation (12) requires solution of integral equation. We present solution of Eq. (12) in the following form: E = E0 + Ei ⎧∞ − py − px ⎪ ∫ ⎡⎣W1 (τ )e sin τ x + W2 (τ )e sin τ y ⎤⎦ dτ , ⎪0 ⎪⎪ ∞ Ei = ⎨ ∫ ⎣⎡W2 (τ )e − py sin τ x + W4 (τ )e px sin τ y ⎦⎤ dτ , ⎪0 ⎪∞ ⎪ ⎡W5 (τ )e py sin τ x + W6 (τ )e − px sin τ y ⎤ dτ , ⎦ ⎪⎩ ∫0 ⎣

(14) x ≥ 0,

y≥0

x ≤ 0,

y≥0

x ≤ 0,

y≤0

(15)

14

S. Apanasewicz and S. Pawłowski / Study of Electromagnetic Field Properties

E0 = E∗ e − jk ( y cos ϕ − x sin ϕ )

(16)

where E* – amplitude of falling wave, ϕ – angle between direction of falling wave and y axis. E0 term represents complex amplitude of falling wave and Ei represents the reaction of the metallic wall. We assume the conditions presented below in order to calculate Wi integrands: a) b)

Continuity of Ei and components of magnetic field H1, H2 on lines x = 0, y ≥ 0 and y = 0, x ≥ 0. Zeroing of E on the metallic surface: E(0, y) = 0, y ≤ 0 and E(x, 0) = 0, x ≤ 0.

We omit detailed calculations and present final result:

W1=W5, W2=W4 p(τ )W1 =

p(τ )W2 =

τ π

∞

τ π

∞

s(W2 − W6 )ds 2 + s2 − k 2

(17)

s (W1 − W3 )ds 2 + s2 − k 2

(18)

∫τ 0

∫τ 0

⎧2 ⎫ τ W6 = E∗ ⎨ + j [δ (τ − k cos ϕ ) − δ (τ + k cos ϕ )]⎬ 2 2 2 π τ k cos ϕ − ⎩ ⎭

(19)

⎧2 ⎫ τ W3 = E∗ ⎨ + j [δ (τ − k sin ϕ ) − δ (τ + k sin ϕ )]⎬ 2 2 2 π τ k sin ϕ − ⎩ ⎭

(20)

where δ – δ-Dirac’s function. After elimination of W3 and W6 functions from expressions (17) and (18), we obtain system of two integral equations: pW1 =

τ π

sW2 ( s )ds ∫0 s 2 + τ 2 − k 2 + ξ1 (τ ) ,

∞

pW2 =

τ π

∞

∫s 0

sW1 ( s )ds + ξ 2 (τ ) +τ 2 − k 2

2

(21)

where: ξ1 (τ ) = −

τ ⎡ τ 2 − k 2 + jk cos ϕ ⎤ ⎢ ⎥ E∗ , π ⎣⎢ τ 2 − k 2 sin 2 ϕ ⎦⎥

ξ1 (τ ) = −

τ ⎡ τ 2 − k 2 − jk sin ϕ ⎤ ⎢ ⎥ E∗ . π ⎣⎢ τ 2 − k 2 cos 2 ϕ ⎦⎥

Instead of system of two equations with two unknowns W1, W2 (21) one can solve two equations with one unknown assuming u1 = W1+W2, u2 = W1–W2. These equations have the following form:

S. Apanasewicz and S. Pawłowski / Study of Electromagnetic Field Properties

pu1 (t ) =

τ π

∞

su1 ( s )ds τ ∫0 s 2 + τ 2 − k 2 + ξ1 (τ ) + ξ2 (τ ) , pu1 (t ) = π

∞

∫s 0

15

su2 ( s)ds + ξ1 (τ ) − ξ 2 (τ ) . +τ 2 − k 2

2

Observed Characteristics of the Field in the Concave and Convex Corners a)

b)

c)

Characteristics of electromagnetic field in the corners are significantly different depending on the sign of curvature. In the presented examples in the concave corner we obtain solutions of adequate equations in the elementary form; however in the event of convex corner solution of these equations causes integral equations, which can be solved only numerically. Infinite values of field in the corners are not observed, what takes place in some other cases (for example if potentials are set on boundaries of the area). But extreme values of field are observed in corners. If corners presented on Fig. 1 are rounded off with use of open arc, it turns out that solution of applied Helmholtz equations (in the case of eddy currents) is easier for the concave corner (centre of the rounding arc is outside of the metallic area) than for the convex case (mentioned centre is inside the metallic area).

16

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-16

Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming Jan ALBERT, Wolfgang HAFLA, André BUCHAU and Wolfgang M. RUCKER Institute for Theory of Electrical Engineering Pfaffenwaldring 47, 70569 Stuttgart, Germany [email protected]

Abstract. This paper deals with the numerical computation of eddy currents and the forces they cause. These effects are of special interest when considering pulsed magnetic forming, which is a technique to form thin metal sheets or pipes. A high transient current in a coil near to the work piece excites eddy currents and the associated Lorentz forces press the work piece into a die.

Introduction Pulsed magnetic forming (PMF) is a quite old technique developed in the first half of the twentieth century. In recent years some new ideas concerning different applications and the use of field formers lead to increased efforts in the research of pulsed magnetic forming. The use of field formers results in an extended lifetime of the excitation coils but also offers the possibility to use the same installation for a large amount of applications when using differently shaped field formers [1]. Numerical simulations take a big part in the development of field formers because they offer a cheap and effective way to show the advantages and drawbacks of the suggested field former constructions and can therefore significantly reduce development costs. In this paper several ways of the numerical computation of the occurring eddy currents and Lorentz forces are considered.

Computation of Eddy Currents The governing equation for eddy current problems neglecting displacement currents and considering only non-ferromagnetic, and conductive materials is 1 ∂ curlcurl A + σ A = J C , μ0 ∂t

(1)

where A is the magnetic vector potential, Jc the impressed current density, σ the conductivity and µ0 the magnetic permeability. Equation (1) can directly be implemented as a formulation for FEM programs. A drawback of simulating PMF with a finite element method (FEM) lies in the large amount of nodes that is required to describe the geometrical problem properly. Particularly meshing the air domain takes much effort.

J. Albert et al. / Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming

17

Other problems result from the fact that the work piece deforms and one has to create a new mesh for every time-step. So, using a boundary element method (BEM), which requires meshing of the surface of the bodies, only, or at least an integral equation method (IEM) with the advantage of an unmeshed air region is suggested. Due to the fact that one has to deal only with nodes on the surfaces, BEM-formulations are more complex than FEM-formulations and have to be solved with a larger effort per node, but the total number of nodes is significantly lower, and the computational costs reduce for large problems especially for bodies with a small relation between surface and volume. Three Examples of Integral Equation Method Formulations In recent years different BEM formulations have been developed. Depending on the considered problem one has to choose between several formulations and has to pick out the one that fits best to the actual requirements. A Direct BEM Formulation Based on Two Field Quantities Equations (2) and (3) are presented as an example for a pure BEM-formulation. In free space the governing equation for the magnetic field is 1 H (re ) + 2 ⎧⎪ ⎫⎪ ⎛ 1 ⎞ Γ∫ ⎨⎪( ns × H t (rs ) ) × grad G(re , rs ) + ⎜⎝ − jωμ0 ns ⋅ curl Et (rs ) ⎟⎠ grad G(re , rs )⎬⎪ d Γ , (2) ⎩ ⎭ =

∫ {J

c

(re ) × grad G (re , rs )} d Ω

Ω

where Г is the surface of the eddy current body and Ω the volume of the coil, Jc is the impressed current density inside the coil. So, on the right side of Eq. (2) we have the Biot-Savart integral, which describes the magnetic field of the exciting coil. For the eddy current region one obtains an equation for the electric field [2] 1 E (re ) − ∫ {( ns × Et (rs ) ) × grad G (re , rs ) + ( − jωμ 0 ) ns × H t (rs )G (re , rs )} d Γ = 0 , 2 Γ (3)

where H and E are the complex magnetic and electric field strengths with Ht and Et as their tangential components, G is Green’s function and n is the normal vector pointing outwards the considered region, re and rs are the coordinate vectors of the evaluation and the source location, respectively. A drawback of this formulation is based on the fact that it deals with problems in the frequency domain, while in PMF one has to consider transient currents. Of course, this problem can be solved with a Fourier transformation at high computation costs.

18

J. Albert et al. / Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming

A Minimum Order Boundary Integral Equation In an indirect formulation developed in the 1970’s a solution vector consisting of the values of the imaginary volume current density J at the surface of the eddy current body and the imaginary magnetic surface charge density σ M at the same surface is computed [3]. ⎛ ⎛ 1 − (1+ j) k rse ⎞ ⎞ e × ⎜ J (rs ) × grad ⎜ ⎟⎟ ⎟⎟ d Γ ⎜r ⎜ Γ ⎝ se ⎠⎠ ⎝ ⎛ 1 ⎞ 1 − σ M (rs )ns × grad ⎜⎜ ⎟⎟ d Γ = −2ns × H coil (re )  ∫ 2π Γ ⎝ rse ⎠ J (re ) +

1 2π

σ M (re ) + 1 + 2π

∫ n

s

1 2π

∫ σ Γ

M

⎛ r ⎞ (rs )ns ⋅ ⎜ se 3 ⎟ d Γ ⎜r ⎟ ⎝ se ⎠

⎛ ⎛ 1 − (1+ j) k rse n ⋅ s Γ∫ ⎜⎜ J (rs ) × grad ⎜⎜ rse e ⎝ ⎝

⎞⎞ ⎟⎟ ⎟⎟ d Γ = −2ns × H coil (re ) ⎠⎠

(4)

(5)

Here, Hcoil is the primary complex magnetic field raised by the exciting coil and com1 puted with Biot-Savart law, k = σμω , rse is the difference beetween rs and re, and 2 the other expressions have the same meaning as in the equations above. The total magnetic field consists of the primary magnetic field and the secondary field Hsec raised by effects in the eddy current body H total = H sec + H coil .

(6)

In free space one obtains H sec, FS = − grad ϕ M , FS ,

(7)

with ϕ M , FS (re ) =

1 4π

∫ Γ

σ M (rs ) dΓ . rse

(8)

The magnetic field in free space can be described by scalar variables σ M, because it is irrotational; σM are the sources of the magnetic field strength, but they should not be mistaken as the magnetic charges in a ferromagnetic body (they are still present even if µr = 1), though both are treated equivalent. In the eddy current body the magnetic field is rotational, and, therefore, must be described by a vector variable, which is the imaginary surface current density J at the

J. Albert et al. / Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming

19

surface of the eddy current region. The magnetic field inside the eddy current region can be obtained by ⎛ 1 H sec, EB (re ) = curl ⎜ ⎜ 4π ⎝

⎛ 1

∫ J (r ) ⎜⎜ r s

Γ

⎝

e

− (1+ j) k rse

se

⎞ ⎟⎟ d Γ ⎠

⎞ ⎟ ⎟ ⎠

(9)

The physical values of the eddy current density can directly be achieved by applying Maxwell’s law J phys , EB = curl( H total ) = curl( H sec, EB + H coil )

(10)

The advantage of this formulation is found in the small size of the matrix, due to the four degrees of freedom. One decisive drawback is the fact that a Laplace transformation is necessary for considering transient problems. A Volume Integral Equation Method for Transient Computations From the induction law follows E+

∂ A−v×B = 0 ∂t

(11)

Under the assumption that motion raised induction can be neglected, and that the total magnetic vector potential A consists of the impressed coil field Acoil and the reduced vector potential Aeddy raised by eddy currents one obtains (12) and (13) E+

∂ ( Acoil + Aeddy ) = 0 , ∂t

⎞ J (re ) ∂ ⎛ μ J (rs ) ∂ + ⎜∫ d Ω ⎟ = − Acoil (re ) . ⎜ ⎟ ∂t ⎝ Ω 4π rse ∂t σ ⎠

(12)

(13)

This formulation has the major advantage that either a time stepping method – a so-called Euler method – or a frequency domain method can be implemented with ease. Unfortunately, this method requires a large amount of boundary and volume elements, but is suited for matrix compression techniques like the fast multipole method [4]. Other enhancements, for example inclusion of motion raised induction effects, are possible but within the first studies motion raised induction shall be neglected, because it is assumed that, in PMF, the induced forces of the first few moments are decisive for the result. At this time the work piece is not really moving yet. Computation of the Occurring Forces Due to the fact that in PMF bodies have to be considered while they are deforming one cannot use Maxwell´s stress tensor, which is the most popular way to compute forces

20

J. Albert et al. / Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming

raised by electromagnetic fields. Instead, a method has to be found, which returns a force density. The direct computation of the Lorentz forces is an accurate way to obtain a force density f, which can be used as an input parameter for a simulation of the structural deformation f = J eddy × B 

(14)

The necessary field values of the eddy current density Jeddy and the magnetic flux density B can directly be obtained from FEM results or from post-processing the BEM results. Conclusion To predict the deformation of a work piece caused by a PMF process one has to couple electromagnetic and structural mechanics simulations. Several FEM, BEM, or hybrid products are applicable to this problem. For small problems FEM simulations are preferred. The larger a problem gets, e.g. when regarding several field formers, BEM simulations become more attractive, due to the reduced number of nodes and the decreased memory requirements. References [1] R. Hahn, Werkzeuge zum impulsmagnetischen Warmfügen von Profilen aus Aluminium- und Magnesiumlegierungen, Dissertation, Berlin, 2004. [2] C. J. Huber, Numerische Berechnung dreidimensionaler elektromagnetischer Felder mit Integralgleichungsverfahren, Dissertation, TU Graz, 1998. [3] I. D. Mayergoyz, “Boundary integral equations of minimum order for the calculation of threedimensional eddy current problems”, IEEE Trans. Mag., vol. 18, no. 2, pp. 536-539, march 1982. [4] G. Rubinacci, A. Tamburrino, S. Ventre, F. Villone, “A fast 3-D multipole method for eddy-current computation”, IEEE Trans. Mag., vol. 40, no. 2, pp. 1290-1293, march 2004.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-21

21

Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in Metals Dagmar FAKTOROVÁ University of Žilina, Faculty of Electrical Engineering, Department of Measurement and Applied Electrical Engineering, Univerzitná 1, Žilina, 010 26, Slovak Republic [email protected] Abstract. The paper deals with non-destructive (NDT) microwave measurement of defects in metal samples exploiting the waveguide features at the defect depth evaluation. In this article some results concerning their evaluations regarding to microwave access are shown. More measurements were performed to evaluate the geometry of defects in metal samples. Apart from established methods two new unusual microwave connections are presented and the results with their use at defects examination are given and compared with the previous results. Their advantages are discussed and some proposals for their utilizations are given.

Introduction Application of microwave technique in defectoscopy at metal materials is relatively a new area, in which the research character of works is prevailing, while new access are emerging and their utilization spectrum is increasing. The testing ability of dielectric materials with microwaves is given by the natural property as their ability to penetrate through such materials and we have devoted the adequate attention in this field, too, [1]. In another works we aimed ourselves to finding geometry and some additional defect properties in metals. After the experimentally confirmation the fact that the defect can be examinate as a special waveguide section we engaged in examination of its impedance properties enabling to obtain more precision information about the defect, [2]. The next experiments were directed at the defect depth settling by utilizing of microwave knowledge. For this purpose were predominately used classical microwave measuring technique exploiting reflected signal properties either registered directly by some elements of the microwave line (e.g. ferrite circulator) or by means of standing wave ratio (SWR) measurement and subsequent adoption of measured values (impedance character, minimum standing wave shift and like that), [2]. As the microwave technique disposes of the extensive range of measuring methods, we directed our attention in next experiments at giving precision to measured results and on the one hand at the shape of the reflected signal from the defect and on the other hand at its depth.

22 D. Faktorová / Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in Metals f=10,1 GHz 100

lossless waveguide

amplitúda [a.u.] amplitude [a. u.]

80

60

40

defect

20

0 0

20

40

60

80

100

120

140

hĺbka defektu ] defect depth[mm [mm]

Figure 1. Dependence of reflected signal amplitude from the defect depth in metal and from the position of the shorted-line piston in lossless waveguide.

Experimental Background For the sake of information of about the link-up this article to our works in the microwave NDT area we will briefly mention the experimental results which led to the achieved results presentation. The basic pieces of knowledge were the measurements aimed at the influence of the defect depth on the reflected signal amplitude, [3]. For this purpose two measurements for comparison were carried out: 1. 2.

measurement of the reflected signal on a lossless waveguide, measurement of the reflected signal on a sample with the defect.

The reflected signal was in both cases recorded through the ferrite circulator and the measurements were performed for such short-lime piston positions on the lossless waveguide which corresponded to the respective defect depth. The comparison of the both measurements is in the Fig. 1. One can concesive of the decreasing amplitude of the reflected signal at the defect λg depth settling at ( 2n + 1) , ( λ g is length of wave in waveguide and n is integral 4 number), distant maxima and consequently also about up to what depths the reliable information about the defect with particular material properties can be obtained at the given measurement precision.

Experimental Results After verifying our theoretical assumptions, [2] we proposed two new methods for the defect evaluation and have also tested them. The first method is based on a special microwave line connection using the balancing principle with the magic T as a passive element from microwave technique with wide possibilities, [4]. More detailed informa-

D. Faktorová / Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in Metals 23

D

matched load

A

input of mw signal

A

B

Figure 2. Magic T.

calibrated load

A

input of mw signal C

C

D

D

open waveguide

B

C

crystal detector

a)

sample with defect

B

b)

Figure 3. Magic T connection for the defect depth measurement.

tion about magic T can be found e.g. also in [4]. We will recapitulate only its properties from the point of view of our measurement. Magic T, Fig. 2 is symmetrical with regard to the plane crossing through the axes of the ports C and D. Microwave generator will be connected to the port C and the indicator to the port D. If the ports A and B are loaded with equal loads, then the energy from generator is divided in port A and B and in the port D energy does not show. If the ports A and B are not the same, then a part of energy in consequence of non symmetry in the fields gets into D and the reading on the indicator will depend from the degree of the load difference in ports A and B. The same is valid when the signals are reflected from loads in ports A and B. We have made use of these properties in two ways: a) b)

for obtaining similar information about the defect as at classical measurements, [3], for obtaining direct information about the defect depth.

The measurements were performed in the connection illustrated in Fig. 3. In the case according to Fig. 3a with the matched load in the port A and the indicator in the port D, the signal indicated in the port D will be proportional to the reflection coefficient from the port B. This way in our measurement port B was terminated with the open-waveguide probe and the sample with a defect (width 0,5 mm, depth 8,5 mm) was shifted in front of it. Corresponding graph is plotted in the Fig. 4, curve a. Measurement in connection according to Fig. 3b was performed for the purpose of getting a facility for simple measuring of the crack depth. To the port A was connected an artificial defect simulating the real one with the continuously changing depth. To the port B were gradually attached samples with known depths and for each depth the bridge was balanced (port D without signal) and corresponding reading recorded. This way the calibration was obtained and it is plotted in the Fig. 5, (dotted line – measured, full line – idealized). Among other things we were interested in the possibility cavity resonator using for the examination of defects in metal samples. For this purpose we have used our experiences with cavity resonators, [2] where also basic theoretical information are presented. We supposed that at suitable resonator binding on the defect as an impedance element it will influence the resonator’s quality factor Q and its resonant frequency.

24 D. Faktorová / Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in Metals f =10,1 GHz

amplitude [a.u.]

120 100 80 60 40 20 0 -15

-10

-5

0

5

10

15

probe position [mm] a

b

c

Figure 4. Dependence of reflected signal amplitude from the defect.

calibration data [mm]

4 3,5 3 2,5 2 1,5 1 0

5

10

15

20

defect depth [mm] Figure 5. Calibration curve for magic T.

For this purpose we have carried out an experiment with the resonator connected on standard microwave line. The resonator was connected on standard microwave line in the transmission way and one resonator aperture was terminated with an open waveguide probe. Resonator was tuned in resonance. In front of the probe was shifted a metal sample with the defect (width 0,5 mm, depth 8,5 mm). The behavior of the resonator was followed by means of a signal brought out through a loop antenna. The measured values are plotted in the Fig. 4, curve b. We have plotted in Fig. 4 for comparison also the curve c obtained by previous familiar way (through the ferrite circulator).

Conclusions Further to the theoretical basis and our previous experiments [2,3] there are in this paper presented new experiments which can provide new possibilities for defect in metal

D. Faktorová / Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in Metals 25

samples evaluation. Especially the “magic T” method seems to be very effective whether the question is the shape of crack determining or very quick method not demanding any computation but only using suitable calibrated aid. From the graph comparing three methods the resonator method is workable in spite of the fact that in Fig. 4 it does not show markable difference. The presented sensitivity was given by the quality factor of the used resonator, and can be improved by using a resonator with high sensitivity (approximately of 15000–20000) and this is not its only feature because also the detuning of the resonator can be very sensitive representation of the crack depth.

Acknowledgement The author would like to thank MSc. Pavol Žirko director of High School for Agriculture and Fishing in Mošovce for technical help at realization of experiments.

References [1] D. Faktorová, Microwave Nondestructive Testing of Dielectric Materials, Advances in Electrical and Electronic Engineering, ISSN 1336-1376, Vol. 5, No. 1-2, pp. 230-233, 2006. [2] D. Faktorová, Using of Microwaves at Investigation of Solid Materials Inhomogenities, Proceedings of APCNDT 2006, 12th Asia – Pacific Conference on Non-Destructive Testing, Auckland, New Zealand, November 5-10, 2006, http://www.ndt.net/article/apcndt2006/index.htm, 4 pp. [3] D. Faktorová, Interaction of Solid Materials Inhomogenities with Microwaves, Proceedings of EDS 2006, Electronic Devices and Systems, Brno, Czech Republic, ISBN 80-214-3246-2, September 14-15, 2006, pp. 388-393. [4] B.M. Maškovcev, K.N. Cibizov, B.F. Emelin, Teorija volnovodov, Moskva-Leningrad: Nauka, 1966.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-26

Electromagnetic Field Energy in Ferromagnetic Barriers Ryszard NIEDBAŁA, Daniel KUCHARSKI and Marcin WESOŁOWSKI The Institute of Electrical Power Engineering, Warsaw University of Technology Abstract. This paper refers to electromagnetic field propagation around highcurrent circuits. The effects of its influence on conducting bodies that are seals and casings of electric devices are discussed. These barriers which are the constructional parts are made of ferromagnetic materials – good absorbers of energy transformed into heat in their structures. Intensity of absorbance depends on the distance between the barrier and source of radiant energy. Limitation of diffusing energy by absorbing it requires a high accuracy quantitative analysis to assure a high value of absorption coefficient and optimal power density of heat sources. The analysis results of basic geometries are shown.

Introduction Production, transmission, processing or reception of electric energy are closely connected with formation and propagation of electromagnetic field (EM) around sources, feeders and receivers of energy. In most cases those disturbances are identified with electromagnetic interferences. From electromagnetic compatibility standpoint, there is a strong tendency to limit the influence of EM interferences. Considering biological and thermal danger for living organisms and mechanisms (especially electrical) caused by electromagnetic field, the most beneficial solution is to clear the environment from EM influences. In this paper high-current systems characterized by slow time-changing fields occurrence in their environment with intensities much exceeding permissible levels will be examined. One of the efficient method to reduce EM energy emitted to the environment is to applicate the conducting barriers fulfilling the function of the casings, bushings or constructional elements of the devices. Those barriers are placed in vicinity of the compact electrical power devices generating EM fields. Effects of attenuation of the electromagnetic energy are particulary intensive in barriers with closed electric circuits [3]. This closure is the basic stipulation for the screens to obtain the best attenuation properties. In those systems the field from induced currents in barriers will counteract the disrupting field. Otherwise, currents in barriers will close themselves in a plane of forced electric field and the energy will not be attenuated [3]. This solution practically does not reduce spreading of EM energy. Most barriers in vicinity of sources of EM waves are made of materials with high value of magnetic permeability. On one hand it raises relativity of barriers sizes, while on the other hand it causes absorption of bigger energy which makes this effect more superficial. Attenuation of EM field in steel and magnetic materials requires special approach to determine power value as very often it is the reason of overheating the constructional elements and protective equipment.

R. Niedbała et al. / Electromagnetic Field Energy in Ferromagnetic Barriers

27

Absorbtion of Electromagnetic Energy by the Ferromagnetic Barriers Application of conducting barriers with accurate matching (material and geometrical) parameters leads to the situation where energy of magnetic field in those barriers is practically attenuated. It is said [1–3] that conducting bodies are good screens of magnetic field, but not the electric ones. However, barriers which are placed in vicinity of generating systems interact with them and that interaction causes absorption of additional energy changed into heat. These losses cause increased load of main circuit what can be the reason of unstable work of electric power systems. Electromagnetic field around high – current circuits can be attenuated practically only by using barriers of thickness bigger than skin depth, what enforce necessity of application massive objects for low frequency. Application of thinner barriers is confined to rare cases of living organisms or measuring equipment protection. Thin barriers can absorb comparatively big portions of energy. It is hard to explicitly determine the conducting barriers properties because of its dependence on geometric and material parameters. These barriers are boundary surfaces separating two regions marked by two different energetic states. Total attenuation rate is defined by equation: k = kr ⋅ ka ⋅ kd

(1)

where kr , ka , kd are reflection, absorption and transmittance coefficients, attached to incident electromagnetic wave. The main component of energy attenuate is absorption in conducting material. In many cases, absorbed energy becomes real thermal hazard to structural parts of casings. Typical casing systems with open circuit conductors [3] are especially exposed to that interaction. Absorbed energy is almost twice in constructions with open electric circuits in comparing to close-circuited, but attenuation effect is much smaller [3]. Close-circuited barriers, described in this paper are characterized by different effects. In these systems reflection coefficient’s (kr ) value equal 1, because of reflection energy effect inside barrier’s structures, hence after multiple reflection, all energy is absorbed by the conducting material. These barriers have a very good attenuation rate for magnetic fields, but not for electric ones. In Fig. 1 the relative magnetic and electric field intensities leaving steel and copper barriers are shown. Relative electric field intensity is similar for both materials, and it depends only on barrier’s thickness. In the case of magnetic field intensity, relative value depends additionally on material properties. Value of attenuation rate is higher for copper but requires larger thickness of conducting body (from several to several dozen larger skin depth). It is necessary to mention that the absorbance of energy strongly depends on physical barrier’s thickness. For thick body, absolute value of absorbed energy is less then in thin one. In Fig. 2 real, reactive, apparent power and power factor distribution versus relative barrier’s thickness are presented. The most popular constructional material is steel. So that, it is necessary to take into account some special parameters of this material, especially nonlinearity of magnetic permeability. There are some methods, based on the idea of isotropic and inhomogeneous material, which has constant parameters during a period. These methods

28

R. Niedbała et al. / Electromagnetic Field Energy in Ferromagnetic Barriers

Figure 1. Magnetic and electric field attenuation vs. relative casings thickness (a/δ), for copper and steel. Thickness is relative to skin-depth of given material. Value of relative field intensity is a proportion between field intensity on both surfaces.

Figure 2. Real, reactive and apparent power, and power factor vs. normalized barrier’s thickness.

allow to take into account the nonlinear property of ferromagnetic material by average energy density during a period time used to construct effective permeability [4–6]. These methods force us to make fast calculations of electromagnetic field, especially important, when one have to couple it with thermal analysis. These methods use the mathematical model described by Helmholtz equation (2). This is not very precisely as it is difficult and/or sometimes impossible to calculate the effective magnetic permeability. ∇ 2 H − jωμγ = 0

(2)

In time-domain analysis one can use the mathematical model described by Eq. (3). It gives the possibility to recalculate current value and magnetic permeability every time step. The power absorbed in barriers and momentary values of inductance currents and voltages will be compared.

R. Niedbała et al. / Electromagnetic Field Energy in Ferromagnetic Barriers

∇ 2 H − μγ

∂H =0 ∂τ

29

(3)

Effect of Nonlinear Permeability for Energy Absorbing In the paper, simple geometry was considered to indicate effect of magnetic permeability for power losses in closed conducting barrier. Therefore calculations of cylindrical ferromagnetic shield surrounding axially placed electrical lines were made. In this case, variability of current and shield distance, have similar effect for electrical power properties. Calculations were made for 0.5 m diameter. Results are presented for a unit length. μ (τ , A ) = A

B − ln H (τ , A )

(4)

Where H is a function of time (τ) and space (A). Magnetic permeability was approximated by Eq. (4) which is exact for magnetic permeability after inflection point. Type of Sources and Their Energetic Result In the example only a fraction of transmitted electrical power would be absorbed by ferromagnetic shield and current-fed is assumed. This is dangerous case because of quadratically rising energy loss in ferromagnetic medium. Inverse behavior is observed in electrical devices that are voltage-fed. In Fig. 3 currents and unit voltage timing diagram in the shield are presented. Voltage un was calculated for magnetic permeability as a function of space and time µ = f(τ,A), but ul was calculated for surface magnetic permeability µ = µs. It results in lower voltage amplitude and greater phase shift and it gives considerably less amount of shield’s absorbed energy. Figure 4 gives a picture of power electric changes, and character of electromagnetic screen load, added to kind of transmission line load.

Figure 3. Effect of nonlinear permeability for voltage with source current 10 kA.

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R. Niedbała et al. / Electromagnetic Field Energy in Ferromagnetic Barriers

Figure 4. Voltage vs. current for different excitation current with constants and variable permeability. Table 1. Power calculation errors for μ = var i μ = const. I[A]

Pn [W]

Pl[W]

ΔP[%]

2500

498.2

419

–19

5000

1504.3

1243

–21

7500

2841.1

2408

–18

10000

4505.6

3918

–15

Figure 5. Comparison of shield absorbed power, calculated with μ = var and μ = const.

Shapes, which area represents value of shield energy absorbed during one period, are shown on graph of voltage vs. current (Fig. 4). Errors of calculation shield’s real power are presented in Table 1. Implementation of function µ = f(τ,A) results in strong deformation of quadratically dependent power with current. This is shown in Fig. 5 (continuous line). Power vs. current for medium with constant permeability, chosen for value of surface magnetic field intensity (µ = µs) and H equal to 0, are shown by dotted lines.

R. Niedbała et al. / Electromagnetic Field Energy in Ferromagnetic Barriers

31

Figure 6. Effect of nonlinear magnetic permeability for time-depending voltage with source current 7500 A.

Figure 7. Shapes of voltage vs. time, for different substitutive magnetic permeability.

Errors for different currents (Table 1) show that one could choose substitutive magnetic permeability which minimized deviations of real power. In Fig. 6 timedependant voltage calculated with magnetic permeability chosen in such way were presented. Although voltage ul shifting is compensated with amplitude, screen load character is different. ⎛P ⎞ ΔP = ⎜ l − 1⎟ ⋅100 P ⎝ n ⎠

(5)

Time-dependant voltage were presented in Fig. 7 for different thicknesses, with the same excitation current. Errors generated with nonlinear magnetic permeability were established for thickness value equal a few skin-depth. For thin ferromagnetic barriers, distribution of power and error of calculated power are presented in Fig. 8. Differences between power errors in function of thickness and current (Table 1) should be emphasized.

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R. Niedbała et al. / Electromagnetic Field Energy in Ferromagnetic Barriers

Figure 8. Surface screen power and their error in relation with screen thickness.

Discussion and Conclusion Conducting barriers surrounding electrical objects that radiate electromagnetic fields are one of most efficient way to reduce their influence on environment. Wrong made casings can absorb great amount of energy, thus dangerously increasing their temperature without much affecting incident wave. It is important to make electrical linking between casings, causing induced current to flow between or outside them. In this case they absorb energy, also they are shielding objects in range of EM source. It is essential to make accurate electromagnetic calculations, due to the fact that absorbed energy, in this kind of shields, often decide on their material and geometric parameters. In the paper, calculations of electromagnetic barriers done with substitutive magnetic permeability were proved not correct. Computations, which give a practically usable results, should be done in time-domain. In this case, value of magnetic permeability could be approximated by magnetization curves, giving more exact results. It was shown that error of calculation depends on the way of approximation of magnetic permeability. This may be function of thickness or time, thus it is causing different rate of error. An interesting alternative for one-layered barrier is compound made of two different physical materials: the first made of nonmagnetic, very good conducting such as copper, and the second – ferromagnetic. This structure is absorbing nearly all energy by nonmagnetic material. This kind of barrier are efficient electromagnetic screens, absorbing less energy, and not exposed to excessive heating.

References [1] H. Waki, H. Igarashi and T. Honma “Analysis of Magnetic Shielding Effect of Layered Shields Based on Homogenization”, IEEE Transactions on Magnetics., vol. 42, No. 4, 2006. [2] J. Čuntala, “Simulation of Electromagnetic Shielding in Comsol Multiphisics Environment”, http://www2.humusoft.cz/www/akce/comsol06/cuntala.pdf. [3] R. Niedbała “Absorpcja Energii Elektromagnetycznej przez Ekrany Magnetyczne”, Przegląd Elektrotechniczny, Number 2, 2003, pp. 127-130.

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33

[4] O. Biro, K. Preis and K. R. Richter, “Various FEM Formulation for the Calculation of Transient 3D Eddy Currents in Nonlinear Media”, IEEE Transactions on Magnetics, vol. 31, No. 3, 1995. [5] K. Preis, L. Bardi, O. Biro and K. R. Richter “Nonlinear Periodic Eddy Currents in Single and Multiconductor System”, IEEE Transactions on Magnetics, vol. 32, No. 3, 1996. [6] G. Paoli and O. Biro “Time Harmonics Eddy Currents in Non-Linear Media”, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Volume 17, Number 5, 1998, pp. 567-575(9).

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-34

The Influence of Temperature on Mechanical Properties of Dielectromagnetics Barbara SLUSAREK, Piotr GAWRYS and Marek PRZYBYLSKI Tele & Radio Research Institute, Ratuszowa 11 St., 03-450 Warsaw, Poland [email protected], [email protected], [email protected] Abstract. Development of technology and notably development of new generation magnetic materials caused substitution of cast magnets with magnets manufactured in the process of powder metallurgy. Development in magnetic materials is observed not only hard magnetic materials, but also soft magnetic materials. Particular intensive development can be observed in powder materials, which more and more often substitute traditional materials, e.g. electrical steel. The main factor deciding about application of soft magnetic elements is its magnetic properties. In many applications mechanic properties are equally important as magnetic parameters. Physical properties of materials change with the change in temperature. The main goal of research is to know changes of mechanical properties of dielectromagnetics with changes of temperature.

Introduction Magnetic circuit is a fragment of space containing interconnected elements executed in ferromagnetic materials, which make a closed way for the flow of magnetic induction flux generated with a permanent magnet or a coil. Magnetic core is the part of magnetic circuit, used to assign the way of magnetic flux generated by a permanent magnet or a coil. Development in magnetic materials causes changes and development in magnetic circuits of electric machinery, and in a broader perspective of electromagnetic transducers. Until recently electric motors were commonly fitted with magnetic circuits employing cast permanent magnets and magnetic core of electrical steel. Development of technology and notably development of new generation magnetic materials caused substitution of cast magnets with magnets manufactured in the process of powder metallurgy. Development in magnetic materials is observed not only in hard magnetic materials, but also in soft magnetic materials. Particular intensive development can be observed in powder materials, which more and more often substitute traditional materials, e.g. electrical steel. First powder magnetic cores were created in the 1880’s. They were executed in crumbled iron, insulated with wax. At the same time however sheet magnetic cores were developed, which was substituted powder cores of worse magnetic parameters for a long time. In recent years we witness a comeback of magnetic cores of powder composites, due to i.e. development of special low power electric machinery. New designs of elec-

B. Slusarek et al. / The Influence of Temperature on Mechanical Properties of Dielectromagnetics

35

trotechnic transducers often require magnetic circuits where application of electrotechnic sheets is difficult for structural or technical reasons. Manufacture of powder magnetic cores with better parameters became feasible thanks to new generations of magnetic materials and new powder metallurgy technologies. Soft magnetic parts of magnetic circuits can be manufactured using two basic processes of powder metallurgy: sintering or binding with plastic. This latter method finds wider and wider application for circuits with alternating magnetic flux, notably of frequency higher than 50 Hz, as this method has many advantages. Binding of powder with plastic method is used in manufacture of permanent magnets too, which allows to obtain integrated magnetic circuits with hard and soft magnetic layers. Application of powder metallurgy has many advantages, such as i.e. low losses of material, low consumption of labor and energy, and resulting low unit cost of product. That is the reason behind intensification of works aiming at improvement of magnetic circuits manufactured with binding magnetic powder with plastic method. Soft magnetic elements manufactured using that method are called dielectromagnetics, whereas hard magnetic elements are called dielectromagnets. Magnetic core manufactured from soft magnetic powder composite allows three dimensional distribution of magnetic flux in magnetic core. Full adaptation of magnetic flux density extends the range of design solutions and allows for better use of powder magnetic material. The strength of such magnetic cores is ability to obtain complex shapes and shaping their physical properties. Presence of dielectric, which acts as a binding and insulating agent, reduces losses due to eddy currents. Thermal isotropy and good thermal conductivity improves dissipation of heat from external surface. Recycling of powder magnetic core of electric machinery is easier than electrical steel. Separation of iron powder from copper is easier in the case of powder magnetic core than in the case of sheet magnetic core. Soft magnetic powder for magnetic circuits should have the following properties: − − − − − − − −

highest possible saturation magnetization highest possible magnetic permeability highest possible resistivity best possible compressibility highest possible mechanic resistance of non-hardened form lowest possible coercive field strength lowest possible magnetic loss lowest possible price.

Since as of present there is no such powder that meets each and every requirement as specified above, designers of electric machinery with magnetic circuits have to decide which feature of given material is most critical to given solution, and match powder parameters to parameters of designed magnetic core at design stage. During first works on manufacture of soft magnetic elements with binding magnetic powder with plastic method powders were applied designated originally for sintering method. As of present leading manufacturers of powder materials manufacture soft magnetic materials designated for magnetic circuits manufactured with powder binding method. The process of manufacturing soft magnetic elements with powder binding method consists in pressing and then hardening the moulding. Physical properties of soft mag-

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B. Slusarek et al. / The Influence of Temperature on Mechanical Properties of Dielectromagnetics

netic element, such as magnetic, mechanic or electric properties, depend mostly on type of used powder, pressing pressure, and therefore density, mixture content, and hardening process parameters, such as temperature or hardening time. It is possible to shape their physical properties, as in the case of bound permanent magnets, through application of mixture of various powder and plastics. That allows proper matching of magnetic core and machine. Soft magnetic powder designated for manufacture of magnetic circuits are acronymized as SMC – Soft Magnetic Composites. They are iron powder isolated with one thin organic or inorganic layer or two. Such barrier reduces losses due to eddy currents. Obviously a decisive parameter for application of soft magnetic element is its magnetic properties. In many applications mechanic properties are equally important as magnetic parameters. In many electric machines magnetic circuit is also a structural element. Application area of appliances with powder magnetic circuits is continuously expanding. They are used in multitude of environmental conditions, often in high temperatures, often also in low temperatures. It is well-known that physical properties of materials change with the change in temperature. It is also the case with magnetic materials. Physical properties of materials as specified in catalogs refer to room temperature. Knowledge of mechanic properties of dielectromagnetics of iron powder designated for magnetic circuits of electric machinery shall allow their designers to consider changes in mechanic properties in temperature at the stage of design. That was the main goal of research [1–7].

Experiments The main goal of experiments is to find correlation between changes of mechanical properties with changes of temperature for dielectromagnetics. The changes of mechanical properties with temperature are shown on compressive, tensile and bending strength of dielectromagnetics. High purity iron powder Somaloy 500 was used in research. Iron powder grains were coated with 0.6% of lubricant and LB1 binder. All test specimens were executed at the same process parameters. Examination of dielectromagnetics samples of bending, compressive and tensile strength were performed using universal testing machine. Samples for bending strength tests were beam-shaped, dimensions 76 × 12 × 6 mm. Samples for compression strength tests were cylindrical in shape, 10mm in diameter, 14 mm in height. Samples for tensile strength tests were 90 mm long and have square cross-section of 5,6 × 5,6 mm and measuring length of 25 mm. They were prepared according with ISO 2740 standard. Compression strength, bending strength and tensile strength tests were performed at temperatures from –40oC to +100oC in a thermal chamber. Magnetic properties were measured on ring samples of inner diameter 45 mm, outer diameter 55mm and 5mm height according to PN EN-604040-6.

Results and Discussion Magnetic properties were measured and are shown on Fig. 1. Figure 1a) represents magnetization curve and Fig. 1b) dynamic amplitude magnetic permeability. Maximum dynamic amplitude magnetic permeability is 230 as Fig. 1 shows.

37

240 220 200 180 160 140 120 100 80 60 40 20 0

1,5 1,4 1,3 1,2 1,1 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0

μ r [-]

B m [T ]

B. Slusarek et al. / The Influence of Temperature on Mechanical Properties of Dielectromagnetics

0

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

Hm [A/m]

Hm [A/m]

Figure 1. a) Magnetization curve for Somaloy 500 + 0,6% LB1, f = 50 Hz b) Dynamic amplitude magnetic permeability for Somaloy 500 + 0,6% LB1, f = 50 Hz.

320 280 Rc, TRS, Rm [MPa]

240 200 160 120 80 40 0 -40

-30

-20

-10

0

10

20

30

40

50

60

70

80

90

100

Temperature [°C] Compression strength - Rc

Bending strength - TRS

Tensile strength Rm

Figure 2. Mechanic properties for Somaloy 500 + 0,6% LB1.

Tests of mechanic properties were performed on specimens of dielectromagnetics. Figure 2 contains results of mechanic properties surveys of tested specimens of dielectromagnetics. Figure 2 presents how temperature influence on mechanic properties such as compressive, tensile and bending strength. As the curve on Fig. 2 shows that compression strength of measured samples decrease with increase of temperature but in negative temperatures compression strength improves. The same phenomenon is observed for bending strength of samples, but the changes of properties are lower. Whereas the influence of temperature on tensile strength of samples of dielectromagnetics is inconsiderable. Figures 3a) and 3b) show for example compressive stress and bending stress in dependence of displacement of cross-bar of universal testing machine for dielectromagnetics at temperature –40°C. Summary As it can be concluded from data presented in Fig. 2 comparison of dielectromagnetics’ compression strength and bending strength proves that those parameters deteriorate

B. Slusarek et al. / The Influence of Temperature on Mechanical Properties of Dielectromagnetics

350

160

300

140

Bending stress [MPa]

Compressive stress [MPa]

38

250 200 150 100 50 0 0

0,2

0,4

0,6

0,8

1

1,2

Displacement [mm]

a)

1,4

1,6

1,8

120 100 80 60 40 20 0 0

0,1

0,2

0,3

0,4

0,5

Displacement [mm]

b)

Figure 3. a) The curve of dielectromagnet sample compression test at temperature –40°C, b) The curve of dielectromagnet sample bending test at temperature –40°C.

with increase in temperature. At the same time decrease in temperature below room temperature causes significant improvement of compression strength and bending strength. Changes of temperature causes not very significant changes in tensile strength of dielectromagnetics. Research proved that decrease in temperature also resulted in decrease in tensile strength. Research is in progress.

References [1] Kordecki A., “Dielektromagnetyki magnetowodów maszyn elektrycznych” Prace Naukowe Instytutu Układów Elektromaszynowych Politechniki Wrocławskiej Nr 44, Seria Monografie Nr 11, 1994 (in Polish). [2] Węgliński B., “Perspektywy rozwojowe w dziedzinie kompozytów proszkowych na magnetowody przetworników elektrycznych” Prace Naukowe Instytutu Układów Elektromaszynowych Politechniki Wrocławskiej Nr 24, Seria Monografie Nr 2, 1977 (in Polish). [3] Węgliński B., “Magnetycznie miękkie kompozyty proszkowe na osnowie żelaza” Prace Naukowe Instytutu Układów Elektromaszynowych Politechniki Wrocławskiej Nr 32, Seria Monografie Nr 5, 1981 (in Polish). [4] Ślusarek B., Długiewicz L., “Application of soft and hard magnetic powders in small electric machines”, Advanced in Powder Metallurgy & Particulate Materials – 2006, ISBN: 0-9762057-6-9, San Diego, June 2006. [5] Hultman L., Persson M., Engdahl P., “Soft magnetic composities for advanced machine design” PM Asia April 2005 Shanghai. [6] Dougan M. J., Torres Y., Mateo L., Llanes L., “The fatigue behaviour of soft magnetic composite powders” Euro PM 2004 Vienna. [7] Gelinas C., Viarouge P., Cros J., “Use of soft magnetic composite materials in industrial applications” Euro PM 2004 Vienna.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-39

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Influence of the Magnetic Anisotropy on Electrical Machines M. HERRANZ GRACIA and K. HAMEYER Institute of Electrical Machines, RWTH Aachen University, Schinkelstraße 4 D-52056 Germany [email protected] Abstract. Non grain-oriented electrical steel has an inherent anisotropy, which is normally neglected in the calculation of electrical machines. Moreover, the magnetic anisotropy is usually measured in small material samples. Due to the cutting effect, the magnetic anisotropy in the machine is not the same as in the sample. In this paper, the magnetic anisotropy is considered as a global problem. A method to measure it is presented and its influence on the electromagnetic and acoustic behavior is considered through the example of an induction motor.

Introduction Non grain-oriented electrical steel and hence electrical machines have an inherent anisotropy due to the variation of the magnetic properties in rolling and perpendicular to the rolling direction. Moreover, the cutting process has a different influence in these two directions. Therefore, the standard procedure of measurement of the anisotropy in small samples of the material [1] and not directly in the machine is not appropriate for a detailed study. The influence of the anisotropy in the magnetic losses of the machine has already been widely studied [2]. But the anisotropy acts also like an eccentricity with double periodicity and, therefore, generates field harmonics in the air gap, which have an influence on the torque and the radial force on the stator i.e. the acoustic behavior of the machine. These effects have been treated only briefly in the literature [2,3]. This paper presents a more global view on the problematic of the magnetic anisotropy. A test setup to measure the anisotropy directly on the stator of the machine is presented. Measurements are conducted in twenty stator samples of an induction motor to study the variability of the magnetic anisotropy. The influence in the electromagnetic and acoustic behavior of the machine is studied then analytically and by FE simulation. The aim of this paper is to predict better the parasitic effects in the machine (torque ripple and radial force on the stator teeth) and so increase the reliability of the machine. Test Setup A variation of the differential method presented in [4] is used here to measure the magnetic anisotropy of the stator of an induction motor. A 2-pole rotor is built in such a way that the magnetic flux passes the stator through the teeth with the same angle to the rolling direction of the iron lamination. The purpose of this test setup is to distin-

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M. Herranz Gracia and K. Hameyer / Influence of the Magnetic Anisotropy on Electrical Machines

(a)

(b)

(c) Figure 1. Rotor variants for the test setup: (a) Variant 1; (b) Variant 2; (c) Variant 3.

guish even small variation on the magnetic properties of the stator teeth. The magnetic anisotropy of the stator yoke is not going to be measured because the magnetic flux flows along 180° on it and, therefore, the magnetic properties for the different angles are averaged. The rotor outer radius is chosen to be equal to the one in the original machine and the rotor winding is dimensioned to generate an air gap flux density as in the original machine. Three different variants (see Fig. 1) for the rotor pole pitch were considered: •

•

•

Variant 1: A rotor pole pitch is equal to a stator tooth pitch. This variant has the advantage that it allows to measure each tooth separately but, as it can be observed on Figure 1, the saturation level on the rotor is so high that the sensibility of the measurement would be strong limited. Moreover, leakage flux would adulterate the results. Variant 2: A rotor pole pitch is equal to a stator pole pitch. This variant produces in the stator teeth a similar magnetization as in the original machine but it has two important drawbacks. Firstly, the yoke of the machine is strong saturated because the original machine has four and not two poles. Secondly, nine teeth would be measured at the same time, what difficults to measure small magnetic variations between the teeth. Variant 3: A rotor pole pitch is three times a stator tooth pitch. In this variant both the yoke of the stator and of the rotor remain unsaturated, so that the magnetization of the teeth is the most important parameter of the system. Three stator teeth are measured simultaneously but this resolution is considered to be sufficient for our study. For this reason this variant is chosen.

M. Herranz Gracia and K. Hameyer / Influence of the Magnetic Anisotropy on Electrical Machines

41

Figure 2. Measurement of the magnetic anisotropy in the space and frequency domain.

The rotor is rotated by a stepper motor at 0.025s–1 and the magnetic anisotropy is measured through the change of the current in the rotor winding, which is fed with a constant 50-Hz voltage. This frequency component is filtered on-line in order to acquire only the changes in the rms. value of the current. The measurement is performed for six voltage levels (5, 10, 15, 20, 25 and 30 V) to study the variation of the magnetic anisotropy at different saturation levels.

Measurements Figure 2 shows the measurement in space and frequency domain for a stator with 36 stator slot and 26 cutting notches. Both orders appear clearly in the measurements but they do not mask the 2nd order, which corresponds to the magnetic anisotropy. For α = 0° the rotor is aligned with the rolling direction, where the magnetic resistance is minimal. Then the inductivity of the rotor winding is at a minimum. Therefore, the current is at a maximum. It can be stated that this test setup is feasible to measure the magnetic anisotropy of the stator core. It is used now to compare the magnetic anisotropy of two different production series (A and B). Ten stators for each series have been measured (see Fig. 3). As comparison parameter, the normalized 2nd order of the current is chosen. Series B shows a near to constant magnetic anisotropy in all the stators of 1% rms. or a peak-to-peak value of 2.8%. This is the expected behavior for a production series. On the other hand, Series A shows a very high variability on the magnetic anisotropy. Some stators have values up to 2.8% rms. (7.9% peak-to-peak) and other ones doe not reach even the value of 0.3% rms. (0.8% peak-to-peak). This fact seems to indicate that either magnetic steel from different origins were used in the same production series or the tools used in the manufacturing process has produces much different cutting effects on the stators because of the erosion.

Effect on Parasitic Effects The measurement results have shown that the magnetic anisotropy can differ largely in a same machine. This fact should be considered for the prediction of the performance of the machine and its parasitic effects to ensure the reliability of the machine. The influence of the magnetic anisotropy on the losses of the machine has been treated ex-

42

M. Herranz Gracia and K. Hameyer / Influence of the Magnetic Anisotropy on Electrical Machines

Figure 3. Comparison of the anisotropy measurements of stator laminations of the series A and B.

tensively in the literature ([3]). Therefore, this work will concentrate in two other parasitic effects: torque ripple and radial forces on the stator teeth. These effects are going to be studied here analytically and through FE Simulation. Thanks to the fact that the studied machine has two pole pairs, it is possible to model the anisotropy only in the stator teeth. Moreover, it is assumed that there is a magnetic difference between teeth but not inside one of them, so that the magnetic anisotropy is modeled through a different magnetization curve for each tooth. This assumption simplifies hugely the simulation. The measurement has shown that the magnetic anisotropy of the stator can be between 0% and 10% (peak-to-peak). In this work, the torque ripple and the radial force in the stator teeth will be compared for simulations of the machine with values of the magnetic anisotropy from 0% to 25%. The magnetic anisotropy acts like a static eccentricity repeated twice on the air gap. The permeability function of the magnetic anisotropy can then be written as Λ ( x, t ) =

μ0 μ 1 , = 0 δ ( x, t ) δ ′′ 1 − Anis ⋅ cos(2 x − ϕ anis )

(1)

where δ″ is the equivalent air gap and Anis the peak-to-peak value of the magnetic anisotropy. This permeability function can be decomposed in its Fourier series as follows. Λ ( x, t ) = Λ 0 +

∑

λ =1,2...

Λ λ cos(λ (2 x − ϕλ ))

(2)

Regarding the first field harmonic, the air gap flux density due to the magnetic anisotropy is

M. Herranz Gracia and K. Hameyer / Influence of the Magnetic Anisotropy on Electrical Machines

43

Figure 4. Frequency spectrum of the torque ripple for different values of the magnetic anisotropy.

b p ± 2 = B p ± 2 cos[( p ± 2) x − (ϕ μ p − ϕλ =1 )],

(3)

where p is the number of pole pairs in the machine. This means, that the magnetic anisotropy generates an air gap flux density with the space harmonic p ± 2. Effect of the Magnetic Anisotropy on the Torque Ripple Figure 4 shows the frequency spectrum of the torque for different values of the magnetic anisotropy. Two different phenomenons can be observed: •

•

The 15th, 21st, 26th, 36th, 62nd and 73rd harmonic order appear with the magnetic anisotropy and the magnitude of them is proportional to the magnetic anisotropy. This new orders appear from the combination of the air gap flux density due to the magnetic anisotropy (Eq. (3)) and other flux density harmonics. For example, the 21st and the 26th order can be easily identified as the combination of the first harmonic of the rotor slots and the magnetic anisotropy. The mean value and the 47th, 49th and 57th order are maximal without magnetic anisotropy and they decrease proportional in magnitude with the magnetic anisotropy. The reason is that the worsening of the magnetic properties in the teeth not parallel to rolling direction acts as an enlargement of the air gap. Therefore, the main component of the magnetic field decreases and also all the orders of the torque ripple caused by it.

In the worst case, the torque mean value is 3% less as without magnetic anisotropy and the torque ripple 8% higher. Effect of the Magnetic Anisotropy on the Radial Forces on the Stator Teeth The electromagnetic radial forces on the stator teeth are the main responsible for the acoustic noise of electrical machines. As well as the magnitude, the frequency and the mode r of these forces are decisive. It has been proved [5] that low modes and specially r = 2 are the most critical ones. Therefore, the analysis of the radial forces on the stator teeth is not done in the time-space domain but after a 2-dimensional FFT in the modeorder domain. Figure 5 shows the radial force with the modes 0, 2, 4 and 6 for different values of the magnetic anisotropy.

44

M. Herranz Gracia and K. Hameyer / Influence of the Magnetic Anisotropy on Electrical Machines

Figure 5. Radial forces on the stator teeth for the modes 0, 2, 4 and 6.

As the torque ripple, the results show two different trends: •

The orders and modes, which are generated from the fundamental component of the field, are weakened with the anisotropy. An example of this is the 21st order with mode 2.

M. Herranz Gracia and K. Hameyer / Influence of the Magnetic Anisotropy on Electrical Machines

•

45

New harmonic orders and modes appear due to the magnetic anisotropy and its magnitude is increased along with the value of the magnetic anisotropy. The 12th and the 36th harmonic order with mode 2 are examples of this. These orders are expected to have a big influence on the acoustic behavior of the machine because mode 2 has mechanically the highest amplification coefficient.

An interesting tendency can be observed for different orders. There is a transfer of force from the original mode rorig to rorig – 2. For example for the 31st order, the electromagnetic force without anisotropy was much higher with mode 6 as with mode 4. As the anisotropy increases, the electromagnetic force with mode 4 increases and with mode 6 decreases. The origin of this transfer is the combination of the original radial force wave with the new component of the flux density caused by the magnetic anisotropy. As it can be seen in Eq. (3), this new wave is invariant in the time and is repeated twice in the space. Therefore, the combination of this wave with the original ones produces waves with the same order as the original waves but with mode ±2.

υmag _ anis = υorig

(4)

rmag _ anis = rorig ± 2

(5)

Conclusions This paper has shown that the magnetic anisotropy of non grain-oriented electrical steel has an influence in the electromagnetic and acoustic behavior of an induction machine, although this effect is neglected regularly. The torque ripple in the machine increases up to 8% due to the emergence of new harmonic orders. The influence of the magnetic anisotropy can be especially critical in the acoustic behavior because the magnetic anisotropy produces radial force waves with the same order as the machine without anisotropy but with smaller modes. Furthermore, a test procedure is presented to measure the magnetic anisotropy directly in the stator of the machine. Measurement has shown that the value of the magnetic anisotropy can vary from near 0 to 10% (peak-to-peak). This variability should be taken into account for the prediction of the parasitic effects i.e. for the reliability studies on the machine. References [1] T. Nakata et al., “Measurement of Magnetic Characteristic along Arbitrary Directions of Grain-Oriented Silicon Steel up to high Flux Densities”, IEEE Transactions on Magnetics, Vol. 29, No. 6, pp. 3544-3548, November 1993. [2] S. Urata, et al., “Magnetic Characteristic Analysis of the Motor Considering 2-D Vector Magnetic Property”, IEEE Transactions on Magnetics, Vol. 42, No. 4, pp. 615-618, April 2006. [3] B. Hribernik, “Influence of Cutting Strains and Magnetic Anisotropy of Electrical Steel on the Air Gap Flux Distribution of an Induction Motor”, Journal of Magnetism and Magnetic Materials, Vol. 41, pp. 427-430, 1984.

46

M. Herranz Gracia and K. Hameyer / Influence of the Magnetic Anisotropy on Electrical Machines

[4] W. Wilczynski, “Influence of Manufacturing Condition of Magnetic Cores on their Magnetic Properties”, Proc. of the Conf. on Soft Magnetic Materials, 20-22 April 1998. [5] C. Schlensok et al., “Structure-Dynamic Analysis of an Induction Machine Depending on Stator-Housing Coupling”, Proc. of the International Electric Machines and Drives Conference, IEMDC 2007, 3-5 May 2007.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-47

47

Analysis of Structural Deformation and Vibration of Electrical Steel Sheet by Using Magnetic Property of Magnetostriction Wataru KITAGAWA, Koji FUJIWARA, Yoshiyuki ISHIHARA and Toshiyuki TODAKA Department of Electric Engineering, Doshisha University 1-3, Tataramiyakodani, Kyotanabe, Kyoto, 610-0321, Japan telephone: +81-774-65-6327, e-mail: [email protected] Abstract. Recently, it is examined with many papers about magnetostriction of electrical steel sheet and magnetostriction of transformer model. In this paper, deformation and vibration of electrical steel sheet by magnetostriction was analyzed and measured. There results were compared and examined. As results, it was reported that natural mode was provoked to force of magnetostriction.

Introduction It is examined that observation of the magnetostriction and the vibration of the electrical steel sheet in some papers [1–4]. However, the displacement analysis of the deformation of the iron core disfigured by magnetostriction has rarely been calculated. The method of preventing the noise and the trouble by the resonance of the vibration is the most general to do the design that the natural frequency of the equipment doesn’t enter ranges of driving vibrations. However, the character frequency exists a lot according to the object equipment, and might not resonate even if the character frequency is a category of driving frequency [5,6]. So transformation by magnetostriction of the electromagnetic steel sheets is analyzed, and correlation with oscillation gets possible to be made clear by confirming form. In this paper, displacement of an electromagnetic steel sheet by magnetostriction was analyzed using structural analysis technique by finite element method and it was compared with measurement value, and the adequacy was inspected, and comparing by eigenvalue analysis and frequency response analysis examined it.

Analysis and Measurement Technique The analysis technique of deformation of electric steel sheet by the magnetostriction which used structural analysis technique by two dimensional finite element method is proposed in this paper. Basic equation of structural analysis is the following equation.

48

W. Kitagawa et al. / Analysis of Structural Deformation and Vibration of Electrical Steel Sheet

Ku = f

(1)

K: Rigidity matrix, u: Displacement vector, f: External force vectoring. The nodal force can be calculated from distortion using the following equation.

p = ∫ B T Dε 0 dS

(2)

S

B = { Bx B y 0} , D=

ε 0 = {ε 0 x ε 0 y 0}

T

0 ⎡1 −ν ν ⎤ E ⎢ ν 1 −ν ⎥ 0 ⎥ {(1 +ν )(1 − 2ν )} ⎢ ⎢⎣ 0 0 (1 − 2ν ) / 2 ⎥⎦

(3)

(4)

E : Young’s modulus㧘ν : Poisson’s ratio. p = [− | p1 y | − | p1x | + | p2 x | + | p2 y | − | p3 y | − | p3 x | + | p4 x | + | p4 y |]T

(5)

p: Nodal force of arbitrary element. The direction of p is established so that volume of constituent keeps uniformity. When magnetostriction is supposed in proportion to square of magnetic flux density, it is following equation.

ε 0 x = α x Bx2 ⎫⎪ ⎬ ε 0 y = α y By2 ⎪⎭

(6)

Figure 1(c’) is shown, the direction of the nodal force p is supposed that the inflation force works in case of the horizontal direction and the compressive force works in case of the perpendicular direction of the magnetic flux because the direction shown in Fig. 1(c) is expand the elements. It is shown by the following Eq. (7). p = [− | p1 y | − | p1x | + | p2 x | + | p2 y | − | p3 y | − | p3 x | + | p4 x | + | p4 y |]T

(7)

As the measurement of frequency response, the sample which it was hanged from ceiling exited using excitation frame like a infinity solenoid. The sample is bound with B coil, and acceleration pickup is glued together, and magnetic flux density and acceleration are detected. Let excitation frequency change with this state, and acceleration in 0.1 T uniformity is measured. As the sample, the thickness of no grain oriented magnetic steel sheet is 50 mm, iron loss is less than 13.00 W/Kg (JIS:50A1300) is used. Results Analysis result be shown Fig. 2, and eigenvalue and natural frequency exist. And it is vibrated by resonant frequency 1656 Hz of the magnetostriction that 2 times of excita-

W. Kitagawa et al. / Analysis of Structural Deformation and Vibration of Electrical Steel Sheet

49

Figure 1. Decision of direction of joint force.

(a) Mode 7 (2410.3 Hz)

(b) Mode 8 (3299.6 Hz)

(c) Mode 9 (3423.1 Hz)

(d) Mode10 (4422.9 Hz)

Figure 2. Natural Mode of Vibration (Analyzed).

Figure 3. Frequency Response (Analyzed).

Figure 4. Frequency Response (Measured).

tion frequency are fundamental wave in the mode 8 from frequency response shown with Fig. 3 is supposed. Natural frequency to measure from calculated consequence is supposed. Acceleration and excitation frequency response of sample are shown in Fig. 4. Measurement result shows that analysis result is agreed, and eigenvalue mode 8 is provoked by 1700 Hz, and it is thought that resonant is provoked. The shape by resonance are shown in Fig. 5.

50

W. Kitagawa et al. / Analysis of Structural Deformation and Vibration of Electrical Steel Sheet

(a) Change of the Sheet by magnetostriction (× 105)

(b) Natural Mode 8 (3299.6 Hz)

Figure 5. Shape by Resonance.

It is able to easily predict which frequency influences resonance and oscillation by structural analysis and natural vibration analysis and be useful for noise prediction of electric steel sheet.

Conclusion Measurement result accords with analysis result well, and eigenvalue mode 8 is guided by 1700 Hz, and it is thought that resonance event is caused. And, as the possible cause that peak of response exists in 3650 Hz, magnetostriction waveform does frequency component of 2 times of excitation frequency with fundamental wave, but shade includes component too of excitation frequency, and natural frequency of existing sample accords in this component and 3400 Hz vicinity, and it is supposed that resonance event produced it. Magnetostriction is converted into force, and structural analysis of electric steel sheets by magnetostriction is possible by structural analysis. It is able to easily predict which frequency influences resonance and oscillation by structural analysis and natural vibration analysis and be useful for noise prediction of electric steel sheets. References [1] A.J. Moses: “Measurement of Magnetostriction and Vibration with Regard to Transformer Noise”, IEEE Trans. Magn., Vol. MAG-10, No. 2, pp. 154-156 (1974). [2] K. Kuhara, S. Kawamura, Y. Hori, and M. Sasaki: “Vibration Analysis of Transformer Core”, 72nd Annual Conference of the Japan Society of Mechanical Engineers, Vol. IV, pp. 109-110 (1998) (in Japanese). [3] Y. Hori, S. Abe, M. Sasaki, and K. Kuhara: “Vibration characteristics of transformer”, The Papers of Technical Meeting on Magnetics, IEE Japan, MAG-99-79, pp. 17-22 (1999) (in Japanese). [4] M. Imamura, T. Sasaki, and H. Nisimura: “AC magnetostriction in Si-Fe single crystals close to (110)”, IEEE Trans. Magnetics., Vol. MAG-19, No. 1, pp. 20-27 (1983). [5] T. Sasaki, S. Takada, S. Saeki, F. Ishibashi, and S. Noda: “Magnetostriction of Erectromagnetic Steel Sheets under AC Magnetization Superimposed with Higher Harmonics”, T. IEEE Japan, Vol. 112-A, No. 6, pp. 539-544 (1992-6) (in Japanese). [6] T. Sasaki, T. Takada, F. Ishibashi, I. Suzuki, S. Noda, and M. Imamura: “Magnetostrictive vibration of electrical steel sheet under non-sinusoidal magnetizing condition”, IEEE Trans. Magn., Vol. MAG-23, No. 5, pp. 3077-3079 (1987).

Chapter A. Fundamental Problems and Methods A2. Methods

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-53

53

Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction J. TUROWSKI a, Xose M. LOPEZ-FERNANDEZ b, A. SOTO RODRIGUEZ c and D. SOUTO REVENGA c a Dept. of Intelligent Information Systems, WSHE – Lodz Poland, consultant for University of Vigo Spain [email protected] b Dep. of Electrical Engineering, University of Vigo – Spain [email protected] c Earlier Socrates Diploma Students from Vigo University in the Technical University of Lodz, Poland; now Engineers in EFACEC Transformer Works, Porto, Portugal Abstract. Rapid design is one of the main imperatives of modern manufacturing, followed from principles of mechatronics [1], and is handy tool of regular optimization of structure. In this work it is presented such rapid design method for specific class of power transformers with sandwich windings. To accelerate the design process an expert system and rapid interactive procedure was applied. At such approach the more scientific and design experience is located into the Knowledge Base of the Expert System, the more, rapid, easier and cheaper is a regular design and optimization of a machine. Thanks to analytical preparation, approximation and linearization coefficients the programming and calculation is discharged from cumbersome iteration and other formal disturbances. Hybrid analyticallyreluctance Network Method three-dimensional RNM−3D [2] has proved here as one of the best for rapid design of such complex structures like modern transformers with extreme electromagnetic filed concentration, its crushing forces, eddy current loss and overheating hazard.

Calculation Model for Rapid Design Rapid design, time to market, innovativeness belong to the main imperatives of modern industry, based on principles of mechatronics [1]. The application of expert system of design is one of the ways to reach these aims (Fig. 1). At such system of programming, the more scientific and design experience is located into the Knowledge Base, the more, rapid, easier and cheaper is a regular design and optimization of a machine. For the solution of problem specified in the title, into the Knowledge Base it was introduced the same scientifically proved analytical formulae, approximation and linearization like in the book [2]. Namely: 1) Basic Turowski’s formula for power losses (W) in solid and/or screened steel walls: ⎡ 2 2 ⎢ pe ∫∫ H ms dAe + pm ∫∫ H ms dAm + a p ∫∫ μ rs H ms Am ASt ⎣⎢ Ae 2 = I (a + bI)

ΔP =

1 2

ωμ0 2γ

2x

⎤ dASt ⎥ = ⎦⎥

(1)

54

J. Turowski et al. / Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction

Figure 1. Block diagram of expert system type (a) design of a machine: 1 – Large portion of introduced knowledge and experience – simple, inexpensive and rapid solution, e.g. 1 sec; 2 – small portion of knowledge and experience – difficult, expensive and labour-consuming solution, e.g. 3 months (see J. Turowski [1]).

where pe << 1, pm << 1 are screening coefficients ([5] p. 200 and 198); Ae, Am, ASt – surfaces covered with corresponding (e-electromagnetic, m-magnetic) screens or not screened (St-steel), Hms – magnetic field strength on a metal surface. 2) Magnetic nonlinearity μ(H) inside solid iron, for field Hms stronger than 5 A/cm, was considered with the help of average linearization coefficients ap ≈ 1,4 for active power and aq ≈ 0,85 for reactive power. 3) Magnetic nonlinearity μ(H) along the steel wall surface was considered with the help of analytical approximation μ r H 2 ≈ cHb and corresponding exponent coefficient x in (1). The value of x varies for different transformers, but is typically between 1,1 and 1,14 ([5], p. 345). 4) Eddy current reaction of solid metal wall has been taken into account with the help of complex reluctances: 2 ωσ – for solid steel or R ≅ α 2 sinhα 2 d ≅ – Rμ1 + jRμ1b = (0,37 + j 0,61) Cu μ ( coshα 2 d − 1) μ 0 d μs when Cu or Al electromagnetic screens are applied. In a first approach one can adopt RCu → ∞. 5) For reluctance of laminated magnetic screens (shunts), in comparison with dielectric and solid metal elements, in a first approach one can adopt value RFe ≅ 0. 6) Excessive heating hazard follows from (1) where value Hms on the steel surface is responsible for the loss density and therefore for a local heating. From a thermal equilibrium equation we can find a permissible tangential field component on the steel surface: Hms,perm = 1962( 1 + 3, 29 ⋅ 10−8 c − 1 )

(2)

for permissible temperature tperm, where c= f(σ,ω,tperm). Over the Hms,perm value excessive local heating hazard (Hot-Spot) of solid steel elements due to induced eddycurrents can appear. Eq. (2) was the basis of electromagnetic overheating criteria pro-

J. Turowski et al. / Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction

a)

b)

55

c)

Figure 2. Cross section of transformers windings: a) Core type, cylindrical winding φσm – maximum stray flux, Bm(x) – stray flux density distribution, lav/π – average winding diameter, I1N1, I2N2 – ampere-turns of HV and LV windings. b) Stray flux density distribution in sandwich winding ([7] p.127). c) Sandwich winding of furnace transformer 16 MVA, 35 kV/ 384, 342, 225, 243 V at different tap connection. At connection 7-6 – Delta, at 7-8 – Y ([8] p. 101).

posed at 1972 CIGRE Plenary Session [4] and used until now for possible Hot-Spot localisation, also in packages RNM-3D [3] for intelligent localisation of hot-spot. Design of adequate calculation model and selection of proper calculation method also belong to the crucial components of the Knowledge Base, which decide on the success. Physics of it is based on principals of technical application of Maxwell’s electromagnetic theory [5].

Transformers with Sandwich Windings After testing different available calculation programs, the hybrid, three-dimensional, analytically-numerical software of the RNM−3D class [2] has been selected. It has proved [6] as one of the best and popular tools for rapid design, broadly used (J. Turowski [3], tab. V) in most of transformer works, world over. In most popular cases of three-phase transformers RNM-3D calculate stray field and losses in CPU time shorter than 1 second, whereas FEM-3D – from 30 to 300 hours [1,3] and is very complicated in regular industrial design. However, the RNM−3D is devoted only to common transformers with cylindrical windings (Fig. 2a). At the same time exist an important group of transformers with sandwich windings (Fig. 2b,c), which needs similar rapid design method for the 3D leakage field region. Main objective of this is such design of winding coils and tank screening in order that to reduce of short circuit destruction forces, eddy current concentrations as well as stray loss, and excessive local heating hazard. Such rapid design programs are especially important in present power system reliability problems and acute market competition, when one needs to design new version of construction and examine its reliability. It reveals particularly, when special transformers are needed with a specific parameters and unique construction, like furnace, low impedance and other “tailor made” solutions.

56

J. Turowski et al. / Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction

a)

b)

Figure 3. Model of single-phase sandwich transformer: a) Design drawing, b) RNM model.

For this task new package “RNM-3Dsandw” was developed, which is basis for such regular rapid design. Theoretical basis is the same like in the equivalent reluctance method RNM-3D [2,3] i.e., where in linear region elementary reluctances were calculated from the geometry Ri =

li and in metal parts – from the Maxwell equations: μ osi

∇ x Hm = σEm and ∇ x Em = – jωμ Hm and complex Poynting’s Vector S = P1 + jQ1 in [VA/m2] considering iron linearization coefficients, eddy current reaction, electromagnetic wave interference inside electromagnetic screens, etc. Only model investigated was different (Fig. 3), as element of total cross-section from Fig. 2b. For power loss calculation the formula (1) was used. Hence from (1) thanks to model symmetry total loss in tank Ptotal = 8 Ptotal(1/8), 2 ⎧⎪ ⎛ B ⎞ ⎛ A⎞ ⎛ B Ptotal (1/ 8) = Pυ a 0 ⎨ ⎜ ⎟ arctg ⎜ ⎟ + ⎜ 2 2 ⎝B⎠ ⎝ A +B ⎩⎪ ⎝ 2 ⎠

⎛B⎞ ⎛ c ⎞⎤ ⎫ − arctg ⎜ ⎟ − arctg ⎜ ⎟⎥ ⎬ ⎝ A⎠ ⎝ A ⎠⎦ ⎭

⎞⎛ A ⎞ ⎡ ⎛B⎞ ⎟ ⎜ ⎟ ⎢ arctg ⎜ ⎟ − 2 ⎝ A⎠ ⎠⎝ ⎠ ⎣

(3)

where A, B, c are geometrical parameters from the model in Fig. 3a. The package of “RNM-3Dsandw” (Fig. 3b) has the same structure like in RNM-3D. Calculation (Fig. 4) is partly automatised like in RNM-2Dexe. Results As a result it was obtained new rapid semiautomatic tool of regular analysis and design in few seconds per one design variant of: Graphs of magnetic field Hms (x,y) – Fig.5,

J. Turowski et al. / Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction

57

Figure 4. Developed interactive design structure of RNM-3Dsandw with Java Solver.

Figure 5. Distribution of Hms on the tank surface.

Figure 6. Power Losses P1 on the tank Surface.

loss density P1 (x, y) – Fig. 6 and hot-spots if Hms > Hms,perm (2) on the tank surface; Graphs of magnetic flux density Bm (x,y) for calculation eddy-loss and forces in windings. Total loss P = ∫∫A P1(x,y,z = 0) dx dy on the whole surface A of the tank for screening or shunting optimisation, considering cost 3000 to 10 000 US$/kW of capitalized power losses.

References [1] J. Turowski, Mechatronics Impact upon Electrical Machines and Drives. Proceedings of Internat. Aegean Conference on Electric Machines and Power Electronics – ACEMP’04. May 26-28, 2004. Istanbul, Turkey, pp. 65-70. Invited plenary lecture. [2] J. Turowski, Reluctance Networks. Chapter 4, pp.145-178 in the book “Computational Magnetics”. Chapman & Hall. London 1990, editor J. Sykulski (Extended translation from Polish: J. Turowski editor, “Ossolineum”, Wroclaw, 1990). [3] J. Turowski, “Stray Losses, Screening, and Local Excessive Heating Hazard in Large Power Transformers”. Chapter in CD book “Transformers in Practice”, ISBN: 978-84-609-7515-9, © 2006 Xose M. Lopez-Fernandez. [4] M. Kozlowski, J. Turowski, Stray losses and local overheating hazard in Transformers. CIGRE. Paris 1972. Rep.12-10. [5] J. Turowski, Elektrodynamika Techniczna. Warszawa, WNT 1993. [6] J. Turowski, I. Kraj, K. Kulasek, Industrial Verification of Rapid Design Methods in Power Transformers. International Conference Transformer’01, 5-6.09.2001, Bydgoszcz, Poland. [7] E. Jezierski, Transformers. Theoretical bases (in Polish), WNT Warszawa 1965. [8] E. Jezierski et al., Transformers. Construction and design (in Polish), WNT Warszawa 1963.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-58

Application of Logarithmic Potential to Electromagnetic Field Calculation in Convex Bars Stanisław APANASEWICZ Technical University in Rzeszów, Chair of Electrodynamics and Electrical Machinery Systems

Description of method of calculation of electromagnetic field induced by sinusoidal alternating current (or direct current) flowing in the infinitely long cylinder with convex intersection is aim of this paper. Potential of simple layer and Newtonian potential are applied. Adequate integral equations were introduced for calculation of integrands in these potentials. In the event of alternating current impedance boundary condition is considered.

List of More Significant Designations Used (s , τ) – curvilinear co-ordinate system on the cross-section of bar; s – length of the arc, τ – perpendicular to the L line, s0 – length of the L curve. L : x = x0 ( s ), y = y0 ( s ) – parametric description of the L line; x0′2 + y0′2 = 1 R12 ( x, y, s) = ( x − x0 ( s )) 2 + ( y − y0 ( s)) 2 R22 ( x, y, x′, y ′) = ( x − x ′) 2 + ( y − y ′) 2   R 0 = ( x0 (t ) − x0 ( s), y0 (t ) − y0 ( s) ) , R0 = R 0   S = ( x0′ , y0′ ) , N = ( y0′ , − x0′ ) – vectors: tangential and normal (in the point (x0, y0)) to the L line.   R 0 ⋅ N ( x0 (t ) − x0 ( s) ) y0′ (t ) − ( y0 (t ) − y0 ( s ) ) x0′ (t ) = W0 ( s, t ) = 2 2 R02 ( x0 (t ) − x0 (s) ) + ( y0 (t ) − y0 ( s) )

Calculation of Electromagnetic Field in the Case of Sinusoidal Current We take the following assumptions to solve depicted problem: a) b)

Depth of field penetration in the metal is small in comparison with radius of curvature of the bar surface. Form of the bar cross-section is symmetrical in relation to the x axis (please see the explanatory drawing enclosed)

S. Apanasewicz / Application of Logarithmic Potential to Electromagnetic Field Calculation

c)

59

 Vector potential (complex amplitude) A = (0, 0, A) has only one component A = A( x, y ) in Cartesian coordinates.

This function fulfils Laplace’s equation or Helmholtz equation: ⎧α 2 A , In the metallic area ΔA = ⎨ ⎩= 0 , outside of it

(1)

First assumption causes that application of simplified boundary condition called impedance condition is possible. So, in the metallic area, one can introduce curvilinear coordinate (s, τ); locally, function A fulfils equation like in the case of Cartesian coordinates: ΔA = Ass + Aττ = α 2 A

(2)

In this equation, similarly like in the skin area (in the boundary layer), derivatives ∂ ∂2 ∂ ∂2 , are small in comparison with derivatives , . 2 ∂s ∂s ∂τ ∂τ 2 So, Eq. (2) reduces itself and we have: Aττ = α 2 A

(3)

Solution of this equation has the following form: A = A* ( s)e −ατ

(4)

Tangential component Bs of magnetic induction and magnetic field strength Hs have the following form: Bs = −α A* ( s )e −ατ ,

Hs = −

α A* ( s)e −ατ μ

(5)

B( s,τ )

τ

Ω

s x

0

A(0,τ )

L Figure 1. Diagram of the studied system.

60

S. Apanasewicz / Application of Logarithmic Potential to Electromagnetic Field Calculation

on the bar surface τ = 0 magnitudes Hs and A* are continuous, so at the side of air the following condition must be fulfilled: Hs = −

s0

α A, μ

∫ H ds = i s

(6)

0

0

Continuity of function A ensures continuity of tangential component of electric field. Solution of Laplace equation (1) is sought in the form of logarithmic potential of simple layer. s0

A( x, y ) = ∫ g ( s) ln 0

1 ds R1

(7)

We determine unknown integrand g = g(s) from the boundary condition (6).  Known function A enables calculation of magnetic induction vector B =  ( B1 , B2 , 0) = rot A : s0

B1 = Ay = − ∫ g ( s ) 0

y − y0 ( s ) ds , R12

s0

B2 = − Ax = ∫ g ( s ) 0

x − x0 ( s ) ds R12

(8)

On the bar surface in the point (x0(t), y0(t)) normal component Bn and tangential component Bs of magnetic induction can be calculated in accordance with the following formulas:  ∂A Bs = B1 x0′ + B2 y0′ = Ay x0′ − Ax y0′ = −grad A ⋅ N = − , ∂n

Hs = −

1 ∂A μ ∂n

 Bn = B1 y0′ − B2 x0′ = Ay y0′ + Ax x0′ = grad A ⋅ S

(9)

(10)

As a result of that, the boundary condition (6) achieves the following form: s

0 ∂A α ∂ α 1 = A ≡ ∫ g ( s ) ln ds = ∂n μ r ∂n R0 μr 0

s0

1

0

0

∫ g (s) ln R

ds

∂ 1 ln in accordance with known characteristic of a potential ∂n R0 of simple layer is determined by the following formula:

Normal derivative

  s s0 ∂A 0 ∂ 1 R0 ⋅ N = ∫ g ( s ) ln ds = ∫ g ( s ) ds − π g (t ) ∂n 0 ∂n R0 R02 0

As a result of that, the boundary condition (6) achieves the following form:

(11)

S. Apanasewicz / Application of Logarithmic Potential to Electromagnetic Field Calculation

61

s0

∫ g (s)W (s, t )ds = π g (t )

(12)

0

W ( s, t ) = W0 ( s, t ) −

1 α ln μ r R0

g integrand is a periodical function with s0 period and is on the strength of assumption even in relation to s. We look for it in the form of the Fourier series. ∞

g ( s ) = g0 + ∑ g n cos n =1

2nπ s s0

(13)

At the same time, free term g0 is known; namely: in the large distance from the bar A function from the formula (7) has the following form s

A( x, y ) = A0 ( x, y ) =

s

0 0 i0 μ0 1 1 1 = g ds = g ( s)ds ln ln 2 ln 2 ∫ ∫ x +y R R 4π 0 0

therefore: s0

i0 μ0 = g 0 s0 , 2π

∫ gds = 0

g0 =

i0 μ 0 2π s0

(14)

We obtain from Eqs (12) and (13): s

∞ 1 0⎡ 2nπ ⎢ g 0 + ∑ g n cos ∫ π 0⎣ s0 n =1

∞ ∞ ⎤ 2k π s ⎥ W ( s, t )dt = g0 + ∑ g k cos t = g0W0 (t ) + ∑ g nWn (t ) s0 n =1 n =1 ⎦ (15)

s

Wn =

1 0 2nπ W ( s, t ) cos sds ∫ π 0 s0

From Eqs (15) we obtain the final system of equations for determination of gn coefficients: g0 =

1 s0

∞ ⎡ ⎤ g W g nWn ,0 ⎥ + ∑ 0 0,0 ⎢ n =1 ⎣ ⎦

gk =

2 s0

∞ ⎡ ⎤ g W g nWn, k ⎥ + ∑ 0 0, k ⎢ n =1 ⎣ ⎦

s0

Wn , k = ∫ Wn (t ) cos 0

2π k tdt = s0

s0 s0

∫ ∫ W (s, t ) cos 0 0

(16) 2π k 2π n t cos s ds dt s0 s0

62

S. Apanasewicz / Application of Logarithmic Potential to Electromagnetic Field Calculation

In order to carry out numerical calculations, we reduce existing infinite series to L first terms and obtain system of algebraic equations: ⎧ ⎡ W0,0 ⎤ 1 L ⎪ g 0 ⎢1 − ⎥ = ∑ g nWn ,0 s0 ⎦ s0 n =1 ⎪ ⎣ ⎨ L −1 ⎪g = 2 ⎡g W + g W ⎤ , n n, k ⎥ ⎪ k s ⎢ 0 0, k ∑ n =1 ⎦ 0 ⎣ ⎩

(17) k = 1, 2,..., L − 1

Calculation of Electromagnetic Field in the Case of Direct Current If direct current i = i0 = const flows in the bar, component of A potential will fulfil Poisson and Laplace’s equation (instead of Eq. (1)): μ i0 ⎧ , in the metallic area ⎪− μ J = − ΔA = ⎨ S , outside of it ⎪⎩0

(18)

Solution of the above equation can be accepted in the form of two logarithmic potentials (potential of simple layer and Newtonian potential): A=

μJ 2π

1

∫∫ ln R Ω

2

dx′dy ′ + ∫ g ( s ) ln L

1 ds R1

(19)

In accordance with properties of logarithmic potentials, A function in accordance with (18) is continuous function on the whole plane (x, y). At the same time, all derivatives of the first term are continuous too; however normal derivative of the second term is discontinuous on the L line (formula (11)). So, normal component of induction is continuous on the L line. One should select function g(s) in (19) in such a way to achieve the continuous tangential component of the magnetic field strength Hs. on the L. We may look for g function in the form of Fourier series (13). g0 constant can be determined analogically as in the previous case (formula (14)). Namely, we have for great values (x, y): ⎡μJ ⎤ 1 1 ⎛ μi ⎞ A( x, y ) = ⎢ dx′dy ′ + ∫ g ( s )ds ⎥ ln = ⎜ 0 + g 0 ⎟ ln ∫∫ 2 2 2 ⎝ 2π ⎠ x +y x + y2 L ⎣ 2π Ω ⎦ iμ 1 = 0 0 ln 2 2π x + y2

hence: g0 =

i0 μ 0 (1 − μ r ) 2π

(20)

S. Apanasewicz / Application of Logarithmic Potential to Electromagnetic Field Calculation

63

On the basis of formula (19) and use of described characteristics of potentials on the line L, we will obtain the condition of continuity of tangential component of magnetic field on the line L. This condition is expressed in the following way: ⎛ 1 ⎞⎡μJ ⎜ − 1⎟ ⎢ ⎝ μr ⎠ ⎣ 2π

⎤

⎛ 1

∫∫ W dx′dy′ + ∫ g (s)W (s, t )ds ⎥⎦ = − ⎜⎝ μ ∗

0

L

Ω

r

⎞ + 1⎟ π g (t ) ⎠

(21)

Equality (21) after taking into consideration (13) can be brought to the following form: μr − 1 ( μ r + 1)π

∞ ∞ 2nπ ⎡ ⎤ W t g u t g u t g g n cos ( ) + ( ) + ( ) = + ∑ ∑ k k 0 0 0, 0 ∗∗ ⎢ ⎥ s0 k =1 n =1 ⎣ ⎦

(22)

s

0 2k π μJ ′ ′ W dx dy , = u ( t ) W0 ( s, t ) cos s ds . ∗ 0, k ∫∫ ∫ s0 2π Ω 0 From equality (22) we obtain final system for determination of unknowns gk:

where W∗∗ (t ) =

∞ ⎡ ⎤ g 0 (1 − pu0,0 ) = p ⎢ h0 + ∑ g k vk ,0 ⎥ k =1 ⎣ ⎦

(23)

∞ ⎡ ⎤ g n = 2 p ⎢ hn + g 0 u0, n + ∑ g k vk , n ⎥ k =1 ⎣ ⎦

where: p=

μr − 1 , s0π (1 + μ r )

s0

hn = ∫ W∗∗ (t ) cos 0

2nπ t dt , s0

s0

vk , n = ∫ u0, k (t ) cos 0

2nπ t dt . s0

System (23) is equivalent of system (16). On the basis of systems (21) and (22) one can state that function g(s) ≡ 0 for μr = 1. It means that in this case Newtonian logarithmic potential (first term in the expression (19)) is a solution of the whole problem.

64

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-64

Multi-Frequency Sensitivity Analysis of 3D Models Utilizing Impedance Boundary Condition with Scalar Magnetic Potential Konstanty Marek GAWRYLCZYK and Piotr PUTEK Szczecin University of Technology, 70-310 Szczecin, Sikorskiego 37, E-mail: [email protected], [email protected] Abstract. In this work the inverse problem solution with iterative Gauss-Newton algorithm and Truncated Singular Value Decomposition (TSVD) is shown. For the goal function a norm l2 was chosen. To solve the inverse problem, which consists of the identification of conductivity distribution in a 3D model, the multifrequency sensitivity analysis has been applied. The correctness of sensitivity calculation has been proved utilizing three different methods, namely Tellegen’s method of adjoint model, differentiation of stiffness and mass matrix, as well as sensitivity approximation by means of difference quotient. Regarding the effectiveness of those methods, the first one is preferred because of shortest computational time.

Introduction The objective of this work is to solve the inverse problem of crack shape recognition arising in eddy-current method of non-destructive testing. For this purpose, the GaussNewton algorithm (GN) with TSVD (Truncated Singular Value Decomposition) based on sensitivity information obtained with the finite element method was applied. The inverse job itself consists in iterative optimization of pre-defined goal function. While this function depends on measured field values, the goal is to find such material parameters distribution, that result in numerical simulation converge with these of measurement. To carry-out the iterative optimization one needs to calculate the Hessian matrix and the gradient of goal function. This may be obtained with sensitivity analysis of electromagnetic field versus material parameters.

Eddy Current Nondestructive Testing Testing with eddy-currents belongs to non-destructive and contact-less quantity methods. Most important is the recognition of crack shape in conducting materials on the surface or inside the material. For example, they may find application in the recognition of crack shape either in conducting or ferromagnetic materials. At first the conception of method relies on placing the tested, electrically conducting object on an area of variable electromagnetic field, and then processing the obtained in this way information, which is included in measured signal and on model

K.M. Gawrylczyk and P. Putek / Multi-Frequency Sensitivity Analysis of 3D Models

65

Figure 1. Absolute probe above conducting plate with search area.

Figure 2. Computation model.

structure. The sinusoidal current at considerably different frequency is field source in case of multi-frequency method. The authors propose solving the above-mentioned problem in an iterative way using the gradient method. However, the information on gradient provides sensitivity analysis. The Considered Test Problem In the case of 3D problems the squared model of absolute probe was applied. The size of the used detector was shown in the Fig. 1. The model of the analyzed object, after providing the spatial discretization using first order tetrahedral finite elements, was presented in Fig. 2. The lift-off parameter was equal to lf = 1 mm. The interaction between the detector and the conducting area was simulated by means of SIBC (Surface Impedance Boundary Condition) [5].

66

K.M. Gawrylczyk and P. Putek / Multi-Frequency Sensitivity Analysis of 3D Models

Figure 3. Model of coil winding.

Figure 4. Distribution of T0z (x,y).

For calculating the field distribution in the considered model, the quantity of coil flow equal to Θ = n ⋅ I = 1 mA was assumed. Described model of probe took a possibility of applying just one component of electrical vector potential T, what one may define such as [2] ⎧ −5⋅106 x +1⋅106 [A/m] in area S x ×( 0, b ) ⎪ 1z T0 ( x, y ) = ⎨ −5⋅106 y +1⋅106 [A/m] in area S y ×( 0, b ) ⎪ 1⋅106 [A/m] in area S0 ×( 0, b ) , ⎩ ⎧ ∂T0z 6 2 ⎪ 1x ∂y = −5⋅10 [A/m ] in area S x ×( 0, b ) ⎪ ∂T ⎪ J S = ⎨ −1y 0z = 5⋅106 [A/m 2 ] in area S y ×( 0, b ) ∂x ⎪ 0 [A/m 2 ] in area S0 ×( 0, b ) , ⎪ ⎪ ⎩

(1)

where symbols such as Sx × (0,b), Sy × (0,b), S0 × (0,b) accruing in Eq. (1) were explained in Fig. 3 and Fig. 4. According to the assumed symmetry in the analyzed model of used nondestructive testing system in practical computation, only half of the space of the model was concerned, for which the symmetry plane is x0z. For simulation of the measurement process, as well as the reconstruction of thee analyzed conducting plane, the same mesh was used. Hence, the applied mesh consisted of NE = 4800 finite elements, ND = 9471 nodes, and a band of mass and stiffness matrix was equal to SB = 360. The calculation was provided on a computer, which was fitted out by processor Intel Centrino 1.86 GHz, and memory stick RAM 512 MB. The time of a single calculation for a fixed pick-up position averaged about 50 s. The score of simulated measurement process, during which the absolute probe was moved along the x axis with the size of movements Δk = 0.5 mm for 25 number of

K.M. Gawrylczyk and P. Putek / Multi-Frequency Sensitivity Analysis of 3D Models

Figure 5. Compensated voltage U0 for different positions of absolute probe.

Figure 7. Voltage U0 in case of noise level –20 [dB].

67

Figure 6. Voltage U0 in case of noise level –30 [dB].

