WlLEY SERIES IN MICROWAVE AND OPTICAL ENGINEERING KAI CHANC, Editor Texas A & M University
INTRODUCTION TO ELECTROMAGNETIC COMPATIBILITY Clayton R. Paul OPTICAL COMPUTING: AN INTRODUCTION Mohammad A. Karim and Abdul Abad S. Awwal COMPUTATIONAL METHODS FOR ELECTROMAGNETICS AND MICROWAVES Richard C.Booton, )r. FIBER-OPTIC COMMUNICATION SYSTEMS Covind P. Agrawal OPTICAL SIGNAL PROCESSING, COMPUTING, AND NEURAL NETWORKS Francis T. S. Yu and Suganda lutamulia MULTICONDUCTOR TRANSMISSION LINE STRUCTURES j . A. Brand50 Faria MICROWAVE DEVICES, CIRCUITS, AND THEIR INTERACTIONS Charles A, Lee and C.Conrad Dalman MICROSTRIP CIRCUITS Fred Cardiol HIGH-SPEED VLSl INTERCONNECTIONS: MODELING, ANALYSIS, AND SIMULATION A. K. Goel MICROWAVE SOLID-STATE CIRCUITS AND APPLICATIONS Kai Chang HIGH FREQUENCY ANALOG INTEGRATED CIRCUIT DESIGN Ravender Coyal ANTENNAS FOR RADAR AND COMMUNICATIONS Harold Mott ANALYSIS OF MULTICONDUCTOR TRANSMISSION LINES Clayton R. Paul
Analysis of Multiconductor Transmission Lines CLAYTON
R. PAUL
Department ol Electrical Engineering University of Kentucky, Lexington
A WILEY-INTERSCIENCE PUBLICATION
JOHN WlLEY & SONS NEW YORK
/ CHICHESTER / BRISBANE / TORONTO / SINGAPORE
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Library of Coagres Catafoghg In P~~fflcation &tar Paul, Clayton R. Analysis of multiconductor transmission lines / Clayton R. Paul. p. cm,--(Wiley eeries in microwave and optical engineering) "A Why-hterdence publication." Includes bibliographical mfamnas and index, ISBN 0-471.02080-X (alk. paper). 1. Multiconductor transmission lines. 2. Electric circuit analysis-Data proassing. I. Title. 11, Series. TK7872.T74P38 1994 621.3 19'2-4~20 94-5704 Printed in the United States of America
10 9 8 7 6 5
To the humane and compassionate treatment ol animals
A man is truEy ethical only when he obeys the compulsion to help all lite which he is able to assist, and shrinks irom injuring anything that lives. Albert Schweitzer It ill becomes us to invoke in our daily prayers the blessings ot God, the compassionate, if we in turn will not practice elementary compassion towards our lellow creatures. Gandhi We can judge the heart ol man by his treatment of animals. Jmmanuel Kant
Contents
PREFACE 1 Introduction
1.1 Examples of Multiconductor Transmission-Line Structures 1.2 Properties of the Transverse ElectroMagnetic (TEM) Mode of
Propagation
XV
1 4 7
1.3 Derivation of the Transmission-Line Equations for
2
Two-Conductor Lines 1.3.1 Derivation from the Integral Form of Maxwell’s Equation 1.3.2 Derivation from the Differential Form of Maxwell’s Equations 1.3.3 Derivatipn from the Per-Unit-Length Equivalent Circuit 1.3.4 Properties of the Per-Unit-Length Parameters 1.4 Classification of Transmission Lines 1.5 Restrictions on the Applicability of the Transmission-Line Equation Formulation 1.5.1 Higher-Order Modes 1.5.1,l The Infinite Parallel-Plate Transmission Line 1.5.1.2 The Coaxial Transmission Line 1.5.1.3 Two-Wire Lines 1.5.2 Transmission-Line Currents vs. Antenna Currents References Problems
30 30 30 36 37 37 40 42
The Mulliconductor Transmission-Line Equations
46
2.1 Derivation from the Integral Form of Maxwell’s Equations 2.2 Derivation from the Per-Unit-Length Equivalent Circuit
46 56
13 13 21 22 24 26
vii
viii
CONTENTS
2.3 Summary of the MTL Equations 2.4 Properties of the Per-Unit-Length Parameter Matrices L,C,G
References Problems 3 The Per-Unit-iength Parameters
57 58 62 62 64
3.1 Definitions of the Per-Unit-Length Parameter Matrices
L,c, G
The Per-Unit-Length Inductance Matrix, L The Per-Unit-Length Capacitance Matrix, C The Per-Unit-Length Conductance Matrix, G The Generalized Capacitance Matrix, Q 3.2 Multiconductor Lines Having Conductors of Circular Cylindrical Cross Section 3.2.1 Fundamental Subproblems for Wires 3.2.1.1 Magnetic Flux Due to a Filament of Current 3,2.1.2 Voltage Due to a Filament of Charge 3.2.1.3 The Method of Images 3.2.2 Exact Solutions for Two-Conductor Wire Lines 3.2.2.1 Two Wires 3.2.2.2 One Wire Above an Infinite, Perfectly Conducting Plane 3.2.2.3 The Coaxial Cable 3.2,3 Wide-Separation Approximations for Wires in Homogeneous Media 3.2.3.1 (n 1) Wires 3.2.3.2 n Wires Above an Infinite, Perfectly Conducting Plane 3.2.3.3 n Wires Within a Perfectly Conducting Shield 3.2.4 Numerical Methods for the General Case 3.2.4.1 Applications to Inhomogeneous Dielectric Media 3.2.5 Computed Resufts: Ribbon Cables 3.3 Multiconductor Lines Having Conductors of Rectangular Cross Section 3.3.1 Method of Moments (MOM) Techniques 3.3.1.1 Applications to Printed Circuit Boards 3.3.1.2 Computed Results: Printed Circuit Boards 3.3.2 Finite Difference Techniques 3.3.3 Finite Element Techniques 3.4 Miscellaneous Additional Techniques 3.4.1 Conformal Mapping Techniques 3.1.1 3.1.2 3.1.3 3.1.4