Figure 8. Voltage U0 in case of noise level –15 [dB].

movements and 5 – number of harmonic signals from 75 kHz to 200 kHz with varied level of noise equal –30, –20, –15 [dB]1 was presented in Fig. 5, Fig. 6, Fig. 7 and Fig. 8. Measurement voltage using absolute probe Up includes information on defect U0, but also the information without defect Un. Mathematically, one could describe it as U0 = UP − Un

(2)

According to the supposition in flaw’s presence on analyzed model, the measurement signal’s amplitude was bigger for the higher frequency of excitation current. The vector of the measurement voltage consisted of 150 complex type samples corresponding to the position of the absolute probe for the assumed spectrum of frequencies.

1

To add the noise level the function available in toolbox Matlab™ Communication was applied.

68

K.M. Gawrylczyk and P. Putek / Multi-Frequency Sensitivity Analysis of 3D Models

Adjoint Model in Tellegen’s Method For the used field problem formulation by means of coupled potentials T – φ, derivation of the adjoint model from the Lorenz principle was done in [1,2]. The final score in the form of sensitivity equation is given by [6]

∫J

∫

S 2 ∂E1dV

= E1 E2 ∂γ dV ,

V

(3)

V

with appropriate assumption of boundary conditions (Neumann and Dirichlet) to the aim of vanishing of integrals

∫ ( H

2 ×∂E1

+ E2 ×∂H1 )1n dS = 0,

(4)

S

where Js2 is the current density in the original model, E1, E2, mean electric intensity vectors in both models, original and adjoint, respectively. Moreover, using distribution of delta function, the left hand side of Eq. 4 can be rearranged to the form

∫J

S 2 ∂E1dV

=

V

∫ ∫ δ( x − x , y − y , z − z )∂E dS 0

0

0

1

p dl

L Sp

∫

= ∂Edl = ∂U , L

(5)

whereas the right hand side using classical conception of SIBC can be described as

∫ E E ∂γ dV = −∂γ jωμγ ∫ H H dV . 1

2

1

V

2

(6)

V

Hence, Eq. 6 after simple rearranging allows one to calculate sensitivity from a more useful form ∂U α 2 ⎛ ∂ϕ1 ∂ϕ 2 ∂ϕ1 ∂ϕ2 ⎞ −2α ( z − z0 ) = + dV ⎟e 2 ⎜ e ∂y ∂y ⎠ ∂γ γ e V ⎝ ∂x ∂x

∫

( )

= = α=

α2

(γ )

e 2

∫

α

( )

2 γ

∞

⎛ ∂ϕ1 ∂ϕ2 ∂ϕ1 ∂ϕ2 ⎞ e −2α ( z − z0 ) + dz , ⎜ ⎟ dS e ∂x ∂x ∂y ∂y ⎠ z0 Se ⎝

e 2

∫

(7)

⎛ ∂ϕ1 ∂ϕ2 ∂ϕ1 ∂ϕ2 ⎞ e + ⎟ dS , ∂x ∂y ∂y ⎠ Se

∫ ⎜⎝ ∂x

jωμγ e = (1 + j) / δ e , δ e = 2 / ωμγ e , e = 1, 2,...E ,

where H is the magnetic intensity vector, inside conducting plate defined as ∂ϕ ⎛ ∂ϕ H ( z )1,2 = ⎜ 1x 1,2 − 1y 1,2 ∂x ∂y ⎝

⎞ −α z ⎟e , ⎠

(8)

and φ1,2 means scalar magnetic potential, γe is conductivity in considered finite element of search area.

K.M. Gawrylczyk and P. Putek / Multi-Frequency Sensitivity Analysis of 3D Models

69

Gauss-Newton Algorithm with TSVD The way of calculation of induced voltage sensitivity versus conductivity in every finite element of search area decides on numerical efficiency of the proposed identification algorithm. From this point of view, the described Tellegen’s method of adjoint model is favourable because of the possibility to calculate in one cycle the voltage sensitivity versus all conductivities. This information is essential to the implementation of the Gauss-Newton’s optimization method. Sensitivity information is a crucial component of a goal function gradient. The goal function in that case is in the classical norm l2. The Sensitivity Analysis for the Specified Goal Function Definition of the goal function is essential for the optimization process due to the use of the deterministic gradient method. In the presented algorithm the following goal function has been assumed using a vector of complex referenced voltages: FG1( U (ξ )) =

1 2

∑ (U(ξ ) j

i

− Uoi )(U (ξ )i − Uoi )*

(9)

where: U (ξ ) j is the jth component of voltage modeled by algorithm, Uo j means the referenced voltage for the jth position of measurement coil, ξi relates to the optimized quantity (the conductivity of the ith finite element). First, integrating voltage sensitivity versus each specified parameter ξi (i = 1…I) on area of the measurement coil for every position (j = 1…J), and then determining quantity sij according to formula (7) one can obtain: ⎡ ΔU1 ⎤ ⎡ s11 ⎢ ⎥ ⎢ ⎢ ΔU 2 ⎥ = ⎢ s21 ⎢ ... ⎥ ⎢ ... ⎢ ⎥ ⎢ ⎣⎢ ΔU J ⎦⎥ ⎣⎢ s J 1

s12 s22 ... sJ 2

... s1I ⎤ ⎡ Δξ1 ⎤ ... s2 I ⎥⎥ ⎢⎢ Δξ 2 ⎥⎥ ... ... ⎥ ⎢ ... ⎥ ⎥⎢ ⎥ ... sJI ⎦⎥ ⎣⎢ Δξ I ⎦⎥

(10)

where: quantity ΔU j =| U j | − | Uo j | denotes the difference of U (ξ ) j = U j . In the events of Jacobian of optimized function having form of rectangular matrix ( J ≥ I ) with disadvantageous features, that is − the singular values decay gradually to zero, − the ratio between the largest and the smallest nonzero singular values is large [4], the identification of model parameters belongs to the wide class of the discrete illposed problems, and requires the application of the special regularization method e.g. TSVD (Truncated Singular Value Decomposition). To combine TSVD with the proposed algorithm the filtering function [3,4] removing the singular vectors corresponding with small singular values σi, was defined ⎧0 σ ≤ δ fTSVD (σ ) = ⎨ ⎩1 σ > δ .

(11)

70

K.M. Gawrylczyk and P. Putek / Multi-Frequency Sensitivity Analysis of 3D Models

Numerical Examples a)

b)

c)

d)

γ [S/m]

Figure 9. Course of identification for the noise level equal –30 [dB]: a) the assumed conductivity distribution, b) the conductivity distribution in 2th iteration, c) the conductivity distribution in 6th iteration, d) the conductivity distribution after 16 iterations.

a)

b)

c)

d)

γ [S/m]

Figure 10. Course of identification for the noise level equal –20 [dB]: a) the assumed conductivity distribution, b) the conductivity distribution in 2th iteration, c) the conductivity distribution in 6th iteration, d) the conductivity distribution after 10 iterations.

a)

b)

c)

d)

γ [S/m]

Figure 11. Course of identification for the noise level equal –30 [dB]: a) the considered/known conductivity distribution, b) the conductivity distribution in 3th iteration, c) the conductivity distribution in 5th iteration, d) the conductivity distribution after 9 iterations.

K.M. Gawrylczyk and P. Putek / Multi-Frequency Sensitivity Analysis of 3D Models

71

Conclusions In this work the simple algorithm for solution of 3D inverse problems of conductivity estimation basing on scalar magnetic potential and impedance boundary condition was proposed. For the solution of current distribution inside conductors the full 3Dformulation with four degree of freedom at each node is necessary. The proposed methods of sensitivity evaluation is applicable also in this case. This work was supported by the Polish Government (No. of awarded grant 3 T10A 045 28). References [1] Gawrylczyk K.M.: Adaptiven Algorithmen auf der Basis der Methode der Finiten Elemente, Szczecin University of Technology, 1992. [2] Putek P.: Phd dissertation, The defects’ identification in conducting material basing on the multifrequency sensitivity method in finite element method, Szczecin University of Technology, Szczecin 2007. [3] Gawrylczyk K.M., Putek P.: Multi-frequency sensitivity analysis in FEM application for conductive materials flaw identification, ISEF’05, Sept. 15-17, Baiona, Spain. [4] Gawrylczyk K.M., Putek P.: Adaptive meshing algorithm for recognition of material cracks, COMPEL Vol. 23, No. 3, 2004, pp. 677-684. [5] Deeley E.M.: Surface Impedance Near Edges and Corners in Tree-Dimensional Media, IEEE Transaction on Magnetics, Vol. 26, No. 2, March 1990, pp. 712-714. [6] Dyck D.N., Lowther D.A.: A Method of Computing the Sensitivity of Electromagnetic Quantities to Changes in Material and Sources, IEEE Trans. On Mag., Vol. 30, No. 5, 1994, pp. 3415-3418.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-72

Very Fast and Easy to Compute Analytical Model of the Magnetic Field in Induction Machines with Distributed Windings Manuel PINEDA, Jose ROGER FOLCH, Juan PEREZ and Ruben PUCHE Department of Electrical Engineering, Universidad Politécnica de Valencia Cno de Vera s/n. 46022 Valencia, Spain [email protected], [email protected], [email protected], [email protected] Abstract. Torque and e.m.f. of an induction motor can be derived from the air-gap flux density. The paper shows a new method for computing the flux density distribution of constant air-gap width machines, neglecting magnetic saturation, by making use of very efficient techniques widely used in the field of discrete signals processing: the Fast Fourier Transform (FFT) and the Discrete Circular Convolution. The mutual inductances between the phases of the machine are obtained with a single, very simple formula, in terms of the machine’s windings distribution and the geometric dimensions, which is solved with the FFT. As the method can handle arbitrary winding conductor distributions, it is highly suitable to the analysis of the magnetic field and electromagnetic torque in machines with stator or rotor faults, such as inter-turn short circuits or broken bars.

Introduction Transient analysis of rotating electrical machines by use of the well established d-q model neglects the harmonic contents generated by phase windings. In [1] a model of the electrical machine which takes into account the effect of the spatial harmonics is presented. In [2] the Multiple Coupled Circuit Model of a symmetrical, general m−n induction machine is established based on the phase self and mutual inductances, derived on harmonic bases. The accuracy of the analysis depends on the number of harmonics included in the calculation. In [3] the Winding Function Approach (WFA) for the calculation of inductances in machines with small and constant air gap, taking into account the space harmonics, is presented, and has been used for modeling of induction machines [4] and fault analysis in [5–7]. In [8] the effect of slot skewing and the linear rise of the air gap MMF across the slot are introduced in the WFA. WFA is a method of general validity for calculating the inductances of rotating electrical machines, but has some drawbacks: to account for coil pitch, slot skewing or the rise of the air gap MMF across the slot, different winding functions must be used in each case. Besides, the winding function of a phase must be assembled from the winding functions of the coils that constitute the phase. Complex integrals must be solved in this process, which may be very cumbersome in the case of arbitrary winding distributions. As it is stated in [3], this task typically consumes a high amount of time, so that only discrete curves of inductance versus rotor position are calculated and linear interpolation is applied at intermediate rotor positions. In this paper, a completely different method of solving problem is undertaken, characterized by:

M. Pineda et al. / Very Fast and Easy to Compute Analytical Model of the Magnetic Field

1.

2.

3.

73

The conductor, instead of the coil, is used as the basic element to compute both the air gap MMF and the flux linkage of a phase, so that arbitrarily complex windings layouts can be easily modeled. There is no need of coils or phase winding functions. Only the physical distribution of the conductors of the phases and a single, characteristic function of the system (the yoke flux of a conductor placed at the origin and fed with a unit current) are needed to establish a single resulting equation, which gives the mutual inductance of two phases for every relative position between them. The resulting equation is expressed as a discrete circular convolution, which is computed in the spatial frequency domain in a very fast way using the FFT.

It is a very general and rather unconscious assumption to associate the FFT exclusively to signal analysis in the time domain. Nevertheless, it proves to be also extremely powerful when applied to the treatment of quantities in the spatial domain, such as the air gap MMF and the yoke flux. The expressions derived for these quantities have a mathematical structure analogous to the ones found in the analysis of time signals, so that the tools used in this field, as the FFT, can be successfully applied to compute phase inductances. This approach results in an extremely fast method: the computation of the mutual inductance between two phases, in 3600 different relative positions, considering 1800 harmonics, skewing and linear rise of the air gap MMF across the slot is computed in less than a tenth of a second, on a 2800 MHz Pentium IV processor. And this time is independent of the complexity of the windings layout.

System Equations The following equations system can be written for an induction machine with m stator and n rotor phases with arbitrary layout (that is, even with winding fault conditions like inter-turn short circuits or broken bars): [U S ] = [ RS ] [ I S ] + d [Ψ S ] dt

(1)

[0] = [ Rr ] [ I r ] + d [Ψ r ] dt

(2)

[Ψ s ] = [ Lss ] [ I s ] + [ Lsr ] [ I r ]

(3)

[Ψ r ] = [ Lsr ]T [ I s ] + [ Lrr ] [ I r ]

(4)

[U S ] = [us1 us 2 ... usm ]T

(5)

[ I S ] = [is1 is 2 ... ism ]T

(6)

[ I r ] = [ir1 ir 2 ... irn ]T

(7)

Te = [ I S ]T

∂ [ Lsr ] [Ir ] ∂θ

(8)

74

M. Pineda et al. / Very Fast and Easy to Compute Analytical Model of the Magnetic Field

Te − TL = J

dΩ d 2θ =J 2 dt dt

(9)

where [U] is the voltage matrix, [I] is the current matrix, [R] is the resistance matrix, [Ψ] is the flux linkage matrix and [L] is the matrix of inductances. Subscripts s and r stand for stator and rotor. Te is the electromechanical torque of the machine, TL is the load torque, J is the rotor inertia, Ω is the mechanical speed and θ is the mechanical angle. To compute (3), (4) and (8), self and mutual phase inductance matrices must be calculated. Due to the presence of the derivatives in (1), (2) and (8), it is necessary to achieve a very good accuracy in this process (especially if different fault conditions are to be detected and diagnosed in a sure way), so that air gap MMF harmonics must be considered. End turn and slot leakage inductances need to be pre-calculated, and are treated as constants in (3) and (4), as usual in the technical literature.

Proposed Method for Computing the Mutual Inductance Between Two Phases via Discrete Circular Convolution and FFT The inductance between two phases, A and B, is calculated in this paper through the following process: 1. 2.

Phase A is fed with a constant unit current, and its yoke flux is obtained, as given in [9]. Flux linkage of phase B due to the yoke flux of phase A is determined, which corresponds to the mutual inductance between the phases. (If B = A, we get the phase magnetizing self inductance).

Both steps are treated with the same mathematical tool, a circular convolution computed with the FFT, giving the mutual inductances between phases A and B for every relative angular position in a single equation. Yoke Flux Produced by an Arbitrary Winding The distribution curve of the air gap MMF, F(φ), is defined by the relation: F (ϕ) = H r (ϕ)· g

(10)

where φ is the angular coordinate, Hr(φ) is the mean value of the radial component of the magnetic field intensity in the air gap at φ and g is the air gap width. The air gap MMF F0(φ) (Fig. 1) that produces a single conductor placed at φ = 0 and fed with a unit current, with of infinite iron permeability, as given in [10]: F0 (ϕ) =

1⎛ ϕ⎞ ⎜1 − ⎟ 2⎝ π⎠

(11)

In the case of a constant and small air gap width, the yoke flux Φy0(φ) generated by (11) is (Fig. 2a):

75

M. Pineda et al. / Very Fast and Easy to Compute Analytical Model of the Magnetic Field

Figure 1. Air gap MMF F0(φ) generated by a conductor placed at the origin and fed with a unit current.

(a)

(b)

Figure 2. Yoke flux generated by a conductor fed with a unit current placed at (a) the origin (b) coordinate α.

⎧ μ 0 ··r π ⎪ ∫ F0 (ϕ)·d ϕ 0 ≤ ϕ < π Φ y 0 (ϕ) = ⎨ g ϕ ⎪ Φ (ϕ − π) π ≤ ϕ < 2π y0 ⎩

(12)

where  is stack’s length, r is the average radius of the air gap, and g its width. Substitution of (11) gives: Φ y 0 (ϕ) =

μ 0 ··r ·(ϕ − π) 2 4πg

(13)

If the conductor is placed at another angular position α, its yoke flux, Φyα(φ), is obtained by shifting the curve Φy0(φ) to the new position, Φ yα (ϕ) = Φ y 0 (ϕ − α) , as shown in Fig. 2b. The yoke flux ΦyA(φ) generated by a phase A with an arbitrary conductors distribution nA(φ), fed with a unit current, can now be obtained from (13), by applying linear superposition. Φ y A (ϕ) =

2π

∫Φ 0

y0

(

)

(ϕ − α )·nA (α)· d α = Φ y 0 ⊗ nA (ϕ)

(14)

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M. Pineda et al. / Very Fast and Easy to Compute Analytical Model of the Magnetic Field

The particular mathematical structure in (14) is very well known in the control and signal theory. It is called a circular convolution, represented with the symbol ⊗. It must be evaluated at every angular coordinate φ, which can be very cumbersome for a complex winding layout, nA(φ). Nevertheless, there is an alternative and very fast way to compute it, based on the following property of the Fourier Transform (FT): the FT of the convolution of two functions is equal to the product of their FTs, that is, FT(f ⊗ g) = FT(f) · FT(g). Applying this property to (14) gives the following algorithm: FT ˆ 0 (ξ) ⎫⎪ IFT Φ 0 (ϕ) → Φ ˆ 0 (ξ) ·nˆ (ξ) = Φ ˆ 0 (ξ) → Φ A (ϕ) ⎬Φ Á FT nA (ϕ) → nˆ A (ξ) ⎪⎭

(15)

This algorithm reduces the computation of integral (14) to a simple product of two functions in the spatial frequency domain, but it has a serious drawback: two FTs and one IFT must be computed. However, there is an extremely efficient algorithm to obtain the FT of a function, the Fast Fourier Transform (FFT), and its inverse (IFFT), but they can only be applied to discrete sequences, not to continuous functions. To convert the functions in (15) into discrete sequences, the circular length of the machine air gap is divided into N equally spaced intervals, each of them spanning an angle Δφ = 2π/N. A high value of N is necessary to achieve a good accuracy (for example, N = 3600 yields a precision of 0.1º). These discrete sequences are represented as column vectors of N elements, generated as follows: •

The discrete sequence of function Φ0(φ), Φ0, is generated by sampling at the beginning of each interval. μ ··r ·π ⎛ 1 k ⎞ Φ y0 [k ] = 0 ·⎜ − ⎟ g ⎝2 N ⎠

•

2

(16)

The discrete sequence corresponding to the distribution of conductors nA(φ), nA, is generated by assigning to each interval the sum of all the conductors that it contains: nA [k ] = ∫

( k +1)·Δϕ

k ·Δϕ

nA (ϕ)·d ϕ

0≤k < N

(17)

Algorithm (15) can now be formulated in terms of these discrete sequences as: Φ yA = IFFT ( FFT (Φ y 0 ).* FFT ( nA ) )

(18)

where the symbol .* denotes an element by element product of two vectors. It must be remarked that, in (18), the sequence Φy0 and its FFT are the same for every induction machine, except for a constant factor that depends on its geometrical dimensions. Besides, the sequence nA can easily represent any arbitrary winding layout. Slot width of skewing, for example, can be taken into account just by distributing the conductors uniformly in the intervals that spans the slot opening of the skew slot, respectively.

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M. Pineda et al. / Very Fast and Easy to Compute Analytical Model of the Magnetic Field

Figure 3. Flux linkage of an arbitrary coil: a) actual coil b) replaced by two equivalent annular coils.

Flux Linkage of a Phase with an Arbitrary Distribution of Conductors The flux linkage of a phase with an arbitrary distribution of conductors is obtained by simply adding up the values of the yoke flux at the yoke sections corresponding to each one of its conductors. Figure 3 shows the basis of this method: the flux linkage Ψab of an arbitrary coil a-b can be calculated by replacing the coil by two equivalent annular ones, (a-a’, b-b’) and summing up the yoke flux that crosses them. The flux linkage ΨBA of a phase B with an arbitrary conductors’ distribution, nB(φ), due to the yoke flux generated by another phase A, ΦyA(φ), is obtained by applying the aforementioned method to each of its coils: ψ BA =

2π

∫n

B

0

(ϕ)· Φ y A (ϕ)· d ϕ

(19)

If phase B is now ideally or actually (for instance, due to a change in the rotor position) shifted by an angle ε with respect to A, (21) can be applied by simply using a shifted distribution of the conductors of phase B ψ BA (ε) =

2π

∫n

B

0

(ϕ − ε)· Φ y A (ϕ)· d ϕ

(20)

If phase A is fed with unit current, (20) is the mutual inductance between phases A and B as a function of ε: LBA (ε) = Ψ BA (ε) =

2π

∫n

B

0

(ϕ − ε)· Φ y A (ϕ)· d ϕ

(21)

The integral in (21) can be difficult to calculate for complex conductors distributions, but it can be computed with an algorithm similar to (18):

(

LBA = IFFT FFT ( Φ yA ) .* ( FFT ( nB ) )

*

)

(22)

where the superscript * stands for complex conjugate. LBA is a column vector, whose kth element is the mutual inductance of phases A and B, for a relative angle between

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M. Pineda et al. / Very Fast and Easy to Compute Analytical Model of the Magnetic Field

IR (A)

IR (A)

T (Nm)

time (s)

time (s)

time (s)

(a)

(b)

(c)

Figure 4. Transient current of a stator phase during start-up (a) simulated (b) experimental, and computed electromagnetic torque (c).

them equal to k·2π/N. Furthermore, (18) and (22) can now be combined, yielding a single equation

(

LBA = IFFT FFT ( nA ) .* FFT ( Φ y 0 ) .* ( FFT ( nB ) )

*

)

(23)

which can be very easily computed with modern scientific software. For example, in MATLAB, it can be written as Lba = ifft( fft(na) .* fft(Yf0) .* conj( fft(nb))).

Analytical Evaluation of the Currents and Electromagnetic Torque The proposed method has been applied to compute the inductances, starting current and electromagnetic torque of an 11 Kw induction motor, fully characterized in [11]. Figure 4 shows the starting current (simulated and experimental) during a start-up transient, with a constant 1.5 Nm load torque, and the computed electromagnetic torque for that transient.

Conclusion A new and completely different approach for the calculation of winding inductances in induction machines has been presented in this paper. The election of the conductor as the winding basic element, the yoke flux as the main flux quantity, and the formulation of inductances in terms of a discrete circular convolution, computed with the FFT, are the key elements of the new method. After discretization of the air gap into N equally spaced intervals, the mutual inductances of two phases corresponding to N relative angular positions, taking into account the first N/2 air gap MMF harmonics, are obtained simultaneously with a single equation, solved via FFT. The method involves only three discrete sequences, namely the distributions of the conductors of the two phases and a characteristic function of the machine: the yoke flux generated by a conductor placed at the origin with a unit current flowing through it. The computation of the mutual inductance of two phases, for N = 3600, takes less than a tenth of a second on a 2600 MHz Pentium IV processor. Arbitrary, complex winding layouts, the linear rise of the air gap MMF across the slot, and slot skewing can be easily modeled and taken into account without increasing at all this computing time. As the method can handle arbitrary phase

M. Pineda et al. / Very Fast and Easy to Compute Analytical Model of the Magnetic Field

79

conductor distributions, it is highly suitable to the analysis of machines with stator or rotor faults, such as inter-turn short circuits or broken bars.

References [1] F. Taegen and E. Hommes, “Das allgemeine Gleichungssystem des Käfigläufermotors unter Berücksichtigung der Oberfelder. Teil I: Allgemeine Theorie”, Archiv für Elektrotechnik, vol. 55, no. 1, pp. 21-31, Jan. 1972. [2] H. R. Fudeh and C. M. Ong, “Modeling and Analysis of Induction Machines containing Space Harmonics. Part I”, IEEE Transactions on Power Apparatus and Systems, vol. 102, no. 8, August 1983. [3] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a Concentrated Winding Induction Machine for Adjustable Speed Drive Applications Part 1 (Motor Analysis)”, IEEE Trans. Energy Convers., vol. 6, no. 5, pp. 679-683, Dec. 1991. [4] X. Luo, Y. Liao, H. A. Toliyat, A. El-Antably, and T. Lipo, “Multiple Coupled Circuit Modeling of Induction Machines”, IEEE Trans. Ind. Appl., vol. 31, no. 2, pp. 311-318, March/April 1995. [5] H. A. Toliyat and T. A. Lipo, “Transient Analysis of Cage Induction Machines under Stator, Rotor Bar and End Ring Faults”, IEEE Trans. Energy Convers., vol. 10, no. 2, pp. 241-247, June 1995. [6] J. Milimonfared, H. M. Kelk, A. Der Minassians, S. Nandi, and H. A. Toliyat, “A Novel Approach For Broken Rotor Bar Detection in Cage Induction Motors”, IEEE Trans. Ind. Appl., vol. 35, no. 5., pp. 1000-1006, Sept./Oct. 1999. [7] S. Nandi and H. A. Toliyat, “Novel Frequency-Domain-Based Technique to Detect Stator Interturn Faults in Induction Machines using Stator-Induced Voltages after Switch-Off”, IEEE Trans. Ind. Appl., vol. 38, pp. 101-109, Jan./Feb. 2002. [8] G. Joksimovic, M. Djurovic, and A. Obradovic, “Skew And Linear Rise of MMF Across Slot Modeling. Winding Function Approach”, IEEE Trans. Energy Convers., vol. 14, pp. 315-320, Sept. 1999. [9] L. Serrano-Iribarnegaray, Fundamentos de Máquinas Eléctricas Rotativas. Barcelona: Marcombo, 1989, p. 195. [10] B. Hague, The Principles of Electromagnetism Applied to Electrical Machines”. New York: Dover Publications, Inc., 1929. [11] M. Pineda-Sánchez, “Máquinas Eléctricas con Armónicos de Devanado: Desarrollo y Comparación de Distintos Métodos de Análisis, de Complejidad Gradualmente Creciente, hasta Incluir Permeabilidad del Hierro Finita, Ranurado, Excentricidad y Desplazamiento De Corrientes”, Ph.D. dissertation, Dept. Ing. Elec., Universidad Politécnica de Valencia, Valencia, 2004.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-80

Coupling Thermal Radiation to an Inductive Heating Computation Christian SCHEIBLICH, Karsten FRENNER, Wolfgang HAFLA and Wolfgang M. RUCKER Institute for Theory of Electrical Engineering, Pfaffenwaldring 47, 70569 Stuttgart, Germany, [email protected] Abstract. Inductive ovens heat with eddy currents due to a low frequency electromagnetic field. Within a first approximative approach, the effects of thermal conduction and thermal radiation are taken into account. Therefore, a finite element method for calculating the thermal conduction is coupled to a boundary element method for calculating the resulting thermal radiation. For the presented application, a steel tube surrounded by an axisymmetric coil is chosen and the normalized thermal radiation distribution resulting from the thermal conductive transfer and the eddy current density is examined.

Introduction To heat materials in a very fast and efficient way, the principle of an inductive oven is of first choice. The main physical effects are eddy currents in case of an induced electromagnetic field and a resulting object-based heat flow due to thermal conduction. Treating temperatures higher than 500° Celsius, heat transfers caused by thermal radiation cannot be neglected. Therefore, thermal radiative transfers should be examined and displayed. For the presented application, the setup of a steel tube surrounded by an axisymmetric coil is chosen (Fig. 1). The tube model consists of a volume mesh of 2014 first order tetrahedrons and a boundary mesh of 758 first order triangles. The coil is meshed with 297 first order tetrahedrons and generates an electromagnetic field of low frequency. The eddy currents in the tube appear mainly in the influence area of the coil and do not intrude into the inner regions of the tube at the chosen frequency. One can compute the heating sources from the eddy current density and the electric conductivity. The heating sources represent the boundary conditions for the thermal diffusion equation. Accordingly, heat sources as boundary conditions for the thermal radiation equation, are also available within this step. Finally, it is possible to obtain the thermal radiation distribution of tube’s surface.

Inductive Heating The problem considered are eddy currents at a low frequency ω , displacement currents are ignored. The magnetic permeability μ and the electric conductivity κ are

C. Scheiblich et al. / Coupling Thermal Radiation to an Inductive Heating Computation

81

Figure 1. The model of a steel tube surrounded by a coil.

assumed to be constant. The mathematical model for time harmonic quasi-static eddy current problems can be derived from Maxwell’s equations resulting in

curl

1 curl A + jωκ A = J c , μ

(1)

where A is the complex magnetic vector potential and J c is the current density of the source coil. The eddy current density jωκ A = − J e is computed with a finite element method (FEM) coupled to a boundary element method (BEM) [1]. Therefore, no volume mesh to model the air is necessary. In Fig. 2 the arising eddy current density in the tube by a maximum value of 2.72 kA per square meter – in the mid of the tube – and by the minimum value of 50 A per square meter – dark, at the end of the tube – is shown. The values appear at a frequency of 500 Hz, one turn for the coil and a current density of 100 kA per square meter feeding the coil. For one single time step, the magnitude of the eddy current density J e determines the heat source distribution,

Pedd =

Je κ

2

.

(2)

The presented heat source distribution for the thermal conduction process is evaluated from one selected time step of the eddy current simulation from which a temperature distribution can be retrieved.

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C. Scheiblich et al. / Coupling Thermal Radiation to an Inductive Heating Computation

Figure 2. Arising eddy currents at tube’s surface.

Conductive Heat Transfer To receive a temperature distribution Tdif one has to solve Fourier’s thermal diffusion equation,

∂Tdif ∂t

= α divgrad Tdif + Pedd + Prad .

(3)

Pedd and Prad are the heat source distributions from the eddy current density and the thermal radiation. The thermal diffusivity α can be retrieved by the specific thermal conductivity relative to the ability of thermal energy storage [2]. Solving equation (3) one can obtain radiation sources at the boundaries from the temperature distribution Tdif to compute the thermal radiation distribution. Denote that in this first approach a single time step from the conductive heat transfer is taken and, further, the temperature distribution and the associated thermal radiation distribution are normalized to one. Radiative Heat Transfer The thermal radiation in an inductive oven is limited to grey diffuse reflectors. Emission, absorption, and reflection are the considered physical effects. To compute the radiosity B j a boundary element method is used [3], where the surfaces of the field

C. Scheiblich et al. / Coupling Thermal Radiation to an Inductive Heating Computation

83

Figure 3. Thermal radiation displayed for tube’s half boundary.

problem have to be considered. This is tackled by the radiosity equation, which is well known from graphics applications,

σε iTi 4 = ∑ B j (δ ij − ρi Fij ) ,

(4)

j

4 where σε iTi is the emitted energy for a given temperature Ti , σ and ε i are the

Stefan-Boltzmann constant and the emission factor, δ ij is the Kronecker symbol, respectively. The reflectance ρi multiplied by the possible rate of energy Fij B j scattered from surface element Ai towards surface element Aj represents the incident flux of a surface. Fij is known as the view factor,

Fij =

1 Ai

∫∫

Ai A j

cos θi cos θ j π rij

2

Vij d Ai d Aj .

(5)

The view factor includes a visibility function Vij that results either in zero for no sight or one for sight between two focused surface elements Ai and Aj . The resulting radiosity B j is the thermal radiation Pj , rad at each surface element Aj and can be directly added to the thermal diffusion equation (3) as a heating source at the boundary of the body. The overall thermal radiation shown in Fig. 3 is represented by the colour scheme at the surface of the steel tube. It was normalized to one, therefore, the result-

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C. Scheiblich et al. / Coupling Thermal Radiation to an Inductive Heating Computation

ing values range from 0.9 due to the loss – dark, outside the tube – to 1.22 – white, inside the tube. Using a time step based process these results can affect the diffusion equation (3) and display the influence of thermal radiation processes to the temperature distribution. Conclusion Within the presented approach, eddy currents are computed for a three-dimensional model. When obtaining thermal heat sources at higher temperatures due to eddy currents, one has to consider thermal conductive and thermal radiative processes. Appearing radiative heat transfers are computed for one time step of the conductive transfer. This allows denoting the thermal radiation distribution for any area of the focused model and can be used to influence the next time step. Further, it is possible to calculate the heat flow to other objects and so, the heat distribution in non-conductive objects. Outlook To obtain the thermal radiation in the presented application it is necessary to solve a large fully populated system matrix originated from the BEM. Therefore, an enormous amount of physical memory is required while BEM applications increase the memory requirements with the second power to the number of unknowns. To reduce this growth a compression of the system matrix may be examined. Due to the smoothness of the kernel function (5), a wavelet-based compression of separated areas of the computed model may reduce the growth of the BEM system matrix (4) nearly to linear complexity and allows for solving problems with higher precision. References [1] V. Rischmüller, J. Fetzer, M. Haas, S. Kurz, W. M. Rucker, Computational efficient BEM-FEM coupled analysis of 3D nonlinear eddy current problems using domain decomposition, Proceedings of 8th International IGTE Symposium, Graz, 1998. [2] F. P. Incropera, D. P. DeWitt, Fundamentals of Heat and Mass Transfer, Fifth Edition, John Wiley & Sons, 2002. [3] C. Scheiblich, K. Frenner, W. M. Rucker, Computation of Radiative Heat Transfer with The Boundary Element Method for Inductive Heating, Proceedings of 12th International IGTE Symposium, Graz, 2006.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-85

85

Consideration of Coupling Between Electromagnetic and Thermal Fields in Electrodynamic Computation of Heavy-Current Electric Equipment Karol BEDNAREK Institute of Electrical Engineering and Electronics, Poznan University of Technology, Piotrowo 3a, 60-965 Poznan, Poland E-mail: [email protected] Abstract. The paper presents a model of electrodynamic computation (current density, power loss, temperatures) of three-phase heavy current busways with the use of integral equation method. The computation makes allowance of coupling between the electromagnetic and thermal fields. Results of calculation are shown and compared with the measurement trials performed with physical objects.

Introduction An important factor of the procedures of heavy-current electric equipment designing is accurate definition of the electrodynamic parameters decisive for their proper operation. A very important element conducive to improving accuracy of electromagnetic calculation consists in precise consideration of physical factors affecting the process, e.g. by appropriate account of coupling between the electromagnetic and thermal interactions. The paper presents a model of electrodynamic calculation of three-phase heavycurrent lines (i.e. power busways) with consideration of coupling between electromagnetic and thermal fields. More accurate calculation of electrodynamic parameters obtained in consequence of the above is confirmed by measurement of the physical objects. The software package developed this way was used in designing of the process of heavy-current busways. This allows for significant financial savings at the stages of their manufacturing and exploitation processes. Taking into account the symmetry of the system and radial heat transmission the use of a 3D system would be unjustified (no significant variations along the path). Therefore, a 2D system was used for computation purposes, that is important for the optimization process (with many times repeated calculations).

The Model of Electrodynamical Calculation A three-phase heavy-current screened busway composed of three oval conductors (Fig. 1) of cross section Sc was considered. The conductors were disposed symmetri-

86

K. Bednarek / Consideration of Coupling Between Electromagnetic and Thermal Fields

Figure 1. Geometry of the considered system (with marked sub-areas).

cally each 120º apart, in a screen of inner radius RS and outer radius RO. Another characteristic dimensions are: the large a and small b diameter of the conductor, thickness g of its walls, height h of the conductor suspension. The basic parameter required for purpose of further electrodynamic analysis is the distribution of the currents of phase conductors and the ones induced in the screen. As a basis for determining the distribution the relationships of the magnetic vectorial potential are used, written down for particular sub-areas of the system. In case of the considered 2D system the magnetic vectorial potential A(r,ϕ,z) has only one component in z-axis, depending solely on the (r,ϕ) coordinates [2,4]. In consequence, A(r,ϕ,z) = 1z A(r,ϕ) and meets the following relationships in particular areas (Fig. 1): − in the I area (inside the screen), i.e. for 0 ≤ r ≤ RS:

A I (r,ϕ) = A1 (r,ϕ) + A 2 (r,ϕ)

(1)

According to the relationship A1(r,ϕ) originates from the currents flowing in the phase conductors and may be expressed by the formula:

A1 (r,ϕ) =

3 μO 4π

∞

∫ J(r',ϕ') ∑ [a i sin i(ϕ − ϕ') + bi cos i(ϕ − ϕ')]

SC

i=1

xi r' dr' d ϕ ' i (2)

The potential A2 (r,ϕ) is due to the currents induced in the screen and fulfills the Laplace equation: ∇ 2 A 2 ( r, ϕ ) = 0

−

(3)

in the II area (material of the screen), i.e. for R S ≤ r ≤ RO: ∇ 2 A II (r,ϕ) = j ω μO μS γS A II (r,ϕ)

(4)

K. Bednarek / Consideration of Coupling Between Electromagnetic and Thermal Fields

−

87

in the III area (outside the screen), i.e. for r ≥ RO: ∇ 2 A III (r,ϕ) = 0

(5)

In the above relationships: Sc – is the cross-section area of a single phase conductor; J – current density of R-phase; ω – pulsation (ω = 2πf); μ0 – magnetic permeability of vacuum; μS – relative magnetic permeability of the screen material; γS – conductivity of the screen material. The coefficients xi, ai, and bi take the values [2]: ⎧r for r ≤ r' ⎪ x = ⎨ r' r' ⎪ for r ≥ r' ⎩r

,

⎧ 0 for i = 3l ⎪ a i =⎨ j for i = 3l −1 ⎪ − j for i = 3l −2 ⎩

,

⎧0 for i = 3l bi = ⎨ ⎩1 for i ≠ 3l

(6)

for l = 1, 2, 3, …. Moreover, the following boundary conditions should be satisfied inside particular areas: − for r = RS: AI(r,ϕ) = AII(r,ϕ) −

and

HIϕ(r,ϕ) = HIIϕ(r,ϕ)

(7)

and

HIIϕ(r,ϕ) = HIIIϕ(r,ϕ)

(8)

for r = RO: AII(r,ϕ) = AIII(r,ϕ)

Distribution of current density J(r,ϕ) in the phase conductor is derived from an approximate solution of integral equations [2,4] obtained in result of the use of known relationships for the electromagnetic field E = –jωA and J = γE: J(r, ϕ) − J(rO , ϕO ) +

3 jωμ O γ C ∫ J(r', ϕ ') [ K(r', ϕ ', r, ϕ) − K(r', ϕ ', rO , ϕO )] dr' dϕ ' = 0 4π S (9)

∫ J(r', ϕ ')r' dr' dϕ ' = I S

(10)

where (ro,ϕo) is a reference point, γC is conductivity of conductor material, I – intensity of the current flowing in the R-phase, while K(r',ϕ',r,ϕ) – is a kernel of the integral equation. The coefficients occurring in equations are described in [2]. It results from symmetry of the system that distribution of current density of two remaining phase conductors (S and T) is the same as in R but shifted by +120° and –120°, respectively. The presented system of integral equations may be solved in approximate manner using a moment method, being a variation of Ritz method [2,4]. In order to apply this method the cross-section S of the conductor is divided into N elements of the areas ΔSm (with m = 1,2,…,N).

88

K. Bednarek / Consideration of Coupling Between Electromagnetic and Thermal Fields

Current density is expanded in functional space: J (r,ϕ) =

N

∑

m=1

Jmfm

(11)

while Jm = J (rm,ϕm) = const (for m = 1, 2, …, N) and fm are base functions defined as follows: ⎧ 1 for ΔSm fm = ⎨ ⎩0 for the remaining elements

(12)

In result, Jm is an approximate value of current density in a ΔSm – element. Making use of the moment method the system of integral equations is then replaced by a system of N linear algebraic equations in the form: l1,2 l1,3 ...... l1,N ⎤ ⎡ l1,1 ⎢l ⎥ ⎢ 2,1 l2,2 l 2,3 ...... l2,N ⎥ ⎢ l3,1 l3,2 l3,3 ...... l3,N ⎥ ⎢ ⎥ ⎢......... ......... ......... .....................⎥ ⎢ l N-1,1 l N-1,2 l N-1,3 ...... l N-1,N ⎥ ⎢ ⎥ ⎣⎢ ΔS1 ΔS2 ΔS3 ...... ΔSN ⎦⎥

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣⎢

J1 ⎤ ⎡ 0 ⎤ J 2 ⎥⎥ ⎢⎢ 0 ⎥⎥ J3 ⎥ ⎢ 0 ⎥ ⎥ =⎢ ⎥  ⎥ ⎢  ⎥ J N-1 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥ J N ⎦⎥ ⎢⎣ I ⎥⎦

(13)

where: l m,n = δ m,n − δ N,n +

3 j ω μ O γ C ∫ [ K(r',ϕ ',rm ,ϕm ) − K(r',ϕ ',rN ,ϕ N ) ] dr' dϕ ' 4π ΔSm

(14) for: m = 1,2,3,...,N–1 and n = 1,2,3,…,N, δm,n is Kronecker delta, K(r',ϕ',r,ϕ) – a kernel of the integral equation defined in the form presented in [2]. Solution of the system (13) provides approximate values Jm (for m = 1, 2, 3,…, N) of current density in particular elements of cross-section ΔS1, ΔS2, …, ΔSN. Total current I flowing in the phase conductor may be considered as a set of m conductors transmitting the currents, Im = JmΔSm, (m = 1,2,…,N). Distribution of the current JS induced in the shield is obtained with the use of the analogical relationships. Knowledge of approximate distribution of the current density vector enables determining the value of active power losses in conductors and shield. The losses of active power in phase conductors PC (falling to unit length) and in the shield PS may be determined e.g. from the Joule law [2,4]: PC =

3 γC

∫ J(r', ϕ') S

2

r'dr'dϕ' ,

PS =

3 γS

∫J

Ss

2

S

(r', ϕ') r'dr'dϕ'

(15)

K. Bednarek / Consideration of Coupling Between Electromagnetic and Thermal Fields

89

Taking into account the approximate distribution of current density, the relationship takes a form: PC =

3 γC

N

∑

m=1

J m (rm ,ϕm )

2

ΔSm ,

1 γS

PS =

N

∑

m=1

JSm (rm ,ϕm )

2

ΔSm

(16)

Knowledge of the active power losses and the distribution of power density emitted in the conductors and in the shield is necessary for determining thermal conditions of the system. Distribution of power density output in the shield is expressed by the relationship: ρ(r, ϕ) =

JS (r',ϕ')

2

(17)

γS

Results of many calculations made for aluminium shield of 3-mm thickness have shown that ρ(r,ϕ) is a function strongly dependent on ϕ and symmetrically distributed every 120° [2,3], while the variable r only slightly affects ρ. Total thermal power emitted from phase conductors dissipates radially, approximately uniformly [1–6]. At the outer surface of the shield the following boundary condition is met: PC ∂T = −λS 2π R S ∂r

for

r = RS

(18)

Inside the shield the temperature meets Poisson equation: ∇ 2 Τ( r,ϕ) =

ρ( r,ϕ) λS

(19)

From outer surface of the shield the thermal power is emitted to the environment in result of convection and radiation: λ

∂ T(r,ϕ) ∂r

= − α CR [ T(r,ϕ) − TO ]

r = RO

for

(20)

with surface film conductance: αCR = αC + αR, and αC – surface film conductance resulting from convection, αR – surface film conductance resulting from radiation, TO – temperature of the environment. Temperature excess above that of the environment meets the Poisson equation [1,3]: λC

∂ 2 T(l) 1 = − p(l) , where p(l) = 2 ∂l g

n

∑P j=1

j

δ (l − lmj ) −

2α T(l) g

(21)

90

K. Bednarek / Consideration of Coupling Between Electromagnetic and Thermal Fields

is a difference between the power emitted from the branch through n inner sources, and the power carried away to the environment with constant α coefficient, δ(l – lmj) is delta function, g is the thickness of conductor wall. Methods of solving the above questions and determining the coefficients occurring in the equations are discussed in detail in the works [1–6]. The calculation model used here makes allowance for coupling between the electromagnetic and thermal fields. The coupling occurs on the conductance of the conductors and shield. The temperature changes cause the changes in conductivity of the shield and conductors and in result power losses, and vice versa. These dependencies may be expressed as: T = f (ρ,γ,λ)

and

ρ = f (T,γ)

(22)

The calculations must be carried out with iterative methods. Before the calculations an assumption must be made about temperature of the conductors and the shield. The assumption must be checked at the end of the calculations. Once the error exceeds 0.5 K a new assumption must be made and the calculation repeated [1,2,5,6].

Results of Calculation and Measurement The thermal analyses are performed for an air-insulated screened three-phase heavycurrent busway. Temperature distribution is determined in aluminum conductors and the shield. The following data were assumed for the calculation purposes: rated current 1.6 kA, a = 78 mm, b = 50 mm, g = 11.5 mm, h = 92 mm, RS = 247 mm, RO = 250 mm, conductivity: of shield material 28⋅106 S/m and of conductor material 31⋅106 S/m, emissivity coefficient 0.4, temperature of the environment 297.5 K, thermal conductivity 220 W/(m⋅K). Results of the calculation are shown in Fig. 2a. The calculated temperature distribution is presented with the accuracy of two digits after decimal point, only with a view to depicting the scale of temperature differences between particular points of the system. Such accuracy is useless in case of practical temperature computation as the temperatures so determined depend on many factors (significantly affecting the results and difficult to consider), like the condition of the conductor and shield surfaces (influencing the values of surface film conductance), arrangement of the busway (even slight deviation from horizontal arrangement may affect the convection), or external conditions (random air motion in proximity of the busways or consideration of insolation in case of open areas). The calculation accuracy achieved in practice considerably depends on the accuracy of the coefficients used for computation (provided by manufacturers of the materials and depending, for example, on the time-varying surface conditions of the elements) and, in the same time, on some random factors (e.g. random motion of the air around). Temperatures were calculated for various current values and the variants with and without the coupling of the electromagnetic and thermal fields. The results of the calculation are compared to the measurements performed on physical objects, that is shown in Fig. 2b. The software package of electrodynamic computation has been used in the optimization process of the heavy-current busways. Results of the optimization calculation are presented in the papers [2,3].

K. Bednarek / Consideration of Coupling Between Electromagnetic and Thermal Fields

a)

91

b)

Figure 2. Results of calculation and measurement: a) temperature distribution T[K] of the system (results of calculation), b) temperatures T[K] of the conductors changes in the system as functions of the currents I[A].

Conclusion Consideration of the coupling between the electromagnetic and thermal fields in the mathematical model of electrodynamic calculation for the heavy-current busways improves accuracy of the results. In case of the calculation carried out without the coupling the calculation error is smaller than 2.5 per cent, while consideration of the coupling reduces the error below 1 percent. Accuracy of the thermal calculation depends on many factors, among which the following might be mentioned: correct choice of the assumed mathematical model, the accuracy of the coefficients determining the type of the material and the condition of the elements’ surfaces (e.g. the colour or roughness change due to external factors), and proper consideration of the external conditions (e.g. random air motion, consideration of insolation or other heat sources in the environment). Better accuracy of the results may be achieved by the use of 3D models, allowing for consideration of any design details. Such calculation models are conducive to remarkable growth of dimensions of equation systems obtained this way and the increase in duration of the computation process. Nevertheless, such an approach becomes burdensome in case of optimization processes in which the calculation is many times repeated (in order to repeated determination of the objective function). It should be noticed that for purposes of initial approximation of a system to be optimized very high accuracy of the parameters is not required. Moreover, in symmetrical systems the electrodynamic parameters in the third dimension very often remain nearly unchanged. Hence, the 3D approach is unjustified.

References [1] G. F. Hewitt, G. L. Shires, T. R. Bott, Process heat transfer, New York CRC Press, Boca Raton, 1994. [2] K. Bednarek, Electrodynamical and optimization problems of oval three-phase heavy current lines, Boundary Field Problems and Computer Simulation, 46th thematic issue, series 5: Computer Science, Scientific Proceedings of Riga Technical University, Riga 2004, pp. 6-18. [3] K. Bednarek, Heat and optimization problems in heavy current electric equipment, ISEF’2005, Baiona (Spain), September 2005, pp. CFSA-2.17_1-4.

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[4] K. Bednarek, Determination of temperature distribution in oval three-phase shielded heavy current lines, UEES’2004, International Conference on Unconventional Electromechanical and Electrical Systems, Alushta (Ukraine), September 2004, pp. 649-655. [5] M. F. Modest, Radiative heat transfer, ed. II, Academic Press, N. York, Oxford, Tokyo, 2003. [6] Y. Jaluria, K.E. Torrance, Computational Heat Transfer. Hemisphere Pub. Corp., Washington, N. York, London, 1986.

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93

Force Computation with the Integral Equation Method Wolfgang HAFLA, André BUCHAU and Wolfgang M. RUCKER Institute for Theory of Electrical Engineering, Pfaffenwaldring 47, 70569 Stuttgart, Germany [email protected] Abstract. There is a great multitude of different approaches to compute magnetic forces, each of them giving different results. Therefore, in this paper three popular force computation methods are investigated for nonlinear magnetostatic problems with the integral equation method. Computed results are compared with measurements from two setups. First, the recently presented TEAM Workshop Problem No. 33.a is used since it allows for verification of force densities. Second, the computed total magnetic force acting on a body is verified with the TEAM Workshop Problem No. 20.

Introduction In recent years the integral equation method (IEM) has become applicable to solve magnetostatic problems since it has been used in combination with matrix compression techniques, such as the fast multipole method. This approach leads to a reduction of computational costs from O ( N ) 2 to approximately O ( N ) , where N is the number of unknowns. Since computation of magnetic forces is often required, e.g. during the design process of electrical machines. There is a large number of different approaches to compute these forces, and unfortunately, many of them give different results. We, therefore, investigated three different force computation formulations with respect to their accuracy and applicability when using them with the IEM. With the investigated methods, a magnetic pressure p is computed. The total force is then obtained by integrating p . By investigation of the TEAM Workshop Problem 20 [1] it is shown that it all formulations are of approximately the same accuracy. This is due to the high permeability of this setup. Fortunately, the new TEAM Workshop Problem No. 33.a [2] has been presented lately. The unique property of this setup is that it contains a sample with an extremely low permeability. It is shown that with this setup it is possible to decide which force computation formulation is the most applicable for magnetostatic problems.

IEM and Force Computation Formulations Magnetostatic field problems as shown in Fig. 1 are considered. Free currents J within domain ΩJ cause a source field that magnetizes magnetic material in the domain ΩM

94

W. Hafla et al. / Force Computation with the Integral Equation Method

ΩM

μr J

ΩF μr Ω0

ΩJ Figure 1. Considered field problems.

and ΩF . The air domain is Ω0 . The magnetostatic problem is solved with the IEM formulation [3]

ψ ( r ) + μ0

∫ ( χ ( r ' ) ⋅∇ψ ( r ' ) ) ⋅∇' G ( r ,r ' ) dV ' = ψ

J

(r ) ,

(1)

ΩM +ΩF

where ψ is the total magnetic scalar potential, ψ J is the magnetic scalar potential of the source field H J , χ is the magnetic susceptibility, r and r ' are observation and source points, and G is Green’s function of free space. The IEM is accelerated with the fast multipole method [4]. Nonlinear problems are tackled by direct iteration [5]. After ψ has been solved with (1) the magnetic field H is computed within ΩM and ΩF with H = −∇ψ

(2)

and in the air domain with

H (r ) =

1 4π

∫ (μ

ΩM +ΩF

r

⎛ ∇ψ ( r ' ) ∇ψ ( r ' )( r - r' ) ⎞ − 1) ⎜ −3 ⎟ dV ' + H J ( r ) . 3 5 r - r' ⎝ r - r' ⎠

(3)

The force that acts on ΩF can be computed from the magnetic pressure p . It is obtained by evaluation of Maxwell’s stress tensor. For evaluation points r ∈ Ω0 \ Ω F , ⎡ ( H ( r ) )2 ⎤ p1 ( r ) = μ0 ⎢( n ( r ) ⋅ H ( r ) ) H ( r ) − n⎥ ⋅ n . ⎣ ⎦ 2

(4)

If r ∈ ∂ΩM p2 ,3 ( r ) = pn ( r ) ± pt ( r )

with

(5)

W. Hafla et al. / Force Computation with the Integral Equation Method

95

pn ( r ) =

1 ⎛ 1 μ0 ⎜1 − 2 ⎝ μr

⎞ 2 ⎟ Bn , ⎠

(6)

pn ( r ) =

1 μ0 (1 − μ r ) H t2 , 2

(7)

where the plus and minus sign is obtained when (5) is derived from Maxwell’s stress tensor method and the energy principle, respectively. H t denotes the tangential component of H in the air domain. For inhomogeneous materials, additional volume forces 1 f ( r ) = − H 2 ∇χ 2

(8)

have to be considered. This gives three different force computation methods. First, the force F can be obtained by integrating p over a closed surface A placed in Ω0 that encloses the body

F2 ,3 ( r ) =  ∫ p1 ( r ) dA .

(9)

A

Also, the surface A = ∂ΩM can be used F2 ,3 ( r ) = ∫ p2 ,3 ( r ) dA + A

∫

f ( r ) dV .

(10)

ΩM

Since the force on a body is well-defined, either p2 or p3 has to give wrong results. In practice, however, this is hard to decide. This is due to the fact that magnetic setups are usually made of highly permeable ferromagnetic materials. In these cases the magnetic field is almost perpendicular on the material’s surface and therefore pt  pn , i.e. (6) and (7) give almost the same results. Numerical Results The TEAM workshop problems No. 20 and No. 33a which are shown in Figs 2a and 2b, respectively, have been investigated. Both field problems were discretized with second-order tetrahedrons. TEAM Workshop Problem No. 20 The force acting on the center pole of the setup has been computed with the three investigated formulations and compared with measurements. Hereby, different ampereturns were used in order to investigate the accuracy of the formulations at different saturation states of the setup. For computation of F1 the integration surface that sur-

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W. Hafla et al. / Force Computation with the Integral Equation Method

A)

B)

Figure 2. Investigated setups: (A) TEAM Workshop Problem No. 20 and (B) TEAM Workshop Problem No. 33a.

rounds the center pole shown in Fig. 2a has been used. It was discretized with 2868 second-order triangles. The results shown in Fig. 3 indicate that the two Maxwell formulations F1 and F2 give the most accurate results. Especially at higher ampereturns the energy formulation F3 gives less accuracy than F1 and F2 . The two formulation F2 and F3 can obviously be used for this setup though in principle, they are valid for linear materials only. TEAM Workshop Problem No. 33a Due to the high permeability of the above setup, the results obtained with F2 and F3 are of the relatively same accuracy. Therefore the TEAM Workshop Problem No. 33a has been investigated. Hereby, the relative permeability of the sample is only 2.5. This leads to large tangential field strength components and a vanishing volume force density. Since the magnetic pressures p2 and p3 differ therefore significantly, this setup allows for deciding which formulation is more applicable for force computation. The results are depicted in Fig. 4. Interestingly, there is no need to compare the result with measurements since the magnetic pressure computed with the energy formulation is obviously wrong. The pressure doesn’t cause a pulling but a pushing force. This is, because due to the large values of H t the pressure pt is larger than pn so when (5) is computed p3 < 0 . Conclusions The integral equation method is applicable for force computation with nonlinear magnetostatic problems. Depending on whether a force formulation is based on Maxwell’s stress tensor or on the energy principle, different analytical results are obtained. Since

W. Hafla et al. / Force Computation with the Integral Equation Method

97

Figure 3. Forces with TEAM workshop problem No. 20 computed with different formulations.

A)

B)

Figure 4. Magnetic pressures computed with A) Maxwell formulation ( p2 ) and B) energy formulation ( p3 ) .

the magnetic force that is acting on a magnetizable body is well-defined it is clear at least one of the formulations has to give wrong results. By investigating the setup with extremely low relative permeability it has been shown that only the Maxwell formulation seem to give accurate results. References [1] O. Bíró, “Solution of TEAM Benchmark Problem #20 (3-D Static Force Problem),” in Proceedings of the Fourth International TEAM Workshop, pp. 23-25, Miami, USA, 1993. [2] O. Barré, P. Brochet, M. Hecquet, “Experimental validation of magnetic and electric local force formulations associated to energy principle,” IEEE Trans. Mag, vol. 42, issue 4, pp. 1475-1478, April 2006. [3] L. Han, L. Tong, “Integral equation method using total scalar potential for the solution of linear or nonlinear 3D magnetostatic field with open boundary,” IEEE Transactions on Magnetics, 30(5): 2897-2900, 1994. [4] A. Buchau, W. M. Rucker, O. Rain, V. Rischmüller, S. Kurz, and S. Rjasanow, “Comparison Between Different Approaches for Fast and Efficient 3D BEM Computations,” IEEE Transactions on Magnetics, 39(3):1107-1110, 2003. [5] W. Hafla, A. Buchau, F. Groh, and W. M. Rucker, “Efficient Integral Equation Method for the Solution of 3D Magnetostatic Problems,” IEEE Transactions on Magnetics, vol. 41, no. 5, pp. 1408-1411, 2005.

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Chapter B. Computer Methods in Applied Electromagnetism B1. Computation Methods

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101

Numerical Simulation of Non-Linear Electromagnet Coupled with Circuit to Rise up the Coil Current a

Slawomir STEPIEN a, Grzegorz SZYMANSKI a and Kay HAMEYER b Chair of Computer Engineering, Poznan University of Technology, Poznan, Poland b Institut fur Elektrische Maschinen, IEM RWTH – Aachen, Aachen, Germany Abstract. In this paper a numerical model and simulation of non-linear field – circuit system dynamics is presented. The system includes electromagnet coupled with two RC circuits. The RC circuit performs a current rise acceleration and improves the electromagnet dynamics. In proposed system a value of capacitance C impact on electromagnet dynamics is numerically determined.

Introduction The applications of electromagnet drives are widespread. They are used in electronic and electromechanical consumer products, telecommunication, automatics, computer technology, etc. The electromagnet actuators can be excited from impulse voltages generated by special electronic drivers. Then device response as a plunger displacement can be controlled by width or frequency modulation of input voltages [4]. In several cases an improvement of electromagnet dynamics is necessary [7]. This improvement is related with reduction of the motion time. Then solution with external circuit composed of serial resistance and parallel capacitor connected to actuator winding is proposed. The analysis is based on field – circuit – movement model [1,3,5,6]. Figure 1 shows cross section of the cylindrical electromagnet geometry and configuration of the external circuit part. The RC circuit is connected to each coil and significantly changes an input impedance of the system. The inductivity and electromotive force change in time during plunger motion. The RC circuit connected to coil input makes rise of the coil current faster when additional resistance R and capacitance C are well matched and voltage excitation is increased. In this article the problem consists of finding relationship between value of capacitance C and time needed to set plunger in extreme position. Thus modeling and nuR

R

spring

u1

C

C

plunger

coil 1

coil 2

Figure 1. An electromagnet geometry with coil input circuits.

u2

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S. Stepien et al. / Numerical Simulation of Non-Linear Electromagnet

merical simulation of mentioned system is presented. There are given advantages and disadvantages of the proposed method to improve the device dynamics. Also as a new contribution an interesting direct coupled field – circuit model of the device is widely presented and discussed. Modeling Technique The actuator is modelled in 3D domain using Maxwell equations to formulate the field model. FEM approximation is used to discretise the domain. Magnetic vector potentials and electric scalar potentials are used as variables of electromagnetic field [2,7]. The boundary value problem is defined by Eqs (1)–(2):

⎞ ⎛1 ⎛ ∂A ⎞ ∇ × ⎜⎜ ∇ × A ⎟⎟ + σ ⎜ + ∇V ⎟ − σ (v × (∇ × A )) = j ⎝ ∂t ⎠ ⎝μ ⎠

(1)

∂A ⎛ ⎞ ∇ ⋅ ⎜ σ (v × (∇ × A )) − σ − σ∇V ⎟ = 0 ∂t ⎝ ⎠

(2)

where A is a magnetic vector potential defined in mesh nodes, V is a electric scalar potential defined in nodes of conductive regions, j is a current density, σ represents conductivity, μ represents permeability and v is a velocity of moving body. To define external electric circuit connected to the actuator windings, an electric circuit equations are defined by Eqs (3)–(4):

RC

∫

∫

l1

RC

(

)

d2 d Adl + Adl + RL1 + R ⋅ i L1 = u1 dt 2 dt d2 dt 2

l1

∫ Adl + dt ∫ Adl + (RL 2 + R )⋅ iL 2 = u2 d

l2

(3)

(4)

l2

where R is a resistance of external circuit, C is capacitance of the external circuit, RL1 and RL 2 represent resistance of windings, i L1 and i L 2 are winding currents, u1 and u are a circuits excitations. 2 Obtained second order differential equations shows that in each circuit exist two elements which are able to magazine energy and can transfer it between themselves. These second order equations can be reduced to first order system of equations by using additionally defined scalar variables w. In this way a state space formulation of circuit equations [1,5] is given by:

d Adl = w1 dt

∫

l1

dw1 dt

=−

1 w − RC 1

(5)

R L1 + R RC

i L1 +

1 u RC 1

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S. Stepien et al. / Numerical Simulation of Non-Linear Electromagnet

d Adl = w2 dt

∫

l2

(6)

1 =− w dt RC 2

dw2

−

RL 2 + R RC

iL 2

1 u + RC 2

In above equations a voltage sources have an impact on winding currents i and L1

i L 2 , winding voltages w1 and w2 ,

dw1

dw2

which can be defined as velocity of dt dt windings voltage change or linkage flux acceleration related to the windings. After discretisation in time and space a global matrix system is obtained. The boundary value equations (1)–(2) produces matrices related to vector potential A and scalar potential V. In non-linear case the matrix related to A also depends on μ . Discretisation of (5)–(6) produces additional rows in global matrix system. The equations (1)–(2) and (5)–(6) have one common variable A and for this reason the field equations can be coupled with circuit equations directly. Then current density j is eliminated from (1), because winding currents are a functions of vector potential A. Taking above into account a coupled field – circuit system of matrix equations becomes: ⎡C( μ ) + D E ⎢ F H G + ⎢ ⎢ L1 0 ⎢ ⎢ 0 0 ⎢ ⎢ L2 0 ⎢ ⎢ 0 0 ⎢⎣

0 0

0 0

0 − Δt R Δt L1 + R + 1 Δt RC RC 0 0 0

0

and

0 0

0 0

⎤ t + Δt ⎤ ⎥ ⎡A ⎥ ⎢ V t + Δt ⎥ ⎥ ⎥⎢ 0 0 ⎥ ⎢ w1t + Δt ⎥ ⎥ ⎢ t + Δt ⎥ = 0 0 ⎥ ⎢ i L1 ⎥ ⎥ ⎢ wt + Δt ⎥ 0 − Δt ⎢ 2 ⎥ R L 2 + R ⎥ ⎢ t + Δt ⎥ Δt ⎥ ⎣ iL2 ⎦ + 1 Δt RC ⎥⎦ RC

⎤ ⎡ D ⋅ At ⎥ ⎢ t F⋅A ⎥ ⎢ ⎥ ⎢ L1 ⋅ A t ⎥ ⎢ ⎢ Δt u t + Δt + w t ⎥ 1⎥ ⎢ RC 1 ⎥ ⎢ t L2 ⋅ A ⎥ ⎢ ⎢ Δt t + Δt t⎥ u w + 2⎥ ⎢⎣ RC 2 ⎦

(7) where matrices C(μ), D, E, F, G, and H are obtained from discretisation of following terms: ⎞ ⎛1 1 1 ∇ × ⎜⎜ ∇ × (..) ⎟⎟dΩ − σ(v × (∇ × (..)))dΩ , σ (..)dS , σ (..)dΩ , σ∇(..)dΩ , μ t Δ Δ t ⎠ ⎝

∫

∫

Ω

Ω

∫ σ∇(..)dS S

tion of

∫

l1

∫

Ω

∫

Ω

∫ S

and − σ (v × (∇ × (..) ))dS . Rows L and L are derived from discretisa2 1

Adl and

∫ S

∫ Adl .

l2

The matrices and vectors defined in equation system (7) are composed of real coefficients and have following dimensions: A ∈ R n , V ∈ R m , C, D ∈ R n×n , E ∈ R n×m ,

F, H ∈ R m×n , L1 , L 2 ∈ R1×n and i L1 , i L 2 , u1 , u 2 , w1 , w2 ∈ R1 .

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This complex system of equations is non-symmetric and solved using bi-conjugate gradient method (BiCG) for large and sparse matrix equations. The non-linearity problem is solved using Newton-Raphson method based on B-H curve continuos approximation [7]. At each time step, the magnetic force is determined using Maxwell stress tensor. The force is a non-linear function of potentials A and calculated locally in each element of discretisation. The total force which acts on movable body is a sum of local forces [7]. The force density is given by following formula:

f = ∇⋅T

(8)

where T denotes Maxwell’s stress tensor. Then total force is defined as:

∫

F = f dΩ

(9)

Ω

[

The total force is determined as a vector of three components F = F r

Fϕ

Fz

]. T

As shown in Fig. 1, the motion of movable armature takes place only along axis z, thus to solve the 1 DOF problem displacement s and velocity v are chosen as state of z

z

mechanical motion and component F is used as excitation. z

d ⎡s z ⎤ ⎡ 0 ⎢ ⎥=⎢ k dt ⎣v z ⎦ ⎢ − ⎣ m

1 ⎤ ⎡s ⎤ ⎡ 0 ⎤ b ⎥ z + ⎢ 1 ⎥F − ⎥ ⎢v ⎥ ⎢ ⎥ z m ⎦⎣ z ⎦ ⎣ m ⎦

(10)

where k is a spring stiffness and b is a damping coefficient. Discrete form of equations (10) is obtained using Eulers recurrence method. − Δt ⎤ ⎡ s t + Δt ⎤ ⎡ ⎡ 1 b ⎥⎢ z ⎥ = ⎢ ⎢ k ⎢⎣ Δt m 1 + Δt m ⎥⎦ ⎣⎢v tz+ Δt ⎦⎥ ⎢v tz ⎣⎢

⎤ s tz ⎥ 1 t + Δt ⎥ + Δt Fz m ⎦⎥

(11)

Summing up this mathematical description of analysed system, in each iterative step take place calculation of field – circuit equation system (7), force (9) and motion equation (11). Numerical Analysis The developed field – circuit model of presented system has been used to investigate the relationship between capacitance of the external RC circuit and time of the plunger motion. The external circuit resistance is assumed R = 2Ω and capacity is taken from set C = {100 nF, 1 μF, 10 μF, 47 μF, 100 μF, 470 μF, 1000 μF}. Presented device is made from Armco. The specific gravity of armature is equal to 7800 kg/m3, substitute stiffness k = 800 N/m of springs, and damping coefficient b = 2 Ns/m. Voltage square-wave

S. Stepien et al. / Numerical Simulation of Non-Linear Electromagnet

105

 Figure 2. Mesh of analysed actuator.

Figure 3. Measured Armco B-H curve.

excitation is applied on device, where actuator windings have 424 turns and resistance RL = 2,05 Ω. Bellow is presented the B-H curve of the Armco. To demonstrate the dynamics improvement of analysed actuator, two kind of simulations are compared: firstly is examined system free of external circuit where the voltages with amplitude u = 12 V and frequency f = 250 Hz are applied directly on actuator windings and next actuator is equipped with external circuits (as shown in Fig. 1). In the second case voltages which control the plunger motion are twice as big and guarantee the same windings current flow in steady state. The voltages in both cases are shown bellow. Firstly the voltage u cause a current flow in right winding and produces force 2 which moves plunger in right direction (s = 3 mm). When u is zero, then voltage u 2 1 controls the plunger in left position (s = –3 mm). Finally when both voltages are zero then movable armature should reach middle position, in other starting point of plunger motion. The results of analysis two extreme cases (without external circuit and with external circuit for C = 100 nF) are given bellow.

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S. Stepien et al. / Numerical Simulation of Non-Linear Electromagnet

a)

b)

Figure 4. An input square-wave voltages: a) without RC circuit b) with RC circuit.

Figure 5. Actuator responses on applied voltages.

As compared in Fig. 5 time needed to move plunger from position s = 0 mm to s = 3 mm is decreased two times by usage external RC circuit. This rule also is being in force when plunger is moved from right s = 3 mm to left s = –3 mm. In this case the time is somewhat larger then previously. Unfortunately, when actuator is autonomous, in other free of the excitations, an energy concentrated in capacitance does not allow to plunger return in position s = 0 mm immediately. The return is delayed. The situations were examined for various capacities C = {100 nF, 1 μF, 10 μF, 47 μF, 100 μF, 470 μF, 1000 μF}. To estimate capacity impact on motion time of the plunger during actuator starting to position s = 3 mm is examined as function ts = f(C). As can be seen in above plots, the usage of external circuits improves the electromagnet dynamics during starting phase. The reason is due to the fact that the inertia of input circuits is reduced. However, the dynamics also depends on input capacity. As shown in Fig. 6, the motion time reduction is the best for small values of capacity. Greater capacitance delays plunger motion. So, the delays of motion can be controlled by value of capacity, too.

S. Stepien et al. / Numerical Simulation of Non-Linear Electromagnet

a)

107

b)

Figure 6. Electromagnet starting: a) displacement b) capacity – stating time relationship.

Conclusion This work presents numerical analysis of field – circuit system considering motion and magnetic non-linearity. The simulation experiment shows that additional circuit with capacitance improves dynamics of an electromagnet. However there exist an exact relationship between value of capacity and plunger displacement. In this paper as a new contribution a strongly coupled field – circuit model which occurs with state space description suggests that presented methodology works in calculations. The proposed method can be used to design a electromagnetic drives and may contribute to the improvement of the drives dynamics. References [1] A. de Oliveira, R. Antunes, P. Kuo-Peng, N. Sadowski and P. Dular, Electrical machine analysis considering field – circuit – movement and skewing effects, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 23, No. 4, pp. 1080-1091, 2004. [2] D. Lowther, Automating the Design of Low Frequency Electromagnetic Devices – A Sensitive Issue, COMPEL – International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 22, No. 3, pp. 630-642, 2003. [3] K. Hameyer, Field – circuit coupled models in electromagnetic simulation, Journal of Computational and Applied Mathematics, Vol. 168, pp. 125-133, 2004. [4] L. Nowak, J. Mikolajewicz, Field-circuit model of the dynamics of electromechanical device supplied by electronic power converters, COMPEL – International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 23, No. 4, pp. 977-985, 2004. [5] H. De Gersem, R. Mertens, D. Lahaye, S. Vandewalle and K. Hameyer, Solution Strategies for Transient, Field-Circuit Coupled Systems, IEEE Trans. Mag., Vol. 36, No. 4, pp. 1531-1534, 2000. [6] S. Lepaul, J. Sykulski, C. Biddlecombe, A. Jay, and J. Simkin, Coupling of motion and circuits with electromagnetic analysis. IEEE Trans. Mag. Vol. 33, No. 3, pp. 1602-1605, 1999. [7] S. Stępień, A. Patecki, Modeling and Position Control of Voltage Forced Electromechanical Actuator, COMPEL – International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 25, No. 2, pp. 412-426, 2006.

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Numerical Calculation of Power Losses and Short-Circuit Forces in Isolated-Phase Generator Busbar a

Dalibor GORENC a and Ivica MARUSIC b Koncar – Electrical Engineering Institute, Fallerovo setaliste 22 Zagreb, Croatia [email protected] b HEP Production Ltd.PA HPP SOUTH Split, Zakucac HPP, Croatia [email protected] Abstract. Metal enclosed generator busbars are applied in electric power plants for the transmission of electric power from generator to transformer. The losses in conductors and enclosures due to rated current are the basis for the calculation of steady state temperature-rises. On the other hand, the maximum force on the conductor due to short-circuit current is an important parameter in determination of mechanical stresses in the conductors and supports. In this paper, using a program based on finite element method (FEM), power losses and short-circuit forces are calculated in arrangement of isolated-phase generator busbar.

Introduction In isolated-phase busbar (IPB), each phase conductor is enclosed by an individual metal enclosure, separated from adjacent conductor enclosures by air (Fig. 1, Fig. 2). The enclosures are electrically continuous and short-circuited at both ends, and grounded at one end. The conductor currents induce longitudinal currents in the enclosures of almost the same magnitude but of the opposite direction to the conductor currents [1]. Thus, only a small percentage of the total magnetic field extends outside the bonded enclosures. This results in a considerable reduction of the electromagnetic forces between phases under short-circuit conditions [2,3] and avoids eddy currents in neighboring steel structures under normal operating conditions.

Field Equations The calculation of power losses and short-circuit forces is based under the following assumptions: − − −

longitudinal dimension of busbar is significantly greater then the cross section and a two-dimensional field analysis may be applied; characteristics of materials are constant; displacements currents are neglected.

The electromagnetic field, under described assumptions, is described by the following equations:

D. Gorenc and I. Marusic / Numerical Calculation of Power Losses and Short-Circuit Forces Phase conductor

Short-circuit plate

109

Enclosure phase conductor phase A

phase B

phase C

y enclosure x

Figure 1. IPB with continuous enclosures.

Figure 2. Cross section of IPB.

  ∇× H = J

(1)

  ∂B ∇× E = − ∂t

(2)

 ∇⋅B = 0

(3)

    where H is the magnetic field, J is the current density, E is the electric field, and B is the magnetic flux density. For linear materials, the constitutive equations are:   B = μ⋅H

(4)

  J =σ ⋅E

(5)

where μ is permeability, and σ is conductivity. It is convenient to define a vector mag netic potential A by:   B = ∇× A

(6)

These equations can be combined into vector Helmholtz equation with the magnetic vector potential as the unknown variable:    1 ∂A ∇ × ∇ × A = Js −σ μ ∂t

(7)

 In Eq. (7), the current density has been split into prescribed sources J s and the in duced currents σ ∂A / ∂t . In two-dimensional problems only z components of A and J exist. Vector Helmholtz equation can be simplified to scalar one:

−∇

∂A 1 ∇Az = J s − σ z ∂t μ

(8)

Using a FEM based program MagNet, Eq. (8) is solved numericaly in a frequency or time domain, depending on whether the problem is steady-state or transient.

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D. Gorenc and I. Marusic / Numerical Calculation of Power Losses and Short-Circuit Forces

Table 1. Main geometry data of the analyzed model Width of conductor (mm) 260

Thickness of conductor (mm) 12.5

Outer diameter of enclosure (mm) 720

Thickness of enclosure (mm) 4

Centre-line distance between conductors (mm) 1000

Figure 3. Model of busbars.

Figure 4. Circuit connection of conductors and enclosures.

Model of Busbar The model of busbar is given on Fig. 3. Each phase conductor consists of an octogonal duct made up of two sections held together at regular intervals by welded spacers, thus maintaining a constant clearance. In order to calculate the current distribution in each conductor section, it is assumed that they are electrically separated along the z-axis. The phase conductor is concentrically enclosed in a welded circular enclosure. The material of conductors and enclosures is aluminum with conductivity of 34.2 m/Ωmm2, and permeability μ0. The conductors are connected to the external current source while the enclosures are short circuited at both ends and grounded at one end (Fig. 4). Total

D. Gorenc and I. Marusic / Numerical Calculation of Power Losses and Short-Circuit Forces

111

Figure 5. Finite element mesh.

current in each phase conductor is known and is given by expression (9)–(11) or (12)–(14). The domain of computation is the rectangular area with busbar and enclosing air, dimensions 4.4 × 2.4 m. The boundary condition was A = 0 on the borders of the area (Dirichlet) and the number of triangular elements was 165064.

Calculation of Power Losses Under normal operating conditions, driving currents in phase conductors are varying sinusoidally in time with a frequency of 50 Hz. In a frequency domain they can be expressed as phasors: i1 = I 2 ⋅ e j 0

i2 = I 2 ⋅ e − j ⋅120

(9) 

i3 = I 2 ⋅ e− j ⋅240



(10) (11)

Here, I is the RMS-value of the rated current in conductors. Results Figure 6 shows the distribution of losses (W/m3) in upper part of conductor and enclosure of phase B for the rated current of I = 7000 A. Total losses in conductors and enclosures per meter of busbar are presented in Table 2. Ratio of ac to dc resistance shows that the skin effect factor for the analyzed conductor design is equal to 1.12.

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D. Gorenc and I. Marusic / Numerical Calculation of Power Losses and Short-Circuit Forces

Figure 6. Distribution of losses in conductor and enclosure of phase B (W/m3). Table 2. Total losses in conductors and enclosures (W/m) A-phase conductor

B-phase conductor

C-phase conductor

A-phase enclosure

B-phase enclosure

C-phase enclosure

167

167

167

155.3

159.2

163.5

Calculation of Short-Circuit Forces Three-phase symmetrical short-circuit was analyzed because it causes the greatest dynamic stress. Under short-circuit condition, the fault currents in phase conductors are function of time as follows [3]: ⎛ −t ⎞ ⎡ ⎤ ⎜ ⎟ i1 (t ) = I k ⋅ 2 ⋅ ⎢cos (ϕ ) ⋅ e⎝ T ⎠ − cos (ω t + ϕ ) ⎥ ⎢⎣ ⎥⎦

(12)

⎛ −t ⎞ ⎡ ⎤ ⎜ ⎟ i2 (t ) = I k ⋅ 2 ⋅ ⎢cos (ϕ − γ ) ⋅ e⎝ T ⎠ − cos (ω t + ϕ − γ ) ⎥ ⎢⎣ ⎥⎦

(13)

⎛ −t ⎞ ⎡ ⎤ ⎜ ⎟ i3 (t ) = I k ⋅ 2 ⋅ ⎢cos (ϕ + γ ) ⋅ e⎝ T ⎠ − cos (ω t + ϕ + γ ) ⎥ ⎣⎢ ⎦⎥

(14)

where: I k is the RMS-value of the steady state short-circuit current, ϕ is the switching angle, T is the time constant of direct current component and γ is the phase angle between currents (γ = 120°). Results In the case of unshielded conductors (enclosures conductivity σ = 0), the maximum force acts on the central conductor, and appears about 10 ms after the beginning of

D. Gorenc and I. Marusic / Numerical Calculation of Power Losses and Short-Circuit Forces

113

short-circuit (Fig. 7). The switching angle for the maximum force is ϕ = 0°. In the case of IPB, maximum force acts also on the central conductor but for switching angle equal to ϕ = 30° [3]. Due to enclosure shielding effect, the maximum force appears 95 ms after the beginning of short-circuit (Fig. 8) and is reduced by the factor of 7 as compared with arrangement without enclosures. However, two section of the B-phase conductor will be mutually attracted by the force which is of the same order of magnitude as the conductor force without enclosures. This fact should be considered in determination of distance between welded spacers along the phase conductor. Forces on enclosures have almost the same magnitude as corresponding conductor forces but act in the opposite direction.

Figure 7. Force on conductor B, without enclosures (Ik = 50 kA, T = 45 ms and ϕ = 0°).

Figure 8. Force on conductor B, with enclosures (Ik = 50 kA, T = 45 ms and ϕ = 30°).

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D. Gorenc and I. Marusic / Numerical Calculation of Power Losses and Short-Circuit Forces

Figure 9. Force on conductor A, with enclosures (Ik = 50 kA, T = 45 ms and ϕ = 30°).

Figure 10. Force on conductor C, with enclosures (Ik = 50 kA, T = 45 ms and ϕ = 30°).

Conclusion Presented numerical procedure enables fast and accurate calculation of power losses and short-circuit forces for any geometry of metal enclosed busbars, thus taking into account the following factors: − − − −

detailed shapes and dimensions of the conductors and the enclosures, distributions of current among multiple conductors in one phase, magnetic field interaction between conductors and enclosures, proximity effect with conductors in other phases.

Similar methodology can be applied for three-phase segregated and nonsegregated type of busbars in which all three phase conductors are in a common metal enclosure with or without metal barriers between phases. In this type of busbars, the enclosure

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115

has a shape of rectangle and the phase conductor may consist of one or more conductors of different shapes and dimensions.

References [1] W.F. Skeats, N. Swerdlow, Minimizing the Magnetic Field Surrounding Isolated-Phase Bus by Electrically Continuous Enclosures, AIEE Transactions PAS 81, pp. 655-667, February 1963. [2] W.R.Wilson, L.L.Mankoff, Short-Circuit Forces in Isolated-Phase Busses, AIEE Transactions PAS, April 1954. [3] P. Dokopoulos, D. Tambakis, Analysis of Transient Forces in Metal Clad Generator Buses, IEEE Trans. on Energy Conversion, Vol. 6, No. 3, September 1991. [4] M.R. Shah, G. Bedrosian, J. Joseph, Steady-State Loss and Short-Circuit Force Analysis of a ThreePhase Bus Using a Coupled Finite Element+Circuit Approach, IEEE Transactions on Energy Conversion, Vol. 14, No. 4, December 1999.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-116

Numerical Methods for Calculation of Eddy Current Losses in Permanent Magnets of Synchronous Machines Lj. PETROVIC a, A. BINDER a, Cs. DEAK a, D. IRIMIE c, K. REICHERT d and C. PURCAREA b a Inst. of Electrical Energy Conversion, Darmstadt University of Technology, Landgraf-Georg-Strasse 4, D-64283 Darmstadt, Germany E-mail: [email protected], [email protected], [email protected] b Inst. of Power Electronics & Control of Drives, Darmstadt University of Technology, Landgraf-Georg-Strasse 4, D-64283 Darmstadt, Germany c Faculty of Electrical Engineering, Technical Univ. Cluj-Napoca, Str. C. Daicoviciu 15, 400020, Romania d emeritus ETH Zuerich, now: Schartenfelsstr. 1B, CH-5430 Wettingen, Switzerland Abstract. Eddy current losses in rotor permanent magnets (PM) of synchronous machines are calculated for sinusoidal stator currents and for PWM inverter supply. Three calculation methods are compared in the FE environment: a) timestepping method, b) quasi-static method, c) semi-analytical post-processing. These 2D methods are with end effect coefficients, and they consider the time variation of currents and of the rotor position. Whereas method a) includes the variation of flux-density over the magnet cross section and the reaction field of the eddy currents, method b) is neglecting the reaction field. Method c) in several variants features either neglecting of the eddy current reaction field or an averaging of the flux density along the magnet width or height. Neglecting the reaction field is possible for materials with low conductivity and low permeability like rare-earth magnets for low to medium frequencies up to several kHz. The quasi-static methods need less computation time, but depend on the machine geometry like stator MMF wave length, slot pitch, segmented vs. massive magnets and small or big magnet height. The comparison of methods a), b), c) is given for two different stator geometries of permanent magnet synchronous machines with open vs. semi-closed slots and surface-mounted vs. buried magnets.

Introduction In PM synchronous machines (PMSM) the stator slot openings cause already at no-load a flux variation in the rotor magnets, which leads to an induced voltage and eddy current losses. At load the stator magneto-motive force (MMF) space harmonics, excited by the sinusoidal stator current, increase these losses. The additional stator current ripple due to PWM inverter supply amplifies this effect. For calculating these losses, a two-dimensional (2D) finite element method (FEM) may be used, if the 3D end effects, like the axial segmentation, are considered by an appropriate end effect coefficient. a)

The time-stepping method solves the Maxwell equations for an arbitrary current waveform per stator phase and a moving rotor, neglecting only the dis-

Lj. Petrovic et al. / Numerical Methods for Calculation of Eddy Current Losses

b)

c)

117

placement current term in the first Maxwell equation (Ampere’s law), which is of no relevance in the frequency range of interest of up to several kHz [3]. The magneto-static finite element method has a shorter computation time, but neglects the eddy current reaction field. In rare earth magnets the low permeability and conductivity in the considered frequency range gives rather small eddy current densities, hence justifying the method. This method can be faster and is less expensive. In case that a quasi-static 2D finite element code is not offering a current density calculation acc. to b), an additional semi-analytical post-processing is possible to evaluate these losses. These semi-analytical methods are considered in several variants in this paper and are compared with the results of a) and b) [1].

Comparison of Calculation Methods In the following for simplification rectangular magnets are considered with a magnet width bM in x-direction, a height hM in y-direction, and a magnet length lM in z-direction (Fig. 1a). The results can be generalized for more arbitrary magnet geometries like shell or bread-loaf shapes in an axial-symmetrical cylindrical coordinate system. For 2D calculations the axial end effect must be taken into account. The “equivalent” conductivity is smaller than real conductivity of the material, if one considers the eddycurrents to be caused by the “voltage forced” situation [2]. The normal component (y-component) of the air gap magnetic flux density, excited e.g. by a sinusoidal varying current time harmonic Iˆ ⋅ cos(ω s t ) , shows as a Fourier series of different field waves, which penetrate the magnet, beside the synchronously moving wave usually a dominant harmonic wave By ( y ) ⋅ cos( x ⋅ 2π / λ − ω s t ) , which causes most of the eddy currents. Considering its wave length λ in relationship to the magnet width bM and the axial length of the magnet lM, the equivalent decrease of the magnet conductivity κM due to the longer eddy current path in 3D is given with the end effect coefficient k as κ M,eq = κ M ⋅ k , 0 ≤ k ≤ 1 [4]. k = 1−

⎛ π ⋅ lM ⎞ 2ζ ⋅ th ⎜ ⎟ π ⋅ lM ⎝ 2ζ ⎠

ζ = λ / 2 for λ / 2 ≤ bM or ζ = bM for λ / 2 > bM

(1) a) The time-step calculation solves the first three Maxwell equations and the con stitutive laws within the magnet in a 2D x-y-coordinate system with H ( x, y, t ) ,       J ( x, y, t ) , B ( x, y, t ) , E ( x, y, t ) and H = ( H x , H y , 0) , B = ( Bx , By , 0) , J = (0, 0, J z ) ,  E = (0, 0, Ez ) as     rotH = J rotE = −∂B / ∂t

 divB = 0

  J = κ M,eq E

  B = μM H ,

(2)

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Lj. Petrovic et al. / Numerical Methods for Calculation of Eddy Current Losses

resulting

in

 divJ = 0

b

and

the

losses

per

magnet

h

l T M M PM = M ∫ dt ∫ dx ∫ dy ⋅ T 0 0 0

J z2 ( x, y , t ) / κ M,eq , where T = 2π / ω is the longest time period of the calculated eddy

current density. For the time-step calculation the FE program FLUX2D was used [3].    b) The quasi-static method is neglecting rotH = J . With the vector potential A   and B = rotA the eddy-current density in each finite element of the mesh is derived from (2) as   J = −κ M,eq ⋅ dA / dt .

(3)

c) The semi-analytical post-processing of a magneto-static finite element field solution can be done in different ways. c) (i) A): For “small” magnets (e.g. segmented magnets) with bM < λ/2 the variation of the flux density along the magnet width is replaced by its average value b 1 M B y ( y, t ) = By ( x, y, t )dx . Then the flux variation Φ ( y, t ) = B y ( y , t ) ⋅ bM lM at each bM ∫0 coordinate plane y = const for 0 ≤ y ≤ hM can be given as a Fourier sum Φ ( y, t ) = ∞

∑Φ ( y) ⋅ sin(kω t ) . The eddy current loss formula for a plane material of thickness bM k =1

k

including the eddy current reaction field (4), (5) is applied [5]. If one takes the result only at the top of the magnet y = hM, where the y-component of the field is maximum, one overestimates the losses by a +60% [1], but by averaging with Simpson’s formula (6) the results agree well with methods a) and b). The losses due to Bx are very small, as the flux density in rotor magnet is oriented mainly radial.

PM ( y ) = hM ⋅ bM3 ⋅ lM ⋅

K m,k =

κ M,eq 24

3 shξ k − sinξ k ⋅ ξ k chξ k − cosξ k

∞

⋅ ∑ (kω ) 2 By , k ( y ) 2 K m,k k =1

By , k ( y ) = Φk ( y ) /(bM ⋅ lM ) (4)

ξ k = bM ⋅ μ M ⋅ κ M,eq ⋅ k ⋅ ω / 2

P M = ( PM ( y = 0) + 4 PM ( y = hM / 2) + PM ( y = hM )) / 6

(5)

(6)

c) (i) B): As alternative to (4) for “narrow” magnets the variation of By ( y, t ) = Φ ( y, t ) /(bM ⋅ lM ) with time is taken directly into account, neglecting the influence of the reaction field of eddy currents: 3

PM ( y ) = lM ⋅ hM ⋅

bM ⋅ (d B y ( y, t ) / dt ) 2 ⋅ κ M,eq 12

(7)

c) (ii): For “broad” magnets with bM > λ/2 (e.g. massive magnet pieces per pole) the averaging of the flux density will lead to wrong results. Hence method b) is

Lj. Petrovic et al. / Numerical Methods for Calculation of Eddy Current Losses

a)

b)

119

c)

Figure 1. a) Cross-section of magnet, b) Cross section of Motor A and c) Motor B.

adopted, but as several magneto-static FEM solvers do not give direct access to the vector potential for post-processing, Az had to be reconstructed by x

Az ( x, y, t ) = ∫ By ( x, y, t ) ⋅ dx → J z ( x, y, t ) = −κ M,eq ⋅ dAz ( x, y, t ) / dt ,

(8)

0

before the losses can be evaluated according to the above noted method. In our case the loss evaluation was again done on three circumferential levels 0, hM/2 and hM. Method (ii) appears to be the most exact one of the methods c), because (i) does not give correct values for bM > λ/2. For methods b) and c) the FE program FEMAG was used [6], and for c) alternatively also FLUX2D.

Investigated Machine Topology Two PMSM, called Motor A and Motor B (Fig. 1 b, c), with tooth-coil three-phase winding for 45 kW, 1000/min, 430 Nm, which can operate at constant voltage 400 V up to 3000/min via field weakening with negative d-current, were investigated [1]. With tooth-coil windings the content of MMF harmonics in PMSM is increased, causing increased magnet losses at load. Both machines are water-jacket cooled, and have been built in our lab, featuring 16 poles, a laminated stator and rotor iron stack of the length lFe = 80 mm and NdFeB rotor magnets. Per pole 7 segmented magnets in circumference and 6 in axial direction are used (dimensions bM × lM = 3.6 x 30 mm) with a magnet height hM = 4.8 mm (Motor A) and 4.7 mm (Motor B). The magnet conductivity is κM = 7·105 S/m, the permeability μM = 1.05μ0. Motor A has open stator slots, q = ¼ slots per pole and phase with profiled copper coils on the wider teeth and buried magnets in the rotor. Motor B has semi-closed stator slots, q = ½ slots per pole and phase with round wire copper coils on each tooth of identical width and surface mounted magnets. A second rotor with surface mounted magnet shells of magnet height 4.7 mm and four rows of axial shell length 45 mm with the same pole coverage 0.78 was built, fitting to the stator of Motor B, called Motor B′. For Motor A a rotor with one massive magnet per pole (dimensions bM × lM = 25.2 × 45 mm, four axial magnet rows) was designed (Motor A′), but was only simulated, not manufactured. In FLUX2D the massive magnets per pole are modelled by redefining the face regions of the 7 magnet segments per pole to be 1 entity (Fig. 2 a, b). For Motors A and B κM,eq = 6.25·105 S/m was used, for Motors A′ and B′ κM,eq = 4.37·105 S/m.

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Lj. Petrovic et al. / Numerical Methods for Calculation of Eddy Current Losses

a)

b)

Figure 2. Calculated eddy current density with Method a) in the magnets at full load 45 kW, 1000/min: a) for Motor B (segmented magnets), b) for Motor B′. Here the 7 magnets are regarded as one magnet entity by the FEM program. Eddy current density range: a) J = –1.3 … + 0.8 A/mm2; b) J = – 1.8 … + 1.1 A/mm2.