+
65 65 69
72 73 77 77 77 80 82 83 83 89 90 92 93 93 96 97
103 109
113 115 124 135 140 146 153 154
CONTENTS
3.4.2 Spectral-Domain Techniques 3.5 Shielded Lines 3.6 Incorporation of Losses; Calculation of R, L,,and G 3.6.1 Calculation of the Per-Unit-Length Conductance Matrix,
G
3.6.2 Representation of Conductor Losses 3.6.2.1 Surface Impedance of Plane Conductors 3.6.2.2 Resistance and Internal Inductance of Wires 3.6.2.3 Internal Impedance of Rectangular Cross Section
Conductors
3.6.2.4 Approximate Representation of Conductor
Internal Impedances in the Frequency Domain References Problems 4
iX
155 156 157 158 161 162 164 168 177 180 182
Frequency-DomainAnalysis
186
4.1 The MTL Equations for Sinusoidal Steady-State Excitation 4.2 Solutions for Two-Conductor Lines 4.3 General Solution for an (n 1)-Conductor Line 4.3.1 Analogy of the MTL Equations to the State-Variable
186 189 194
+
Equations
195
Transformations
200
Parameter Matrix
205 206 209 210 213 214 216 219 229 222 224
4.3.2 Decoupling the MTL Equations by Similarity 4.3.3 Characterizing the Line as a 2n Port with the Chain 4.3.4 Properties of the Chain Parameter Matrix 4.3.5 Incorporating the Terminal Conditions 4.3.5.1 The Generalized Th6venin Equivalent 4.3.5.2 The Generalized Norton Equivalent 4.3.5.3 Mixed Representations 4.3.6 Approximating Nonuniform Lines
4.4 Solution for Line Categories 4.4.1 Perfect Conductors in Homogeneous Media 4.4.2 Lossy Conductors in Homogeneous Media 4.4.3 Perfect Conductors in Inhomogeneous Media 4.4.4 The General Case: Lossy Conductors in Lossy
Inhomogeneous Media
4.5 4.6 4.7 4.8
4.4.5 Cyclic Symmetric Structures
Lumped-Circuit Iterative Approximate Characterizations AI ternative 2n-Port Characterizations Power and the Reflection Coefficient Matrix Computed Results
225 225 23 1 234 236 238
x
CONTENTS
References Problems
239 241 246 246
5 Time-Domain Analysis
252
4.8.1 Ribbon Cables 4.8.2 Printed Circuit Boards
5.1 Two-Conductor Lossless Lines 5.1.1 Graphical Solutions 5.1.2 The Method of Characteristics (Branin’s Method) 5.1.3 The Bergeron Diagram 5.2 Multiconductor Lossless Lines 5.2.1 Decoupling the MTL Equations 5.2.1.1 Lossless Lines in Homogeneous Media 52.1.2 Lossless Lines in Inhomogeneous Media 5.2.1.3 Incorporating the Terminal Conditions via the
SPICE Program 5.2.2 Extension of Branin’s Method to Lossless
253 257 266 270 275 277 279 280
283
Multiconductor Lines in Homogeneous Media
288 292
Characterizations
295 295 309 311 317 320 323 324
5.2.3 Time-Domain to Frequency-Domain Transformations 5.2.4 Lumped-Circuit Iterative Approximate 5.2.5 Finite Difference-Time Domain (FDTD) Methods 5.2.6 Computed Results 5.2.6.1 Ribbon Cable 5.2.6.2 Printed Circuit Board
5.3 Incorporation of Losses 5.3.1 Two-Conductor Lossy Lines 5.3.1.1 Lumped-Circuit Approximate Characterizations 5.3.1.2 Time-Domain to Frequency-Domain
Transformations
325
Methods
326
Transform
334
a Two Port 5.3,2 Multiconductor Lines 5.3.3 Computed Results 5.3.3.1 Ribbon Cable 5.3.3.2 Printed Circuit Board References Problems
337 343 347 348 35 1 35 1 354
5.3.1.3 Finite Difference-Time Domain (FDTD) 5.3.1.4 Direct Solution via Inversion of the Laplace 5.3.1.5 TimaDomain Characterization of the Line as
CONTENTS
6
XI
Literal (Symbolic) Solutions for Three-Conductor Lines
359
6.1 Frequency-Domain Solution 6.1.1 Inductive and Capacitive Coupling 6.1.2 Common-Impedance Coupling
363 367 369 37 1 373 375 378 382 383 383 388 393 394
6.2 Time-Domain Solution 6.2.1 Explicit Solution 6.2.2 Weakly Coupled Lines 6.2.3 Inductive and Capacitive Coupling 6.2.4 Common-Impedance Coupling 6.3 Computed Results 6.3.1 A Three-Wire Ribbon Cable 6.3.2 A Three-Conductor Printed Circuit Board References Problems 7 Incident-Field Excitation of the Line
7.1 Derivation of the MTL Equations for Incident-Field
Excitation 7.1.1 Equivalence of Source Representations 7.2 Frequency-Domain Solutions 7.2.1 Solution of the MTL Equations 7.2.1.1 Simplified Forms of the Excitations 7.2.2 Incorporation of the Terminal Conditions 7.2.2.1 Lossless Lines in Homogeneous Media 7.2.3 Lumped-Circuit Iterative Approximate Characterizations 7.2.4 Uniform Plane-Wave Excitation of the Line 7.2.5 Two-Conductor Lines 7.2.5.1 Uniform Plane-Wave Excitation of the Line 7.2.5.2 Special Cases 7.2.5.3 One Conductor Above a Ground Plane 7.2.5.4 Electrically Short Lines 7.2.6 Computed Results 7.2.6.1 Comparison with Predictions of the Method of Moments Codes 7.2.6.2 A Three-Wire Line in an Incident Uniform Plane Wave 7.3 Time-Domain Solutions 7.3.1 Two-Conductor Lossless Lines 7.3.1.1 The General Solution via the Method of Characteristics
395 395 402 405 406 407 410 413 415 416 423 425 426 429 433 435 435
440 444 446 446
xi;
CONTENTS
7.3.1.2 The General Solution via the Frequency Domain 7.3.1.3 Uniform Plane-Wave Excitation of the Line 7.3.1.4 Electrically Short Lines 7.3.1.5 A SPICE Equivalent Circuit 7.3.1.6 Computed Results 7.3.2 Multiconductor Lines 7.3.2,l Decoupling the MTL Equations 7.3.2.2 A SPICE Equivalent Circuit 7.3.2.3 Lumped-Circuit Iterative Approximate Characterizations 7.3.2.4 Time-Domain to Frequency-Domain Transformations 7.3.2.5 Finite Difference-Time Domain Methods 7.3.2.6 Computed Results References Problems 8 Transmission-line Networks
8.1 Representation with the SPICE Model Representation with Lumped-Circuit Iterative Models Representation via the Admittance or Impedance Parameters Representation with the BLT Equations Direct Time-Domain Solutions in terms of Traveling Waves References Problems
8.2 8.3 8.4 8.5
448 453
459
460 463 466 467 470 475 475 477 480 486 487 489 492
492 494 508
517
522
523
Publications by the Author Concerning Transmission lines
525
Appendix A Description of Computer Software
53 1
A.l Programs for Calculation of the Per-Unit-Length Parameters A.1.1 Wide-Separation Approximations for Wires: WIDESEP.FOR A.1.2 Ribbon Cables: RIBBON.FOR A. 1.3 Printed Circuit Boards: PCB.FOR, PCBGALFOR A.1.4 Coupled Microstrip Structures: MSTRP.FOR, MSTRPGAL.FOR A.2 Frequency-Domain Analysis A.2.1 General: MTLFOR A.3 Time-Domain Analysis
532
533 536 539 541 542 542 543
CONTENTS
A.3.1 Time-Domain to Frequency-Domain Transformation : TIMEFREQ.FOR A.3.2 Branin's Method Extended to Multiconductor
Lines: BRANIN.FOR
543 544
A.3.3 Finite Difference-Time Domain Method:
FINDIF.FOR
544
A.3.4 Finite Difference-Time Domain Method:
FDTDLOSS.FOR A.4 SPICE/PSPICE Subcircuit Generation Programs A.4.1 General Solution, Lossless Lines:
SPICEMTL.FOR
544 545 545
A.4.2 Lumped-pi Circuit, Lossless Lines:
SPICELPLFOR
545
A.4.3 Inductive-Capacitive Coupling Model:
SPICELC.FOR 547 547 A S Incident Field Excitation A.5.1 Frequency-Domain Program: 1NCIDENT.FOR 547 A.5.2 SPICE/PSPICE Subcircuit Model: 548 SPICEINC.FOR A 5 3 Finite DiKerence-Time Domain (FDTD) 550 Model: FDTDINCFOR 551 References INDEX
553
Preface
This textbook is intended for a senior or graduate-level course in an Electrical Engineering curriculum on the subject of the analysis of Multiconductor Transmission Lines (MTL’s). It will also be a useful reference on the subject for industrial professionals. The term MTL typically refers to a set of (n + 1) parallel conductors that serve to transmit electrical signals between sources and loads. The dominant mode of propagation in a MTL is the Transverse ElectroMagnetic or TEM mode of propagation where the electric and magnetic fields surrounding the conductors lie solely in the transverse plane orthogonal to the line axis. This structure is capable of guiding waves whose frequencies range from dc to where the line cross-sectional dimensions become a significant fraction of a wavelength. At higher frequencies, higher-order modes coexist with the TEM mode and other guiding structures such as waveguides and antennas are more practical structures for transmitting the signal between a source and a load. There are many applications for this wave-guiding structure, High-voltage power transmission lines are intended to transmit 60 Hz sinusoidal waveforms and the resulting power. In addition to this low-frequency power frequency, there may exist other, higher-frequency components of the transmitted signal such as when a fault occurs on the line or a circuit breaker opens and recloses. The waveforms on the line associated with these events have high-frequency