Table 1. Calculated losses in the most left magnet segment of a pole for Motor A at 1000 rpm, rated sinusoidal current, 45 kW. Results of Method c) (i) A) with field variation at the top of the magnet (T) and averaged for top, middle and bottom (TMB) of the magnet in comparison to Method a) Method

c) (i) A) T

c) (i) A) TMB

a)

Losses PM,left [W]

0.190

0.097

0.121

Eddy Current Loss Calculation Results in the Permanent Magnets Investigation of Radial Variation of Flux Density Over a Magnet Segment Methods a) and c) (i) A) are compared for Motor A in order to clarify the influence of the change of flux density in radial direction within the “narrow” magnets on the eddy current losses, using the FEM program FLUX2D. The Fourier sum of Method c) (i) A) considered 47 flux harmonics, but already 5 harmonics would have been sufficient. The sinusoidal currents for operation at load are impressed into the stator winding via three current sources. The initial phase angle of the current was adjusted with respect to the initial rotor position to get the rated torque 430 Nm. In Table 1 the calculated losses PM,left are given for the most left magnet segment per pole, for which either the flux variation at the top (label: T) y = hM, or the average of the loss calculation for three levels y = hM (top), y = hM/2 (middle), y = hM (bottom) (label: TMB) were considered. Considering the field variation of By only at the top of the magnet gives by +60% too large losses, as the decrease of the field to the bottom of the magnet is neglected. Therefore the presented loss values in [1] are by about 60% too big. Considering “TMB”, the losses are by 20% too small, as the influence of the field component Bx is neglected. This influence is seen in the central magnet in Fig. 2a. The eddy current losses in all magnets PM of Motor A are shown in Table 2 for no-load (Is = 0) at 1000/min and 3000/min, which vary with n2, hence with a factor 9 in difference. At rated power 45 kW at 1000/min (Isq-current operation) and at 45 kW, 3000/min (field weakening, Isd-Isq-current operation) the losses differ only by a factor 2.4 due to the weakened field at 3000/min. In the same way the losses were evaluated with Method c) (i) B), showing very good coincidence with Method c) (i) A), which proves also numerically, that the influence of the self field of the eddy currents on the losses in segmented magnets with

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Lj. Petrovic et al. / Numerical Methods for Calculation of Eddy Current Losses

Table 2. As Table 1, but calculated losses for all magnet segments of Motor A Method n [1/min] PM [W]

no-load load

c) (i) A) TMB

c) (i) A) T 1000 0.42

3000 3.8

36

89

a)

1000

3000

1000

3000

0.24 21

2.2 51

0.19 24

1.7 57

Figure 3. Motor B at no load, 1000 rpm: Calculated radial component of flux density in each of seven magnet segments: a) 10 values per magnet segment (dark line), b) 1 average value per magnet segment (light line).

bM < λ/2 is small. Each magnet segment of Motor B was subdivided into 10 finite elements in x-direction to evaluate the flux density profile By(x, t) for a given rotor position (Fig. 3). For example, at no-load Motor B has the dominating wave length of the field harmonic as the slot pitch λ = τQ, which is obviously bigger than bM by a factor of at least 2. The average value By ( y, t ) = Φ ( y, t ) /(bM ⋅ lM ) , which is used for the calculation, is also shown in Fig. 3. Eddy Current Losses in “Broad” Magnets For “broad” magnets bM > λ/2 like in Motor A′ and Motor B′ the calculation Methods c) (i) (either A) or B)) will not give correct results, but yield 40 … 50 (!) times bigger results. This is due to the different eddy current distribution in massive magnets, which is clearly visible in Fig. 2b, where the seven magnet segments are regarded as one piece bM hM

(bM = 25.2 mm, hM = 4.7 mm) with the condition

∫ ∫J 0

z

( x, y ) ⋅ dx ⋅ dy = 0 . Hence only

0

Method a) and b) can be used for the calculation. Both Methods a) and b) consider the influence of the Bx- and By-component on the eddy current losses, but method b) neglects the self field of the eddy currents. Hence it gives about 30% too big results for “broad” magnets (Motor B′) at no-load (Table 3). Alternatively to Method b) the simplified Method c) (ii) may be used, which neglects the influence of Bx and is only averaging the variation of A in the planes y = hM (top), y = hM/2 (middle), y = hM (bottom). At load both Methods a) and b) give nearly the same results. As Method b) was implemented in the FEM program FEMAG [6] and Method a) in the program FLUX2D, there is also a slight influence of the program package itself on this difference. For “narrow” magnets the results of the above described Method c) (i) B) are also close to

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Lj. Petrovic et al. / Numerical Methods for Calculation of Eddy Current Losses

Table 3. Calculated eddy current magnet losses for Motor B and B′ at no-load (zero current) and at sinusoidal current operation 45 kW; with Method a), Method b) and Methods c) (i) B) and c) (ii) PM [W] Method Motor B no-load Motor B′ no-load Motor B load 45 kW Motor B′ load 45kW

1000 /min c) (ii) c) (i) B) 5.8 7.2 34 – 19 22 223 –

b) 7.6 35 28 207

a) 8.4 27 28 211

b) 68 319 83 454

3000 /min c) (ii) c) (i) B) 52 65 307 – 59 70 489 –

a) 76 240 89 486

Table 4. Inverter-caused losses in the Motors A and A′ for nominal power 45 kW at 1000 /min; Method a) Stator winding supply PM / W

Motor A/Motor A′

Sinusoidal

Inverter (Case 1)

Inverter (Case 2)

25.5/784

33.0/984

32.5/–

Table 5. Fourier analysis of the phase current for inverter-operation of Motor A at 45 kW, 1000 /min fk/Hz

Case

133

665

931

1463

1729

2261

2527

3857

4123

Iˆk /A

1/2

166/157

5.4/12.9

2.2/8.1

1.4/5.1

6.2/7.3

5.1/3.6

1.4/3.8

0.8/2.0

0.9/2.0

the results of Method a). Method c) (ii) gives similar results as Method b), but with a deviation of up to 30%. Due to the reconstruction of the vector potential A from the flux density B it is more inaccurate. Nevertheless this method is helpful, if the used FEM program does not feature any eddy current loss calculation for magnets. Inverter-Caused Eddy Current Losses in the Magnets In the case of inverter supply, the feeding by current sources can be applied only, if the current ripple due to the switching of the inverter is known in advance. If only the inverter data are given, a coupled circuit FE simulation must be used, which is done with Method a). Hence the IGBT transistors of the inverter are modelled in FLUX2D as low resistances in the “on-state” and with high resistance in the “off-state”. Two extremely different sets for these resistance values both for the IGBT transistors (T) and the freewheeling diodes (D) were applied to investigate the influence of transistor data on the eddy current losses in Motor A and Motor A′. The switching frequency was 2 kHz for both cases. Hence the dominant current harmonics occur at frequency side bands around 2 kHz and 4 kHz, but also over-modulation influence is visible. The harmonic spectrum of calculated currents for both cases is given in Table 5 for Motor A. For Motor A′ it was nearly identical. The smaller 1st harmonic in Case 2 influences the result more than the bigger higher harmonics. An increase of eddy current losses due to inverter-caused harmonics of 25 … 30% occurs in both cases. The massive magnets suffer from 30 times (!) higher eddy current losses. Case 1: RON,T = RON,D = 0.1 mΩ, ROFF,T = ROFF,D = 1 MΩ Case 2: RON,T = 50 mΩ, RON,D = 10 mΩ, ROFF,T = 0.1 MΩ, ROFF,D = 50 kΩ. Conclusions The time-stepping Method a) is time-consuming, but allows with a coupled circuit calculation a reliable determination of the eddy current losses in permanent magnets also

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123

at inverter supply. The quasi-static position-stepping (Method b)) neglects the influence of the self field of the eddy currents in the magnets. Due to the low conductivity and permeability the penetration depth at the interesting frequencies is bigger than the magnet dimensions. So it delivers nearly the same loss results. In addition three semianalytic post-processing methods to determine eddy current losses are presented (Methods c) (i) A), (i) B), and (ii)). The first two are only valid for “narrow” magnets, where the variation of flux density over the magnet width is small, whereas the third is also useful for “broad” magnets, as long as the self field of the eddy currents is negligible. The influence of the chosen calculation method on the resulting losses is shown for four permanent magnet synchronous machines with a rating 45 kW, 1000/min and tooth coils.

Acknowledgements The authors acknowledge the support of German Research Foundation (DFG) for financing the project FOR575.

References [1] Deak, C; Binder; A.; Magyari, K.: “Magnet Loss Analysis of Permanent-Magnet Synchronous Motors with Concentrated Windings”, ICEM 2006, Chania, Greece, 2006, 6 pages, CD-ROM. [2] Atkinson, G.; Mecrow, B.; Jack, A.; et al.: “The Analysis of Losses in High-Power Fault-Tolerant Machines for Aerospace Applications”, IEEE Trans. Ind. Appl. 42, 2006, p. 1162-1170. [3] FLUX 9.30 User’s Guide, April 2006, www.cedrat.com. [4] Russell, R. L.; Norsworthy, K. H.: “Eddy currents and wall losses in screened-rotor induction motors”, Proc. IEE, p. 163-175, April 1958. [5] Schuisky, W.: Die Berechnung elektrischer Maschinen, Spinger, Wien, 1960. [6] Reichert, K.: FEMAG, interactive program to calculate and analyze 2-dimensional and axis-symmetric Magnetic and Eddy-Current fields, User’s Manual, February 2007, http://people.ee.ethz.ch/~femag.

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3-D Finite Element Analysis of Interior Permanent Magnet Motors with Stepwise Skewed Rotor Yoshihiro KAWASE a, Tadashi YAMAGUCHI a, Hidetomo SHIOTA a, Kazuo IDA b and Akio YAMAGIWA c a Department of Information Science, Gifu University, 1-1, Yanagido, Gifu, 501-1193, Japan E-mail: [email protected] b Daikin Industries, Ltd., 1000-2, Ohtani, Okamoto-cho, Kusatsu, Shiga, 525-8526, Japan E-mail: [email protected] c Daikin Air-Conditioning And Environmental Laboratory, LTD., 1000-2, Ohtani, Okamoto-cho, Kusatsu, Shiga, 525-8526, Japan E-mail: [email protected] Abstract. In this paper, the effects of the stepwise skew on the torque waveform of an interior permanent magnet motor are analyzed by using the 3-D finite element method. The usefulness of the stepwise skew is confirmed through the calculated torque waveforms and measured ones.

Introduction The interior permanent magnet motors (IPM motors) are widely used as high-efficiency motors in various usage. It is important for the IPM motors to reduce the noise and vibration as well as to improve efficiency [1,2]. In the IPM motors, it is thought that the torque ripple is one of the reasons for the noise and vibration. There are some techniques like the skew of rotor in order to reduce the torque ripple. The interlaminar gap should be considered in the 3-D finite element analysis for skewed IPM motors. Because the axial component of flux density vectors in the cores is computed very large when the interlaminar gap in the cores is not taken into account. In this paper, we analyzed the effects of the stepwise skew of IPM motors on the torque waveform by using the 3-D finite element method (3-D FEM) with gap elements to take the interlaminar gap in the rotor and stator cores into account. The usefulness of the computation is confirmed through the comparison between the calculated torque waveforms and measured ones [3].

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125

Analysis Method Magnetic Field Analysis The fundamental equation of the magnetic field can be written using the magnetic vector potential A as follows: rot( ν rot A ) = J 0

+ ν 0 rot M

(1)

where ν is the reluctivity, J0 is the exciting current density, ν0 is the reluctivity of the vacuum, M is the magnetization of permanent magnet. Gap Elements It is necessary to take into account the interlaminar gap between the electrical steel sheets, which is very small, in order to calculate axial component of the flux density caused by the stepwise skew accurately. However, if the very small air gap is divided by the conventional meshes, it costs a lot of physical memory and CPU time. Therefore, we take into account the interlaminar gap using the gap elements [4]. The weighted residual Gi of Galerkin’s method for the gap elements are given by: G i = D ∫∫

Ss

rot N i (ν 0 rot A ) dS

(2)

where D is the length of the gap, Ss is the region of the gap element, and Ni is the interpolation function. Nodal Force Calculation Nodal force method is to calculate a local magnetic force in the 3-D FEM. The force Fn on each node n can be calculated as follows [5]: Fn = −

∫

V

(3)

(T grad N n ) d V

where V is the total volume of elements related the node n, T is the Maxwell stress tensor, and Nn is the interpolation function of elements related the node n. The torque Tm is given as follows: Tm =

R×r ∑ ( Fn ⋅ λ r ) , λ = Ω

R×r

(4)

where Ω is all the nodes contained in the rotor region (the rotor and the half of air gap), λ is the unit vector of a rotary direction, r is the directional vector towards the node n, and R is the axis directional vector [6].

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Y. Kawase et al. / 3-D Finite Element Analysis of Interior Permanent Magnet Motors

(a) rotor core and magnets

(b) stator core and coil

Figure 1. Photographs of an IPM motor.

(a) no-skewed rotor model

Figure 2. Analyzed model (no-skewed rotor model).

(c) skewed rotor model B (4 magnets)

(b) skewed rotor model A (2 magnets)

Figure 3. Appearances of stepwise skew.

Figure 4. 3-D finite element mesh (except coil and air).

Table 1. Analysis conditions Number of poles Exciting current (A) Coil Number of turns (turn/slot) Magnetization of magnet (T) Frequency of coil current (Hz) Revolution speed (min–1)

4 14 22 1.2 60 1,800

Analyzed Model and Conditions Figure 1 shows the photographs of an IPM motor. Figure 2 shows the analyzed model of an IPM motor. Figure 3 shows the appearances of stepwise skew. Figure 3(a) shows the no-skewed rotor model. Figures 3(b) and 3(c) show the stepwise skewed rotor model. Figure 4 shows the 3-D finite element mesh. The gap elements are inserted in each x-y plane in the stator and rotor cores. It is assumed that the space factor of the electrical steel sheets of the stator and rotor core is 96%. The analyzed region is 1/4 of the whole region because of the symmetry and the periodicity. Table 1 shows the analysis conditions.

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127

Figure 5. Distributions of flux density vectors (o-a section).

Results and Discussions Figure 5 shows the distributions of flux density vectors in o-a section. The distributions of flux density vectors of Figs 5(i) and 5(ii) in each model look almost the same by comparing between no-skewed rotor model and with skewed one. Therefore, Fig. 5(iii) shows the difference between Figs. 5(i) and 5(ii). From Fig. 5(iii), it is found that the axial components of flux density vectors between stepwise skewed magnets of rotor core are very large. Figure 6 shows the waveforms of cogging torque with and without gap elements. In the no-skewed rotor model, the waveforms of cogging torque with gap elements and without one is about the same. On the other hand, in the skewed rotor model, it is found that the peak of cogging torque is different between with gap elements and without one. Consequently, it is found that the usufulness of considering gap elements is clarified.

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Y. Kawase et al. / 3-D Finite Element Analysis of Interior Permanent Magnet Motors

Figure 6. Waveforms of cogging torque with and without gap elements.

Figure 7. Waveforms of torque. Table 2. Discretization data and CPU time Number of elements

814,716

Number of nodes

141,470

Number of edges

967,355

Number of unknown variables

939,928

Number of time steps

180

Computer used: Pentium 4 (3.0GHz) PC

Figure 7 shows the waveforms of torque, which is normalized by the average of the torque of no-skewed rotor model. From Fig. 7, it is found that the calculated waveforms of torque agree very well with measured ones. It is also found that the torque ripples of skewed rotor models A and B are reduced to about 1.5% of the no-skewed rotor model by the stepwise skew in both calculation and measurement. Table 2 shows the discretization data and CPU time.

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Conclusion The effects of the stepwise skew of IPM motors on the torque waveforms were analyzed using the 3-D FEM with gap elements to take the interlaminar gap in the rotor and stator cores into account. It was found that the ripples of torque can be reduced by the stepwise skew. It was also found that there is ample effect for the stepwise skew by 2 magnets to reduce the torque ripple in this model. The validity of the stepwise skew was confirmed through the calculations and measurements.

References [1] Y. Kawase, T. Yamaguchi, S. Sano, M. Igata, K. Ida and A. Yamagiwa, “3-D eddy current analysis in a silicon steel sheet of an interior permanent magnet motor”, IEEE Trans. on Magnetics, vol. 39, no. 3, pp. 1448-1451, May, 2003. [2] Y. Kawase, N. Mimura and K. Ida, “3-D electromagnetic force analysis of effects of off-center of rotor in interior permanent magnet synchronous motor”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 1858-1862, July, 2000. [3] A. Yamagiwa, K. Nishijima, Y. Sanga, Y. Kawase, T. Yamaguchi and T. Yano, “Reduction of Motor Vibration by Stepwise Skewed Rotor”, Japan Industry Applications Society Conference, No. 3-92, 2005. [4] T. Nakata, N. Takahashi, K. Fujiwara and Y. Shiraki, “3-D magnetic field analysis using special elements”, IEEE Trans. Magn. vol. 26, no. 5, pp. 2379-2381, 1990. [5] A. Kameari, “Local force calculation in 3D FEM with edge elements”, International Journal of Applied Electromagnetics in Materials, vol. 3, pp. 231-240, 1993. [6] Y. Kawase, H. Kikuchi and S. Ito, “3-D Nonlinear Transient Analysis of Dynamic Behavior of the Clapper Type DC Electromagnet”, IEEE Trans. Magn. vol. 27, no. 5, pp. 4238-4241, 1991.

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Advance Computer Techniques in Modelling of High-Speed Induction Motor Maria DEMS and Krzysztof KOMĘZA Institute of Mechatronics and Information Systems, Technical University of Lodz, ul. Stefanowskiego 18/22, 90-924 Lodz, Poland E-mail: [email protected], [email protected] Abstract. In the paper, circuit and field-circuit analyses of high-speed small size induction motors are presented. The circuit analysis is possible only for the first harmonic of the supply voltage. For the real shape of the voltage which has many higher harmonics accurate field – circuit analysis is necessary, but this method is very time consuming. The circuit and field-circuit analyses were done for 2-D structure of the motor and for all values of applied frequencies. The results of calculation of magnetizing current are compared with the measurement.

Introduction Nowadays the high-speed induction motors are widely used in many industrial installations and also in aircraft industry. Many of them are designed as converter-fed induction machines. Some electrical drives with not so sophisticated speed control have voltage shape which has many higher harmonics. The field-circuit method makes accurate computation of high-speed induction motors characteristics possible. Unfortunately, this method is very time consuming. When the motor is supplied by a PWM inverter the length of time step must be smaller than time given by carrier frequency of the inverter. Furthermore the Newton-Raphson iteration is carried out at each time step to consider the magnetic saturation. Therefore in design and optimisation process of these motors improved classical circuit methods are very interesting. In the paper, the different field-circuit and circuit methods are presented and the results are compared with the measurement.

Object of Investigation A high-speed construction of the small induction motor was designed basing on the classical structure of the four-pole induction motor model size 80. The supply voltage was 230 V for the frequency 200 Hz and stator windings were delta connected. The number of series turns of stator windings was 216. This motor had stator core shape with cut of parts making stator yoke width not constant; the external maximal diameter of the stator core was Dse max = 120 mm, and external minimal diameter Dse min = 114 mm [ ]. In the field – circuit analysis the real shape of the stator core was taken into account, but in the circuit analysis the calculations were made for the average value of the external stator diameter Dse av = 117 mm, The motor construction is with closed rotor slots.

M. Dems and K. Kom˛eza / Modelling of High-Speed Induction Motor

131

Field-Circuit Analysis of the Motor The field-circuit analysis of the high-speed induction motors can be made using different levels of accuracy. For each circuit, including the parts described with the field equations and external elements described with the resistance Rz and inductance Lz, the circuit equation has the following form: u = R zi + Lz

di + ∑ d j ΔV j dt j

(1)

Applying this to the combination of the presented equations leads to the following system: ⎡G H ⎢ ⎢0 W ⎢⎣ 0 D T

0 ⎤⎡ A ⎤ ⎡ Q ⎥⎢ ⎥ ⎢ D ⎥ ⎢ΔV ⎥ + ⎢H T R ⎥⎦ ⎢⎣ i ⎥⎦ ⎢⎣ 0

0

0⎤ ∂ 0 0 ⎥⎥ ∂t 0 L ⎥⎦

⎡ A ⎤ ⎡J⎤ ⎢ΔV ⎥ = ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ i ⎥⎦ ⎢⎣U ⎥⎦

(2)

where

Gij = ∫

1 ∇N i ∇N j dS μ

H ij = ∫ γ N i dS k

Wkk = ∫ γ dS k

Qij = ∫ γ N i N j dS

J i = ∫ J 0 N i dS (3) R and L are diagonal matrixes of the resistances and external inductances. The equations system can be presented in a more general form: RX + S

∂X =B ∂t

(4)

For quasi-static model the assumption that all field variables are varying sinusoidally is made. Since the potential and the currents are varying sinusoidally, they can be expressed as the real part of complex functions. The equations system now becomes: RX + iSX = B

(5)

This system can be solved using complex arithmetic. The quasi-static solvers calculate element permeability using amplitude of the magnetic flux density. This can introduce some errors in highly saturated small machines despite the transient calculation of the magnetisation current needed. In this case, the system can be solved using differential schema with time step equal to Θ

S⎤ S⎤ ⎡ ⎡ ⎢ R (1 − Θ ) − Δt ⎥ X n + ⎢ RΘ + Δt ⎥ X n +1 + (1 − Θ )B n + ΘB n +1 = 0 ⎦ ⎦ ⎣ ⎣

(6)

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M. Dems and K. Kom˛eza / Modelling of High-Speed Induction Motor

Figure 1. Vector plot and distribution of magnitude of magnetic flux density for 200 Hz supply. current [A] 3 calculated 2 measured 1

0 0,1632 -1

0,1642

0,1652

0,1662

0,1672

0,1682

0,1692 time [s]

-2

-3

Figure 2. Comparison of calculated and measured current values versus time for 200 Hz supply.

Nowadays, the economy of the production causes the use of the construction with not round stator core in motor manufacturing. This type of construction has very high saturation of the motor stator yoke parts with decreased width. Additional problems appeared when the motor has been supplied by the PWM inverter. In this case the supply voltages have complicated shape which depends on the type and algorithm used by the control system. In this case two methods are available: the first when only the first harmonic of the supply voltage is taken into account and the second when the real voltage shape is used. The last method is of course the most accurate but is very time consuming according to the fact that the transient state has to be considered before the steady state is achieved. Figure 1 shows the vector plot and magnitude of magnetic flux density calculated by mentioned methods for 200 Hz supply. It can be noticed that the magnitude of magnetic flux density for this case is rather small despite the fact that above 120 Hz the voltage is constant. Figure 2 presents the calculated terminal current values versus time for real voltage shape at the motor compared with measured values, for 200 Hz supply. The main

M. Dems and K. Kom˛eza / Modelling of High-Speed Induction Motor

133

Figure 3. Vector plot and distribution of magnitude of magnetic flux density for 100 Hz supply, for the first time instant.

Figure 4. Vector plot and distribution of magnitude of magnetic flux density for 100 Hz supply, for the second time instant.

source of differences is the influence of capacitances they are not incorporated in fieldcircuit model. In Figs 3 and 4, two different times instant for 100 Hz supply are presented. From these it can be seen that saturation of the motor changes significantly in time depend on the resultant flux position. In Fig. 5 the voltage and current versus time, calculated and measured, for 100 Hz supply are presented. In this Figure, the first harmonic of the supply voltage is also presented. Values of no-load currents, calculated with the first harmonic supply, by field-circuit and circuit methods correspond roughly. The comparisons with the measurement show that the current values calculated with this assumption are very narrow.

134 800

M. Dems and K. Kom˛eza / Modelling of High-Speed Induction Motor 5

voltage [V]

current [A]

measured

4

600

3 400

2 200

1

0 0,18

0,19

0,2

0,21

0,22

0,23

-200

time [s]

0 0,066 -1

0,068

0,07

0,072

0,074

0,076

0,078

0,08

-2

-400

calculated

-3 -600 time [s] -800

-4 -5

Figure 5. Voltage and first harmonic of the voltage and comparison of calculated and measured current values versus time for 100 Hz supply. Table 1. Flux density in the motor core for different values of the frequency 50 Hz

100Hz

150 H

200Hz

Flux density in the stator yoke [T]

Frequency

1,674

1,709

1,376

1,031

Flux density in the stator tooth [T]

1,481

1,476

1,185

0,892

Flux density in the rotor tooth [T]

1,478

1,473

1,182

0,890

Flux density in the rotor yoke [T]

0,828

0,825

0,662

0,497

effective voltage [V] 300

magnetizing current [A] 2,5

250

2,0

200

measured

1,5

first harmonic of the voltage

150

1,0

measured

calculated for first harmonic of the voltage

100 0,5

50

frequency [Hz]

frequency [Hz] 0,0

0 0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

Figure 6. Effective voltage and magnetizing current vs. frequency measured and calculated for first harmonic of the supply voltage.

Calculation of Magnetizing Current of the Motor Using Circuit Method For this motor the calculation using improved circuit method enables the calculation for higher frequencies of the magnetising current for different conditions of the supply of the motor. From 10 Hz to 120 Hz the linear increase of the supply voltage and frequency was made, and in result we obtain the value of flux density, and torque equal constant. For the frequency higher than 120 Hz the value of the voltage was constant. The values of magnetic flux density in the motor core calculated using the circuit method (STAT) are shown in Table 1. Figure 6 shows the effective value of supply voltage, measured, and the first harmonic of this voltage, which was used for calculations. In this figure also magnetizing current of the motor, measured and calculated for first harmonic of the supply voltage was shown.

135

M. Dems and K. Kom˛eza / Modelling of High-Speed Induction Motor power factor 0,8

current [A] 6,0

100Hz, 120Hz

0,7

5,0 4,0

0,5

3,0

30Hz

50Hz

0,4

0,2

1,0

0,1 relative power output 0,25

0,50

0,75

1,00

1,25

1,50

0,0 0,00

0,50

0,75

1,00

1,25

1,50

0,10

100Hz, 120 Hz

0,09

0,7

0,08

0,6

0,07

0,5

50Hz

0,4

30Hz

0,06

30H

0,05 0,04

0,3

0,03

0,2

100Hz

120Hz

50Hz

0,02

0,1 0,0 0,00

relative power output 0,25

slip

efficiency 0,9 0,8

100Hz, 120Hz

0,3

2,0

0,0 0,00

50H 30H

0,6

relative power output 0,25

0,50

0,75

1,00

1,25

1,50

0,01 0,00 0,00

relative power output 0,25

0,50

0,75

1,00

1,25

1,50

Figure 7. Current, power factor, efficiency and rotor slip versus output power for the high-speed induction motor, for different values of the frequency, for linear increase supply voltage.

Computing of Operating Curves Using Circuit Model For this induction motor the parameters and curves of current, power factor, efficiency and rotor slip versus output power were computed, for different values of the frequency. The results of these calculations for the different values of the frequency for linear increase of the supply voltage are shown in Fig. 7. Figure 8 shows the same curves for the frequency higher than 120 Hz and constant value of the supply voltage. From Fig. 7 and Fig. 8 we can state that the highest value of the efficiency and power factor we obtain for the frequency equal to 200 Hz. It is caused by the lowest value of the magnetising current and in result also total stator current of the motor.

Conclusion In the calculations of the high-speed small power induction motors supplied by inverter the higher harmonics and also nonlinear phenomena can be taken into account only in the field-circuit models but they are still very time consuming. Therefore, in design and optimisation process of these motors improved classical circuit methods are very interesting, but their use is possible for the first harmonic of the supply voltage only.

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M. Dems and K. Kom˛eza / Modelling of High-Speed Induction Motor

current [A]

power factor 0,8

6,0

120Hz

0,6

4,0

0,5 0,4

3,0 150H

2,0

0,50

0,1

0,75

1,00

1,25

1,50

100Hz

relative power output 0,25

0,50

0,75

1,00

1,25

1,50

0,030

150Hz

0,025

0,7 0,6

200Hz 150Hz

0,020

0,5

200Hz

0,4

0,015

0,3

0,010

0,2

120Hz

0,005

0,1 0,0 0,00

0,0 0,00 slip

efficiency 0,9 0,8

0,2 relative power output

0,25

120Hz

150Hz

0,3

200Hz

1,0 0,0 0,00

200Hz

0,7

5,0

relative power output 0,25

0,50

0,75

1,00

1,25

1,50

0,000 0,00

relative power output 0,25

0,50

0,75

1,00

1,25

1,50

Figure 8. Current, power factor, efficiency and rotor slip versus output power for the high-speed induction motor, for different values of the frequency, for constant value of the supply voltage.

References [1] Dems M., Komęza K, Wiak S., Stec T., Kikosicki M., Application of circuit and field-circuit methods in designing process of small induction motors with stator cores made from amorphous iron, COMPEL, The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 25, No. 2, 2006, s. 283-296. [2] Dems M., Modeling of electromechanical transient processes in induction motors with closed slots of the rotor, Archives of Electrical Engineering, nr. 3, 1997, pp. 333-353. [3] Dems M., Komęza K., A comparison of circuit and field-circuit models of electromechanical transient processes of the induction motor with power controller supply, Proceedings COMPUMAG’2001, Lyon– Evian, France, 2-5 July, 2001, pp. 206-207. [4] Dems M., Rutkowski Z., “STATz_F Software for calculation of electromagnetic parameters and characteristics of induction motors”, Technical University of Lodz, Poland. [5] Gąsiorowski T., “Experiences of FSE “BESEL” S.A. in production of induction motors supplied from frequency inverters”, Proceedings of VII Symposium PPEE 99, Ustroń 22-25 mars 1999, pp. 2-6. [6] PC OPERA-2D – version 11, Software for electromagnetic design from Vector Fields, 2006.

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137

Computation of the Equivalent Characteristics of Anisotropic Laminated Magnetic Cores E. NAPIERALSKA-JUSZCZAK, D. ROGER, S. DUCHESNE and J.-Ph. LECOINTE Laboratoire Systèmes Electrotechniques et Environnement, Technoparc Futura, 62400 Béthune, France [email protected] Abstract. The method presented in this paper allows computing the equivalent characteristic of anisotropic steel sheets used to stack the magnetic circuits. The reluctivity of the anisotropic sheets is a function of the flux density B and the angle between the flux density vector and the rolling direction. This paper focuses particularly the problem of modeling different kinds of overlaps and apparent air-gaps is solved by the homogenization technique which is based on the assumption that, in the layer structure, the magnetic field energy tends to achieve its minimum. The presented method has been verified by comparing the computational results with the measurements of real sets. The paper presents the analysis of magnetic properties of overlaps taking into account the magnetic characteristics of the steel.

1. Introduction The effects caused by magnetic fluxes in anisotropic cores are difficult to handle, both theoretically and numerically. Modeling of the overlapping regions of thin laminations, of thickness 0.2–0.3 mm and surface area of tens or hundreds of square centimeters, taking account of the insulation of micrometer thickness, is practically impossible for realistic systems and thus it poses significant challenges. The authors have undertaken research related to such anisotropic magnetic cores used in transformers and electrical machines. In particular, they are addressing the issues of the influence of the position and pattern of equivalent air-gaps, under different overlapping arrangements, on iron losses and noise due to magnetostriction. This research programme necessitates creation of several mathematical models capable of simulating the transfer of flux between laminations under different anisotropy angles and various air-gap positions. This paper presents the results from the first stage of the project. A homogenization technique has been developped to approximate the overlapping of the laminations. This has enabled to replace the real three-dimensional structures by far simpler homogenous two-dimensional models. The approach relies on the natural tendency of the energy of the magnetic field to achieve a minimum in the non-homogenous laminated structure [1–4]. When calculating the flux density vector in each lamination using the minimisation principle, the functional has been assumed in terms of the energy in the whole structure, while the constraint is the relationship between the flux density vector in the whole structure and the relevant vectors in every lamination. At last, the distribution (sub-division) of the total flux density vector between component laminations of a particular overlapping structure will lead to:

138

E. Napieralska-Juszczak et al. / Characteristics of Anisotropic Laminated Magnetic Cores

• • • •

a method of calculating the equivalent (homogenised) reluctivity at any point of the laminated sheet and/or the air-gap; a method of establishing equivalent characteristics for various structures; a method of calculating the magnetic field distribution in a transformer core under different overlapping schemes; a method of calculating the magnetic field distribution in an electrical machine, when the core is anisotropic, at various anisotropy angles.

2. Method Presentation Using the homogenization technique makes it possible to replace real 3D structures by simpler homogeneous 2D structures. In the homogenization technique, concepts of macrostructure and microstructure are of importance. In the considered problem, the macrostructure is a complete assembly of layers, while the microstructure is a repeatable structure of two or more layers made of sheets with different rolling directions. In the case of apparent air gaps, the microstructure is a set of sheet layers and air layers. The usual procedure consists in defining a basic volume V able to replace the whole structure [5,6]. Thus, the main purpose in this paper is to replace the set of microstructures by the equivalent volume V without changing the flux distribution in each layer. The first step of the method is to define the components of the resulting flux den   sity vector B . The relation between B and the flux density vectors b in the repetitive structures representing a full set of the overlapped layers is given by formula (1) where V is the macrostructure basic volume.

 1  B = ∫ bdv VV

(1)

If the integral is replaced by the sum for the basic volume dx*dy*H, Eq. (2) is obtained. Denotes n is the number of different not repetitive layers in the structure, hi is the thickness of the layer i and H is the thickness of the macrostructure.  B=

1 dx ⋅ dy ⋅ H

n



∑ B ⋅ dx ⋅ dy ⋅ h i =1

i

i

(2)

 The components Bix and Biy (Eq. 3a and 3b) of the flux density B in point P of a multilayer set representing overlapping of sheets (i = 1, 2, 3,…n) are computed. The calculation is based on the total flux penetrating through the limiting surfaces of the volume V. Bix =

Biy =

φix′ + φix′′ 2 ⋅ dy ⋅ hi φiy′ + φiy′′ 2 ⋅ dx ⋅ hi

(3a)

(3b)

E. Napieralska-Juszczak et al. / Characteristics of Anisotropic Laminated Magnetic Cores

139

φ’’y

φ’x

φ’’x φ’y

axis x

Figure 1. Fluxes going inside and outside the volume V.

φix′ and φix′ are the fluxes going inside the volume; φix′′ and φix′′ are the fluxes going outside the volume (Fig. 1). The fluxes going through the limiting surfaces of the volume V have to be equal to the sum of fluxes in all particular layers. Therefore the components Bx and By of the resulting flux density vector are given by Eqs (4a) and (4b). The flux density component in the direction z is smaller then 10% of Bx and By, so this component can be omitted. n

Bx =

∑ φ ′ + φ ′′ i =1

ix

ix

(4a)

2 ⋅ dy ⋅ H n

By =

∑ φ ′ + φ ′′ i =1

iy

iy

(4b)

2 ⋅ dx ⋅ H

In order to replace the three-dimensional system by a 2D system, it is assumed that the analyzed volume is made of an equivalent homogenous material in which the distribution of magnetic field is identical to the resulting distribution in the real structure. The second step of the method is to define the goal function to minimize the magnetic energy. The total energy of the macrostructure results from the sum (Eq. 5) of the  magnetic energy Wµi stored in each volume Vlay of the layer i (Eq. 6). µi and Bi are respectively the permeability and the flux density vector of the layer i.  Wµ min = ∑ Wµi µi , Bi

(

  1 Wμi = Vlay Bi  H i 2

)

(5)

(6)

140

E. Napieralska-Juszczak et al. / Characteristics of Anisotropic Laminated Magnetic Cores

To calculate flux density vectors in the particular layer, a minimization method is applied. The applied goal function is the minimum of the magnetic energy in the volume V, while the relationship between the resultant flux density vector and each vector of the flux density in particular layers is the restriction. The goal function depends on the computed structure and it has to be prepared taking into account the interlacing, the insulation between the laminations and the air-gaps. On the one hand, the air-gaps depend on the inaccuracy of the sheet overlapping. On the other hand, it is possible to take advantage of the air gaps to get the required orientation of the magnetic flux. Indeed, apparent air-gaps exist on the edges and inside the structure made of overlapped sheets. Air gaps and sheets form a non-homogenous layer structure. That is why two kinds of materials – steel sheets with permeability μFe and air with permeability μ0 – appear in parallel, forming together the apparent air gap. The division of the resulting  flux density B between the steel and air depends as well on the rule of minimum of energy stored in magnetic field. The resulting flux density vector in apparent air-gap is   given by the expression (7) were B0 is the flux density in the air, BFe is the flux density in the sheet.

   Bres = BFe + B0

(7)

The magnetic energy stored in the structure results from the sum of the energy stored in the air gaps and in the sheets. If Vlay denotes the volume of the simple layer, nFe the number of sheet layers, n0 the number of air layers, ν0 and νFe the reluctivity respectively in air and in sheets, then the goal function for the structure is given by the formula (8).   Wµ min BFe , B0

(

)

2 ⎡Vlay nFeν Fe BFe + Vlay n0ν 0 B02 ⎤ =⎢ ⎥ 2 ⎣⎢ ⎦⎥ min

(8)

Third step of the method consists in establishing the family of the anisotropic characteristics of the whole structure. Successive values of the flux density from 0 to the steel saturation are imposed. Every point of the characteristics is determined with the following algorithm. First the flux density vectors for the structure are calculated with the minimization task. Then, the expression (9) makes it possible to calculate the reluctivity of macrostructure. At last, the field intensity H = ν B is calculated.  n ν B2 + n ν B2 ν B,Wµ min = Fe Fe Fe 2 0 0 0 nB

(

)

(9)

3. Comparison of Calculation and Measurement Results Calculations for two types of different structures are made. The proposed method is composed the following steps: • •

the homogenized reluctivity at any point of the laminated sheet or the air gap is calculated, the equivalent characteristics are established,

E. Napieralska-Juszczak et al. / Characteristics of Anisotropic Laminated Magnetic Cores

141

Figure 2. Family of characteristics B(H) for the different anisotropy directions.

•

• • •

the properties of the overlapped structure are compared with the magnetic properties of the steel forming these overlaps. It gives, in few seconds, a first idea of the quality of the overlapping without doing a Finite Element (FE) simulation which requires a long computation time, the magnetic field distribution in every tested structure is calculated using a FE method, taking into account a nonlinear anisotropic reluctivity, the division of the total flux density vector between the component laminations is analysed, the results of simulations are compared with measurements.

The first tested structure is made of layers alternatively shifted of 90°: the first layer is placed in the rolling direction (anisotropy angle 0°), the second layer has an anisotropy angle of 90°. This structure corresponds to the conventional two-cycle interlacing (90° transformer core overlap). Figure 2 shows the family of the equivalent magnetizing characteristics B(H). The anisotropy angle α of the whole structure is arbitrary defined as the angle of between the resulting flux density vector and the rolling direction of the first layer. The characteristics of the equivalent structure for α = 0°, 30°, 45°, 60° and 90° are presented at Fig. 2. The characteristic of the anisotropic steel used to form the structure are also presented for the anisotropy angles 0°, 60°, 90°. It makes possible to estimate the quality of the whole structure.  Figure 3 presents the total flux density B at the point p(x,y) of the structure (solid  line) and the division of B between the layer 1 and the layer 2 (interrupted lines) versus ωt (with ω the grid pulsation). Figure 4 gives the comparison of the calculated results and measurement results for both layers at p(x,y) [8–10]. The calculations were made using measured vectors of the resultant flux density. A second structure makes it possible to test the influence of the air gaps. Its geometry is presented at Fig. 5. Every layer is made of a sheet in the rolling direction (anisotropy angle of 0°), an air gap and a sheet placed perpendicularly to the rolling direction (anisotropy angle of 90°). The position of the air gap and the repartition of the two kinds of magnetic materials are different for every layer. The characteristics calculated for every layers of this structure and compared with the characteristics of the anisotropic steel are shown at Fig. 6. Figure 7 shows the family of the equivalent magnetizing characteristics B(H) for the layer 4. One can observe that the equivalent layer material has anisotropic properties, like the elementary steel

142

E. Napieralska-Juszczak et al. / Characteristics of Anisotropic Laminated Magnetic Cores

Figure 3. Resultant flux density in the structure divided between layers.

Air gap

90°

Figure 4. Comparison of measurements and calculations.

0°

Layer 1 Layer 2 Layer 3 Layer 4

Figure 5. Presentation of the structure.

Figure 6. Family of characteristic B-H for the directions 0° for the 4 layers.

sheets. The characterization is different for the low and high values of the flux density and the numerical board between them is about 1.1 T. The properties of the equivalent layer are worse than the sheet properties for α between 0° and 45°. However, equivalent layer properties are better if α is superior to 45°. To check the proposed method, not only the simple structure has been tested. Different types of transformers and complex systems have been also studied, for example a transformer supplying a converter system during normal working or working under different kinds of faults.

E. Napieralska-Juszczak et al. / Characteristics of Anisotropic Laminated Magnetic Cores

143

Figure 7. Family of characteristics B(H) for the layer 4 for the different anisotropy directions.

4. Conclusion The measurement results confirm the good accuracy of the proposed method. The average fractional error betweens calculation and measurements is between 5–10%, depending of the structure. This method makes it possible to substitute the complex 3-D structures by far simples two-dimension structures. For example, it gives the influence of the joints and air-gaps on the required magnetic flux distribution. The method can be applied to model all kinds of transformer core overlapping. It allows the calculation of the flux density vectors in any layer of the transformer cores with different overlapped limb and yoke sheets. It should provide a considerable help for the designers since it allows them to arrange cores of the same dimensions but of optimum structure. The presented method can be also applied to design rotating electrical machines equipped with a rotor or a stator made of anisotropic materials. References [1] J. Gyselinck, R.V. Sabariego, P. Dular, “A nonlinear time-domain homogenization technique for laminated iron cores in three-dimensional finite-element models”, IEEE Trans. on mag., Vol. 42, Issue 4, April 2006, pp. 763–766. [2] Hiroyuki Kaimori, Akihisa Kameari, Koji Fujiwara, “FEM Computation of Magnetic Field and Iron Loss in Laminated Iron Core Using Homogenization Method”, IEEE Trans. on mag, Vol. 43, Issue 4, April 2007, pp. 1405–1408. [3] A.J. Bergqvist, S.G. Engdahl, “A homogenization procedure of field quantities in laminated electric steel”, IEEE Trans. on mag, Vol. 37, Issue 5, Part 1, Sept. 2001, pp. 3329–3331. [4] L. Krahenbuhl, P. Dular, T. Zeidan, F. Buret, “Homogenization of lamination stacks in linear magnetodynamics”, IEEE Trans. on Mag, Vol. 40, Issue 2, Part 2, March 2004, pp. 912–915. [5] A. De Rochebrune, J. Dedulle, J. Sabonnadiere, “A Technique of homogenization applied to the modeling of transformers”, IEEE Trans. on Mag., vol. 26, No2, 1991, pp. 520–523. [6] J.M. Dedulle, G. Meunier, A. Foggia, J.C. Sabonnadiere, D. Shen, “Magnetic fields in nonlinear anisotropic grain-oriented iron-sheet”, IEEE Trans. Mag., Vol. 26, N°. 2, 1990, pp. 524–527. [7] E. Napieralska-Juszczak, M. Pietruszka, “Semi-analytical method of modelling the magnetising curves for anisotropic sheets”, 4th Int. Workshop on Electric and Magnetic Fields, Marseille France, 1998, pp. 451–456. [8] M. Pietruszka, E. Napieralska-Juszczak, “Lamination of T-joints in the transformer core”, IEEE Trans. on Mag, Vol. 32, Issue 3, Part 1, May 1996, pp. 1180–1183. [9] A.J. Moses, “Rotational magnetization-problems in experimental and theoretical studies of electrical steels and amorphous magnetic materials”, IEEE Trans. on Mag, Vol. 30, N°2, 1994, pp. 902–906. [10] M. Pietruszka, “A method to compute the magnetic field in anisotropic 3-phase transformer cores with arbitrary overlapping structures”, D.Sc Thesis, Poland, 1995 (ISSN 0137-4834).

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-144

Improving Solution Time in Obtaining 3D Electric Fields Emanated from High Voltage Power Lines Carlos LEMOS ANTUNES a,b, José CECÍLIO b and Hugo VALENTE c Lab. CAD/CAE, Electrical Engineering Dept., University of Coimbra, Pólo II, 3030 – 290 Coimbra, Portugal b APDEE – Assoc. Port. Prom. Desenv. Eng. Electrotécnica, Rua Eládio Alvarez, Ap. 4102, 3030 – 281 Coimbra, Portugal c REN – Rede Eléctrica Nacional, Av. Estados Unidos da América 55, 1749 – 061 Lisboa, Portugal E-mail: [email protected]; [email protected] a

Abstract. In this paper it is presented an algorithm to reduce the computational time in obtaining the electric field distribution in a plane of analysis due to High Voltage Power Lines. It is used a two dimensional interpolation based on a spline function using as known nodal values, the field solution at nodes of a coarser plan grid.

Introduction The LMAT_SIMEL [1] is a software program that calculates the 3D electric field distribution on specified nodes, emanating from general 3D Line(s) configurations. The electric field is calculated using a 3D integral numerical approach and makes use of the image method. The conductors are considered filamentary wires of arbitrary geometric configuration with known imposed voltages: phase-earth or zero if it corresponds to the guard conductor and the catenary is approximated by straight lineal segments. The electric field can be calculated along any path or on any plane. The earth is considered as a perfect conductor at zero voltage reference value and its influence is taken into account using the method of images. The influence of vegetation and terrain elevations is not taken into consideration. The grid discretization of the solution plane is very important to obtain a good or smoother solution for electric field distribution which may lead to a considerable computational time. To reduce this computational time, we have used a two dimensional interpolation function to estimate the field solution at intermediate nodes, from the field solution obtained in a coarser plane grid.

Formulation The phasor electric field Eˆ at any point P(x, y, z) due to a Line, is calculated by

Eˆ =

C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields

145

ˆ (s) 1 λ 1 NS i ⋅ ∑ Li ⋅ ∫ ⋅ ds ⋅ aˆ 2 0 4πε 0 i =1 r − r'

(1)

where the point P(x, y, z) is defined by r and the phasor charge density λˆi in the seg-

(

)

ment i is located at r ' , with aˆ as the unit vector in direction r − r ' . It is seen that the phasor linear charge density has to be previously calculated for all the Line(s) and their images. For each line segment the charge distribution is approached by a cubic spline polynomial as (2).

λˆ ( s ) = c0 + c1s + c2 s 2 + c3 s 3

(2)

where ⎛ 3s 2 2 s 3 ⎞ ⎛ ⎛ 3s 2 2 s 3 ⎞ ⎛ s 2 s3 ⎞ 2s 2 s 3 ⎞ c0 = ⎜1 − 2 + 3 ⎟ , c1 = ⎜ s − + 2 ⎟ , c2 = ⎜ 2 − 3 ⎟ and c3 = ⎜ − + 2 ⎟ L1 L1 ⎠ L1 L1 ⎠ L1 ⎠ ⎝ ⎝ ⎝ L1 ⎝ L1 L1 ⎠ s is an adimensional parameter ( s = 0 at the beginning of the segment and s = 1 at the end of that segment). For each Line node i it is required the calculation of the phasor charge density λˆi and its derivative λˆ' , which is based on the following equation: i

VˆP =

t 4πε 0

∫

⎡ˆ ⎣ λ0 1

0

⎡k ( s)⎤ λˆ1 ⎦⎤ ⋅ ⎢ o ⎥ ⎣ k1 ( S ) ⎦ ds + t 4πε 0 r − r'

∫

1

0

⎡ ˆ' ⎣ λ0

⎡ k0' ( s ) ⎤ λˆ1' ⎦⎤ ⋅ ⎢ ' ⎥ ⎢⎣ k1 ( S ) ⎦⎥ ds r − r'

(3)

It was used a two dimensional interpolation function [2,3] to estimate the intermediate values from two known values. The interpolation is essential to obtain a smoother or good representation of electric field distribution, between two known field values. Given a rectangular grid { xk , yl } and the associated set of numbers zkl which correspond to the known field values, with 1 ≤ k ≤ m, 1 ≤ l ≤ n , we have to find a bivariate function z = f ( x, y ) that interpolates the data (field solution), i.e., f ( xk , yl ) = zkl for all values of k and l . The grid points must be sorted monotonically, i.e. x1 < x2 < ⋅⋅⋅ < xm with a similar ordering of the y-ordinates. To generate a bivariate interpolation on the rectangular grids and calculate the value in the points specified in the arrays xi and yi it is used a spline interpolation, like (4), for example for x: P ( x ) = a ⋅ x3 + b ⋅ x 2 + c ⋅ x + d

(4)

The corresponding mathematical spline must have a continuous second derivative and satisfy the same interpolation constraints. The breakpoints of a spline are also referred to as knots.

146

C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields Field distribution in the plane

Field distribution In the plane

4000

5000 4000

3000

Eef

Eef

3000 2000

1000

2000 1000

0 100

0 100 100 80

50

60

100 80

50

60

40 y

0

40

20 0

20 x

Figure 1. Electric Field smoother solution (Case 1).

y

0

0

x

Figure 2. Electric Field smoother solution (Case 2).

The first derivative P' ( x ) of our piecewise cubic function is defined by different formulas on either side of a knot xk . Both formulas yield the same value d k at the knots, so P' ( x ) is continuous.

Case Studies It is presented as illustration examples two case studies regarding the electric field emanated from High Voltage Power Lines. Case 1 corresponds to a single Line and Case 2 corresponds to two Lines orthogonally placed to each other. For both cases the electrical conditions are the same, with 220 kV and 1140 A per phase conductor and the catenary of the Line(s) approached by 30 straight lineal segments. Both Lines have 100 m length. The solution plane is defined by a span of Line and it was considered as reference, a grid defined by one meter space between nodes in the solution plane. This discretization corresponds to a grid with 10000 nodes and produces a smoother solution of the electric field. It is seen in Fig. 1 the electric field distribution (smoother solution) in the solution plane for Case 1, and in Fig. 2 for Case 2. To obtain the electric field for Case 1 and Case 2, with 10000 nodes a considerable computational time was required namely 32.49 ×103 sec and 6.51× 104 sec ≈18.2 hours respectively. The computational time to obtain the charge density and the corresponding derivatives has to be added to this time for obtain the total computational time. The idea was then to produce an electric field solution as accurate as possible but with considerable much less computational time.

Results Three different grids with two meter space, five meter space and ten meter space between nodes were used. The field solutions at these nodes were exactly the same as for the finer grid and the derived field solution for the other nodes of the finer grid were processed by the interpolation function. It is shown in Fig. 3 the different computational time t versus the nº of nodes n for Case 1.

C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields

147

4

3.5

x 10

3

t=Computational time [s]

2.5

2

1.5

1

0.5

0

0

1000

2000

3000

4000 5000 6000 n=nº of nodes

7000

8000

9000 10000

Figure 3. Computational time (Case 1). Error (%) - 10x10 grid 100

2.52e-001

90 2.10e-001 80 70 1.68e-001

y

60 50

1.26e-001

40 8.41e-002 30 20 4.21e-002 10 0

0

10

20

30

40

50 x

60

70

80

90

100

0

Figure 4. 2D projection of the error distribution in plane (Case 1).

The computational time function t = t ( n ) can be approximated by Eq. 5. t (n) = 3.2425 ⋅ n + 60.53

(5)

To access the accuracy of the field solution, a nodal local error parameter ε n [ % ] was calculated, as: ε n [%] =

En − Eref Eref

(6)

where En is the electric filed value obtained at the nodes for the coarser grid and Eref is the corresponding electric field value obtained with the reference grid (one meter space between nodes). In Fig. 4 it is shown in the form of coloured plot the visualization of a 2D projection of the error distribution at the nodes in the plane of analysis for one grid defined by 10 meter space between nodes for Case 1.

148

C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields Error in x direction for y=70 0.25 2x2 5x5 10x10

0.2

Error [%]

0.15

0.1

0.05

0

0

20

40

60 x

80

100

120

Figure 5. 2D error distribution in plane (Case 1). -3

5

Error in y direction for x=70

x 10

2x2 5x5 10x10

4.5 4 3.5

Error [%]

3 2.5 2 1.5 1 0.5 0

0

20

40

60 x

80

100

120

Figure 6. 2D error distribution in plane (Case 1).

In the form of graphic it is shown in Fig. 5 the variation of ε n [ % ] for a line y = 70 m for the three different grid discretization, and in Fig. 6 it is shown the variation of ε n [ % ] for a line x = 70 m.

As it is seen the error ε n [ % ] varies in the range of [0, 0.252]%, and the computa-

tional time to get the electric field solution is 384.7 sec, thus 84 times lower than the time to obtain the field solution for 10000 nodes. For Case 2, it is shown in Fig. 7 the computational time t versus the nº of nodes n corresponding to these different grids. The computational time function t = t ( n ) can be approximated by Eq. 7. t (n) = 6.4917 ⋅ n + 198.2292

(7)

In Fig. 8 it is shown the visualization of a 2D projection of the error distribution in plane of analysis for one grid defined by 10 meter space between nodes for Case 2.

C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields

149

4

7

x 10

6

t=Computational time [s]

5

4

3

2

1

0

0

1000

2000

3000

4000 5000 6000 n=nº of nodes

7000

8000

9000 10000

Figure 7. Computational time (Case 2). Error (%) - 10x10 grid 100

1.00e+001

90 2.79e-001 80 70 2.23e-001

y

60 50

1.67e-001

40 1.12e-001 30 20 5.58e-002 10 0

0

10

20

30

40

50 x

60

70

80

90

100

0

Figure 8. 2D projection of the error distribution in plane (Case 2).

In this case the error ε n [ % ] varies in the range of [0, 10]%. As this error is high, the effective value varies in the range of [0, 500] V/m, it was analysed the error distribution in plane of analysis for one grid defined by 5 meter space between nodes (400 nodes). The 2D projection of this error distribution it is shown in Fig. 9. In this case the error ε n [ % ] varies in the range of [0, 1] %. The nodal error is bigger than for Case 1, but still very low and quite acceptable. The computational time to get the electric field solution is 2.7949 ×103 sec , thus 19 times lower than the time to obtain the field solution for 10000 nodes. The computational time to get the electric field solution for 100 nodes is lower than the time to obtain the field solution for 400 nodes, but the error is bigger. The authors suggest that the user should take as minimal grid configuration, the 5 meter space between nodes (400 nodes) to obtain a quite acceptable field accuracy solution.

150

C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields Error (%) - 5x5 grid 100

9.96e-001

90 2.77e-002 80 70 2.21e-002

y

60 50

1.66e-002

40 1.11e-002 30 20 5.53e-003 10 0

0

10

20

30

40

50 x

60

70

80

90

100

0

Figure 9. Error distribution in plane for one grid defined by 5 meter space between nodes (Case 2).

Conclusions It was presented an algorithm to reduce the computational time in obtaining the electric field distribution in a plane of analysis due to High Voltage Power Lines. The nodal error in the field solution is quite negligible when comparing solution obtained with finer grids in plane of analysis. This algorithm is implemented in the LMAT_SIMEL software, which is part of a more complete package LMAT_SIMX that allows the analysis and simulation of Electrical and Magnetic Fields emanated from very High Voltage Power Lines, developed by the authors.

Acknowledgments The authors gratefully acknowledge REN-Redes Energéticas Nacionais SGPS, SA for the financial support received under Project COIMBRA_EMF.ELF.

References [1] Carlos Lemos Antunes, José Cecílio, Hugo Valente, “LMAT_SIMEL – The Electric Field numerical calculator of the package LMAT_SIMX for Very High Voltage Power Lines”, accepted for presentation at ISEF 2007 – XIII International Symposium on Electromagnetic Field in Mechatronics, Electrical and Electronic Engineering, Prague, Czech Republic, September 13-15, 2007. [2] www.mathworks.com/access/helpdesk/help/techdoc/ref/index.html. [3] www.mathworks.com.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-151

151

Thermal Distribution Evaluation Directly from the Electromagnetic Field Finite Elements Analysis A. DI NAPOLI, A. LIDOZZI, V. SERRAO and L. SOLERO University ROMA TRE, Via della Vasca Navale 79 – Roma, Italy [email protected] Abstract. This paper deals with the thermal field evaluation achieved directly from the finite elements representation of the relative electromagnetic field. The proposed strategy can be applied to any generic electrical device; in this work it has been proved on a permanent magnets electrical machine. This methodology has been implemented in two different ways. At first laminar and turbulent motions have been considered from thermal convection point of view and then convection has been reduced to pure conduction heat transfer.

Introduction A finite element evaluation algorithm devoted to analyze both electric machines and power switches has been implemented by means of ANSYS@ software. At first the electromagnetic field inside the structure has been evaluated and plotted. The structure has been considered as a discretized surface and both boundary condition and currents have been set. Induction and eddy currents values have been achieved and then dc-current Joule effect losses and the additional losses have been computed. Thermo-electric analysis has been accomplished considering convection and conduction heat transmission. Two different methodologies have been implemented. At first, thermal convection has been studied under the assumption of both laminar and turbulent motion. After that, convection heat transfer has been simulated under laminar motion, where a particular coefficient has allowed reducing the heat exchange to pure conduction, simplifying the simulation model and making the simulation faster. When convection is studied as normal conduction, simulation software ANSYS allows achieving directly the temperature distribution starting from the electric currents, so avoiding previous steps concerning the evaluation of the electromagnetic field. In this manner the simulation time is strongly reduced together with memory occupancy.

Magnetic Analysis Proposed finite elements analysis has been applied to a permanent magnet synchronous machine, where main data are shown in Table 1 and Table 2. Maxwell Eq. (1) written

152

A. di Napoli et al. / Thermal Distribution Evaluation

Table 1. PM machine data Outer diameter Air-gap diameter Air-gap thickness PM residual induction (NdFeB)

D Di

m m m T

δ Bres

0.0125 0.0068 0.001 1.15

Table 2. PM machine evaluated data I

rms

(A) 75 125

Torque (Nm) 20 40

Figure 1. Permanent magnet machine representation and boundary conditions.

Id (A) 69 110

Iq (A) 30 58

Figure 2. Permanent magnet machine mesh.

 using the vector A has allowed studying the system as a plane if the observation is far enough from the machine end regions.

  μ ∂ A  ∇ A= + μJ ρ ∂t 2

(1)

Concerning the boundary conditions, magnetic induction field (B) paths are supposed to be tangent to both inner and outer circumferences shown in Fig. 1 as the segment BD and AC respectively. Finite elements analysis has been implemented to only one by six of the machine section; having the machine three pole-pairs it shows an odd symmetry. Avoiding any saturation effect being the amplitude of the magnetic field quite small, a linear analysis has been performed. Figure 2 shows the achieved mesh useful to evaluate machine inner magnetic fields. Inside the regions where the parasitic currents phenomena are absent, the current density from the previous magnetic fields evaluation has allowed to evaluate the dccurrent losses. In the other regions the heat sources has been accomplished from the eddy currents.

A. di Napoli et al. / Thermal Distribution Evaluation

153

Thermal Analysis Thermal analysis is based on the heat sources which can be defined for every mesh point. Machine geometry and its mesh must be kept unchanged when both magnetic and thermal analysis are based on the PLANE77 solver. The data achieved by the magnetic analysis are then used to perform the thermal investigation. Collected data, especially heat sources, have been stored in a data-base and used in the machine mesh. Heat transmission is controlled by the following expression ⋅

q 1 ∂T ∇ T+ = λ α ∂t 2

(2)

where ρ is the mass density [kg/m3], Cp is the specific heat [J/(kgK)], λ is the thermal conductivity [W/(m2K)], q is the sources heat power density [W/m3] and finally α is the thermal diffusion. Along lines AC and BD shown in Fig. 2, boundary conditions have been defined as the expression reported in Eq. (3): −λ

dT = h ( T − T∞ ) dn

(3)

where h takes into account the convection heat transfer from stator outer surface to ambient which is supposed to be T∞ = 30 °C. Surfaces AB and CD, still shown in Fig. 2, have been considered adiabatic. Rotor position affects both power losses and mesh thermal characteristic; temperature is accomplished by the averaging different rotor position since thermal time constant is greater than rotor revolution time. The most difficult task is determining the convection coefficients; these parameters describe heat and mass exchange within the fluid. To this purpose, non-dimensional numbers achieved by analytical and experimental results explain fluid behaviors. Under the assumption of natural convection, convection coefficient can be evaluated starting from the Nusselt number, which can be achieved from Prandlt and Grashof numbers. Convection heat transfer between stator outer surface (BD line in Fig. 3) and airgap has been deeply analyzed where the first surface is usually common with still surfaces. Outer Surface When the heat transfer is based on natural convection, which is true if the stator outer surface and the free air heat exchange is taken into account, main heat transfer numbers are shown as follow. Pr =

Cp ⋅μ λ

= 0.710

(4)

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A. di Napoli et al. / Thermal Distribution Evaluation

Figure 3. Map of magnetic flux, Id = 68.7 A and Iq = 30 A.

Gr =

gβ ( T − T∞ ) D3 ν2

where β =

Figure 4. Map of magnetic flux, Id = 110 A and Iq = 57,9 A.

1 =30 °C T∞

Nu = C ⋅ (Pr⋅ Gr) n

(5)

(6)

where Pr is Prandlt’s number, Gr is the Grashof’s number, Nu is the Nusselt’s number and D is the machine diameter. The operating point has been selected as C p = 1.007 [KJ/KgK], μ = 19.2010–6 [Kg/sm], λ = 27.16*10–3 [W/mK]. Under the assumption of laminar motion where Pr·Gr is lower than 109 and turbulent for higher values, Nusselt equation parameters can be determined as follow: 102 < (Pr⋅ Gr) < 104 104 < (Pr⋅ Gr) < 109 10 < (Pr⋅ Gr) < 1012

C = 0.85, n = 0.188 C = 0.53, n = 1/4 C = 0.13, n = 1/3

After that, the convection coefficient h can be simply achieved considering a smooth motor outer surface, and it allows the defining of the outer surface boudary condition. h=

Nu ⋅ λ = 10 W/m2K D

Air-Gap Both convection and conduction within the air-gap are affected by the roughness of the rotor surface (K. Ball, B. Farouk e V. Dixit. [1]), the stator system and rotor rotational speed. In case of laminar motion, Reynolds number is small and the air-gap thermal conductivity coefficient is near to the still air. In case of turbulent motion, conduction coefficient is replaced by a new one that takes into account also convection transfer.

λ eff = 0, 0019 ⋅ η⋅−2,9084 ⋅ Re0,4614⋅ln(3,3361⋅η)

(7)

A. di Napoli et al. / Thermal Distribution Evaluation

155

where η is shown in Eq. (8) η=

ro Ri

(8)

with 0.4 < η < 1. r0 is the rotor outer radius and Ri is the stator inner radius. Reynodls number is given by Re = r0 ⋅ ωm ⋅ δ / ν

(9)

where ωm is the rotor angular speed [rad/s], δ is the air-gap lenght [m] and ν kinetic viscosity [m2/s]. When smooth surface is assumed, the transition between laminar motion and turbulent motion is given by the Taylor’s number, ω2m rm δ3 ν2

Ta =

(10)

where rm =

δ is the average logarithmic radius ⎛r ⎞ ln ⎜ o ⎟ ⎝ ri ⎠

(11)

ro and ri are respectively the stator inner and outer radius. Following Becker and Koye theory, when Ta is lower than 1700 heat transfer is mainly devoted to laminar motion and Nusselt number is equal to 2, otherwise the following expression should be used:

h ⋅ 2g =2 λ N u = 0.128 ⋅ Ta 0.367 Nu =

N u = 0.41 ⋅ Ta 0.241

for

Ta < 1700 and g / ri → 0

for

1800 < Ta < 12000

for

12000 < Ta < 4 ⋅106

(12)

Air-gap heat transfer coefficient includes both conduction and convection and it Nu ⋅ λ can be written as h = . 2g Machine under test has 3 pole-pairs and it is fed with 50 Hz electrical, then Ω = 2π ⋅ f / p = 104.6 rad / s , rm = 67.5 m (5.14); g = 0.001 m; ν = 172.6 10–7 m2/s. Taylor number is equal to Ta = 2479, therefore turbulent motion in the air-gap determines the heat transfer which a Nusselt number equal to 2.25 and then h = 29.25 W/m2K. Concerning the air-gap, the data shown in [4] have been used, when the laminar motion and thermal conductivity coefficient is closer to quiescent free air.

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A. di Napoli et al. / Thermal Distribution Evaluation

Figure 5. Thermal performance of stator and rotor structure.

Figure 6. Thermal performance of stator structure.

Figure A1. Example of the structure under study.

Appendix In this section the comparison of the two proposed methodologies is shown. As example, it has been applied to the study of the thermal behavior of a simple structure shown in Fig. A1, where areas A1 and A3 are composed by iron (thermal conductivity λ Fe = 45 [W/mK] and size 0.03 × 0.03 [m]) and A2 area is air (adduction coefficient λ aria = 0.026 (T = 30°C) [W/mK] and size 0.03 × 0.03 [m]). Boundary condition are T1 = 40°C and T2 = 30 °C. At first, the air has been considered as a solid material with its thermal conductivity. After that, air has been considered as a fluid with its phisical characteristics. The achieved results have been compared. Once the materials, the boundary conditions and the mesh have been selected, thermal analysis has been carried out considering pure conduction heat transfer. The second step was the introduction of the thermal convection, fluid characteristics as well as the density temperature dependence. A comparison between the proposed methodologies is shown in Figs A.2 and A.3, where air temperature close to the borders is the same in both cases. Heat transfer is mainly due to thermal conduction being temperatures very close. Thermal convection

A. di Napoli et al. / Thermal Distribution Evaluation

Figure A.2. Solution of the thermal analysis by means of the adduction coefficient.

157

Figure A.3. Air-iron temperature.

heat transfer can be taken into account by means of the adduction coefficient since the convection motion speed is quite low.

References [1] Christos Mademlis, Nikos Margaris, and Jannis Xypteras, Magnetic And Thermal Performance Of A Synchronous Motor Under Loss Minimization Control, Proc. Of the IEEE International Symposium On Industrial Electronics July 1995. [2] Y.K. Chin, D.A. Staton, Transient Thermal Analysis using both Lumped Circuit Approach and Finite Element Method of a Permanent Magnet Traction Motor, AfriCon 2004, pp. 1027–1035. [3] Ansys, Ansys Thermal Analysis Guide, November 2004. [4] J.M. Owen, Fluif Flow and Heat transfer in Rotating Disk Systems, Proc. Heat and Mass Transfer in Rotating Machinery, pp. 81–116, Springer Verlag, 1984.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-158

Coordination of Surge Protective Devices Using “Spice” Student Version Carlos Antonio França SARTORI, Otávio Luís DE OLIVEIRA and José Roberto CARDOSO Escola Politécnica da Universidade de São Paulo, Departamento de Engenharia Elétrica PEA/EPUSP, Av. Prof. Luciano Gualberto Trav. 3, 158. O5508-900. São Paulo, SP. Brazil [email protected], [email protected], [email protected] Abstract. This paper presents a methodology concerning Surge Protective Device (SPD) Coordination Studies; taking into account the available tools of the student version of the “Spice”. As an approach to satisfy the related limitations of this version, preliminary analytical studies are carried out; allowing us to selected a list of the available Spice SPD models to be applied in the simulations. Some applications regarding residential installations were chosen, and their results are presented and compared with the ones presented in literature.

1. Introduction Nowadays, the electrical and electronic equipment and systems have been used in many branches of our modern society. On the other hand, the electromagnetic phenomena that they are exposed presents resulting effects which characteristics can be higher than their immunity levels, representing potential sources of Electromagnetic Interference (EMI). Concerning the electrical phenomena related to surges; it should be mentioned that suitable protection systems have to be designed. In particular, the adoption of Surge Protection Devices (SPD) is recommended, and the SPD Coordination studies are a project requirement: These studies have as objective to guarantee the suitable energy to be dissipated by SPD, besides the clamped voltages to satisfy the equipment immunity levels. Moreover, the improper usage of protective devices can result in several fails or damages to the electrical systems and equipment. In fact, the selection of those devices is not a simple task, and it requires many parameters to be considered like the kind of the surges waveform, its intensity and the associated energy, frequency of occurrence, the SPD configuration, the equipment immunity levels, etc. [1–5]. Based on the aforementioned scenario, it should be mentioned that the use of a single analytical methods can result in a complex and time-consuming work, and the application of computational tools appear as an interesting and a suitable approach. There are a lot of computational tools that can assist in solving electric circuits, but many of them are relatively expensive. For this reason, we have proposed, as part of the methodology, the use of a student version of a circuit simulator. Due to the wide use and literature availability, the so-called “Spice” was the software that the authors have chosen to be applied [6,7].

C.A.F. Sartori et al. / Coordination of Surge Protective Devices Using “Spice” Student Version

159

Figure 1. Methodology Flowchart.

2. Methodology Basically, the proposed methodology presents as main feature, preliminary analytical evaluations that allows obtaining pre-defined SPD models [3]. These SPD models will be, then, taken into account in the further Spice simulations. The details of the SPD, equipment models, and of the full method are briefly detailed in this section. Regarding the general aspects of the method, the following steps can describe it: 1. 2. 3. 4. 5. 6. 7. 8.

Definition of the surge characteristics according to Lightning Protection Zones (LPZ) [8]; Representation of the electrical system to be protected; Definition of the immunity level of equipment [9]; Definition of the SPD characteristics; Definition the pre-selected SPDs and configuration to be used in the computational simulation [3]; Implementation of the devices selected in the previous step in the “Spice” simulator; Computational simulation of the electrical system in study; Verification of the Surge Coordination.

The flowchart concerning the proposed methodology is presented in Fig. 1. 2.1. Definition of the Electromagnetic Environment The IEC 61312-3 standard presents the parameters that helps us to classify the electromagnetic environment of a structure to be protected [8]. These aforementioned areas or electromagnetic environments are called Lightning Protection Zone (LPZ). The Fig. 2 shows this principle concerning a pre-selected structure. It should be mentioned that the

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C.A.F. Sartori et al. / Coordination of Surge Protective Devices Using “Spice” Student Version

Figure 2. Lightning Protection Zone division.

maximum value of voltage and current surges, the test wave-form, are defined according to the LPZ. 2.2. SPD Characteristics Regarding the SPD, it could be mentioned the following characteristics [2,3]: 1. 2. 3. 4. 5.

Under the rating voltage, the device will not conduct, although we can observe a small current, called leakage current, in this condition; For higher voltage, an electric current flows through the device, but the voltage across it will not increase significantly (Clamp Voltage); The energy capability of the SPD must be compatible with the surges energy level of the electrics systems; After the suppression of the surge, the SPD returns to the condition described in 1; The resulting clamp voltage must be smaller than the required immunity level of the equipment.

A typical device used in this study is the well-known varistor. 2.3. “Spice” Varistor Model The SPD models can be built based on the aforementioned characteristics and obtained directly from the available manufacturer technical literature. Regarding the varistor models, they can be considered as an association of 4 components, representing the physical phenomena that occur in the real varistor. Figure 3 presents an electric model of these devices. In the proposed varistor “Spice” model, initially, a series resistance (Rserie) is assumed, whose value is constant and equal to 100 nΩ. The components, Lserie and Cparalelo, represent series inductance and parallel capacitance of the device. These values change in accordance with the real model of the device. The variable re-

C.A.F. Sartori et al. / Coordination of Surge Protective Devices Using “Spice” Student Version

161

Figure 3. Spice varistor Model.

sistance (Rvariavel) is related to representation of the nonlinear characteristic of them. The dependence between voltage (v) and current (i) is simulated by a voltage generator, controlled by the current, and modeled by: log (V ) = p1 + p2 + p3 + p4

(1)

where, p1 = b1; p2 = b2 log(i); p3 = b3 exp (–log(i)); p4 = b4 exp (–log(i)). The parameters b1, b2, b3, b4 are related to specific characteristics of each varistor. 2.4. Details of the SPD Model Implementation in Spice Although the aspects concerning the software Spice have already been presented in various scientific papers, it should be emphasized some aspects of it. The Spice can be understood as formed by sub-routines, each one related to a specific part of the circuit simulation task. Firstly, it should be emphasized the structure and which files are used; in order to allow the model implementation in its library, like the varistor ones, in order to be available though the Schematics. In this case, the first file of interest is the “VAR.SLB” that contains the information regarding the graphical part of the devices: name, nodes, electrical parameters, mathematical characteristics, etc. This file should be associated to “PSpice” through the program Schematics, and it does not present any restrictions for the student versions. In fact, this file defines only the generic device of the component, called, e.g., “VAR”. The second file of interest is the “VAR.LIB”. This file has the mathematical definition of the characteristics of the component. It is where the parameters of each type of varistor (Cparalelo, Lserie, b1, b2, b3, and b4) are defined. This file should also be associated to “PSpice” through the program Schematics, and

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C.A.F. Sartori et al. / Coordination of Surge Protective Devices Using “Spice” Student Version

SPD 1

SPD 2

Figure 4. Schematic representation of the circuits.

now the restrictions of the student version take an important role on this process. The Spice student version admits no more than 15 definitions of components in “VAR.LIB”. Preliminary analytical studies are proposed, as an approach to satisfy the related constrains of this version, allowing us to select a list of the available Spice SPD models to be applied in the simulations. Thus, the file “VAR.LIB” can be edited, and the non-selected device definitions deleted from the available list of components. 3. Application and Results Several results regarding different residential circuits were obtained. In order to validate the proposed SPD coordination approach, a configuration presented in literature was selected [4]: A residential one with eight branch circuits, which circuits were modeled as Transmission Lines (Z0 = 100 Ω, C = 50 pF/m and L = 0,5 µH/m). Figure 4 shows a schematic representation of circuits, and the corresponding Spice models. A surge waveform 20 kA (10 × 350 μs) was assumed in the simulations. Some results, related to different SPD arrangements, and energy dissipation, are presented in Table 1. Six different cases were simulated varying the configuration of the SPD, using four types of devices (S20K130, S20K150, S20K250 and S20K625). The energy dissipated on the SPD was compared with the device withstand energy, as well as the resulting clamp voltages were checked with the equipment immunity levels. Concerning the convergence simulation requirements, an analysis of sensitivity for the parameters named “ABSTOL”, “RELTOL” and “Step Ceiling” were carried out in order to opti-

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163

Table 1. Resulting SPD energy SPD 1 Model

SPD 2 Model

Energy SPD 1 (J)

Energy SPD 2 (J)

0

-

1

20K625

-

-

-

-

23216

2

-

20K625

20K130

1255

496,261

3

20K150

-

1565,7

-

4

20K150

20K250

1467,2

89,566

5

20K150

20K130

1242,4

260,802

Configuration #

Table 2. Parameters used in simulations Case

Print Step Final Time Step Ceiling ABSTOL 1pA

RELTOL

0

0.1us

500us

0.1us

0.001

1

0.1us

4.5ms

0.1us

1pA

0.5

2

0.1us

1.6ms

0.1u

1pA

0.5

3

0.1us

3ms

0.1us

1pA

0.5

4

0.1us

3ms

0.1us

1pA

0.5

5

0.1us

3ms

0.01us

1pA

0.5

mize it. Notice that these parameters can affect the simulations, since they are directly related to approach used, like the Modified Node Analysis of the circuits [6,7]. The parameters used in each simulation are presented in Table 2.

4. Discussion Case 0 can be considered as a “control case”. It represents a residence without a surge protection system, and it is suitable for making comparisons with the cases in which the SPDs were adopted. Note that the results here presented are focused on the SPD energy dissipation. Cases 1 and 3 represent single SPD systems, whose SPD are positioned in the entry of the energy distribution system. The Case 2 counts with a second SPD, and it can be observed that the energy wasted in device S20K625 are smaller when compared to the Case 1 and Case 3. Case 4 presents the best alternative among the proposed protection configuration, presenting suitable energy coordination. The same conclusion can be observed in [4]. Case 5 does not present a good coordination between the SPDs. The Case 5 SPDs have a relatively low and close operation voltage rating, and the amount of energy dissipated in SPD 2 has increased when compared to the previous SPD configurations. Some differences on the results were observed when they are compared with the ones given in [4]: For example, the values of energy obtained in the Case 2 are smaller than the ones presented in this reference. On the other hand, in the Case 5, the energy observed on SPD 1 is lower, while the value of energy found for SPD 2 shows a value that is slightly above of the one presented in [4]. These differences can be attributed to the fact that was not possible to assume the same SPD characteristics in these works. As an example, the one of the devices used in the reference work that presents a nominal voltage equal to 200V was not available in the library of components that we have used, and it was substituted by another SPD (S20K130),

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C.A.F. Sartori et al. / Coordination of Surge Protective Devices Using “Spice” Student Version

which Vn is 205 V. This can affect significantly the results due to the nonlinear behavior of those devices. That is, depending on the point of operation of the SPD, concerning its characteristic curve, small variations of terminal voltage can result in a wide difference in the resulting current. Besides that, as mentioned before, the parameters assumed to satisfy the convergence requirements, can also explain some of the differences on the results.

5. Conclusion An approach to satisfy the limitations of the Spice student version were presented based, preliminary, on analytical studies that allows us to selected a list of the available Spice SPD models to be applied in the simulations. Some applications regarding residential installations were chosen, and their results were here presented and discussed. Although some adequacies were adopted to satisfy the convergence constraints of the numeral method, the present methodology appears as a potential one to be used in the studies related to low voltage SPD coordination. It is emphasized the importance of this study, due to the relative low immunity levels of the equipments that is used in all branches of our society. An analysis of the influence of different load and circuit models are proposed as part of the future development of this work.

References [1] Lai, J.S.; Martzloff, F.D. “Coordinating Cascaded Surge Protection Devices: High-Low versus LowHigh”, IEEE Transaction on Industry Applications, Vol. 29, No. 4, pp. 680-687, July/August 1993. [2] Paul, D.; Srinivasa I.V. “Power Distribution System Equipment Overvoltage Protection”, IEEE Transaction on Industry Applications, Vol. 30, No. 5, pp. 1290-1297, September/October 1994. [3] Paul, D. “Light Rail Transit DC Traction Power System Surge Overvoltage Protection”, IEEE Transaction on Industry Applications, Vol. 38, No. 1, pp. 21-28, January/February 2002. [4] Standler, Ronald B. “Calculations of Lightning Surge Currents Inside Buildings”, 1992 IEEE International Symposium on EMC, Proceedings, pp. 195-199, Aug. 1992. [5] Standler, Ronald B. “Transient on the Mains in a Residential Environment” IEEE Transactions on Electromagnetic Compatibility, Vol. 31, No. 2, pp. 170-176, May 1989. [6] Tuinenga, P.W. “Spice: A guide to Circuit Simulation & Analysis using PSpice”, Prentice-Hall, 1988. [7] Herniter, Marc E. “Schematic Capture with PSpice”. Macmillan College Publishing Company, 1994. [8] IEC/TS 61312-3: 2000. International Electrotechnical Commission – “Protection against lightning electromagnetic impulse – Part 3: Requirements of surge protective devices (SPDs)”. [9] IEC 61000-4-5: 2001. Electromagnetic compatibility (EMC) of electrical and electronic equipment – Part 4: Testing and measurement techniques – Section 5: Surge Immunity test.

Chapter B. Computer Methods in Applied Electromagnetism B2. Numerical Models of Devices

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-167

167

Nonlinear Electromagnetic Transient Analysis of Special Transformers Marija CUNDEVA-BLAJER, Snezana CUNDEVA and Ljupco ARSOV Ss. Cyril & Methodius, Faculty of Electrical Engineering and Information Technologies, Karpos II b.b, POBox 574, R. Macedonia E-mail: [email protected] Abstract. In the paper original methodology for nonlinear electromagnetic transient analysis of special transformers will be given. A universal nonlinear transformer model will be developed by using the finite element method study results. The electromagnetic field analysis will be done by applying the original program package FEM-3D developed at the Faculty of Electrical Engineering and Information Technologies-Skopje (FEIT). The FEM results will experimentally be verified on a resistance welding transformer through actual test results recorded in a laboratory. The same methodology and model will be used for the metrological transient analysis of 20 kV combined current-voltage instrument transformer.

Introduction The special transformers, like resistance welding transformer (RWT) or combined current-voltage instrument transformer (CCVIT) are complex non-linear electromagnetic systems which operate in transient working regimes. The transients of the RWT are introduced by the nature of the welding process. The CCVIT must comply with the rigorous metrological specifications of the IEC 60044-2 standard [1] during the transient regimes. The RWT and CCVIT electromagnetic phenomena are described by the voltage equilibrium equations: 1 dψ1 ⋅ 1 1 ω dt b

u = i ⋅r + 1

u '2 = i '2 ⋅ r '2 +

1 dψ '2 ⋅ ωb dt

(1)

(2)

where, ψ 1 = ω b ⋅ λ1 , ψ 2 = ω b ⋅ λ 2 , λ1 and λ2 are the resultant fluxes created by the primary and secondary winding currents, respectively and ωb is the basic (industrial) frequency at which the reactances will be calculated by using the FEM. The RWT and CCVIT are non-linear bounded electromagnetic systems with prescribed boundary conditions and the electromagnetic field distribution is most suitably expressed by the system of non-linear partial differential equations of the Poisson’s type:

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M. Cundeva-Blajer et al. / Nonlinear Electromagnetic Transient Analysis of Special Transformers

    ∂ ⎛   ∂A ⎞ ∂ ⎛   ∂A ⎞ ∂ ⎛   ∂A ⎞ ⎜ν B ⎟ + ⎜ν B ⎟ + ⎜ν B ⎟ = − j ( x, y , z ) ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠

( )

( )

( )

(3)

  where the magnetic vector potential A as an auxiliary quantity is introduced, B is the   magnetic flux density, ν the magnetic reluctivity and j ( x, y, z ) the volume current density. The analyzed devices are heterogeneous and (3) can be solved by numerical methods, only. The magnetic field analysis is done by an original and universal program package FEM-3D developed at the FEIT-Skopje, [2]. The results of the FEM analysis will be input data in a dynamic transformer model for transient analysis. The core saturation will be incorporated in the model as in [3] by using the relationship between measured saturated and unsaturated values of mutual flux.

Transformer Model for Transient Study The transient performance of transformers is influenced by a number of factors with most notable the exponentially decaying dc component of the primary current. Its presence influences the build-up of core flux, a phenomenon which is likely to cause saturation and subsequently substantial errors in the magnitude and phase angle of the generated signals. Core saturation mainly affects the value of the mutual inductance and, to a much lesser extent, the leakage inductances. Though small, the effects of saturation on the leakage reactances are rather complex and would require construction details of the transformer that are not generally available. In the FEM numerical approach transformer leakage inductance can be calculated by computing the normal flux at no load along a contour defined over the winding i.e. the normal leakage flux Φσ' . Multiplying the normal flux with the actual transformer length in the z direction lz yields:

Φσ = Φσ' ⋅ l z

(4)

Then the total leakage flux is defined as:

ψ σ = Φσ ⋅ N

(5)

where N are the number of turns excited. The leakage inductance is calculated by:

Lσ = ψ σ / I 0

(6)

where I0 is the no load current flowing through the primary winding. In many dynamic simulations, the effect of core saturation may be assumed to be confined to the mutual flux path. Different iron core models are described in the literature and their summary is described in [4]. In this paper a nonlinear SIMULINK transformer model has been developed and it is presented in Fig. 1. The effects of core saturation in the dynamic simulations have been incorporated using the relationship between saturated and unsaturated values of the mutual flux linkage as described in [3].

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M. Cundeva-Blajer et al. / Nonlinear Electromagnetic Transient Analysis of Special Transformers

Initialize and plot

y To Workspace

Scope1

Scope

Plots FFT FFT

uslovi

Mux

Mux5

1

Clock

Out_psi1

Mux

wb*(u[2]-(r1/xl1)*(u[3]-u[1]))

s psi1_

Fcn

v1

Mux

psi1

1

i1

f(u)

3

Mux4

Scope4

Out_i1

Fcn4

Mux psim Mux

V

xM*(u[1]/xl1+u[2]/xpl2-u[3]/xm)

2

PQ I

Scope3 Active & Reactive Power

Mux3

Dpsi Memory1

v2p•

Mux

Out_psim

Fcn3

wb*(u[2] -(rp2/xpl2)*(u[1]-u[3]))

s psi2'_

Fcn2

Dpsi=f(psisat)

psi2'

1

Mux1

Mux

Mux2

(u[1]-u[2])/xpl2

i2'

Fcn5

4 Out_i2'

Load Module

Figure 1. SIMULINK transformer model for transient analysis. Table 1. FEMM numerical results

Φσ [Vs/turns]

ψσ [Vs]

Lσ [H]

magnetic field energy W [J]

5,656⋅10–6

531,6⋅10–6

118,13⋅10–6

1,877

normal leakage flux

total leakage flux

Leakage inductance

Magnetizing inductance Lm [H] 0,185

Study Case-RWT: Experimental Verification of the Model Prior to setting up a transformer model suitable for transient studies, a set of test results for commercial RWT were performed in the laboratory at FEIT. According to the manufacturer data and the measurements performed, the resistance welding transformer has the following rated data: primary voltage 380 V; secondary no-load voltage (1,41–4,63) V; conventional power 24 kVA; rated frequency 50 Hz; thyristor controlled switching; number of primary tap positions 9. The transformer is a single phase with shell type core. Core saturation has been determined from the open circuit magnetization curve of the investigated transformer. The leakage reactances of the RWT were calculated by using the FEMM program [5,6] and they served as input in the dynamic model shown in Fig. 1. The results from the numerical finite elements calculation are presented in Table 1. The derived SIMULINK results have been experimentally verified on the resistance welding transformer through actual test results recorded in a laboratory. A small selection of the derived results is presented in Table 2. The test results are satisfactory with the exception of a slightly different magnitude of the simulated iron losses Pfe and simulated reactive power Qfe. The simulation results for the currents are nearly identical to the laboratory tests. With these results the validity of the derived model has been proved. The transient performance of the RWT has been modeled by defining zero transition of the winding voltage β = 0. Sample results for the transient performance of the

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M. Cundeva-Blajer et al. / Nonlinear Electromagnetic Transient Analysis of Special Transformers

Table 2. Comparison of the RWT experimental and simulated results Tap position 5 Experiment Simulation 0,65 0,80 55,4 58,0 78,3 83,0 115 100 217 285

I0 [A] noload current I1 [A] primary rms current I1max [A] primary magnitude current Pfe [W] active core losses Qfe [VA] reactive core losses

Tap position 8 Experiment Simulation 5,26 5,40 125 134 177 187 298 180 1529 2000

Figure 2. RWT switching transients versus time [s] at most rigorous phase angle β = 0.

analyzed RWT at nominal position 8 are shown in Fig. 2. From the results the exponentially decaying dc component of the primary current can be clearly observed. The current waveform displays large peak at the beginning (up to 100 times the rated value). Study Case-CCVIT: Application of the Verified Model The developed transformer model verified on the RWT study case is applied for transient analysis of the combined instrument transformer CCVIT with (voltage measurement core VMC ratio

20000 V

3

: 100 V

3

and current measurement core CMC ratio

100 A: 5 A). The CCVIT is with a complex electromagnetic construction and its geometry has been given in details in [7]. The electromagnetic parameters of the CCVIT are most important for its transient analysis, as they are the input data in the above developed SIMULINK transformer model. The CCVIT quasi-steady-state electromagnetic field analysis has been done by the original and universal program package FEM-3D, [2], developed at the Faculty of Electrical Engineering and Information Technologies in Skopje. The detailed CCVIT FEM-3D analysis has been given in [8]. The main electromagnetic characteristics of the CCVIT derived by FEM-3D, necessary for the transient analysis are given in Figs 3–8. In Fig. 3 the main flux characteristic per turn ϕmu in the upper middle cross-section of the VMC magnetic core via the VMC

M. Cundeva-Blajer et al. / Nonlinear Electromagnetic Transient Analysis of Special Transformers 40,0

30,0 20,0 10,0 0,0 0,0 0,5 1,0 1,5 Relative VMC input voltage U u /U ur [r. u.]

only VMC Ii/Iir=0 Ii/Iir=0,2 Ii/Iir=0,4 Ii/Iir=0,6 Ii/Iir=0,8 Ii/Iir=1,0 Ii/Iir=1,2

Figure 3. CCVIT main flux characteristics in [μVs] in the upper middle cross-section of the VMC magnetic core via the VMC input voltage and the CMC input current as a parameter.

Main flux per turn

Main flux per turn

40,0

30,0

only CMC Uu/Uur=0 Uu/Uur=0,2 Uu/Uur=0,4 Uu/Uur=0,6 Uu/Uur=0,8 Uu/Uur=1,0 Uu/Uur=1,2

20,0 10,0 0,0 0,0

1,0

1,5

Figure 4. CCVIT main flux characteristics in [μVs] in the upper middle cross-section of the CMC magnetic core via the CMC input current and the VMC input voltage as a parameter.

30

25 20

only VMC Ii/Iir=0 Ii/Iir=0,2 Ii/Iir=0,4 Ii/Iir=0,6 Ii/Iir=0,8 Ii/Iir=1,0 Ii/Iir=1,2

15 10 5 0 0,0

0,4

0,8

Leakage flux per turn

35 30

25 only VMC Ii/Iir=0 Ii/Iir=0,2 Ii/Iir=0,4 Ii/Iir=0,6 Ii/Iir=0,8 Ii/Iir=1,0 Ii/Iir=1,2

20 15 10 5 0

1,2

0,0

Relative VMC input voltage U u /U ur [r. u.]

0,0

0,4

0,8

1,2

0,8

1,2

only CMC Uu/Uur=0 Uu/Uur=0,2 Uu/Uur=0,4 Uu/Uur=0,6 Uu/Uur=0,8 Uu/Uur=1,0 Uu/Uur=1,2

Figure 6. VMC secondary winding (120 turns) leakage flux characteristics per turn in [μVs] via the VMC relative input voltage and the CMC relative input current as a parameter.

Leakage flux per turn

70 60 50 40 30 20 10 0

0,4

Relative VMC input voltage U u /U ur [r. u.]

Figure 5. VMC primary winding (24000 turns) leakage flux characteristics per turn in [μVs] via the VMC relative input voltage and the CMC relative input current as a parameter.

Leakage flux per turn

0,5

Relative CMC input current I i /I ir [r. u.]

40 Leakage flux per turn

171

25 20 15 10 5 0 0,0

0,4

0,8

1,2

only CMC Uu/Uur=0 Uu/Uur=0,2 Uu/Uur=0,4 Uu/Uur=0,6 Uu/Uur=0,8 Uu/Uur=1,0 Uu/Uur=1,2

Relative CMC input current I i /I ir [r. u.]

Relative CMC input current I i /I ir [r. u.]

Figure 7. CMC primary winding (6 turns) leakage flux characteristics per turn in [μVs] via the CMC relative input current and the VMC relative input voltage as a parameter.

Figure 8. CMC secondary winding (120 turns) leakage flux characteristics per turn in [μVs] via the CMC relative input current and the VMC relative input voltage as a parameter.

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Figure 9. Switching transient primary CCVIT current of the VMC via time [s] at most rigorous phase angle β = 0.

Figure 10. Switching transient primary CCVIT current of the CMC via time [s] at most rigorous phase angle β = 0.

Figure 11. Time dependence of the VMC primary current (RMS value) at rated load of the both cores and β = 0.

Figure 12. Time dependence of the CMC primary current (RMS value) at rated load of the both cores and β = 0.

relative input voltage Uu/Uur and the CMC relative input current Ii/Iir as a parameter are displayed. In Fig. 4 the main flux characteristic per turn ϕmi in the upper middle crosssection of the CMC magnetic core via the CMC relative input current Ii/Iir and the VMC relative input voltage Uu/Uur as a parameter are displayed. The CCVIT transient analysis is done by coupling with the FEM-3D results. The magnetic field distribution results, e.g. leakage reactances characteristics are input data into the non-linear mathematical model of the CCVIT. The complex non-linear analysis has been done for rated loads of the both measurement cores and rated frequency of 50 Hz. The input voltage phase angle is β = 0. The CCVIT is a measurement device therefore its metrological parameters are of greatest interest. By using the SIMULINK transformer model the most important metrological CCVIT parameters have been calculated for the most rigorous moment the first forth of the signal period at the worst, from metrological point of view, regime at β = 0: the VMC voltage error pu = –17,5% and CMC current pi = –19%.