spectral content. Cables in modem electronic systems such as aircraft, ships and vehicles serve to transmit power as well as signals throughout the system. These cables consist of large numbers of individual wires that are packed into bundles for neatness and space conservation. The electromagnetic fields surrounding the individual wires interact with each other and induce signals in all the other adjacent circuits. This is unintended and is referred to as crosstalk. This crosstalk can cause functional degradation of the circuits at the ends of the cable. The prediction of crosstalk will be one of our major objectives in this text. There are numerous other similar structures. A printed circuit board (PCB) consists of a planar dielectric board on which rectangular cross section conductors (lands) serve to interconnect digital devices as well as analog devices. Crosstalk can be a significant functional problem with these PCB’s as can the degradation of the intended signal transmis-
xvi
PREFACE
sion through attenuation, time delay, and other effects. Signal degradation, time delay and crosstalk can create significant functional problems in today’s high-speed digital circuits so that it is important to understand and predict this effect. It has been said that optical fibers will eliminate many of these problems associated with metallic conductors such as crosstalk. Although this is true to a large degree, full implementation of fiber optic transmision paths will occur well into the future because of the present low cost and significant use of metallic-conductor lines. The analysis of a two-conductor line (n = 1) is a standard and well-understood subject in all Electrical Engineering curricula. However, the analysis of MTL’s consisting of three or more coupled conductors is not as well known. The purpose of this text is to provide a compact and complete description of the existing mathematical techniques for analyzing MTL’s. The assumption of the TEM mode of propagation on the line results in a set of coupled partial differential equations. These are referred to as the transmission-line equations. The sole purpose of this text is to investigate ways of solving these transmission-line equations for MTL’s and incorporating the constraints imposed by the terminations into that general solution. If one looks at the research literature, one finds a seemingly unbounded number of methods for analyzing MTL’s. However, there actually exist a small number of standard techniques which we will elucidate in this text. Understanding the primary, fundamental analysis techniques given in this text will allow the reader to understand and categorize the myriad of seemingly new analysis techniques that appear in the literature. Our focus will be on two important interference mechanisms in MTL’s--crosstalk and the effects of incident electromagnetic fields on MTL’s. In the case of crosstalk, the driving signals are in the termination networks and produce the electromagnetic fields of the line which result in intended as well as unintended reception in the terminations. In the case of incident field illumination of the MTL, the driving signals are produced by distant sources and can also create interference effects in the MTL. These driving signals can be characterized in thefiequency domain (single frequency sinusoidal signals) and the time domain (general time variations). It is convenient to break our analyses into these two classes. If the terminations are linear, the time domain results can be obtained from the frequency domain results by superposition. For nonlinear terminations, the general time-domain results must be obtained directly. The text is divided into eight chapters. Considerable thought has gone into the organization of the text. The author is of the strong opinion that organizationof subject material into a logical and well thought out form is perhaps the most important pedagogical technique in a reader’s learning process. This logical organization is one of the important attributes of the text. Chapter 1 discusses the background and rationale for the use of MTL’s. The general properties of the TEM mode of propagation are discussed, and the transmission-line equations are derived several ways for two-conductor lines. The various classifications of MTL’s (uniform, lossless, homogeneous medium) are discussed along with the restrictions on the use of the TEM model. Chapter 2 provides a derivation of the MTL equations along with the general properties of the per-unit-length parameter matrices in those equations. A key ingredient in all MTL characterizationsis the per-unit-length parameter matri-