M. Cundeva-Blajer et al. / Nonlinear Electromagnetic Transient Analysis of Special Transformers

173

0,10

I1umax [A]

0,08

only VMC Ii/Iir=0.0 Ii/Iir=0.2 Ii/Iir=0.4 Ii/Iir=0.8 Ii/Iir=1.0 Ii/Iir=1.2

0,06 0,04 0,02 0,00 0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

Relative VMC primary voltage U u /U ur [r. u.]

Figure 13. Maximal VMC primary plug-in current via the VMC relative voltage, the relative CMC current is a parameter at β = 0 and rated loads of both cores.

120 100

I1imax [A]

80 60 40 20

only CM C Uu/Uur=0.0 Uu/Uur=0.2 Uu/Uur=0.4 Uu/Uur=0.8 Uu/Uur=1.0 Uu/Uur=1.2

0 0,0 0,5 1,0 1,5 Relative to rated steady- state regime CM C primary current I i /I ir [r. u.]

Figure 14. Maximal CMC primary plug-in current via the CMC relative current (through the CMC steadystate regime current), the relative VMC voltage is a parameter at β = 0 and rated loads of both cores.

Conclusions The developed transient performance transformer model has been verified as accurate on the resistance welding transformer and it has been further applied for transient analysis of a combined instrument transformer. The confirmed transformer model has been coupled with finite element method results. The methodology in the paper is universal and can be applied for other complex electromagnetic devices.

References [1] IEC (International Electrotechnical Commission) 60044-2, 1980: Instrument transformers, Part 3: Combined transformers, Geneve, 1980. [2] M. Cundev, L. Petkovska, The Weighted Residuals Method for Electromagnetic Field Problems in Electrical Machines, Proc. of the 32nd UPEC’97 Conference, Vol. 2, UMIST, pp. 934-937, 1997.

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[3] C.M. Ong “Dynamic Simulation of Electric Machinery Using MATLAB/SIMULINK”, Prentice Hall PTR, Upper Saddle River, New Jersey, 1998. [4] Working Group C-5 of the Systems Protection Subcommittee of the IEEE Power System Relaying Committee, “Mathematical models for current, voltage, and coupling capacitor voltage transformers”, IEEE Transactions on power delivery, Vol. 15, No. 1, Jan. 2000, pp. 62-72. [5] D. Meeker, Finite Element Method Magnetics – User’s Manual 3.0, 1998-2000. [6] S. Cundeva, L. Petkovska., V. Filiposki., A Methodology for Coupled Steady-State ElectromagneticThermal Modeling of Resistance Welding Transformer, Proceedings of XI International Symposium on Electromagnetic Fields in Electrical Engineering ISEF’03, Maribor, Slovenia, 2003, pp. 751-756. [7] M. Cundeva, L. Arsov, G. Cvetkovski, Genetic Algorithm coupled with FEM-3D for metrological optimal design of combined current-voltage instrument transformer, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23, No. 3, 2004, pp. 670-676. [8] M. Cundeva-Blajer, L. Arsov, FEM-3D for Metrological Optimal Design and Transient Analysis of Combined Instrument Transformer, Przeglad Elektrotechniczny, R. 83 NR 7-8/2007, pp. 96-99.

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Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine a

Y. TAMTO a, A. FOGGIA a, J.-C. MIPO b and L. KOBYLANSKI b LEG, UMR5529 INPG/UJF-CNRS, BP46, 38402 Saint Martin d’Hères Cedex, France b VALEO Equipements Electriques & Moteurs, 2 Rue André Boulle-BP150, 94017 Créteil Cedex, France Abstract. To increase claw pole machine efficiency, we evaluate leakages in the magnetic circuit. These leakages are mainly due to end windings, air gap and slot leakages. Thus we obtain an equivalent circuit model. This model will be helpful to simulate the machine under load conditions and to make possible its optimization. The aim of this paper is then to present some methods of determination of leakage inductance, by computation and under test.

Introduction Methods for leakage inductance determination under test and computation, with the short circuit, the open-circuit and the zero-power-factor characteristic, to build the Potier diagram are well known. These methods were developed mainly for high power alternator. In the case of claw pole machine, zero-power-factor test and computation are difficult to run because of the value of inductances needed to simulate zero-powerfactor. This is why we explore other ways of leakage evaluation: we used the singular configuration of the claw pole alternator under load conditions, and the stator alone, without the rotor.

1. The Potier Method This model takes into account the magnetic saturation and is used in the special case of alternators with smooth poles. Figure 1 represents the Potier diagram in the general case. Under load, there is not phase difference between V and I because the load is a diode rectifier bridge charging a battery. We then compute λ with load characteristic, by solving Potier vector equations (1) and using finite element computation. Tests results will validate the parameters previously obtained. Ε 2r = (V + RI ) 2 + (ωλ I ) 2 I 2fr = I 2f + (α I ) 2 − 2α I cos(γ ) γ =

π λω I ) + arctg ( 2 V + RI

(1)

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Y. Tamto et al. / Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine

Er: Voltage behind leakage inductance λ : Leakage inductance α : Potier coefficient of equivalence Ifr: Load excitation corresponding to Er I f: Load excitation V: voltage /phase I: Current/phase R: Resistance/phase Figure1. Vector diagram of Potier.

2. Example The main characteristics of one of the claw pole machine used for tests are on the following table Winding Number of pole pairs Number of turns of rotor winding Rotor Resistance Number of phases Number of turns per phase Resistance per phase Number of stator slots Air-gap thickness

star 6 380 2,66 ohms 3 6 27 milliohms 36 0.365 mm

No load characteristic allow to express Ifr from Er, the parameter α is the slope of short circuit graph. Then with load tests we find λ by using electrical values at one phase boundaries at different speeds. Results for Potier parameters are: α = 0, 033 λ = 48μ H

The proposed method is more accurate for this type of alternator because test under load is more easy to carry out than the use of zero factor characteristic. But the computation time is long. So, we try to evaluate leakages by the removed rotor method.

3. Removed Rotor Method The rotor is removed and the stator is supplied by a three-phase current source, the amplitude of current varies from a very low value to the nominal one. The idea is that by removing the rotor, we take the main flux off; it only remains the leakage flux. The leakage inductance is then deduced from the electric quantities at the terminals of the stator windings. Another calculation of the leakages is done by using the electromagnetic energy stored.

Y. Tamto et al. / Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine

177

Removed Rotor Test A tree phase current source is needed to feed stator phases. We made frequency and module variable. For each set of variables, the temperature of the windings is measured to evaluate the right resistance. Xa = V Z = I

Z

2

− R

2

(2)

Xa = ω λ: Leakage reactance; Z: Phase impedance; V: Voltage; I: Current; R: Resistance of the phase at T1 temperature R T1: = R T0. (1 + α.(T1 – T0)); α = 4 10–3 °C–1 for copper.

Removed rotor tests

Leakage inudctance (uH)

96

48

0 50

250

450

650

Freque ncy( Hz)

Figure 2. Leakage inductance.

Finite Element Computation The machine to computaee has three phases and 6 pole pairs. We choose the area under the stator teeth with tangent property for the magnetic field. A triplet of current is imposed in the phases [A, B, C] and they are distributed as follows: [IA, –IA/2, –IA/2]. Then leakage inductance is calculated by (3).

λ=

(4/ 3)*E Ι 2

A

E: electromagnetic energy. IA: current in phase A.

(3)

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Y. Tamto et al. / Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine

Figure 3. Design without Rotor.

Results IA (A) 5 10 20 30 40 50 60 70 80 90

Energy (J) 9,01E-04 3,61E-03 1,45E-02 3,26E-02 5,80E-02 9,07E-02 1,31E-01 1,78E-01 2,32E-01 2,94E-01

Leakage Inductance (μH) 48,1 48,2 48,3 48,3 48,4 48,4 48,4 48,4 48,4 48,4

We can notice that value of leakage doesn’t change with phase current and are the same for removed rotor computations and tests.

Conclusion The values of leakage inductances obtained by method exposed are nearly equals, this study validates removed rotor method for determination of leakage inductance and we can determinate alternator outputs with an accuracy of 99%. This method is speed and easy to test and to computate. In the aim of an optimization for example it’s useful to simplify the claw pole machine by its electrical model of Potier, this model allow to create a virtual test bench of the claw pole environment with numerical language.

References [1] IEEE Std 115-1995, pp. 50-66. [2] J. Dos Ghali, Essais Spéciaux sur les Machines Electriques.

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179

Reduction of Cogging Torque in Permanent Magnet Motors Combining Rotor Design Techniques Andrej ČERNIGOJ a, Lovrenc GAŠPARIN a and Rastko FIŠER b Iskra Avtoeletrika d.d., Polje 15, 5290 Šempeter pri Gorici, Slovenia [email protected], [email protected] b University of Ljubljana, Faculty of Electrical Engineering, Tržaška cesta 25, 1000 Ljubljana, Slovenia [email protected] a

Abstract. High performance motor drive applications require permanent magnet synchronous motors (PMSM) that produce smooth torque with low torque ripple components. This paper quantifies various sources of torque ripple and is focused on rotor PMSM design techniques that can be used for reducing the cogging torque. For each chosen design technique a validation with finite element method (FEM) analysis is given. Finally the comparison and evaluation of design principles and their combination are presented and commented.

Introduction The ability of PMSM to produce smooth torque and high power density is important in high performance motor drive applications. Nevertheless, due to the mechanical construction and material properties there are various causes responsible for producing undesired torque ripple like: reluctance torque, cogging torque, and harmonics in the back emf. Several well-known design techniques [1–4] can be used to minimize this parasitic harmonic torque components, but most of them consequently reduce the output torque as well. Presented paper discusses and quantifies compromises among reducing cogging torque in connection with preserving the main output torque at high value in order to optimize the PMSM design for a specific drive application.

Instantaneous Torque of PMSM Instantaneous torque of a PMSM has two components: constant component T0 and periodic component Tr(α), which is a function of an electric angle α and presents torque pulsations called torque ripple (Fig. 1). T (α ) = T0 + Tr (α )

(1)

ˇ A. Cernigoj et al. / Reduction of Cogging Torque in Permanent Magnet Motors

180

Figure 1. The instantaneous torque of a PMSM motor T(α).

There are three origins of torque ripple in PMSMs: • • •

the difference between permeances in the air-gap in the d and q magnetic axis produces reluctance torque, cogging effect is the interaction between rotor magnetic flux produced by permanent magnets and variable permeance of the air-gap due to stator slots and produces cogging torque, distortion of sinusoidal distribution of the magnetic flux density in the air-gap produces field harmonic electromagnetic torque.

Rotor Design Techniques for Cogging Torque Reduction Elimination of cogging torque using available design techniques, while keeping the output torque at the same level is a challenging task. At the beginning the proper slot/pole combination must be selected for effective reduction of cogging torque. Beside this, for a given motor, the following rotor design techniques can be considered respectively or in any mutual combination: magnet span variation, magnet pole shifting, magnet shape and magnetization pattern optimisation, and magnet step skew or skewing. Because of the complexity of the given task, the full parametric FEM model of 36-slot and 6-pole PMSM was chosen. Due to slot/pole combination basic design will express a significant cogging torque, thus different rotor design techniques for cogging torque reduction can be studied and evaluated. Selection of Magnet Span with Magnet Pole Shifting Among several possibilities an effective way to reduce cogging torque, while maintaining the output torque, is to optimize the magnet span αm and shift magnetic pole angle γ, as shown in Fig. 2. To simplify the representation and enhance clearness only curves of maximal cogging torque Tcog max versus magnetic pole shift angle γ for several magnet spans αm are shown in Fig. 3. The minimal cogging torque values appear around the shift angle γ = 56° and are two to three times smaller than in the case of symmetrical magnetic pole distribution at γ = 60°. The influence of magnet span αm on cogging torque values can also be observed and it is obviously that the minimum appears at αm = 50°, thus optimizing of the PM motor design is an iterative process with many variable parameters.

ˇ A. Cernigoj et al. / Reduction of Cogging Torque in Permanent Magnet Motors

181

Figure 2. Principle of magnetic poles shifting γ ≠ δ (left), and 2D FEM model of PM rotor (right).

Figure 3. Maximal cogging torque versus magnet pole shift angle γ.

Shape and Magnetization Pattern of Permanent Magnets Air-gap flux density distribution is strongly dictated by the shape and the magnetization pattern of applied permanent magnets. Furthermore this has a substantial influence on cogging torque, harmonic contents and magnetic saturation. Figure 4 shows various shapes and magnetization patterns of arc magnets usually used for surface mounted PMSMs. The influence of shape and magnetization pattern on the air-gap flux densitiy distribution is presented in Fig. 5. Notice that a bread loaf (lens-shaped) magnet shape flux densitiy distribution is the closest to the sinusoidal distribution. The analysis of shape and magnetization pattern of permanent magnets on cogging torque was carried out for all three shapes of PMs. Using a bread loaf magnet shape a minimal value of cogging torque compared to the constant component of the output torque T0 is achieved, but the output torque T0 is also considerable reduced in respect to the initial basic PMSP model, as presented in Fig. 6. Shifted magnet poles with selection of magnet span at the same time results in efficient reduction of cogging torque while keeping the level of the output torque T0 high (Table 1). Such approach represents more efficient solution in cogging torque minimization.

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ˇ A. Cernigoj et al. / Reduction of Cogging Torque in Permanent Magnet Motors

Figure 4. Various shapes and magnetization patterns of PM: surface radial (left), surface parallel (middle), and bread loaf (right).

Figure 5. Flux densitiy along the centre of the air-gap.

Figure 6. Maximal cogging torque and output torque for various shapes and magnetization patterns of PMs versus magnet span αm.

Conclusion Numerous parametric FEM calculations and laboratory measurements have proved that majority of rotor design techniques are very efficient in reducing the parasitic cogging

ˇ A. Cernigoj et al. / Reduction of Cogging Torque in Permanent Magnet Motors

183

Table 1. Gathered results of rotor design techniques Design technique Basic PMSM model (αm = 56°) Optimal magnet span (αm = 40°) Magnet pole shifting with magnet span (γ = 56°, αm = 50°) Bread loaf magnet shape

Tcog max (Nm) 4,42 2,10 0,99 0,024

Tcog max / T0 17,7% 9,4% 4,0% 0,11%

T0 / T0 basic mod. 100% 89% 98% 89%

torque in PMSMs. If they are combined with several additional stator design techniques (optimization of slot openings, additional notches in stator teeth), the decreasing of the cogging torque Tcog max could be even improved, while maintaining the output torque T0 of the PMSM at the same level. Presented ascertainments are already brought into use in mass-production of PMSM motors for high demanding special applications.

References [1] J. Gieras, M. Wing, Permanent Magnet Motor Technology, New York, 1997, Marcel Dekker Incorporation. [2] N. Bianchi, S. Bolognani, Design techniques for reducing the cogging torque in surface-mounted PM motors, IEEE Trans. Industry Application, Vol. 38, No. 5, September/October 2002, pp. 1259–1265. [3] R. Lateb, N. Takorabet, F. M. Tabar, J. Enon, A. Sarribouete, Design technique for reducing the cogging torque in large surface mounted magnet motors, ICEM 2004 International Conference on Electrical Machines, Proceedings ICEM 2004 CD-ROM, Krakow, Poland, 5-8 Sept 2004. [4] M.S. Islam, S. Mir, T. Sebastian, Issues in Reducing the Cogging Torque of Mass-Produced PermanentMagnet Brushless CD Motor, IEEE Tran. Industry Application, Vol. 40, No. 3, May/June 2004, pp. 813820. [5] M. Furlan, A. Černigoj, M. Boltežar, A coupled electromagnetic-mechanical-acoustic model of a DC electric motor, Compel, Vol. 22, No. 4, 2003, pp. 1155-1165.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-184

Optimum Design of Linear Motor for Weight Reduction Using Response Surface Methodology Do-Kwan HONG, Byung-Chul WOO, and Do-Hyun KANG Korea Electrotechonolgy Research Institute, P. O. Box 20, Changwon, 641-120, Korea Tel: +82-55-280-1395, Fax: +82-55-280-1547, E-mail: [email protected]

Abstract. This paper presents an optimum design procedure of linear motors to reduce the weight of the machines with the constraints of thrust and detent force using response surface methodology (RSM). RSM is well adopted to make the analytical model of the minimum weight with constraints of thrust and detent force, and it enables objective functions to be easily created, and a great deal of computation time can be saved. Therefore, it is expected that the proposed optimization procedure using RSM can be easily utilized to solve the optimization problem of the linear motors.

Introduction In many applications were the motion is essentially linear. It is possilble to use linear motors instead of rotary motors. Linear motors are electromagnetic devices developing mechanical thrust without mechanical slider-crank system mechanism. Advantages of the linear motors include low noise, reduced operating cost, and incearsed flexibility of operation due to gearless feature [1]. The linear motors, however, have some practical limitations. One of the major reasons of the limitations is that inherently large air gap causes low power density. In order to increase the power density, permanent magnet (PM) type longitudinal flux linear motors (LFLMs) can be considered for the application of the linear motors in high power density systems. Since LFLMs can produce high magnetic thrust and reluctance thrust with relatively small air gap. There are several practical examples of LFLM in [2,3]. In this paper we consider the development of a LFLM for use in linear compressor applications. Figure 1 shows the structure of the developed LFLM. It has two important electromechanical components, a linear actuator and springs for refrigeration application. For short stroke applications like in this situation in odrer to recover the energy at the end of the displacement mechanical springs are used. By controlling the operating frequency of the actuator around mechanical resonance frequency, the system has higher efficiency than conventional rotary type compressors. In order to increase the performance of the system it was mandatory to consider a method of optimization. RSM is recently receiving attention for its modeling ability of electromagnetic devices performance by using statistical fitting method, given that RSM is well adopted to develop analytical models for the complex problems. With this analytical model, an objective function with constraints can be easily created, and computation time can be saved.

D.-K. Hong et al. / Optimum Design of Linear Motor for Weight Reduction

185

Figure 1. PM type LFLM for compressor.

This paper presents the optimum design of longitudinal flux actuator for linear compressor using RSM. Its design goal is to reduce the weight of the machine with the constraints of thrust and detent force with respect to the initial machine. At first step, most influential design variables and their levels should be determined and be arranged in a table of orthogonal array. Each response value is determined by 3D finite element method (FEM). With the use of reduced gradient algorithm (RGA), finally, the most desired set is determined, and the influence of each design variables on the objective function can be obtained. The weight can be reduced by 10.09%, thrust force improved by 3.06% and detent force improved by 4.15% of initially designed PM type LFLM. Optimum Design for Longitudinal Flux Linear Motor (LFLM) RSM Method RSM seeks for the relationship between design variables and response through statistical fitting method. A polynomial approximation model is commonly used for a secondorder fitted response (u) and can be written as follow k

k

j =l

j =i

k

u = β 0 + ∑ β j x jj + ∑ β jj x j + ∑ β ij xi x j +ε 2

(1)

j =l

β : regression coefficients, x : design varaibles, ε : random error, k : number of design variables. The least squares method is used to estimate unknown coefficients. Matrix notations of the fitted coefficients and the fitted response model can be written as:

βˆ = ( X ′X ) −1 X ′u

uˆ = X βˆ

(2)

It should be evaluated at the data points. βˆ , where βˆ is the vector of the unknown coefficients which are usually estimated to minimize the sum of the squares of the error term. RSM method is applied in connection with FEM and the response actually represent FEM output values. Figure 2 presents the principal steps of RSM procedure.

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D.-K. Hong et al. / Optimum Design of Linear Motor for Weight Reduction

Figure 2. RSM.

2.5

B (T)

2.0

S23

1.5 1.0 0.5 0.0

Laminated steel

0

10

20

30

H (kA/m)

40

50

60

Figure 3. B-H curves of the used materials. Table 1. Specifications of LFLM analysis model Item Stator/Iron material Permanent Magnet Nominal air gap Nominal current MMF

Unit

Specification

mm A

S23 NdFe35H 0.5 1 600

AT(Ampere Turn)

Design Variables and Levels Table 1 shows the specification of the PM type LFLM for a compressor. Figure 3 shows the B-H characteristic of the S23 material which was used for the active parts of the LFLM. The variables represent geometrical dimentsions which are completely determining the geometry of the LFLM no material variable was considered. Figure 4 shows the design variable of the PM type LFLM. Table 2 shows the design variable

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D.-K. Hong et al. / Optimum Design of Linear Motor for Weight Reduction

Figure 4. Design variable of PM type LFLM. Table 2. Design variable and level Design Variable Level 1(–1) 2 (0) 3 (1)

DV1

DV2

DV3

DV4

DV5

DV6

DV7

8.449 9.94 11.431

5.95 7 8.05

1.9975 2.35 2.7025

4.25 5 5.75

5.95 7 8.05

1.833 3.666 5.499

16.15 19 21.85

and level. The table of orthogonal array is shown in Table 3 and corresponding simulation result (nominal current value 600 AT). The table of orthogonal array is determined by considering the number of design variables and each level of them. After obtaining the experimental data from 2D FEM, the function necessary to draw response surface is extracted. In order to determine the equations of the response surface, several experimental designs are developed to establish the approximate equation using the smallest number of experiments. The first level of DV6 and the second level of the other design varaibles have the values of the initially designed LFLM in Table 2. Optimum Design Result Table 2 represents the mixed orthogonal array, which is determined by considering the number of the design variables and each level of them. After getting the experimental data by 2D FEM, the function to draw response surface is extracted. In order to determine the equations of the response surface, several experimental designs are developed to establish the approximate equation using the smallest number of experiments. The purpose of this paper is to minimize the object function (weight) with constraints of thrust and detent force. Table 3 and Table 4 show optimum solution and comparison result of initial model and optimum model, respectively. The two fitted second order polynomial of the object functions for the seven design variables are as follows.

Weight = 1.44767 + 0.078 DV 1 + 0.07477 DV 2 + 0.01819 DV 3 + 0.05701DV 4 + 0.08289 DV 5 − 0.006 DV 6 + 0.00346 DV 7 − 0.00091DV 12 + 0.000465 DV 2 2 + 0.00196 DV 32 + 0.0024 DV 4 2 − 0.0006 DV 52 + 0.00335 DV 6 2 + 0.00129 DV 7 2 (3)

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D.-K. Hong et al. / Optimum Design of Linear Motor for Weight Reduction

FThrust = 57.8049 + 4.6812 DV 1 + 3.2581DV 2 + 3.5414 DV 3 − 0.1728DV 4 + 0.0871DV 5 + 0.196 DV 6 − 0.5666 DV 7 + 0.0336 DV 12 − 0.0492 DV 22 2

2

2

2

− 0.2469 DV 3 − 0.1479 DV 4 + 0.0906 DV 5 + 0.2294 DV 6 − 0.0434 DV 7

(4) 2

FDetent = 0.38872 + 0.09841DV 1 + 0.03092 DV 2 + 0.07204 DV 3 − 0.03208DV 4 − 0.03093DV 5 + 0.12183DV 6 − 0.12523DV 7 + 0.05814 DV 12 − 0.03928 DV 42 (5) − 0.03498 DV 52 + 0.03957 DV 6 2 + 0.06569 DV 7 2

The adjusted coefficients of the multiple determination R2adj for three responses are weight (99.6%), FThrust (99.9%) and FDetent (98.1%). In Table 3 and 4, the optimal point is searched to find the point of less than 10.09% of the weight, 4.15% of detent force, and greater than 3.06% of the thrust force of the initially model. Table 4 and Table 5 show the optimum solution and the comparison result of initial model and optimum model, respectively. Figure 5 shows the 2D flux line and flux density of the initially model. Pareto chart of thrust force, detent force and weight are presented in Fig. 6. Also from this figure the most sensitive design variables can be identified. The response surface of thrust force according to change of the most influential design variables are shown in Fig. 7. Figure 8 shows an interaction plot of means for thrust force between design variables. Figure 9 shows a comparison between initial and optimum model. The weight can be reduced 10.09%, thrust force improved by 3.06% and detent force improved by 4.15% of the initially designed PM type LFLM. Figure 10 shows the detent force and thrust force profile of the optimum model by mmf. Table 3. Table of mixed orthogonal array L18(21×37) Exp.

DV1

DV2

DV3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

8.449 8.449 8.449 9.94 9.94 9.94 11.431 11.431 11.431 8.449 8.449 8.449 9.94 9.94 9.94 11.431 11.431 11.431

5.95 7 8.05 5.95 7 8.05 5.95 7 8.05 5.95 7 8.05 5.95 7 8.05 5.95 7 8.05

1.9975 2.35 2.7025 1.9975 2.35 3 2.35 2.7025 1.9975 2.7025 1.9975 2.35 2.35 2.7025 1.9975 2.7025 1.9975 2.35

DV4 DV5 DV6

DV7

4.25 5 5.75 5 5.75 4.25 4.25 5 5.75 5.75 4.25 5 5.75 4.25 5 5 5.75 4.25

16.15 19 21.85 21.85 16.15 19 21.85 16.15 19 16.15 19 21.85 19 21.85 16.15 19 21.85 16.15

5.95 7 8.05 7 3 5.95 8.05 5.95 7 7 8.05 5.95 5.95 7 8.05 8.05 5.95 7

1.833 3.666 5.499 5.499 1.833 3.666 3.666 5.499 1.833 3.666 5.499 1.833 5.499 1.833 3.666 1.833 3.666 5.499

Avg. thrust force(N) 46.72 53.23 59.41 50.60 58.30 64.42 58.80 66.76 61.65 53.27 49.92 55.72 54.53 60.47 57.85 62.72 57.81 66.71

Max.detent force(N) 0.33 0.33 0.4 0.37 0.34 0.46 0.39 0.97 0.36 0.53 0.38 0.25 0.46 0.33 0.49 0.43 0.33 0.91

Weight (kg) 1.16 1.37 1.62 1.36 1.6 1.41 1.49 1.46 1.66 1.37 1.37 1.37 1.35 1.43 1.59 1.57 1.49 1.54

Table 4. Optimum level and optimum size Design variable Level Optimum level Optimum size

DV1

DV2

DV3

DV4

DV5

DV6

DV7

0.508 10.70

–1 5.95

1 2.703

–1 4.25

–1 5.95

–1 0.4116 1.833 20.173

D.-K. Hong et al. / Optimum Design of Linear Motor for Weight Reduction

Table 5. Comparison of initial model and optimum model Model Initial Optimum (predicted RSM) 2D FEM (optimum simulation) Variation between initial and 2D FEM model (%)

Weight (kg) 1.4575 1.3099 1.3104

Avg. Thrust force (N) 57.8900 60.0000 59.6664

Peak. Detent force (N) 0.3594 0.3609 0.3445

–10.09

3.06

–4.15

(a) Flux line

(b) Flux density

Figure 5. Flux density and flux line in LFLM using FEM (initial model).

(a) Thrust force

(b) Weight

(c) Detent force Figure 6. Pareto chart of thrust force(Avg.), detent force and weight.

189

190

D.-K. Hong et al. / Optimum Design of Linear Motor for Weight Reduction

Figure 7. Surface plot of thrust force(Avg.).

Figure 8. Interaction plot of means for thrust force (avg).

0

0

600

Detent, thrust force (N)

60 50 40

Optimum (avg.) : 59.6664 N Initial (avg.) : 57.89 N

10

detent

120

thrust

100

initial

30 20

MMF(AT) :

600

optimum Initial (peak) : 0.3594 N Optimum (peak) : 0.3445 N

0

Detent, thrust force (N)

MMF (AT) :

0

300

600

900

1200

Optimum model

80 60 40 20 0

-3

-2

-1 0 1 Position (mm)

2

3

Figure 9. Comparison between initial and optimum model.

-3

-2

-1 0 1 Position (mm)

2

3

Figure 10. Detent and thrust force profile.

D.-K. Hong et al. / Optimum Design of Linear Motor for Weight Reduction

191

Conclusion In this paper, an optimum design procedure is introduced to design LFLM to reduce its weight and to improve thrust force and detent force of the initially designed LFLM with many shape design variables, Also, Optimization design by RSM and table of orthogonal array are presented in detail for the LFLM in this paper. The performance of optimized LFLM is improved as compared with the initial model. Based on this method the weight of optimized LFLM is reduced by 10%. Therefore, when this proposed approach is applied, it can efficiently raise the precision of optimization and reduce the number of iterations of experiments in the optimization design by RSM. References [1] H. Lee, S. S. Jeong, C. W. Lee and H. K. Lee, “Linear Compressor for Air-Conditioner,” International Compressor Engineering Conference at Purdue, pp. 1-7, 2004. [2] T. Mizuno, M. Kawai, F. Tsuchiya, M. Kosugi, and H. Yamada, “An Examination for Increasing the Motor Constant of a Cylindrical Moving Magnet-Type Linear Actuator,” IEEE Trans. Magn., Vol. 41, No. 10, pp. 3976-3978, October, 2005. [3] P. Zheng, A. Chen, P. Thelin, W. M. Arshad, and C. Sadaranani, “Research on a Tubular Longitudinal Flux PM Linear Generator Used for Free-Piston Energy Converter,” IEEE Trans. Magn., Vol. 43, No. 1, pp. 447-449, January, 2007.

192

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-192

Analytical Evaluation of Flux-Linkages and Electromotive Forces in Synchronous Machines Considering Slotting, Saliency and Saturation Effects Antonino DI GERLANDO, Gianmaria FOGLIA and Roberto PERINI Dipartimento di Elettrotecnica – Politecnico di Milano, Piazza Leonardo da Vinci, 32 – 20133 Milano, Italy [antonino.digerlando, gianmaria.foglia, roberto.perini]@etec.polimi.it Abstract. An analytical approach to the evaluation of flux linkages and e.m.f.s in salient-pole synchronous machines is developed, capable to accurately allow for the actual winding structure, the stator and rotor air-gap geometry (slotting and saliency), under any saturated, steady-state or transient operating condition. The method is based on a Park d-q decomposition of the stator m.m.f. distribution (preserving the spatial harmonics) and on the use of FEM identified saturation functions. A relevant feature is that the self and mutual inductances, evaluated in unsaturated conditions, are simply corrected by using the saturation functions. Several transient FEM simulations validate the method.

Introduction An analytical procedure for the evaluation of the air-gap field, of the inductances, of the e.m.f.s and of the electromagnetic torque of salient-pole, three-phase synchronous machines was previously developed, considering anisotropy, slotting and actual winding structure [1–3]: the procedure showed good accuracy features, but it was affected by the significant limitation of the operation in unsaturated conditions. Starting from that theory, the method is here extended to any saturated operating condition: this generalisation is based on a d-q approach to the calculation of the airgap field distribution, made possible by the use of FEM identified saturation functions [4] and by a d-q decomposition of the stator m.m.f. distribution [5]. In this paper, the expression of the coil and of the phase flux linkage and e.m.f. is obtained, while the electromagnetic torque evaluation is performed in another paper [6]. Based on the developed method, various simulations have been carried out, in different operating conditions, comparing the results with those obtained by corresponding transient FEM calculations [7]: the accuracy level is investigated and discussed, for the validation of the described analytical method. In the model, for now the presence of the rotor damper cage is not considered.

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces

Stator phase global currents: i1s(t), i2s(t), i3s(t)

Park transformation on the rotating frame

Park current components:

Distinct antitransformation of the Park components

iPd(t) iPq(t)

193

Stator phase d, q current components: i1sd(t), i2sd(t), i3sd(t) i1sq(t), i2sq(t), i3sq(t)

Figure 1. Flow chart illustrating the d-q decomposition of the stator phase currents.

The d-q Decomposition of m.m.f. and Flux-Density Air-Gap Distributions As more completely described and illustrated in [5], by using the Park transformation, each stator phase current can be expressed in terms of phase d, q instantaneous components: i k ( t ) = i kd ( t ) + i kq ( t )

k = 1s, 2s, 3s.

(1)

The meaning of (1) is resumed by the flow-chart of Fig. 1. Thus, being ξ the generic angular position along the stator, the instantaneous m.m.f. distribution ms(ξ, i1s(t), i2s(t), i3s(t)) produced by the stator three-phase winding actual structure [1] can be decomposed as follows:

(

)

ms ( ξ,i1s ( t ) ,i 2s ( t ) ,i3s ( t ) ) = ms ( ξ,i1sd ,i 2sd ,i3sd ) + ms ξ,i1sq ,i2sq ,i3sq = = msd ( ξ, t ) + msq ( ξ, t ) .

(2)

As can be verified, the msd(t) and msq(t) distributions, produced by the d-axis and q-axis instantaneous current terns, act along the d, q axis respectively, showing the same stepped waveform features of the total m.m.f. [5]. Therefore, the model described by (1) and (2) represents a more general transformation than the classical Park one, usually applied to decompose sinusoidally distributed m.m.f.s only. As shown in [5], the instantaneous saturation functions are expressed by σd ( t ) = σd ( i r ( t ) ,id ( t ) ) ,

(

)

σq ( t ) = σq i u ( t ) ,iq ( t ) ,

(3)

where: id and iq are 1/ 3 times the current Park vector components iPd, iPq; ir is the rotor current; iu is an equivalent d-axis current corresponding to the resultant d-axis m.m.f. mu: i u ( t ) = m u N f = i r ( t ) + α Pd ⋅ id ( t ) ,

(4)

αPd is the classical d-axis Potier coefficient, Nf is the field winding turn number per pole. Called ζ = ζ(t) the rotor angular mechanical position, mr(ξ − ζ(t), ir(t)) the rotor m.m.f. distribution, βs and βr the stator slotting and rotor saliency functions respec-

194

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces

tively [1], the air-gap flux density distribution (radial component, measured along the circumference at half the minimum air-gap width [5]) can be expressed as follows in saturated operation: b ( ξ, ζ ( t ) , t ) = ( μo g ) ⋅βs ( ξ ) ⋅βr ( ξ − ζ ( t ) ) ⋅

(

(

)

)

⋅ σd ( t ) ⋅ msd ( ξ, t ) + m r ( ξ − ζ ( t ) ,i r ( t ) ) + σq ( t ) ⋅ msq ( ξ, t ) = = σd ( t ) ⋅ bd.ns ( ξ, ζ ( t ) , t ) + σq ( t ) ⋅ bq.ns ( ξ, ζ ( t ) , t ) = = bd ( ξ, ζ ( t ) , t ) + bq ( ξ, ζ ( t ) , t ) .

(5) where bd.ns(ξ, ζ(t), t) and bq.ns(ξ, ζ(t), t) are the non-saturated d and q flux density field distributions. As can be observed in the first formulation of (5), the saturation functions are originally applied to the axis m.m.f. distributions: in fact, they can be interpreted as the factors that, along each axis, express the ratio between the distribution of the magnetic potential difference (m.p.d.) at the air-gap and the distribution of the m.m.f. The second line formulation of (5) shows that the d and q axis flux density distributions can be evaluated as the unsaturated ones, times the corresponding saturation functions.

The Development of the Flux-Linkage Expression in Saturated Conditions The flux linkage ψk of the phase k includes the main flux linkage (ψmk) and the leakage flux linkage (ψℓk): ψ k ( t ) = ψ mk ( t ) + ψ k ( t ) .

(6)

As regards the leakage flux linkage ψℓk, its rigorous analytical evaluation is difficult to be performed, as like as its FEM evaluation [3]. However, for the leakage model, here the well known assumptions of the classical theory will be adopted: − −

the leakage flux paths exhibit wide portions developing in air: thus, leakage can be considered as a substantially unsaturated phenomenon; the leakage flux linkage is assumed to be independent on the rotor position.

Considering the typical structure of the stator winding, in which the shorted coil pitch is usually employed, in general the following constructional properties occur: some slots contain conductors of different phases; moreover, some end-windings of different phases are close each others. As a consequence, the leakage flux linkage exhibits self and mutual terms, all due to the stator currents only; however, considering the three-phase structural symmetry, the phase self leakage inductances are all equal (Lℓself), as like as the phase mutual leakage inductances (Lℓmutual). Therefore, the leakage flux linkage of the phase 1s can be written as follows:

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces

ψ 1s ( t ) = Lself ⋅ i1s ( t ) + Lmutual ⋅ i 2s ( t ) + Lmutual ⋅ i3s ( t ) = = ( Lself − 2 ⋅ Lmutual ) ⋅ i1s ( t ) = Ls ⋅ i1s ( t ) ,

195

(7)

where the “service” equivalent leakage inductance Lℓs has been introduced, thanks to the link: i1s ( t ) + i 2s ( t ) + i3s ( t ) = 0 ,

(8)

consequence of the hypothesis of the stator winding insulated neutral point. Of course, the leakage flux linkages of the other two phases have expressions similar to (7): ψℓk = Lℓs⋅ik, (k = 1s, 2s, 3s). As regards the main flux linkage ψmk, it can be expressed as a function of the teeth fluxes. From the expression (5) of the flux density distribution, the tooth flux in the j-th stator tooth [2] can be expressed as follows: ξfj b ( ξ, ζ, t ) ⋅ dξ = ξij

ϕ t j ( ζ, t ) =  ⋅ R ⋅ ∫

⎛ ⎞ ξ ξ =  ⋅ R ⋅ ⎜ σd ( t ) ⋅ ∫ fj bd.ns ( ξ, ζ, t ) ⋅ dξ + σq ( t ) ⋅ ∫ fj bq.ns ( ξ, ζ, t ) ⋅ dξ ⎟ = ξ ξ ij ij ⎝ ⎠ = σd ( t ) ⋅ ϕtd.ns j ( ζ, t ) + σq ( t ) ⋅ ϕtq.ns j ( ζ, t ) ,

(9) where ξij and ξfj are the j-th tooth initial and final angular positions (along the axes of the slots adjacent to the j-th tooth itself), ℓ the lamination stack length, R the average radius at the minimum air-gap (along the d axis). Equation (9) shows that also the tooth flux can be expressed as the sum of the unsaturated d and q axis components, ϕtd.nsj(ζ(t), t) and ϕtq.nsj(ζ(t), t), each multiplied by the corresponding saturation function. The main flux linkage ψmk of the phase k can be expressed as a function of the tooth fluxes: ψ mk ( t ) = Σ jΓ jk ⋅ ϕt j ( t ) ;

(10)

in (10) Γjk is an integer number, called linkage coefficient [2]: the value of Γjk expresses how many times the j-th tooth flux is linked with the k-th phase; the sign of Γjk depends on the winding direction of the phase winding coils, compared with the positive radial direction adopted for the generic tooth flux. Thanks to (9), the main phase flux linkage can be considered as the sum of the d and q components; moreover, each axis main flux linkage term can be expressed as the corresponding unsaturated main flux linkage component, multiplied by the suited saturation function; therefore, from (10) it follows: ψ mk ( t ) = ψ mdk ( t ) + ψ mqk ( t ) = σd ( t ) ⋅ ψ md.ns.k ( t ) + σq ( t ) ⋅ ψ mq.ns.k ( t ) . (11)

196

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces

On the other hand, each non-saturated main flux linkage component (ψmd.ns.k(t) and ψmq.ns.k(t)) can be expressed as proportional to the corresponding current components (i1sd, i2sd, i3sd, ir and i1sq, i2sq, i3sq respectively), multiplied by the non saturated self and mutual inductance functions [2]: ψ md.ns.k ( t ) = L kr ( ζ ) ⋅ i r ( t ) + ∑ Lkv ( ζ ) ⋅ i vd ( t ); v

ψ mq.ns.k ( t ) = ∑ Lkv ( ζ ) ⋅ i vq ( t ); v, k = 1s, 2s, 3s.

(12)

v

The expressions of ψmd.ns.k(t) and ψmq.ns.k(t) of (12) can be considered as the limit of the main flux components ψmdk(t) and ψmqk(t) given by (11), when the saturation is negligible (i.e., when σd, σq → 1). Considering (7) and (12), (6) becomes (again with: v, k = 1s, 2s, 3s): ⎛ ⎞ ψ k ( t ) = σd ( t ) ⋅ ⎜ Lkr ( ζ ) ⋅ i r ( t ) + ∑ Lkv ( ζ ) ⋅ i vd ( t ) ⎟ + ⎜ ⎟ v ⎝ ⎠ ⎛ ⎞ + σq ( t ) ⋅ ⎜ ∑ L kv ( ζ ) ⋅ i vq ( t ) ⎟ + Ls ⋅ i k ( t ) . ⎜ ⎟ ⎝ v ⎠

(13)

The structure of (13) confirms the previous remarks concerning the possibility to express a saturated quantity as the product between the unsaturated one, times a saturation function; however, in this case, this property is particularly important, because it allows to evaluate the non saturated self and mutual inductance functions once and for all, as a function of the air-gap geometry and of the rotor position only, regardless of the current amplitude. Considering the heaviness of the inductances and inductance derivatives calculation process [2], the cited property is crucial for an acceptable application of the machine analytical model in saturated operating conditions. In order to illustrate the accuracy of the described flux linkage evaluation method, Fig. 2 shows the phase flux-linkage waveforms, in ideally unsaturated and saturated conditions, in steady-state, loaded, balanced operation, with d and q reaction components, for the 8-pole, fractional slot winding machine considered in [4] (waveforms evaluated analytically by (13) and by FEM transient simulations [7]). As can be seen, the calculation accuracy in saturated conditions is similar to that obtained in unsaturated ones, thus confirming the soundness of the saturation model: of course, the filtering effect of the winding distribution makes the waveforms almost sinusoidal. If the effect of saturation on more distorted quantities is to be examined, it is useful to analyse the coil flux linkage, shown in Fig. 3: in it, the waveforms correspond to those of Fig. 2, for the same machine and in the same loaded condition: as can be observed, the agreement between analytical and FEM results remains acceptable. The steady-state operating conditions of Figs 2 and 3 correspond to constant values σd and σq in (13). In order to evaluate the soundness of the analytical method in all the d and q axis saturation conditions, the most suited way is to consider an holding torque test; in this

197

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces 6

[Wb]

0.20

ψph.ns

4

ψc

0.50

0

0 −0.50

−2

−0.10

−4 −6

ψc.ns

0.10

ψph

2

[Wb]

0.15

−0.15 t [ms] 0

2

4

6

8

10

12

14

16

−0.20

18 20

Figure 2. Phase flux linkage at load (machine data: [4]): ψph.ns = unsaturated operation; ψph = saturated operation; solid lines = analytical; dotted lines = FEM.

0.1

4

6

ψc.ht

8

[ms]

10

12

14

16

18

20

ψph.ht.ns

[Wb]

4

ψph.ht

2

0

0

−0.1

−2

−0.2

−4

−0.3

−6 −8

−0.4

−10

−0.5 −0.6 0

2

Figure 3. Coil flux linkage waveforms; same machine and loaded operating conditions as in Fig. 2; solid lines = analytical results; dotted lines = FEM results.

6

ψc.ht.ns

[Wb]

t 0

−12

t [ms] 5

10

15

20

Figure 4. Coil flux linkage, in ideally unsaturated and in saturated operation, during rotation with constant ir, is1, is2, is3 values (holding torque test operation): solid line = analytical results; dotted line = FEM results.

−14

t [ms] 0

5

10

15

20

Figure 5. Phase flux linkage, in ideally unsaturated and in saturated operation, in the same conditions of Fig. 4 (holding torque test operation): solid line = analytical results; dotted line = FEM results.

condition, all the rotor and stator currents are maintained constant (I r, I1s, I2s, I3s) and the rotor is supposed to be driven at constant rotational speed Ω. So, the stator current Park vector has a constant amplitude, but its d, q components Id and Iq change with the rotor positions, in such a way to produce all the possible reaction situations (magnetizing (Id > 0), demagnetizing (Id < 0), generating (Iq < 0) and motoring (Iq > 0) conditions). Such operating condition should be considered as a virtual test, because difficult to be actually performed in case of large rating machines. The simulated results of this test are shown in Figs 4, 5, for the coil and phase winding flux linkage respectively: as can be observed, the correctness of (13) is confirmed. It should be noted the important flux linkage reduction due to saturation, that is unsymmetrical, depending on the magnetizing or demagnetizing reaction.

198

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces

60

1.5

ec0.ns

[V] 40

[kV] 1.0

ec0

20

0.5

0

0

−20

−0.5

−40

−1.0

−60

t 0

5

10

eph0.ns

[ms] 15

20

Figure 6. Coil e.m.f. in no-load operation: ec0.ns = non saturated core; ec0 = saturated core; solid lines = analytical results; dotted lines = FEM results.

−1.5

eph0

t 0

5

[ms]

10

15

20

Figure 7. Phase e.m.f. in no-load operation: eph0.ns = non saturated core; eph0 = saturated core; solid lines = analytical results; dotted lines = FEM results.

The Development of the e.m.f. Expression in Saturated Conditions By performing the time derivative of (13), and posing Ω = dζ/dt, it follows:

(

)

e k = ekΩ + ekt + eks = Ω ⋅ σd ⋅ dLkr dζ ⋅ i r + ∑ v dLkv dζ ⋅ i vd +

(

)

+ Ω ⋅ σq ⋅ ∑ v dL kv dζ ⋅ i vq + σd ⋅ Lkr ⋅ di r dt + ∑ v Lkv ⋅ di vd dt +

+ σq ⋅ ∑ v L kv ⋅ di vq dt + Ls ⋅ di k dt + dσd dt ⋅ ∑ v ( Lkv ⋅ i vd + L kr ⋅ i r ) + + dσq dt ⋅ ∑ v L kv ⋅ i vq

v, k = 1s, 2s,3s.

(14) As can be noted, in addition to the two classical components (the speed e.m.f. ekΩ(t) and the transformer e.m.f. ekt(t)), a third term arises (that can be called “saturation” e.m.f.), eks(t), proportional to dσd/dt and to dσq/dt. It is interesting to analyze some particular operating conditions, again for coil and phase winding waveforms. Figures 6 and 7 refer to the steady-state, no-load, operation, with constant values of the speed Ω and of the rotor current Ir; in this condition, (14) reduces to the following no-load speed term: e k o = e kΩo = Ω ⋅ σd ( Ir , 0 ) ⋅ dL kr dζ ⋅ I r .

The following remarks can be made:

(15)

199

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces

− −

as expected, the phase winding waveform is less distorted than the coil one, thanks to the filtering effect of the adopted winding structure (fractional slot, with shorted pitch coils); the waveform analytical reproduction is accurate, both in unsaturated and in saturated operation.

In steady state, loaded operation, the field current Ir is constant, while the stator currents i1s, i2s, i3s are balanced, sinusoidally time dependent quantities; considering that the d, q Park current components have constant values (Id, Iq), σd and σq can be evaluated once and for all; from (14), we obtain (again with v, k = 1s, 2s, 3s):

(

)

e k load = Ω ⋅ σd ( I r , Id ) ⋅ dLkr dζ ⋅ Ir + ∑ v dLkv dζ ⋅ i vd +

(

)

+ Ω ⋅ σq Iu , Iq ⋅ ∑ v dL kv dζ ⋅ i vq + σd ( Ir , Id ) ⋅ ∑ v ( Lkv ⋅ di vd dt ) +

(

)

+ σq Iu , Iq ⋅ ∑ v L kv ⋅ di vq dt + Ls ⋅ di k dt .

(16) Figures 8 and 9 show the coil and phase winding e.m.f.s respectively, in steady state, loaded, balanced operation, both in ideally unsaturated conditions and in saturated ones; the following remarks can be proposed: − − − − −

the saturation reduces significantly the waveforms amplitude, roughly maintaining their shape; a good agreement can be observed between analytically evaluated waveforms and FEM calculated ones, both in ideally unsaturated conditions and in saturated operation; the coil e.m.f. waveforms are highly distorted, because of the loaded operation; the phase winding e.m.f. waveforms are less distorted, thanks to the filtering effect due to the winding factor; however, the armature reaction affects also the distortion level of the phase e.m.f. waveform, mainly because of the third harmonic components; in fact, the line to line e.m.f. appears more sinusoidal.

The last operating condition here considered is the holding torque test, already examined in Figs 4 and 5 as concerns the coil and phase flux linkages; here, the coil and phase e.m.f.s are analyzed, both in ideally unsaturated operation and in saturated conditions. Considering that, in this case, the time derivatives of the currents are zero, (14) reduces to:

(

)

e k ht = ekΩht + eksht = Ω ⋅ σd ( Ir , Id ( t ) ) ⋅ dL kr dζ ⋅ Ir + ∑ v dL kv dζ ⋅ I vd +

(

)

+ Ω ⋅ σq I u ( t ) , Iq ( t ) ⋅ ∑ v dL kv dζ ⋅ i vq +

+ dσd dt ⋅ ∑ v ( L kv ⋅ I vd + L kr ⋅ I r ) + dσq dt ⋅ ∑ v L kv ⋅ I vq ; k, v = 1s, 2s,3s (17)

200

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces

80

2.0

ec.ns

60 40

1.0

ec

0.5

20

[V]

0

0

−20

−0.5

−40

−1.0

−60

−1.5

−80

t 0

5

10

[ms]

15

20

−2.0

eph [kV]

t 0

5

10

3

[V]

ec.ht.ns

20

[kV]

2

50

eph.ht.ns

1 0

[ms]

15

Figure 9. Phase e.m.f. in loaded operation: eph.ns = non saturated core; eph = saturated core; solid lines = analytical results; dotted lines = FEM results.

Figure 8. Coil e.m.f. in loaded operation: ec.ns = non saturated core; ec = saturated core; solid lines = analytical results; dotted lines = FEM results.

100

eph.ns

1.5

eph.ht

0

ec.ht

−1

−50

−2 −100

−3 −150

t [ms] 0

5

10

15

20

Figure 10. Coil e.m.f., in ideally unsaturated and in saturated operation, during rotation with constant ir, is1, is2, is3 values (holding torque test operation): solid line = analytical results; dotted line = FEM results.

−4

0

5

10

t [ms] 15

20

Figure 11. Phase e.m.f., in ideally unsaturated and in saturated operation, in the same conditions of Fig. 9 (holding torque test operation): solid line = analytical results; dotted line = FEM results.

Figures 10 and 11 show the coil and phase winding e.m.f.s corresponding to the holding torque test, evaluated in ideally unsaturated conditions (i.e., when σd, σq → 1) and taking into account the saturation, on the basis of (17); the following remarks are valid: − −

the analytically evaluated waveforms are fairly similar to those evaluated by FEM transient simulations; the saturation has an important influence on the waveforms amplitude and shape;

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces

−

201

the validity of (17) is substantially confirmed, even if some local discrepancies can be observed between analytical and FEM calculated waveforms, in the saturated operation.

Conclusion An analytical approach to the evaluation of the phase flux linkage and e.m.f. waveforms of salient-pole synchronous machines has been developed, able to take into account the actual structure of the stator winding, the slotting and anisotropy features under saturated operating conditions. The saturation has been modelled by using suited saturation functions, that are p.u. quantities, dependent on the amplitudes of the d-q m.m.f. main sinusoidal components: this saturation model allows to evaluate the actually saturated operating quantities as the corresponding unsaturated ones multiplied by the saturation functions. The most important consequence of this approach is that the self and mutual inductances, and the corresponding derivatives with respect to the rotor position, can be evaluated once and for all in unsaturated conditions, subsequently including the saturation effects corresponding to the actual conditions. Several operating situations have been analysed, comparing the waveforms calculated by the developed analytical approach with those obtained by means of corresponding transient FEM simulations: in general, the agreement appears satisfactory, showing the soundness of the developed method. References [1] A. Di Gerlando, G. Foglia, R. Perini: “Analytical Calculation of the Air-Gap Magnetic Field in SalientPole Three-Phase Synchronous Machines with Open Slots”, ISEF 2005 – XII Int. Symp. on Electromag. Fields in Mechatronics, Electrical and Electronic Eng., Baiona, Spain, Sept. 15-17, 2005, Proc. on CD, ISBN N° 84-609-7057-4, paper EE-3.14. [2] A. Di Gerlando, G. Foglia, R. Perini: “Calculation of Self and Mutual Inductances in Salient-Pole, ThreePhase Synchronous Machines with Open Slots”, ibidem, paper EE-3.15. [3] A. Di Gerlando, G. Foglia, R. Perini: “E.M.F. and Torque Analytical Calculation in Salient-Pole, ThreePhase Synchronous Machines with Open Slots”, ibidem, paper EE-3.16. [4] A. Di Gerlando, G. Foglia, R. Perini: “FEM identification of d-q Saturation Functions of Salient-Pole Synchronous Machines”, ISEF 2007 – XIII International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering, Prague, Czech Republic, September 13-15, 2007. [5] A. Di Gerlando, G. Foglia, R. Perini: “Analytical Model of the Air-Gap Magnetic Field in Synchronous Machines considering slotting, saliency and saturation effects”, ISEF 2007. [6] A. Di Gerlando, G. Foglia, R. Perini: “Analytical Calculation of the Electromagnetic Torque in Synchronous Machines considering slotting, saliency and saturation effects”, ISEF 2007. [7] Maxwell 2D FEM code, Version 10, Ansoft Corporation, Pittsburgh, PA, USA.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-202

Radiation in Modeling of Induction Heating Systems Jerzy BARGLIK a, Michał CZERWIŃSKI b, Mieczysław HERING c and Marcin WESOŁOWSKI c a Silesian University of Technology, Krasińskiego 8, 40-019 Katowice, Poland E-mail: [email protected] b The Industrial Institute of Electronics, Długa 44/50, 00-241 Warsaw, Poland E-mail: [email protected] c Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland E-mail: [email protected], [email protected] Abstract. The paper presents analysis of third type boundary conditions applied for numerical simulation in more frequently used induction heating systems. A special emphasis was put on analysis on radiation heat transfer with taking into consideration multiple reflections phenomenon. Obtained results confirmed necessity of usage of multiple reflections model for analysis of high temperature induction heating systems.

Introduction Induction heating seems to be modern, environment-friendly industrial technology quite well explored from theoretical and practical point of view. Usage of the induction heating technologies leads to significant energy savings, distinct shortening the time of heating and consequently to the growth of a total efficiency. A powerful tool for designing and optimization is a suitable computer model based on a mathematical modeling of the task. The model is typically based on a system of non-linear second order partial differential equations for coupled electromagnetic and temperature fields. Development of the mathematical modeling of induction heating processes as well as the computations with required accuracy by means of professional software and sometimes also by some user codes has reached a high level. However there are some exceptions. One of them seem to be modeling of high temperature induction heating systems, for instance surface induction hardening of steel bodies. During such processes a radiation heat transfer plays an important role. The phenomenon is connected not only with high temperatures of the workpiece, but also with a typical arrangement of the system characterized by heated to high temperature charge and many surrounding elements with distinctly lower and often various temperatures. Majority of papers on induction heating present classical approach: taking into account radiation in a simplified way only without considering multiple reflections phenomenon. There are two reasons of the such the approach. One of them seems to be a false idea that multiple reflections do not influenced strongly on accuracy of calculations. The more important is the second reason: lack of professional software having precise, well done algorithms for such calculations. There are of course some packages having well prepared procedures for model-

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ing of radiation heat transfer with multiple reflections for instance TAS (Thermal Analysis System) but in that case software is not well prepared for electromagnetic calculations. The paper presents analysis of coupled electromagnetic and temperature in typical induction heating system with taking into consideration radiation heat transfer with multiple reflections.

Model of Radiation Heat Transfer in a System with Many Elements Let us consider an induction heating system with N surfaces of different temperatures T. For such a system the energy balance for each surface is given by Eq. (1), describing energy losses from the inductively heated charge to external surfaces [1]: N

⎛ δ k ,i 1− εi − ϕ k ,i ⋅ ε εi i =1 ⎝ i

∑⎜

⎞ Pi N 4 ⎟ ⋅ =∑ (δ k ,i − ϕk ,i )σ0Ti S ⎠ i i =1

(1)

where δ k ,i denote Kronecker delta defined as by (2): ⎧1 for k = 1 ⎫ δ k ,i = ⎨ ⎬ ⎩0 for k ≠ 1⎭

(2)

and ε i – effective emissivity of surface i, ϕ k ,i – view factor between two surfaces k and i, Pi – energy losses of surface i, σ 0 = 5.67·10–8 W/(m2·K4) – Stefan–Boltzmann constant, Si – area of surface i. Usually Si are elementary surfaces obtained which are lines or faces of a single elements of mesh. Main difficulty to solve the system of equations (1) is to calculate view factors ϕ k ,i , defined as the fraction of total radiant energy that leaves surface k which arrives directly on surface i. Another complication is that a radiating surfaces are a gray diffuse body, so total emissivity are less than one. Radiant energy that leaves surface k which arrives directly on surface i is not total absorbed on it, but according to Lambert law, is partly reflect and diffuse [1]. This effect of energy balance between all radiating surfaces in one enclosure called a reflection effect.

Formulation of Technical Problem and Illustrative Example Let us consider an axi-symmetric model of induction hardening system of cylindrical steel workpiece by three-turns cylindrical inductor. Main parameters of the induction heating system are as follows: Workpiece: diameter d = 100 mm; length l = 200 mm; thermal conductivity λ = 18 W/(m·K); density ρ = 8000 kg/m3; specific heat c = 500 J/(kg·K); effective emissivity ε = 0.8; conductivity γ = 7·106 S/m; relative magnetic permeability μr = 1;

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Figure 1. Dependence of temperature of workpiece on time of heating with and without taking into consideration multiple reflections.

Inductor: number of turns n = 3; inner diameter Di = 130 mm; outer diameter De = 154 mm; width of one turn w = 16 mm; gap between turns Δw = 8 mm; effective emissivity of internal surface ε = 0.3 ÷ 0.8; Supply source: current density within the inductor J = 3·107 A/m2; frequency f = 1000 Hz. In order to determine influence of radiation heat transfer on accuracy of temperature calculations, in the first stage of computations a phenomenon of convection was neglected. Third type boundary condition having only radiation component was applied. We modeled the radiation heat transfer in induction heating process for several values of inductor temperature, its emissivity, distance between inductor and workpiece and shape of conductor. These elements have strongly influence on radiation heat transfer and on temperature profile in the workpiece. In order to simplify calculations steady-state analysis of temperature field was used. However before start the simulation it was necessary to solve also a transient problem. The obtained results (Fig. 1) show that maximal temperature differences are noticed in steady state. So in order to simplify the calculations in the further part of the paper only steady-state analysis will be provided. So the analysis could be done for that state. For calculations of weakly coupled electromagnetic and temperature fields Quick Field 2D (QF) and TAS packages were use. Electromagnetic calculations were made by means of QF (number of nodes is equal to 10428). Based upon results of specific power density released in the workpiece taken from QF steady-state temperature field

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was calculated by means of TAS 3D. with 18373 nodes, 17800 elements, 17680 volume heat sources, 1720 radiation surfaces. Total power released in the workpiece is equal to 6702 W. Some results showing temperature distribution within the surface of the workpiece (Fig. 2) and in its longitudinal (Fig. 3) and transversal (Fig. 4) crosssections were presented below. O

C

Figure 2. Steady-state temperature field in workpiece heated by cylindrical inductor. O

Figure 3. Steady-state temperature field in a longitudinal cross-section of the workpiece.

C

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J. Barglik et al. / Radiation in Modeling of Induction Heating Systems O

C

Figure 4. Steady-state temperature field in a longitidinal cross-section of the workpiece.

Temperature distribution in steady-state along length of the workpiece surface is presented in Fig. 5. Two cases were solved: with and without taking into consideration multiple reflection effect. The calculations confirmed that if multiple reflection phenomenon was neglected temperature of the workpiece is much lower (about 100 oC). The dependence between maximal temperature and total emissivity of the workpiece was shown in Fig. 6. For the case without multiple reflections (not shown in Fig. 6) maximal temperature is equal to 1197 oC. The calculations, that apply to inductor’s temperature influence on maximum temperature of charge proves, that he’s faint and can be entirely omitted. This is true in case of heating systems characteristic for hardening processes. In systems used in plastic forming of metals processes situation could be different. Assuming, that the workpiece transfer heat by natural convection, according to generally accepted formulas [2], the value of convection heat transfer coefficient changes for considered system and temperature range from α = 4.5 to 8 W/(m2·K), with practically constant value for temperatures bigger than 530 oC. Results of calculations are shown in Fig. 7. It follows from them, that participation of convection in heat transfer process is in comparison to radiation is rather small. It proves necessity to build model for temperature calculations taking into account radiation heat transfer with multiple reflections effect.

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ε = 0.3

ε = 0.8

no reflections

1400 1300

Temperature, 0C

1200 1100 1000 900 800 700 600 0

25

50

75

100

125

150

175

200

Distance, mm Figure 5. Steady state temperature distribution on the workpiece surface of charge for two different values of emissivity and without multiple reflection phenomenon.

1300 1290

Temperature,0C

1280 1270 1260 1250 1240 1230 1220 1210 0.2

0.3

0.4

0.5

0.6

0.7

0.8

Emissivity, Figure 6. Maximum temperature of workpiece in function of total emissivity of inductor.

0.9

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with convection

no convection

1400 1300

Temperature, 0C

1200 1100 1000 900 800 700 600 0

20

40

60

80

100

120

140

160

180

200

Distance, mm Figure 7. Steady state temperature on the surface of charge for inductor emissivity 0.3 with and without convection losses.

Modeling in ANSYS Environment Further simulations were made in ANSYS environment in 2D coordinate system. They include analysis of dependence of charges temperature in function of distance from inductor and its geometry. ANSYS is one of the rare packages, that allow simulation of radiation heat transfer with multiple reflection effect in both 2D and 3D coordinate systems. Such operation is impossible for TAS, that realizes this operation in 3D only. In purpose of objectivity in all cases the same volume power distribution in charge was maintained, as in earlier calculations. In purpose of comparison of simulation results obtained in different environments in Fig. 8, was shown temperature field received from ANSYS, with the same conditions and body loads, as presented previously (obtained with usage of QF+TAS packages). Figures 9 and 10 present temperature distribution on the surface of the workpiece for three different distances between inductor and the workpiece, with constant emissivity of inductor and workpiece and with the same power distribution within the workpiece. It follows, that value of emissivity has big influence on reflected radiation effect. This is frequently in classical models. On mentioned reflection effect essential influence has distance between inductor and workpiece L. The influence of the inductor can be omitted only for big values of distance inductor-workpiece L, that is avoided due to system efficiency.

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NODAL SOLUTION APR 27 2007 14:23:54

STEP=1 SUB =4 TIME=1 TEMP (AVG) RSYS=0 SMN =50 SMX =1287

MX

MN

50

421.156

235.578

606.733

792.311

977.889

1163

1287

Figure 8. Temperature field with 15mm distance of inductor from charge and emissivity 0.3.

no reflections

L = 15 mm

L = 35 mm

L = 55 mm

1300

Temperature, 0C

1200 1100 1000 900 800 700 600 0

20

40

60

80 100 120 140 160 180 200 220 Distance, mm

Figure 9. Temperature distribution on the charge surface of with constant emissivity ε = 0.8 and different distances of inductor from charge (L).

Last series of simulations concerned on influence of inductor geometry on temperature field distribution in the workpiece. Results are presented in Fig. 11. Change of the geometry in analyzed case was reduced to replacement of rectangular cross section of the conductor into circular one. It was assumed, that the surface of conductor cross section does not change and their geometrical centers are located in same place. The biggest influence of inductor geometry was observed with its small values of emissivity. In this case differences reached level of about 25 oC. For bigger values of emissivity this difference significantly decreased.

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no reflections

L = 15 mm

L = 35 mm

L = 55 mm

1300

Temperature, 0C

1200 1100 1000 900 800 700 600 0

20

40

60

80 100 120 140 160 180 200 220 Distance, mm

Figure 10. Temperature distribution on the surface of charge with constant emissivity ε = 0.3 and different distance between inductor and charge (L).

circular turn ε = 0,8

circular turn ε = 0,3

rectangural turn ε = 0,8

rectangural turn ε = 0,3

1300

Temperature, 0C

1200 1100 1000 900 800 700 600 0

0.02

0.04

0.06

0.08 0.1 0.12 Distance, m

0.14

0.16

0.18

0.2

Figure 11. Temperature on the surface of workpiece in function of emissivity and for two different shapes of conductor.

Conclusions Obtained results confirmed necessity of consideration of multiple reflection effects in majority of induction heating systems. Errors caused by neglecting of multiple reflec-

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tions phenomena may reach the order of 100 oC. For analyzed systems radiation takes a crucial part in heat transfer to surroundings. Convection is significantly smaller. Inductor is a kind of mirror, that reflect part of radiation and in result increasing temperature of charge (especially in close to surface area, that is most interesting in hardening processes).

Acknowledgement This work was financially supported by the Polish Ministry of Science and Higher Education (Grant Projects 9T08C 04678 and 503 /G/1041/0744/006).

References [1] Siegel R., Howell J.: Thermal radiation heat transfer. New York. McGraw-Hill 1972. [2] Hering M.: Termokinetyka dla elektryków. Warszawa. WNT 1980.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-212

Time-Domain Analysis of Self-Complementary and Interleaved Log-Periodic Antennas A.X. LALAS, N.V. KANTARTZIS and T.D. TSIBOUKIS Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, GR-54124, Thessaloniki, Greece E-mail: [email protected] Abstract. A systematic near- and far-field analysis of dual polarisation logperiodic antennas is presented in this paper. The investigation of these demanding structures is performed by means of a 3D finite-difference time-domain (FDTD) methodology, properly tailored to tackle their geometrical details and abrupt discontinuities. In particular, the analysis delves into the design parameters of selfcomplementary, interleaved and trapezoidal-toothed structures and conducts a thorough examination of their radiation characteristics. Furthermore a new discretisation concept involving very flat cells which enhance algorithmic performance is introduced. Numerical verification addresses an extensive set of realistic applications with diverse parameter setups as well as instructive comparisons which indicate the merits of the proposed formulation.

Introduction Log-periodic antennas, introduced by DuHamel and Isbell [1], are typically used in wideband systems because their structure exhibits a practically independent behaviour with regard to operating frequency. As many applications require dual polarisation arrangements, potential choices comprise two different designs, i.e. the selfcomplementary (SC) and the interleaved (IL) configuration [2–5]. To this direction, one may also add an alternative design; the trapezoidal-toothed (TT) one, which shares similar attributes. Due to the inherent broad frequency spectrum, such antennas are essentially encountered in radar and measuring systems, attaining high-Q resonance properties. Amid the degrees of freedom of log-periodic antennas, one can discern their electrically thin ground-plane-backed dielectric substrate. So, prospective bandwidth advancement may be accomplished by increasing the substrate thickness or reducing its dielectric constant. However, the former issue is regularly accompanied by inductive impedance offsets and augmentation of the surface-wave effect. Thus, it becomes apparent that a meticulous procedure should be followed for the design of these particular radiators and the most significant: prior to any fabrication process to avoid expensive construction costs and totally misleading products. Towards the preceding deductions, the impact of design parameters on the radiation characteristics of log-periodic antennas has been comprehensively explored via several measurement attempts [6]. Moreover, various frequency-domain computational investigations have been presented, chiefly via commercial software packages [7]. Nevertheless, to our best knowledge, only a limited amount of time-domain attempts

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have been discussed. Therefore, it is the goal of this paper to introduce a FiniteDifference Time-Domain (FDTD) algorithm [8] for the efficient modelling and accurate simulation of 3D self-complementary, interleaved as well as trapezoidal-toothed antennas and derive reliable guidelines of their radiation characteristics. Special emphasis is drawn on the geometrical peculiarities of the devices along with their periodically-repeated attributes. Pursuing the improvement of the standard FDTD discretization rationale, a novel flexible method, incorporating very flat cells, is devised. In this manner, the domain is divided into more robust lattices devoid of erroneous mechanisms, oscillatory vector parasites or late-time instabilities. Additionally, open boundaries are truncated through diverse versions of the Perfectly Matched Layer (PML) absorber [9], which offer notable wave annihilation rates without any other non-physical conventions. To substantiate these qualities, the proposed approach is successfully applied to various real-world configurations, concerning all three structures, and numerical outcomes are carefully compared to extract possible similarities or prominent discrepancies. Structural Description of Log-Periodic Antennas The design parameters of a log-periodic antenna are α, β, Rmax, Rmin, the number of teeth N, geometrical ratio τ, and width ratio χ, as shown in Fig. 1. Specifically, τ and χ are given by τ=

Rn r = n −1 Rn +1 rn

and

χ=

rn Rn +1

(1)

and therefore, Rn+1 and rn are terms of geometrical progressions Rn +1 =

Rmin τ n +1

and

rn =

rmin τn

Figure 1. Geometry of a log-periodic planar element.

(2)

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A.X. Lalas et al. / Time-Domain Analysis of SC and IL Log-Periodic Antennas

(a)

(b)

(c)

Figure 2. Geometry of (a) a self-complementary, (b) an interleaved, and (c) a trapezoidal toothed structure.

During the design process, τ and χ can be acquired from τ =N

Rmin Rmax

and

χ= τ

(3)

given the rest of the parameters. Taking into account that the size of the antenna is fixed, we use (3) and then (2) to resolve the dimensions of metal surfaces. This is exactly the process followed throughout this paper. In particular, the self-complementary and interleaved structures are schematically depicted in Figs 2a and 2b, respectively. The antennas are realised through the use of two metallic sheets, properly shaped and mounted on a dielectric board, while the feed sections are constructed by integrated tapered-microstrip baluns. Two out of four branches are located on the upper side of the board and two are placed at its bottom. An alternative implementation of a dual polarisation device is the trapezoidal-toothed log-periodic radiator, whose geometry is illustrated in Fig. 2c. Accurate Analysis via the 3D FDTD Methodology The length and width of the antennas is set to 90 mm and their height is 6.67 mm. Rmax and Rmin are 40 mm and 5 mm, respectively, with these values kept fixed throughout the paper. The computational domain is divided into 144 × 144 × 44 cells with Δx = Δy = Δz = 0.8333 mm and Δt = 1.458 ps. Furthermore, termination of the unbounded space is attained by a 6-cell PML absorber. The two ports are excited by hard sources, appropriately phase-shifted to ensure circular polarisation. Our interest principally focuses on the variation of α, N and the relative dielectric permittivity εr of the dielectric board. From near and far-field data, acquired by our simulations, we can estimate the normalised (in dB) current distribution upon the surface of the structures and their radiation patterns. Some indicative results are illustrated in Figs 3a to 3c. It is emphasised that current values outside the outline of metal surfaces have no physical meaning, since they are merely a side-effect of the processing algorithm. As observed, maximum values of current distributions appear over the edges of the metal structures and hence the inner material is obsolete. This notification confirms previously-reported

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(a)

(b)

(c) ο

ο

Figure 3. Current distribution of (a) an SC antenna when N = 5, α = 80 , β = 10 , (b) an IL antenna when N = 4, α = 160ο, β = 10ο, and (c) a TT antenna when N = 5, α = 80ο, β = 10ο. 90

90

0 dB

120

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60

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30

120

0

150

60

30

120

60

90

90

90

2.0028 GHz

2.1949 GHz

2.2909 GHz

(a)

(b)

(c)

Figure 4. Radiation pattern at x-plane (blue line for Eθ; red line for: Eφ) of (a) an SC antenna when N = 4, α = 80ο, β = 10ο, (b) an IL antenna when N = 5, α = 160ο, β = 10ο, and (c) a TT antenna when N = 5, α = 80ο, β = 10ο. 90

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0 dB 60

120 −10

60 −10

φ [deg]

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0 dB

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−30

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300

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120

0 180

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300

0

210

330

240

300

270

270

270

2.0028 GHz

2.1949 GHz

2.2909 GHz

(a)

(b)

(c)

Figure 5. Radiation pattern at z-plane (blue line for Eθ; red line for: Eφ) of (a) an SC antenna when N = 4, α = 80ο, β = 10ο, (b) an IL antenna when N = 5, α = 160ο, β = 10ο, and (c) a TT antenna when N = 5, α = 80ο, β = 10ο.

measurements [6] and leads to the wire structures. In addition, radiation patterns for each antenna, over the x- and z-plane, are presented in Figs 4 and 5, respectively.

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A.X. Lalas et al. / Time-Domain Analysis of SC and IL Log-Periodic Antennas 90

90

0 dB 60

120

60

θ [deg]

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30

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0 dB

120

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θ [deg]

−10

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120

30

120

60

60

90

90

90

2.0028 GHz

2.0028 GHz

2.0028 GHz

(a)

(b)

(c) ο

ο

Figure 6. Investigation (x-plane) on the variation of (a) N when α = 80 , β = 10 (blue line for N = 3; red line for N = 4, green line for N = 5; black line for N = 6), (b) α when N = 4, β = 10ο, (blue line for α = 20ο, red line for α = 40ο, green line for α = 60ο; black line for α = 80ο), and (c) εr when N = 5, α = 80ο, β = 10ο (blue line for εr = 2.8; red line for εr = 3.38, green line for εr = 4.4).

90

90

0 dB 60

120 −10

60 −10

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90

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120

−30

30

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0

330

240

300

300

270

270

270

2.0028 GHz

2.0028 GHz

2.0028 GHz

(a)

(b)

(c) ο

ο

Figure 7. Investigation (z-plane) on the variation of (a) N when α = 80 , β = 10 (blue line for N = 3; red line for N = 4, green line for N = 5; black line for N = 6), (b) α when N = 4, β = 10ο (blue line for α = 20ο; red line for α = 40ο, green line for α = 60ο; black line for α = 80ο), and (c) εr when N = 5, α = 80ο, β = 10ο (blue line for εr = 2.8; red line for εr = 3.38, green line for εr = 4.4).

The impact of N, α variation and that of the relative dielectric permittivity εr on the radiation characteristics is next examined. Radiation patterns for an SC structure, over the x- and z-plane, are shown in Figs 6 and 7. More specifically in Figs 6a and 7a, an investigation on the behaviour of N is provided. Because of the fixed size of the antenna, any change of N is actually translated to a corresponding change of τ and therefore teeth dimensions vary when a supplementary one is inserted. As a result, teeth resonate in different frequencies for each case but the overall performance of the antenna is similar for small modifications. On the other hand, Figs 6b and 7b give an investigation on the variation of α. As α increases, a more bidirectional behaviour is revealed. This is because for smaller angles, dimensions of the teeth are smaller too and thus can not resonate. Subsequently in Figs 6c and 7c, an examination on the effect of relative dielectric permittivity εr is depicted. The attitude of the antenna is, now, more bidirectional for smaller εr. In this context, for a substrate with a high dielectric constant, the effective length of the teeth

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A.X. Lalas et al. / Time-Domain Analysis of SC and IL Log-Periodic Antennas 90

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30

120

0

90

(a)

(b)

30

120

60

90

0

150

30

120

60

180

60 90

(c)

Figure 8. Comparison of radiation patterns (x-plane) for different (a) frequencies, when N = 5, α = 80ο, β = 10ο (blue line for f = 2.675 GHz; red line for f = 2.867 GHz, green line for f = 3.6352 GHz), (b) structures (blue line for SC; red line for IL, green line for TT), and (c) distances of metallic surfaces at 2.0028 GHz, when N = 5, α = 80ο, β = 10ο (blue line for Δz = 0.8333 mm; red line for Δz = 0.2525 mm, green line for Δz = 0.1263 mm).

90

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120

0 180

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330

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300

0

210

330

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300

270

270

270

(a)

(b)

(c)

Figure 9. Comparison of radiation patterns (z-plane) for different (a) frequencies, when N = 5, α = 80ο, β = 10ο (blue line for f = 2.675 GHz; red line for f = 2.867 GHz, green line for f = 3.6352 GHz), (b) structures (blue line for SC; red line for IL, green line for TT), and (c) distances of metallic surfaces at 2.0028 GHz, when N = 5, α = 80ο, β = 10ο (blue line for Δz = 0.8333 mm; red line for Δz = 0.2525 mm, green line for Δz = 0.1263 mm).

is decreased. Thus, when increasing the value of εr, lower frequencies can not resonate [10,11]. Next, some additional comparisons are conducted. Radiation patterns of an SC structure, for different frequencies, are shown in Figs 8a and 9a. Bi-directional behaviour is observed in each case. Moreover, in Figs 8b and 9b a comparison of the structures presented earlier is depicted. Their behaviour is satisfactory in the sense of being bi-directional. The differences on the amplitude of the radiation patterns are explained taking into account the mismatching of the ports. In order to avoid them a more accurately modelling of the ports is needed. It is noteworthy to observe the performance of the antenna in the case of reducing the distance between its two metallic surfaces. To model these intricate cases, a new flat-cell discretization approach, namely Δx = Δy >> Δz, is developed. We examine two cases. In the first, the height of the antenna is 2.0202 mm and the FDTD domain is divided into 148 × 148 × 90 cells with Δx = Δy = 0.8333 mm, Δz = 0.2525 mm,

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and Δt = 0.703 ps. In the second case, the height of the antenna is 1.0101 mm. The FDTD domain is divided into 148 × 148 × 146 cells with Δx = Δy = 0.8333 mm, Δz = 0.1263 mm, and Δt = 0.374 ps. Again, termination of open boundaries is attained by a 6-cell PML along x- and y-axis and a larger PML setup towards z-axis. This is deemed necessary to ensure that the thickness of the absorber will be the same at every direction in the grid. As we can notice from Figs 8c and 9c when the distance between the two metallic surfaces decreases, the behaviour of the antenna at a specific frequency follows a more omni-directional pattern, basically due to loss of antenna resonance. Moreover, lower frequencies are suppressed at the entrance of the tapered-microstrip baluns owing to the smaller height of these waveguiding structures. Consequently, they can not excite the metallic parts of the antenna. Overall, the preceding investigations prove that bandwidth enhancement may be conducted either by augmenting the thickness of the substrate or by decreasing its dielectric constant.

Conclusion A consistent 3D FDTD technique for the rigorous analysis of log-periodic antennas has been presented in this paper. Three different structures with respect to their periodical features are investigated, whereas for improved precision a new flat-cell approach is developed. Numerical validation involves a variety of simulations and comparisons between different antenna types. Particularly, the influence of the teeth number and spanning angle on the antenna overall performance are extensively explored and combined with a study on the effect of the relative dielectric constant and the distance of the basic metallic surfaces. Results confirm the benefits of the proposed time-domain method and demonstrate the potential to be employed as a promising tool for logperiodic antenna characterization. Future aspects involve a more detailed modelling of the ports in order to obtain higher levels of accuracy.

Acknowledgement This work was supported by the National Scolarships Foundation of Greece (IKY).

References [1] R.H. DuHamel, and D.E. Isbell, Broadband logarithmically periodic structures, Record of 1957 IRE National Convention, Part 1, 99, pp. 119-128. [2] A.L. Van Hoozen, et al., Conformal log-periodic antenna assembly, US Patent 6,011,522, 4 Jan. 2000. [3] A.L. Van Hoozen, et al., Bidirectional broadband log-periodic antenna assembly, US Patent 6,018,323, 25 Jan. 2000. [4] D.A. Hofer, et al., Compact multipolarised broadband antenna, US Patent 5,212,494, 18 May 1993. [5] D. Campbell, Polarised planar log-periodic antenna, US Patent 6,211,839 B1, 3 April 2001. [6] R.H. DuHamel, and F.R. Ore, Logarithmically periodic antenna designs, Record of 1958 IRE National Convention, Part 1, Vol. 100, pp. 139-151. [7] K.M.P. Aghdam, R. Faraji-Dana, and J. Rashed-Mohassel, Compact dual-polarisation planar logperiodic antennas with integrated feed circuit, IET Microw. Antennas Propag., Vol. 152, pp. 359-366, 2005. [8] A. Taflove, and S. Hagness, (3rd ed.), Computational Electrodynamics: The Finite-Difference TimeDomain Method, Boston: Artech House, 2005.

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[9] J.P. Berenger, Three-dimensional perfectly matched layer for the absorption of electromagnetic wave, J. Comp. Phys., Vol. 127, pp. 363-379, 1996. [10] E. Avila-Navarro, J.M. Blanes, J.A. Carrasco, C. Reig, and E.A. Navarro, A new bi-faced log periodic printed antenna, Microw. Optical Technol. Lett., Vol. 48, pp. 402-405, 2006. [11] B.L. Ooi, K. Chew, and M.S. Leong, Log-periodic slot antenna array, Microw. Optical Technol. Lett., Vol. 25, pp. 24-27, 2000.

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New Spherical Resonant Actuator Y. HASEGAWA a, T. YAMAMOTO a, K. HIRATA a, Y. MITSUTAKE b and T. OTA b a Department of Adaptive Machine Systems, Osaka University, Yamadaoka, 2-1, Suita-city, Osaka 565-0871, Japan [email protected] b Advanced Technologies Development Laboratory, Matsushita Electric Works, Ltd., 1048, Kadoma, Osaka 571-8686, Japan [email protected] Abstract. This paper proposes the new spherical resonant actuator. The basic construction and the operating principle of the actuator are described. The torque characteristics of the actuator are computed by the 3-D FEM analysis. The geometry of the mover is investigated to improve the torque characteristics and the effectiness is clarified by both of the computation and the measurement of a prototype. Futhermore, the dynamic characteristics of the improved model are also confirmed by the measurement.

1. Introduction Recently, multi-dimensional actuators are a topic of great interest because of solution for vibration, noise, size constraints and limitations on operating speed [1]. Particularly, spherical actuators [2] are studied as the application to the joints and eyeballs for robots because they can be freely rotated in every axis direction. Due to the computer progress, the computer simulation becomes an effective tool to design electric devices and actuators. Authors have been studying the analyzed method employing the 3-D FEM to apply to multi-dimensional actuators with complicated magnetic structure [3]. In this paper, the new spherical actuator is proposed and the torque charactersistics are computed through the FEM analysis. The validity of the computation is verified by the comparison with the measurement of a prototype. The resonance characteristics around two rotation axes are confirmed through the measurement.

2. Basic Structure and Operating Principle Figure 1 shows the basic construction of the proposed spherical resonant actuator. It has the hybrid magentic structure [4] so that the magnetic flux by the current can not flow through the permanent magnet because permanent magnet has large magnetic resistance. The mover has four magnetic poles made of cross-shaped iron, permanent magnets (Br = 1.42 T), and spherical iron cores. The stator has four spherical magnetic poles with exciting coils of 100 turns. The air-gap between both spherical faces is

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Figure 1. Basic construction of the proposed model.

Figure 2. Magnetic circuit of the basic model.

Figure 3. Torque characteristics of the basic model.

0.3 mm. The mover is connected to four common resonance springs to be operated in resonance frequency. Figure 2 shows the cross section of x-z plane of the basic model. The magnetic flux by permanent magnet flows along the solid line, and the mover keeps balance at the center. When the coils are excited as shown in this figure, the flux flows along the dotted line, and the flux in the air-gap becomes unbalanced, and torque is generated. The mover can be rotated around arbitrary axis by changing the amplitude and direction of four coil currents.

3. Static Torque of the Basic Model Figure 3 shows the computed torque characteristics of the basic model employing the 3-D FEM when the coil A and C are excited, and the mover is rotated from 0 to 5 degree in step of 1 degree around y-axis. The average torque constant is 13.9 × 10–2 mN·m/A. Figure 4 shows the distribution of magnetic flux vectors with the coil excitation of 0 and 100 A. When coils are not excited, the magnetic flux by the magnet flows around the mover ploes and the upper parts of stator poles. On the other

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(a) 0AT

(b) 100AT (Coil A + Coil C) Figure 4. Distribution of flux density vectors.

hand, when both of coil A and coil B are excited, the magnetic flux by the current flows through the yoke of whole stator and the mover. It is found that the magnetic path by the current is completely divided with the magnetic path by the magnet.

4. Static Torque of Improved Model Figure 5 shows the improved model, which has the same structure as the basic model mentioned above except the mover. The mover of this model has the full-scale spherical yoke in order to keep the facing area between the mover and the stator, and has the ring-shaped permanent magnet (Br = 1.42 T) at the center. Figure 6 shows the comparison between the measured and the calculated torque characteristics with the coil excitation of 0 and 100 A. As can be seen, both results are in good agreement. This actuator has the stable position at the rotation angle of 0.0 degree. The computed average torque constant is 27.4 × 10–2 mN·m/A. It is twice as large as the basic model. Figure 7 shows the distribution of the flux density vectors, Fig. 8 shows the contours of the flux density in the neighborhood of facing area. The magnetic flux density of the improved model becomes stronger than that of the basic model, because the magnetic flux from whole of the ring-shaped magnet flows into the stator poles. As a result, the average torque constant becomes large. And the cogging torque characteristic of the improved model shows the linearity versus rotation angle compared with the basic model.

5. Dynamic Characteristics of Improved Model Figure 9 shows the prototype of the proposed model, which has the gimbal mechanism to be operated in the spherical surface with 0.3 mm gap between mover and stator, and has four resonant springs on the upside of the mover. Figures 10 and 11 show the measured frequency characteristics for x- and y- directions while it is operated at the same time. When the input voltage of 2.4 V (peak to peak) is applied with resonance frequency of 173 Hz for x-direction, the, the maximum rotation angle is 4.3 degree (peak

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(a) overall view

(b) x-z section Figure 5. Construction of the improved model.

to peak) and the average current is 0.4 A. On the other hand, when the same voltage is applied with resonance frequency of 127 Hz for y-direction, the maximum rotation

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Figure 6. Comparison between measured and calculated torque characteristics of improved model.

(a) Basic model

(b) Improved model Figure 7. Distribution of flux density vectors.

angle is 6.4 degree (peak to peak) and the average current is 0.5 A. The difference of resonance frequencies for x- and y- directions is due to the inertia of the gimbal mechanism. Figure 12 shows the trajectory of the mover when it is operated at resonant frequencies of x- and y-directions (173 Hz and 127 Hz). As shown, it is found that this actuator can be operated in arbitrary direction.

Y. Hasegawa et al. / New Spherical Resonant Actuator

(a) Basic model

225

(b) Improved model

Figure 8. Contours of flux density in the neighborhood of facing area.

Figure 9. Prototype.

6. Conclusions This paper presentedthe new spherical resonant actuator. The torque characteristics were computed through the 3-D FEM analysis. The effect of mover geometry on the torque characteristics was investigated. And, it was found that average torque of the improved model was twice as large as the basic model. The validity of the computation was verified by the comparison with the measurement of a prototype. Furthermore, the dynamic characteristics versus rotation angle were confirmed through the measurement. As a result, it was found that the proposed actuator was operated in spherical surface. This research was supported in part by “Special Coordination Funds for Promoting Science and Technology: Yuragi Project” of the Ministry of Education, Culture, Sports, Science and Technology, Japan.

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Figure 10. Measured frequency characteristics (x-direction).

Figure 11. Measured frequency characteristics (y-direction).

Figure 12. Trajectory of multi-motion. (x direction: 173 Hz, y direction: 127 Hz).

References [1] A. Tanaka, M. Watada, S. Torii and D. Ebihara, “Proposal and Design of Multi-Degree-of-Freedom Spherical Actuator”, 11th MAGDA Conference, PS2-3, pp. 169-172, 2002. [2] E.h.M. Weck, T. Reinartz, G. Henneberger and R.W. De Doncker, “Design of a spherical motor with three degrees of freedom”, Annals of the CIRP, Vol. 49, pp. 289-294, 2000.

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[3] K. Hirata, T. Yamamoto, T. Yamaguchi, Y. Kawase and Y. Hasegawa, “Dynamic Analysis Method of Two-Dimensional Linear Oscillatory Actuator Employing Finite Element Method”, IEEE Transaction on Magnetics, Vol. 43, No. 4, pp.1441-1444, 2007. [4] K. Hirata, Y. Ichii and Y. Kawase, “Novel Electromagnetic Structure with Bypass Magnetic Path for Reset Switch”, IEEJ Trans. IA, Vol. 125, No. 3, pp. 293-296, 2005.

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Chapter C. Applications C1. Electrical Machines and Transformers

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Influence of the Correlated Location of Cores of TPZ Class Protective Current Transformers on Their Transient State Parameters Elzbieta LESNIEWSKA and Wieslaw JALMUZNY Department of Applied Electrical Engineering & Instrument Transformers, Technical University of Lodz, ul. Stefanowskiego 18/22, 90-924 Lodz, Poland [email protected], [email protected] Abstract. A method of avoidance of mutual influences between secondary windings of multi-core type in a designed protective current transformer is presented in this paper. The analysis was performed for different location of cores with secondary windings. The mutual influence is determined on the basis of 3D field distributions obtained by the numerical field method. Some results of computation were compared with test results.

Introduction Protective current transformers are very important parts of electric power systems. They are indispensable for the proper functioning of a system, because they are elements of the protection system. There are two kind of protective current transformers; class P to protection at steady state and TP to protection at transient state. The multicore type current transformer is composed of a number of cores with individual secondary windings and a common primary bar in the same casing. During a transmission line short circuit, the primary current takes on an exponential component resulting in core saturation and a deformation of the secondary current. The cores of measuring current transformer and protective class P should be saturated during a transmission line short circuit. Therefore the measuring current transformer has a core without air gaps. The TPZ class protective current transformer has a core with air gaps which guarantee linearity of magnetic characteristic of the core at an assumed value of primary short circuit current Ipsc. Behaviour of protective current transformer at a transient state is very important because it influences the proper functioning of the protection system. The mutual influence between windings can occur through the magnetic field. The aim of research was to estimate the influence of the correlated location of cores on current error and the phase displacement at rated state and transformation errors at transient state. During the design process of the multi-core type current transformer it is important to predict the mutual coupling between the current transformers and then to avoid it.

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A Mathematical Model The field-and-circuit method for steady state, based on the solution of Helmholtz equation for 3D electromagnetic field determines secondary voltage and compares it with the circuit equation of the same secondary voltage. It is impossible to solve the transient problem using the complex method, since the wave-shapes of field quantities are considerably deformed. Joining field-and-circuit method and space-time 3D analysis allows computing the secondary current vs. time assuming a non-sinusoidal wave-shape of the primary current. For 3D analysis a full set of time-dependent differential equations must be solved instead. ⎛ ∂A ⎞ curl (ν curl A ) = σ ⎜ − − gradV ⎟ ⎝ ∂t ⎠

(1)

⎛ ⎛ ∂A ⎞⎞ div ⎜ σ ⎜ − − gradV ⎟ ⎟ = 0 ⎠⎠ ⎝ ⎝ ∂t

(2)

Using the commercial software 3D based on the numerical finite element method enables solving this equation and estimating the secondary current vs. time while the primary current has an exponential component. The non-linear magnetic characteristic of cores was taken into account. The boundary conditions were A × n = 0 and

V=0

(3)

at the boundary of the whole system with the surrounding air. The applied time stepping was 0.0001 s. The mesh of 616043 elements for 3D model was result of accuracy analysis. Further mesh refinement does not change the solution. Computational Results As an example, the three toroidal protective current transformers TPZ class 1200 A/1 A with eight air gaps of δ = 3 mm were considered. The total length of air gaps is 24 mm. The air gaps are rotated one by one at 45 degrees.

Figure 1. Multi-core type current transformer composed of three cores with individual secondary windings and a common primary bar.

E. Lesniewska and W. Jalmuzny / Protective Current Transformers

a)

233

b)

Figure 2. Cores with gaps in the same position and the cores are turned at 22.5 degrees with relation to each other.

The neighbouring cores with secondary windings can be located in a different way. Two border cases were considered in work; one when the cores with gaps are in the same position and the second when the cores are turned at 22,5 degrees to each other. The analysis was performed for rated steady state and transient state. The current error and the phase displacement indicate the accuracy class of designed current transformer. The computations of current error and phase displacement were performed at rated state for load R = 5 Ω. The measurements were performed in the same conditions using the type Φ5304 measuring bridge with comparator produced by the company POCTOK. The test for the single TPZ class protective current transformers 1200 A/1 A gives the following results: the current error –0.92% and the phase displacement 398minutes. The results of 3D analysis carried out for the individual TPZ class protective current transformers 1200 A/1 A were: the current error –0.88% and the phase displacement 346 minutes. Table 1 shows results of computation for two different positions of cores. Both constructions give convergent results of errors. The middle protective current transformer has somewhat better conditions if the cores are turned at 22,5 degrees to each other. The computations of the instantaneous error current vs. time were performed at transient state for load R = 5 Ω, specified primary time constant Tp = 50 ms and rated symmetrical short-circuit current factor Kssc = 25. The primary current was equal ip = 30 2 (cos314.16t–e–20t) kA. In Fig. 5 the results of the computed instantaneous error current vs. time at a transient states are presented for three protective TPZ current transformers. All curves are very close. For both case the magnetic field linked between current transformers is practically negligible. In Fig. 4 we can observe that the both constructional solution give practically negligible magnetic flux density in the neighbouring core caused by the other protective current transformers.

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a)

b)

Figure 3. Distribution of magnetic flux density [T] of protective current transformer at rated steady state at the same conditions for two different positions of cores. Table 1. The current error and the phase displacement of the TPZ current transformer at rated state Rb = 5 Ω TPZ class protective current transformers 1200A/1A (δ = 24 mm) gaps in the same position core: current error ΔI [%] phase displacement δi [min]

outside –0.887 342

middle –0.883 347

outside –0.880 342

gaps turned at 22.5° outside middle outside –0.828 –0.800 –0.825 341 334 341

Peak instantaneous alternating current errors have been determined on the basis of obtained curves (Fig. 5)

εˆac =

iˆε ac ×100% 2 I psc

(4)

where Ipsc = KsscIpn = 25 * 1200 = 30 kA, îεac – maximum instantaneous error of the alternating current component and is equal 9.7%, and have been determined on the basis of test is equal 11.4%. Test was carried out using the d.c. saturation method.

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a)

b)

instantenous error current (A)

Figure 4. Distribution of magnetic flux density [T] of protective current transformers at transient state for two different positions of cores at the same conditions and time t = 0.054 s.

25 20 15 10 5 0 -5

0

0,05

0,1

0,15

0,2

time (s)

outside

middle

outside

Figure 5. Instantaneous error current vs. time for three protective current transformers (in terms of secondary winding).

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a)

b)

Figure 6. Distribution of magnetic flux density [T] of the two P class protective current transformers and one TPZ class protective current transformer in the middle at rated steady state a) three current transformers together b) only the TPZ class protective.

The next problem was recognizing the influence of the two neighbouring protective current transformers type 10P 45 1200 A/5 A on operation of TPZ clas protective current transformer 1200 A/1 A. The same computation was performed, for the case with different class of current transformer. The computations of current error and phase displacement were performed at rated state of the TPZ class protective current transformer for load R = 5 Ω and the 10P class protective current transformers of the accuracy limit factor ALF = 45 for load S = 15 kV and cosφ = 0.8. In Fig. 6 can be observe that the cores of both P class protective current transformers are more saturated but the TPZ class protective current transformer with core with air gaps works in normal condition. Its current error and the phase displacement practically did not change.

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Table 2. The current error and the phase displacement of different class of current transformers at rated state class of current transformer:

10P 1200 A/5 A

current error ΔI [%] phase displacement δi [min]

–0.099 +7,70

TPZ 1200 A/1 A ° –0.836 +343

10P 1200 A/5 A –0.099 +7,70

a)

b)

Figure 7. Distribution of magnetic flux density [T] of the two P class protective current transformers and one protective current transformer TPZ class in the middle at transient state at the same conditions and time t = 0.054s a) three current transformers together b) only TPZ class protective.

Research shows significant difference between protective current transformers class P and TP (Fig. 7). Nevertheless that the accuracy limit factor of them is very big (45) and symmetrical short circuit current factor Kssc during test is 25, P class protective current transformer cores are saturated and the current transformation are incorrect, in opposite to TPZ class protective current transformer (Figs 8, 9). It causes by an exponential component of the primary current.

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secondary current (A)

130 80 30 -20

0

0,02

0,04

0,06

0,08

0,1

-70 -120 -170 -220 time (s) TPZ

P

P

Figure 8. Secondary currents vs. time of TPZ and P class protective current transformers.

instantaneous error current (A)

25 20 15 10 5 0 0

0,02

0,04

0,06

0,08

0,1

-5 time (s)

instantaneous error current (A)

sigle TPZ

TPZ nearby P

350 300 250 200 150 100 50 0 -50 0

0,02

0,04

0,06

0,08

0,1

time (s) Type P

Figure 9. Instantaneous error current vs. time for two type of protective current transformers a) a comparison curves for separated TPZ class current transformers and working in neighbourhood of P class protective current transformers b) for P class protective current transformers.

Conclusions The transient state 3D analysis of protective current transformers performed with the application of the field-and-circuit method can determine the correlated location of cores, which guarantees a magnetically separated operation of each core with individual secondary winding. The computing results show if correlation location of cores has an influence on steady state and transient errors. To eliminate the mutual interactions during the design

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process, the construction may be redesigned if the steady state and transient state errors are excessive. The tests show that in these cases the TPZ class protective current transformers can operate as independent devices. This means that the neighbouring current transformers have only a slight influence on their errors.

References [1] E. Lesniewska, Applications of the Field Analysis During Design Process of Instrument Transformers, Transformers in Practice, Vigo Spain 2006, pp. 227-251. [2] E. Lesniewska and J. Ziemnicki, Transient State Analysis of Protective Current Transformers at Different Forced Primary Currents, Przegląd Elektrotechniczny 5’2006, pp. 57-60. [3] W. Jalmuzny, D. Adamczewska, I. Borowska-Banas, Analysis of Current Difference Test Arrangement Operation for Measuring Class TP Current Transformers, Pomiary Automatyka Kontrola (PAK), no 10bis/2006, pp. 102-110. [4] E. Lesniewska and W. Jalmuzny, Influence of the Number of Core Air Gaps on Transient State Parameters of TPZ Class Protective Current Transformers, Compumag 2007, Aachen. This work was supported by the Polish Ministry of Science and Higher Education (Project No. 3 T10A 004 30).

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Machine with a Rotor Structure Supported Only by Buried Magnets Jere KOLEHMAINEN ABB Oy, Motors, P.O. Box 633, FI-65101 Vaasa, Finland E-mail: [email protected] Abstract. A buried magnet rotor structure, which is supported only by permanent magnets, is proposed for medium speed permanent magnet machines. A machine utilizing the construction is built, tested and compared to another machine with traditional V-shaped poles. The machine is also simulated using Finite Element Method and the results are compared to tested values. The obtained results demonstrate the feasibility of the construction.

Introduction Permanent magnet synchronous machines (PMSM) with buried magnets have been considered in a wide range of variable speed drives. A buried magnet design has many advantages compared to designs with surface mounted and inset magnets. With a buried magnet design flux concentration can be achieved, which induces higher air gap flux density [1,2]. That, in turn, gives a possibility to increase torque of a machine. The typical way of manufacture a buried PM rotor is to assemble a stack of punched rotor disks with rectangular holes and insert magnets into these holes. The rotor poles between the magnets are fixed to rest of the rotor structure with thin iron bridges. The disadvantage of the supporting bridges is the leakage flux, the magnitude of which depends on the thickness of the bridges. In low speed applications this is not a problem, since the centrifugal forces acting on the poles are relatively small and the bridges can be kept thin. However, as the tangential speed of the rotor surface in medium speed applications (4000…8000 1/min) exceeds 60 m/s (corresponding to 4000 1/min in machine size IEC250) the stress in the bridges will exceed the yield strength of the electrical steel (typically 300 MPa for grade M400-50A). The problem can be countered by increasing the thickness of the bridges, however, this increases the leakage flux, which in turn increases the amount of magnet material needed to get the required torque. However, there exists a solution with thinner bridges, where magnets are partly used to support the pole structure [3]. In this paper we go further and study a solution on how to get mechanically more robust rotor structures without using iron bridges. In the solution the tensile stress is geometrically converted into compressive one and only the magnets are used to support the pole structure. The new solution is compared to a traditionally used solution with V-shaped poles. The comparison is done using time stepping and static calculations using Finite Element Method (FEM) [4]. Machines with both the rotor designs are built and tested. The machine with the new dovetail pole design is analyzed further and results are compared to simulations.

J. Kolehmainen / Machine with a Rotor Structure Supported Only by Buried Magnets

241

Figure 1. Designs with dovetail and V shaped poles with flux lines created by the flux of the magnets.

Machine Designs An 8-pole machine with V-shaped rotor poles is used as an example for comparison to a machine with the new dovetail design without supporting bridges. The machine has shaft height 250 mm, nominal power 110 kW, voltage 370 V, and speed 4800 1/min. The only difference of the two machines is in their rotor structure as it can be seen in Fig. 1. An 8-pole machine has 8 symmetry sections in V-pole design, but in the dovetail design the rotor has magnets in every second pole [1]. With both designs, volume of the magnets and dimensions of the magnets seen by stator are same. With the V shape design, total magnet width and length in one pole are 2 × 7.3 = 14.6 mm and 52 + 52 = 104 mm and with the dovetail design, these are 1 × 14.6 = 14.6 mm and 26 + 52 + 26 = 104 mm. Length of the both rotors is 120 mm. Magnetically, there are two major differences which affect to electrical properties of machines. The dovetail design has not magnetic bridges between poles so the leakage flux is reduced especially with low saturation of flux. Every second pole of the dovetail design has a different tangential air gap length so electrical properties with high load angles are expected to be slightly worse than with the V shape design.

Manufacturing Two machines with both rotor types are manufactured. The general method to manufacture the rotor with a V shaped design is to assemble a stack of disks, compress it using bolts and nonmagnetic end plates and shrink fit the stack on the shaft. Then, the magnets are inserted into their holes using glue. The rotor with the dovetail design is manufactured with a different method, which is to assemble five (one central body and four small poles) stacks of disks, compress them using bolts and nonmagnetic end plates and shrink fit the central body stack on the shaft. Next, the magnets are fixed to four pole stacks using glue. Resulting poles are axially inserted to central body stack using glue.

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Figure 2. Manufacturing rotor with dovetail design. In the left there is a pole with magnets, in the middle there is the rotor without the pole, and in the right the pole is inserted to rotor. Table 1. Open Circuit Voltages Quantity

Dovetail design

V-shape design

311.1 365.5

330.5 347.6

Measured voltage (V) Calculated voltage (V)

Results The machines with the both designs are tested and analyzed. For all load tests, as for our industrial cases, the direct torque control strategy with software for permanent magnet AC machines is used with frequency converter ACS600 [5]. All electromagnetic calculations are done with time stepping Finite Element Method [4]. In load calculations, voltage source is used, the form of the voltage is sinusoidal and amplitude is kept the same. Simulations are started with various rotor angles without initial solution and stopped after 41 electric periods when transient oscillations have totally died away. Constant rotor speed is used. Iron losses are calculated from the equation 2

PTOT = kh Bm2 f +

1 T ⎡ d 2 ⎛ dB ⎞ dB ( t ) ⎟ + ke ⎛⎜ ( t ) ⎞⎟ ⎢σ ⎜ ∫ 0 T ⎢⎣ 12 ⎝ dt ⎠ ⎝ dt ⎠

32

⎤ ⎥ k f dt , ⎥⎦

(1)

where Bm is the maximum flux density at the node concerned, f is the frequency, σ is the conductivity, d is the lamination thickness, k h is the coefficient of hysteresis loss and k e is the coefficient of excess loss. Open Circuit Voltage Measured open circuit voltages of both machines, at speed 4800 1/min, are compared to the calculated ones in the Table 1. The measuring and the calculating temperature has been 20 oC. In this case, the magnets have a remanence flux density of 1.1 T and energy product 230 kJ/m3. The dovetail design has 5.2% larger calculated open circuit voltage than the V shape design. The measured open circuit voltage is 4.9% smaller than calculated for the V shape design and it is 14.9% smaller for the dovetail design. The measured open circuit voltages were expected to be smaller, because small rotor length and diameter

J. Kolehmainen / Machine with a Rotor Structure Supported Only by Buried Magnets

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Figure 3. Open voltage harmonics and torque ripples with speed 4800 1/min of dovetail and V shape designs.

Figure 4. Calculated torque and current (left) and efficiency and power factor (right) as a function of electric load angles with dovetail and V shape design.

ratio (120/289 = 0.42) causes remarkable leakage fluxes in ends of rotors. With the dovetail design, also axial deviations of the rotor will add leakage fluxes (the rotor body is slightly longer than the poles). The machine with the dovetail design has also different harmonic distribution of open circuit voltage, as can be seen in Fig. 3. The fifth harmonic is almost same with both designs. The seventh, eleventh and thirteenth harmonics are larger with the dovetail design while the fifth, seventeenth and nineteenth harmonics are larger with the V shape design. This is caused by different air gap forms and different sizes of every second pole with dovetail design. Practically, voltage and torque ripples are on the same level. Calculated Electrical Properties with Different Loads The calculated electrical properties as a function of electric load angle are compared in Fig. 4. With the dovetail design the torque is sinusoidal; the reluctance torque is negligible and the maximum torque 382 Nm with load angle 90 degrees is smaller than torque (414 Nm) with the V shape design, because of asymmetric pole pairs. With the V shape design, maximum torque is 426 Nm at load angle 102 degrees. Furthermore, maximum reluctance torque is 49 Nm. In addition, the current behaves differently, because of different saturation. With the dovetail design, power factor is larger with electric load angles under 55 degrees and with higher load angles, it is slightly smaller. However, efficiencies are slightly better with the V shape design.

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Figure 5. Calculated and measured input and output power with dovetail (left) and V shape (right) designs as a function of current. Table 2. Comparison of Nominal Load Results Quantity Shaft Power (kW) Torque (Nm) Voltage (V) Current (A) Efficiency Power Factor Total Losses (kW) Copper Losses (kW) Total – Copper (kW) Iron Losses (kW) Other Losses (kW)

Measured Dovetail design

Calculated Dovetail design

Measured V-shape design

Calculated V-shape design

110.2 219.1 370 229.7 0.947 0.791 6.26 1.66 4.60

110.4 219.6 370 194.0 0.950 0.935 6.26 1.19 5.08 1.50 3.58

111.1 221.0 370 212.9 0.950 0.858 5.90 1.30 4.60

110.3 219.5 370 202.8 0.946 0.898 6.03 1.18 4.85 1.27 3.58

Comparisons with Different Loads Measured input and output powers as a function of current are compared to calculated ones in Fig. 5. Generally, measured powers are smaller than calculated powers. The possible reason is the same than with the case of voltages; leakage fluxes in the ends of rotors. Difference is also bigger with the dovetail design. Same differences can be seen in comparison of nominal load results. Nominal Load The measured and the calculated nominal load results of the two designs are compared in the Table 2. Used stator winding temperature is 75 oC. Approximated “Other Losses” contains all other losses except friction and additional losses. Iron losses are calculated with Eq. (1). All efficiencies have the same magnitude. With the dovetail design, the calculated power factor is 4.1% better and measured power factor is 7.8% worse than with the V shape design. Torque oscillations are compared with different electric load angles in Fig. 6. The oscillations are larger with the dovetail design and calculated load angles over 8 degrees. Oscillation is studied further with load angle 40 degrees in Fig. 6. Clear sixth order torque harmonics can be seen with the dovetail design. With the V shape design, remarkable twelfth order torque harmonic reduces total oscillation.

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Figure 6. Calculated torque oscillations as a function of electric angle (left) and torque as a function of time with electric load angle 40 degrees (right) with dovetail and V shape designs.

Figure 7. Von Mises stress with speed 4800 1/min. The stresses are greatest in dark grey areas.

Strength of Structures Stress Analysis The rotor with the new dovetail design has a totally different stress distribution compared to the V-pole rotor. In the V-pole rotor, all of the shear and tension stresses are in the iron bridges whereas in the dovetail design, most of stresses are compression stress in the magnets and shear stress near corners of magnets. Von Mises stresses in the dovetail and in the V shape designs with are shown in Fig. 7. Computation is done using the centrifugal force associated with the speed of 4800 1/min. The largest stress in electric steels of the dovetail design, 130 MPa, is locally in the corners of sheets. With the V shape design, average stress in the inner bridges is 90 MPa and the largest stress in electric steels, 200 MPa, is also locally in the corners of sheets. These values are below the yield strength (305 MPa) of the steel. With the dovetail design, in center of the smaller magnets the stress is 50 MPa. It is well below the maximum compressive strength of the magnets. The calculated maximum stress in magnets is 381 MPa (located in corners). Using magnets to compose the structure it becomes robust enough for the speed of 4800 1/min.

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Stability of Rotor with Dovetail Design The mechanical durability test of the motors consists of two load runs with speed 4800 1/min. First, the motor is driven with different loads and operating temperatures for one and half hours. After this, the motor was cooled over night. Finally, temperature test has done for four and half hours. Measured vibration levels remained same thought all tests. Visual check was done after test. The magnets remained solid. The glue seam between inner magnets and rotor body was separated. In the sides of smaller magnets, glue was changed its color from grey to light grey. This indicate that glue was deformed, not separated. Hence, stability of the rotor remained, but more tests should be done to see whether stability remains with longer period.

Conclusion The prototype machine with a dovetail-shaped magnet poles exhibits a significant increase in mechanical stability over the conventional V-pole design. By converting the tensile stress in the iron bridges into a compressive stress in the magnets by redesigning the pole geometry, a very robust construction can be achieved. The electrical properties and the consumption of magnetic material can be kept on the same level as in the V-pole design.

References [1] Kolehmainen J., “Finite Element Analysis of Two PM Motors with Buried Magnets”, ICEM, Krakow, Poland, 2004, SPRINGER MONOGRAPH “Recent Developments of Electrical Drives”, Nov. 2006. [2] Ohnishi T., Takahashi N., “Optimal design of efficient IPM motor using finite element method”, IEEE Trans. Magn., vol. 36, no. 5, 3537-3539, Sep. 2000. [3] Kolehmainen J., Ikäheimo J., “Motors with Buried Magnets for Medium Speed Applications”, IEEE Trans. Energy Convers., to be published. [4] Flux2D software – www.cedrat.com. [5] ACS 607-0400-5, frequency converter – www.abb.com.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-247

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FEM Study of the Rotor Slot Design Influences on the Induction Machine Characteristics Joya KAPPATOU, Kostas GYFTAKIS and Athanasios SAFACAS University of Patras, Department of Electrical and Computer Engineering, Electromechanical Energy Conversion Laboratory, 26500 Rion Patras, Greece E-mail: [email protected] Abstract. A library of parameterized Squirrel Cage Induction Machine models has been constructed regarding the rotor slot design and is used to investigate the influences of the rotor bars shape on the machine characteristics. The slot shapes chosen correspond to standard induction machines used for specific applications. The calculations were conducted using the Finite Element Model (FEM) of the machine and the torque and current waveforms against speed, as well as the field distribution and the copper losses in the rotor bars for every model at starting have been obtained. Useful conclusions about the influences of the slot design on the Induction Machine behavior, mainly at starting are derived.

Introduction As known, the rotor of Induction Machines has stronger effect than the stator on the performance characteristics of the machine and especially on its starting performance. On the other hand, there are specific limits of the machine variables, e.g. maximum value for the starting current and minimum values for the starting torque, pull-up torque and breakdown torque, depending upon its ratings. Thus, different design classes for cage Induction Motors are set, each one suitable for different performance requirements and specific applications. These design classes refer to the rotor design and specially to the geometrical characteristics of the rotor slots. Furthermore the interest of both the designers and the users is directed to the optimization of the machine performance and the increase of the efficiency. For that reason several papers have been published during the last decades on the analysis and optimization of cage Induction Motors and specially on the suitable design of the rotor slots, which is critical for the starting performance and the shape of the torque-speed curve [1–5]. In this paper in order to compare more accurately the performance of cage Induction Machines of different rotor slot design, the 2-d Finite Element Method is used. Four different rotor slot designs are used, which are presented schematically in Fig. 1 and they refer to standard squirrel cage motors used for specific applications. The geometrical variables of the rotor cross section have been parameterized and a FEM code, which is used to model the motor has been developed. In order to obtain a more flexible and user friendly code, a library of four parameterized models has been constructed, which differ between them as for the shape of the rotor slots. As the shape of the slot remains the same in each model, the slot dimensions can easily and accurately be modi-

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J. Kappatou et al. / FEM Study of the Rotor Slot Design Influences

a)

b)

c)

d)

Figure 1. Characteristic rotor slots designs.

fied. In all cases the stator model, the air-gap length, the stator voltage and frequency, as well as the number of rotor slots are the same in order to compare the influences of the rotor slots design on the operational characteristics of the machine. Besides, as the rotor core is usually slightly saturated and also the skin effect plays an important role, specially in the deep-bar design, the same B-H curve of the ferromagnetic material and a suitable number of Finite Elements in the slot area are used. In all these cases the distribution of the field in the core and the waveforms of the torque and current as functions of the speed are calculated and useful results of the rotor slots shape effect on the machine behavior are derived. The use of the models library results in a flexible and more friendly code, which can be used for the application of optimization techniques for every specific design type, as fewer design geometrical variables are used than in a model of arbitrary slot shape.

FEM Analysis The numerical analysis is based on a magnetic vector potential formulation and a commercial package, OPERA of VF, has been used for the Finite Elements Analysis. The model can include non-linear materials as the magnetic saturation plays an important role, mainly in the rotor of the machine. In order to model the skin effect, specially in the cases of deep bars and double cage a suitable mesh has been constructed in the bars area. In the above cases, high currents are induced in the rotor bars, specially at starting and the skin effect must be taken into account. The stator windings are fed from sinusoidal voltage sources, which are connected via external circuits to the model. A library of four parameterized models of Asynchronous machines has been constructed, which differ between them as for the shape and the dimensions of the cage bars. Each design from now on will be reported with the letters a,…,d according to Fig. 1. All models have the same stator with 36 slots and 2 pole pairs, the same value and frequency of the supply voltage and 48 rotor slots. Although the dimensions of the rotor bars were selected arbitrarily for the various models, some parameters which affect the motor performance are computed and some useful qualitative conclusions regarding the influence of the bars design on the motors behaviour can be derived. These are calculated mostly for the starting, which is critical for the machine performance.

Simulation Results As known the design of the rotor bars affects strongly the motor performance and specially its starting. An arbitrary bar design can result to higher torque and less current

J. Kappatou et al. / FEM Study of the Rotor Slot Design Influences

a)

b)

c)

d)

249

Figure 2. Flux lines and current density in the bars at starting for the various models of the slots according to Fig. 1: a) bar model a, b) bar model b, c) bar model c, d) bar model d.

Figure 3. Flux lines and current density in the bars near synchronous speed for the model of the double cage, bar model d.

during starting, but to much greater slip and consequently to a reduction of the efficiency at nominal load. In this work the dimensions of the rotor bars were selected arbitrarily for the various models and we can not achieve general conclusions. However some parameters which affect the motor performance, mainly at starting, are computed and some useful qualitative conclusions regarding the advantages and disadvantages of each model can be derived. In Fig. 2 the flux lines at starting on a cross section of the machine for the various models of the slots, according to Fig. 1, are presented. In the same plot the current density in the bars is superimposed also. As the bar becomes deeper, Figs a and d, the leakage of the rotor increases and less flux is penetrating into the rotor body. At starting the skin effect plays an important role because the skin depth is at its smallest value. The distribution of the current density in the bars, specially in the case of the double cage, Fig. 2d, shows clearly the skin-effect. In Fig. 3 the flux lines and

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β

γ

Figure 4. Spatial distribution of the amplitude of current density along the axis (j-k) of a rotor bar, in A/mm2, at starting for the different bars used: α) the type of bars a and b, β) the bar type c for two values of the conductivity of the bars, γ) the bar type d for the linear and the non-linear B-H curve.

the distribution of the current density in the bars is presented for the case of the double cage near the synchronous speed (1 Hz) for comparison with the starting, Fig. 2d. For a better display of the skin effect, the spatial distribution of the current density along the axis j-k of a rotor bar at starting is demonstrated in Fig. 4. In all models the same bar has been selected for the calculations. The value of the electrical conductivity of the rotor bars has been taken equal to 60 kS/ mm, except for the bar design c, where a second case of less conductivity, 40 kS/mm, has been analyzed also for comparison. The current density increases in the top of the slot, specially for the case of double cage, Fig. 4γ, as a result of the skin-effect. In Fig. 4β the influence of the electrical conductivity of the bars on the current density in the bar type c is demonstrated. When a non-linear B-H curve is used in the model of the motor with double cage, bar type d, the current density is gradually increasing along the axis of the bar comparing to the linear case, essentially in the top of the slot near the air-gap. The effect of the bar design on the waveforms of the torque and stator current against speed is presented in Fig. 5. The motor design c, which develops the highest starting torque is a high slip motor, while the smallest value of starting current appears in the motor design d. Although the dimensions of the rotor bars were selected arbitrarily for the various models, some more motors parameters which affect the motor performance are computed and some useful qualitative conclusions regarding the advantages and disadvantages of each model can be derived. These are calculated at starting and are presented in the following Table 1. From Fig. 5 and Table 1 the following useful conclusions are obtained: •

The starting current is lower in the case of the double cage, type d, comparing with the motor of bar type c. This can be explained from the higher value of

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J. Kappatou et al. / FEM Study of the Rotor Slot Design Influences

35

c

d

a

b

c

d

50

a

b

30 40

30

20

I(A)

M (Nm)

25

15

20

10 10

5 0

0 0

200

400

600

800

1000

1200

1400

1600

0

200

400

600

800

1000

1200

1400

1600

n (rpm)

n (rpm)

a

b

Figure 5. Waveforms for the various designs of the bars of: a) the torque, b) the stator current. Table 1. Performance characteristics for the different rotor slots design at starting

0.268 0.282 0.679 1.012 0.853

Amplitude of current in the stator (A) 45.352 48.4 35.63 29.8354 19.86

Total ohmic losses in the bars (W)/1bar resistance (mΩ) 5856.302/0.0938 6812.75/0.09609 8907.686/0.2354 9202.5/0.3527 4526.206/0.425

Mean ohmic losses per surface in the bars (W/mm2) 2.8632 3.7955 14.944 15.4385 3.372

1.079

22.5

5863.27/0.425

4.368

Bar type according to Fig. 1

Torque (Nm)

Torque/Is (Nm/A)

a b c, conduc. = 60 kS/mm c, conduc. = 40 kS/mm d, linear B-H

12.18 13.68 24.2 30.212 16.957

d, non-linear

24.29

•

•

the rotor leakage and consequently the leakage reactance in the first case, as observed in Fig. 2 c and d. Although the starting torque is greater in case c, the ratio of starting torque per current presents higher value in the bar type d, particularly when the magnetic saturation is taking into account in the model, the above ratio increases more. The decrease of the conductivity of the bars in the bar design c, which results in higher bar resistance, increases the starting torque and limits the starting current. Particularly the ratio of the starting torque per ampere increases from 0.679 to 1.012. The mean ohmic losses per surface in the bars in the design c are much greater than in any other case. In Table 1 also appears the resistance of a cage bar, as calculated, for every model.

Conclusions A library of four parameterized Squirrel Cage Induction Motors models of different bars shapes has been constructed and an investigation of the effect of the bars design on the characteristics of the motor has been carried out. The waveforms of torque and current as function of speed are calculated and compared for all models. Moreover the distribution of the field and some more parameters at starting, like the variation of the

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current density along the axis of a bar, as well as the ohmic losses in the cage have been computed, which give us some useful qualitative conclusions regarding the advantages and disadvantages of each model.

References [1] M. Nurdin, M. Poloujadoff, A. Faure, “Synthesis of squirrel cage motors: A key to optimization”, IEEE Trans. on Energy Conversion, Vol. 6, No. 2, pp. 327-335, June 1991. [2] S. Williamson, C. McClay, “Optimization of the geometry of closed rotor slots for cage Induction motors”, IEEE Trans on Industry Applications, Vol. 32, No. 3, pp. 560-568, May/June 1996. [3] M.R. Feyzi, H.V. Kalankesh, “Optimization of Induction motor design by using the finite element method”, CCECE/CCGEI 2001. [4] C. Grabner, “Investigation of Squirrel cage Induction motors with semi-closed and closed stator slots by a transient electromechanical finite element technique”, ISEF 2005, Baiona, Spain, conference proceedings record, September 2005. [5] Min-Kyu Kim, Cheol-Gyun Lee, Hyun-Kyo Jung, “Multiobjective optimal design of three-phase Induction Motor using improved evolution strategy”, IEEE Trans on Magnetics, Vol. 34, No. 5, pp. 2980-2983, September 1998.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-253

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Concentrated Wound Permanent Magnet Motors with Different Pole Pair Numbers Pia SALMINEN, Hanne JUSSILA, Markku NIEMELÄ and Juha PYRHÖNEN Lappeenranta University of Technology, Department of Electrical Engineering, P.O. Box 20, 53851 Lappeenranta, Finland [email protected] Abstract. The study addresses the torque production capabilities and losses of concentrated wound permanent magnet machines. Different permanent magnet synchronous motors are modelled with both semi-closed slots and open slots. With open slots, the coils can be easily assembled to the stator, thus making it attractive to study the performance or benefits of the concentrated wound PM machines with open slots. Different slot and pole combinations are considered on the basis of the finite element analysis (the Flux2D program package by Cedrat being employed in the computations).

Introduction This work examines the performance of different model designs for a machine with a rated torque of 1075 Nm, a frame size of 225, and a rated speed of 400 rpm. These values represent ratings for machines used for example in the paper making industry. To compare the machines, the electromagnetic losses are calculated to estimate the efficiency of each machine. Particular attention is paid to the iron losses, Joule losses (I2R losses), and eddy current losses caused by the permanent magnets of concentrated wound PM machines. Concentrated wound machines are one type of fractional slot wound machines. The number of slots per pole and per phase q ≤ 0.5. In a concentrated wound machine, each coil is wound around one tooth to achieve as short end windings as possible. This reduces the amount of copper and leads to low Joule losses by virtue of the short end winding. It is also possible to utilize short end windings by inserting a longer stator stack compared to the stack of integer slot wound machines in the same frame size. Consequently, longer active parts will give more torque. Figure 1 illustrates a concentrated wound prototype machine, constructed within a size 225 frame. In this case, the end windings are so short that it is possible to make the stator stack 30% longer compared to the stack length of a four pole integer slot wound induction motor [1–3].

Finite Element Analysis A set of finite element analyses (FEA) is performed to estimate the pull-out torque and the losses of each design. The pull-out torque is the maximum torque that a motor can sustain at synchronous operation. The finite element computations are based on Cedrat Flux2D [4] program package using transient analysis for model concentrated wound

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end windings

270 mm

end windings

D

Stator stack

Figure 1. Concentrated wound PM machine in a size 225 frame. The short end windings leave more space inside the frame; hence, the stator stack may be designed longer in the axial direction.

a)

b)

Figure 2. Concentrated wound PMSM a) with a low pole pair number and b) with a high pole pair number.

machines with different pole pair numbers, p. Figure 2 (a) shows a machine with a low pole pair number p = 4, and 12 slots. For this machine, the number of slots per pole and per phase q = 0.5. In Fig. 2 (b), the pole pair number is high (p = 15), there are 36 slots, and q = 0.5. It has been noticed in previous studies [5] that with q = 0.5 the torque production capability is higher than with q < 0.5. When q = 0.5, the cogging torque and torque ripple may be higher than with other possible concentrated wound combinations, while it is still possible to minimize torque ripples by optimising the magnet width [6]. Semi-Closed Slots The machines under investigation have surface mounted magnets as shown in Fig. 2. Most of the models are constructed with semi-closed slots in the stator, except for certain special cases that have open slots. The boundary conditions for machine dimensions are determined by the size 225 frame; the outer diameter 364 mm and the machine length 270 mm. The rated speed is 400 rpm and the rated torque is 1075 Nm. The total losses of concentrated wound PM machines calculated are presented in Fig. 3. The total losses comprise the iron losses, Joule (I2R) losses, and eddy current losses caused by the permanent magnets. It can be seen in Fig. 3 that the Joule losses dominate by a large margin, but their proportion of the total losses decreases when p is large. Consequently, if low Joule losses are an important design parameter, it is advisable to avoid low pole pair numbers. However, the iron losses increase as the number of poles increase, which is also expected as the frequency increases. The proportion of

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255

Figure 3. Iron losses, Joule losses, and eddy current losses caused by permanent magnets for machines with surface mounted magnets having semi-closed slots. The results are obtained by the FEA.

iron losses to the machines in this study rises from 0.3 kW to 1 kW as the frequencies increase from 27 to 140 Hz. Please note that the minimum net losses occur when p = 10 (see Fig. 3). The machine with p = 10 has about 25% lower net losses than p = 4 and p = 21 machines. The model parameters for the magnet material are based on the values for Neorem 495a100, the remanence flux density of which is Br = 1.05 T and the coercivity HC = 800 kA m–1. The resistivity of the sintered magnets permits remarkably large eddy current losses if the flux density may pulsate in the magnet during the motor running. This phenomenon makes concentrated wound machines quite vulnerable to large eddy current losses in the magnets. In integer slot wound machines with semi-closed slots the flux density pulsation in the magnets may be negligible compared to fractional slot machines with concentrated windings. The aim here is to investigate what kind of a role the permanent magnet eddy current losses have at low speed (e.g. 400 rpm). Usually, a machine with geometrically wide magnets generates high eddy current losses, while small or narrow magnets generate lower eddy current losses [7]. Figure 4 shows the eddy current losses caused by permanent magnets for several surface-mounted PM machines. As expected, the machines with low pole pair numbers have high eddy current losses because of their large magnets in the rotor. Geometrically large surfaces are harmful with respect to the eddy current losses. Semi-Closed Slots Compared with Open Slots Next, the effect of the slot opening width on the losses of PM machines is examined. The above semi-closed machines are now modified so that they have open slots. The results obtained by the FEA are shown in Fig. 5. The results demonstrate that only the iron losses are smaller with the open slot structures than with the semi-closed slot structures. The Joule losses turn out to be higher with open slots than with semi-closed slots, because more winding turns are needed in the stator slots to induce sufficient back electromotive force in the system. With open slots, the eddy current losses caused by permanent magnets are approximately double compared with those of motors using

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500 12 Slots 24 Slots 36 Slots

Losses (W)

400 300 200 100 0 0

4

8

12

16

Pole pair number, p Figure 4. Eddy current losses caused by permanent magnets for machines with surface-mounted magnets having semi-closed slots. Results are obtained by the FEA.

Figure 5. Iron losses, Joule losses and eddy current losses caused by the permanent magnets for machines with surface mounted magnets having semi-closed slots and open slots. Results are obtained by the FEA.

semi-closed slots. For a machine with a low pole pair number (e.g. p = 5), the eddy current losses with semi-closed stator slots are calculated to be 400 W, while with open slots, the losses rise to 950 W. This result suggests that the eddy current losses in the case of concentrated wound PM machines should be evaluated carefully even at low speeds. With high pole pair numbers, the magnet dimensions are small, and therefore the eddy current losses are low. The calculated pull-out (maximum synchronous operating) torques for four pairs of model motor designs are shown in Fig. 6. Each of the designs has either open or semi-closed stator slots. For both types, when p = 21, the available torque is relatively

P. Salminen et al. / Concentrated Wound Permanent Magnet Motors

257

3

Pull-out torque (p.u.)

Torque

2

1

0 Semi

Open

12-slot-10-pole

Semi

Open

24-slot-16-pole

Semi

Open

36-slot-24-pole

Semi

Open

36-slot-42-pole

Figure 6. Pull-out torque for machines with surface mounted magnets having semi-closed slots and open slots. Results are obtained by the FEA.

small and not enough for appropriate motor operation. The machines with 16 poles and 24 poles have a high torque, and also their efficiency shows good performance. Open slot motors achieve slightly higher torque levels than the ones with semi-closed slots, except when p = 10; however, the efficiency with open slots is lower than with semiclosed slots.

Conclusion By applying finite element analytical methods, this paper examines several factors that affect the performance and efficiency of permanent magnet electric motors. The authors have, however earlier experimental data (e.g. [2]) that confirms the validity of the calculation process. Joule heating accounts for most of the losses; however, also eddy current effects play a significant role and must therefore be included in the analysis. For machines with rotor surface magnets, the electromagnetic energy losses are higher in the machines with an open-slot stator than in the machines with a semi-closed slot structure, especially when the pole pair number is low. This is a consequence of the flux pulsation in magnets and the large magnet surface area. It is possible to achieve a high pull-out (maximum) torque with both open slots and semi-closed slots by using an intermediate number of poles and slots; these designs also exhibit the lowest Joule losses. Since Joule losses are the dominant factor in the machines (machine design), the further studies will focus on embedded magnet motors. It is hoped that in these motors, the Joule losses will be low even when the number of pole pairs is large.

References [1] Cros J., Viarouge, P., Carlson, R. and Dokonal, L.V. 2004. Comparison of brushless DC motors with concentrated windings and segmented stator. Proceedings of the International Conference on Electrical Machines. ICEM 2004, Krakow, Poland. CD-ROM.

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[2] Salminen, P. 2004. Fractional slot permanent magnet synchronous motor for low speed applications. Dissertation, Acta Universitatis Lappeenrantaensis 198, Lappeenranta University of Technology. 151 p. [3] Libert, F. and Soulard, J. 2004. Investigation on Pole-Combinations for Permanent-Magnet Machines with Concentrated Windings, Proceedings of the International Conference on Electrical Machines, ICEM 2004, Krakow, Poland. [4] Cedrat 2007. Software solutions: Flux®. [Online] Available from http://www.cedrat.com/ [Date accessed 26.6.2007]. [5] Salminen, P., Jokinen, T. and Pyrhönen, J. 2005. The Pull-Out Torque of Fractional-slot PM-Motors with Concentrated Winding, Electric Power Applications, IEE Proceedings, Vol. 152, Iss. 6, pp. 1440–1444. [6] Salminen P., Pyrhönen J., Libert F., and Soulard J. 2005. Torque Ripple of Permanent Magnet Machines with Concentrated Windings. 15–17 September 2005, International Symposium on Electromagnetic Fields in Mechatronics, ISEF 2005, Electrical and Electronic Engineering, Baiona, Spain. [7] Zhu, Z.Q., Ng, K., Schofield, N. and Howe, D. 2004. Improved analytical modelling of rotor eddy current loss in brushless machines equipped with surface-mounted permanent magnets. Electric Power Applications, IEE Proceedings. Vol. 151, Issue 6, 7 Nov. 2004 Page(s):641–650.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-259

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Air-Gap Magnetic Field of the Unsaturated Slotted Electric Machines Ioan-Adrian VIOREL, Larisa STRETE, Vasile IANCU and Cosmina NICULA Electrical Machines Department, Technical University of Cluj-Napoca, Daicoviciu 15, 400020, Romania [email protected], [email protected] Abstract. This paper deals with the air-gap magnetic field analytical calculation in the case of unsaturated slotted electric machines. Different analytical estimations of the flux density variation versus circumferential coordinate in the machine airgap are considered. The obtained results are compared between them and with the two dimension finite element method (2D-FEM) calculated values, where the iron core material nonlinearity is fully considered.

1. Introduction The air-gap flux density provides valuable information in evaluating slotted electric machine performance. Any method to design an electric machine, regardless its type, requires knowledge on the air-gap magnetic flux density to calculate the main dimensions, the necessary mmf, the torque (average, maximum, starting, cogging or ripple), the back emf value and shape and the main inductances, which may be dependent on the rotor position. It is clear by now that the common, and the most accurate way of obtaining the air-gap flux density is based on finite element method (FEM) calculation, but this is still time consuming even on powerful computers and it is difficult to use FEM in iterative design optimizing procedure, or to implement FEM results directly in on line control systems. The electric machines’ air-gap flux density calculation was in the researchers’ attention for a very long time. Important results in the domain, concerning the air-gap magnetic field in synchronous and respectively induction machine were published in the twenties of the last century, as were the works of Weber [1], Spooner [2], Carter [3] and Wieseman [4]. Heller’s book, [5], represents a synthesis of Heller entire work, containing also the most important other contributions published until the sixties last century concerning the air-gap field in the induction machine. In the last years, two types of electric machines were in the attention of the researchers, the synchronous permanent magnet (PM) machine with PMs in the air-gap [6–8], and the switched reluctance machine (SRM) [9,10]. In the case of rotating or linear transverse flux reluctance motors, the air-gap magnetic field calculation is important too and some results are given in [11,12], but the problem is not too different from the case of SRM’s. The calculation of the air-gap permeance of the double-slotted electric machines was also done, results being published, as in [13–15]. Since the case of the machines with PMs on the surface of rotor and stator was quite extensively studied recently, [6–8] for instance, in this paper will be considered

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only the case when there are no PMs in the air-gap, or buried inside the stator or rotor core. The stator and rotor core surfaces are considered as equipotential ones; therefore the magnetic field in the air-gap can be analytically described by a one-dimension (1D) model. The air-gap flux density variation versus circumferential coordinate, calculated via 2D-FEM analysis, is, as expected, continuous and smooth, without local points where its first derivative became zero on the entirely domain, half of the slot pitch, but at the domain extremities. Due to this important observation, the air-gap flux density variation requires a simple analytical estimating function in the cases when only Carter’s factor, torques or back emf have to be calculated. When main phase inductance depends on the rotor position, SRM for instance, the air-gap flux density variation must be approximated by an analytical function which gives values as closed as possible to the actual ones, since a simple estimating function might introduce errors. Different analytical estimations of the air-gap flux density variation versus circumferential coordinate are considered in the paper. The obtained results are compared between them and with the 2D-FEM calculated values on a simplified machine model. The Carter’s factor, which represents a criterion for the estimation accuracy, is calculated too and compared with that obtained via 2D-FEM analysis. The best fitted estimations for different cases are discussed and pertinent conclusions regarding the analytical estimations of the air-gap flux density are presented.

2. 2D-FEM Analysis The 2D-FEM analysis was carried on a simple model. The model’s structure contains two slots, a coil on the stator and a non-salient rotor. For symmetry reasons the stator length is equal to two tooth pitches. The notations contain the suffixes S, R for stator and rotor. The topology was parameterized in a way to make possible adequate variation of the most important dimensions, only the tooth pitch, which is the circumferential length, was kept constant. A linear layout was considered, but it does not affect the generality. In the air-gap were considered six layers in order to obtain a more homogenous distribution of the vector magnetic potential affected by the magnetic permeability difference between the core and the air-gap domain. The flux density, calculated in the middle of the air-gap, has a smooth and continuous variation, Figs 2 and 3. If the ratio between the tooth width and the double of the stator yoke is small, then the iron-core is unsaturated even if the air-gap flux density has large values, Table 1, via 2D-FEM analysis, when t/g = 37.5, wt/2hyS = 0.667, wt/2hyR = 0.333. Based on 2D-FEM analysis obtained values, the Carter’s factor is calculated as the ratio between the peak and the average value of the air-gap flux density, K C = Bg max / Bgav

(1)

The saturation coefficient Ksat results as: K sat = Bg max unsat / Bg max

(2)

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Table 1. Saturation Ksat and Carter KC factors Bgmax Ksat Kc

0.8546 1.011 1.3597

1.16 1.015 1.362

1.433 1.042 1.368

1.526 1.183 1.372

Figure 1. Machine model with nonsalient rotor.

where the saturated peak air-gap flux density value Bgmax is obtained in the tooth axis via 2D-FEM analysis and the unsaturated air-gap flux density value is: Bg max unsat = μ 0 F / 2 g

(3)

with the following usual notations, Fig. 1, t – tooth pitch, wt – tooth width, ws – slot width, hyS, hyR – stator and rotor yoke radial length, g – air-gap radial length, F – coil mmf in ampere turns, µ0 – air-gap permeability (4π10–7H/m). The air-gap flux density average value Bgav is given by: Bgav =

2 t

t/2

∫B

g

( x)dx

(4)

0

where Bg(x) are the air-gap flux density values obtained at equidistant points via 2D-FEM analysis, the integral being numerically calculated. Examples of the air-gap flux density variation versus the circumferential coordinate x ∈ [0,t/2], calculated by FEM, are given for two values of mmf in each case, in Figs 2 and 3. The values of the actual mmf, the pole pitch to air-gap length ratio t/g, the slot width to double air-gap length ws/2g ratio, peak and average flux density Bgmax, Bgav, Carter and saturation factors KC, Ksat for all cases presented in Figs 2 and 3 are shown in Table 2. Two important remarks should be made considering the curves shown in Figs 2 and 3, and the values given in Table 2:

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Table 2. Representative values for the curves given in Figs 2 and 3 Figure

t/g

ws/2g

Fig. 2

37.5

6.25

Fig. 3

75

25

F[A] 2200 3800 1000 1800

Figure 2. Flux density variation, t/g = 37.5, ws/2g = 6.25.

Bgmax[T] 0.855 1.433 0.776 1.383

Bgav[T] 0.628 1.047 0.304 0.539

KC 1.36 1.368 2.556 2.568

Ksat 1.011 1.042 1.011 1.022

Figure 3. Flux density variation, t/g = 75, ws/2g = 25.

i) The air-gap flux density variation versus circumferential coordinate is represented by smooth and continuous curves. ii) Even at important peak flux density values, the saturation is not important for the case considered, wt/2h y < 1 and consequently the Carter’s factors are not dependent on the mmf. 3. Air-Gap Flux Density Variation One of the analytical approximation of the air-gap flux density variation quite intensively employed is based on the air-gap variable equivalent permenace, a method which was first introduced in the case of induction motor [5,15] and was later extended to other motors, SRM [16] or transverse flux motor (TFM) [12], for example. The air-gap variable equivalent permeance can be easily defined and allows the calculation of the air-gap magnetic field in the case of double slotted machine. Its accuracy depends on the accuracy of the estimation for the air-gap flux density variation over a tooth pitch. If Bgmin is the minimum value of the air-gap flux density, then: Bg ( x) ≅ Bgav + 0.5( Bg max − Bg min ) cos

2x π, t

x ∈ [0, t / 2]

(5)

The variable equivalent air-gap permeance is defined as: P( x) =

ΔBg 1 Bg ( x) ⎛ Bgav 2x ⎞ 1 =⎜ + cos π ⎟ ⎜ g Bg max ⎝ Bg max Bg max t ⎟⎠ g

(6)

I.-A. Viorel et al. / Air-Gap Magnetic Field of the Unsaturated Slotted Electric Machines

263

where the air-gap topology coefficient β is [6]: Bg max − Bg min

β=

2 Bg max

=

ΔBg Bg max

⎛ 1 β = 0.5 ⎜1 − ⎜ 1 + ( ws / 2 g )2 ⎝

⎞ ⎟ ⎟ ⎠

(7)

The air-gap variable equivalent permenace, function of the circumferential coordinate x, comes: P( x) =

1 1 ⎛ 2x ⎞ ⎜ 1 + pr cos π ⎟ g KC ⎝ t ⎠

(8)

where the permeance coefficient pr = KCβ or, as proposed in [15], pr =

γ =

⎛γ gπ ⎞ 4 β K C sin ⎜ ⎟ π ⎝β t 2⎠

(9)

2 ⎡ ⎛w ⎞ ⎤ w 4 ⎢ ws a tan( s ) − ln 1 + ⎜ s ⎟ ⎥ 2g π ⎢ 2g ⎝ 2 g ⎠ ⎥⎦ ⎣

(10)

Some equation, eventually more accurate than (5), can be developed, such as: i) The one proposed by Weber [1] t − ws ⎛x⎞ Bg ( x) = Bg max (1 − 2β sin 2 a ⎜ ⎟ π ) , a = , ws ⎝t⎠

x ∈ [0, t / 2]

(11)

ii) The one proposed by Heller [5] Bg ( x) = Bg max, ,

x ∈ [0, β wt )

Bg ( x) = Bg max (1 − β + β cos y ) , y=

x ∈ [ β wt , t / 2] ,

πβ wt π x− t / 2 − β wt t / 2 − β wt

(12)

iii) A nonsinusoidal variation, considering a simplified flux lines topology in the air-gap, [7] Bg ( x) = Bg max, ,

x ∈ [0, wt / 2] −1

⎡ ⎤ π Bg ( x) = Bg max, ⎢1 + ( x − wt / 2) ⎥ , 2 g ⎣ ⎦

x ∈ ( wt / 2, t / 2]

(13)

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Figure 4. Air-gap flux density variation, referred to its peak value, FEM and proposed approximations, t/g = 37.5, ws/2g = 6.25, F = 3800 A.

Figure 5. Air-gap flux density variation, referred to its peak value, FEM and proposed approximations, t/g = 75, ws/2g = 25, F = 1000 A.

iv) An exponential approximation, considering x = 0 in a slot axis, obtained through a curve fitting procedure, given by: Bg ( x) = Bg max

1 , 1 + 10( β ws − x )

x ∈ [0, t / 2]

(14)

A comparison between the FEM analysis and different proposed approximations of the air-gap flux density referred to its peak value is given in Figs 4 and 5. In Fig. 4 the calculated values obtained in the case of t/g = 37.5, ws/2g = 6.25 and F = 3800 AT, as in Fig. 2 and in Fig. 5 in the case of t/g = 75, ws/2g = 25 and F = 1000 AT as in Fig. 3, are shown. From all the cases considered were chosen these two since there is an important difference between tooth pitch to air-gap length and respectively slot width to double air-gap length ratios, which allows for a large degree of generality of the results and conclusions. 4. Carter’s Factor Basically, the Carter’s factor is defined by (1), the average air-gap flux density being calculated accordingly to (4). Only the air-gap flux density variation on a half of a tooth pitch is considered due to the symmetry against the tooth axis. Carter’s factor can be calculated using different equations such as, [8]: kC 1

⎡ ⎤ 1 = ⎢1 − ⎥ ⎢⎣ t / ws ( 5 g / ws + 1) ⎥⎦

kC 2

⎡ 2w = ⎢1 − s tπ ⎢ ⎣

−1

2 ⎛ ws g ⎛ 1 ⎛ ws ⎞ ⎞ ⎞ ⎥⎤ ⎜ a tan( ) − ln ⎜1 + ⎜ ⎟ ⎟ ⎟ ⎜ 2 g ws ⎜⎝ 4 ⎝ g ⎠ ⎟⎠ ⎟ ⎥ ⎝ ⎠⎦

(15)

−1

(16)

I.-A. Viorel et al. / Air-Gap Magnetic Field of the Unsaturated Slotted Electric Machines

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Table 3. Carter’s factor values, t/g = 150 ws/2g KC FEM (15) (16) (17)

25

31.25

37.5

43.75

50

1.446 1.435 1.424 1.433

1.637 1.628 1.61 1.622

1.886 1.882 1.854 1.87

2.225 2.231 2.187 2.209

2.718 2.739 2.666 2.669

Table 4. Carter’s factor values, t/g = 37.5 ws/2g KC FEM (15) (16) (17)

6.25

7.813

9.375

10.938

12.5

1.359 1.313 1.311 1.338

1.515 1.461 1.455 1.49

1.714 1.652 1.639 1.685

1.978 1.904 1.88 1.94

2.341 2.25 2.207 2.293

⎡ w 4 g ⎛ π ws ⎞ ⎤ kC 3 = ⎢1 − s + ln ⎜1 + ⎟⎥ t tπ ⎝ 4 g ⎠⎦ ⎣

−1

(17)

In Table 3 and 4 the calculated values of Carter’s factor for unsaturated machine, Ksat<1.04, based on Eqs (15)–(17) in comparison with the values obtained via 2D-FEM analysis are given. As one can see, the differences are not that important, and this is valid for a large number of values of the ratios t/g and ws/2g. The adopted notations for the Carter’s factor are: KCFEM – computed via FEM, KCW(0.5) – computed for Weber approximation (11) when a = 0.5, KCH – computed for Heller approximation (12) and KCN, KCE – computed for nonsinusoidal (13) and exponential (14) approximations. In Tables 5 and 6 are also given the calculated values of the air-gap topology coefficient β(7). As it can be seen from Tables 5 and 6, all the proposed approximations, but Heller’s, give accurate values for the Carter’s factor, very closed to the FEM based calculations. In the case of Heller’s approximation the differences are quite important due to the fact that the resulting average value of the air-gap flux density is too large compared to the FEM calculations, which can be considered very close to the actual value. 5. Conclusions Five analytical approximations of the air-gap flux density variation versus circumferential coordinate are considered and the calculated values are compared with each other and with the 2D-FEM analysis results, computed on a simple machine model with slots only on the stator. The Carter’s factor for all considered approximations and for the 2D-FEM analysis are calculated too. The following remarks should be made concerning the air-gap flux density analytical estimation: i) The approximation based on the air-gap variable equivalent permeance method is very simple but less accurate.

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Table 5. Carter’s factor, t/ws = 1.5, Weber (11), Heller (12) approximations, a = 0.5 ws/2g β KCW(0.5) KCFEM KCH

50 0.49 2.66 2.72 1.69

33.3 0.48 2.61 2.63 1.68

25 0.48 2.574 2.556 1.676

20 0.47 2.53 2.49 1.66

16.67 0.46 2.46 2.43 1.65

12.5 0.46 2.41 2.34 1.64

Table 6. Carter’s factor, t/g = 150 nonsinusoidal (13) and exponential (14) approximation ws/2g

50

43.75

37.5

31.25

25

β KCN KCE KCFEM

0.49 2.699 2.885 2.720

0.488 2.209 2.303 2.225

0.486 1.87 1.923 1.886

0.484 1.622 1.655 1.637

0.48 1.433 1.442 1.446

ii) Weber’s approximation is not so simple and gives better results for larger values of the ratio between the tooth width and the slot opening. It shows an important dependency on the air-gap topology coefficient β calculated values. iii) Heller’s approximation depends on β too, is a function defined on two intervals and gives a too large air-gap flux density average value, consequently a too small Carter’s factor. iv) The nonsinusoidal (13) and exponential (14) approximations are accurate and produce values very closed to that calculated by a 2D-FEM analysis. When only the Carter’s factor has to be calculated then one can use Eqs (15)–(17) or can calculate it based on nonsinusoidal (13) or exponential (14) approximations. The exponential approximation reproduces quite well the 2D-FEM computed characteristic but its slope is a bit too large. The exponential or the nonsinusoidal approximations can be employed even when the main inductance variation has to be calculated. As an overall conclusion, it must be said that there are good analytical approximations of the air-gap flux density variation and that they can be usefully implemented in designing procedures or in on line control packages. References [1] C.A.M. Weber, F.W. Lee, “Harmonics due to slot openings”, A.I.E.E. Trans., vol. 43, 1924, pp. 687-693. [2] T. Spooner, “Tooth pulsation in rotating machines”, A.I.E.E. Trans., vol. 44, 1925, pp. 155-160. [3] F.W. Carter, “The magnetic field of the dynamo-electric machine”, The Jour. of the I.E.E., vol. 64, 1926, pp. 1115-1138. [4] R.W. Wiesseman, “Graphical determination of magnetic fields”, A.I.E.E. Trans., vol. 46, 1927, pp. 141-148. [5] B. Heller, V. Hamata, “Harmonic field effects in induction machines”, New York, Elsevier, 1977. [6] Z.Q. Zhu, D. Howe, “Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part III: Effect of stator slotting”, IEEE Trans. on Magnetics, vol. 28, no. 1, January 1993, pp. 143-151. [7] A.B. Proca, A. Keyhani, A. El-Antably, W. Lu, M. Dai, “Analytical model for permanent magnet motors with surface mounted magnets”, IEEE Trans. on Energy Conversion, vol.18, no. 3, September 2003, pp. 386-391.

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[8] D.C. Hanselman, “Brushless permanent-magnet motor design”, second edition, Writes’ Collective Cranston, Rhode Island, SUA, 2003. [9] S.A. Hossain, I. Husain, “A geometry based simplified analytical model of switched reluctance machines for real-time controller implementation”, IEEE Trans. on Power Electronics, vol.18, no. 6, pp. 1384-1389, Nov. 2003. [10] H.-P. Chi, R.-L. Lin, J.-F. Chen, “Simplified flux linkage model for switched reluctance motors”, IEE Proc. – Electr. Power Appl., vol. 152, no. 3, pp. 577-583, 2005. [11] J.-H. Chang, D.-H. Kang, I.-A. Viorel, Larisa Strete, “Transverse flux reluctance linaer motor’s analytical model based on finite element method analysis results”, IEEE Trans. on Magnetics, vol. 43, no. 4, April 2007, pp. 1201-1204. [12] I.-A. Viorel, G. Henneberger, R. Blissenbach, L. Löwenstein, “Transverse flux machines. Their behaviour, design, control and applications”, Mediamira, 2003, Cluj-Napoca, Romania. [13] G. Qishan, Z. Jimin, “Saturated permeance of identically double-slotted magnetic structures”, IEE Proc.-B, vol. 140, no. 5, pp. 323-328, 1993. [14] G. Qishan, E. Andersen, G. Chun, “Airgap permeance of vernier-type, doubly-slotted magnetic structures”, IEE Proc.-B, vol. 135, no. 1, pp.17-21, 1988. [15] I.-A. Viorel, M.M. Radulescu, “On the calculation of the variable equivalent air-gap permeance of induction motors” (in Romanian), EEA-Electrotehnica, vol. 32, no. 3, 1984, pp. 108-111. [16] I.-A. Viorel, A. Forrai, R.C. Ciorba, H.C. Hedesiu, “Switched reluctance motor performance prediction”, Proc. of IEE-IEMDC, Milwaukee, USA, 1997, TBI-4.1-4.3.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-268

Dynamic Simulation of the Transverse Flux Reluctance Linear Motor for Drive Systems Ioan-Adrian VIOREL a, Larisa STRETE a and Do-Hyun KANG b Electrical Machines Department, Technical University of Cluj-Napoca, Daicoviciu 15, 400020, Romania [email protected], [email protected] b Korean Electro-technology Research Institute, Changwon 641-120, South Korea [email protected] a

Abstract. Three different models of transverse flux reluctance linear motor (TFRLM) are considered – a simple one which neglects the saturation, one based on the look-up table technique and an analytical model; the last two fully considering the nonlinearities and based on finite element method (FEM) analysis. The models are compared concerning the TFRLM dynamic regime when the motor is fully controlled in a specific drive system.

1. Introduction The paper’s goal is to simulate the dynamic regime of the Transverse Flux Reluctance Linear Motor (TFRLM) when the motor is fully controlled in a specific drive system. Three variants of models are considered – a simple linear one, neglecting the saturation, a look-up table technique based on FEM results and an analytical model which fully considers the nonlinearities and is also based on FEM results. The machine analyzed is a basic transverse flux reluctance linear motor (TFRLM). Some of the advantages of linear motors come from the fact that they deliver the required linear motion directly, without the need of an intermediate system to transform rotation in translation. TFRLM has a double salient structure and a passive mover. It has the same number of poles on both parts on the stator and on the mover, and each of TFRLM’s phase is constructed as an independent module, Fig. 1, [1,2]. Basically the transverse flux reluctance motor (TFRM), in a rotating or linear construction, is similar to the switched reluctance motor (SRM) except for the armature winding which is of ring type with homopolar features. The force/torque in both cases, TFRM or SRM is produced by the tendency of the movable part to reach a position where the inductance or the flux linkage of the stator phase winding is maximized. Due to this similitude, equivalence between these two machines can be defined [3]. The transverse flux machine (TFM) is capable to deliver high power densities due to its homopolar topology as it allows an increased number of poles without reducing the MMF per pole [1]. Any attempt to describe, or to predict TFRLM dynamic behavior relays on its mathematical model, which, as in the case of SRM, is not a simple one since both the

I.-A. Viorel et al. / Dynamic Simulation of the TFRLM for Drive Systems

269

Figure 1. TFRLM topology.

stator and the mover have salient poles and the iron core should be quite highly saturated in order to obtain good performances. The TFRLM mathematical models, or the SRM ones which are basically similar, presented in the literature start from a simplified idealized one [4] and continue with different models, mostly based on numerical field calculation. Some of the models [5,6] use directly the 2D-FEM analysis results, others only build analytical models based on 2D-FEM obtained values [7–10]. In this paper three, quite representative models are considered: i) An analytical simplified one which does not consider the saturation effect, which lays on the idea formulated in [4]. ii) An analytical one based on 3D-FEM analysis obtained values which combines, in an original way, the models presented in [7–9]. iii) A model employing the look-up table technique, the values being obtained directly via a 3D-FEM analysis as in [5]. After a brief presentation of the sample TFRLM considered in the paper, each model will be described and representative comparative results will be shown. Some conclusions concerning the accuracy and versatility of the models used to study TFRLM dynamic behavior will end up the paper.

2. TFRLM Construction TFRLM has a double-salient topology and a passive mover, with the same number of poles on both parts, Fig. 1. The flux is generated by the current flowing through the coils and the thrust is based on the minimum reluctance principle; the movable part is acting to achieve a position where the inductance and the flux linkage of the stator excited winding are maximized. The moving direction is perpendicular to the magnetic flux. The transverse flux reluctance linear motor is simple, robust and low cost. Due to the latest improvements of its construction, design and control, TFRLM has become a valuable option for variable speed linear drives. A sample TFRLM was designed to deliver a 1000[N] thrust. The rated current is 50[A] and the number of coil turns per phase is N = 80. The motor is a long one with

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I.-A. Viorel et al. / Dynamic Simulation of the TFRLM for Drive Systems

four phases and 30 pole pairs per phase. The air-gap g = 1 mm and the pole length on y direction is wp = 20 mm. The stator and the mover core are made from the same material, S23, which has the initial relative permeability µri = 2023 and a saturation flux density value of Bs = 2.1 T.

3. Mathematical Models The machine behavior is fully described by the following equations: V ph = R ⋅ i ph +

∂λ dx ∂λ di ph dλ = R ⋅ i ph + + ∂x dt ∂i dt dt

λ = f (i, x)

(1) (2)

i

∂λ di 0 ∂x

F=∫

F =m

d2x dx + cf 2 dt dt

(3)

(4)

where Vph, iph, λ, R, F are the phase voltage, current, flux linkage, resistance and produced thrust and m, cf are the mover mass and friction coefficient. It means that the motor behavior can be obtained by solving (1)–(4) if the parameters (resistance and inductance) can be calculated or measured and if the flux linkage dependence on the current and mover position can be defined analytically or based on FEM analysis, respectively on tests. The TFRLM dynamic behavior, studied by using MATLAB/Simulink environment, is described by: ∂λ = V ph − R ⋅ i ph ∂t d2x 1 ⎛ dx ⎞ = ⎜ F − cf ⎟ dt 2 m ⎝ dt ⎠

(5)

the flux linkage and the thrust being given by (2), (3) once f (i, x) is defined. In the following, the way f (i,x) is obtained in the three cases considered is detailed. 3.1. Linear Model In the linear model the flux leakage is neglected and the flux linkage is considered like varying linearly with the current and mover position. It simply means that: λ=N

μ ⋅ wp F = N2 x ⋅ i ph ℜm 2g

(6)

I.-A. Viorel et al. / Dynamic Simulation of the TFRLM for Drive Systems

Figure 2. Aligned and unaligned flux linkage variation versus phase current for TFRLM.

271

Figure 3. Flux linkage vs. displacement.

where the magnetic reluctance ℜm is: ℜm =

2g , μ ⋅ Ap

Ap = w p ⋅ x

(7)

the core reluctance being neglected against that of the air-gap since the core relative permeability is considered infinite; in fact it is at most greater than 10 3, but smaller than 104. 3.2. Nonlinear Model Based on 3D FEM Analysis TFRLM flux linkage depends on the current and on the mover position and has its maximum value in aligned position and its minimum value in unaligned position. A typical TFRLM phase flux linkage variation function of phase current for aligned and unaligned mover position is shown in Fig. 3. In Fig. 2 the following notations are made: λ0al, λ0un – aligned, unaligned unsaturated flux linkage (λ0al = L0al . Iph; λ0un = L0un . Iph) λal, λun – aligned, unaligned saturated flux linkage L0al, L0un – aligned, unaligned unsaturated phase main inductance. The unsaturated flux linkage in aligned λ0al and in arbitrary mover position λ0 is: λ0 al = L0 al ⋅ i,

λ0 = L0 ⋅ i

(8)

where L0 is unsaturated value of the phase inductance in an arbitrary mover position. The saturated flux linkage in the same positions is: λal = λ0al/ksal,

λ = λ0/ks

(9)

and, by combining (8) and (9) results the saturated flux linkage in an arbitrary mover position:

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I.-A. Viorel et al. / Dynamic Simulation of the TFRLM for Drive Systems

λ = λal ⋅

l0r ksr

(10)

where the ratios:

l0 r =

L0 , L0 al

ksr =

ks ksal

(11)

should be calculated by using aligned, average and unaligned flux linkage characteristics obtained via 3D-FEM analysis [11]. Both function l0r and ksr should contain a cosinusoidal function dependent on the mover displacement since the thrust produced by a phase must be zero in aligned, respectively unaligned position [11]. For the sample motor considered, in Fig. 3 are given in comparison the variation of the phase flux linkage against displacement for different currents, values calculated via 3D-FEM respectively analytical model. In the case of the considered sample TFRLM the analytical model proposed gives for the flux-linkage the following expression: λ (i, α ) = ( 0.19925 ⋅ cos(α ) + 0.80075 ) ⋅

1 al (i ) + bl (i ) cos(α )

i ⋅ 0.03675 ⋅ i + 24.58 ⋅ i + 431.1 2

(12)

al (i ) = 3.361 ⋅ e −0.1385⋅i + 0.8566 ⋅ e −0.0004712⋅i bl (i ) = 0.1457 ⋅ e0.002302⋅i − 3.293 ⋅ e −0.1375⋅i α = π ⋅ x / τ , where τ is the pole pitch.

3.3. Look up Tables Based on 3D FEM Analysis In Figs 4 and 5 the developed thrust and phase flux linkage calculated via 3D FEM versus mover displacement for different values of phase mmf are presented:

Figure 4. Thrust vs. displacement at different current, 3D FEM.

Figure 5. Flux linkage vs. displacement at different current, 3D FEM.

I.-A. Viorel et al. / Dynamic Simulation of the TFRLM for Drive Systems

273

Figure 6. Extended thrust force vs. displacement.

Figure 7. Extended flux linkage vs. displacement.

Figure 8. Phase inductance versus mover position, constant current.

Figure 9. Phase developed thrust versus position, constant current.

From these data, the extended plots covering two pole pitches and the entire domain for phase ampere turns were obtained, as shown in Figs 6 and 7. These extended plots will be introduced in MATLAB/Simulink and TFRLM dynamic characteristics will be calculated based on the look up table technique which are widely used nowadays and there is no reason to extend the explanations here.

4. Computed Results From all simulations performed in MATLAB/Simulink environment some are shown in the following, two distinct regimes are presented: i) A steady state regime when the phase is supplied with constant current, the rated value of 50 A. ii) A dynamic regime when all the phases are supplied and the mover speed is imposed at 2 m/s, the phase current being controlled function of the speed error. The steady state regime characteristics in Figs 8 and 9 show a good agreement between the obtained and the expected results. In Fig. 8 the phase inductance variation

274

I.-A. Viorel et al. / Dynamic Simulation of the TFRLM for Drive Systems

Figure 10. Phase current versus mover position, imposed speed.

Figure 11. Phase flux linkage versus mover position, imposed speed.

Figure 12. Phase developed thrust versus mover position, imposed speed.

Figure 13. Resulting four phase thrust, controlled current at constant speed.

versus mover position is given when the analytical model and the look-up table technique are employed. In the case of the linear model the inductance is varying linearly with the displacement x and therefore it is not shown. In Fig. 9 the phase developed thrust is presented and, as expected, the values calculated via analytical model and look-up table technique are quite the same, both algorithms starting from 3D-FEM analysis. In the case of the linear model the phase developed thrust is constant and has larger values. In Figs 10, 11 and 12 the variation of the phase current, phase flux linkage and phase developed thrust versus mover position are shown in comparison for the three considered models in the case of a dynamic regime at constant speed and controlled current. A dynamic regime of entire four phases motor, at the same constant speed as before was performed and the values obtained for the resulting thrust variation function of mover position, all three models, are presented in Fig. 13. It is clear that the look-up table technique and the analytical model lead to quite the same results, much different from that obtained via linear model.

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275

5. Conclusions Three mathematical models for a TFRLM were considered: a linear one, a look-up table technique based on 3D-FEM calculated values and an analytical model based also on 3D-FEM obtained values. The models are briefly presented and TFRLM steady state, respectively dynamic regime was simulated via MATLAB/Simulink environment. Some results are shown and commented. Three are the most important conclusions that occur from the study: i) The linear model, even if it is very simple, is far of being accurate and it does not offer pertinent information ii) The proposed analytical model is fully validated by the fact that the values calculated by employing it are very close to that obtained via look-up table technique, a well known one iii) The proposed analytical model requires the 3D-FEM calculations only for three mover positions and consequently shorter computation time than the look-up table technique The overall conclusion is that a good analytical model, which is fully considering the nonlinearities, can be a very good solution when TFRLM dynamic behavior study has to be done.

References [1] Viorel, I.-A., Henneberger, G., Blissenbach, R., Löwenstein, L., “Transverse flux machines. Their behavior, design, control and applications”, Mediamira Publishing House, Cluj, Romania, 2003. [2] Henneberger, G., Viorel, I.-A.: “Variable reluctance electrical machines”, Shaker Verlag, Aachen, Germany, 2001. [3] Crivii, M., Viorel, I.-A., Jufer, M., Husain, I., “3D to 2D equivalence for a transverse flux reluctance machine”, Proc of ICEM’02, Brugge, Belgium, on CD-ROM:254.pdf. [4] Kang, D.H., “Transversalflussmaschinen mit permanenter Erregung als Linear Antriebe im schienengebundened Verkehr”, Dissertation, 1997, Germany. [5] Soares, E., Costa Branco, P.J., “Simulation of a 6/4 switched reluctance motor based on MATLAB/ Simulink environment”, IEEE Trans Aerosp. Electrom. Syst. vol. 37, no. 3, pp. 989-1009, 2001. [6] Viorel, I.-A., et al., “A new approach to the computation of the switched reluctance motor dynamics”, Proc of ICEM’2000, Helsinki, Finland, pp. 1605-1608. [7] Chang, J.H., Kang, D.H., Viorel, I.-A., Larisa, S., “Transverse Flux Reluctance Linear Motor’s Analytical Model Based on Finite Element Method Analysis Results”, IEEE Trans. on Magnetics, vol. 43, no. 4, April 2007, pp. 1201-1204. [8] Chi, H.-P., Lin, R.-L., Chen, J.-F., “Simplified flux linkage model for switched reluctance motors”, IEE Proc. – Electr. Powe Appl., vol. 152, no. 3, pp. 577-583, 2005. [9] Chang, J.H., Kang, D.H., Viorel, I.-A., Tomescu, Ilinca, Strete, Larisa, “Saturated double salient reluctance motors analytical model”, Proc of ICEM’06, Chaina, Greece, on CD-ROM. [10] Viorel, I.-A., et al., “Transverse flux machine mathematical model”, Rev. Roumanie Sci. Tech., Electrotechn. Energ., vol. 48, no. 2-3, pp. 369-379, 2005. [11] Viorel, I.-A., Strete, Larisa, “Switched reluctance motor analytical flux-linkage model”, (Sent for publication).

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-276

Influence of Air Gap Diameter to the Performance of Concentrated Wound Permanent Magnet Motors Pia SALMINEN, Asko PARVIAINEN, Markku NIEMELÄ and Juha PYRHÖNEN Lappeenranta University of Technology, Department of Electrical Engineering, P.O. Box 20, 53851 Lappeenranta, Finland [email protected] Abstract. The study addresses the torque production capabilities of concentrated wound permanent magnet machines in the motor frame size 225. The main target of the study is to investigate how much more torque can be achieved by increasing the air gap diameter of concentrated wound machines, using the pole pair number as a variable in the study and keeping the machine stator frame external diameter and stack length as fixed parameters. First, the torque dependence on the pole pair number with a constant air gap is studied.

Introduction Concentrated wound machines belong to the group of fractional slot wound machines; however, the number of slots per pole and per phase q ≤ 0.5. In a concentrated wound machine, each coil is wound around one tooth. The purpose of the paper is to explicate the selection principles of the pole pair number with concentrated wound permanent magnet machines. The paper provides useful information about the influence of the pole number on the performance of a motor designed to a standardized IEC motor frame, Fig. 1. The torque T is proportional to the product of the rotor surface average tangential force F and the rotor radius r (T = Fr). As the rotor radius and the length of the machine are kept constant, also the rotor surface area producing torque remains equal. The effect of the number of poles is studied. Increasing the number of the poles gives an opportunity to increase the rotor radius, which also allows employing quite a thin yoke.

Figure 1. IEC die cast iron housing for an electric motor [1].

P. Salminen et al. / Influence of Air Gap Diameter

a) 12 slots 10 poles, q = 0.4, D = 254 mm

b) 36 slots 30 poles, q = 0.4, D = 254 mm

277

c) 36 slots 42 poles, q = 0.286, D = 279 mm

Figure 2. Machine topologies of concentrated wound PM machines a) 12-slot-10-pole, b) 36-slot-30 pole, and c) 36-slot-42 pole. For the machine with 12 slots and 10 poles q = Q/2pm = 12/(2·5·3) = 0.4. For the 36/30 machine q = 0.4, too and for the 36/42 machine q = 0.286.

Different structures were designed for a machine with a rated torque of 1075 Nm, a frame size of 225 mm, and a rated speed of 400 rpm; these are values typically applied for instance in paper machines. The machines studied here have a 364-mm stator stack outer diameter as a mechanical constraint. In the study, the magnetic loading of the stator yoke and the teeth as well as the current density in the stator windings were kept constant. Also the stack length was fixed to the value of 270 mm, even though a high pole pair number also brings a possibility of increasing the stack length, while the end windings become smaller. The main concerns related to the use of high pole-pair numbers in a selected frame size are the mechanical stiffness of a thin stator yoke and the efficiency aspects of the motor associated with the increased iron losses as the pole number increases. Figure 2 shows three types of concentrated wound PM machines. The flux diagrams are obtained by the finite element analysis FEA (Cedrat Flux2D) for a) 12-slot-10-pole and b) 36-slot-30-pole machines with an air gap diameter Dδ of 254 mm, and c) 36-slot-42-pole machine with approximately 10% larger air gap diameter of 279 mm. From the magnetic saturation point of view, the stator yoke may be thinner in machines with a higher pole number, and therefore some of the high pole number machines were also computed using air gap diameters larger than 254 mm.

Torque with a Constant Air Gap The pull-out torque equation for a non-salient pole machine (if direct and quadrature inductances are approximately equal) may be written as 3 p U ph EPM Tˆ = . 2πf 2πfLd

(1)

Rewriting the frequency f = np in Eq. (1) we get 3 p U ph EPM 3 U ph EPM Tˆ = = 2 2 . 2πnp 2πnpLd 4π n pLd

(2)

278

P. Salminen et al. / Influence of Air Gap Diameter 2.5

Pull-out torque (p.u.)

2

1.5

1

0.5

0

q

0.25

0.25

0.25

0.29

0.29

0.29

0.36

0.4

0.4

0.4

0.43

0.5

0.5

0.5

p

24

12

8

21

7

14

11

5

15

10

7

4

12

6

8

Q

36

18

12

36

12

24

24

12

36

24

18

12

36

18

24

0.5

Figure 3. Pull-out torques of concentrated wound machines with different numbers of slots per pole and per phase q.

In this study, the supply phase voltage Uph, the induced phase voltage EPM and the speed n [1/s] were constants, and therefore the torque is inversely proportional to the number of pole pairs p and the synchronous inductance Ld 1 Tˆ ∝ . pLd

(3)

When the number of pole pairs p is large and the number of slots per pole and per phase q is small, the magnetizing inductance of a PM machine is small compared with the leakage inductance, and hence the leakage inductance may dominate. In such a case, we may write the synchronous inductance Ld and the pull-out torque Tˆ as [2] Ld ∝

1 pq

and

Tˆ ∝ q.

(4)

In other words, the pull-out torque is proportional to the number of slots per pole and phase [2]. For concentrated wound machines, in which the coil is wound around one tooth, the highest slots per pole and per phase number q is 0.5. In this study, this machine type gives the highest torque values. Figure 3 shows the pull-out torques computed with the FEA for several concentrated wound machines when q varies from 0.25 to 0.5. The machines have the same air gap diameter of 254 mm. Other fixed parameters are the supply phase voltage Uph, the induced phase voltage EPM and the speed n [1/s]. It can be seen that machines having q = 0.25 give the smallest pull-out torques. Higher torques are produced when q is increased. In the study, the machines with q = 0.5 give the highest pull-out torque values of 2.1 p.u. Figure 3 indicates the validity of q when concerning the torque production capability of concentrated wound machines. One may investigate the machines having 8 pole pairs (in Fig. 3): As q equals to 0.25 the maximum torque is about 1 p.u., but when q is doubled to the value of 0.5 also the torque is doubled to the value of 2 p.u. The results verify the correlation of T and q in Eq. 4.

P. Salminen et al. / Influence of Air Gap Diameter

279

Figure 4. Flux routes of a 24-slot 28-pole machine.

With high pole pair numbers, the magnet leakage fluxes are increased to a relatively high level, and thereby the inductance increases causing a reduction in the torque production capability. Some methods to estimate the leakage fluxes are provided in [6]. In some cases of concentrated wound PM machines, the slots are unfavourably aligned with the magnets, so that the flux available will decrease. Consequently, the flux generated by the permanent magnets in the rotor surface cannot be used completely, because there is no suitable route for the flux to travel to the stator iron. Therefore the magnet flux leakage can be large, which can be seen in Fig. 4 illustrating the flux routes of a 24-slot 28-pole machine. We can see that some of the flux lines do not circulate around the stator slots, and hence, they are not generating torque. For machines designed into the same frame size, the same speed and the power, it is found out that the pull-out torque is roughly proportional to the number of slots per pole and per phase q. This correlation may assist the motor designer to estimate the desired pull-out torque for concentrated winding PM motors.

Torque with a Larger Air Gap Diameter As the motor pole pair number p is increased, the magnetic loading of the stator yoke may allow to reduce the yoke thickness. Correspondingly, one may apply larger air gap diameters (note also that the length of the stator teeth may be reduced, because a larger air gap diameter yields a reduced number of coil-turns. The slots will be smaller). Thus, some high pole number machines were computed also using a larger air gap diameter. When the stator yoke was redesigned, also the slot dimensions were varied. The slot height was reduced and the teeth were designed thinner. When the air gap diameter Dδ is increased by 10%, the air gap flux Φ increases in the same proportion, and therefore the torque T may increase by 20%, as can be seen in Eq. (5), since the torque-producing radius increases and the inductances decrease. ˆ Bˆcosζ T = (π 4)Dδ2 LA

(5)

ˆ is the peak value of the fundamenIn Eq. (5), L is the length of the stator stack, A ˆ tal of the linear current density, and B is the peak value of the air gap flux density fundamental, and ζ the angular displacement between the air gap flux density and the stator linear current density.

280

P. Salminen et al. / Influence of Air Gap Diameter

Losses The study addresses the computation of losses both analytically and by applying the finite element analysis. Flux2D is employed in computations. The iron losses can be calculated in a magnetic region during the analysis. The losses, computed with the FEA, include the hysteresis losses, the Joule losses and the excess losses. In the periodic state (time stepping magnetic applications over one complete period), the iron losses are defined as T

T

1 1 PFe (t )dt = k h bˆ 2 f kf + ∫ kf ∫ T 0 T 0

1.5 ⎡ d 2 ⎛ db(t ) ⎞ 2 ⎛ db(t ) ⎞ ⎤ + k ⎢σ e⎜ ⎜ ⎟ ⎟ ⎥dt , ⎝ dt ⎠ ⎥⎦ ⎢⎣ 12 ⎝ dt ⎠

(6)

where bˆ is the maximum flux density in the element concerned, f the frequency, σ the conductivity, d the lamination sheet thickness, kh the coefficient of hysteresis loss, ke the coefficient of excess loss and kf is the filling factor. The factors depend on the steel material used. In the computations, magnetic steel M600-50 is used. The parameters in the computations for M600-50 are σ = 4⋅106 (1/ Ω m), d = 0.5 mm, kh = 152 (Ws/T2/m3), ke = 2.32 (W/Ts-1)1.5/m3 and kf = 0.98. [3] The Joule losses may be defined by PCu = mRph I n2 ,

(7)

where m is the number of phases, Rph the phase resistance and In the rated current. In permanent magnets, the Joule losses caused by the eddy currents are evaluated in the FEA according to PPM = ∫∫ ρ J 2 dV , V

(8)

where ρ is the material resistivity, V the volume and J the current density. The friction and bearing losses are estimated to be 20 W. Table 1 reports the results for motors with 28, 30 and with 42 poles, when the air gap diameter and the stator slot dimensions are varied. In Table 1, we can see that in the case of a 24-slot-28-pole machine, the pull-out torque increased only by 5% when increasing the air gap diameter by 10%. In the case of 36-slot-42-pole and 36-slot-30-pole machines, the air gap diameter was 10% larger, which increases the pull-out torque by 20%. When the air gap Dδ is increased by 10%, the air gap flux increases and therefore the torque may increase by 20%. The torque is proportional to Dδ2. From the results shown in Table 1 and in Fig. 5 one may see that the losses are lower – leading to higher efficiency – with the machines having larger air gap diameters. Figure 5 shows how the losses are divided into Joule losses, iron losses and eddy current losses caused by the permanent magnets. It can be seen in Fig. 5 that with a larger air gap diameter the Joule losses were somewhat higher and the iron losses clearly smaller than when using the smaller air gap of 254 mm.

281

P. Salminen et al. / Influence of Air Gap Diameter

Table 1. Computation results with different air gap diameters Slots, Pole pair Q number, p

q

Air gap Air gap Rated diameter diameter current (mm) increase (A) (%)

Iron losses (W)

Freq. (Hz)

Pull-out Efficiency Pull-out torque at rated torque (p.u.) load increase (%)

24

28

0.286

254

0

86

540

93

1.29

0.93

24

28

0.286

274

+10%

87

420

93

1.33

0.94

36

30

0.4

254

0

92

720

100

1.58

0.93

36

30

0.4

274

+10%

94

560

100

1.86

0.93

36

42

0.286

254

0

92

1100

140

1.0

0.92

36

42

0.286

279

+10%

96

720

140

1.2

0.92

4000

Losses (W)

3000

+20%

+20%

36-slot-42-pole

36-slot-30-pole

3500

+5%

24-slot-28-pole

2500 2000 1500

Eddy losses (W) Iron losses (W) Joule losses (W)

1000 500 0 254 mm

274 mm

254 mm

274 mm

254 mm

279 mm

Air gap diameter (mm)

Figure 5. Joule losses, iron losses and eddy current losses caused by the permanent magnets.

Mechanical Analysis The reduction in the stator yoke thickness may influence the mechanical rigidity of the stator yoke. To demonstrate the mechanical behaviour, the natural frequencies of different designs were compared by applying analytical methods according to [4,5] as well as an FE analysis based on the Autodesk Inventor Professional™ finite element module. Owing to the space limitations, the analytical equations to compute the natural frequencies are not reported here. In the analytical approach, only the vibration mode (2,0) is of interest. Thus, the teeth and windings do not notably contribute to the stiffness of the stator. With the vibration mode (2,0), the influence on the rotary inertia of the stator core is also insignificant [4,5]. Figure 6 reports the results obtained from the finite element analysis and from the analytical computations for 36 slot stators. In the analysis, the slot cross-sectional area is considered to be equivalent between the designs, i.e. the stator yoke thickness and the thicknesses of the stator teeth are variables. The reported results show that the natural frequency of the stator yoke appears to be even lower than the line frequency as the yoke thickness is reduced below 15 mm.

282

P. Salminen et al. / Influence of Air Gap Diameter

(a) 254 mm 264 mm

500

274 mm

400 300 200

30

28

26

24

22

20

18

16

14

8

6

4

0

12

100

10

Stator natural frequency [Hz]

600

Stotor yoke thickness [mm]

(b) Figure 6. (a) The lowest shape mode for the stator at the frequency of 290 Hz. The stator studied has 36 slots and an air gap diameter of 274 mm. (b) Natural frequencies of the stators when the yoke thickness is a variable (the slot area is fixed). Curves for air gap diameters of 254 mm, 264 mm and 274 mm.

This may be considered the major concern for constructions such as 36-42 in the selected frame size.

Conclusions The paper analyzes the performance improvement of concentrated wound permanent magnet machine achieved by the change of the air gap diameter. The results show that in the selected frame size, the most attractive solution is a 36-slot 30-pole machine with an air gap diameter of 254 mm. A slightly higher pull-out torque is achieved with the air gap diameter of 274 mm, and the efficiency shows to slightly improve compared with the solution with the air gap diameter of 254 mm. The analysis shows also that the selection of 36 slots and 42 poles is questionable, because the motor performance appears to weaken significantly. The efficiency and torque production capabilities of this machine are weaker compared to other investigated designs. Further, the line frequency and the stator natural mechanical frequency are very close to each other causing problems in the mechanical behaviour of the structure.

P. Salminen et al. / Influence of Air Gap Diameter

283

Generally, the results show that the stator yoke thickness in the frame 225 should be 15 mm or above in order to operate on a safe regime even though the magnetic loading of the yoke allowed to use a thinner yoke. Because of this limit, the practical improvement achieved by the change of the air gap diameter in the machine design is quite negligible in the frame size studied.

References [1] Komponenten für Elektromotoren. Kurt Maier Motor-Press GmbH. CD-ROM catalogue, 2007. [2] Salminen, P., Jokinen, T., and Pyrhönen, J., 2005. The Pull-Out Torque of Fractional-slot PM-Motors with Concentrated Winding, Electric Power Applications, IEE Proceedings. Vol. 152, Iss. 6, pp. 1440– 1444. [3] Salminen, P., 2004. Fractional slot permanent magnet synchronous motor for low speed applications, Dissertation, Acta Universitatis Lappeenrantaensis, ISBN, Lappeenranta, 151 p. [4] Yang, S.J., 1981. Low-Noise Electrical Motors. Oxford, Clarendon Press, 101 p. [5] Parviainen, A., 2005. Design of axial-flux permanent-magnet low-speed-machines – and performance comparison between radial-flux and axial-flux machines, Dissertation, Acta Universitatis Lappeenrantaensis, ISBN 952-214-029-5, Lappeenranta, 153 p. [6] Qu, R., and Lipo, T.A., 2004. Analysis and Modeling of Air-Gap and Zigzag Leakage Fluxes in a Surface-Mounted Permanent Magnet Machine. IEEE Transactions on Industry Applications. Vol. 40, No. 1, pp. 121–127.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-284

Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries V. FIREŢEANU POLITEHNICA University of Bucharest, EPM_NM Lab., Bucharest, Romania [email protected] Abstract. This paper deals with a particular design of squirrel-cage rotor slots and copper bars of high power induction motors able to ensure high values of both starting torque and breakdown torque. The innovation consists in a rotor with two different geometries of slots, respectively of bars cross-section shape. Along the rotor periphery, bars with cross-section of rectangular shape are followed by bars of stepped shape cross-section and vice-versa.

Introduction A good design of classical induction motors with respect to the electromagnetic torque must answer two contradictory requirements, respectively high value of starting torque and high value of breakdown torque. In case of squirrel-cage induction motors, these characteristics are very dependent on the rotor slot geometry [1,2]. Starting from the results presented in the reference [3] this paper studies a new configuration of the squirrel cage able to realize a good compromise between a high value of motor starting torque and a high value of breakdown torque. A three-phase squirrel-cage induction motor with rated power Pn = 500 kW, synchronous speed 750 rpm, supplied at 6 kV, 50 Hz is studied. Two shapes in Fig. 1 of the rotor bars cross-section, rectangular shape and respectively stepped shape are considered. Independently of the value of h2/a2 and h1/h2 parameters, the rotor bars have the same cross-section area.

Simulation Results Analysis Starting Torque, Breakdown Torque and Energetic Parameters In case of rectangular shape of bar cross-sections, Fig. 1a, the increase of ratio h 2/a2 between the bars height and thickness ensures the increase of per unit starting torque (Mp/Mn), Fig. 2, and the decrease of per unit breakdown torque (Mmax/Mn). The two electromagnetic torque – slip characteristics (M-s) in Fig. 3, correspond to the lower (h2/a2)min and upper (h2/a2)max values which characterize the rectangular bars cross-section geometry, Fig. 1a, considered in this study. These values correspond to an almost square bar, with a2 = 15 mm and h2 = 14.92 mm, respectively, a deep rectangular bar, with a2 = 4 mm and h2 = 55.93 mm.

V. Fire¸teanu / Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries 285

a)

b)

Figure 1. Slot geometries and different shapes of rotor bars cross-section a) rectangular shape; b) stepped shape.

Figure 2. Starting torque (Mp/Mn) and breakdown torque (Mmax/Mn) versus ratio h2/a2.

In case of stepped shape bars, Fig. 1b, the height h1 of the upper bar step is the parameter of bar cross-section geometry and h2 is the total height of the stepped bar. The upper step of the bar has the thickness, a1 = 3 mm and the lower step has square shape. The dependence of starting torque Mp on the ratio (h1/h2) in Fig. 4 relieves an optimal shape of stepped bars for the value (h1/h2)opt = 18/31.03 = 0.58. After this value, both starting torque and breakdown torque decrease.

286 V. Fire¸teanu / Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries

Figure 3. Electromagnetic torque versus slip for rectangular bars.

Figure 4. Starting torque (Mp/Mn) and breakdown torque (Mmax/Mn versus (hl/h2) for stepped shape bars.

The (M-s) characteristics in Fig. 5 correspond to the minimum value (h1/h2)min = 10/23.92 = 0.418 and to the optimal value (h1/h2)opt. The comparison of these curves with those in Fig. 3 shows the following: −

−

if the almost square shape bars, with the ratio (h2/a2)min = 14.92/15 = 0.994 are replaced with bars of stepped shape cross-section with (hl/h2)min = 10/23.92 = 0.418, the starting torque increases with 58.4% and to the breakdown torque diminishes with 41.3%; if the deep rectangular bars with (h2/a2)max = 55.92/4 = 13.98 are replaced with stepped bars of optimal cross-section, with (hl/h2)opt = 0.58, the starting torque increases with 32.6% and the breakdown torque increases also with 9.2%. Both starting torque and breakdown torque increase when changing the deep rectangular shape bars with optimal stepped shape bars.

V. Fire¸teanu / Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries 287

Figure 5. Electromagnetic torque versus slip for stepped shape bars.

Figure 6. Squirrel-cage rotor with intercalated bars.

The high value of the starting torque in Fig. 5, corresponding to the ratio (h1/h2)opt = 18/31.03 = 0.58 in case of stepped shape bars, and the high value of breakdown torque corresponding to the value (h2/a2)min = 14.92/15 in case of rectangular bars, suggest the idea of an innovative squirrel-cage, where the (h 1/h2)opt bars are intercalated with (h2/a2)min bars. It results the rotor geometry in Fig. 6, called squirrel-cage with intercalated slots of different geometries. The (M-s) characteristic for the new rotor geometry in Fig. 7 and the values of starting and breakdown electromagnetic torques, Table 1, reflect a compromise between high values of starting torque, which can be obtained with stepped shape bars – green colour in Figs 6 and 7, and high values of breakdown torque that characterise the rectangular bars of almost square shape – magneta colour in the same figures.