PREFACE
XVii
ces in the MTL equations. All structural dimensions of MTL’s are contained in these per-unit-length parameter matrices and nowhere else. If one intends to obtain predictions of the response of a MTL without actually constructing it, one must not only solve the MTL equations but also determine the per-unit-length parameters. Solving the MTL equations without determining the per-unit-length parameters is of no use. Chapter 3 details the general techniques for determining these per-unitlength parameters for a MTL. Analytical as well as numerical methods will be discussed. The general solution of the MTL equations begins with the frequency-domain analysis in Chapter 4. Chapter 5 examines the direct, time-domain solution of the MTL equations. The solution techniques in the previous chapters for a MTL require matrix methods and numerical solution. Chapter 6 gives closed-form solutions in terms of the line parameters for the case of a three-conductor line. This serves to elucidate the general behavior that is common to all MTL’s and gives useful design formulae. Chapter 7 examines the effects of an electromagnetic field incident on the MTL. The effect of this incident field is to yield sources distributed along the line rather than existing solely in the termination networks. The general solution of the MTL equations for this case is examined. Chapter 8 provides the application of these techniques to interconnections of transmission lines: Transmission- Line Networks. There are a number of important learning aids included in this text. A limited but representative set of end-of-chapter problems are provided. Computed data are obtained in each chapter to illustrate the methods, and experimental results are provided to illustrate their accuracy or limitations. These are provided for two practical applications: a ribbon cable and a PCB. It is important for the reader to obtain a feel for the prediction accuracy and limitations of the MTL model. These computed and experimental results provide that insight. And finally, several FORTRAN codes were written to implement the methods of this book. Each of these codes implements one of the important analysis techniques described in the text. They are written in ANSI Fortran 77 language and may be compiled with any standard FORTRAN compiler. They are suitable for use on personal computers. Thus the reader can immediately begin testing each technique for practical structures of hidher choosing. These computer codes can be downloaded from the Wiley ftp site at ftp ://ftp .wiley.com/public/sci-tech_med/multiconductor-~ans~ssio~
There are many colleagues who have contributed substantially to the author’s understanding of this subject. Frederick M. Tesche and Albert A. Smith, Jr. are among those to whom the author owes a debt of gratitude for many insightful discussions. CLAYTON R.PAUL
Lexington, KY July 1993
CHAPTER ON€
Introduction
This text concerns the analysis of transmission-line structures that serve to guide electromagnetic (EM) waves between two points. The analysis of transmission lines consisting of two parallel conductors of uniform cross section is a fundamental and well-understood subject in electrical engineering. However, the analysis of similar lines consisting of more than two conductors is not as well understood. The purpose of this text is to provide a concise, yet complete, description of the formulation and analysis of the transmission-line equations for lines consisting of more than two conductors (multiconductor transmission lines or MTLs). The analysis of MTLs is somewhat more difficult than the analysis of two-conductorlines but the applications cover a broad frequency spectrum and extend from power transmission lines to microwave circuits CB.1, 1-16]. However, matrix methods and notation provide a straightforward extension of many, if not most, of the aspects of two-conductor lines to MTLs. Many of the concepts and performance measures of two-conductor lines require more elaborate concepts when extended to MTLs. For