288 V. Fire¸teanu / Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries

Figure 7. Electromagnetic torque versus slip for intercalated rotor bars. Table 1. Motor characteristics Rotor bars geometry

Mp/Mn

Mmax/Mn

Ip/In

Pj2/Pn

Rectangular bars with (h2/a2)min Stepped bars with (h1/h2)opt Intercalated bars: rectangular bars (h2/a2)min + stepped bars(h1/h2)opt

0.545 1.423

2.958 1.597

6.101 4.187

1.164 1.278

η [%] 95.43 95.15

cosϕ 0.859 0.804

1.044

2.320

5.167

1.216

95.30

0.833

In case of the new rotor with intercalated bars the starting torque increases with 91.6% and the breakdown torque decreasees with only 21.6% with respect the rotor variant with almost square bars ((h2/a2)min). In comparison with with rotor variant with stepped bars with (h1/h2)opt, the starting torque of the new rotor variant decreases with 26.6% but the breakdown torque increases with 45.3%. As waited, the starting current, the rotor Joule losses and the motor energetic parameters – efficiency and power factor, Table 1, have values between those obtained when all rotor slots have only one of the two geometries in Fig. 6. The magnetic field lines for rated load operation of the motor with intercalated bars and the chart of current density in rotor bars at motor start-up are presented in Fig. 8, respectively Fig. 9. For rated load operation, the Joule effect generates 0.704 kW in each rectangular bar and 0.783 kW in each stepped shape bar of the new rotor variant. When the rotor contains only rectangular bars with (h2/a2)min, this power is 0.745 kW/bar and when the rotor contains only stepped bars with (h1/h2)opt , 0.749 kW/bar. For motor start-up, the Joule power is 5.85 kW in each rectangular bar and 11.09 kW in each stepped shape bar of the new rotor variant. When the rotor contains only rectangular bars with (h2/a2)min this power is 4.93 kW/bar and when the rotor contains only stepped bars with (h1/h2)opt , this power is 12.7 kW/bar.

V. Fire¸teanu / Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries 289

Figure 8. Magnetic field lines for motor rated load operation: rotor with intercalated bars, rectangular bars (h2/a2)min + stepped bars(h1/h2) opt.

Figure 9. Chart of current density in rotor bars for motor start-up: intercalated bars, rectangular bars (h2/a2)min + stepped bars(h1/h2) opt.

Time Variation and Harmonics of Electromagnetic Torque at Rated Load The time variation of electromagnetic torque for rated load operation of the new motor with intercalated bars is presented in Fig. 10, where the numerical values corresponds to the half on the motor, which was considered for the electromagnetic field computation domain. The mean electromagnetic torque of the motor, Fig. 10 a), has the value 3247.24 × 2 = 6494.5 Nm. The most important harmonics of the electromagnetic torque, Fig. 10 b), with amplitudes of around 30 Nm have the frequency 300 Hz and respectively 700 Hz.

290 V. Fire¸teanu / Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries

a)

b) Figure 10. Time variation and harmonics of the electromagnetic torque at motor rated load.

Transient Time Variation of Motor Velocity The three curves in Fig. 11 ilustrate the transient variation of velocity for motor noload start in cases: rotor with rectangular bars with (h2/a2)min, rotor with stepped bars with (h1/h2)opt and the new rotor with intercalated bars. The time of motor start in 6.30 seconds in the first case, 5.08 seconds in the second case and 5.18 seconds in the third case. The three curves in Fig. 12 show the transient variation of velocity when double of rated load is applied after no-load start of the motor in three previously defined cases. Since in the second case – stepped bars with (h1/h2)opt, the breakdown torque is not high enough, when apply a charge double with respect rated one, the motor is not able to reach a new steady state operation regime, with a lower value of the rotor speed that hapens in the first and the third case of rotor bar cross-section geometry.

V. Fire¸teanu / Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries 291

Rotor with rectangular bars with (h2/a2)min

Rotor with stepped bars with (h1/h2)opt

New rotor with intercalated bars Figure 11. Transient time variation of velocity at motor start-up.

292 V. Fire¸teanu / Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries

Rotor with rectangular bars with (h2/a2)min

Rotor with stepped bars with (h1/h2)opt

New rotor with intercalated bars Figure 12. Transient time variation of velocity at motor start-up

References [1] M. Brojboiu, Concerning the influence of the rotor bar geometry on the induction motor performances, Proc. of 5th TELSIKS’01 International Conference, Sept. 2001. [2] J.L. Kirtley Jr., Designing Squirrel Cage Rotor Slots with High Conductivity, Proc. of ICEM’04 Conference, Sept. 2004.

V. Fire¸teanu / Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries 293

[3] O.A. Turcanu, T. Tudorache, V. Fireteanu, Influence of Squirrel-Cage Bar Cross-Section Geometry on Induction Motor Performances, Proc. of SPEEDAM’06, May 2006. [4] V. Fireteanu, T. Tudorache, O.A. Turcanu, Optimal Design of Rotor Slot Geometry of Squirrel-Cage Type Induction Motors, Proc. of IEMDC’06 Conferrence, May 2007.

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Analysis and Performance of a Hybrid Excitation Single-Phase Synchronous Generator a

Nobuyuki NAOE a, Akiyuki MINAMIDE a and Kazuya TAKEMATA b Kanazawa Technical College 2-270 Hisayasu Kanazawa, Ishikawa Japan 921-8601 b Kanazawa Institute of Technology 7-1 Oogigaoka Nonoichi, Ishikawa Japan 921-8501 Corresponding author e-mail: [email protected] Abstract. This paper describes a hybrid excitation-type single-phase synchronous generator. The generator has a permanent magnet and wound fields on the same shaft. The finite-element method (FEM) is applied to analyze the no-load characteristic of the generator. The test results obtained on a 0.5-kVA prototype machine demonstrate its capability for field regulation.

1. Introduction Among the many advantages of a permanent magnet (PM) synchronous generators are their simple rotor structures and high efficiency. However, because the PM flux is constant, the air gap flux is difficult to control. In wound rotor synchronous generators, on the other hand, the air gap flux is controlled by the field current. Therefore, a synchronous generator with features of both PM and wound-rotor synchronous generators would display enhanced practicality. A hybrid excitation-type synchronous machine with both PM poles and excitation poles on the rotor has been studied [1–6]. The authors propose a hybrid excitation-type single-phase synchronous generator (HESSG) with a two-part rotor that has both PM and wound rotors, retaining a conventional stator armature winding. The generator contains the features of both PM and wound-rotor synchronous machines. The PM part is highly efficient, and the wound rotor provides field regulation and allows field-strengthening operation. In this paper, the finite element analysis of the HESSG is described, and its basic characteristics are demonstrated.

2. Structures The structure of the HESSG is shown in Fig. 1. Unlike other types of synchronous machines, the proposed HESSG has one PM and one wound field separately mounted on the same shaft. A 0.5-kVA, 60-Hz, two-pole prototype machine was developed. The field winding is wound around the rotor for easy production. In the PM part, which is 5-mm thick, Nd-Fe-B permanent magnets are embedded inside the core. The rotor is a salient pole without damper windings. The volume of the PM rotor part is within 50%

N. Naoe et al. / Analysis and Performance of an HESSG

295

ARMATURE WINDING STATOR PM

FIELD WINDING

SHAFT

ROTOR

Figure 1. Structure of the HESSG.

of the volume of the rotor core. The volume of the wound rotor part is within 50% of the volume of the rotor core. The stator of the prototype machine is similar to that of a conventional synchronous machine. The magnetic circuits of the PM and the wound rotors are independent of each other. Moreover, the flux produced by the field current does not pass through the PM because it has a large degree of reluctance.

3. Performance The finite element (FE) analysis was carried out using software Maxwell 2D provided by ANSOFT. Through FE analysis, the no-load characteristics of the generator were examined. The two-dimensional FE analysis was adopted. To determine the effectiveness of the proposed model, we used software Maxwell 2D. The magnetic circuits of PM and wound rotors are independent of each other. Moreover, the flux produced by the field current does not pass the PM because it has a large reluctance. Therefore, the EMFs of PM and wound rotors are simply added.

Figure 2 shows the cross-section configuration with the no-load PM rotor magnetic flux distribution obtained from the FE analysis. Figure 3 shows the computed EMF of the phase voltage with the PM rotor. Figure 4 shows the cross-section configuration with no-load wound rotor magnetic flux distribution obtained from FE analysis. Figure 5 shows the computed EMF of phase voltage with the wound rotor. The measured terminal voltage is compared with the calculated one as shown in Fig. 6. It can be seen that the calculated results agree with the measured results. The voltage regulation of the resistive load is shown in Fig. 7. As is evident in Fig. 7, the armature EMF can be controlled by adjusting the field current If.

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N. Naoe et al. / Analysis and Performance of an HESSG

Figure 2. Cross section and magnetic flux distribution in the PM rotor.

Figure 3. Computed EMF of the phase voltage in the PM rotor.

Figure 4. Cross section and magnetic flux distribution in the wound rotor.

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N. Naoe et al. / Analysis and Performance of an HESSG

Figure 5. Computed EMF of the phase voltage in the wound rotor.

Figure 6. Calculated and measured terminal voltages.

with If

If[A]

Vla

Vla[V]

without If

If Resistive load

Ia[A] Figure 7. Measured terminal voltage and field current waveforms.

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N. Naoe et al. / Analysis and Performance of an HESSG 10

100

Teminal Voltage

8

Field Current

60

6

40

4

20

2

0

0

-20

-2

-40

-4

-60

-6

-80

-8

-100 0.000

0.005

0.010

0.015

Field Current[A]

Terminal Voltage[V]

80

-10 0.020

Time[s]

Figure 8. Measured terminal voltage and field current.

The measured terminal voltage and the field current waveforms are shown at a resistive load in Fig. 8. The field current has confirmed double harmonic EMF for the negative-phase-sequence field of the armature reaction. The field current of double harmonic EMF can be used for the excitation of the field winding.

4. Conclution The FE analyses of the HESSG with a two-part rotor are described in this paper. A 0.5kVA prototype machine was built, and its basic characteristics were demonstrated. The FE analyses of the HESSG carried out a no-load EMF. The computed EMF clearly agrees with the measured one. The test results clearly show that the proposed HESSG has the capacity for field regulation. The field current of double harmonic EMF can be used the excitation of the field winding.

References [1] X. Luo and T.A. Lipo, “A synchronous/permanent magnet hybrid ac machine”, IEEE – International Electric Machines and Drives Conference ’99, pp. 19-21,1999. [2] J.A. Tapia, F. Leonardi and T.A. Lipo, “Consequent-Pole Permanent-Magnet Machine with Extended Field-Weakening capability”, IEEE Tran. IA Vol. 39, No. 6, pp. 1704-1709, 2003. [3] Y. Amara, J. Lucidarme, M. Gabsi, M. Lecrivain A. Ahmed and A.D. Akemakou, “A New Topology of Hybrid Synchronous Machine”, IEEE Tran. IA Vol. 37, No. 5, pp. 1273-1281, 2001. [4] N. Naoe and T. Fukami, “Trial Production of a Hybrid Excitation Type Synchnous Machine”, IEEE – International Electric Machines and Drives Conference’01, pp. 545-547, 2001. [5] K. Matsuuchi, T. Fukami, N. Naoe, R. Hanaoka, S. Takada and A. Miyamoto, “Performance Prediction of a Hybrid-Excitation Synchronou-Machine with Axially Arranged Excitation Poles and PermanetMagnet Poles”, Trans. IEE Japan, Vol. 123-D, No. 11, pp. 1345-1350, 2003. [6] N. Naoe, T. Fukami, A. Minamide and K. Takemata, “Steady-State Performance of a Hybrid Excitation single-Phase Synchronous Generator”, XVII International Conference on Electrical Machines, PSA2-28, p. 5, 2006, CD-ROM.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-299

299

Numerical Calculation of Eddy Current Losses in Permanent Magnets of BLDC Machine a

Damijan MILJAVEC a and Bogomir ZIDARIČ b University of Ljubljana, Faculty of electrical engineering, Trzaska 25, 1000 Ljubljana, Slovenia E-mail: [email protected] b TECES, Maribor, Slovenia

Abstract. The aim of the paper is the numerical calculation and analyze of eddy currents in permanent magnets (PM) of brush-less direct current (BLDC) machines. The main source of these induced eddy currents is in reluctance variation due to stator geometry. The study is carried out on outer rotor BLDC machine. The 3-D time-stepping finite-element analyze is used to clarify this phenomenon. The induced eddy-current distribution in the PM and the resulting power losses are calculated. Also, the study of this parasitic effect is curried out by 2D time-stepping finite-element analyze. The induced eddy current losses in PM due to different stator geometries are calculated and analyzed.

Introduction The main object of presented study is the outer rotor BLDC machine (Fig. 1). The permanent magnets (PM) are placed around the inner surface of rotor. Inside of rotor is laminated stator. The rare-earth permanent magnets are not laminated and are electrically good conductors. Any kind of magnetic flux density change induces the eddy currents and with them power losses. These losses are shown in supplementary heating of PM. With this, the temperature dependant working characteristic of PM (second quadrant of B-H curve) is moving toward coordinate origin point. The main study is based on induced eddy currents in PM just due to stator geometry. Stator slot openings are causing the magnetic flux density changes in permanent magnets (Fig. 1) while they rotate around the stator. In such a study, the stator windings must not be connected to outer source and further more, no currents have to flow in these windings. To achieve these working conditions the BLDC machine must work in generator mode with no-load.

3D Numerical Analyze of Eddy Current Losses in Permanent Magnets The outer diameter of studied ten-pole surface-mounted-PM BLDC machine is 160 mm. Other geometric parameters of the machine: the air-gap length is 1mm, the thickness of the PM is 5 mm, the stator diameter is 126 mm, stator slot opening is

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D. Miljavec and B. Zidariˇc / Numerical Calculation of Eddy Current Losses

Permanent magnets

Outer rotor

Stator

Figure 1. BLDC machine geometry with finite elements.

Figure 2. Magnetic flux density distribution in PM at 4000 rpm with stator in shown position.

7 mm and the stack length of stator is 40 mm. Figure 1 shows in a 3-D view analysis model. Using the 3-D time-stepping finite-element analyzes [1] all upper described working conditions were achieved. The stator ferromagnetic material was presented with its B-H curve (M600-35A). The main concern of this study was to analyze the induced eddy currents in permanent magnets so the conductivity of stator and rotor iron was set to zero. The FeNdB permanent magnets were described with their magnetic and electric characteristics (Br = 1.24 T, Hc = 900 kA/m, ρ = 0.15∗10−5 Ωm). In form of color scale the magnetic flux density distribution in PM at 4000 rpm is shown in Fig. 2. The changes of magnetic flux density in PM’s (Fig. 2) rotate with

D. Miljavec and B. Zidariˇc / Numerical Calculation of Eddy Current Losses

301

Eddy current loss in magnets (W)

Figure 3. Induced eddy current density in PM at 4000 rpm with stator in shown position.

250 200 150 100 50 0

0

1000 2000 3000 4000 5000 6000 Speed (rpm)

Figure 4. Eddy current losses in all PM calculated with the 3-D time-stepped finite-element analyzes.

4000 rpm and cause the induction of eddy currents. They are presented in Fig. 3. The presentation of these currents (Fig. 3) stands at certain moment of time. The values of induced losses mainly depend on the speed of rotation, stator geometry, PM geometry and its conductivity. The induced power losses are dissipated in all magnets and this means additional heating of magnets. The power losses for presented geometry of BLDC machine (Fig. 1) in function of speed are presented in Fig. 4. The penetration of induced eddy currents into PM presented in form of cross section slices are shown in Fig. 5. The form and depth of penetration into PM mainly de-

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D. Miljavec and B. Zidariˇc / Numerical Calculation of Eddy Current Losses

Figure 5. Penetration of induced eddy current density (axial component) into second PM (from left side in Fig. 3) at 4000 rpm.

Figure 6. Eddy current density lines in second and third PM (from left side in Fig. 3) at 4000 rpm.

pends on stator slot opening and on rotor rotational speed. Both parameters cause the magnetic flux density changes which are penetrating into PM. Figure 6 shows the eddy current density lines in second and third PM (from left side in Fig. 3) at 4000 rpm. These lines present the points in PM with equal eddy current densities. Higher density of these lines means higher value of induced eddy cur-

D. Miljavec and B. Zidariˇc / Numerical Calculation of Eddy Current Losses

303

Figure 7. Eddy current density lines in PM (magnets are axial segmented into 3 pieces) at 4000 rpm.

Figure 8. Induced eddy current density in PM (magnets are axial segmented into 4 pieces) at 4000 rpm.

rents. The vectors of eddy current density are always tangential to the presented eddy current density lines. One way to reduce the induced power losses is to axially segment the PM [3–7]. Figure 7 shows the eddy current density lines in PM when they are segmented in three levels. The result shown in Fig. 6 is valid at 4000 rpm and at presented stator position. The same situation is presented in a form of color palette, but for 4 axial segments per PM.

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D. Miljavec and B. Zidariˇc / Numerical Calculation of Eddy Current Losses

Eddy current loss in magnets (W)

120 Peddy , n = 2000 rpm Peddy , n = 4000 rpm

100 80 60 40 20 0

1

2 3 4 Number of segments per magnet

5

Figure 9. Power losses in PM depending on number of PM’s axial segments at 2000 rpm and 4000 rpm.

Permanent magnets

Rotor Slot openings

Stator

Figure 10. 2-D outer rotor BLDC machine geometry with finite elements.

The study was curried out for different number of axial PM segments. The eddy current losses depending on number of PM segments at 2000 rpm and at 4000 rpm are shown Fig. 9.

2-D Numerical Analyze of Eddy Current Losses in Permanent Magnets On the base of the 2D finite element method with time-stepped finite-element analyze [1,2], in the introduction described working conditions were achieved with BLDC machine model shown in Fig. 10. Normally, the eddy currents are not included in 2D finite element calculations. To have them taken into account, the PM were presented to FE analyze as solid conductors with their electric and magnetic properties. Using solid conductors in calculation of eddy currents the end-effect is neglected, because the 2-D FEM assumes that solid conductors are infinitely long and currents flow only in axial direction.

D. Miljavec and B. Zidariˇc / Numerical Calculation of Eddy Current Losses

305

magnet boundary

Figure 11. 3D view of induced eddy current density distribution in PM due to slot opening at 4000 rpm.

Stator slot openings: 7 mm original 6 mm 5 mm 4 mm

Eddy current loss in magnets (W)

250 200 150 100 50 0

0

1000 2000 3000 4000 5000 6000 7000 Speed (rpm)

Figure 12. Eddy current losses versus speed and stator slot opening.

The Fig. 11 shows the induced eddy current density distribution on PM magnet cross-section as consequence of magnetic flux density changes only due to slot opening at rotor speed of 4000 rpm. The eddy current losses in PM, calculated with described 2-D FEM analyze, as a function of the rotor speed and different stator slot openings are presented in Fig. 12. Good agreements between results from 2-D (Fig. 12, 7 mm slot opening) and 3-D (Fig. 4) analyses are mainly due to relatively thick rotor beck iron. This thickness allows the PM’s magnetic flux to distribute itself in the rotor beck iron in a quite the same manner, so in 2-D and 3-D analyses.

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D. Miljavec and B. Zidariˇc / Numerical Calculation of Eddy Current Losses

Conclusion The eddy current losses in permanent magnets (PM) of outer rotor BLDC machine has been analyzed both 3-D and by 2-D time stepping finite-element method. The eddy current losses in PM have been induced only by reluctance variation due to stator air gap side surface. It has been shown that the reduction of these induced losses can be achieved by axial segmentation of PM and by narrowing the stator slot openings.

References [1] FLUX2D, software for electromagnetic design from CEDRAT, 2006. [2] D. Maga, R. Hartansky, “Numerical solutions (Numericke riesenia)”, University of Defence, Brno, Czech republic, 2006, ISBN 80-7231-130-1. [3] Yacine Amara, Jiabin Wang, David Howe, “Analytical Prediction of Eddy-Current Loss in Modular Tubular Permanent-Magnet Machines”, IEEE Trans. On Energy Conversion, vol. 20, No. 4, December 2005. [4] Hiroaki Toda, Zhenping Xia, Jiabin Wang, Kais Atallah, David Howe, “Rotor Eddy-Current Loss in Permanent Magnet Brushless Machines”, IEEE Trans. On Magnetics, vol. 40, No. 4, July 2004. [5] W.N. Fu, Z.L. Liu, “Estimation of Eddy-Current Loss in Permanent Magnets of Electric Motors Using Network-Field Coupled Multislice Time-Stepping Finite Element Method”, IEEE Trans. On Magnetics, vol. 38, No. 2, March 2002. [6] D. Nedeljkovic, R. Fiser, V. Ambrozic, “Time-optimal magnetization of inductors with permanent magnet cores”, Inf. MIDEM, 2004, Vol. 34, No. 1, pp. 32-36. [7] Kinjiro Yoshida, Yasuhiro Hita, Katsumi Kesamaru, “Eddy-Current Loss Analysis in PM of SurfaceMounted-PM SM for Electric Vehicles”, IEEE Trans. On Magnetics, vol. 36, No. 4, July 2000.

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307

Analysis of High Frequency Power Transformer Windings for Leakage Inductance Calculation Mauricio Valencia FERREIRA DA LUZ a and Patrick DULAR b GRUCAD/EEL/CTC, C.P. 476, 88040-900, Florianópolis, SC, Brazil E-mail: [email protected] b University of Liège, Institut Montefiore, Sart Tilman, B28, BE-4000, Liège, Belgium E-mail: [email protected] a

Abstract. This paper deals with the analysis of high frequency power transformer windings. The current and magnetic flux densities are determined by the finite element method. The considered system is a pot core transformer, to which the axisymmetric magnetic vector potential formulation is applied. The leakage inductance obtained by the finite element method is compared with the analytical one for different types of winding. The contribution of this paper is focused on the development of the analytical equations to calculate the leakage inductance.

Introduction In the last two decades energy supply systems have undergone some major changes, although this is not always visible from outside. Power electronics plays a growing role in the grid through power flow conditioners and converters of electricity. High frequency power transformers are the main components of the modern power electronics. The investigation of high frequency power transformers for switching mode power supply application is very important for power electronic area. Winding structure of high frequency transformers is the major factor to determine the performance of the transformer. The coil winding is often the most delicate part of a finite element method (FEM) model, since it is used within a loss computation and coupled to a thermal FEM model. This part of the device has significant temperature dependence. In general, two types of coil windings are distinguished: stranded coil windings and massive or solid conductor coil windings. The fundamental distinction between both types is the occurrence of significant internal eddy currents, being a frequency determined phenomenon. Therefore, it is possible that the same coil construction is represented as stranded for relatively low frequencies and as solid for the higher frequencies. When a transformer transfers energy, large currents flow through its windings and cause Joule losses. Depending on the skin depth relative to the winding strand size, internal skin and proximity effects occur, increasing the losses. Part of the leakage flux passes through conductive parts, including tank, inducing eddy currents, and thus additional losses. The magnetic flux distribution and the currents flowing through the windings are two important characteristics to be considered for high frequency transformer design.

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Not all the magnetic flux created by primary winding of transformer follows the magnetic circuit and links the other windings. In addition to the mutual flux, which does link both of the windings, there is the leakage flux that leaks from the core and returns through the air and consequently causes imperfect coupling. Therefore, the leakage inductance is a very important factor for transformer design; it can cause overvoltage in power switch at turn-off action. It is important to make a very clear distinction between leakage field (or flux) and stray field (or flux). The leakage is formed by the flux that links one winding and does not link the other winding. It can be measured as a voltage drop at the transformer terminals. The leakage flux does not necessarily escape the transformer. Stray fields necessarily escape the transformer. The stray flux can link one or two of the windings. Stray flux can exist in the air without adding to the leakage inductance. The stray flux is not measured as a voltage drop at the terminals. It can be measured with a coil in the neighborhood of the transformer. A portion of the leakage flux can also be stray flux when it escapes the transformer boundaries.

Magnetic Vector Potential Formulation and Types of Coil Windings A bounded domain Ω of the two or three-dimensional Euclidean space is considered. Its boundary is denoted Γ. The equations characterising the magnetodynamic problem in Ω are [1]: curl h = j , b = μ h + br ,

curl e = − ∂ t b ,

div b = 0 ,

(1a-b-c)

j= σe ,

(2a-b)

where h is the magnetic field, b is the magnetic flux density, br is the permanent magnet remanent flux density, e is the electric field, j is the electric current density, including source currents js in Ωs and eddy currents in Ωc (both Ωs and Ωc are included in Ω), μ is the magnetic permeability and σ is the electric conductivity. The boundary conditions are defined on complementary parts Γh and Γe, which can be non-connected, of Γ, n×h Γ = 0 , h

n .b Γ = 0 , e

n×eΓ = 0 ,

(3a-b-c)

e

where n is the unit normal vector exterior to Ω. Furthermore, global conditions on voltages or currents in inductors can be considered [1]. The a-formulation, with a magnetic vector potential a and an electric scalar potential v, is obtained from the weak form of the Ampère equation (1a) and (2a-b) [1], i.e. (ν curl a, curl a' ) Ω − (ν b r , curl a' ) Ω + < n × h s , a' > Γh + (σ ∂ t a, a' ) Ω c + (σ grad v, a' ) Ω c − ( js , a' ) Ω s = 0,

∀a'∈ Fa (Ω),

(4)

M.V. Ferreira da Luz and P. Dular / Analysis of High Frequency Power Transformer Windings

309

where n × h s is a constraint on the magnetic field associated with boundary Γh of the domain Ω and ν = 1 / μ is the magnetic reluctivity. Fa(Ω) denotes the function space defined on Ω which contains the basis and test functions for both vector potentials a and a'. (. , .)Ω and <. , .>Γ denote a volume integral in Ω and a surface integral on Γ of products of scalar or vector fields. Using edge finite elements for a, a gauge condition associated with a tree of edges is generally applied. In FEM model, two types of coil windings are distinguished [2]: Stranded coil windings: the coils are made of strands having a radius smaller than the skin depth. It is not required to discretize every individual conductor and the entire coil cross-section is meshed as a whole. The stranded winding circuit model corrects resistance and losses by the filling factor. The losses due to the leakage flux have to be considered in the model by an external calculation. Massive or solid conductor coil windings: when the skin effect is significant, the individual conductors have to be modelled and meshed separately so eddy currents can be included. This implies that the elements cannot be larger than the skin depth at the outskirts. This is the case with thick wire coils and foil (sheet) windings. It depends on the relative sizes of the conductor and the conductor’s insulation whether this individual model is merely a set of touching conductive regions or a detailed conductor/insulation composite. All kinds of Joule and eddy current losses are automatically included in the winding model.

Analytical Prediction of Leakage Inductance The leakage inductance is a very important factor for transformer design. It can be estimated via analytical methods. For example, the leakage inductance for winding arrangements can be calculated by [3]: L1 = μ o N12

lw 1 b M2

⎞ ⎛ ∑h ⎜ + ∑ hg ⎟ , ⎟ ⎜ 3 ⎠ ⎝

(5a-b) L 2 = μ o N 22

⎞ l w 1 ⎛⎜ ∑ h + ∑ hg ⎟ , 2 ⎟ ⎜ b M ⎝ 3 ⎠

where L1 and L2 are the primary and secondary leakage inductances, respectively. N 1 is the number of primary turns, N2 is the number of secondary turns, M is the number of section interfaces, lw is the mean length, h is the height and b is the breadth of the winding. Figure 1 shows the winding arrangements for calculation of leakage inductance.

Application In order to demonstrate and validate the proposed method, we consider a pot core transformer. It is a single-phase transformer with single layer, a primary winding located on the top section of three copper wires and a secondary winding arranged at the bottom

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Figure 1. Winding arrangements for calculation of leakage inductance.

(a)

(b) f = 1 kHz

(d) f = 100 kHz

(c) f = 10 kHz

(e) f = 1 MHz

Figure 2. (a) Mesh 2D and axisymmetric studied domain. (b), (c), (d) and (e) Zoom of the magnetic flux distribution of an axisymmetric pot core transformer.

half section. Figure 2(a) shows the axisymmetric studied domain and the 2D mesh. Figures 2(b)–(e) show the magnetic flux distribution. The primary winding and the secondary winding are totally separated. At the operating frequency of 1 kHz, the magnetic flux mostly goes through the core structure and couples both windings. At operating frequencies of 10 kHz and 100 kHz, the phenomena of leakage flux happen and the majority of flux goes out from the magnetic core into the gap between the primary and secondary windings. In the last picture, at the operating frequency of 1 MHz, almost all the flux goes out of the core and only a very low flux couples the primary and secondary windings. This leakage flux generates eddy

M.V. Ferreira da Luz and P. Dular / Analysis of High Frequency Power Transformer Windings

311

Figure 3. (a) Pot core transformer with separated winding and (b) pot core transformer with interweaving windings.

Table 1. Leakage inductance of two winding structures at the frequency of 1 MHz Winding Structures Separated windings Interweaving windings

Primary Secondary Primary Secondary

Leakage inductance – analytical equation 226.504 μH 226.516 μH 3.712 μH 3.713 μH

Leakage inductance – MEF 229.684 μH 229.963 μH 4.828 μH 4.831 μH

currents flowing inside the primary winding and the top section of the secondary winding. Therefore, no magnetic flux can penetrate the secondary winding as the operating frequency further increases. The phenomenon of leakage flux inside a high frequency operated transformer can be explained by the eddy current flowing in the windings. According to these numerical simulations, the transformer can be defined not working well the operating of 10 kHz or above. Figure 3 shows an axisymmetric pot core transformer with (a) separated windings and (b) interweaving windings. The interweaving windings have a winding structure with the most number of section interfaces. If the number of section interfaces increases, then, the leakage inductance decreases. The relationship between them is that leakage inductance is inversely proportional to the square of the number of section interfaces of the winding. Each wire of primary winding is separated by secondary wire, as shown in Fig. 3. The leakage inductance of this winding structure is much less than the leakage inductance of the separated winding structure. The comparison of the leakage inductance of these transformers (Fig. 3) is shown in Table 1.

Conclusions In this paper, the leakage inductance was numerically calculated with a magnetodynamic formulation and with adapted techniques for considering stranded and massive conductors. The comparison of the results between the analytical equation and the FEM simulation was satisfactory. An axisymmetric pot core transformer with separated windings and interweaving windings was analyzed. The leakage inductance of the interweaving winding structure is very small compared with the separated winding structures. This structure can increase the operating frequency of the transformer. However,

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M.V. Ferreira da Luz and P. Dular / Analysis of High Frequency Power Transformer Windings

it is not easy to wound circular copper wires in a perfect square arrangement by hand or by machine. Therefore new structure of windings or new structure of whole transformers is urgently needed for modern power electronics.

References [1] M.V. Ferreira da Luz, P. Dular, N. Sadowski, C. Geuzaine, J.P.A. Bastos, “Analysis of a permanent magnet generator with dual formulations using periodicity conditions and moving band”, IEEE Transactions on Magnetics, Vol. 38, No. 2, pp. 961-964, 2002. [2] J.P.A. Bastos, N. Sadowski, “Electromagnetic modeling by finite elements”. Marcel Dekker, Inc, New York, USA, 2003. [3] P.L. Dowell, “Effect of eddy currents in transformer windings”, Proc. IEE, Vol. 113, No. 8, pp. 13871394, 1966.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-313

313

Influence of the Stator Slot Opening Configuration on the Performance of an Axial-Flux Induction Motor a

Asko PARVIAINEN a and Mikko VALTONEN b AXCO-Motors Oy, Laserkatu 6, 53850 Lappeenranta, Finland b Lappeenranta University of Technology, Skinnarilankatu 34, 53850 Lappeenranta, Finland [email protected], [email protected]

Abstract. Axial-flux induction machines are found to be an attractive solution in some specific industrial applications. However, the rotor construction of an axialflux induction motor is sensitive to harmonic losses caused by permenace harmonics and time harmonics in the supply current. The paper concentrates on studying the possibility of improving the machine efficiency by adjusting the stator slot opening configuration.

Introduction Axial-flux induction machines are found to be an alternative solution to radial flux machines in several industrial applications. However, because of the limited number of machines manufactured in the past, their design principles are not very well known and documented. In particular for medium speed, i.e., machines with a rated speed of 4000 min–1–10 000 min–1, solid-rotor-core axial flux machines may be considered a totally new machine topology. The target of this work is to find out whether the stator slot opening configuration has an influence on the performance of a solid-rotor-core axial flux induction motor. It was assumed that rotor losses, caused by the stator permenace harmonics, will be reduced if a totally enclosed slot opening is employed. Figure 1 shows the typical loss density distribution, caused by the stator slot openings on the rotor surface. Further, it was expected that a change in the stator leakage inductance will reduce the losses caused by the time harmonics in the frequency converter use. Based on these assumptions, three different slot opening configurations were studied by FE analysis and by measurements for the actual machines. The paper is a continuation for the work commenced by the authors in [1] and [2].

Computation Model The performance characteristics of the axial-flux solid-rotor-core induction motor were evaluated by using a two-dimensional, non-linear, time-stepping FEA, i.e., the magnetic saturation, skin effect and the movement of the rotor with respect to the stator were taken into account. The analysis was carried out by applying FLUX2D™ finite

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A. Parviainen and M. Valtonen / Influence of the Stator Slot Opening Configuration

Figure 1. Loss density distribution on the rotor surface caused by the stator slot opening.

Figure 2. Slot opening shapes.

element software. Most of the spatial harmonics in the air gap field are generated by the variation of the air gap permeance caused by the stator slot opening. In the electrically conducting ferromagnetic rotor material, the losses caused by eddy currents may be significant. The power loss in the solid-rotor-core materials can be computed as P = ∫∫∫ ρ J 2 dV V

(1)

where ρ is the resistivity of the material, J the current density, and V the volume of the rotor conducting material

Slot Opening Configurations Figure 2 illustrates the slot opening configurations analyzed in this study. The structure (a) is a conventional semi-open slot configuration, while the structure (b) depicts a conventional semi-open slot opening, but the slot opening is filled with semi-magnetic slot wedge material. The structure (c) is a totally enclosed slot opening configuration. The corresponding stator was manufactured by adjusting the punch line parameters in the stator manufacture so that there remained a narrow steel lamination bridge in the slot opening towards the air-gap.

A. Parviainen and M. Valtonen / Influence of the Stator Slot Opening Configuration

315

Figure 3. Stator with open slot.

Figure 4. Test setup.

Prototype Machines Three different equal stators were manufactured corresponding to the slot opening shapes (a)–(c). The corresponding machines are 45 kW/6000 min–1 single-sided axial flux machines. Test Setup For each stator, an input-output-based load test was performed. The stator configurations were measured using the same housing and the same rotor in order for the test setups to be equal between the designs. The machine air-gap length and the parameters of the frequency converter were also kept the same in the tests. Figure 4 provides the test configuration employed.

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A. Parviainen and M. Valtonen / Influence of the Stator Slot Opening Configuration

semi-closed slot 0,94

totally enclosed slot open slot

Efficiency

0,9

0,86

0,82

0,78

0,74

0,7 25

30

35

40

45

50

55

60

65

70

75

Torque [Nm]

Figure 5. Efficiency of the motor as a function of torque. The motor rated torque is 75 Nm.

1

150

0.9 120

0.7 0.6

90

0.5 0.4

60

0.3

cosfii measured

cosfii calculated

0.2

efficiency measured

efficiency calculated

current measured

current calculated

Current [A]

Power factor / Efficiency

0.8

30

0.1 0

0 enclosed slot

semi-closed

open slot

enclosed slot

semi-closed

open slot

Slot opening shapes

Figure 6. Effects of the slot opening shapes on the efficiency, power factor, and current at the nominal point of the motor.

Results Figures 5 and 6 provide the results of the test runs. A comparison with the computed values is also provided in Fig. 6.

A. Parviainen and M. Valtonen / Influence of the Stator Slot Opening Configuration

317

Discussion The measurement results show some unexpected behavior: according to the measurements, a totally enclosed slot opening yields a poorer performance in terms of motor efficiency than semi-closed slot opening. Also the effect of the totally enclosed slot on the motor power factor is considerable. The weakest performance, in terms of efficiency, is for the open-slot stator; this is an expected result. Considering the stator manufacturing issues, a totally closed slot causes some major problems in the winding in serial production. Therefore, the practical relevance and applicability of this configuration are questionable.

Conclusions In this paper, the influence of the slot opening shape on the performance of a solidrotor-core medium-speed axial flux induction motor is studied. The results from the computations and measurements show some interesting results. In the studied slot opening configurations, the totally enclosed slot opening shows no benefits over the semi-closed slot provided with semi-magnetic slot wedge material. When the obtained result is combined to the time-consuming winding process of the stator with a totally enclosed slot openings, this slot opening shape cannot be recommended in machines that are aimed to be manufactured for industrial purposes.

References [1] M. Valtonen, A. Parviainen and J. Pyrhönen, Determination of the Rotor Losses in an Inverter Supplied Axial-Flux Solid-Rotor Core Induction Motor by Using 2D FEM, ICEMS 2006 CD-Proceeding, 2006. [2] M. Valtonen, A. Parviainen and J. Pyrhönen, Inverter Switching Frequency Effects on the Rotor Losses of an Axial-Flux Solid-Rotor Core Induction Motor, PowerEng 2007 CD-Proceeding, 2007.

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Characteristics of Special Linear Induction Motor for LRV a

Nobuo FUJII a, Kentaro SAKATA a and Takeshi MIZUMA b Kyushu University /Dept. of Electrical and Electronic Systems Eng., Japan [email protected] b National Traffic Safety & Environment Laboratory, Japan [email protected] Abstract. The special linear motor with function of linear transformer is studied analytically by using a special integral equation method (IEM) and a finite element method (FEM). The basic configuration is a wound-secondary type of linear induction motor (LIM), and comprises the primary winding of single-phase concentrated coil and the secondary winding with two-phase distributed coil. The computed results by IEM and FEM agree well in the flux density and thrust respectively. The estimate of thrust by IEM is easy, especially for dynamic analysis. The thrust of apparatus has about 2.1 kN/m and an enough value for the propulsion of LRV.

Introduction For a future public transportation which is in harmony with environment, a new type of light rail vehicle (LRV) is hoped. The overhead-wireless and non-contact power collection for on-board power is the key to the development. The linear induction motor (LIM) will be one of the key devices [1–4]. The authors have proposed a new apparatus with functions of propulsion and noncontact power collection [5]. The apparatus changes to the transformer and the LIM respectively by only the signal of converter connected to the onboard winding. The basic configuration is a wound-secondary type of LIM. In standstill condition in the section of guide way with the ground winding, the batteries on board are charged by the non-contact power collection. For the acceleration and the deceleration of vehicle in the section with ground winding, the vehicle is driven by the LIM operation using the ground commercial power. In the section without the ground winding, the vehicle is run by the power supplied from the onboard battery source. In the paper, the characteristics of linear motor are studied analytically by using the special integral equation method (IEM) and the finite element method (FEM).

Basic Configuration and Analytical Model Figure 1 shows the basic configuration. Figure 2 shows the dimensions of model, in which the real secondary winding has the 16 poles. In the analytical model, the length of longitudinal (x-) direction is extended infinity as the periodic model of Fig. 3 is adopted. The primary winding set on the ground is the concentrated winding of single-

319

N. Fujii et al. / Characteristics of Special Linear Induction Motor for LRV

z a

Secondary (Vehicle)

−a

+

-

b

x

−b

Primary (Ground)

Linear motor

Onboard converter (Two-phase inverter operation)

Figure 1. Special wound type of linear induction motor and onboard converter for LRV.

x

Primary

-a -a -b -b a a b b

2 34

x=0

140

20 100 12

y

35 10

Secondary

30 11

z

300

Figure 2. Dimensions of analytical model.

(a) Primary coil Figure 3. Model for IEM or FEM.

(b) Secondary coil

Figure 4. Coil shape for IEM or FEM.

phase current as shown in Fig. 4(a), which is supplied from commercial power source with a fixed frequency. The secondary winding equipped on the vehicle is the doublelayer winding as shown in Fig. 4(b) and two-phase arrangement for linear motor operation, and is changed to the single-phase arrangement for the transformer operation. The secondary current and frequency are controlled by an onboard inverter. The secondary frequency is the slip-frequency for the vehicle speed with a slip. The slip power is charged to the onboard battery. The thrust for propulsion force of vehicle is obtained by the interaction between the alternative flux of primary winding and the shifting magnetic field of secondary winding. The special integral equation method (IEM) named ELF/MAGIC is used for the analysis, which has no mesh in the air region. To check the validity the FEM analysis is also used. The analytical model is shown in Figs 3 and 4. The model is for a pair of pole and the periodic method is used. In the design, one coil of the primary winding has

N. Fujii et al. / Characteristics of Special Linear Induction Motor for LRV 0.4

IEM FEM

0.2 0 -0.2 -0.4

z = 1 mm -0.1

0 x-position x(m)

x-component flux density Bx (T)

x-component flux density Bx (T)

320

0.1

0.4

IEM FEM

0.2 0 -0.2 -0.4 z = 11 mm -0.1

0 x-position x(m)

0.1

1.0 0.5

z = 1 mm

0 IEM FEM

-0.5 -1.0 -0.1

0 x-position x(m)

z-component flux density Bz (T)

z-component flux density Bz (T)

Figure 5. Distribution of x-component flux density in air gap. 1.0 0.5

z = 11 mm

0 -0.5

IEM FEM

-1.0 -0.1

0.1

0 x-position x(m)

0.1

0.20

z = 6 mm

ωt = 0, 30, 60, ..., 180°

0 -0.20 -0.1

0 x-position x(m)

(a) x-component

0.1

z-component flux density Bz (T)

x-component flux density Bx (T)

Figure 6. Distribution of z-component flux density in air gap. 1.0 0.5

ωt = 0, 30, 60, ..., 180°

z = 6 mm

0 -0.5 -1.0

-0.1

0 x-position x(m)

0.1

(b) z-component

Figure 7. Flux density distribution with time at standstill.

32 turns with rated current of 113 A, and one coil of the secondary winding is 8 turns with rated current of 110 A. Flux Density in Air Gap Figures 5 and 6 show the x- and z-component air-gap flux density respectively. The position of z = 1 mm means 1mm height from the surface of primary core, and z = 11 mm means 1mm height from the surface of secondary core. The shape of flux density distribution is varied by the position in the air gap, because the slot pitch and the phase between currents of the primary and the secondary are different respectively. The solid line represents the value computed by IEM, in which the tooth of 11 mm width is divided into two in the direction of x, and the broken line represents the value computed by FEM, in which the tooth width is divided into 11. There is little difference between two computed results. Figure 7 shows the distributions of flux density with time at middle of air gap (z = 6 mm). Although the shifting magnetic field generates by the secondary winding, the point of zero flux is present because the influence of alternative magnetic field of primary winding with single-phase current is large.

321

N. Fujii et al. / Characteristics of Special Linear Induction Motor for LRV 100

z = 1 mm

IEM FEM

50 0 -50

-0.1

0 x-position x(m)

Thrust fx (kN/m)

Thrust fx (kN/m)

100

z = 11 mm

0 -50

0.1

IEM FEM

50

-0.1

0 x-position x(m)

0.1

Figure 8. Thrust distribution.

IEM FEM 2D current sheet

20

z = 6 mm

0 -0.1

0 x-position x(m)

40 Thrust fx (kN/m)

Thrust fx (kN/m)

40

z = 6 mm

0 -20

0.1

ωt = 90, .... , 180°

20

ωt = 0, .... , 90° -0.1

0 x-position x(m)

0.1

δ = 90°

6 4

1 -180

2 0

2

IEM FEM

90

Time ω t (deg.)

180

Figure 11. Thrust (whole in model) with time.

-90

Thrust Fx(kN/m)

Thrust fx (kN/m)

Figure 9. Comparison among thrust distributions of Figure 10. Thrust distribution with time at standstill. three types of analytical methods.

0 -1

2D current sheet model IEM FEM

90

180

Phase ϕ (deg.)

-2

Figure 12. Comparison among thrust (time average) of three types of analytical methods.

Thrust at Standstill Condition Figure 8 shows the computed thrust distribution at near surface of primary member and surface of secondary member respectively. Although there are large ripple in the distributions, the computed results of IEM and FEM are agree well. Figure 9 is the thrust distribution at the middle position of air gap. The 2D current sheet model means the two-dimensional electromagnetic theoretical method with smooth cores, in which Carter’s coefficient for both cores is used to correct the real air gap. At this position, it is suitable to estimate the spatial average thrust as the ripple is small. The computed value for the 2 divided model of IEM is almost equal to that for 11 divided model of FEM. That is, the convergence of force in the IEM is very well compared to FEM. At standstill condition, the thrust distribution doesn’t move in longitudinal direction and only the magnitude varies with time, as shown in Fig. 10. The whole thrust in a pair of pole length changes with time as shown in Fig. 11. The value of current sheet model is about 13% smaller than that of IEM as shown in Fig. 12.

322 1.0 0.5

ωt = 0, 30, 60, ..., 180°

z = 6 mm

0 -0.5 -1.0

-0.1

0 x-position x(m)

1.0

z-component flux density Bz (T)

z-component flux density Bz (T)

N. Fujii et al. / Characteristics of Special Linear Induction Motor for LRV

0.5

z = 6 mm

ωt = 0, 30, 60, ..., 180°

0 -0.5 -1.0

0.1

(a) Observed on the primary member

-0.1

0 x-position x(m)

0.1

(b) Observed on the secondary member

Figure 13. z-component flux density distribution with time at moving speed of 21.6 km/h.

z = 6 mm

6

ωt = 90, .... , 180°

Thrust fx (kN/m)

Thrust fx (kN/m)

40 20 0 -20

ωt = 0, .... , 90° -0.1

0 x-position x(m)

0.1

v =0,

v =21.6km/s

4 2

0

180

360 540 Time ω 1t (deg.)

720

Figure 14. Thrust distribution with time at moving Figure 15. Comparison between thrusts at standstill speed of 21.6 km/h. and moving speed of 21.6 km/h.

Thrust in Moving Condition As the linear motor has the special winding configuration, the dynamic characteristics are studied by using IEM, which is especially convenient because there is no need the meshing in air region. Figure 13 shows the distribution of z-component flux density with time when the vehicle with secondary winding is running at speed of 21.6 km/h. As the flux by primary winding is alternative at fixed position and the flux by secondary winding with moving speed of 21.6 km/h is shifting field with 28.6 Hz, the wave of flux distribution varies at the observed position. The distribution observed from the primary member varies in waveform for the moving effect as shown in Fig. 13(a). The flux observed from the secondary moves and varies as shown in Fig. 13(b), which is much different from Fig. 7(b) and represents the effect of moving clearly. On the thrust shown in Fig. 14, the effect of moving appears. Figure 15 shows the variation with time in the whole thrust for a pair of pole length. For the moving time of ω1t = 840° the vehicle moves by the distance of 0.28 m which is the length of two poles. Although there is difference between the thrusts at 21.6 km/h and standstill, those averages are almost equal. The spatial and time average thrust is about 2.1 kN/m and an enough value for practical use.

Conclusions 1. 2.

The validity of computed flux density in the air gap and the thrust respectively is confirmed from the agreement of results of IEM and FEM analysis. The thrust of proposed apparatus has about 2.1 kN/m and an enough value for the propulsion of LRV.

N. Fujii et al. / Characteristics of Special Linear Induction Motor for LRV

3. 4.

323

The convergence of solution of IEM is well for a division of elements in a three-dimensional analysis. The estimate of thrust (force) by IEM is easier compared with the FEM, especially for dynamic analysis.

References [1] B. Yang, M. Henke, H. Grotstollen, “Pitch Analysis and Control Design for the Linear Motor of a Railway Carriage”, IEEE IAS Annual Meeting (IAS2001), pp. 2360-2365, 2001, Chicago. [2] B. Yang, H. Grotstollen, “Application, Calculation and Analysis of the Doubly Fed Longstator Linear Motor for the Wheel-on-Rail NBP Test Track”, EPE-PEMC 2002, 2002 Dubrovnik, Croatia. [3] R. Shindoh, T. Mizuma, “Evaluation of Air Suspended LIM Driven Transit System”, Proc. of 6th Int. Sympo. on Magnetic Suspension Technology, pp. 345-348, Oct. 2001, Turin, Italy. [4] N. Fujii, I. Hirata, K. Kawamura, K. Nishimura, “Investigation of High Performance Drive of Linear Motor Train for Urban Transit”, Trans. IEE of Japan, vol. 114-D, pp. 910-917, Sep. 1994 (in Japanese). [5] N. Fujii, T. Mizuma, “Device with Functions of Linear Motor and Non-contact Power Collector for Wireless Drive”, Trans. IEE of Japan, vol. 126-D, No. 8, pp. 1113-1118, Aug. 2006 (in Japanese).

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Electromagnetic Computations in the End Zone of Power Turbogenerator M. ROYTGARTS a, Yu. VARLAMOV b and А. SMIRNOV a a OJSC “Power Machines” Branch ELECTROSILA Telephone: +7 (812) 387 47 88, Fax: +7 (812) 388 18 14, E-mail: [email protected] b St.-Petersburg State Polytechnic University Telephone: +7 (812) 348 91 77, Fax: +7 (812) 388 18 14, E-mail: [email protected] Abstract. The mathematical model and computations results of electromagnetic fields, eddy currents and losses in the construction of the power turbogenerators end zone are presented. Effect of the stator and rotor size relations, geometry of the skewed core end part, shape, sizes and arrangement of stator core screens and pressure plate by the method of numerical experiment are investigated. The results of numerical modeling in designs of powerful turbogenerators are implemented. Computational and test data are compared.

1. Introduction At designing and exploitation modern high loaded turbogenerators inevitably there is a problem of limitation of heating and increase of use reliability of end zone. The heating of structural element is determined both efficiency of ventilating, and intensity of allocated losses dependent on affecting electromagnetic fields, characteristics of used materials, features of design [1,2]. Aim of this paper consists in numerical calculation and analysis of electromagnetic fields, eddy currents and losses in the most loaded end zone of non-salient pole synchronous machines in steady state mode of operation. This problem is essentially threedimensional in view of complicated geometry of a field sources – currents in stator and rotor windings, and also composite configuration of computational area including end surfaces of the stator and a rotor cores, screens, casing, end shields. The full threedimensional analysis of a field is time-consuming even at using of modern computer technology, therefore in practice the introducing of the justified assumptions is expedient.

2. The Mathematical Model Application of Fourier series for harmonic presentation of currents and fields in the direction of electric machine rotation made it possible to combine analytical approach

M. Roytgarts et al. / Electromagnetic Computations in the End Zone of Power Turbogenerator

325

Figure 1. Computation model of turbogenerator Figure 2. Computation model of turbogenerator 160 MW with air cooling. 800 MW with water cooling.

to the problem in the direction of angular coordinate with numerical analysis in the plane of rotational axis. In this case geometry of conductors, configuration and electromagnetic properties of the core, end plate and shield of the stator, rotor geometry, space-time law of variation of currents in the stator and rotor windings have been taken into account. The computation model covers the end zone and a part of active length of the stator and rotor adjacent to it (Figs 1, 2). The field sources are densities of the rotor and stator currents in the singled out sub-areas of the slot and end portions of the winding, including the linear, bent parts and heads. Two-layer type of the winding, shortening of the stator winding pitch, bending of the end portions, phase shift of currents in the bars and in the rotor winding have been taken into account. Amplitudes of current densities were determined by the formulas well-known in the electric machine theory with the only difference that the winding factors were assigned by the distribution factors and the shortening of pitch varying along the length of the end portions was automatically taken into account at computation of the field in the end zone. The analytical representation of the field sources as rotating waves as well as uniformity or periodicity of the properties of the machine construction in the direction of rotation made it possible to obtain (in the quasi-linear statement of problem) the results being also in the form of superposed rotating waves, as follows: X (r , ϕ , z , t ) = X m (r , z , ) exp[ j (ωt − vϕ + ψ x )]

(1)

where X m (r, z) – complex amplitude, r, z, ϕ – cylindrical coordinates, v – harmonic number, ψx – starting phase. Scalar magnetic potential um out of eddy currents zone is defined by the following equation: div 2 μ grad 2 u m − (v 2 / r 2 ) μ u m = div 2 μ H 0 m

(2)

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M. Roytgarts et al. / Electromagnetic Computations in the End Zone of Power Turbogenerator

where μ is the medium magnetic permeability, H0m is the current vector function amplitude, index «2» with differential operator means differentiation in the (r, z) plane. In the eddy currents zone Maxwell’s equations are directly integrated expressed through electric field strength E: rot

1 rotE + jωγE = 0 μ

rotE =

(3)

jv [ E × eϕ ] + rot 2 E r

(4)

where ω – circular frequency, γ – medium conductivity, eφ– unit vector in the direction of rotation. For subdomain adjoint boundaries classic boundary conditions of continuity of magnetic intensity tangent components and normal component of magnetic flux density as well as non-linear surface impedance boundary conditions, which simplify significantly the solution with sharply expressed surface effect, have been applied [3,4]. Z ⎛⎜ ∂ 2um ⎞⎟ Z ⎛⎜ ∂ 2um v 2 ⎞⎟ ∂um = − + 2 um = ⎟ ∂n − jωμ 0 ⎜⎝ ∂n 2 ⎟⎠ − jωμ 0 ⎜⎝ ∂τ 2 r ⎠

(5)

where Z is a wave impedance of conducting ferromagnetic body thickness and curvature radius of which are much larger than electromagnetic field penetration depth, n – normal to the boundary surface, directed inside of conducting medium. With the help of the impedance boundary conditions the geometrical dimensions, shape and electromagnetic properties of the turbogenerator ferromagnetic housings and end shields were taken into account [5,6]. For simulation of non-uniform magnetic properties of the stator core along and across the rolled steel, for account of insulation intervals between the stacked core laminations as well as for account of magnetic properties of the slot and tooth zone, the equivalent anisotropic magnetic permeability has been used. The averaging was made under condition of keeping the magnetic flux. When non-linear case was considered, for each of iteration the average magnitude of the magnetic permeability tensor has been re-computed in each point of the computation area. The average value of magnetic permeability tensor components are obtained as μ re = 1 + kϕ (μ r − 1)

[

(

)

]

μ ϕe = 1 + k ϕ k z / μ ϕ / μ ϕ − 1 − k ϕ , μ ze = 1 + kϕ k z / [μ z / (μ z − 1) − k z ]

where kφ and kz are iron factors in direction of rotation and axial coordinate.

(6)

M. Roytgarts et al. / Electromagnetic Computations in the End Zone of Power Turbogenerator

327

Table 1. Losses in the end zone of loaded turbogenerator Power factor 0.85 lag 1.0 0.95 lead

Screen end 1 1.78 2.48

Whole screen 1 1.37 1.83

Plate end 1 1.32 1.74

Whole plate 1 1.12 1.38

End shield 1 0.58 0.4

Housing 1 0.6 0.44

3. Results of Calculations The computations of the end zone for air-, water- and hydrogen-cooled turbogenerators 63 to 1000 MW have been done according to the programs developed. The scalar potential and three components of the function of current in the area free of eddy currents as well as three components of field strength, eddy currents and losses in the nonmagnetic end plate, electromagnetic shield, fan screen, housing and end shields have been determined. 3.1. Load Changing Using developed mathematical model the computations of the turbogenerator end zones at no-load, sustained short-circuit as well as resistive-inductive and resistivecapacity load were made. Phase shift of magnetic excitation field and armature reaction field vectors is changed with load. Losses in the end zone are changed correspondingly. The results of calculations (p.u.) of additional losses in the end zone structural components of the air cooled turbogenerator at rated resistive-inductive, resistive and resistive-capacity load according to the capability curve are given in Table 1. Essential increase of losses in the resistive-capacity load mode confirms that numerical analysis of the end zone is required for successful design of powerful highutilized turbogenerators. 3.2. Phase Shift Between Current Layers of Stator Winding In two-layer windings, one side of coil lays in the upper layer, another – in the lower. Due to the shortening of pitch lower side of coils are shifted respectively upper ones, phases of current load rotating waves are shifted correspondingly. If coils begin in the upper layer, current wave of lower layer remains behind, if coils begin in lower layer, current wave of upper layer remains behind. Due to the presence of steel tooth cores with high magnetic permeability in the machine active length, for machine operation it doesn’t practically matter which coil side is a right or which is left one. In the end zone of the electrical machine the value of additional losses is determined by the total magnetic field essentially depending from the current layer being the next to the design component. As one can see from vector diagram and calculation results (Figs 3, 4), having air gap induction constant, losses in the stator end zone reduce by 10% when stator winding lower layer wave is lagging, losses in the fan shield reduces when stator winding lower layer wave is leading.

328

M. Roytgarts et al. / Electromagnetic Computations in the End Zone of Power Turbogenerator

A. Stator winding lower layer wave is leading.

B. Stator winding lower layer wave is legging.

Figure 3. Magnetic field strength for turbogenerator 160 MW at rated load.

M. Roytgarts et al. / Electromagnetic Computations in the End Zone of Power Turbogenerator AA ZZ BB XX CC YY A ZZ BB XX CC YY A

A ZZ BB XX CC YY A AA ZZ BB XX CC YY

329

E Eδ Ia

A

X Direction of rotation

A

X

Iδ

Ia

If

Direction of rotation

Ia

1

Ia

2

(1-β)π

Figure 4. Winding design and current vector diagram.

Figure 5. Radial component of induction on horisontal part of end plate. Turbogenerator 800 MW at rated load. o – experimental results.

4. Test Results The numerical computation results were compared with the experimental data. When conducting the experimental tests using JSC Electrosila test rig the end zone of turbogenerators was equipped with three-component induction measuring coils, Rogovsky’s sensors and temperature detectors. Indications of the temperature detectors depend, to a considerable degree, on cooling rate in the area of the sensors installed while the induction measuring coils and Rogovsky’s sensors make it possible to measure directly the field strength, eddy currents and to determine the density of power losses released. Figures 5, 6 present the numerical calculations results and experimental values of magnetic field induction on the cylindrical surface of the end plate with water cooling tubes for 800 MW completely water cooled turbogenerator.

330

M. Roytgarts et al. / Electromagnetic Computations in the End Zone of Power Turbogenerator

Figure 6. Axial component of induction on horisontal part of end plate. Turbogenerator 800 MW at no-load conditions. Î – experimental results.

5. Conclusions The developed technique of calculation provides the analysis of an electromagnetic condition of the end zone of turbogenerators in all operational conditions, allows selecting of electromagnetic loads, efficiency of cooling, shape, sizes and placement of structural components, including stator and rotor cores with windings, screen, pressure plate, housing and end shields. Mathematical modeling of powerful turbogenerators has shown essential increase of losses in stator end zone when resistive-inductive load is changing to resistivecapacity one. Selection of the housing and end shields material, geometry and electromagnetic loading should be done taking into account the losses released and cooling system efficiency. On the basis of on numerical calculations and study of winding design and vector diagram it is shown that when the beginnings of coils are laid in the upper winding layer, losses in the stator end zone are reduced. Numerical experiment has shown essential increase of external magnetic leakage fields and concerned additional losses in saturated turbogenerators has been established when using anisotropic steel with radial direction of rolling in stator core. The comparison of the numerical and analytical computations for the test problems as well as numerical computations and results obtained during the experimental studies of electromagnetic fields in the end zone of powerful turbogenerators shows the correctness of obtained assumptions. Now technique has improved by the consideration of inhomogeneous properties of a design in direction of rotation. Eddy currents and losses in the end part of stator core are determined for real geometry of steel segments.

M. Roytgarts et al. / Electromagnetic Computations in the End Zone of Power Turbogenerator

331

References [1] Sameh R. Salem and Menoj R. Shah. Electromagnetic design practices of turbo-generator end region, International Conference on Electrical Machines, ICEM 2002, 25–28 August, Brugge-Belgium, 2002. Conference record. [2] R.D. Stancheva and I.I. Iatcheva. Numerical determination of operating chart of large turbine generator, International Conference on Electrical Machines, ICEM’2002, 25–28 August, Brugge-Belgium. Conference record. [3] M. Roytgarts, V. Chechurin. Electromagnetic field computation at the strongly expressed skin effect. – In book: Methods and means of boundary problem solution, L.: LPI, 1981, pages 68-76. [4] V. Chechurin, Yu. Varlamov, M. Roytgarts. Surface impedance for electromagnetic field computing in large turbogenerators. International Conference on Electrical Machines, Helsinki, 28–30 Aug. 2000, ICEM 2000 Proceedings Vol. 2, pp. 1035-1037. [5] V. Chechurin, I. Kadi-Ogly, M. Roytgarts, Yu. Varlanov. Computation of Electromagnetic Field in the End Zone of Loaded Turbogenerator. IEMDC 2003. Proceedings of the International on Electrical Machines and Drives Conference, Madison, Wisconsin, USA, June 1–4, 2003. [6] Yu. Varlamov, M. Roytgarts, V. Chechurin. Numerical electromagnetic analysis of the end zone of power turbogenerators. In book: Problems of creation and exploitation of new types of power equipment. Russian Academy of Sciences, Department of electrical power problems, Issue 6, St.-Petersburg, 2004, pp. 60-78 (In Russian).

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Chapter C. Applications C2. Actuators and Special Devices

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-335

335

The Impact of Magnetic Circuit Saturation on Properties of Specially Designed Induction Motor for Polymerization Reactor Andrzej POPENDA and Andrzej RUSEK Technical University of Czestochowa, Al. Armii Krajowej 17, 42-200 Czestochowa, PL [email protected] Abstract. The impact of magnetic circuit saturation on properties of specially designed induction motor driving the mixer of polymerization reactor is studied in the paper. The frequency characteristic of Butterworth’s low-pass filter was proposed to represent a nonlinear magnetization function in mathematical model of induction machine. Digital simulation of polymerization reactor drive has been made on the basis of presented mathematical model of specially designed induction motor and the examples of transient responses and trajectories are shown.

Description of Driving Unit The polymerization reactors play an important role in production of polyethylene. The drives for polymerization reactors work under extraordinary conditions because of necessity of keeping a constant temperature in reactor chamber and ethylene atmosphere and working pressure of 2800 × 105 Pa [1]. A driving motor adapted for vertical work has specific dimensions because of socket fixing in upper part of reactor chamber. Supply systems are often damaged due to exceptional working conditions, including feed of the motor via specially designed electrodes. A non-typical driving set is characterized by the following extremely difficult operating conditions: (1) location of 55 kWinduction motor in closed tubular seat with diameter of 302 millimeters and total length of 919 millimeters, (2) impossibility of application of both ventilation in the motor and standard power supply due to location of working motor directly in reactor chamber with the pressure of 2800 × 105 Pa, (3) vertical single-point suspension of working motor together with the mixer; upper bearing aligns the rotor in stator, (4) the work of vertically suspended motor with application of self-aligning non-lubricate slide bearing containing the large-size rings made of sintered carbides, etc.

The extreme working conditions of motors working in polymerization reactor chambers resulted in the necessity of developing the new prototypes of specially designed induction motors being more resistant.

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A. Popenda and A. Rusek / The Impact of Magnetic Circuit Saturation

Figure 1. The cross-sections of rotor and stator slots.

Parameters of Specially Designed Induction Motor The geometrical dimensions of rotor and stator slots of specially designed induction motor made as a prototype in the frame of purposeful grant no. Nr 6 T10 2003C/06105 are given in Fig. 1 [1]. This figure depicts cross-sections of both slots. Additional calculations made on the basis of considered slot dimensions and measurements of both a non-loaded motor and motor with blocked rotor allow estimating the substitute diagram parameters of induction motor as well as the operating parameters of induction motor [1]. The following values of magnetization reactance related to unsaturated magnetic circuit and magnetization reactance at nominal voltage supplying the motor were assumed for calculations: 10.7 Ω and 5.34 Ω. The one-sided displacement of current from squirrel-cage rotor bars occurs during starting the motor. This phenomenon is known as a skin effect [2]. The depth σ of current penetration into rotor bar is measured from external end of slot and determines the working surface of bar cross-section for passage of current at given f2 frequency of current in rotor bars. Therefore, the rotor resistance depends on σ. The skin effect disappears during the work of motor with nominal rotational speed.

Approximation of Magnetization Curve for Induction Motor The frequency characteristic of Butterworth’s low-pass filter was proposed in order to represent a nonlinear magnetization function [3]. In dependence describing Butterworth’s low-pass filter the longitudinal flux ψ in main magnetic circuit of motor (main flux) instead of frequency is taken as an argument. A few modifications of Butterworth’s polynomial allow obtaining the optional gradient of non-loaded motor curve in a range of main flux proportionality to magnetizing current im: k ( ψ) =

Lm b = , q Lmn a (ψ ψn ) + 1

ψ = Lm im ,

ψ n = Lmn imn ,

(1)

where: Lm is magnetization inductance relating to main magnetic circuit of motor, Lmn is magnetization inductance at nominal voltage supplying the motor, ψ is longitudinal flux in main magnetic circuit of motor, ψn is longitudinal flux at nominal voltage supplying the motor. The following values of coefficients in (1) derived from estimated parameters of motor substitute diagram: a = 3, b = 2, q = 4.

A. Popenda and A. Rusek / The Impact of Magnetic Circuit Saturation

337

Figure 2. Approximation of magnetization curve and corresponding to that a non-loaded motor curve.

Figure 3. The schematic diagram of supply system for drive of mixer in polymerization chamber.

Assuming that ratio of any longitudinal flux and nominal longitudinal flux is approximately equal to ratio of respective voltages the following dependence may be derived: im 1 ψ ψ = = imn k ( ψ) ψ N bψ N

⎛ ψ a ⎜⎜⎜ ⎝⎜ ψ

q

⎞⎟ U ⎟⎟⎟ + 1 ≈ bU N ⎠ N

⎛U a ⎜⎜⎜ ⎝⎜U

q

⎞⎟ ⎟⎟⎟ + 1 . N ⎠

(2)

The proposed approximation and corresponding to that a non-loaded motor curve are depicted in Fig. 2. The linearization of non-loaded motor curve through the nominal saturation point is also shown. Despite the fact that there are some differences between approximated and measured curves, the proposed representation is enough precise. The difference between experimentally obtained and approximated magnetization reactance does not exceed few percents in the range of magnetizing current variability from zero to rated value of motor current [3]. A simple representation of the nonlinear magnetization curve was achieved at the cost of the abovementioned differences.

The Supply System The schematic diagram of supply system for induction motor is depicted in Fig. 3. The induction motor drives the mixer in polymerization reactor chamber. The supply system for induction motor consists of switchhouse P17 segmented into two fields. The field no. 13/14 allows directly connecting the motor to the grid and it consists of the circuit breaker and the fuses and contactor. The field no. 25/26 allows supplying the motor via

338

A. Popenda and A. Rusek / The Impact of Magnetic Circuit Saturation

frequency converter and contains: the circuit breaker with fuses and frequency converter and contactor that is located behind the converter. In the design of supply systems for electrical machines and electrical equipments it should not be omitted that mentioned machines and equipments exert influence on power grid and local electric power systems. The drops and variations of mains voltage as well as the harmonics generated by electric machines and other disadvantageous phenomena from power quality point of view occur as a result of this influence. The author of [4] pays attention to mentioned phenomena and he proposes number of prognosis analyses concerning power demand for individual consumers or groups of consumers as well as determined areas connected with work of selected sets of local consumers in power systems.

The Vector Equations of Induction Machine The equations of induction machine mathematical model without derivation of magnetization inductance Lm are more advantageous in case of taking into consideration the magnetic circuit saturation. The zero value of rotor voltage u r = 0 should be taken into consideration in a case of squirrel-cage motor. The equations in stationary coordinate system 0αβ using spatial vectors [5,6] are given as follows: d ψ = −Rs i s + u s , dt s

d ψ = −Rr i r + jpb ωm ψ r dt r

(3)

where: ψ s = ψ sα + jψ sβ , ψ r = ψ rα + jψ rβ , i s = isα + jisβ , i r = irα + jirβ are flux and current vectors of both stator and rotor, u s = usα + jusβ is stator voltage vector, Rs, Rr are stator and rotor windings resistances, ωm is angular velocity of rotor, pb is number of couple pairs. The Eqs (3) should be completed with following flux-current dependencies (4) and equation of motion (5): is =

Lr ψs − L m ψr Ls Lr − L

2 m

,

ir =

d 1 ωm = ( M e − M t − M m ) , θm dt

where:

L s = (1 + j ) Lσs + L m ,

L s ψr − Lm ψs 2

Ls Lr − Lm

,

(4)

M e = pb Im ( ψ s ⋅ i s ) *

L r = (1 + j ) Lσr + L m ,

L m = Lmα + jLmβ

(5)

are self-

inductance vectors of both stator and rotor windings and magnetization inductance vector, θm is moment of rotor inertia, Me is output torque of motor, Mm is load torque of motor, Mt is moment of friction in lower slide bearing of motor. The components of magnetization inductance vector Lm are as follows:

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A. Popenda and A. Rusek / The Impact of Magnetic Circuit Saturation

Lmα =

2 Lmn 4

3( ψ α ψ n ) + 1

,

Lmβ =

2 Lmn 4

3 ( ψβ ψ n ) + 1

,

(6)

where: ψα = ψ sα − Lσs isα , ψβ = ψ sβ − Lσs isβ are components of main flux vector. The following auxiliary variables are defined in order to obtain the appropriate results of numerical analysis: ia = 2 3isα ,

ib = 1 2isβ − 1 6isα ,

I s = is2α + is2β ,

ic = − 1 2isβ − 1 6isα ,

(7)

n = 9,55ωm,

(8)

where: ia, ib, ic are phase currents of motor, |Is| is absolute value of stator current vector, n is rotational speed of rotor.

Examples of Transient Responses and Trajectories A digital simulation of polymerization reactor drive has been made on the basis of presented mathematical model of specially designed induction motor. The examples of transient responses and trajectories for selected working conditions of drive are shown in Figs 4–9. The magnetic circuit saturation is taken (a) and not taken (b) into consideration. Connection of Motor to the Power Grid 500

500

b)

ia [A]

250 0

250 0 -250

-250

-500

1600

1600

1200

1200

M e [Nm]

-500

800 400

800 400

0

0

1600

1600

1200

1200

n [rpm]

n [rpm]

Me [Nm]

ia [A]

a)

800 400 0 0,0

0,1

0,2

t [s]

0,3

0,4

0,5

800 400 0

0,0

0,1

0,2

t [s]

0,3

0,4

0,5

Figure 4. Transient responses of motor including time functions of stator phase current ia, output torque Me and rotational speed n.

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A. Popenda and A. Rusek / The Impact of Magnetic Circuit Saturation

a)

b)

1600

1200

Me [Nm]

Me [Nm]

1200

1600

800 400

800 400

0

0 0

300

|Is| [A]

600

0

300

|Is| [A]

600

Figure 5. Trajectories of motor output torque as a function of absolute value of current vector.

a)

b)

600

600 300

isβ [A]

isβ [A]

300 0

0

-300

-300 -600 -600 -300

0

-600 -600 -300

300 600

0

300 600

isα [A]

isα [A] Figure 6. Trajectories of current vector on phase plane.

Starting the Motor Supplied via Frequency Converter Applying Constant Ratio E/f Method 400

b)

200

ia [A]

ia [A]

a)

0

200 0 -200 -400

600

600

Me [Nm]

-400

400 200

400 200

0

0

1600

1600

1200

1200

n [rpm]

n [rpm]

Me [Nm]

-200

400

800 400 0 0,0

0,2

0,4

t [s]

0,6

0,8

800 400 0 0,0

0,2

0,4

0,6

0,8

t [s]

Figure 7. Transient responses of motor including time functions of stator phase current ia, output torque Me and rotational speed n.

341

A. Popenda and A. Rusek / The Impact of Magnetic Circuit Saturation

a)

b)

750

Me [Nm]

Me [Nm]

500

750

250

500 250 0

0 0

200

|Is| [A]

0

400

200

|Is| [A]

400

Figure 8. Trajectories of motor output torque as a function of absolute value of current vector.

a)

b)

400

400 200

isβ [A]

isβ [A]

200 0

-200 -400

0 -200 -400

-400

0

isα [A]

400

-400

0

isα [A]

400

Figure 9. Trajectories of current vector on phase plane.

Conclusion The mathematical model together with mathematical and numerical studies of drive for polymerization reactor with specially designed induction motor is presented in the paper. The mathematical model considers number of real phenomena e. g. saturation of motor magnetic circuit, the skin effect occurring in bars of squirrel-cage rotor, the moment of friction in lower slide bearing of motor as a function of rotational speed of rotor and dependence of load torque resulting from polymerization process. The saturation of main magnetic circuit of motor causes additional distortions of both transient phase currents and transient output torque (Figs 4a, 6a, 7a, 9a). The higher harmonics occur in mentioned variables as a result of a nonlinear magnetization curve taking into consideration. The saturation of main magnetic circuit taken and not taken into consideration causes minimal differences between extreme values of both output torque and absolute value of current vector during starting the motor (Figs 5, 8). The differences may depend on assumed initial conditions. Significant increase of magnetizing current when nominal value of supply voltage is exceeded is additional disadvantageous of magnetic circuit saturation.

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A. Popenda and A. Rusek / The Impact of Magnetic Circuit Saturation

References [1] A. Rusek, A. Popenda, Transient states of polymerizer drive including real load of specially designed induction motor, 17th Int. Conf. on Electrical Machines, Chania, Crete Island, Greece, (CD ROM), 2006. [2] F. Kohlrausch, Praktische Physik (in German), Bd. 2. Stuttgart, Teubner 1968. [3] A. Popenda, The mathematical model of induction machine with variable mutual inductance, Int. Conf. PCIM Proceedings (CD ROM), Nuremberg Germany, June 2006. [4] T. Popławski, Application of the Takagi-Sugeno (TS) fuzzy logic model for load curves prediction in the local power system, III-rd International Scientific Symposium Elektroenergetika 2005, Stara Lesna Slovak Republic, 2005. [5] K.P. Kovacs, J. Racz, Transiente Vorgange in Wechselstrommaschinen (in German), Ung. Akad. D. Wiss., Budapest 1959. [6] J.P. Kopylow, Elektromechaniczeskije preobrazowatieli energii (in Russian), Energia, Moscow 1973.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-343

343

Electromagnetic Design of VariableReluctance Transducer for Linear Position Sensing J. CORDA and S.M. JAMIL School of Electronic and Electrical Engineering University of Leeds Leeds LS2 9JT United Kingdom E-mail: [email protected] Abstract. The linear position transducer based on variable reluctance principle is a contactless device distinguished by a simple construction which allows accomplishment of a large sensing range without adverse effects on accuracy and sensitivity. The goal in the electromagnetic design of the transducer is to achieve its output directly proportional to position, which is of particular relevance for precise real-time position feedback control of linear drives under dynamic conditions. Such a requirement is resolved through a combined consideration of magnetic and electrical arrangements, which includes magnetic circuit optimisation and investigation of the effects of core losses and motional e.m.f. on the transducer output.

Introduction Linear servo drive systems usually include a position transducer which is most commonly of Linear Variable Differential Transformers type – LVDT [1]. In linear systems with long displacement this contactless transducer has major drawbacks due to an increase of the absolute error with measuring range and a reduced sensitivity expressed in terms of output voltage per unit length. Other types of contactless transducers [2–4] which alleviate the above drawbacks are however a major cost component of a linear drive system. To overcome the above drawbacks, a research has been carried out on an alternative contactless linear position transducer of variable-reluctance (VR) type [5,6]. Such transducers rely on repetitive inductance variations of its sensors with position, which are produced by changes of the reluctance between magnetic saliencies of the outer part, here referred to as the sensors, and the inner part having a form of transversely slotted rod, here referred to as the mover. These transducers were originally developed for the use in electronic commutation of phases of the linear tubular switched reluctance motor (LTSRM) [7] where it is essential for the mover to be unrestricted in terms of displacement. In other words, when the transducer is used in conjunction with LTSRM, the mover of the latter interacts with the transducer. The transducer design can be adapted for the use with other types of linear motors with a toothed ferromagnetic structure on the mover but without permanent magnets.

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J. Corda and S.M. Jamil / Electromagnetic Design of VR Transducer for Linear Position Sensing

Figure 1. VR transducer with E-core sensors.

Figure 2. Axial magnetic field plot.

To provide precise continuous position sensing under dynamic conditions of operation of LTSRM, it is desirable that the transducer output is directly proportional to the position. So, the objective of the electromagnetic design presented here is to accomplish the output of the VR type transducer similar to the one available from a linear encoder where the output varies with displacement in a linear fashion.

Transducer Construction and Operating Principles The linear position transducer of VR type shown in Fig. 1 comprises two identical sensors with E-shaped magnetic cores interacting with a slotted ferromagnetic shaft. Each E-core is formed of low-loss laminations assembled into a thin stack. The central saliency of the core, holding a primary coil, has a width of a full mover pitch, λ. Each of the two outer saliencies holds a secondary coil and has a width of λ/2. The slots between the saliencies are also λ/2 wide. Thus, when one of the outer saliencies is in full alignment with the mover tooth, the other one is in full misalignment. The two E-core sensors, diametrically opposite to each other and mutually shifted along the shaft axis by λ/4, are held by a nonmagnetic shell supported on the slotted mover by means of slide bearings. Provided that the radial gap between the salient surfaces of the E-core and the shaft is small compared to the tooth width and that the excitation is below saturation level, the flux of the central saliency produced by an excitation current is nearly independent of the mover position and it is equal to the sum of the two position dependent fluxes of the outer saliencies plus a small amount of flux leakage. The interaction between the fields of the two sensors is insignificant (Fig. 2) because the excitation of their primaries is arranged so to produce magnetic polarities of identical senses. Variations of the two secondary flux linkages of each E-core against displacement are in anti-phase. When considering all four outer saliencies at a given excitation current in the primaries of two sensors, the four secondary flux linkage waveforms against displacement and hence the waveforms of mutual inductances between each secondary and the corresponding primary, hereinafter referred to as the ‘mutual inductance profile’, are spatially shifted from one another by λ/4, as illustrated in Fig. 3. When the primary coils are fed with sinusoidal current of constant amplitude and carrier frequency, the amplitude of each secondary voltage is modulated with the corresponding variation of mutual inductance with displacement (Fig. 4). The transducer output (Fig. 5) is formed from the envelope differences of induced secondary voltages and therefore the waveform of the output voltage variation with position is principally determined by the mutual inductance profile.

345

Secondary voltage [v]

J. Corda and S.M. Jamil / Electromagnetic Design of VR Transducer for Linear Position Sensing

Position [mm]

Output signal

Figure 3. Flux-linkage (mutual inductance) vs. position Figure 4. Modulated waveform of induced waveforms. secondary voltage.

V24

V31

V42

V13 Position

λ/4

λ/4

λ/4

Figure 5. Output voltage vs. position waveform.

The requirement for a linear output addresses, at first, the need to optimise proportions of the magnetic circuit so that the mutual inductance profile has a high level of linearity over the rising and falling segments in the waveform. These segments are indicated in Fig. 3 and are hereinafter referred to as ‘λ/4-segments’. Secondly, the effects of eddy currents and motional e.m.f., which have adverse impact on linearity and dynamic error, need to be minimised. Magnetic Circuit Optimisation At a given geometry of the mover, which was selected while optimising the design of the LTSRM, the main parameters affecting the transducer’s mutual inductance profile are the airgap length between the sensor and the mover, and the ratio of the sensor’s pole to slot widths. The reduction of the airgap increases the span of mutual inductance variation, i.e. the transducer sensitivity, but at the same time it enhances the effect of mechanical imperfections on the waveform profile and increases the cost of manufacturing. For the mover diameter of 40 mm, the airgap of 0.2 mm is considered as an optimum. The effect of changing the sensor pole/slot width ratio on the mutual inductance profile was examined by magnetostatic field analysis using FE field solver and the results are summarised in Table 1. The example is related to the mover pitch length of 10 mm, a tooth/slot ratio of 4/6 and a slot depth of 6 mm. The parameter values were constrained with regard to achieve a segment of linear variation of at least λ/4 length on

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J. Corda and S.M. Jamil / Electromagnetic Design of VR Transducer for Linear Position Sensing

Table 1. Effects of sensors proportions at fixed mover tooth/slot ratio 4/6 and λ = 10 Sensor pole/slot width ratio 5.0/5.0 4.0/6.0 3.5/6.5 3.0/7.0

Span of mut. ind. variation [mH] 2.29 2.10 2.06 2.02

Nonlinearity [%]

Maximum deviation [mH] 0.006 0.008 0.011 0.018

i

0.26 0.38 0.53 0.89

iw

iμ

M1

vS

r vS M2 Figure 6. Variation of mutual inductances against displacement for optimised E-core sensor.

Figure 7. Equivalent circuit of one E-core sensor.

both rising and falling part of mutual inductance waveform, which ensures that any displacement is correlated with a repetitive linear segment. The criterion used for selecting the best λ/4-segment of the mutual inductance waveform was the minimal nonlinearity, which is expressed as a percentage ratio of the maximum deviation from the linear least-square fit line to the span of variation over the λ/4-segment. Equivalent Circuit Model Figure 7 illustrates an approximate equivalent circuit representing one E-core sensor when the primary is subjected to a current-fed excitation of constant amplitude. The core losses caused by eddy-currents and hysteresis are coarsely represented by a resistive branch. Under a current-fed sinusoidal excitation the primary current is unaffected by changes of the circuit parameters. In an ideal case without core losses (r = ∞), the secondary voltage is given by

v s = ω M I m cos ω t

(1)

and its magnitude is modulated by the variation of mutual inductance in a direct linear relationship

Vs = ω M I m

(2)

If the core losses are taken into account, the instantaneous secondary voltage equation is

J. Corda and S.M. Jamil / Electromagnetic Design of VR Transducer for Linear Position Sensing

vs = M

v d ( I m sin ω t − s ) dt r

347

(3)

and the relationship between the secondary voltage magnitude and the mutual inductance has a nonlinear form given by

Vs =

ωM Im 1 + (ω M / r ) 2

(4)

So, when the mutual inductance rises linearly with displacement the gradient of secondary voltage envelope will not remain constant but will decrease progressively. This undesirable effect becomes more enhanced at higher excitation frequencies. The resistance representing the core losses rises with frequency but at a progressively slower rate. For the prototype E-core sensor the estimated values of r at 1, 2, 5 and 10 kHz are respectively 0.75, 1.3, 1.9 and 2 kΩ, and the nonlinearities calculated on the basis of Eq. 4 are 0.1, 0.2, 0.4 and 1.5%. Nonetheless, a higher excitation frequency increases the secondary voltages which is desirable from a viewpoint of the reduction of the noise-to-signal ratio. In addition, as discussed below, a higher excitation frequency is required if the transducer is to be used at higher running speeds. The model considered above assumed that the induced e.m.f. was caused simply by the time-varying current excitation. Beside this transformational e.m.f. there is another e.m.f. component caused by the mutual inductance variation during motion. Likewise the core losses, this component e.m.f. (hereinafter referred to as ‘the motional e.m.f.’) distorts the simple linear relationship (2) between the secondary voltage and mutual inductance. A simple estimate of the ratio of amplitudes of the motional (undesirable) to the transformational (desirable) e.m.f. can be made if these e.m.f.s are considered in isolation of the core losses. The amplitudes of the two e.m.f.s are given by (dM /dt)Im and ωM Im respectively, and their ratio is

E motional υ 1 dM f w λ 1 dM = ⋅ ⋅ = ⋅ ⋅ ⋅ Etransform. ω M dp f 2π M dp

(5)

where dM /dp is the gradient of linear λ/4-segment, υ denotes the mover speed and fw /f is the ratio between the frequency of the envelope waveform and the excitation frequency. So, the percentage error caused by the motional e.m.f. is directly proportional to the ratio fw /f . For the transducer having mutual inductance profile shown in Fig. 5, the e.m.f. ratio is not greater than 0.65 fw /f. So, for instance, at the speed of 0.1 m/sec i.e. fw = 10Hz, the error caused by the motional e.m.f. at the excitation frequency of 1kHz is not larger than 0.65%. If the excitation frequency is increased to 5 kHz, the error is not larger than 0.13%. Under running conditions the transducer error is caused by effects of both the core losses and motional e.m.f., and its minimisation requires a compromise in selecting the excitation frequency.

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J. Corda and S.M. Jamil / Electromagnetic Design of VR Transducer for Linear Position Sensing

Figure 8. Output signal measured at the speed of 0.1 m/s and excitation frequency of 5 kHz.

Prototype Transducer An experimental prototype transducer was designed in conjunction with a 4-phase LTSRM and it interacts with the mover which is constructed as a shaft made of mild steel with a 40 mm diameter, a 6 mm slot depth, a 10 mm pitch length, and the tooth/slot ratio of 4/6. The magnetic core of each sensor was made of low-loss laminated steel and has the outer dimensions of 30 × 13 mm, the stack thickness of 5 mm and was fixed at 0.2 mm from the shaft. The central and outer saliencies are 10 and 5 mm wide respectively. The primary and secondary coils have 200 turns each. The sensors’ primaries are connected in series and fed by a sinusoidal current provided from a linear amplifier. The amplitude variations of each full-wave rectified secondary voltage are tracked by a sample-and-hold circuit. The transducer output is formed from linear λ/4-segments which are extracted from the four differential voltage waveforms using a comparativelogic circuit [5]. The transducer was tested first statically at various excitation frequencies. The amplitude of excitation current was accordingly adjusted to ensure that the amplifier operated in the linear region and that there was no magnetic saturation in the sensors. At excitation frequency of 1kHz the nonlinearity on each λ/4-segment was within 0.35%. At 5 kHz, the effect of core losses was still low causing a relatively small increase of nonlinearity to 0.5%. However, when the frequency was increased to 10 kHz, the impact of core losses became apparent, and the nonlinearity rose to 4.5%. The action of the sample-and-hold circuit and the associated low-pass filter in the processing unit generates a slight shift between the electronically derived envelope and the actual secondary voltage envelope. The error caused by this shift is speed dependent and can be assessed on the basis of positional resolution of the output from sampleand-hold circuit. The positional resolution is given by 0.5υ/f and at excitation frequency of 5 kHz and speed of 0.1 m/s the absolute error due to the resolution is not larger than 0.01 mm which corresponds to 0.4% of a λ/4-segment. When combined with the error caused by the motional e.m.f., assessed in the previous section, the total dynamic error is not larger than 0.53%. Figure 8 shows the measured output from the prototype transducer at operating speed of 0.1 m/s and excitation frequency of 5 kHz.

J. Corda and S.M. Jamil / Electromagnetic Design of VR Transducer for Linear Position Sensing

349

The important feature of the transducer is that the position error occurring over individual λ/4-segments are not accumulative, which allows accomplishment of a large sensing range without adverse effects on accuracy and sensitivity.

Conclusions The paper has presented the electromagnetic design of a linear position transducer based on repetitive reluctance variations between saliencies of two longitudinally shifted E-core sensors and a transversely slotted ferromagnetic shaft which in the studied case represented the mover of a linear tubular switched reluctance motor. The transducer with an optimised magnetic circuit is distinguished by the linear segments in the cycle of inductance variation against position, where each segment covers one quarter of the shaft pitch. The inductance variations were utilised to give the transducer output linearly proportional with position. This was achieved by applying a current-fed sinusoidal excitation to the primary coils, and by using a processing unit for tracking the amplitude variations of the sensors’ induced voltages and for the identification of the envelope linear segments. Exciting the transducer sensors with a high frequency enables more accurate tracking of the amplitude variation of secondary voltages, a better sensitivity and a relative reduction of the undesirable impact of the motional e.m.f., but because of the increase of core losses it has an adverse impact on the profile of the secondary voltage envelope. A prototype transducer was accomplished for a linear drive application with a 1000-mm long displacement at a speed range up to 0.1 m/s. Its absolute position error is not larger than 0.026 mm which corresponds to a non-linearity of 1.03% occurring over a 2.5-mm long repetitive segment. (The position error occurring over individual λ/4-segments are not accumulative.) This is a substantial improvement compared to a typical LVDT transducer with a non-linearity of 0.5%, which at a full stroke of ±250 mm is equivalent to a position error of 1.25 mm.

References [1] E.O. Doebelin, “Measurement Systems – Application and Design”, McGraw-Hill, 4th edition, 1990. [2] Y. Kano, S. Hasebe, C. Huang and T. Yamada, “New Type Linear Position Differential Transformer Position Transducer”, IEEE Transaction on Instrumentation and Measurement, Vol. 38, No. 2, April 1989, pp. 407-409. [3] Y. Kano, S. Hasebe, C. Huang, T. Yamada and M. Inubuse, “Linear Position Detector with Rod Shape Electromagnet”, IEEE Transaction on Magnetics, Vol. 26, No. 5, September 1990, pp. 2023-25. [4] A.E. Bennemann and R.L. Hollis, “Magnetic and Optical – Fluorescence Position Sensing for Plannar Linear Motors”, IEEE International Conference on Intelligent Robots, Vol. 3, 1995, pp. 101-107. [5] J. Corda, J.K. Al-Tayie and P. Slater, “Contactless linear position transducer based on reluctance variation”, IEE Proceedings – Electric Power Applications, Vol. 146, No. 6, Nov. 1999, pp. 151-158. [6] J. Corda and J.K. Al-Tayie, “Enhanced performance variable reluctance transducer for linear position sensing”, IEE Proceedings – Electric Power Applications, Vol. 150, No. 5, Sept. 2003, pp. 623-628. [7] J. Corda and E. Skopljak, “Linear switched reluctance actuator”, IEE Publication No. 376, International Conference on Electrical Machines and Drives, Oxford, 1993, pp. 535-539.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-350

The Influence of the Matrix Movement in a High Gradient Magnetic Filter on the Critical Temperature Distribution in the Superconducting Coil Antoni CIEŚLA and Bartłomiej GARDA AGH University of Science and Technology, al. Mickiewicza 30, 31-059 Kraków, Poland [email protected], [email protected] Abstract. The subject of the paper is an analysis of dynamical state of work of the High Gradient Magnetic Separator. HGMS requires high values of the magnetic field in big spaces. Superconducting coils are used for that field excitement. Separator works in two states. First state – static – magnetic filtration, and dynamic – exchanging the matrix. Matrix is made of ferromagnetic wool. So it influences on the field excited by the coil. During dynamical state, the matrix is taking out the filter and it affects on the field distribution in the coil windings. Unfortunately the level of temperature required to maintain the superconductivity properties locally might be dramatically lowered and it could be the cause of the serious accidents. Magnet designers have to take into consideration also the dynamical state of the magnetic filter.

1. High Gradient Magnetic Separator In this paper a prototype superconducting magnet is considered (Fig. 1). The winding is made of a composite multifibre conductor with a NbTi superconductor placed in a copper matrix and cooled by liquid helium [1]. This electromagnet provides the source of magnetic field in a High Gradient Magnetic Separator (HGMS). HGMS is a device that can separate a solid particles and slurries. Effectiveness of that action is proportional to the value of the magnetic field and “nonhomegenity” of the field in the working space of that filter (high gradient methods). That strong magnetic field is usually excited by a superconducting coil. High nonhomegenity is made using a matrix which is usually made of the stainless magnetic wool. The magnetic fraction of the feed passes through the matrix and is attached to the ferromagnetic elements. The non-magnetic particles are collected out side of the matrix. So while that matrix is filled with those magnetic particles it has to be removed from the working space of the filter (Fig. 2). Because the field is excited by the superconducting winding it is not easy to turn it off and exchange the matrix without the field. Superconducting coils works in the closed circuit. Such a coil has gathered the big energy amount While matrix is taken out the filter first some force appears due to the energy change [2] and it also influences on the field distribution in the coil windings. That field is linked with the temperature distribution of the

A. Cie´sla and B. Garda / Matrix Movement in a High Gradient Magnetic Filter

canister with slurry

351

cryostat

water

matrix of the filter z [m]

ferromagnetic filling of the canal

superconductor winding

filtration product

Figure 1. High Gradient Magnetic Filter. b Le Superconducting Solenoid a2 Bm a1

High Gradient Magnetic Filter

direction of matrix movement

z

Bn, B0, zo

Figure 2. Removing of the matrix from the working space of the coil.

J Jc;t Tp T

Bp

Bc0

B

Tc0

Figure 3. Semi-linearized critical surface of superconductor.

coil what could be the cause of quench. In the paper authors present an analysis of critical temperature distribution change while moving out the matrix from the filter.

2. Material Properties The winding is to be made of a composite multifibre conductor with NbTi. To simplify the calculations, the critical surface of the superconductor has been semilinearized. On Fig. 3 authors present this critical surface. It is the surface that binds three critical values magnetic field, current density and the temperature. All those values must not be exceeded. All the superconducting properties was calculated using a reduced state model widely described in [3].

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A. Cie´sla and B. Garda / Matrix Movement in a High Gradient Magnetic Filter

Table 1. Superconductor characteristic Jc = f(B) for fixed temperature Tc0 = 4,2 K B [T] Jc [A/m2]

3 2.588

4 2.235

5 1.882

6 1.544

7 1.147

8 0.735

1,5

ε =0.15 ε =0.10 ε =0.05

1,4

μr

1,3

1,2

1,1

1,0 0

1

2

3

B [T]

4

Figure 4. Magnetic relative permeability in the function of magnetic induction.

To obtain semi-linearized critical surface first one have to find the relation between the field, current and temperature: Tc ( B, J ) = α1 B + α2 J + α3

(1)

where: T ⎡α1 ⎤ ⎢α ⎥ = ⎡− Tc 0 ; T p Bc 0 − ( Bc 0 − B p )Tc 0 ;T ⎤ ⎥ ⎢ c0 ⎢ 2⎥ B Bc 0 J c;t (B p ) ⎥⎦ ⎢⎣α 3 ⎥⎦ ⎢⎣ c 0

(2)

and Tc0 – is the critical temperature for B and J = 0, Bc0 – critical magnetic field induction for J and T = 0, and Jc;t is usually given in the table as the function of Bp for fixed temperature Tp. Usually all those values are provided by the superconductor producers. In our case authors used for model description an NbTi wire produced by Vacuumschmelze GmbH from Hanau with the superconducting properties presented in Table 1. Coil designers usually construct magnets minimizing the superconductor volume fixing field value and the shape of that field in the working space. When the coil is calculated it has to be proved that the all values are lower than the critical surface, if not, our possible solution must be recalculated. The field is to be excited in the area where the ferromagnetic wool is placed (matrix). That wool is described by so called “packing factor” defined as the ratio of value of the ferromagnetic wool and the total value of the matrix. In our model authors applied an pacing factor on the level of 7% what is the average value of the real matrix models (3÷15%). Another problem is that ferromagnetic wool has nonlinear magnetic properties. Special model was used which compares an ferromagnetic wool magnetic properties and pacing factor of the matrix. On the Fig. 4 authors present an μr as an function of the field B.

A. Cie´sla and B. Garda / Matrix Movement in a High Gradient Magnetic Filter

353

Magnet designers calculate the coil when the matrix is placed directly in the center of the working space. All the critical values are fulfilled and the superconducting winding is in “no danger” of the quench. But unfortunately when the matrix is being moved out of the windings local unstable points might appear and what is worst the superconductor might lose its superconducting properties.

3. Simulation of the Dynamical State Authors made some simulations with the matrix movements calculating the critical temperature distribution for different position of the matrix. Using FEM model and having calculated an field distribution and applying Eq. (1) authors could calculate critical temperature distribution on the cross-section of the coil. Having that data authors made some animation. Figure 5 presents the some important situation. Figure 5a presents the situation when matrix is being placed in the center of the coil (static state of work of the filter). Figures 5 from b to f shows distribution during the dynamical state of work of the filter. It can be seen that position of the matrix has the great influence on the critical temperature distribution. Starting from Fig. c to Fig. e critical temperature is lower than temperature of the liquid helium so the superconductivity can not be maintain using liquid helium as cooling factor. On Fig. 6 it can be seen the influence of the matrix movement on the minimal critical temperature in the whole area of the coil. Critical temperature is divided by the 4.2 K (the temperature of liquid helium), so when the locally temperature is lower then 1 it means that at that point we could have problem to maintain that low temperature and the superconductive state might be lost. From the figure it can be seen that during static state of the filter the maximum critical temperature is on the level of 1,2÷1,3 of the liquid helium, but if the matrix is being moved the critical temperature is being lowered rapidly, and the dangerous quench situation appears. Above dangerous situation might appear when middle of the matrix is between 0,12÷0,34 m and the critical temperature is lower then 4,2 K.

4. Conclusions Movement of matrix made of the ferromagnetic wool has an great influence on critical temperature in the coil windings. Unwanted accidents might happen during dynamical state of the work of the filter. The problem is getting harder when the shape of the coil cross-section is not simply rectangular. There are two solutions to avoid the problems. First is to lower the current in the coil just before the matrix is being taken out the filter. It is a little problematic action because of big energy change. Otherwise one can lower the input current in the coil, but then also field value in the filter is lowered furthermore the filtering effectiveness is decreased dramatically.

354

A. Cie´sla and B. Garda / Matrix Movement in a High Gradient Magnetic Filter

a) Tmin rel = 1,24, z0 = 0 m

b) Tmin rel = 1,01, z0 = 0,11 m

c) Tmin rel = 0,96, z0 = 0,12 m

d) Tmin rel = 0,79, z0 = 0,21 m

e) Tmin rel = 0,95, z0 = 0,32 m

f) Tmin rel = 1,16, z0 = 0,5 m

Figure 5. Influence of the matrix movement on the critical temperature. Pictures made for some different positions of the matrix.

355

A. Cie´sla and B. Garda / Matrix Movement in a High Gradient Magnetic Filter 1,30

Line of the border of the coil

T rel [/]

1,20

1,10 Liquid Helium Line 4,2K

1,00

0,90

0,80 z [m]

0,70 0

0,1

0,2

0,3

0,4

0,5

0,6

Figure 6. Influence of the position of the middle of the matrix z0 (Fig. 2.) on the minimal critical temperature of superconductor of the coil.

Acknowledgement The work presented in this paper was supported by the Polish State Committee for Scientific Research, Warsaw, in the frame of project Internal Research (“Badania własne”) contract No. BG10.10.120.561/III/9/p.

References [1] A. Cieśla, Superconductor Electromagnet as a DC Machine (the Constructional and Exploitations Peculiarities), Proceeding of 43rd ISCT University of Ilmenau, September 21–24, 1998. [2] A. Cieśla, B. Garda, Analysis of the force acting on the matrix of a superconducting filter in high magnetic field, proceedings of ISEF 2001, pp. 247-250, September 20–22, 2001. [3] M.A. Green, Calculating the Jc, B, T surface for Niobum Titanum using a reduced state model, IEEE Transactions on Magnetics, Vol. 25, No. 2, March, pp. 2119-2122; 1989.

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Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-356

Macro- and Microscopic Approach to the Problem of Distribution of Magnetic Field in the Working Space of the Separator Antoni CIEŚLA AGH – University of Science and Technology, Department of Electrical Engineering, al. Mickiewicza 30, PL 30 – 059 Kraków, Poland, Tel: +48 12 617 39 86, E-mail: [email protected] Abstract. The subject of this paper is field distribution in a magnetic separator containing ferromagnetic matrix where the magnetic field is generated by a dc superconducting magnet. To develop invariable conditions for the extraction of particles from the slurry in a filter matrix, it is necessary to create a homogenous magnetic field within the working space of the device. The source of the field is usually a solenoidal coil winding with superconducting wire and, in order to achieve the design objective of field uniformity, various configurations have been considered using optimisation techniques (macroscopic model). Also distribution of the magnetic field around single ferromagnetic filament of the steel wool placed in homogenous field of the separator is considered (microscopic model). Analysis of the single filament is essential to describe particles movement in the matrix. Some simulation results are presented and the most promising solutions are highlighted.

Principle of Magnetic Separation in Matrix Separator When fine particles are dispersed in air, water, sea water, oil, organic solvents, etc., their separation or filtration by using a magnetic force is called magnetic separation. (Fig. 1).

Figure 1. High Gradient Magnetic Separator.

A. Cie´sla / Macro- and Microscopic Approach to the Problem of Distribution of Magnetic Field

357

In the magnetic field generated by this superconducting solenoidal winding, a matrix is positioned in which particles from the slurry flowing through a separator are extracted. The matrix is a canister filled with gradient-generating elements such as chips or ferromagnetic wool, to which particles of specific magnetic properties are attracted. The separator matrix should be placed in a homogenous magnetic field to develop consistent conditions during the technological process (the filtration of suspended solids). Therefore, the shape and dimensions of the superconducting winding that generates a magnetic field of required homogeneity must be carefully considered.

Distribution of Magnetic Field in the Working Space of the Separator A circular solenoid of rectangular cross-section (Fig. 2) is the most common coil shape used in magnetic separation. The shaded region is where a cylindrical HGMS matrix is located. The coil is characterized by the parameters, collected in Table 1. The basic relationships that may be applied to the solenoid winding are [1]: B0 = μ 0 a1 JK 0 (α,β ) K 0 (α,β ) = β ln

(1)

α + α2 + β2

(2)

1+ 1+ β2

V = 2πa13β (α 2 −1)

(3)

solenoid

Lm

b

Bm matrix

B0

Bn

a1

a2

z

Figure 2. Configuration of the matrix and solenoid of the separator. Table 1. Parameters characterizing coil windings GEOMETRIC PARAMETERS 2 a – inside diameter of solenoid, 2 a – outside diameter of solenoid,

ELECTRIC PARAMETERS J – averaged density of current in solenoid, B – magnetic flux density in the geometric centre

2 b – length of solenoid, Le – length of matrix. The above parameters are interconnected through the following relationships:

of the solenoid, Bm – maximum value of flux density on the winding external surface, in its middle plane, Bn – value of flux density on the end of matrix.

1

0

2

a2 = α; a1

b = β; a1

Le = βe ; a1

b − Le = Δβ. a1

358

A. Cie´sla / Macro- and Microscopic Approach to the Problem of Distribution of Magnetic Field

a)

1,0

z c2 c1 a1

0,9 0,8

Bz/Bzmax

0,7 0,6

b

0,5

b = 100 mm b = 160 mm b = 200 mm b = 250 mm

0,4 0,3

b)

MATRIX

0,2

J3

0,1 -150 -125 -100

-75

-50

-25

0

3 2d

25

50

75

100

125

150

J2

J1

d

d

J2

J3

b

z [mm]

z

Figure 3. Distribution of Bz (z component of magnetic Figure 4. Proposed designs: a) coil with additional turns of the winding, b) coil with variable flux density) for r = 0 for the case depicted by Fig. 2. current density in the cross-section (J1, J2, J3); β1 = Le (a1 − c1 ), β2 = Le (a1 − c2 ) .

v (α,β) =

V = β (α 2 −1) 2πa13

(4)

where V is the volume of the winding. To determine the shape of the winding, the relationships (2) and (4) should be applied. In Eqs (2) and (4), four variables occur: v, K0, α, and β; two of them being independent ones. Figure 3 presents computed distributions of the relative value of the z component of magnetic flux density on the symmetry axis as shown in Fig. 2. In this case an analytical expression for the magnetic field distribution on the symmetry axis can be achieved: 2 2 ⎞ ⎛ a2 + a22 + (b − z ) a2 + a22 + (b + z ) ⎟⎟ I⎜ + (b + z ) ln H = ⎜⎜⎜(b − z ) ln ⎟⎟ 2 2 ⎟ 2 2 2 ⎜⎜ a1 + a1 + (b − z ) a1 + a1 + (b + z ) ⎠⎟⎟ ⎝

(5)

The analysed magnetic field distribution is for the value of z = ±150 mm and is related to the length of the matrix (2 Le = 300 mm). The author [1] proposes considering other possibilities of windings for magnetic field excitation in HGMS, which promises improved homogeneity of the magnetic field in the working space of the separator. Two of the possible improved designs are presented in Fig. 4. Field distributions for the designs of Fig. 4 have been predicted numerically using finite elements modelling. For convenience, when comparing the results, the inhomogeneity factor ε has been introduced and defined as ε = Bz max where Bz max is the maximum value of Bz, and Bz ave Bz ave is the mean (average) value of Bz on the desired length, in our case – length of the matrix. Figures 5 and 6 present distribution graphs of a relative value of z component of magnetic flux density on the symmetry axis for z = ±150 mm for the two proposed new designs (Figs 5 and 6 should be compared with Fig. 3). The author proposes the shaping of the magnetic field using the construction from Fig. 4b, where the coil is divided into small sections with different current density. This

A. Cie´sla / Macro- and Microscopic Approach to the Problem of Distribution of Magnetic Field 1,00

1,00

0,95

0,95

Bz/Bz max

Bz / Bz max

359

0,90

b b b b

0,85

= = = =

250 200 160 100

b = 250 [mm] J2/J1=1,042; J3/J1=1,063

MATRIX

MATRIX -50

-25

0

z [mm]

b = 200 [mm] J2/J1=1,057; J3/J1=1,134

0,80

0,75

0,75 -75

b = 160 [mm] J2/J1=1,013; J3/J1=1,430

0,85

[mm] [mm] [mm] [mm]

0,80

-150 -125 -100

b = 100 [mm] J2/J1=0,9524; J3/J1=1,9048

0,90

25

50

75

100

125

150

-150 -125 -100 -75

-50

-25

0

25

50

75

100 125 150

z [mm]

Figure 5. Distribution of Bz (z component of the mag- Figure 6. Distribution of Bz (z component of the netic flux density) for r = 0 for the design from Fig. 4a; magnetic flux density) for r = 0 for the design from Fig. 4b. (β1 = 5 and β2 = 3,75).

a)

b)

c)

Figure 7. Matrix of the separator (a), matrix filling with stainless steel magnetic wool (wool seen with a magnifier) (b), one ferromagnetic filament of the steel wool treatment as a collector (c).

solution leads to both technical and economical considerations. The technical aspect consists in a good usage of superconductor. From the economical point of view, the variable cross-section method makes it possible to minimize the volume of the superconductor used. However, there may be a problem of supplying different currents to different sections, as in the case of using the superconductor this would require using several current leads. This could cause an increase in liquid helium evaporation from the cryostat. Motion of the Particles in Matrix Separator The separator matrix in which magnetic separation occurs is placed in the magnetic field being induced by solenoidal winding described in the previous chapter. Figure 7 shows matrix of the High Gradient Magnetic Separator [2]. As Fig. 7b indicates, there are a few arrangements possible between the directions of magnetic field and particle flow velocity with respect to the ferromagnetic wire. Three of the configurations are shown in Fig. 8 [3]. Since it is assumed that the fibre is

360

A. Cie´sla / Macro- and Microscopic Approach to the Problem of Distribution of Magnetic Field y

particle

b

H0 v0

Fs

Rk

Fg x collector

Figure 8. Representation of three geometrical configura- Figure 9. Particle with radius b in the magnetic tions. field around the collector with radius Rk.

always orthogonal to the external magnetic field, the fluid flow can be either parallel (longitudinal (L) orientation) or perpendicular (transverse (T) orientation) to the direction of magnetic field. The third alternative configuration, when the initial fluid flow is parallel to the wire axis is known as the axial (A) configuration. Analysis of the particle movement around the collector will concern configuration L and T. The particles in the suspension that flows through the matrix are under the influence of the magnetic force and move in the direction of the collector (which is a single fibre of ferromagnetic wool). The distribution of the magnetic field around the collector shown in Fig. 9 is described by he following equation [4]:

⎡⎛ K ⎞  ⎛ K ⎞ ⎤  H = A0 H 0 ⎢⎢⎜⎜⎜1 + 2c ⎟⎟⎟ cos θ1r − ⎜⎜⎜1− 2c ⎟⎟⎟ sin θ1θ ⎥⎥ ⎜ ra ⎟⎠ ra ⎠⎟ ⎝⎜ ⎣⎢⎝ ⎦⎥ b for 1 < ra < Rk   b H = H0 for < ra < ∞ Rk

(6)

where: A0 = 1 (1− ε K c ) , ν = μwk μwo

K c = (ν −1) / (ν + 1) , and

ra = r Rk .

ε0 – the packing factor of the matrix, µwk, µwo – relative magnetic susceptibility of the collector and medium, respectively. Motion of a single particle in nonhomogeneous magnetic field in the vicinity of a collector is analyzed (see Fig. 9). Svoboda [3] showed the complexity of the process of particles deposition on a collector and proposed another formulation of the problem. Accepting the above, we are going to limit our considerations only to such particle size for which the magnetic interaction is decisive. Under these conditions the following equation is assumed to be valid:

mc

  dvc = ΣF dt

(7)

A. Cie´sla / Macro- and Microscopic Approach to the Problem of Distribution of Magnetic Field

361

Figure 10. Steps of particles’ movement in the vicinity of the collector in the matrix separator.



where ΣF is the sum of forces affecting particles in the matrix. The complete equation of the particle motion including most important forces occurring in the matrix is as follows [4]:

ρcVc

 ⎛1   ⎞ d vc    = (ρc − ρ0 )Vc g+ 6πηb ( v 0 − v c ) + χcVc grad ⎜⎜ H⋅ B0 ⎟⎟⎟ ⎜⎝ 2 ⎠ dt

(8)

Particles affected by the magnetic force move towards the collectors and settle on their surface. Particles outside the capture zone that is determined by the border trajectory, will not be captured by the collector. Deposition takes place up to moment when the balance of holding magnetic force and shear force is achieved. Subsequent steps of a grain movement in the vicinity of the ferromagnetic element of the matrix – collecting grains of particular magnetic properties – are presented in Fig. 10. Figure 11 shows simulation of the particles trajectory for one collector (for configuration L from Fig. 8) and for five collectors (for configuration L and T from Fig. 8).

Conclusions The high force separation capabilities of superconducting magnets and their application for the most difficult separation problems of paramagnetic or low susceptibility materials are now recognized. Recent advances in superconducting technology mean that technology, once limited to the laboratory research, can be successfully used on a large scale. Theoretical model presented in this paper makes it possible to determine basic variables that characterize extraction particles from slurry in High Gradient Magnetic Separator.

362

A. Cie´sla / Macro- and Microscopic Approach to the Problem of Distribution of Magnetic Field

a)

b)

c)

d)

e)

f)

Figure 11. Results of the particles trajectories for different conditions: one collector (configuration L from) (a, b); five collectors (configuration L) (c, d); five collectors (configuration T ) (e, f).

Acknowledgement The work presented in this paper was supported by the Polish State Committee for Scientific Research, Warsaw, in the frame of the project No 4 T12A 027 28 (2005–2007).

References [1] A. Cieśla, B. Garda, J. Sykulski: Shaping of Magnetic Field Distribution in a High Gradient Magnetic Filter. Archiwum Elektrotechniki, Vol. LI, No 4, pp. 403–415, 2002. [2] A. Cieśla: Use of the superconductor magnet to the magnetic separation. Some selected problems of exploitation. Int. Journal of Applied Electromagnetics and Mechanics 19 (2004), pp. 327–331, IOS Press. [3] J. Svoboda: Magnetic Methods for the Treatment of Minerals. Elsevier Science Publishers B.V., 1987. [4] Mayuree Natenapit, Wirat Sanglek: Capture radius of magnetic particles in random cylindrical matrices in high gradient magnetic separation, Journal of Applied Physics, Vol. 85, No 2, pp. 660–664, 1999.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-363

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Electric Field Exposure Near the Poles of a MV Line D. DESIDERI, A. MASCHIO and E. POLI Università di Padova, Dipartimento di Ingegneria Elettrica, via Gradenigo 6/a, Padova, Italy [email protected], [email protected], [email protected] Abstract. In this work the electric field on the head of a human body under a medium voltage line is studied, close to, at a short distance from, and far from a reinforced concrete pole. Two models of the body have been studied numerically: a more complete one and another simplified. The numerical data obtained with both models were found equivalent, in order to compute the field on the top of the head. The simplified configuration has been experimentally validated. The results show a reduction of the field, close to the pole, compared with the value generally calculated neglecting the poles.

Introduction The distribution of the electric field on a human body (“body” in the following) standing under a power transmission line is basic for the calculations of the induced current density, subjected to recommendations and standards giving limits for human exposure to electromagnetic fields. The calculations are generally performed on a conductive surface, representing a body, by using numerical techniques such as Finite Element Method, Boundary Element, and Charge Simulation Method [1–4]. The computations are especially important on the top of the head, where the field is maximum [4]: when doing such computations, the ambient field prior to the presence of the body (“unperturbed field” in the following) is generally assumed to be vertical and uniform [2–4]. Moreover, the presence of the poles is generally neglected. Close to a reinforced concrete pole (“pole” in the following) of a medium voltage (MV) line, these assumptions on the field are no longer valid: moreover, near to the pole, the unperturbed field can be much higher than that in the midpoint of the span [5]. The aim of this work is to investigate the variation of the field on the head of a body when approaching the pole of a MV line.

Numerical Analysis Aim of the numerical analysis was to investigate whether two different models of the body, the first one closer to the actual body shape and the second one more suitable as regards the construction of a simple, cheap and easily manageable experimental set-up, gave equivalent results for the evaluation of the electric field on the head of the body.

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Figure 1. MV line with a pole-top-pin construction.

Figure 2. Numerical models of the body: (a) complete model (case A); (b) simple model (case B).

The Model A MV overhead line supported by poles with a pole-top-pin construction has been considered. The line model is shown in Fig. 1, where both Cartesian (x, y, z) and cylindrical (r, θ, z) coordinates are indicated. Three straight cylindrical conductors, with the same diameter of 0.01 m, whose axes are parallel to each other and to the ground, represent the three wires. In agreement with typical spacings used for the analyzed construction [6], the axes of the two lateral conductors are as high as a = 10 m, while the axis of the central top conductor is at a + b = 10.8 m over the ground: the distance 2c between the lower conductors is 1.52 m. The pole is assumed to be a cylindrical conductor at zero potential, with its vertical axis on the z axis: its height is a = 10 m and the radius of its cross-section is 0.17 m. The xz vertical plane containing the axis of the central overhead conductor is at y = 0. In a first representation (case A, Fig. 2-a), a model of the body complete silhouette has been implemented. It is composed by two parts, both at zero potential: i) a vertical conducting cylinder put on the ground, with a diameter of 0.30 m and a height of 1.55 m; ii) a conducting sphere with a diameter of 0.20 m, placed on the cylinder, with the centre on the axis of the cylinder. The chosen dimensions are very close to the range reported in [3]. A second simple model (case B, Fig. 2-b) was also used: a conducting sphere, at zero potential, with a diameter of 0.25 m, with the top placed at 1.75 m, i.e. at the same height of the model of case A. The analyses performed in the following refer only to a symmetrical configuration, i.e. with the axis of the cylinder and the centre of the spheres on the xz plane at y = 0.

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Figure 3. Charge configuration: (a) one couple of charged segments inside the equipotential surface of an overhead conductor, (b) one set of 10 segments inside the equipotential surface of the pole.

The Charge Simulation Method The electric field around the MV power line was computed by using the charge simulation method (CSM). In this method conductors and their surface charges are replaced by simulating charges placed inside the equipotential contours of the conductors [7–8]. The presence of the ground plane is taken into account by a corresponding set of image charges. The magnitude of the simulating charges is initially unknown but is readily calculated when potentials are imposed on a number of points of the contours. Then potentials and electric fields can be computed everywhere. Furthermore, the accuracy of the method is tested by calculating potentials on control points of the conductors. A section of the MV line, extending 30 m apart from the pole on both sides, was considered. For the representation of the overhead conductors and of the pole, segments with a constant charge density were used. The equipotential contour of each overhead conductor was simulated by means of 55 couples of segments. The segments of each couple are on a xy plane, parallel to the axis of the conductor, on opposite sides, one twentieth of the radius apart from it along y (Fig. 3-a). The couples differ one from another for their length: the start point of each segment varies from couple to couple, while the end point is, for all the couples, the end of the conductor. The pole was simulated by 69 sets of 10 segments, regularly placed all around the axis of the pole, on a circumference with a radius of one twentieth of the radius of the pole (Fig. 3-b). The lengths of the segments of the same set are equal, while they are different from set to set; all the segments end at the top of the pole. It is worth noting that the presence of the body induces, on the surface of the pole, charges that are higher on the side where the body is closer to the pole, i.e. the charge density can vary strongly with the angle θ around the pole (Fig. 1). Thus, the above-stated charge configuration has been adopted for the pole, in order to correctly reproduce the equipotential contour of the surface of the pole. For the body representation, rings with a constant charge density along the circumference and charged points were used. Inside the spheres (case A and case B), 7 rings with different radii were placed on xy planes: the major radius was for the ring located in the middle of the sphere. Moreover, 2 charged points were placed near the top and the bottom of the spheres. On each ring, distributed along the circumference, 15 charged points were added. Therefore each sphere has been replaced by 7 rings and 107 charged points.

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Inside the cylinder (case A), 34 rings of the same radius were placed on xy planes. Furthermore, for each ring, distributed along the circumference, 9 charged points were added. Therefore the cylinder has been replaced by 34 rings and 306 charged points. Finally, the body induces on the overhead conductors a very low charge, and therefore the above mentioned representation with simple couples of charged segments has been used. The accuracy was tested. With 1 kV imposed on the central phase conductor and –0.5 kV on the other ones, on the control points a deviation, on the potential, lower than 2 V was found: on the pole and on the top of the head a difference of less than 1 mV was computed. It is worth noting that the charge simulation method was chosen in this study because it allows to compute the electric field with a small number of simulating charges: for the calculations, the largest matrix to be solved was only of 1434 × 1434. Consequently, each computation run requires about 70 seconds on a PC with 504 MB RAM, 1.4 GHz Intel Celeron M. Electric Field Computation The way to compute, by the CSM, the rms value of the three components of the electric  field is simple. It is well known that a sinusoidal electric field E ( P,t ) can be expressed as the combination of two terms:    E ( P,t ) = E1 ( P ) cos(ωt ) + E2 ( P ) sen(ωt )

(1)

and the rms values of the three components result: Ei ( P) =

E12i E22i + 2 2

(2)

with i indicating one of the coordinates (x, y, z). In a simple way, the calculation of   E1 ( P ) and of E2 ( P ) is done respectively setting the potentials of the three overhead conductors at t1 = 0 and at t2 = π/(2ω). Any choice of the potential configuration at t1 = 0 is valid: different values of the potential configuration result in different expres  sions for E1 ( P ) and E2 ( P ) . Since the MV power line is fed by a set of balanced three-phase voltages, to take profit of the symmetry of the whole system (the pole-top-pin configuration, the models used for the body and their location on the plane y = 0), the following two configurations have been chosen. The first one is with the central top conductor at maximum potential, and the two lateral conductors at one half (and opposite) of the maximum; the second one, shifted of π/(2ω) in time, is with the central top conductor at zero potential, and the two lateral conductors subjected to potentials, with opposite sign, equal in magnitude to the maximum multiplied by 3 /2. With this choice, it can be noticed that, in the plane y = 0, the x and z components of the electric field, with the values indicated for the second configuration, are zero.

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Ez [V/m]

800 700 600 500 400 300 200 100

0

0

2

4

x [m]

6

8

10

Figure 4. Vertical component of the electric field (Ez) at y = 0 on the top of the head for: case A (continuous line) and case B (dashed line).

A line voltage of 20 kV rms has been considered. In Fig. 4 the vertical component Ez of the electric field on the top of the head, at y = 0, against the distance of the center of the body from the axis of the pole, is reported, both for case A (continuous line) and for case B (dashed line).

Experimental Validation The case B configuration has been realized. A hollow sphere, in brass, with a diameter of 0.25 m, bonded to the ground by a thin copper thread, has been used, with a wooden tripod as supporting structure. The top of the sphere has been placed at 1.75 m over the ground, under the central conductor of the MV line, in three different measuring positions (with the centre at 0.32 m, 1.14 m and 6.67 m from the axis of the pole). The electric field has been measured by using a PMM EHP-50C, usually located on the top of the sphere, except near to the pole where it was located also by the sphere’s side. Figure 5 shows the experimental set-up placed on-site. In Table 1, with reference to the vertical component of the electric field (E z) on the top of the head, the numerical data obtained with the two depicted representations (case A and case B) are reported and compared with the experimental values. The values computed with the simple sphere are in good agreement with those obtained with the complete model (a difference less than 8%). Moreover the numerical analysis relative to the case B configuration has been experimentally validated: the difference between numerical and experimental data is of about 20%. In Table 1 it is also apparent the large drop of Ez on the top of the head when shifting the body from 6.67 m to 1.14 m and, even more, to 0.32 m. It can be seen that the Ez variation with the distance is the same for both measured and computed fields. Finally, in the position at 0.32 m, on the side of the sphere opposite to the pole, the measured electric field parallel to the overhead conductors was 266 V/m, in agreement with the numerical data. The field is higher than on the top: that happens close to the pole, due to the local enhancement of the unperturbed field observed in [5].

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Figure 5. Case B configuration: on-site experimental set-up. Table 1. Ez on the top of the body: numerical results (case A and case B) and experimental data position

case A

case B

measured

at 0.32 m

144 V/m

132 V/m

168 V/m

at 1.14 m

350 V/m

330 V/m

412 V/m

at 6.67 m

681 V/m

650 V/m

789 V/m

Conclusions A comparison between two different models of the body, centered on the vertical plane of symmetry along the line, has been performed in this work. A first conclusion is that, when only the evaluation of the field on the top of the head is required, a simple sphere can be used, instead of choosing among the different shapes or sizes of the model of the body used in literature. This solution is much easier to be handled in the numerical analysis and to be implemented as experimental apparatus. A second conclusion is that the presence of the pole strongly reduces the field on the top of the head near the supporting structure. Close to the pole, a severe overestimation of the field results, when the line supporting structures are neglected in the computations.

References [1] E. Poli, Magnetic and electric field computation in proximity of power-line towers and supported busbar, CIGRE 33-96 (WG-07) IWD 19, 1996. [2] O. Bottauscio, R. Conti, Magnetically and electrically induced currents in human body models by ELF electromagnetic fields, 10th ISH, pp. 5-8, 1997.

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[3] A.B. Mahdy, H.I. Anis, Field exposure modeling using charge simulation, 5th ISH, paper 33.20, 1987. [4] CIGRE WG 36-01, Electric and magnetic fields produced by transmission systems. Description of phenomena - practical guide for calculation, Paris 1980. [5] D. Desideri, A. Maschio, E. Poli, Environmental analysis of the electric field due to MV overhead lines supported by concrete poles, 4th International Workshop on Biological Effects of EMFs, pp. 876-883, 2006. [6] Standard Handbook for Electrical engineers, 10th ed., D. G. Fink and J. M. Carroll Eds., McGraw-Hill, 1968. [7] N. H. Malik, “A review of the charge simulation method and its applications,” IEEE Trans. Electr. Insulation, vol. 24, pp. 3-20, Feb. 1989. [8] E. Poli, “The use of image charges in the charge simulation method: a parallel-plane dielectric plate covering a conductor,” IEEE Trans. Magn., vol. 28, pp. 1076-1079, Mar. 1992.

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Study on High Efficiency Switched Reluctance Drive for Centrifugal Pumping System Jian LI, Junho CHA and Yunhyun CHO Depart of Electrical Engineering, Dong-A University, 840 Hadan 2 dong, Saha-gu, Pusan, 604-714, Korea [email protected], [email protected], [email protected] Abstract. Variable speed drives are widely used in pump system for the energy efficiency potential. In this paper a switched reluctance motor and variable speed drive is designed. Considering pumping system’s requirements, ‘base speed’ and rated torque is carefully decided to give a high efficiency in wide speed range. Rotor’s shape is also studied to give good dynamic performance. The motor’s geometry was optimized for high efficiency.

1. Introduction In last decades, the pumping system driven by variable frequency controlled induction motors are widely used in industry. But switched reluctance motor has advantages such as high efficiency, high reliability, and low cost construction. More importantly, submerged pumps are in slender shape and SRMs can be much more easily manufactured in slender shape than induction motors. So the demand of SRM application in pumping systems is getting stronger. Variable Speed Drive (VSD) converts everyday pumps into sensor-controlled pumps that automatically adjust speed in real time to changes in pressure of pumping system. As more flow is required, the pump motor speeds up to provide smooth and consistent performance. The pump’s operating characteristics is show in Fig. 1. In a constant pressure piping system, system curve changes from 1 to 2 when large flow rate is required, so the motor’s speed increases from n3 to n2 . In our project, the pump’s speed range is from 2000 rpm to 3600 rpm. When design a switched reluctance motor, the rated torque should first satisfies pump’s shaft torque. It’s important to decide the ‘base speed’ since motor’s dynamic performance is much better below base speed [1] but higher efficiency can be obtained above ‘base speed’. These two cases are shown in Fig. 2. Another problem is that switched reluctance motor’s efficiency gets lower when operating with smaller load, and usually this is common in variable speed driven pumps. 2. Preliminary Design and Flux Linkage Analysis Design of switched reluctance motor is a complicate procedure due to its nonlinearities because of core saturation and salient poles. Many guides and considerations were

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371

 Figure 1. Pump performance curves.

 Figure 2. Motor and pump torque curves.

given in literature [1,2], but the flux-linkage was first further studied here to give a high efficiency design. Firstly, the rated power, operating speed range are derived from the application’s requirements. These data together with dimensional constraints are used to derive cross shape detentions using equations from [3]. The motor studied in this paper has 6 stator poles and 4 rotor poles and the dimensions of core are given in Table 1. The well-known expression for inductance calculation will be used: L=

Tph2 R

(1)

where Tph is the number of the turns of the excited phase, R is the reluctance of the magnetic circuit in which the inductance is calculated. Since it is a complicated magnetic circuit, consisting of parts, such as stator pole, stator yoke, air gap, rotor pole, rotor yoke, the total reluctance will be the sum of single reluctance along the magnetic

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Table 1. Specifications of 6/4 SRM for 2.2 kW centrifugal pump Item Core material Stator outer diameter (mm) Rotor outer diameter (mm) Shaft diameter(mm) Stack length (mm) Stator yoke width (mm) Rotor yoke width (mm) Stator pole arc (degrees) Rotor pole arc (degrees) Air-gap length (mm) DC voltage (V) Rated Torque (N.m) Rated Speed (rpm)

Quantity Non-oriented silicon steel 500N60 150 80 24 80 12.1 13.2 30 32 0.4 310 5.8 3600

Figure 3. Magnetic flux tubes to calculate the inductance at unaligned position.

path. The reluctance will be calculated by means of geometrical dimensions and magnetic permeability: R=

l Hl = S μ BS

(2)

where l is the length of the magnetic path, S is the area which is penetrated by the magnetic flux, magnetic permeability μ is given by the values of B and H in the B/H curve of the lamination material. According to the operation modes, the angle between the unaligned and aligned positions which is the equal to half the rotor pole pitch can be divided into three regions: a) fully unaligned to start of the pole overlap, b) starting of the pole overlap to full pole overlap, and c) full pole overlap to fully aligned condition of stator and rotor poles. Each region is then modeled by several flux tubes [4] (as shown in Fig. 3), the inductance of which is calculated at given phase current. Assume a certain flux density in the stator pole, and then flux densities in other parts of the machine such as rotor pole, rotor back iron, stator yoke, and air gap are derived as the areas of cross sections of these parts for assumed flux paths are obtained from the machine geometry and the assumed stator pole flux density. From the flux densities in various parts of the machine and the flux density vs. magnetic field inten-

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373

Figure 4. Flux-linkage versus current and rotor position.

sity characteristics of the lamination material, corresponding magnetic field intensities are obtained. Given the magnetic field intensities and length of the flux path in each tube, their product gives the magneto motive force (mmf). The mmfs for various parts are likewise obtained, and for the magnetic equivalent circuit and stator excitation Ampere’s circuital law is applied. If error between the applied stator mmf and that given equivalently by various parts of the machine reveals a discrepancy, then that error is used to adjust the assumed flux density in the stator pole and entire iteration continues until the error is reduced to a set tolerance value. Once the phase inductance L(θ , i ) is known, it can be multiplied with the phase current i (θ ) to get the phase flux linkage λ (i,θ ) as shown in Fig. 4, where θ is the angle of displacement of the rotor from the fully unaligned position. The energy conversion can be modelled by the flux linkage information. In order to maximize the power density at a fixed inverter volt-ampere rating, the SRM must be designed with a sufficiently narrow air gap [5]. After obtaining the flux-linkage from the analytical method, the curves can be described by linear and polynomial fit for unaligned and aligned position: λu (i ) = Au * i

(3)

λ a (i ) = B5 * i 5 + B4 * i 4 + B3 * i 3 + B2 * i 2 + B1 * i1 + A

(4)

Equation (4) is a linear equation where λu is the flux linkage at unaligned position and Au is linear coefficient, inductance for this position actually. Equation (5) is a five order polynomial which accurately determines the nonlinear relationship between flux linkage and current. Define F (i ) = λa (i ) − λu (i ) = B5 * i 5 + B4 * i 4 + B3 * i 3 + B2 * i 2 + ( B1 − Au ) * i + A

(5)

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λ

0.4

a

ΔW ΔW C

ΔW ΔW C

C

λ

C

u

Increment of co-energy (J)

Flux linkage Wb

λ

0.35 0.3 0.25 0.2 0.15 0.1 0.05

Δi

Δi

ΙP Δ i Current [A]

Δi

0

0

1

2

3

4

(a)

5

6

7 8 9 10 11 12 13 14 15 16 17 Current (A)

(b)

Figure 5. (a) Variation of co-energy increment with respect of exciting current. (b) the increment of the coenergy according to current evaluated by the proposed method.

And take derivative of it dF (i ) =0 di

(6)

By solving the above equation we can get the current I p at which co-energy increment ΔWc is maximum for the same current increment Δi as shown in Fig. 2. This value is defined as maximum co-energy increment current (MEIC), which is the optimum operating point of the motor. The increment of coenergy was shown in Fig. 5 (b). When the motor is under deep saturation, the increment of co-energy does not increase but decreases. This is because magnetic circuit at aligned position is under deep saturation and the flux- linkage only increases slightly but at unaligned position magnetic circuit is far from saturation so flux-linkage increases linearly with current. Excitation current should be at or a little larger than MEIC to take full use of co-energy increment when we design the motor. The advantage of this criterion is verified in the following sections. After the cross section of motor was designed, the saturation effect is the similar with same winding turns while stack length varies. This is illustrated by the increment of co-energy vs. current curves at various stack lengths in Fig. 3. The MEICs are almost constant in spite of stack length caused by saturation effects under same electric loading. The dimension of cross section is given in Table 1 and the winding per pole has 80 turns. The MEIC I p = 13.5 A is derived from Eqs (5) and (6). Comparing the data in Table 2, the stack length 90 mm with winding of 80 turns is mostly suitable to the rated torque 5.8 N.m. 3. Modeling and Analysis of Switched Reluctance Drive When considering the performance of the designed motor, there are many criteria such as efficiency, maximum output power, thermal effect and so on among which efficiency seems to be mostly critical especially in the continuous operating apparatus.

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375

Increment of Co-energy (J)

0.5 0.4 0.3 70mm 80mm 90mm 100mm

0.2 0.1 0.0

0

5

10

15

20

25

30

35

Current (A)

Figure 6. Increment of co-energy with respect of the variation of current for different stack length. Table 2. SpeciMotor stack length and power at MEIC Turns per Pole

MEIC (A)

Saturate Current (A)

Co-energy (W)

Torque Tc (N.m)

60 70 80 90

18.1 15.4 13.5 12.1

9.03 7.74 6.52 5.48

3.1 3.01 3.08 3.13

5.81 5.75 5.78 5.90

Dynamic system is simulated by a computer program or a standard modeling package, such as Spice and Simulink, which usually ignore the effects of the multi-phase currents on the saturation, mutual inductance effects, electromagnetic transient phenomenon and its accuracy depends on the fitting of the static magnetization. A more accurate model was developed using FE method coupling with electrical circuit as shown in Fig. 7. The control algorithm and inverter are modelled in Matlab/Simulink environment and FEM model of motor is developed in commercial software package Flux 2D. The two models are connected by a coupling block supported by Flux 2D. In the control algorithm, the turn on angle is set to give enough current rising time for I p before overlap position and conducting angle is constant with 30°. The current and torque waveforms from simulation results are shown in Fig. 8. Accurate efficiency estimation needs more investigation on losses. The power losses in the electromechanical devices are mainly of three types: iron losses, copper losses and the mechanical losses. Copper losses can be easily calculated using phase current waveforms from simulation results (7), PCu = mRs

1 T i (t )2 dt T ∫0

(7)

where m is number of phases and Rs is resistance of winding. The iron loss is calculated by LS model introduced in [7] which is based on dynamic B-H characteristics associated with transient finite element simulation. It is more accurate compared with the method in [6].

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Figure 7. Dynamic FEM simulation coupling with external circuit.

Figure 8. Phase currents and torque from simulation.

J. Li et al. / Study on High Efficiency Switched Reluctance Drive for Centrifugal Pumping System

377

Figure 9.

The motor in Table 1 was analyzed by the above method and the efficiency of motor is 89.7%. Then the prototype of the designed motor was manufactured and tested. The motor and inverter are shown in Fig. 9. The mechanical losses were obtained by turning off the inverter at rated speed and the efficiency achieves 88.2% which is excluding mechanical losses. This is a little lower than simulation result because of the losses in the inverter. 4. Conclusions The design and analysis of high efficiency switched reluctance motor for pumping system have been proposed in this paper. The maximum co-energy increment current (MEIC) was defined after modeling and analyzing the flux linkage curves. The motor’s geometry was optimized according to the mechanical requirements of pumping system. then dynamic FEM simulation which can estimate the iron loss using a more accurate LS models was used to analyze both the motor and drive. And the performance was verified by experiments. References [1] M.N. Anwar, Iqbal Husain, and Arthur V. Radum, “A Comprehensive Design Methodology for Switched Reluctance Machines”, IEEE Trans. Industrial Electronics, vol. 37, No. 6, February 2001. [2] T.J.E. Miller, “Optimal Design of Switched Reluctance Motors” IEEE Trans. Industrial Electronics, vol. 49, No. 1, February 2002. [3] R. Krishnan, Switched Reluctance Motor Drives: Modeling, Simulation, Analysis, Design, and Applications. Boca Raton, CRC: 2001, pp. 79–120. [4] N.K. Shen and K.R. Rajagopal, “Calculation of the flux-Linkage characteristics of a switched reluctance motor by flux tube method” IEEE Trans. Magnet, vol. 41, No. 10, October 2005. [5] Arthur V. Radun, “Design Considerations for the Switched Reluctance Motor”, IEEE Trans. Industrial Applications, vol. 31, No. 5, September/October 1995. [6] Peter N. Materu and Ramu Krishnan, “Estimation of Switched Reluctance Motor Losses,” IEEE Transactions on Industry Applications, vol. 28, No. 3, pp. 668–679, May/June 1992. [7] FLUX® 8.10 2D Application New features.

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Chapter C. Applications C3. Special Applications

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Power Quality Effects on Ferroresonance Luca BARBIERI, Sonia LEVA, Vincenzo MAUGERI and Adriano P. MORANDO Politecnico di Milano, Dipartimento di Elettrotecnica Piazza L. da Vinci, 32-20133 Milano (Italy) luca1.barbieri; [email protected] sonia.leva; [email protected] Abstract. Ferroresonance is a complex phenomenon very dangerous for electric power system. In this paper, the power quality effect on ferroresonant circuit is for the first time investigated by using nonlinear dynamics theory. In particular, the software package AUTO is used to determine the period solutions of the differential equation describing a typical voltage transformer circuit, supplied by distorted voltage, in which ferroresonance can occur.

Introduction Ferroresonance is a complex phenomenon – due to the interaction between a non linear inductance and a capacitance – very dangerous for electric power system. Research involving in transformers has been conducted over the last 80 years. The word ferroresonance is first seen in the literature in 1920 [1], although papers on resonance in transformers appears as early as 1907 [2]. Early analysis was done using graphical method, which appeared in American literature as early as 1938 [3]. More exacting and detailed works was done later by Hayashi in the 1950s [4]. Several works focuses on improvement of system and transformer models used with software such as EMTP [5]. Today the term ferroresonance is firmly established in the power system engineer's vocabulary and is used to not only describe the jump to higher current fundamental frequency state but also bifurcations to subharmonic, quasi-periodic and even chaotic oscillations in any circuit a non linear inductor. The connection of ferroresonance to nonlinear dynamics and chaos was established in 1988 and published in 1992 [6]. The application of nonlinear dynamics and chaotic theory in studying of a ferroresonant circuits is first seen in literature in 1990 [7] and in 1994 [8]. In case of ferroresonance, remnant magnetization of core, voltage at the time switching, and amount of charge on the capacitance are all initial conditions which determine the steady state response of the non linear dynamical system. Even with minutely small differences in initial condition, it is possible that subsequent initiations of ferroresonance may result in very different voltage waveforms. Bifurcation theory has proved to be the adequate mathematical framework for the study of non linear dynamical systems. This theory implies the calculation of a solution of system of ODE with respect to a parameter. The critical values of this parameter, where the type or number of solution of system changes, are called the bifurcation values. Different approaches exist to solve the system, amongst them the Galerkin method, which transforms the problem into frequency domain [9]. In this present work the software package AUTO [10] was used.

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L. Barbieri et al. / Power Quality Effects on Ferroresonance

CCB

DS1

DS 2

circuit breaker

e (t )

CBB

VT

Figure 1. Typical Voltage Transformer circuit arrangement.

e (t )

Csh

φ

i (t )

Cs

R

v (t ) =

dφ dt

i

Figure 2. Basic ferroresonance circuit.

There is an increasing interest in Power Quality (PQ) topics among both customers and utilities. The effects of the disturbances, mainly harmonics, interharmonics and unbalances, have been deeply investigated and in particular their negative effects have been highlighted in the technical literature in this field. Power losses, thermic effects, degradation of the electric insulation, interferences with electronic, control systems and telecommunication lines, voltage fluctuations and flicker are the most common subjects of interesting and well-established studies in the PQ bibliography. This paper investigates – for the first time – the influence of these disturbances that could decrease the quality level of the system on ferroresonance circuit. Our attention will be focused on single-phase electrical network in which is present a Voltage Transformer (VT). Figure 1 shows the typical VT circuit arrangement where CCB is the circuit breaker capacitance and CBB is the total bus bar capacitance to earth. Ferroresonance conditions occurred upon opening the circuit breaker with DS1 closed and DS2 open, leading to failure of the transformer primary winding [11].

System Configuration The basic ferroresonance equivalent circuit [11] subject of the present study is schematically depicted in Fig. 2. The resistor R represents transformer core losses, Csh is total phase-to-earth capacitance of the circuit and Cs is the circuit breaker grading capacitance. The φ-i characteristic of the transformer (Fig. 2) is simulated by the following seventh-order polynomial: i = aφ + bφ 7

(1)

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where a = 3.42, b = 0.41, i is the current in p.u. value and φ is the flux in the transformer core in p.u. value. The time behaviour of the ferroresonant circuit is described by the differential equation: Cs dv = dt Csh + Cs

2 E (ω cos (ω t ) + αωn cos (ωn t )) −

dφ =v dt

v

R ( Csh + Cs )

−

i ( Csh + Cs )

(2)

where E is the RMS of the fundamental harmonic supply phase voltage, α is the weight of the nth harmonic components supply phase voltage and ω is the angular supply frequency (ωn=nω). The software package AUTO computes and continuates the solutions of systems of algebraic and autonomous differential equations. The computation of periodical solutions can be treated as boundary value problem. A periodically forced system of order k can be transformed into an autonomous system by adding a stable non linear oscillator with the desired pulsation ωn:

( x − y(x

) +y )

x = x + ω n y − x x 2 + y 2 y = y − ω n

2

2

(3)

the solution of these equations are: x = sin(ωnt) and x = cos(ωnt). By coupling (3) to the system (2), a sixth-order system is formed that can be solved by AUTO. Simulation Results The system defined in (2) presents three different oscillation modes (three different periodical attractors): a normal response plus two possible types of ferroresonance conditions. Csh is taken as bifurcation parameter. All the results are obtained by detailed numerical simulation, and preliminary analyses are based on mathematical theories and method of [12]. At first, the supply voltage is supposed sinusoidal then a distortion harmonic component is considered. In the following analysis, instead of using actual values of circuit parameter E, ω, Cs, Csh, etc., system equations are made dimensionless by employing per unit value. Sinusoidal Supply Voltage In this first case n = 0 and α = 0 in (2) and (3). First of all, the bifurcation diagram is given in Fig. 3 by using AUTO [10]. There are seven types of bifurcations and three different branch of solutions, one for each attractors. The branch A is obtained from the continuations of the normal sinusoidal response (Fig. 4). At Csh* = 0.03 nF a Limit Point (LP) bifurcation is detected. In this point the stable periodic solution (Fig. 3 full line) coalesces and obliterates with unstable periodic solution (Fig. 3 dashed line).

384

L. Barbieri et al. / Power Quality Effects on Ferroresonance 3.1

3.5

B

C 3

3.05 D

max ( flux ) [p.u.]

max ( flux ) [p.u.]

2.5

LP

2

1.5

PD

3 C PD

2.95

1

2.9

A

0.5

0

0

0.5

1

1.5

Csh [nF]

2

2.5

3

2.85 -0.05

0

0.05

0.1 Csh [nF]

0.15

0.2

Figure 3. Bifurcation diagram at Cs = 0.5 nF, R = 225 MΩ (full line for stable solution, dashes line for unstable solution).

0.1

0.1 phi [p.u.]

0.2

v [p.u.]

0.2

0 -0.1 -0.2 12.66

0 -0.1

12.67

12.68 time [s]

12.69

12.7

-0.2 12.66

12.67

12.68 time [s]

12.69

12.7

Figure 4. Normal sine wave response. Cs = 0.5 nF, R = 225 MΩ, Csh = 3 nF, E = 1 p.u. 4

3 2

2 phi [p.u.]

v [p.u]

1 0

0

-1

-2 -2 -3 12.66

12.67

12.68 time [s]

12.69

12.7

-4 12.66

12.67

12.68 time [s]

12.69

12.7

Figure 5. Fundamental frequency ferroresonance. Cs = 0.5 nF, R = 225 MΩ, Csh = 3 nF, E = 1 p.u.

From the continuations of fundamental frequency ferroresonance are not detected bifurcations (Fig. 3 attractor B). The solution remains stable for all range of Csh value. Operation in ferroresonance region is demonstrated by the high amplitude of transformer voltage waveform and by the presence of high frequency harmonic components in the voltage waveform power density spectrum (Fig. 5). The branch C is obtained from the continuations of the subharmonic ferroresonance (Fig. 6), the resulting waveform is still periodic, but with a period twice the period of the supply cycle. We found four limit point bifurcation and two period doubling

385

3

3

2

2

1

1

phi [p.u.]

v [p.u.]

L. Barbieri et al. / Power Quality Effects on Ferroresonance

0

0

-1

-1

-2

-2

-3 12.62

12.64

12.66 time [s]

-3 12.62

12.68

12.64

12.66 time [s]

12.68

Figure 6. Subharmonic ferroresonance. Cs = 0.5 nF, R = 225 MΩ, Csh = 0.19 nF, E = 1 p.u. 0.6

3.5

B

C 3

0.5 0.4

LP

2

α

max ( flux ) [p.u.]

2.5

0.3

1.5

0.2 1

0

0.1

A

0.5

0

0.5

1

1.5

Csh [nF]

(a)

2

2.5

3

0 0.03

0.031

0.032 0.033 Csh* [nF]

0.034

0.035

(b)

Figure 7. (a) Bifurcation diagram at Cs = 0.5 nF, R = 225 MΩ (full line for stable solution, dashes line for unstable solution); (b) Bifurcation diagram in two-parameter space Cs-α.

bifurcation (PD). In this point the branch D (Fig. 3) of periodic-doubled solution emerges, and the original branch of stable periodic continues as a branch of unstable solution at the post-bifurcation. From Fig. 3 it is also possible note that the bifurcation diagram can be subdivided in two different region based on Csh value. In the first region, where Csh < 0.03 nF, only ferroresonance oscillations can occur. In the second region, for Csh > 0.03 nF, both the normal and the ferroresonant oscillation can occur. In such cases the steady state oscillation mode will be determined by initial conditions. Distortion Supply Voltage For what concern the presence of harmonic components, different cases changing the harmonic order (n = 2 and n = 7) and the weight of the considered harmonic (α = 0.01, 0.1, 0.5) have been analyzed. For each one has been calculated the bifurcation diagram by using AUTO. In Figs 7 and 8 the n = 7, α = 0.1 case is shown. By analyzing the obtained results it is possible to note that the bifurcation diagram is substantially equal to the one obtained in the sinusoidal case. Also in this case there are seven types of bifurcations and three different branch of solutions, one for each attractors. The three A, B and C branches have the same meaning that in the sinusoidal case:

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L. Barbieri et al. / Power Quality Effects on Ferroresonance

0.2

0.2

0.15

0.15

flux [p.u.]

0.1 0.05

v [p.u.]

0.1 0.05 0

0

-0.05

-0.05

-0.1

-0.1

-0.15

-0.15 -0.2

-0.2 12.66

12.665

12.67

12.675

12.68 12.685 time [s]

12.69

12.695

12.66

12.7

12.665

12.67

12.675

12.68 12.685 time [s]

12.69

12.695

12.7

12.665

12.67

12.675

12.68 12.685 time [s]

12.69

12.695

12.7

(a) 3

4 3

2

2 1 flux [p.u.]

v [p.u.]

1 0

0 -1

-1

-2

-2 -3 12.66

-3

12.665

12.67

12.675

12.68 12.685 time [s]

12.69

12.695

-4 12.66

12.7

3

2

2

1

1 flux [p.u.]

v [p.u.]

(b) 3

0

-1

0 -1

-2

-2

-3 12.62

12.63

12.64

12.65

12.66 time [s]

12.67

12.68

-3 12.62

12.69

12.63

12.64

12.65

12.66 time [s]

12.67

12.68

12.69

(c) Figure 8. (a) Normal wave response Cs = 0.5 nF, R = 225 MΩ, Csh = 3 nF, E = 1 p.u.; (b) Fundamental frequency ferroresonance. Cs = 0.5 nF, R = 225 MΩ, Csh = 3 nF, E = 1 p.u.; (c) Subharmonic ferroresonance. Cs = 0.5 nF, R = 225 MΩ, Csh = 0.19 nF, E = 1 p.u.

− − −

A represents the normal wave response: the voltage v has the same waveform of the power supply, v and φ have low amplitude. B represents the fundamental frequency ferroresonance: v and φ have high amplitude with the same period of the power supply. C represents the subharmonic ferroresonance: v and φ have high amplitude but with a period twice the period of the power supply.

Furthermore, the flux and voltage waveforms (see Fig. 8) – distorted for the presence of harmonic components – have amplitude only a few different from the previous waveforms.

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387

As AUTO has the possibility of freeing a second parameter, the bifurcation point in a two-parameter space may be determinate. This can be applied to the continuation of LP represented in Fig. 7a: the goal is analyze the behaviour of the capacitance value Csh* as a function of α between 0 and 0.6. The result of the continuation of this LP is shown in Fig. 7b. Also in this case it is possible to note that the presence of harmonic components does not give big difference. It is possible conclude that concerning on the ferroresonance phenomenon the presence of harmonic components does not give any contribution. Conclusions The influence of the PQ-disturbances decrease the quality level of the voltage and current of the system but on the point of view of ferroresonance phenomenon, the harmonic components change only a few the bifurcation diagram. Furthermore, by confirming the limited influence of the harmonic components on the ferroresonance, the Csh value that individuates the two different working region characterised by only ferroresonance oscillations and normal/ferroresonant oscillation change in imperceptible way as a function of the harmonic components RMS value. In the near future this work will be extended to the study the PQ effect on the bifurcation analysis of three-phase ferroresonant circuit. Besides more attention will be spent to the accurate modelisation of voltage transformer and high power transformer. References [1] P. Boucherot, Existence de Deux Régime en Ferrorésonance, R.G.E., pp. 827-828, December 10, 1920. [2] J. Bethenod, Sur le Transformateur et Résonance, L’Eclairae Electrique, pp. 289-296, November 30, 1907. [3] P.H. Odessey and E.Weber, Critical Condition in Ferroresonance, Trans. AIEE, vol. 57, pp. 444-452, 1938. [4] C. Hayashi, Nonlinear Oscillations in Physical System, McGraw-Hill Book Company, New York, NY, 1964. [5] D.L. Stuehm, B.A. Mork and D.D. Mairs, Five-Legged Core Transformer Equivalent Circuit, IEEE Trans. Power Delivery, vol. 4, no. 3, pp. 1786-1793, July, 1989. [6] B.A. Mork, Ferroresonance and Chaos . Observation and Simulation of Ferroresonance in a FiveLegend Core Distribution Transformer, Ph.D. Thesis, North Dakota State University, May, 1992. [7] C. Kieny, Application of the Bifurcation Theory in Studying and Understanding the Global Behavior of a Ferroresonant Electric Power Circuit, IEEE Trans. Power Delivery, vol. 6, no. 2, pp. 866-872, April, 1990. [8] B.A. Mork and D.L. Stuehm, Application of Nonlinear Dynamics and Chaos to Ferroresonance in Distribution Systems, IEEE Trans. Power Delivery, vol. 9, no. 2, pp. 1009-1017, April, 1994. [9] C. Kieny, A. Sbai, Ferroresonance Study Using Galerkin Method with Pseudo-Archlength Continuation Method, IEEE Trans. Power Delivery, vol. 6, no. 4, pp. 1841-1847, 1991. [10] E.J: Doedel and J.P. Kernévez, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, Applied Mathematics Report, Californian Institute of Technologies, 1986. [11] Z. Emin, B.A.T. Al Zahawi, D.W. Auckland, Y.K. Tong: Ferroresonance in Electromagnetic Voltage Transformer: A Study Based on Nonlinear Dynamics, IEE Proc. Gener. Transm. Distrib., vol. 144, no. 4, pp. 383-387, July, 1997. [12] Y.A. Kuznetsov, Elements of applied bifurcation theory, Springer, New York, 1995.

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FEM Computation of Flashover Condition for a Sphere Spark Gap and for a Special Three-Electrode Spark Gap Design Matjaž GABER and Mladen TRLEP University of Maribor, Faculty of electrical engineering and computer science, Smetanova 17, SI-2000, Slovenia [email protected] Abstract. Electrical discharges in gases represent a complex problem in highvoltage techniques. The paper presents the findings which have been used to build a model of flashover in gases designed for use in engineering practice. The model is based on the electrostatic FEM calculation and on the value of the flashover electrical field intensity in air. The model has been used to calculate the flashover for a two-electrode sphere spark gap and a special three-electrode spark gap.

Introduction In classical theory Townsend [1,2] defined the physical picture of flashover by (1), where α, β, γ and d represent the first Townsend ionization coefficient, the electron attachment coefficient, the second Townsend ionization coefficient and the distance between the electrodes. Unfortunately, (1) applies for gases of low pressure only because some coefficients cannot be defined at normal air pressure [3]. γ

α [ exp(α − β )d − 1] = 1 α −β

(1)

As a consequence, it has become necessary to search for new approaches to electrical discharge in gases of pressures near the normal air pressure. These attempts produced a new theory that uses numerical mathematics and advanced computers, and has lately yielded some significant results [4–7]. However, the use of such models for the calculation of flashover is very demanding, so it is limited to scientific research purposes. In engineering practice, the condition for flashover is determined by using the flashover electrical field intensity condition Eb = 3 MV/m only. This means that the flashover takes place if the value of the electrical field intensity exceeds the flashover electrical field intensity at any point of the calculation. This condition often turns out to be insufficient, so we decided to find the criteria that would determine the flashover in air more precisely and be more suitable for engineering practice. The conditions in air vary due to the changes in air humidity, temperature and pressure. As a result, the mean free path of electron λ e ≈ 10−5 m and the ionization energy of the average molecule in air Wion ≈ 5 ⋅10−18 J are slightly changed. Thus the

M. Gaber and M. Trlep / FEM Computation of Flashover Condition for a Sphere Spark Gap

389

flashover electrical field intensity is Eb ≈ 3 MV/m. We decided to test this condition on some examples of standard two-electrode sphere spark gaps.

Computational Determination of Flashover Voltage for a Two-Electrode Spark Gap We selected some standard sizes of sphere spark gaps from [8], modeled them and used the obtained models in our calculations. The diameters of the selected spheres were 2 cm, 10 cm, 50 cm and 200 cm. Having in mind the speed of computations, we decided to use triangles of the first order-degree for the FEM mesh. The mesh contained 20000 to 40000 of elements, depending on the distance between the electrodes. The size of finite elements in the space between the electrodes is proportional to the size of the electrodes, considering the distance between them. This approach simplifies the evaluation of the results obtained with different electrode diameters. Since in our computation we set the relative voltage of the low voltage (LV) electrode to 0 V and the relative voltage of the high voltage (HV) electrode to 1 V as the boundary condition on the electrode surfaces, we could use (2) to calculate the flashover voltage Ub. The maximum electrical field intensity Emax has been determined from the FEM calculation as the electrical field intensity in the finite element in which the calculated electrical field intensity is the greatest. As the flashover electrical field intensity Eb is not precisely defined, we made calculations for three assumed flashover electrical field intensity values of Eb: 2.8 MV/m, 3 MV/m and 3.2 MV/m. Ub =

Eb Emax

(2)

The calculated values were compared with the values defined in the standard for voltage measurements with sphere spark gaps [9]. In addition, we determined the percentage of discrepancy between the flashover voltage Ub and the standard value for the flashover voltage for electrical field intensity Eb = 3 MV/m. Figure 1 shows how the flashover voltage depends on the distance between two electrodes with a diameter of 2 cm. We can see that the calculated values of the flashover voltage are up to 50% lower than the values defined by the standard. The discrepancy decreases slightly with the increasing distance between the electrodes. The percentage of discrepancy of the calculated values shown in Fig. 1 would not change substantially even the 3% measuring uncertainty defined in the standard was considered. In Fig. 2 the same dependence is shown for a sphere spark gap with a diameter of 10 cm. Calculations were made only for some standard distances between the electrodes. The discrepancy between the calculated values and the values defined in the standard is about 10%. The calculated flashover voltage values are again lower than the standard values, and the discrepancy again decreases with the increasing distance between the electrodes. In Fig. 3 we can see the conditions in case of a sphere spark gap with a diameter of 50 cm. In the part where the accuracy of the standard value is 3%, the calculated values deviate from the standard values for less than 2%. In the part where the standard val-

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M. Gaber and M. Trlep / FEM Computation of Flashover Condition for a Sphere Spark Gap

Figure 1. Flashover voltage depending on the distance between the electrodes, sphere diameter is 2 cm.

Figure 2. Flashover voltage depending on the distance between the electrodes, sphere diameter is 10 cm.

Figure 3. Flashover voltage depending on the distance between the electrodes, sphere diameter is 50 cm.

ues are defined at a 5% accuracy, the calculated values deviate up to 10%. In the case of spheres with a diameter of 50 cm, the calculated values exceed the standard values, and the discrepancy increases with the increasing distance between the electrodes.

M. Gaber and M. Trlep / FEM Computation of Flashover Condition for a Sphere Spark Gap

391

Figure 4. Flashover voltage depending on the distance between the electrodes, sphere diameter is 200 cm.

Figure 4 shows the conditions for a sphere diameter of 200 cm. Again the calculated values exceed the standard values. In this case the discrepancy reaches up to 30%, and it increases with the increasing distance between the electrodes. The discrepancy between computational results and the standard values is surprisingly great. One of the observations derived from computational results is that the values of flashover voltage obtained with smaller spheres are always smaller than the values defined in the standard. Also, we can observe that the calculated flashover voltages obtained with large spheres always exceed the standard values. Further observations reveal that the discrepancy decreases with the increasing distance between small electrodes, but in contrast, it increases with the increasing distance in the case of large spheres. It should be noted that the number of finite elements varies little in the computations, so the elements are a hundred times larger if the spark gap is a hundred times bigger (linear dimension increment). At the same time it holds for the latter that the size of finite elements grows if the distance between the electrodes increases. Having in mind these findings, we made a dense-mesh calculation for the sphere with a diameter of 200 cm, which confirmed our assumption that the obtained flashover voltage would be lower. These findings lead to the conclusion that there exists an optimal surface (volume) where the average value of the electrical field intensity and the flashover electrical field intensity are equal, so the effective flashover can take place. If we look at the results we can see that the optimal surface is equal to the size of the finite element whose greatest electrical field intensity has been reached with a sphere with a diameter of 50 cm and a distance of 140 mm between the electrodes. When we are dealing with small spheres and short distances between them the flashover strongly depends on the presence of free electrons in air. In such cases the electrical field is non-homogeneous. As a consequence, ionization can take place only in a small volume (short distances and small diameters) and only a small number of new free electron generations are available. This explains why the number of all electrons depends so much on the number of the initial free electrons in case of flashover. Greater ionization volumes yield a greater number of generations, so the number of all electrons depends less on the number of the initial free electrons. In our calculation the number of the initial free electrons was not considered, but we assumed equal concentration for all cases.

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M. Gaber and M. Trlep / FEM Computation of Flashover Condition for a Sphere Spark Gap

Figure 5. Three-electrode spark gap: to the left the picture of the spark gap (1 – HV main electrode, 2 – LV main electrode, 3 – trigger electrode), to the right the results of computation.

Figure 6. Shapes of trigger electrode edges.

Flashover Voltage of a Three-Electrode Spark Gap The same condition as applied for calculating the flashover voltage was used to analyze the operation of a special three-electrode spark gap (Fig. 5) used in a current impulse generator [9]. If the trigger electrode is properly placed, this kind of spark gap has a wider bandwidth (3) than a spark gap with the trigger electrode placed in the main electrode [10]. ΔUW, UW max, UW min, Ub, Ub1 and Ub2 represent the bandwidth, the maximum working voltage, the minimum working voltage, the flashover voltage between the main electrodes, the flashover voltage between the upper main electrode and the trigger electrode, and the flashover voltage between the lower main electrode and the trigger electrode. It turns out that the widest bandwidth is obtained if Ub1 = 2Ub2 [11], which actually is the equipotential line. ΔUW =

UW max − UW min U b − U b 2 = UW max Ub

(3)

We took the basic dimensions for the three-electrode spark gap model from [10] and increased them linearly until the diameter of the main electrode was 10 cm and the distance between the main electrodes 27.8 mm. Calculations were made for an electrode with a sharp inner edge (Fig. 6, left) and an electrode with a rounded inner edge (Fig. 6, right). The obtained results were similar, the only difference was observed in the size of the bore in the electrode: the size of the bore in the electrode with a rounded edge is smaller (10 to 60 mm) than the bore in the electrode with a sharp edge (40 to 100 mm). Therefore, the conditions for the sharp-edged electrode will not be presented, except in Fig. 7. The impact of the shape of electrode edges on the conditions in the spark gap is shown in Fig. 7, where the diameter of the bore in the rounded electrode is

M. Gaber and M. Trlep / FEM Computation of Flashover Condition for a Sphere Spark Gap

393

Figure 7. Flashover voltage depending on the percentage of trigger electrode voltage, for rounded and sharpedged electrodes.

Figure 8. Flashover voltage depending on trigger electrode bore diameter for different voltages on trigger electrode.

33.3 mm and the bore in the sharp-edged electrode 88.9 mm. Despite the difference in the size of bores, a relatively good agreement of curves is observed. It is a consequence of high electrical field values on sharp edges. When the electrode with a rounded edge is placed in the middle of the distance between the main electrodes and the trigger voltage is 50% of the flashover voltage, we can see that at a certain bore size the flashover voltage stops changing with the distance (Fig. 8). In this case the flashover voltage is approximately 70 kV. If we look up in Fig. 2 we can see that the calculated flashover voltage of the spark gap without the trigger electrode is app. 70 kV. The conclusion is that in this case it is not reasonable to increase the bore diameter over 45 mm. If we did so, we would get a direct flashover between the main electrodes. Provided that the amplitude of negative voltage signal is high enough, the spark gap can nevertheless be used in certain cases, but the shrinkage of the effective bandwidth makes such use senseless. Figure 9 also shows the conditions in case of a rounded trigger electrode placed in the middle of the distance between the main electrodes. Here we can see that it is the best to use a trigger electrode with a bore diameter of about 45 mm because in this case the greatest possible part of flashover voltages is covered, which means 32 kV out of 70 kV at a bore diameter of

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M. Gaber and M. Trlep / FEM Computation of Flashover Condition for a Sphere Spark Gap

Figure 9. Flashover voltage depending on percentage of flashover voltage on trigger electrode for different trigger electrode bore diameters.

Figure 10. Flashover voltage depending on the actual trigger electrode voltage for different trigger electrode bore diameters.

44.4 mm. It should be noted that with respect to Fig. 2 the flashover voltages for the given case are about 10% lower than the expected measured values. The data presented in Fig. 9 are not suitable for laboratory use, so we recalculated the percentage record of the trigger electrode voltage into the actual trigger electrode voltage. The results are presented in Fig. 10. In all the cases calculated so far the trigger electrode was placed in the middle of the distance between the main electrodes. Figure 11 presents the results for the case where the trigger electrode is shifted 4.6 mm downwards. The peaks of curves are at different values of the trigger electrode voltages, which is explained by the fact that the position of the trigger electrode changes in relation to the equipotential line while its bore is increasing.

Conclusion The condition for flashover in air can be simply acquired with FEM-based electrostatic calculations where the size of finite elements has to be adjusted to the size of the spark

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395

Figure 11. Flashover voltage depending on the percentage of trigger electrode voltage for different diameters of trigger electrode.

gap. If the size of finite elements in the critical region is appropriate, the results can be corrected by comparison with the standard. In addition to the presence of the initial free electrons, the flashover depends on the pressure, humidity and temperature of the gas, which is included in the standard in the calculation of correction factors. The presented results will be confirmed by measurements.

References [1] M. J. Druyvesteyn, F. M. Penning, “The Mechanism of electrical discharges in gases of low pressure,” Reviews of modern physics, Vol. 12, No. 2, pp. 104-120, April 1940. [2] A. Pedersen, “Criteria for spark breakdown in sulfur hexafluoride”, IEEE Transactions on power apparatus and systems, Vol. PAS-89, No. 8, Nov./Dec. 1970. [3] J. M. Meek, “A theory of spark discharge,” Physical review, Volume 57, April 1940. [4] A. Fiala, L. C. Pitchford, J. P. Boeuf, “Two-dimensional, hybrid model of low-pressure glow discharges,” Physical review E Vol. 49, No. 6, June 1994. [5] J. P. Boeuf, “Characteristics of dusty nonthermal plasma from a particle-in-cell Monte Carlo simulation,” Physical review A, Vol. 46, No. 12, Dec. 1992. [6] Yu. V. Serdyuk, A. Larsson, S. M. Gubanski, M. Akyuz, “The propagation of positive streamers in a weak and uniform background electric field,” J. Phys. D: Appl. Phys.3,4 pp. 614-623, 2001. [7] M. Akyuz, A. Larrson, V. Cooray, G. Strandberg, “3D simulations of streamer branching in air,” Journal of electrostatics, No. 59, pp. 115-141, 2003. [8] IEC 60052, “Voltage measurement by means of standard air gaps,” Geneva, november 2002. [9] M. Gaber, J. Pihler, “Design of current impulse generator,” Electrotechnical Review, Vol. 73(1), pp. 53-58, 2006. [10] P. Osmokrović, N. Arsić, Z. Lazarević, D. Kušić, “Numerical and experimental design of threeelectrode spark gap for synthetic test circuit,” IEEE Transactions on Power Delivery, Vol. 9, No. 3, July 1994, pp. 1444-1450. [11] J. Dams, K. F. Geibig, P. Osmokrović, A. J. Schwab, “Computer aided design of three-electrode spark gaps,” 4th ISH, Athens 1983.

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Recent Developments in Magnetic Sensing Barbaros YAMAN a, Sadık SEHIT b and Ozge SAHIN c Dokuz Eylul University, Department of Electrical and Electronics Eng. 35160 Buca, Izmir-Turkey a [email protected], b [email protected], c [email protected] Abstract. Measurement systems must have some properties such as sensitivity, accuracy, stability and low power consumption. New sensor types are continuously being developed according to new needs of applications besides improving measurement systems. In this study, recently developed magnetic sensors are searched. Fundamentals of new magnetic sensor types, their advantages and application areas are presented and compared in a table to form a guide for magnetic sensor users.

Introduction Magnetic sensors sense magnetic field and generate electrical output according to its input. Basically, sensing capability and application areas of the sensor is used to classify the sensors. In industry, generally magnetic sensors are used in control and measurement systems, especially current sensing and position sensing. Indeed these sensors can not be used for special applications because of their sensitivity, size, power consumption and stability. At recent years, with development of the silicon technology, new sensor types are developed to be used at scientific researches and contribute to the new developed technologies.

SQUID Superconductor materials have a property as resistance decreases when temperature became lower. SQUID (Superconducting Quantum Interferometer Devices) is the most sensitive type of the magnetic sensors. SQUID magnetometers have an ultimate combination of field and spatial resolution [1]. Unsurpassed magnetic field sensitivity and wide bandwidth of the SQUID sensors allows creating a large variety of measurement systems with unique resolution for different applications [2]. A SQUID can be realized as: • It can measure magnetic flux on the order of one flux quantum. A flux quantum can be visualized as the magnetic flux of the Earth’s magnetic field (0.5 Gauss = 0.5 × 10–4 Tesla) through a single human red blood cell (diameter about 7 microns). • It can measure extremely tiny magnetic fields. The energy associated with the smallest detectable change in a second, about 10–32 Joules, is about equivalent to the work required to raise a single electron 1 millimeter in the Earth’ gravitational field.

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397

Figure 1. Schematic of SQUID.

A SQUID is essentially a magnetic flux to voltage transducer (Fig. 1). SQUIDs are amazingly versatile, being able to measure any physical quantity that can be converted to flux. DC SQUIDs are structures involving two Josephson junctions connected by a low inductance superconducting loop. The basic phenomena governing the operation of the SQUID devices are flux quantization in a superconducting loop and the Josephson Effect. SQUID sensors have been used in various applications like SQUID magnetometer for physical property measurements of small samples, non-destructive evaluation (NDE) of sub-surface defects, bio-magnetic evaluation of human heart and brain, geomagnetic prospecting, detection of gravity waves etc. [1,2].

High-Resolution, Chip-Size Magnetic Sensor Arrays Arrays of very small sensing device in a single chip are not new phenomena. This technology was used at the late of 70’s in the auto-focus mechanism of the 35mm cameras. Photodiodes were used at these systems. But in deed used in magnetic sensors in single chip array is a new phenomena. These nano-technology arrays of micron-sized magnetic sensors and sensor spacing on a single chip can be used to detect very small magnetic fields with very high spatial resolution [2]. The older magnetic sensor technologies as Hall-effect and Anisotropic Magnetoresisitive (AMR) were not able to be applied in these applications either due to size, high power consumption or low sensitivity issues. Giant magneto resistive (GMR) and Spin-Dependent Tunneling (SDT) makes it possible the production of these devices [3]. This kind of the devices are used at the credit cards, magnetic biosensors, magnetic imaging and application which are needed measurement of very small magnetic field and to detect the change in magnetic field. This kind of integrating technique reduces the effect of the noise and simplifies the sensor/signal processing interface [3]. The development of Giant Magnetoresistive (GMR) and Spin Dependent Tunneling (SDT) materials has opened up a new era of miniature, solid-state, magnetic sensors. These deposited, multi-layer materials exhibit large changes in resistance in the presence of a magnetic field. Their thin film nature allows the fabrication of extremely small sensors using traditional photolithography techniques from the semiconductor industry. Since these sensitive films can be deposited on semiconductor wafers, integrated sensors can be manufactured that incorporate both sensing elements and signal

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Figure 2. Schematic of MEMS resonant magnetometer.

processing electronics same chip. On-chip integration is especially important for single-chip sensor arrays. A single-chip, sensor array with a large number of sensors requires a correspondingly large number of off-chip connections unless on-chip multiplexing is utilized. The area of the bonding pads can dominate the chip area unless onchip electronics are used [2,3].

MEMS Magnetometers There is a military requirement to measure the magnetic bearing accurately to detect targets and navigational needs. MEMS-based magnetometers are based on the Lorentz force to cause movement of a resonant MEMS device. The change in amplitude of the resonance caused by the Lorentz force is sensed capacitively and related to the magnetic force which caused it [5]. The famous Lorentz equation F=qE+qvxB

(1)

defines the force on a charged particle (charge q) moving at a velocity v through a region where there is an electric field (E) and magnetic field (B) present. The magnetic force is perpendicular to both the local magnetic field and the particle’s direction of motion. No magnetic force is exerted on a stationary charged particle. The flow of an electric current down a conducting wire is ultimately due to the movement of electrically charged particles (in most cases, electrons) along the wire. Thus if we make a MEMS structure and pass a current down a conducting beam, in the presence of a magnetic field the Lorentz force will cause the free structure to move in a perpendicular direction [5]. This basic principle has been be applied to a magnetometer designed using MEMS techniques with appropriate electronic control. It consists of a shuttle mass suspended by two straight suspension arms, with differential capacitive pick-off to detect the position, and electrostatic actuators to force the device into resonance (Fig. 2). There is a positive feedback loop between the pickoff and electrostatic actuator. To measure the

B. Yaman et al. / Recent Developments in Magnetic Sensing

399

Figure 3. LVDT structure.

magnetic field an alternating current is passed down the suspension at the same frequency as the resonance. In a closed loop mode the electrostatic force could be adjusted to null the magnetic force. For the device to operate efficiently it has to be packaged in a vacuum to achieve a high Q [5,6]. MEMS technology can improve magnetic sensors by minimizing the effect of noise. The most successful MEMS products exploit one or more of the following characteristics: − advantageous scaling properties for improved device or system performance, − batch fabrication to reduce size and hopefully cost [5], − circuit integration to improve performance [2].

Spring Specialized LVDT Sensor An LVDT is a device that produces an electrical output proportional to the displacement of a separate movable core. The spring specialized LVDT measures the variations of the magnetic field to determine the change of spring position. In this device, there is no contact between movable coil and coil structure. This gives the LVDT an infinite life. The infinite mechanical life is also important in high-reliability mechanisms and systems found in aircraft, missiles, space vehicles and critical industrial equipment [8,9]. It consists of a primary coil and two secondary coils symmetrically spaced on a cylindrical form [8]. A free-moving, rodshaped magnetic core inside the coil assembly provides a path for the magnetic flux linking the coils (Fig. 3). When the primary coil is energized by an external AC source, voltages are induced in the two secondary coils. These are connected series opposing so the two voltages are of opposite polarity. Therefore, the net output of the transducer is the difference between these voltages, which is zero when the core is at the center or null position [9]. When the core is moved from the null position, the induced voltage in the coil toward which the core is moved increases, while the induced voltage in the opposite coil decreases. This action produces a differential voltage output that varies linearly with changes in core position. The phase of this output voltage changes abruptly by 180° as the core is moved from one side of null to the other [9].

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PLCD (Permanent Magnet Linear Contact-Less Displacement) PLCD-Displacement Sensors (Permanentmagnetic Linear Contactless Displacement Sensors) basically consist of a special soft magnetic core surrounded by a coil, wound around its entire length. On each end of the core there is a second short coil [10]. A permanent magnet guided close to the sensor causes localized magnetic saturation and thereby a virtual division of the core. The position of the saturated area along the sensor axis can be determinated by the coil system. The sensor is supplied by an external electronic module or by an integrated circuit which also produce the output signal. The output signal is linearly dependent on the position of the magnet. The signal can be conditioned by a Standard electronic module, which gives either a current output (4–20 mA) or a voltage output (0–5 V or 0–10 V) [10]. The most significant property of Displacement Sensors PLCD is their operation without any mechanical connection between driving magnet and sensor element. This property enables the driving of the sensor through walls made of nonmagnetic materials. The applications, for which the advantages of this quality are very significant, is the indication of the position of a piston in hydraulik or pneumatic systems. The permanent magnet is simply fixed on the piston and drives the PLCD sensor through the cylinder from e.g. aluminum or brass [8]. PLCD sensors have the following properties: • Continuous, contact less, linear displacement measurement, • No mechanical connection between driving magnet and sensor, • Magnetic operation adaptable to individual applications, • Control through partitions of non-ferromagnetic materials, • Large mounting tolerances [8].

Fiber Optic Magnetometers Some of the principal reasons for the popularity of optical fiber based sensor systems are [2] small size, light weight, immunity to electromagnetic interference (EMI), passive (all dielectric) composition, high temperature performance, large bandwidth, higher sensitivity as compared to existing techniques, and multiplexing capabilities. Moreover, the widespread use of optical fiber communication devices in the telecommunication industry has resulted in a substantial reduction in optical fiber sensor cost As a result, optical fiber sensors have been developed for a variety of applications in industry, medicine, defense and research. Some of these applications include gyroscopes for automotive navigation systems strain sensors for smart structures and for the measurement of various physical and electrical parameters like temperature, pressure, liquid level, acceleration, voltage and current in process control applications. Small, fiber optic-based magnetic field sensors could be used for many applications which are inaccessible to electrical based sensors. A variety of optic fiber based magnetic field sensors have already been developed. But these, in general, are relatively large or have low sensitivity when compared to small electrical based semiconductor Hall effect sensors or magnetoresistive sensors commonly in use.

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Table 1. Development and applications of magnetic sensors Magnetic Sensor SQUID

Development Year 1970–

GMR

1988

LVDT-PLCD

1990–

FIBER OPTIC

1991

MEMS

2001

Application Laboratory instrumentation, biomedical applications (magnetoencephalography, magnetocardiology, etc), military applications (for submarine and mine detection), in geophysics (from prospecting for oil and minerals to earthquake prediction) and in non-destructive evaluations. [2] Imaging of magnetic media Magnetic bioassay Non destructive testing Magnetic couplers Hard Disk Heads A spring specialized LVDT sensor is used at [10]. Automotive: Shock Absorbers Suspension Brake systems Industrial automation Professional balances Dynamometers Load cells Vibrations monitoring Transportation and aerospace [8] PLCD is used for indication of the position of a piston in hydraulic or pneumatic systems. Measuring low level magnetic fields and strain produced in a magnetostrictive material in the presence of an external magnetic field. MEMS technologies include many microsensors (e.g., inertial sensors, pressure sensors, magnetometers, chemical sensors, etc.), microactuators (e.g., micromirrors, microrelays, microvalves, micropumps, etc.), and microsystems (e.g., chemical analysis systems, sensorfeedback- controlled actuators, etc.) [2]

Conclusion Magnetic field measurement has become an essential procedure in many industrial, scientific and defense projects today. Weather satellites routinely map the earth’s magnetic field for scientific information. Airplanes and ships use the earth’s magnetic field to compute direction as well as altitude, in case of airplanes. Measuring magnetic field is one of the primary methods of estimating high voltages and currents in industrial environments. Magnetic field detection is of much interest to defense scientists, particularly in the area of minefield and submarine detection. All the above reasons have made the development of reliable, rugged, extremely sensitive magnetic field sensors, highly essential. This article includes information on recently developed magnetic sensor types. One of them is SQUID, the most sensitive magnetic sensor, is used in very low magnetic field detection. Another type is GMR, development version of the AMR. GMR produced by using silicon technology reduces size. Other types PLCD and LVDT remove the effect of the contact and reduce the mechanical problems and effective using range. MEMS has advantages as scaling properties for improved device or system performance; batch fabrication to reduce size and hopefully cost. MEMS were developed in 2001. These sensor types are compared in Table 1.

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These sensors are generally used in specific systems. In future they may be of more common use. But development of materials technology is important parameter for development of these sensors since materials restrict the performance of devices.

References [1] SQUIDs–Highly sensitive magnetic sensors, Materials Science Division, Indira Gandhi Centre for Atomic Research. [2] James Lenz and Alan S. Edelstein, Magnetic Sensors and Their Applications, IEEE Sensors Journal, Vol. 6, N. 3, June 2006. [3] Dr. Carl H. Smith and Robert W. Schneider, Chip-Size Magnetic Sensor Arrays, NVE Corporation, Eden Prairie, MN. [4] J. Moreland, J. Kitching, P.D.D. Schwindt, S. Knappe, L. Liew, V. Shah, V. Gerginov, Y.-J. Wang and L. Hollberg, Chip Scale Atomic Magnetometers, Time and Frequency Division National Institute of Standards and Technology Boulder, Colorado. [5] D.O. King, K.M. Brunson, A.L. McClelland, R.J.T. Bunyan, MEMS Magnetometers and Gradiometers for Magnetic Compasses with Bearing Correction, IEE seminar on MEMS Sensor Technologies, April 2005. [6] Jack W. Judy and Nosang Myung, Magnetic Materials for MEMS, University of California, Los Angeles, CA, USA. [7] http://www.hitachigst.com/hdd/research/recording_head/headprocessing/index.html. [8] Derek Weber & Enrico Giorgione, New Applications for Magnetic Based Sensors, Plcdand Lvdt Principles and Applications, Inprox Sensors. [9] Schaevitz, LVDT Technology, LVDT Functional Advantages and Operation Principles. http:// www.schaevitz.com/. [10] PLCD Displacement Sensor for Industrial Applications Tyco Electronics. http://www. tycoelectronics.com.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-403

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Modelling of Open Magnetic Shields’ Operation to Limit Magnetic Field of High-Current Lines R. GOLEMAN, A. WAC-WŁODARCZYK, T. GIŻEWSKI and D. CZERWIŃSKI Institute of Electrical Engineering and Electrotechnologies, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, 20-618 Lublin, 38A Nadbystrzycka St.,Poland E-mail: [email protected] Abstract. The paper presents the analysis of magnetic field around high-current line (50 Hz) and the influence of the shields on spatial distribution of the field. The comparison of the systems with open ferromagnetic single shield for each line and the systems with common shield has been made. Open shields are characterised by the air gap located above the conductor or current line. Ferromagnetic shields are made of transformer plate M117-30P 5. The calculations have been made for the shields of defined angles i.e. 90°, 120° and 150° and various thickness.

Introduction In our surroundings human beings have been influenced by various kinds of electric and magnetic fields of different frequencies. Such fields at controlled values of the intensity, frequency and time of treatment may have positive impact on human organisms or they may be used in diagnostics. Low-magnetic fields are neutral for people but in most cases high-magnetic fields are dangerous. Apart from biological influence on living organisms, artificially generated magnetic and electric fields have hazardous impact on a wide range of sensible technical devices that cause disturbances or even make their operation impossible. The protection against the influence of electromagnetic field is difficult in particular in case of lower intensity and low frequency level. Magnetic field generated by transformers and high-current lines rarely exceed the safety level however it may disturb the operation of other technical devices e.g. it may interact on the deflection of the monitors situated in the areas of considerable concentration of field intensity. Such interaction is characterised by lines distortion and vibration of the image displayed by the monitor which make the operation difficult. Shielding buses are usually applied to reduce the influence of the magnetic field generated by currents in the buses on the surroundings. The paper presents the analysis of magnetic field around high-current line and the influence of the shield on spatial field distribution [1–5]. Open ferromagnetic shield systems [5], individual for each bus (Fig. 1) and the systems with one common shield for all buses (Fig. 2) were compared. Open shield are characterised by the air gap above the conductor or current line. Ferromagnetic shields are formed by M117-30P5 transformer sheets. The calculations were carried out for the shields of characteristic angles like 90°, 120°, 150° and different thickness.

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R. Goleman et al. / Modelling of Open Magnetic Shields’ Operation

Figure 1. Section of tested bus-shield system, shielding of particular phases.

Figure 2. Section of tested bus-shield system, common bus shielding for three phases.

Figure 3. Schematic diagram of analysed system with magnetic shield: 1–current buses, 2–shield.

Bus Shielding System with Open Magnetic Shield The system of current buses made of aluminium was selected for the analysis (Fig. 3). The assumption was made that the current intensity in the bus is 3 kA. Vector potential in the system with the shield can be described by the following equations: in current bus

⎞ ⎛ 1 rot⎜⎜ rot A ⎟⎟ = J ⎠ ⎝ μ0

(1)

⎛ 1 ⎞ rot⎜⎜ rot A ⎟⎟ = 0 ⎝ μ0 ⎠

(2)

in air

R. Goleman et al. / Modelling of Open Magnetic Shields’ Operation

405

in magnetic shield

⎞ ⎛ 1 rot⎜⎜ rot A ⎟⎟ = 0 ⎠ ⎝ μ0 μr

(3)

where: A – vector potential, J – current density vector, μ 0 , μ r – magnetic permeability of vacuum and relative permeability of the shield, γ – conductivity, ω – pulsation. Current density in conductive medium, in two-dimensional field is described by the following relation

J = −γ

∂A − γ grad V ∂t

(4)

Assuming that the value of scalar potential V is constant in cross-section of the conductor and taking the relation (4) into consideration as well as the following relation

U = − l ⋅ gradV

(5)

voltage between conductor’s ends is

U = R i + Rγ

∂A

∫ ∂t d s

(6)

s

where: R – conductor’s resistance determined at direct current, l – conductor’s length, s – conductor’s section. According to II Kirchhoff’s law for current circuits the following equation can be written U L1 −U L3 = ( RL + Robc )(iL1 − iL3 ) + RL γ( ∫

∂AL3 ∂AL1 ds−∫ d s) ∂t ∂t s

(7)

U L2 −U L3 = ( RL + Robc )(iL2 − iL3 ) + RL γ( ∫

∂AL3 ∂AL2 ds−∫ d s) ∂t ∂t s

(8)

s

s

Currents in buses fulfil the equation

i L1 + i L2 + i L3 = 0

(9)

At the assumption that current line and the shields are long enough, 2D analysis would be sufficient. The equations (1–9) were numerically solved with the application of FLUX 2D package. Cyclic boundary conditions were taken for the calculation of triple-phase model of shielded buses. They enable to calculate vector potential beyond the region limited by boundary curve taking into account its decay into infinity. Computation results have been presented as maps of magnetic flux density (Figs 4, 5) and charts of maximum values of magnetic flux density (Figs 6–10).

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R. Goleman et al. / Modelling of Open Magnetic Shields’ Operation

Figure 4. Distribution of magnetic flux density around buses with single magnetic shields for ω t = 90°.

Figure 5. Distribution of magnetic flux density around buses with common magnetic shield for ω t = 90°. 0.006 alpha=90 alpha=120 alpha=150

0.005

B, T

0.004 0.003 0.002 0.001 0 0

1000

2000

3000

x, mm

Figure 6. Distributions of maximum value of magnetic flux density in plane A (100 mm) of current line with single magnetic shields of different values of the angle α = 90°, α = 120°, α = 150° and thickness of the wall: 5 mm.

R. Goleman et al. / Modelling of Open Magnetic Shields’ Operation

407

0.0004

B, T

0.0003

0.0002

alpha=90 alpha=120 alpha=150

0.0001

0 0

1000

2000

3000

x, mm

Figure 7. Distributions of maximum values of magnetic flux density in plane A (1000 mm) of current line with single magnetic shields of different values of the angle α = 90°, α = 120°, α = 150° and thickness of the wall: 5 mm. 0.004

B, T

0.003

0.002

0.001 alpha=90 alpha=120 alpha=150

0 0

1000

2000

3000

x, mm

Figure 8. Distributions of maximum values of magnetic flux density in plane A (100 mm) of current line with common magnetic shield of different values of the angle α = 90°, α = 120°, α = 150° and thickness of the wall: 5 mm.

The charts of magnetic flux density were determined in two planes located under the shield marked as A in Figs 4 and 5 distant from conductors’ axis of 100 mm and 1000 mm respectively. Conclusions In the paper the influence of open magnetic shields on magnetic field distributions around three phase buses was investigated. Open magnetic shields can be used to limit

408

R. Goleman et al. / Modelling of Open Magnetic Shields’ Operation 0.01

0.008 without shield single shields common shield

B, T

0.006

0.004

0.002

0 0

1000

x, mm

2000

3000

Figure 9. The comparison of the distributions of maximum values of magnetic flux density in plane A (100 mm) in the systems: non-shielded with individual shields and with common shield (α = 120°, thickness of shields’ wall: 5 mm).

0.0005

0.0004

B, T

0.0003

0.0002 without shield single shields common shield

0.0001

0 0

1000

2000

3000

x, mm

Figure 10. The comparison of the distributions of maximum values of magnetic flux density in plane A (1000 mm) in the systems: non-shielded with individual magnetic shields and common shield (α = 120°, thickness of shields’ wall: 5 mm).

magnetic field only in this part which is directly in front of the shield. In such systems the intensity of the field in the area above the shield increases and the highest values occur at the edges of the shield. The distribution of magnetic flux density under the shield depends on shield type and the distance from its surface. In the system with separate shields for each bus in the protected zone close to the surface of the shield the highest value of magnetic flux density was noticed under central bus however under remaining buses the values are lower of about 25%. Negligible increase of magnetic flux density appears also under shields’ edges of extreme buses.

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409

In the system with common shield, in the area close to the surface of the shield the highest value of magnetic flux density was noticed under shield’s edges whereas under central bus the system is well shielded. Along with the increase of the distance from conductor’s axis the distribution of magnetic flux density under the shields tends to be more steady, the value of magnetic flux density increases and shielding efficiency is lower. The change in the length of shield’s gap which corresponds to the increase of the shield’s angle from 90° to 150°, has reasonable impact on shielding efficiency. In case of individual shield it results in increased shielding efficiency of about 8% at the points located under the shield in the symmetry axis. This influence is noticeable in case of common shield. Assumed change of shield’s angle corresponds to the increase of the shielding efficiency of about 30% at the plane points mentioned above. The change in magnetic shield’s thickness in the range from 5 to 10 mm does not influence magnetic flux density distributions and shielding efficiency since in considered models the shields were not subject to saturation. The simulations revealed that the system with common magnetic shield has better shielding efficiency. It’s shielding coefficient at the points located in symmetry plane of the system and located 25–925 mm from the shield varies from 0,12 to 0,65 and is lower of about 0,2–0,3 compared to the system with individual shields. References [1] K. Bednarek, R. Nawrowski, A. Tomczewski, Electromagnetic compatibility in the neighbourhood of high-current lines, X International Symposium on Electromagnetic Fields in Electrical Engineering, ISEF 2001, pp. 423-428. [2] O. Bottauscio, M. Chiampi, D. Chiarabaglio, M. Zucca, Use of grain-oriented materials in low-frequency magnetic shielding, Journal of Magnetism and Magnetic Materials, Vol. 215-216, pp. 130-132, 2000. [3] R. Goleman, J. Szponder, Reduction of low frequency weak magnetic fields using laminar screens, X International Symposium on Electromagnetic Fields in Electrical Engineering, pp. 267-270, 2001. [4] T. Rikitake, Magnetic and Electromagnetic Shielding, Terra Scientific Publishing Company, Tokyo 1987. [5] K. Wassef, V.V.Varadan, K.K.Varadan, Magnetic field shielding concepts for power transmission lines, IEEE Trans. Magnetics, Vol. 34, No. 3, pp. 649-654, 1998.

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Selected Problems of the Flux Pinning in HTc Superconductors J. SOSNOWSKI Electrotechnical Institute, 04-703 Warsaw, Pozaryskiego 28, Poland Abstract. The applications of the HTc superconductors in electricity depends among other on the current capability of the superconducting materials, especially in the form of wires and tapes. In the paper is investigated the influence of the electromagnetic interaction of the vortices with the nano-sized pinning centers on the critical current of HTc superconductors. The capturing interaction is considered taking into account the change in the free energy density arising during the motion of pinned vortex against it’s equilibrium position. The influence of the pinning interaction on magnetic induction distribution and flux trapped is considered too for ceramic HTc superconductors.

Introduction High temperature oxide superconductors (HTs) were discovered more than twenty years ago. Now it is time therefore for numerous applications of these materials in electric devices. Unique electromagnetic phenomena appearing in them should be taken into account before constructing the superconducting device. In the paper is investigated the influence of the nano-sized pinning centers on the critical current of the HTc superconductors. It is considered too related mechanism of the flux trapping in these materials, which giant value allows to treat HTc superconductors as promising permanent magnets very useful in magnetic levitation process, as magnetic bearing, motors etc. We mention that word record of remanent magnetic moment reached just for HTc superconducting YBaCuO macromolecule attains the giant value of 17 T, even in not so low temperature of 29 K. This temperature is presently frequently obtained both using cryogenic liquids – hydrogen or without any liquids with help of cryocoolers.

Critical Current Analysis As follows from theoretical analysis especially based on the Ginzburg-Landau theory [1] superconducting state is characterized by the order parameter, which means that this state is energetically more favor than normal one. It denotes further that normal phase enhances the energy of system and therefore should be minimized. It is in fact the base of the proposed pinning interaction model. The movement of the captured on nanosized pinning center vortex with normal core of the radius equal to the temperature dependent coherence length ξ(T), at zero temperature ξ0, enhances the normal state energy of the superconductor. From the other side Lorentz force acting on the pinned vortex during current flow and elasticity forces tear off the vortices from the pinning centers, which leads then to their movement and dissipation effects. Critical current

J. Sosnowski / Selected Problems of the Flux Pinning in HTc Superconductors

411

Figure 1. Scheme of the vortex interaction at T = 0 with the nano-sized pinning center.

density is reached, while the potential barrier height against capturing the vortices vanishes. During the movement of the vortex on the distance x against the pinning center of the size d, as it is shown in the Fig. 1, the normal part energy of the superconductor with captured vortex increases initially according to the relation 1 valid for x < xc: U 2 ( x) =

2 ⎛ d ⎞ μ o H c2 l ⎡⎢ 2 d dξ πξ + dx − ξ 2 arcsin − 1− ⎜ ⎟ ⎝ 2ξ ⎠ 2 ⎢ 2ξ 2 ⎣

⎤ ⎥ ⎥ ⎦

(1)

while for larger declinations, it is for x > xc according to the following expression: U 3 ( x) =

μ o H c2 l ξ 2 ⎡⎢ π x x + arcsin + ⎢2 ξ ξ 2 ⎣

2 ⎛ x⎞ ⎤ 1− ⎜ ⎟ ⎥ ⎝ξ⎠ ⎥ ⎦

(2)

Critical value of the declination parameter xc is defined here by the formula: xc = ξ 1 − (

d 2 ) 2ξ

(3)

The pinning potential barrier which should be passed by the vortex during its movement on the distance x for both cases of x smaller or larger than xc is given now respectively by the expressions:

ΔU 2 ( x ) =

2

μ o H c ldx 2

(4)

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J. Sosnowski / Selected Problems of the Flux Pinning in HTc Superconductors

⎛ x⎞ ⎛d⎞ μ H lξ 2 ⎡⎢ x π d x d arcsin − + arcsin + 1− ⎜ ⎟ + 1− ⎜ ⎟ ΔU3 ( x) = o c ⎢ 2 2ξ ξ ξ 2 ⎝ ξ ⎠ 2ξ ⎝ 2ξ ⎠ ⎣ 2

2

2

⎤ ⎥ ⎥ ⎦

(5)

For determination of the total energy balance the potential of the Lorentz force as well as the elasticity energy of the vortex lattice should be taken into account. Capturing of the vortices by the nano-sized pinning centers causes the shift of the vortex from it’s equilibrium position in the regular vortex array, thus leading to an increase in the elasticity energy of the magnetic structure of the vortex lattice. This effect is the function of the declination of the individual vortex from it’s equilibrium position in the lattice. We have taken it into account by assuming that the enhancement of the vortex elasticity energy is proportional then to the square of the length of the vortex deflection from the equilibrium site., with coefficient of proportionality expressed by the value of the parameter α

U el =

2cs π ξ 2( ξ − x)

2

la

= α (ξ − x ) 2

(6)

Parameter cs is the corresponding elasticity shear modulus, and la ≈ l denotes the length on which the magnetic flux lines of vortices are distorted. This model leads therefore finally to the relation describing the potential barrier for flux creep process ΔU in the function of reduced current density I = j/jC, where jC is defined as transport current density j for which potential barrier disappears. ΔU(i) =

μoHc2lξ 2 z +αξ 2 1−i2 1−i2 −2 2

(

)

(7)

Parameter z appearing in Eq. (7) is defined according to the relation: z = arcsin

d d + 2ξ 2ξ

2

⎛d ⎞ 1 − ⎜⎜ ⎟⎟⎟ − i 1 − i 2 − arcsin i ⎜⎝ 2 ξ ⎠⎟

(8)

Hc is critical magnetic thermodynamic field, l pinning center thickness. Parameter α describes the elasticity energy of the vortex lattice, as was stated previously. Let’s insert now the expressions 7–8 into the constitutive relation describing generated electric field in the flux creep process just in the function of the potential barrier height ΔU, what allows us to predict the form of the current -voltage characteristics: ⎡ ⎡ ΔU ⎛ ⎛ ΔU ⎞⎟⎤ j ⎞⎤ 0 ⎜ ⎟⎟⎥ E = −Bω a ⎢⎢exp ⎢⎢− ⎜⎜1 + ⎟⎟⎟⎥⎥ − exp ⎜⎜⎜− ⎜ jC ⎟⎠⎦⎥ ⎝ k B T ⎠⎟⎥⎦⎥ ⎢⎣ ⎣⎢ k BT ⎝

(9)

The dependence of the real critical current i.e. filling the electric field criterion on material’s parameters is determined in this way. An example of the results of the

J. Sosnowski / Selected Problems of the Flux Pinning in HTc Superconductors

413

8

Jc(a.u)

6 4

T=10 K

2T

B=1 T

5T

2 0 0

0.5

1

1.5

2

10T 2.5 d/ξ 3

Figure 2. Influence of the nano-sized pinning centers dimensions d/ξ in reduced units on the critical current versus applied magnetic field.

25 Jc(a.u)

20 15

B=1T

10

B=5T

α (105 J/m2 )

B=10T

5 0 0

5

10

15

20

Figure 3. Influence of the vortex lattice elasticity parameter α on the critical current density versus applied magnetic induction.

calculations the influence of the pinning centers dimensions on the critical current in arbitrary units shown in Fig. 2 indicates that for too small pinning centers the capturing of vortices is not effective, while for larger dimensions critical current related to individual vortex – pinning center interaction saturates. Analogous calculations have been performed concerning the influence of the elasticity parameter of the vortex lattice α on the critical current, while results shown in the Fig. 3 also indicate on the importance of this effect for the critical current density, what should be interesting for superconducting tape technology. Numerical data presented in the Fig. 3 suggest that rigid vortex lattice does not allow for movement of the individual vortex, which stay in their equilibrium position in vortex lattice and therefore flux pinning and critical current density appropriately decrease for such case in an approximation of low concentration nano-sized pinning centers. Flux Trapping Analysis Described previously pinning interaction influences too the magnetic induction distribution and then the flux trapped magnitude. In the case of the free movement of the vortices the magnetization is reversible and therefore none remanent moment arises.

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J. Sosnowski / Selected Problems of the Flux Pinning in HTc Superconductors

Figure 4. The magnetic induction profile in the magnetic field cycle 0 → Bm → 0. The influence of the granular structure of the ceramic superconductors on magnetic induction distribution is shown in this picture.

However for HTc superconductors anomalous giant value of the flux trapped is observed, which allows to classify these materials as potentially very promising permanent magnets. Analysis of this problem is in this chapter performed. Set of Equations has been obtained, describing the dependence of the flux trapped Ftr on the amplitude of external magnetic induction in the cycle 0 → Bm → 0, taking into account the ceramic structure of HTc superconductors. Schematic view of magnetic induction profile in this case is shown in Fig. 4. Tooth-like shape of magnetic induction follows just from the approximation that in ceramic materials the superconducting grains are immersed in the matrix characterized by worse superconducting properties, which effect can be expressed too in the language of an existence weak Josephson’s junctions. Such shape of magnetic induction profiles reflects then appearance of large intra-granular currents and weak Josephson’s currents flowing between grains. As it was mentioned in the introduction the flux trapping value of the high temperature superconductors has essential meaning from the technical point of view. In the present chapter it will be performed an analysis of this material parameter by considering the nature of the generation flux trapped, taking into account especial features of the HTc superconductors. Performed analysis should be helpful therefore for finding necessary conditions for optimizing the magnitude of this very important parameter. In Fig. 4 it is drawn the variation of the magnetic induction profile inside the superconducting ceramics, taking into account the existence of their granular structure, which leads to different critical current values inside the grains and in surrounding matrix. As it is shown here for increasing magnetic field trapped in grains magnetic flux reduces total flux trapped value, while in decreasing one adds to the total trapped flux. Four regions have been considered in dependence on the maximal magnetic induction amplitude Bm in the magnetic field cycle, according to the profiles presented in the Fig. 4. First one has the place for maximal external magnetic field smaller than the first penetration field which is the superposition of the first critical magnetic field B c1/μ0 and magnetic potential barrier Δ. In this region magnetic induction does not penetrate generally inside superconductor and clearly the trapped magnetic flux Ftr vanishes. For higher amplitudes of increasing external magnetic field magnetic induction begins to penetrate the superconducting material and flux trapped per unit volume of the superconducting ceramic arises. This region of non-total magnetic induction penetration is given by the condition:

J. Sosnowski / Selected Problems of the Flux Pinning in HTc Superconductors

γ > Be1 > 0

415

(10)

where it has been introduced definition Be1 = Be – Bc1 – Δ. Parameter γ, which describes the value of the full penetration of magnetic induction inside the superconducting cylindrical sample is equal to:

γ = μ 0 jc R

(11)

Trapped magnetic flux normalized to the sample volume is described by the integral over the superconducting cylindrical sample from the magnetic flux profile in the cross section presented in the Fig. 4 and for the magnetic field range determined by Eq. (10), for time stable magnetic induction distribution is given by Eq. (12). In real cases magnetic induction profiles according to flux creep process are varying during the relaxation effects, which also will be in the paper discussed.

Ftr =

⎤ ( Be1 ) 2 ⎡ Be1 γ − + nBsg ⎥ ⎢ 2 2 2γ ⎣ ⎦

(12)

In Eq. (12) it has been taken into account explicitely, according to the previous considerations existence of the superconducting grains immersed inside the surrounding matrix. The grains concentration is described by the parameter n, while parameter Bsg is normalized to the grain’s cross-section and describes trapped flux in individual cylindrical grain. It is determined therefore by the relation.

Bsg = Bc1g +

γg 3

(13)

Bc1g is the first critical field, while parameter γg describing penetration of magnetic induction into grains, is defined as γ g = μ0 jcg Rg , where the radius of the superconducting grain is Rg, while critical current density inside grains jcg. Relations 12–13 were obtained in an approximation of the linear magnetic induction profiles, known frequently as Bean’s approach. For larger values of the magnetic induction amplitude, it is for the range 2γ ≥ Be1 ≥ γ flux trapped is equal to: Ftr =

3 ⎛ 2 nB sg ⎜⎛ 1 B e1 ⎟⎞ ⎟⎞⎟ γ 2 ( Be1 ) 2 ⎟⎞ 2 γ ⎜⎜ 1 ⎜⎛ ⎟ ⎟ B γ − − + − 1 − ⎜ e ⎜ ⎜ ⎟ ⎟ ⎟ 2 4 ⎠⎟ 3 ⎝⎜⎜ 2 ⎝⎜⎜ 2 γ ⎠⎟ ⎠⎟⎟ γ 2 ⎝⎜⎜

(14)

In the last case of the maximal magnetic field defined as Be1 ≥ 2γ , flux trapped γ + nBsg . Above model of flux trapping allows to deter3 mine remanent moment of the high temperature superconductors, very important parameter for describing the magnetic properties of ceramic superconducting material. An example of the computer calculations of the influence on flux trapped of filling ceramic

reaches constant value Ftr =

416

J. Sosnowski / Selected Problems of the Flux Pinning in HTc Superconductors

Figure 5. The influence of the filling the HTc superconductor with the superconducting grains (parameter n value) on the square root of the flux trapped in the magnetic induction cycle 0 → Bm → 0. Numbers at curves indicate value of the parameter n, defined at Eq. (12).

material with superconducting grains is shown in Fig. 5 and indicates on relevance of this effect. References [1] J. Sosnowski, Superconductivity and applications, Warsaw, Poland, Book Publisher of Electrotechnical Institute, 2003, pp. 1-200, in polish language.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-417

417

The Effect of the Direction of Incident Light on the Frequency Response of p-i-n Photodiodes Jorge Manuel Torres PEREIRA Institute of Telecommunications, Department of Electrical and Computer Engineering, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal E-mail: [email protected]

Abstract. This paper investigates the effects of the direction of the incident light on the transit time limited frequency response of p-i-n photodiodes. The simulation model starts from dividing the absorption region into any desired number of layers and, for each layer, the continuity equations are solved assuming that the carriers’ drift velocities are constant. The frequency response of the multilayer structure is calculated from the response of each layer using matrix algebra. The bandwidth is seen to increase when the light is incident on the p side. In this case the use of constant saturation velocities underestimates the device’s bandwidth.

Introduction In optical communication systems photodetectors are key elements on the receiver’s side. Photodiodes have been widely used because they are easy to fabricate and have good frequency response. The basic photodiode structure is of the p-i-n type and consists in a double heterojunction, which is known to have some advantages over the corresponding homojunction, namely lower dark current and noise. For the longwavelength range, i.e. 1.3–1.6 μm, the most important semiconductor materials are InP, for the n- and p-regions, and one element of the InGaAsP family, for the i-region. The choice of semiconductor for the i-region takes into account the wavelength of the radiation to be detected and makes sure that the semiconductor is lattice matched to InP. Both of these conditions are met by the ternary In0.53Ga0.47As, which is commonly used for the absorption region [1]. The optical system’s performance may be determined by the photodetector’s frequency response and therefore it is necessary to have a very detailed knowledge of its behaviour in the frequency domain. The frequency response of p-i-n photodiodes has been investigated by several authors using different numerical techniques and assumptions [2–5]. Analytical expressions for the frequency response of Si p-i-n photodiodes have been presented by assuming constant values for the carrier velocities [2]. This approximation is not valid for InGaAs/InP photodiodes with large absorption region’s width and/or small reverse bias voltage because the dependence of the carrier’s drift velocity on the electric field should be taken into account. An analytical solution for the frequency response of InGaAs p-i-n- detectors, that includes the influence of the electric field, may be expressed in terms of a frequency response function [3]. Numerical

418

J.M.T. Pereira / The Effect of the Direction of Incident Light

Φ

n+

n-

p+

InP

In0.53Ga0.47As

InP

Φ'

ith layer

i 0

xi-1

xi

a

x

Figure 1. Schematic of the p-i-n structure used in this work. The figure shows the light incident on the n+ (Φ) and on the p+ side (Φ').

approaches involving a finite element calculation [4], and the Ramo’s theorem [5] have also been implemented. A simple and powerful numerical technique that looks at the device as a sequence of spatially uniform layers, each one with analytical solutions, has been developed to obtain the frequency response of multilayer structures [6]. This technique has been applied to several types of devices and situations [7,8] and is used in this paper to investigate the effects of the direction of the incident light on the transit time limited frequency response of n+InP/n–InGaAs/p+InP photodiodes. The influence of the absorption layer width and bias voltage on the frequency response is also considered.

Device Structure and Modelling The p-i-n structure under study is shown in Fig. 1. The contact layers are of highly doped InP semiconductor. Between the contacts there is an absorption layer of In0.53Ga0.47As with width  a . The light is assumed to be absorbed only in the InGaAs layer. Under normal bias conditions the electric field, which extends throughout the absorption region, is responsible for the charge transport of the optical generated electron-hole pairs. In general the carriers’ drift velocities depend on the electric field, which is likely to change spatially in that region. By dividing the absorption region into a certain number of layers enables us to assign to each layer a constant value for the electric field. These discrete values of the electric field are then used to obtain the carriers’ drift velocities in each layer where they are assumed to be constant. This approach may be used for any known electric field profile, the accuracy depending on the number and size of layers. The numerical model starts by solving, for each layer and direction of light, the continuity equations which, in this case, have analytical solutions. In the frequency domain the electron and hole current densities for the ith layer, Jin (x,ω) and Jip (x,ω) respectively, obey the following equations [8]: d J in ( x, ω ) iω J in ( x, ω ) = + Gi ( x, ω ) vin dx

(1)

J.M.T. Pereira / The Effect of the Direction of Incident Light

d J ip ( x, ω ) iω J ip ( x, ω ) = − + Gi ( x, ω ) vip dx

419

(2)

where vin , vip are the electron and hole drift velocities respectively and Gi ( x, ω ) refers to the electron-hole generation rate due to optical absorption in the ith layer given by Gi ( x, ω ) = q αφ1 e −α x

( xi −1 ≤ x ≤ xi ) ,

(3)

for light incident on the n-side, and given by G i ( x, ω ) = q αφ1 e −α ( a − x )

( xi −1 ≤ x ≤ xi )

(4)

for light incident on the p-side. The parameter α is the absorption coefficient, q is the magnitude of the electron charge and φ1 is the amplitude of the sinusoidal input optical flux component. The ith layer may then be represented by a set of linear response coefficients   Ti , Si , Ri and Di which are used to compute the corresponding response coefficients  for the multilayer structure. The quantities Ti , Si are related to the electron and hole current densities through the equations ⎡ J ip ( xi ) ⎤ ⎡ J ip ( xi −1 ) ⎤  ⎢ ⎥ = Ti ⎢ ⎥ + Si ⎢⎣ J in ( xi ) ⎥⎦ ⎢⎣ J in ( xi −1 ) ⎥⎦ ⎡Tipp Ti = ⎢ ⎢⎣Tinp

Tipn ⎤ ⎥ and Tinn ⎥⎦

 ⎡ Sip ⎤ Si = ⎢ ⎥ ⎣ Sin ⎦

(5)

(6)

 The quantities Ri and Di are obtained from the equation

 ⎡ J ip ( xi −1 ) ⎤ pi (ω ) = RiT ⎢ ⎥ + Di ⎣⎢ J in ( xi −1 ) ⎦⎥

pi (ω ) =

∫

i

⎡⎣ J in ( x, ω ) + J ip ( x, ω ) ⎤⎦ dx .

(7)

(8)

 The quantity Di is a scalar whereas RiT = ⎡⎣ Rip Rin ⎤⎦ . The frequency response I(ω) may then be expressed as [6]: I (ω ) = ( D − Rn Sn ) (Tnn  a )

(9)

420

J.M.T. Pereira / The Effect of the Direction of Incident Light

where D, Rn, Sn, Tnn are the coefficients for the whole structure which may be obtained following a simple set of rules

Ti +1,i = Ti +1Ti    Si +1,i = Si +1 + Ti +1Si    RiT+1,i = RiT+1 Ti + RiT   Di +1,i = RiT+1 Si + Di +1 + Di

(10)

where the subscripts (i), (i+1) and (i+1,i) refer to the ith layer, (i+1)th layer and the union of the two layers respectively.   For the ith layer and light incident on the n-side, Ti , Si , Ri and Di are given by [8]: ⎡e−iωτ ip Ti = ⎢ ⎢⎣ 0

(

)

⎡ f iωτ ip ⎤  ⎥ Ri =  i ⎢ ⎢ f ( −iωτ in ) ⎥ ⎣ ⎦

0 ⎤ ⎥ eiωτ in ⎥⎦

(

)

⎡ f iωτ ip − α  i ⎤  ⎥ Si = qαφ1 i e −α xi ⎢ ⎢⎣ − f ( −iωτ in − α  i ) ⎥⎦

(11)

(12)

⎡ f (α  i ) − f ( −iωτ in ) Di = qαφ1 2i eα ( i − xi ) ⎢ − α  i + iωτ in ⎣

(

(13)

)

f (α  i ) − f iωτ ip ⎤ ⎥ − α  i − iωτ ip ⎥ ⎦

with τ ip =  i / vip , τ in =  i / vin and f (θ) = (1 − e −θ ) / θ . When the light is incident from  the p-side the expressions for Ti and Ri do not change but Si and Di are given by: ⎡ f iωτ ip + α  i ⎤  ⎥ Si = qαφ1 i e −α ( a − xi ) ⎢ ⎢⎣ − f ( −iωτ in + α  i ) ⎥⎦

(

)

(14)

⎡ f ( −α  i ) − f ( −iωτ in ) + Di = qαφ1 2i e−α  a e−α (  i − xi ) ⎢ −α  i + iωτ in ⎢⎣ f ( −α  i ) − f iωτ ip ⎤ ⎥ + α  i + iωτ ip ⎥ ⎦

(

)

(15)

In this work a linear electric field profile is assumed and may be expressed as [9],

421

J.M.T. Pereira / The Effect of the Direction of Incident Light 4

Drift velocity (×105m/s)

In0.53Ga0.47As

electron 2

hole

0

0

5

10

Electric field (MV/m)

Figure 2. Electron and hole drift velocities as a function of the electric field for In0.53Ga0.47As. E ( x) =

with U d =

2U d  2a

⎛ U −Ud ⎞ x+⎜ ⎟ ⎝ a ⎠

(U > U d )

(16)

qN d  2a . Ud is called the punchthrough voltage, U is the reverse bias volt2ε n

age, Nd is the residual donor concentration in the absorption region and εn is the InGaAs dielectric constant. The electron and hole drift velocities in the In0.53Ga0.47As are calculated, for each value of the electric field, from two empirical expressions that show very good agreement with experimental results [4]:

(

)(

v n ( E ) = μ n E + β v n E γ / 1 + β E γ

(

v p ( E ) = v p tanh μ p E / v p

)

)

(17)

with μ n = 1.05 m2V–1s–1; μ p = 0.042 m2V–1s–1; v n = 6 × 104 m/s; v pl = 4.8 × 104 m/s; β = 7.4 × 10–15 (m/V)2.5; γ = 2.5 and the electric field E expressed in (V/m). Figure 2 shows the electron and hole drift velocities as a function of the electric field for In0.53Ga0.47As.

Results and Discussion In the calculations we used α = 1.15 μm–1 (λ = 1.3 μm) and ε = 14.1ε0 for the ternary compound [1]. The results were obtained for an absorption region divided into 100 layers. However, due to the electric field profile and the carriers’ drift velocity type of dependence on the electric field, the results seem to stabilize for a number of layers as low as 10.

422

J.M.T. Pereira / The Effect of the Direction of Incident Light

Figure 3. Computed frequency response of two structures with  a = 1.5 μm and 3 μm.

Figure 4. Computed frequency response of one structure with  a = 3 μm for U = 6 V and when the saturation velocities are used.

Figure 3 shows the computed frequency response of two structures of the type described before with  a = 1.5 μm and 3 μm, impurity concentration Nd = 1021m–3, and bias voltage U = 10 V. For both directions of the incident light it is seen that shorter devices have a better frequency response which is mainly due to a decrease in the carriers’ transit time. However, for each structure, better frequency response is obtained when the incident light is on the p+ side. This result agrees with other published results and may be explained in terms of a larger electron-hole optical generation rate closer to the p+ contact so that it is the electrons which must travel across the absorption region [2]. As it is seen from Fig. 2 the drift velocity is higher for electrons than for holes. In Fig. 4, the computed frequency response of a structure with  a = 3 μm, Nd = 1021 m–3, and bias voltage U = 6 V is compared to the results obtained using the carriers’ saturation velocity values. When using the carriers’ saturation velocity the device’s bandwidth is seen to be overestimated or underestimated depending whether the light is incident from the n+ side or p+ side respectively. These results confirm the

423

J.M.T. Pereira / The Effect of the Direction of Incident Light

25.0 Light incident on the p-side

f3dB (GHz)

21.0 U =6V

17.0 13.0

U = 10 V

Light incident on the n-side 9.0 5.0 1.5

U =6V

N d = 1021 m −3

2.0

2.5

3.0

 a (μm) Figure 5. Computed bandwidth as a function of

a

for U = 6 V and U = 10 V.

type of dominant carrier in each case and may be explained in terms of the velocityfield relationship for the electrons and holes. In fact the electron’s average velocity in the absorption region is higher than its saturation velocity whereas the hole’s average velocity is lower than its saturation velocity. Figure 5 shows the bandwidth as a function of  a for U = 6 V and U = 10 V when the light is incident on the n- or p-side. The results show that, for a fixed  a , the bandwidth is always higher when light is incident on the p-side than when light is incident on the n-side. Furthermore, when the bias voltage changes from 6 V to 10 V, the bandwidth increases for light incident on the n-side and it decreases for light incident on the p-side. We may then conclude that the electron’s transit time determines the bandwidth when light is incident on the p-side and that the hole’s transit time determines the bandwidth when light is incident on the n-side.

Conclusions A numerical model has been implemented to investigate the effect of the direction of light on the frequency response of p-i-n photodiodes. This model may be applied to devices with an arbitrary electric field profile in the absorption region and non-uniform illumination. Better bandwidths are obtained when the light is incident on the p side. Bandwidth values obtained with the saturation velocity values are underestimated and overestimated for light incident on the p and n side respectively.

References [1] J.E. Bowers and C.A. Burrus, “Ultrawide-band long-wavelength p-i-n photodetector”, J. Lightwave Technol., vol. 5, pp. 1339-1350, 1987. [2] G. Lucovsky, R.F. Schwarz, R.B. Emmons, “Transit-Time Considerations in p-i-n diodes”, J.Appl. Phys., 35(3), pp. 622-628, 1964. [3] R. Sabella, S. Merli, “Analysis of InGaAs p-i-n photodiode frequency response”, IEEE J. Quantum Electron., vol. 29, pp. 906-916, 1993.

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J.M.T. Pereira / The Effect of the Direction of Incident Light

[4] M. Dentan, B. de Cremoux, “Numerical simulation of the nonlinear response of a p-i-n photodiode under high illumination”, J. Lightwave Technol., vol. 8, nº 8, pp. 1137-1144, 1990. [5] Z. Šušnjar, Z. Djurić, M. Smiljanić, Ž. Lazić, “Numerical calculation of photodetector response time using Ramo’s theorem”, Proceed. MIEL’95, vol. 2, NIS, Serbia, pp. 717-720, 1995. [6] J.N. Hollenhorst, “Frequency response theory for multilayer photodiodes”, J. Lightwave Technol., vol. 8, nº 4, pp. 531-537, 1990. [7] J.M. Torres Pereira, “Frequency-Response Simulation Analysis of InGaAs/InP SAM-APD Devices”, Microwave Opt. Technol. Lett., 48 (4), pp. 712-717, 2006. [8] Jorge Manuel Torres Pereira, “Modeling the frequency response of p+InP/n-InGaAs/n+InP photodiodes with an arbitrary electric field profile”, EPNC’2006, Maribor, Slovenia, vol. 1, pp. 181-182, 2006. [9] J.B. Radunovic, D.M. Gvozdic, “Nonstationary and nonlinear response of a p-i-n photodiode made of two-valley semiconductor”, IEEE Trans. Electron Devices, vol. 40, nº 7, pp. 1238-1244, 1993.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-425

425

3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis Algorithm S. COCO a, A. LAUDANI a, F. RIGANTI FULGINEI b and A. SALVINI b a DIEES, Università di Catania, V. A. Doria 6, I 95125 Catania, Italy b DEA, Università di Roma 3, Via della Vasca Navale, 84, 00146 Roma, Italy asalvini@uniroma3 Abstract. An approach based on a bacterial chemotaxis algorithm dedicated to the 3-D finite element mesh optimization is presented. By starting from an assigned mesh, the ‘bacterial chemotaxis algorithm’ modifies the initial configuration by changing the spatial coordinates of specific mesh nodes (or possibly by canceling them) while all node connections are kept fixed. This approach guarantees an appreciable reduction of the number of elements preserving a high quality of the mesh. This is a fundamental rule for reducing computational costs and preserving the accuracy of the numerical results. The approach has been validated on the meshing of a 3-D Helix structure of a TWT.

1. Introduction Dedicated 3-D numerical simulators are necessary to develop suitable electromagnetic analysis for a correct design of electronic and electric devices. Moreover, the Finite Element (FE) approach is the most employed method implemented. The key of success of the FE is mainly due to its capability to give accurate solutions for several different kinds of electromagnetic problems, but also for its vocation for analyzing 3-D complex geometries. To obtain accurate results with an acceptable computational cost, it is necessary to provide mesh refinement and adaptive mesh generation in critical spatial regions of the device to be studied. In particular 3-D electromagnetic analyses, the generation of the finest mesh, able to guarantee accurate results, is a very difficult task due to the complexity of the device shape (e.g. the helix slow wave structure in a Traveling Wave Tube (TWT)). Usually, the mesh is generated by assembling together elementary small blocks (primitives) having simple shape. In other words, the overall mesh is obtained by applying a special “connect-it” function to all the interface elements between adjacent primitives. Unfortunately, this way to operate may produce a low quality mesh. This aspect is due to the non-optimal choice of the initial shape of the connecting elements. Consequently, a refinement procedure is necessary in order to increase the mesh quality [1]. The aim of this paper is to present an approach for the 3-D finite element mesh optimization based on Bacterial Chemotaxis Algorithm (BCA) [2]. The choice of this specific optimization algorithms can be justified by the particular nature of the problem

426 S. Coco et al. / 3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis Algorithm

under analysis as described in [3]. By starting from an assigned mesh, the BCA modifies the initial configuration by moving the specific mesh nodes (or possibly by canceling them), while all node connections are kept fixed. This approach guarantees an appreciable reduction of the number of elements preserving a high quality of the mesh. The paper is structured as follows: in the Section 2 the main aspects of mesh optimization are presented; in the Section 3 the Bacterial Chemotazis Algorithm and its application to mesh optimization is discussed; two experiments for the approach validation are discussed in Section 4; authors’conclusions follow in Section 5.

2. Mesh Optimization The main motivation of the optimization of a Finite Element mesh is to improve the quality of a poorly shaped elements in order to reduce computation time, also by decreasing the number of elements of the original mesh. Two different approaches can be carried out to optimizing a mesh, local and global. A local approach search a optimal discretization in the proximity of an assigned mesh node (e.g. Laplacian smoothing), while a global mesh optimization is aimed to improve mesh quality for the entire domain. Unfortunately the local approach is not able to guarantee an overall mesh enhancement, thus a global approach, although its more expensive computational cost, is preferable. In addition there are two main problems in the mesh optimization: the former is that the optimal mesh does not exist for 3D domain, since it is impossible to fill an arbitrary space with regular tetrahedra; the second is the choice of a metric for the evaluation of quality of a 3D unstructured mesh is an open problem. In fact it is possible to qualify a mesh according to “geometrical metrics”, related to interpolation error of numerical method, or to “condition number metrics”, related to condition number of the discretized FE matrices. In our optimization approach we have adopted geometric metrics, which will briefly discuss in the next paragraph, since the Bacteria Chemotaxis behaviour can be easily connected to geometrical properties of the mesh. Tetrahedron Shape Measures and Fitness Functions Tetrahedron Shape Measures (TSM) are usually defined in terms of the ratio of two characteristic sizes of the tetrahedron (the volume of the tetrahedron, the length of longest or smallest edge, the radius, the area or the volume of the insphere or circumsphere, etc) and usually vanish with the volume of tetrahedron and achieve the maximum possible quality of 1 only for regular elements [4,5]. Among the different shape measures the usually adopted are: −

the Radius Ratio ρ ρ=3

ρinscribed ρcircumscribed

(1)

where ρinscribed and ρcircumscribed are the radius of the inscribed sphere and circumscribed sphere respectively. − The Mean Ratio η

S. Coco et al. / 3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis Algorithm 427

η=

12 3 9V 2 6

∑ li2

(2)

i =1

where V is the volume of the tetrahedron and li is the length of the i-th edge of the tetrahedron. − The Edge Ratio σ defined as the ratio between the smallest edge over the largest edge of the tetrahedron. − The Aspect Ratio γ γ=

2 6ρinscribed max {li }

(3)

1
where ρinscribed is the radius of the inscribed sphere and li is the length of the i-th edge of the tetrahedron. Clearly in order to evaluate the overall quality of a mesh we should refer to a statistical parameter like the minimum, the average, the maximum of the choosen shape measures of the tetrahedra. Among these quantities the average value is the most important. Infact the maximum is pratically useless since reaches the 1.0 value, while the minimu is less important because should be erroneus evaluate the quality of a mesh by considering just the worst tetrahedra, which often belong to the boundary and cannot successfully be modify during optiomization. In addition as discussed in literature [4,5] there is an equivalence among the tetrahedron shape measures and conseguently a mesh optimized by using as fitness function one of the previuos defined tetrahedron shape measure is close to be optimal for the other tetrahedron shape measures. In the following and in the examples we adopt as fitness function for our test the Edge Ratio σ , however very similar results have been found by using all the other TSMs. 3. BCA The BCA [2] takes inspiration by the motion of particular micro-organisms (bacteria) like in natural life. The bacterium activity is due to the different chemical properties encountered into its habitat. This behavior is called bacterial chemotaxis. In particular, a bacterium is sensitive both to the gradient of the nutritive substance concentrations and to the gradient of harmful substances. Thus, the bacterium motion mechanism is due to the evolution of the bacterium position for achieving survival or, simply, a better life conditions. Thus, all scientific observations of a particular bacterium specie behavior (e.g. Escherichia coli and Salmonella typhimurium), can be effort for heuristic evolutionary computation (virtual bacterium). Thus, from the knowledge of the chemical mechanisms governing the bacterial chemotaxis, mathematical abstractions and implementation of numerical algorithms has been made for optimization problems [2,3]. A mathematical description of a bacterium motion can be developed by the determination of suitable probabilistic distributions referred both to the motion duration and to the velocity vector (speed and direction) of the bacterium. The bacterium motion is performed in a n-dimensional hyperspace in which its velocity vector is made of n speeds and n directions components.

428 S. Coco et al. / 3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis Algorithm

Figure 1. Visualization of the initial mesh (left) and of the optimized one (right).

In the present implemented BCA for mesh optimization, the virtual bacterium motion follows the next listed rules: a) the bacterium population consists of an assigned number individuals; b) each bacterium follows its own path, each consisting of a sequence of rectilinear trajectories; c) each path segment is identified in a n-dimension space which dimension is depending on the number of parameters to be optimized; d) the bacteria speeds and directions are updated for each calculation step according to the Error Index of solutions; e) new speeds and directions are generated by means of a Gaussian probabilistic functions; f) the path duration τ is assumed to be constant The “BCA” optimizes the initial configuration of the primitive blocks by changing the assigned coordinates of specific nodes without changing the connections which do not belong to the boundaries. On the other hand, the BCA can cancel some nodes/elements. Following this approach we obtain a mesh having an higher quality and also a lower number of elements, i.e. it is possible to achieve accurate results with a low computational cost.

4. Approach Validation The BCA mesh optimizer has been tested by using several finite element model. Hereafter two examples are illustrated in order to validate the approach. The first example presented regards a very simple cilindrical geometry. In particular, by starting from a mesh made of 420 points, 1710 tetrahedra, 2483 edges, and having an average TSM (edge ratio σ ) of 0.31, an optimized mesh is obtained after 61 iterations, consisting of 281 points, 893 tetrahedra, 1431 edges, and having an average TSM (edge ratio) of 0.44. The enhancement of the overall quality was of more than 40%, while the number of elements, points and edges is reduced of about 35% with respect the initial one. Figure 1 shows the intial mesh of the cylinder and the otpimized one. Figure 2 shows the histogram of element TSM for the two meshes before and after BCA optimization. The initial mesh have an high percentage (about 75%) of tetrahedra having a TSM of 0.3.

100

50

90

45

80

40

70

35

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S. Coco et al. / 3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis Algorithm 429

60 50 40

30 25 20

30

15

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10

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tetrahedron shape measure

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0

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0.5

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0.7

0.8

0.9

1

tetrahedron shape measure

Figure 2. Histogram of the percentage of elements having an assigned shape measure for the two mesh of helix, before and after the optimization.

Figure 3. Visualization of the optimized mesh of a TWT Helix: the number of elements is about 1/3 of initial mesh.

It is worth noticing that the high percentage of element having a TSM quality of 0.3 in the optimized mesh is relative to elements belonging to the boundary. The second test here presented regards the optimization of an helix slow wave structure of a TWT [6]. The original mesh has 5064 points, 27324 tetrahedra, 33335 edges, an average TSM (edge ratio) of 0.135. The optimized mesh, after 88 iterations, has 1747 points, 8633 tetrahedra, 10951 edges, an average TSM (edge ratio) of 0.361. The enhancement of the overall quality was of more than 150%, while the number of elements, points and edges is reduced to 1/3 of the initial one. Figure 3 shows the optimized mesh obtained. Obviously, we do not show the initial mesh because it is made of

430 S. Coco et al. / 3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis Algorithm 25

50 45

20

40

% of elements

% of elements

35 30 25 20

15

10

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Figure 4. Histogram of the percentage of elements having an assigned shape measure for the two mesh of helix, before (left) and after (right) the optimization.

a high number of elements which makes the mesh not intelligible. In Fig. 4 the histogram of element TSMs for the two meshes before and after optimization is presented, showing the enhancement of the elements quality achieved by BCA optimization.

5. Conclusions The presented approach based on a bacterial chemotaxis algorithm dedicated to the 3-D finite element mesh optimization seems to be an attractive method. By starting from an assigned mesh, the ‘bacterial chemotaxis algorithm’ modifies the initial configuration by changing the spatial coordinates of specific mesh nodes (or possibly by canceling them) while all node connections are kept fixed. This approach guarantees an appreciable reduction of the number of elements preserving a high quality of the mesh. This is a fundamental rule for reducing computational costs and preserving the accuracy of the numerical results.

References [1] S. Coco, A. Laudani, S. Pulvirenti e M. Sergi, An object-orientated 3D finite element deterministic/neural mesh generator, in Software for Electrical Engineering Analysis and Design V, Wessex Institute of Technology, United Kingdom, pp. 27-36. 2001. [2] S.D. Muller, J. Marchetto, S. Airaghi, P. Kournoutsakos, “Optimization Based on Bacterial Chemotaxis”, IEEE Trans. on Evolutionary Computation, Vol. 6, No. 1, February 2002, pp. 16-29. [3] F. Riganti Fulginei, A. Salvini, Comparative Analysis between Modern Heuristics and Hybrid Algorithms, COMPEL, The International Journal for Computation and Mathematics in Electrical and Electronics Engineering, MCB University Press, March 2007, Vol. 26, No. 2, pp. 264-273. [4] J. Dompierre, P. Labbe, F. Guibault and R. Camarero, Proposal of Benchmarks for 3D Unstructured Tetrahedral Mesh Optimization, 7th International Meshing Roundtable, Dearborn, MI, pp. 459-478, October 1998. [5] A. Liu, B. Joe, Relationship between tetrahedron shape measures, BIT 34, pp. 268–287, 1994. [6] S. Coco, A. Laudani, G. Pollicino, R. Dionisio, R. Martorana, A FE tool for the electromagnetic analysis of slow-wave helicoidal structures in Traveling Wave Tubes, IEEE Transactions on Magnetic, Vol. 43, Issue 4, pp. 1793-1796, April 2007.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-431

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Electro-Quasistatic High Voltage Field Simulation of Large Scale 3D Insulator Structures Including 2D Models for Conductive Pollution Layers Daniel WEIDA a, Thorsten STEINMETZ a, Markus CLEMENS a, Jens SEIFERT b and Volker HINRICHSEN c a Helmut-Schmidt-University, University of the Federal Armed Forces Hamburg, 22043 Hamburg, Germany b Lapp Insulators GmbH, 95632 Wunsiedel, Germany c High-Voltage Laboratory, Technische Universität Darmstadt, D-64283 Darmstadt, Germany Abstract. Simulations of large scale 3D insulator structures covered by conductive pollution layers are presented. These conductive pollution layers on the surfaces are thin with respect to the device dimensions. The 3D geometric modeling of these layers is expensive in time and leads to ill-conditioned linear systems of equations due to large aspect ratios. Alternatively, in this paper they are modeled as 2D surface layers coupled to the 3D geometric model in the electro-quasistatic field simulations. Numerical results for realistic 3D high voltage insulator structures covered by both 3D and 2D pollution layers are presented.

1. Introduction High voltage insulator structures are essential for electric power transmissions, where the maximum electric field stress is a crucial design criterion. If the field stress on the surface of the structure is too high, the corrosion of the insulator material due to electric discharge phenomena is intensified, thus reducing its lifetime. During operation, the insulators structures which are exposed to environmental influences are contaminated by dirt. For the analysis of the field stress, the pollution layers covering the insulator structure have to be taken into account. These layers are naturally not included in the CAD design data of the insulators. Hence, for conventional 3D simulations, they have to be provided additionally, which is expensive especially for curved surfaces and – due to the necessarily required finer mesh resolutions – leads to more degrees of freedom (DoF) in the discrete models. Whereas the electric conductivity of the insulator and the ambient air vanishes, a perfect electric conductivity could be used as modeling assumption for the simulation of the electric field in the pollution layers. Perfect conductive material behavior can be considered by projection techniques applied to Krylov-subspace methods [7]. However, with this approach, the conductive current through the pollution layer can not be calculated which prevents the computation of the Ohmic losses eventually required for coupled or subsequent thermodynamic simulations. In this paper, these thin conductive dirt layers positioned on the outer surface of the insu-

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lator are modeled as 2D surfaces, assuming, that the conductive currents normal to the insulator surfaces are negligible. These 2D surfaces coupled into the 3D model save expensive further geometric modeling, they help to avoid additional difficulties in the mesh generation and the introduction of additional, effectively inessential DoF for the thin dirt layers. In order to cope with both capacitive and resistive material behavior, the simulations are performed under the electro-quasistatic (EQS) assumption [1–3], i.e. the omission of the magnetic induction term in Faraday’s law. Thus, a scalar potential function φ(r, t) allows computing the resulting irrotational electric field intensities. The EQS differential equation for the electric scalar potential in the time domain reads −div ((κ + ε ∂ t ) gradϕ (r , t )) = 0 .

(1)

Here, the electric conductivity is denoted by κ, the electric permittivity by ε. Simulations of contaminated insulators have already been reported e.g. in [3], using 3D dirt layers only. In this paper 2D pollution layers with a finite electric conductivity are introduced and the simulation results are compared to the 3D approach.

2. Description of the Model Problem and Discrete Formulation The application of Green’s first integral theorem to the weak form of the EQS differential equation (1) − ∫ div ((κ + ε ∂ t ) grad ϕ ) v d V = 0

(2)

Ω

yields

∫κ

gradϕ ⋅ gradvdV +

Ω

d ∂ϕ ε gradϕ ⋅ gradvdV = ∫  vdΓ . dt ∫Ω ∂n ∂Ω

(3)

At this point the solution function φ and the test function v in (3) are elements of the infinite dimensional function spaces

1

( ) and

H Ω

1

( ) , respectively. Applying

H0 Ω

Galerkin FEM techniques, φ and v can be approximated by finite nodal shape functions. Using Whitney-FEM [5], the geometrical discretization (3) reads Cκ Φ + Pε

ers

d Φ=0. dt

(4)

Here, Φ is the vector of the electric nodal scalar potential. Since the pollution layare very thin compared to the lateral dimensions, the tangential component of the

Γp

total current density (electric + displacement current density) in the pollution layers

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433

outweighs its normal component. Considering these anisotropic material behavior, equation (3) is rewritten as

∫κ

gradϕ ⋅ gradvdV +

Ω

+ ∫ κS Γp

d dt

∫ε

gradϕ ⋅ gradvdV

Ω

d grad S ϕ ⋅ grad S vdΓ + dt

∫ε Γp

S

∂ϕ grad S ϕ ⋅ grad S vdΓ = ∫  vdΓ, ∂Ω ∂ n

(5)

where ε S and κ S are the electric surface permittivity and conductivity and grad S the surface gradient. The discrete formulation of (5) reads

(C κ + C κ ,2 D )

Φ + (Pε + Pε ,2 D )

d Φ=0, dt

(6)

which is an ordinary differential equation (ODE) in the time domain. The matrices Cκ and Pε are assembled element-wise for each volume element, the matrices Cκ ,2D and Pε ,2D for each pollution layer surface element via a strongly modified version of the

FEMSTER C++ class library [6] with nodal shape functions based on SylvesterLagrange interpolation polynomials. 3. Electro-Quasistatic 3D Simulations Transient EQS simulations of high voltage insulators for applications on overhead transmission lines with 21 sheds covered by 3D dirt layers and 2D dirt layers are performed. The electric conductivity and permittivity of the 1mm thick pollution layersis set to κ = 0.03 A/Vm and ε = 8.854e-12 As/Vm (κS = 3.0e-5 A/m and εS = 8.854e-15 As/V), while the insulator and the ambient air feature zero conductivity. A time-harmonic electrode voltage of 105 kV with a frequency of 50 Hz is applied. The time integration of (6) is performed by an implicit Singly-Diagonal-Runge-Kutta SDIRK3(2) method with four internal stages [4] where the resulting linear system of equations are solved by a preconditioned conjugate-gradient iterative solver (PCG). The preconditioning is carried out by an algebraic multigrid method. 3D vs. 2D Pollution Layer In the first example, the top sides of the sheds of a high voltage insulator are halfcovered by conductive pollution layers. These are modeled both as thin 3D volumes and as 2D surfaces, respectively. On the electrodes of the insulator that are assumed to be perfect conductors, the voltage profile depicted in Fig. 1 is applied. The insulator in Fig. 2 is embedded in a vacuum box serving as computational domain. Homogeneous Neumann conditions are assigned to the outer surfaces. The simulation results show a nearly identical distribution of the scalar potential as well as the electric field intensity. However, in addition to avoiding a more complicated

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Figure 1. Ramped time-harmonic sinus voltage impressed on the insulators electrodes.

Figure 2. CAD Model of the insulator with sheds’ upside half-polluted.

geometric modeling of the pollution layer, the following simulation is more efficient using the 2D pollution layers than the 3D dirt layers. In the table below, the number of PCG steps and the solution time required for the four stages of a specific time step solution shown in Fig. 3 are listed. Type of pollution layer 3D 2D

Number of DoFs 386.157 291.093

Total number of PCG steps 471 290

Total solve time 543 sec 271 sec

Entire Pylon with Insulator Covered by 2D Pollution Layers The first example above shows the efficiency and reliability of the 2D pollution layer models. As next model problem the entire pylon of an overhead transmission line is simulated. The configuration shown in Fig. 4 consists of a cylindrical pylon, a traverse, a 21 shed insulator and the central phase cable. The outer surfaces of its surrounding air

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435

Figure 3. Comparison of 3D EQS simulations with 3D (left) and 2D (right) pollution layers, respectively. The Finite-Element mesh, the scalar potential Φ and the electric field magnitude |E| are shown.

Figure 4. Geometry of the entire electricity pylon: the top sides of the insulator sheds are polluted.

region are set to 0 V except the surfaces normal to the cable on which homogeneous Neumann conditions are defined. The top sides of the sheds of the high voltage insulator are covered by conductive pollution layers. The simulation results in Fig. 5 show the effect of the traverse, the insulator and the corona ring on the radial field of the cable. The effect of the conductive pollution layers is obvious in Fig. 6. While the electric field magnitude is smaller at the pollution layers compared to the simulation without pollution layers, it is increasing between the pollution layer and the next shed. The consideration of additional pollution layers results in increased numerical costs as shown in the table below. The number of PCG steps needed to solve the stage systems of equations more than octuplicates for the model problem with 2D pollution layers. However, based on the numerical results achieved for the first example, the use of 3D pollution layers is assumed to raise the numerical costs even more. Particularly, if the more complex modeling, the essential mesh refinement and the larger aspect ratios of the 3D pollution layers are taken into account, the approach containing 2D pollution layers turns out to be more efficient.

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pollution layer None 2D

Number of DoFs 706.007 706.007

Total number of PCG steps 28 230

Total solve time 129,96 sec 508,30 sec

Figure 5. 3D EQS simulation with 2D pollution layers. The scalar potential Φ and the electric field magnitude |E| are shown.

Figure 6. Electric field magnitude |E| along the rod of the insulator. Even with the effect of the pollution layer the EPRI (Electric Power Research Institute) norm (|E| < 4.5 kV/cm) is still fulfilled.

4. Conclusion EQS simulations of high voltage insulator structures covered by thin pollution layers were presented. Modeling these layers as 2D surfaces instead of 3D volume bodies resulted in both a smaller number of DoF and a more economical modeling process, respectively. Thus, the efficiency of the simulation scheme was clearly increased. Numerical results which compared the scalar potential and electric field distributions of the particular approaches as well as the efficiency of the proposed method were shown.

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References [1] R.M. Matias, A. Raizer, Calculation of Electric Field Created by Transmission Lines, by 3D-FE Method Using Complex Electric Scalar Potential, ACES Journal, Vol. 12(1), pp. 56-60, 1997. [2] K. Preis, FEM Numerical Computation of Transient Quasistatic Electric Fields, Proceedings of the 4th International Conference on Computation of Electromagnetics (CEM) 2002. [3] U.v. Rienen, M. Clemens, T. Weiland, Simulation of Low-frequency Fields on High-Voltage Insulators with Light Contaminations, IEEE Transactions on Magnetics, Vol. 32(3), pp. 816-819, 1996. [4] T. Steinmetz, M. Helias, G. Wimmer, L.O. Fichte, M. Clemens, Electro-Quasistatic Field Simulations Based on a Discrete Electromagnetism Formulation, IEEE Transactions on Magnetics, Vol. 42(4), pp. 755-758, 2006. [5] A. Bossavit and L. Kettunen, Yee-like schemes on staggered cellular grids: A synthesis between FIT and FEM approaches, IEEE Transactions on Magnetics, Vol. 36(4), pp. 861-867, 2000. [6] P. Castillo, J. Koning, R. Rieben, M. Stowell, D. White, Discrete Differential Forms: A Novel Methodology for Robust Computational Electromagnetics, 2003. [7] M. Clemens M. Wilke, G. Benderskaya, H. De Gersem, W. Koch, and T. Weiland: Transient electroquasi-static adaptive simulation schemes, IEEE Trans. Magn., Vol. 40(2), pp. 1294-1297, 2004.

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Electromagnetic Aspects of Data Transmission a

Liliana BYCZKOWSKA-LIPIŃSKA a and Sławomir WIAK b Technical University of Lodz Institute Computer Science, ul.Wólczańska 215, Poland [email protected] b Technical University of Lodz Institute of Mechatronics and Information Systems, ul. Sefanowskiego 18/22, Poland [email protected] Abstract. The aim of this paper is to analyze the problems connected with transmission of information and the application of electromagnetic wave in data transmission. The issue of electromagnetic compatibility is discussed. The problem of electromagnetic radiation as a threat to natural environment is presented.

1. Electromagnetic Phenomena in Data Transmission Telecommunication is a branch of science and technology that deals with transmission of information from a transmitter to one or more receivers (either humans or technical appliances appropriately adapted to this purpose) [2,4,5]. In practice, contemporary telecommunication is teleinformatics, as it integrates two technical disciplines: − −

telecommunication, which deals with transmission of information, informatics, which deals with data processing.

The integration of these two disciplines is obvious. However, it is not complete and does not proceed simultaneously in all countries. Currently it concerns: • •

developing new structures of data networks integrating data transmission services into one technical form capable of gaining access to processing systems and commutation between them; developing new algorithms improving the quality and quantity of data transferred, creating computer networks, integrating access times and processing power, as well as development and integration of teleinformatic systems.

The process of information transfer from a transmitter to a receiver in a telecommunication network comprises the following stages: 1.

Processing, which involves the adaptation of information to transfer conditions. This refers mainly to the change of original information into electromagnetic signal and appropriate processing of this signal in order to improve the quality and quantity of information transferred (modulation, analog-digital processing, coding) and information retrieval (demodulation, digital-analog processing, decoding).

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439

Figure 1. Electromagnetic spectrum [10,12].

2.

3.

Teletransmission from one point of telecommunication network to another. Depending on the frequency range, an appropriate transmission channel is used, i.e. copper wire, optical fiber or wireless. Commutation and telecommutation dealing with joining and disjoining the elements of the transfer channel of information carrying signals. This stage of the process comprises also designing, manufacturing, installation and exploitation of telecommutation devices and the issues of movement in telecommunication networks.

In telecommunication, it is the electromagnetic wave that is used as a carrier of original information. Figure 1 shows electromagnetic spectrum [6–10]. The majority of contemporary telecommunication systems is based on broad frequency electromagnetic phenomenon. The following electromagnetic wave ranges can be distinguished (Fig. 1): − − −

electromagnetic waves used in wireless transmission, commonly known as radio waves, electromagnetic waves used in copper wire transmission (electric current), electromagnetic waves used in optical fiber transmission (optical infrared, visible, and ultraviolet radiation).

This is an example of how your paper is to be prepared according to the instructions. When numbering equations enclose the number in parentheses and place it flush with the right hand margin as shown below.

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Table 1. Frequencies and wavelengths of electromagnetic radiation Range Radio waves Optical radiation X-rays Gamma rays

Wavelength λ, [m] 108...10–4 10–4...10–8 10–8...10–11 < 10–11

Frequency f, [Hz] 3...3⋅1012 3⋅1012...3⋅1016 3⋅1016...3⋅1019 > 3⋅1019

Table 2. Subranges of radio waves Designation Decamegametric Megametric Hectokilometric Myriametric Kilometric Hectokilometric Decametric Metric Decimetric Centimetric Milimetric Decimilimetric

Symbol

Wavelength 105 – 104 km 104 – 103 km 1000 – 100km 100 – 10 Km 10 – 1 Km 1000 – 100 m 100 – 10 m 10 – 1 m 100 – 10 cm 10 – 1 cm 10 – 1 mm 1 – 0,1 mm

VLF LF MF HF VHF UHF SHF EHF

Frequency 3 – 30 Hz 30 – 300 Hz 300 – 3000 Hz 3– 30 kHz 30 – 300 kHz 300 – 3000 kHz 3 – 30 MHz 30 – 300 MHz 300 – 3000 MHz 3 – 30 GHz 30 – 300 GHz 300 – 3000 GHz

Table 3. Traditional division of electromagnetic wave ranges Designation Very long Long Medium Intermediate Short Ultrashort Microwaves

Wavelength ≥ 20 km 20 – 3 km 3000 – 200 m 200 – 100 m 100 – 10 m 10 – 1 m

Frequency ≤ 15 kHz 15 – 100 kHz 100 – 1500 kHz 1,5 – 3 MHz 3 – 30 MHz 30 – 300 MHz

≤ 1m

≥ 300 MHz

Table 1 shows frequencies and wavelengths of electromagnetic radiation used for information transfer. Since the properties of electromagnetic waves used in wireless transmission (radio range) are very diverse, the division into subranges was introduced. At present, decimal division of radio waves introduced by Consultative Committee on International Radio (CCIR) and compatible with the Radio Regulations is used (Table 2). The decimal classification is totally formal and does not reflect the properties of particular wave subranges. Therefore, the traditional division is often used, as illustrated in Table 3. Wavelength λ is expressed with (1)

λ=

c , f

(1)

where: f – frequency, c = 3⋅108 m/s – the speed of electromagnetic wave propagation in vacuum.

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441

main side lobe

rear lobe

Figure 2. Characteristics of antenna’s radiation.

Figure 3. Beamwidth.

2. Transmitters and Receivers Of Electromagnetic Wave Contemporary data transmission systems are based to a large extent on the application of electromagnetic wave to wireless information transfer. This refers mainly to intercontinental communication links via satellite. Antenna is a part of a transmitter or receiver, designed to send or receive electromagnetic waves [5]. Depending on the radiation characteristics, three antenna types may be distinguished, namely omnidirectional, one-directional and two-directional [12]. The properties of an antenna are characterized by the following parameters: • •

•

gain – a ratio that indicates how much energy an antenna radiates in a given direction compared to the isotropic antenna and expressed in dBi, beamwidth – antenna’s radiation characterized by the so called lobes, i.e. the main lobe, and smaller side and rear lobes (Fig. 2). Beamwidth is the angle within which the intensity of radiation is one-half the maximal intensity (Fig. 3), front-to-back ratio – the ratio of power radiated to the front of an antenna versus the amount of power radiated to the back of an antenna (Fig. 4),

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L. Byczkowska-Lipi´nska and S. Wiak / Electromagnetic Aspects of Data Transmission

Figure 4. Front-to-back ratio.

•

polarization – depends on the characteristics of electromagnetic field vector; there are two types of polarization: − −

•

• •

linear polarization – depending on the direction of electric field vector, it can be either horizontal or vertical, circular polarization – depending on the direction in which the electric field vector rotates, it can be left-hand-circular or right-hand-circular.

impedance – the ratio between the voltage and current intensity at antenna’s terminal; it is a very important parameter, because its mismatch with the transmitter can result in some fraction of the wave’s energy reflecting back to the source, VSWR (voltage standing wave ratio) – characterizes the energy reflecting back to the source as a result of the mismatch between the antenna and the transmitter, capacity – the frequency range, within which antenna maintains its parameters, e.g. radiation characteristics and input impedance.

3. Technologies Used in Data Transmission The first technologies used for data transmission networks were wire technologies, used in corporate networks. With time, there appeared new solutions using the existing phone lines, wireless transmission and even the existing energetic networks. According to the type of transmission channel and the standards used in transmission of digital signals, data transmission networks (computer networks) can be divided into several groups [1,4,5,7–9,11]. 1.

Wired networks built from scratch: a)

b)

HAVI (Home Audio Video Interoperability) – a standard which allows all manner digital consumer electronics and home appliances to interoperate by means of one controlling appliance; Ethernet 10Base–T (using UTP cat. 5);

L. Byczkowska-Lipi´nska and S. Wiak / Electromagnetic Aspects of Data Transmission

c)

d) 2.

IEEE 1394 Protocol (freeware) – international hardware and software standard that integrates entertainment, communication and computation systems, thanks to which data may be transferred at 100, 200 or 400 Mb/s; UpnP (Universal Plug and Play) – allows devices to exchange data under the control of a controlling device in the network.

Wire networks using existing cables: a) b) c)

3.

443

Home PNA (Home Phone Line Alliance) – computer network standards designed to operate over existing phone lines, PLC (Power Line Communication) broadband network using outdoor electrical cables, PLC (Power Line Communication) narrowband network using indoor electrical cables.

Wireless LANs: a)

b)

c)

Home RF (Home Radio Frequency) – a standard providing high efficiency wireless connectivity for multimedia computer applications according to three versions of SWAP (Shared Wireless Access Protocol); Bluetooth – a set of standards and products that enable devices to find each other and connect seamlessly over a short range. This technology is a universal method of access to existing data networks, IEEE 802.11 WLAN – a Wireless LAN transmission standard that offers two network connection methods: − −

d)

ad-hoc that requires no server and no access point, client/server that requires a central server, coordinating all stations in the network.

IrDA (Infrared Data Association) – infrared digital signals transmission protocol.

In wired networks, the connection of the elements with network wiring system is stationary. The change of localization of a particular device requires new configuration of cable connection. Wireless networks provide full mobility of the workstation within the network.

4. Typology of Wireless Networks A great variety of wireless transmission technologies and devices employed for creating wireless LAN helps to create different kinds of data transmission networks: − − −

fully wireless; wireless with an access to wired network; wireless bridges, used for connecting distant segments of wired networks or connecting individual WLANs.

The above types can be mixed. An example of a fully wireless network is illustrated in Fig. 5.

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range zasięg stacji

stacja docelowa destination

Figure 5. An example of wireless station with multistage transmission [6].

Individual devices belonging to such a network work in a “peer-to-peer” system, which enables them to exchange resources. They are also capable of moving within the scope of the network, and a potential loss of a direct connection between two devices does not have to mean transmission breakdown, because indirect information transfer is possible. Wireless networks with an access to the resources of wired network are the most common type of wireless networks, e.g., LAN. Communication between the two segments (wired and wireless) is held by specialized mediating circuits, called AP (Access Points). They enable the cooperation with Ethernet and Token Ring, in both cases creating a mixed network (Fig. 6).

5. Electromagnetic Compatibility The increase in the number of electronic systems and devices used in various fields of human activity results in greater electromagnetic interference. Electromagnetic compatibility the capability of an object (device, installation or system) to be operated in its intended operational environment without causing electromagnetic interference [14]. The sources of electromagnetic phenomena may be objects emitting electromagnetic waves both intentionally (television and radio stations) and unintentionally (hair dryer). Electromagnetic compatibility is connected with the following notions: 1. Emission of disturbances, 2. Immunity to disturbances. Every piece of electrical equipment that is in operation can induce disturbances of various levels and different character, one of them being the propagation of electromagnetic waves. The issue of limiting the influence of unwanted signals should be taken into consideration in the process of designing, constructing and manufacturing of devices and systems that use electromagnetic field (teleinformatics). These problems should be dealt with in the earliest phase of the design process. Usually, changes introduced in

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445

a)

punkty assessdostępu points

Ethernet sieć Ethernet

b)

accessdostępu point punkt

sieć Token Ring

Figure 6. Examples of mixed networks, a) wireless segment and Ethernet, b) wireless segment and Token Ring [6].

the later stages are not very efficient. Moreover, the number of solutions available decreases with time, and the later they are introduced, the more costly they get. The phenomena occurring in natural environment and human activity result in the emergence of unwanted random that in turn cause disturbances. They occur everywhere, both in technical devices and in natural environment, and their character and intensity depend on many factors. Every device is susceptible to electromagnetic interference and can be a source of disturbance for other device, as well as its own elements. The disturbances can be of either interior or exterior origin. If the source is inside a given device, then interior disturbances occur. If the source is outside a given object, exterior disturbances take place. The sources can be divided into low and high frequency sources, with the high frequency sources being most difficult both from the theoretical and the practical point of view. Classification of sources depends on the criteria, which may include: origin, physical and biological phenomena, and mathematical description of a signal. Classification of disturbances according to different criteria:

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L. Byczkowska-Lipi´nska and S. Wiak / Electromagnetic Aspects of Data Transmission

O th e r b ro a d c as t in g s t a t io ns

A t m o s p h e ric d is c h a rg es

E le c t r ic e q u ip me n t in m e c h a n ic a l v e h ic le s V ib ro a c o u s t ic p h en o men a

E le c t ro s t a t ic d is c h a rg es

En e rg e t ic l in e s a n d su bs t a t io ns

R a d io a n d t e le v is io n re c e iv e rs E le c t r ic a l d o m e s t ic e q u ip m e n t

A t mo s p h e ric p re c ip it a t io n (a g a in s t a n t e n na ’s c o ns t ru ct io n )

Figure 7. Sources of signals disturbing the work of a television receiver.

1. According to the origin: − −

natural (cosmic and terrestrial), human-caused.

2. According to the character of physical phenomena: − − −

vibroacoustic (vibrations, acoustic oscillations), biological, connected with natural environment both animate and inanimate (changes in temperature, humidity, pressure, the presence of fungi, molds and dust), electromagnetic (noises emitted by electronic elements and circuits, signals from radio and television transmitters, switching and ignition signals, signals from energetic lines, light devices and other devices used at home and in industry).

3. According to mathematical description: − −

deterministic disturbances (can be defined in terms of mathematical dependencies), stochastic disturbances (random; they cannot be defined in terms of mathematical dependencies, as the signal observation results are different each time. They are dealt with by means of probability theory).

Figure 7 illustrates an example scheme of disturbances in a television receiver. In practice, electromagnetic compatibility includes two stages. First, the emission channel of disturbing signals is determined, their values are evaluated and compared with the acceptable values. Second, their reliability level in the environment of electromagnetic interference is determined. While designing and operating electronic systems and devices, one should allow for different types of disturbance, as they can influence substantially the way a given system or device works.

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Some phenomena are strongly connected with each other and it is difficult to separate them. What is important is the intensity of disturbances compared to the proper signals of an appliance. To deal with the increasing wave of electromagnetic disturbances, as well as for commercial reasons, the European Union has passed a law that prohibits marketing and installation of devices which do not fulfill the protection requirements.

6. The Influence of Electromagnetic Field on Living Organisms The role of electromagnetic field in data transmission has become a subject of increasing interest among ecologists, who examine how electromagnetic fields (EMF) emitted by telecommunication devices, in particular those using wireless technology, can affect humans. Potential health hazards resulting from mobile phone usage (wireless transmission of information) concern not only the consumers and producers of mobile phones, but also governmental and non-governmental organizations responsible for citizens’ health. The exposure to EMF emitted by mobile telephony is below the acceptable norms. It is important to remember that these norms were developed on the basis of the predicted thermal effects and do not allow for other effects of electromagnetic emission. Since mobile phones emit EMF in close proximity to the head, it is the influence of EMF on central nervous system that arouses most of the concerns. Thus, most experimental research in this field is concerned with the reaction of nervous system. EEG tests did not give unequivocal results either. Part of them did not indicate any changes in brain activity due to the exposure to EMF [14] and some of the works report an increase of beta1 and delta waves 15 min after EMF exposure finished [15]. The examination of ulnar and facial nerves’ conduction did not reveal any vital disorders either during or after the exposure to 900 MHz EMF of a mobile phone [14]. Hormone tests conducted on people who were exposed to mobile phone EMF for 4 weeks (2 h/day, 5 days a week), did not reveal any change in the level of pituitary gland, corticotrophin (ACTH), thyrotrophic (TSH), growth hormone (GH), prolactin (PRL), Latinizing hormone (LH) and follicle stimulating hormone (FSH). It was only noticed that the concentration of TSH fell, but still maintained an acceptable level [15]. While examining circadian rhythm of melatonin secretion, no remarkable changes were noticed

7. Summary It is the electromagnetic phenomenon in all frequency ranges (depending on the transmission channel) that carries data. A vital problem connected with the development of all-type electric devices and teleinformatics is the influence of electromagnetic fields on natural environment and living organisms. The producers of mobile phones, scientists and technologists admit that the radiation of a mobile phone’s antenna penetrates through the head and neck, which increases brain temperature. Contradicting opinions about the influence of this phenomenon on health have been the subject of hot debates all over the world. While in operation, mobile phone is the only device with a radiation source placed in close proximity to the head. The issue of electromagnetic threats posed by telecommunication devices

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is very problematic and should be approached with reason. Further research in the field is needed in order to determine the actual influence of electromagnetic field on the environment.

References [1] Bing B.: Wireless local area networks. New York John Wiley & Sons, Inc. 2002. [2] Byczkowska-Lipińska L., Mandzij B.: Aspekty informatyczne w telekomunikacji. Łódź, ŁTN, 2004, (in Polish). [3] Byczkowska-Lipińska L., Cegielski M.: Architektury systemu informatycznego dla przedsiębiorstw małej i średniej wielkości. Zeszyty naukowe seria: Technologie informacyjne nr 4, Gdańsk, 2004, (in Polish). [4] Derfler F.J.: Poznaj sieci komputerowe. W. MIKOM, 2003, (in Polish). [5] Hallberg B.: Sieci komputerowe – kurs podstawowy. W. „Edition 2000”, 2001, (in Polish). [6] Nowicki K., Woźniak J.: Przewodowe i bezprzewodowe sieci LAN. OWPW, Warszawa 2003, (in Polish). [7] Ogletree T.: Rozbudowa i naprawa sieci. W. Helion, Gliwice 2002, (in Polish). [8] Plumley S.: Sieci komputerowe w domu i biurze. W. Helion, Gliwice 2004, (in Polish). [9] Sprtack M.: Sieci komputerowe. Księga eksperta. W. Helion, Gliwice 2005, (in Polish). [10] Stojmenović I.: Handbook of Wireless Networks and Mobile Computing. John Wiley & Sons, Inc. New York 2002. [11] Wajda K.:Wybrane zagadnienia z budowy i eksploatacji sieci korporacyjnych.WPT, 2002, (in Polish). [12] Wesołowski K.: Systemy radiokomunikacji ruchomej. WKiŁ, Warszawa 2002, (in Polish). [13] Zieliński B.: Bezprzewodowe sieci komputerowe. W. Helion, Gliwice 2005, (in Polish). [14] Hasse L., Krakowski Z., Spiralski L., Kołodziejski J., Konczakowska A.: „Zakłócenia w aparaturze elektronicznej” Radioelektronik Sp. Z o.o., W-awa 1995, (in Polish). [15] Badania Kompatybilności Elektromagnetycznej: http://www.delta.poznań.pl./info/kompatybil.htm (in Polish). [16] Di Barba P., Savini A., Wiak S.: Field Models in Electricity and Magnetism. Springer, 2007, monograph, under publishing procedure.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-449

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Application of the Magnetic Field Distribution in Diagnostic Method of Special Construction Wheel Traction Motors Zygmunt SZYMAŃSKI Institute of Electical Engineering and Automation in Mines, Silesian University of Technology Gliwice, 44-100 Gliwice, str. Akademicka 2, Poland Email: [email protected] Abstract. The paper presents a review of a special construction traction motors applied in electric and hybrid vehicle. Mathematical model of the magnetic fields distribution in traction wheel motor. Dynamic model of PMSM traction motor based on 2D magnetic flux distribution are also presented in the paper. ANSYS, JMAG and FEMM computer programs were applied in calculation of magnetic fields distribution for different faults state of the motors. On the base of magnetic field distributions were analyzed different failures situations and method of limitation their negative effects. Some laboratory experiments was realized for the traction motor.

Introduction Modern drive system of the road traction vehicle should ensure: environmental safety, high reliability and economical speed control in specific duty circumstances. A significant improvement in economical and power indexes can be achieved by: application of new design drive motors (wheel induction motor, permanent magnet motor, or hybrid drive system which contained petrol and electric motor), application of modern voltage converters controlled by microprocessor systems and optimum control of machines and electric vehicles. Special construction wheel traction motors: double rotor motor and cutting magnetic core motor, are presented in the paper. Modern construction of the permanent magnet wheel traction motor is described in the paper. Mathematical model of the magnetic fields distribution in wheel traction motor (equivalent scheme and finite element method) are presented in the paper. Most occur failure in traction motor include: broken rotor bars and end ring connectors, stator faults, eccentricity and bearing faults. The consequences of such faults are: unbalanced phase voltage and line currents, torque pulsations, mechanical vibrations and excessive heating. Ones of the non-invasive technique for the traction motor faults diagnosis is monitoring and processing of the stator currents, to detect spectrum harmonics in motor characteristics for the various types of faults. ANSYS, JMAG and FEMM computer programs were

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Z. Szyma´nski / Application of the Magnetic Field Distribution in Diagnostic Method

applied in analysis of magnetic fields distribution in traction motors with various types of faults. On the base of magnetic field distributions were calculated basic parameters of different type traction motors. Some results of computer calculation are presented in paper. Results of calculation were partially verified in laboratory experiments.

Wheel Traction Motors For electric vehicle applications, wheels motor drives represented a new attractive possible solution for their lightness and compactness. The wheels are directly driven by the electric motor and the gears are not necessary anymore. For axial flux motor applied in electric vehicle must be realized: high power/weight and torque/weight ratio, high efficiency and suitable shape to match constrain space. In the axial flux motors family, the axial flux PM (permanent magnet), axial flux induction motor, and cutting magnetic circuit motor are potential solution [1,3]. Axial flux induction motor Trim has one stator core with two polyphase windings and two rotors with two different shafts which may rotate independently. All the three magnetic cores (the stator and the two rotors) are in the form of discs with slots for the stator windings and the rotor cages. In this case, the motor can not be mounted inside the wheels but between them. Two identical three phase windings are connected in series in such a way, that the stator current flows in the same direction in any back to back stator slot. There is one main flux which links the stator windings and the two rotor cages. The motor has a small stator yoke which reduces the iron core cost and the iron losses, but long ends windings which results in copper losses. Induction traction motor with twin rotor axial flux is a physical combination of two motors into one in such a way, that their magnetic circuits are no longer independent. The rotor is on different shafts and may rotate independently. Different solution is permanent magnet wheel traction motor. The outer rotor may be designed to have permanent magnet NdFeB magnets, with high remanence (1, 1 T) and coercity (1275 kA/m.) on the inside surface, and concentrated windings on the stator. In motor applied in electric vehicle drive systems must be concerned problems: rise temperature to avoid demagnetization effects, exposition to road shocks, spills, dust and particles, and force demands on the wheel and tire. High number of poles reduced the torque ripple and yields a smaller magnetic yoke, decreasing volume and weight. Figure 1 shows a 2D drawing of the FCAFPM (field controlled axial flux permanent magnet) motor, and Fig. 2 left and right side a rotor of that motor [1]. The stator structure is formed by two strip wound or tape wound stator rings, circumferentially wound DC field winding and two sets of 3-phase AC windings. The stator core is divided into two sections in order to place the DC field winding in between, which makes it possible to vary the net air gap flux. The stator core has also slots to accommodate the two sets of 3 phase AC windings which are connected in parallel. The rotor is divided into two parts. The upper permanent magnets of the left rotor, which are mounted on every other pole, are magnetized as N-poles and aligned with the upper permanent magnets of the right rotor which are magnetized as S-poles. The lower section of the pole is a nonmagnet pole of iron core. Similarly, the magnets in the lower sections of the poles are magnetized as S-poles which are again located on every other pole and aligned with the N-pole side magnets.

Z. Szyma´nski / Application of the Magnetic Field Distribution in Diagnostic Method

451

Figure 1. Scheme of FCAFOPM motor [1].

Figure 2. Rotor core of the wheel PM motor [1].

Mathematical Model of Traction Motor For analysis of magnetic field distribution in the wheel traction motors are applied two methods: equivalent scheme method and finite element method. In equivalent scheme method total magnetic circuits is divided for particular elements contain: stator and rotor core, air gap, stator and rotor teeth and slot and yokes. The slots being skewed will produce a small MMF component in the radial direction, but this MMF will en-

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counter high reluctance and very little radial flux will eventuate, and so this effect is also ignored. This analysis is similar to a single rotor machine. Concerning of boundary and border conditions, ensured a flux continuous at all points in the machine system, differential equation described of magnetic field distributions is presented in (1) [4]:

∂F ∂α

C g ∂F μ ∂α

− ∂F ra =

+ T s ∂H tsa + T r ∂H tra +

r ( − ) H H ∂α p rS (ν ) Cg F − F = μ B + T H + T H + pν ( H − H ) rS (ν ) Cg F − F = μ B + T H + T H + pν ( H − H ) s

e

∂α

a

e

0

a

sο

ga

a

∂α

e

e

s

a

raν

gaν

s

tsaν

r

sa

e

traν

r

r

sν

raν

(1)

0

b

sο

b

rbν

gbν

s

tsbν

r

trbν

r

r

sν

rbν

0

where:

S (ν )

=

(−1)(ν + 3) , 2

a =1 − s

z t , = − z t , = −zt a 1 a 1 2πr 2π r 2πr s

zs

r

r

r zs

zr

sr

Full analysis of magnet field distribution, in double rotor induction motor, realized equivalent scheme method is presented in [3,4]. In finite element method total magnetic circuit is divided for digitized elements. Mathematical model of FEA field distribution analysis described equations (2)–(5) [2]. The magnetic scalar potential is used to describe the magnetic field in non-conducting materials.

− div (μ grad Ψ ) =0 grad Ψ ⋅ grad μ + μ ∇2 Ψ =0.

(2)

The magnetic field strength can be expressed as the sum of three components: H = Hs + Hm + H e

(3)

where: Hs – magnetic field obtained as a result of the source’s current flow in the air, Hm – magnetic field existing in ferromagnetic material in the surroundings, He – magnetic field existing of some conducting material in the surroundings. Using the formula Biot-Savart law for the Hs component becomes: Hs =

1 B= μ0

⌠ ⎮ ⎮ ⌡ V

J ×1r dv . 4 π r2

(4)

The Hm, He components can be described with equation (6): H m + H e = − grad Φ .

(5)

Z. Szyma´nski / Application of the Magnetic Field Distribution in Diagnostic Method

453

The magnetic vector potential can describe the electromagnetic field in the conducting area. The magnetic vector potential is defined by: B = rot A , and after evaluation conducted to the equations (6): ⎛1 ⎞ rot ⎜⎜ rot A⎟⎟⎟ - grad ⎟⎠ ⎜⎝ μ

⎛1 ⎞ ⎛∂ A ⎞ + gradV ⎟⎟⎟ ⎜⎜ divA⎟⎟⎟ = − γ ⎜⎜⎜ ⎟⎠ ⎜⎝ μ ⎝ ∂t ⎠

⎛ ∂ A ⎟⎞ =0 div ( γgradV ) + div ⎜⎜ γ ⎜⎝ ∂ t ⎟⎟⎠

(6)

For obtaining the unambiguous solution of the equations (3)–(7) in the whole area, the equations (2)–(6) have to be completed by consideration in equations of conditions of continuity of solution at borders of areas described by different potentials. The Galerkin procedure and the weighted residual method are used to transform the partial differential equation to a digitized set of non-linear algebraic equations. Boundary conditions and interface conditions will also be embedded in these equations.

Diagnostic Model of the Traction Motor Magnetic flux distribution are applied in detection of the failure work state in electrical or mechanical part of wheel vehicle. Most occur failure in traction motor include: broken rotor bars and end ring connectors, stator faults, eccentricity and bearing faults. Analysis of the influence broken bars on traction motor parameters and magnetic fields distributions are presented in the paper. This analysis used a transient eddy current, accounting for the rotation rotor, with considerations of non-linear characteristics of the magnetic materials. Analysis is based on magnetic vector potential formulation (7):

∇×(ν∇×ρ A) = ρ J

s

−σ

∂μ A ∂t

(7)

where: μA – magnetic potential, μ – magnetic permeability, Js – current density, σ – electric conductivity. Stator windings are fed from sinusoidal voltage source Us, which are connected from series stator resistance Rs and leakage inductance Ls. Solving the equation (8) line current is obtained.

U = RI

s

+ Ls

∂I ∂t

s

+

∂φ ∂t

μ

(8)

with

φ

−

μ

−

=∫  A dl and

−

I

s

= ∫∫

J

s

−

dS

ANSYS, JMAG and FEMM computer programs were applied in calculation of magnetic fields distribution, for different faults state of the motors. Results of computer calculations are presented in Figs (3–6).

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Z. Szyma´nski / Application of the Magnetic Field Distribution in Diagnostic Method

Figure 3. Magnetic field distribution in single phase stator winding supply.

Figure 4. FEM analysis of stator magnetic field.

Figure 5. Flux density distribution and lines flux in healthy traction motor [2].

Analysis of magnetic field distribution were realised for different types of wheel traction motors: double rotor induction motor, outer permanent magnet axial motor, surface permanent magnet motor. ANSYS, JMAG and FEMM computer program were applied in calculation of magnetic fields distribution. Each simulation cycle was carried out in the following steps: specification of motor geometry, specification of material properties, specification of boundary condition and excitation sources, generating the solution and graphical postprocessing and analysis. Analysis was performed for two work conditions of the motor: no-load and rated load torque. Computer program enable calculation of different kind of faults: different number broken rotor bars, short-circuit of stator winding. Results of computer calculation are applied to calculation of basic parameters of the different constructions motors, and diagnostic of technical states of the traction motor. In Fig. 3 presented a magnetic field distribution in supplied of single phase winding a stator traction motor, in Fig. 4 are presented application of ANSYS computer program for calculation of magnetic field distribution. Figure 5 presents a flux density distribution and lines flux in healthy traction motor. Figure 6 present a distribution of radial component of the flux density outside the stator traction motor for healthy and for 5 broken bars of the rotor, and in Fig. 7 presented a torque characteris-

Z. Szyma´nski / Application of the Magnetic Field Distribution in Diagnostic Method

455

Figure 6. Distribution of radial component of the flux density outside the stator traction motor for healthy and for 5 broken bars of the rotor.

Figure 7. Traction motor torque characteristics for 5 broken bars.

Figure 8. Potential distribution in the middle of the air gap of the traction motor.

tics for 5 broken bars. Figure 8 presents a characteristic of potential distribution in the middle of the air gap of the traction motor for different fault cases [4]. Analysis of computer calculations realized in different computer program, enable elaborate an original diagnostic an monitoring procedures, which make possible realization of diagnostics systems with prediction of traction motors faults states. Results of application Diagprzem computer program will be presented in next papers. The state of all components is monitored and recorded to Digital Fault Recorders (DFR) [4,5], while the electrical values of every sensors, of the wheel vehicle drive system terminal are measured by installed current and voltage transformers. Mechanical and kinetics values (speed, temperature and strength) are measured by intelligent

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Z. Szyma´nski / Application of the Magnetic Field Distribution in Diagnostic Method

sensors and transducers. From the operator perspective an alarm situation arises when a monitored value exceeds a predefined upper or lower limit, activating a sound or light alert on control panel. An expert operator would handle this situation by first checking the control panel indications, trying then to locate the faulted area, according to the theoretical state of the switching equipment and the current values of the measurement points. ANSYS, JMAG and FEMM computer programs were applied in calculation of magnetic fields distribution for different faults state of the motors. A sophisticated fault diagnosis and monitoring system can detect similar contradictions and point out the optimal restoration sequence. The proposed expert system, use a dedicated module for the topology and state estimation of the wheel vehicle. Some diagnostics algorithms are realized for qualification of failure work state a traction motor and also of the whole wheel vehicle. Results of computer simulations of the different failure state of the traction motor are presented in the paper. Results of calculations were partially verified in laboratory experiments.

Conclusion For analysis of magnetic field distribution can applied different method of analysis: equivalent scheme, finite element and finite boundary method. Knowledge of magnetic field distribution in steady state and in transient state of the motor enables precisely computation of their parameters. FEA in 2D and 3D enables considerate a saturation of magnetic circuit. From the power capability and the principal dimensions, the axial flux PM and induction motors can be mounted into the wheels to realize the driving strategy. The axial flux interior PM and induction motor seem to be best compromise in terms of power/weight ratio, efficiency, compactness capability characteristics. Analysis of computer calculations realized in different computer program, enable elaborate an original diagnostic an monitoring procedures, which make possible realization of diagnostics systems with prediction of traction motors faults states

References [1] M. Aydin, J. Yao, S. Huang, T.A. Lipo: Design consideration and experimental results od an axial flux PM motor with field control. Proceeding of ICEM’04, Cracow, Poland, 2004r, pp. 764-770. [2] C. Bocaletti, C. Bruzzese: A procedure for squirrel cage induction motor phase model parameters identification and accurate rotor faults simulation: mathematical aspect. Proceedings of ISEF’05. September, Vigo, Spain 2005. [3] D. Platt, B.H. Smith: Twin rotor drive for an electric vehicles. IEEE Proceedings-B, vol. 140, no. 2, March 1993r, pp. 131-138. [4] Z. Szymański: Analysis of the field distribution in special construction traction wheel motor in electric and hybrid vehicles. Proceedings of ISEF’05, September 15–17, Baiona, Spain. [5] Z. Szymański: Application of the artificial intelligence methods in diagnostics of mine machine drive system. Proceedings of 20th World Mining Congress, 7–11 November, Tehran, Iran.

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved.

457

Author Index Albert, J. Apanasewicz, S. Arsov, L. Barbieri, L. Barglik, J. Bednarek, K. Binder, A. Buchau, A. Byczkowska-Lipińska, L. Cardoso, J.R. Cecílio, J. Černigoj, A. Cha, J. Cho, Y. Cieśla, A. Clemens, M. Coco, S. Corda, J. Cundeva, S. Cundeva-Blajer, M. Czerwiński, D. Czerwiński, M. de Oliveira, O.L. Deak, Cs. Dems, M. Desideri, D. di Gerlando, A. di Napoli, A. Dolezel, I. Duchesne, S. Dular, P. Faktorová, D. Ferreira da Luz, M.V. Fireţeanu, V. Fišer, R. Foggia, A. Foglia, G. Frenner, K. Fujii, N. Fujiwara, K. Gaber, M. Garda, B. Gašparin, L.

16 8, 58 167 381 202 85 116 16, 93 438 158 144 179 370 370 350, 356 431 425 343 167 167 403 202 158 116 130 363 192 151 v 137 307 21 307 284 179 175 192 80 318 47 388 350 179

Gawrylczyk, K.M. Gawrys, P. Giżewski, T. Goleman, R. Gorenc, D. Gyftakis, K. Hafla, W. Hameyer, K. Hasegawa, Y. Hering, M. Herranz Gracia, M. Hinrichsen, V. Hirata, K. Hong, D.-K. Iancu, V. Ida, K. Irimie, D. Ishihara, Y. Jalmuzny, W. Jamil, S.M. Jussila, H. Kang, D.-H. Kantartzis, N.V. Kappatou, J. Kawase, Y. Kitagawa, W. Kobylanski, L. Kolehmainen, J. Komęza, K. Krawczyk, A. Kucharski, D. Lalas, A.X. Laudani, A. Lecointe, J.-Ph. Lemos Antunes, C. Lesniewska, E. Leva, S. Li, J. Lidozzi, A. Lopez-Fernandez, X.M. Marusic, I. Maschio, A. Maugeri, V.

64 34 403 403 108 247 16, 80, 93 39, 101 220 202 39 431 220 184 259 124 116 47 231 343 253 184, 268 212 247 124 47 175 240 130 v 26 212 425 137 144 231 381 370 151 53 108 363 381

458

Miljavec, D. Minamide, A. Mipo, J.-C. Mitsutake, Y. Mizuma, T. Morando, A.P. Naoe, N. Napieralska-Juszczak, E. Nicula, C. Niedbała, R. Niemelä, M. Ota, T. Parviainen, A. Pawłowski, S. Pereira, J.M.T. Perez, J. Perini, R. Petrovic, Lj. Pineda, M. Poli, E. Popenda, A. Przybylski, M. Puche, R. Purcarea, C. Putek, P. Pyrhönen, J. Reichert, K. Riganti Fulginei, F. Roger Folch, J. Roger, D. Roytgarts, M. Rucker, W.M. Rusek, A. Safacas, A. Sahin, O. Sakata, K. Salminen, P. Salvini, A.

299 294 175 220 318 381 294 137 259 26 253, 276 220 276, 313 8 417 72 192 116 72 363 335 34 72 116 64 253, 276 116 425 72 137 324 16, 80, 93 335 247 396 318 253, 276 425

Sartori, C.A.F. Scheiblich, C. Sehit, S. Seifert, J. Serrao, V. Shiota, H. Slusarek, B. Smirnov, А. Solero, L. Sosnowski, J. Soto Rodriguez, A. Souto Revenga, D. Steinmetz, T. Stepien, S. Strete, L. Szymanski, G. Szymański, Z. Takemata, K. Tamto, Y. Todaka, T. Trlep, M. Tsiboukis, T.D. Turowski, J. Valente, H. Valtonen, M. Varlamov, Yu. Viorel, I.-A. Wac-Włodarczyk, A. Weida, D. Wesołowski, M. Wiak, S. Woo, B.-C. Yamagiwa, A. Yamaguchi, T. Yamamoto, T. Yaman, B. Zakrzewski, K. Zidarič, B.

158 80 396 431 151 124 34 324 151 410 53 53 431 101 259, 268 101 449 294 175 47 388 212 53 144 313 324 259, 268 403 431 26, 202 v, 438 184 124 124 220 396 3 299

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