example, in order to eliminate reflections at terminations on a two-conductor line we simply terminate it in a matched load, i.e., a load impedance which equals the characteristic impdance of the line. In the case of MTLs, we must terminate the line in a characteristic impedance matrix or network of impedances in order to eliminate all reflections. It is not sufficient to simply insert a “characteristic impedance” between each conductor and the reference conductor; there must also be impedances between every pair of conductors. In order to describe the degree of mismatch of a particular load impedance on a two-conductor line, we compute a scalar reflection coefficient. In the case of a MTL, we can obtain the analogous quantity but it becomes a rejection coeflcient matrix. On a two-conductor line there are forward- and backward-traveling waves each traveling in opposite directions with velocity Y. In the case of a MTL consisting of (n 1) conductors, there exist n forward- and n backward-traveling waves each with its own velocity. Each pair of forward- and backward-traveling waves is referred to as
+
1
2
INTRODUCTION
a mode. If the MTL is immersed in a homogeneous medium, each mode velocity is identical to the velocity of light in that medium. Each mode velocity of a MTL that is immersed in an inhomogeneous medium (such as wires with dielectric insulations) will, in general, be different. The governing transmissionline equations for a two-conductor line will be a coupled set of two, first-order partial differential equations for the line voltage, V(z,t), and line current, I(z, t), where the line conductors are parallel to the z axis and time is denoted as t. Solution of these coupled, scalar equations is straightforward. In the case of a MTL consisting of (n 1) conductors parallel to the z axis, the corresponding governing equations are a coupled set of 2n, first-order, matrix partial differential equations relating the n line voltages, V;(z,t), and n line currents, I&, t), for i = 1,2,. ,n. The number of conductors may be quite large, e.g., (n 1) = 100, in which case efficiency of solution of the 2n MTL equations becomes an important consideration. The efficiency of solution of the MTL equations depends upon the assumptions or approximations one is willing to make about the line, e.g., lossless vs. lossy line, homogeneous vs. inhomogeneous surrounding media, etc., as well as the solution technique chosen. Although it is tempting to dismiss the analysis of MTLs as simply being a special case of two-conductor lines thereby not requiring scrutiny, this is not the case. The purpose of this text is to examine the common solution techniquesfor the MTL equations. This makes it clear that a seemingly new solution technique may simply be a version of an existing technique. The analysis of a MTL for the resulting n line voltages, q(z, t), and n line currents, I&, t), is a three-step process.
+
..
+
1: Determine the per-unit-length parameters of inductance, capacitance, conductance and resistance for the given line. All cross-sectional information
STEP
about the particular line that distinguishes it from some other line is contained in these per-unit-length parameters. The MTL equations are identical in form for all lines: only the per-unit-length parameters are different. Without a determination of the per-unit-length parameters for the specificline, one cannot solve the resulting MTL equations because the coefficients in those equations will be unknown. STEP 2 : Solve the resulting MTL equations. For a two-conductor line, the general solution consists of the sum of forward- and backward-traveling waves with 2 unknown coeflcients. For a MTL consisting of (n 1) conductors the general solution consists of the sum of n forward- and n backward-traveling waves with
+
2n unknown coefficients. STEP 3:
Incorporate the terminal conditions to determine the unknown coeflcfents in the general form of the solution. A transmission line will have terminations
at the left and right ends consisting of independent voltage and/or current sources and lumped elements such as resistors, capacitors, inductors, diodes, transistors, etc. These terminal constraints provide the additional 2n equations
INTRODUCTION
3
(n for the left termination and n for the right termination) which can be used to explicitly determine the 2n undetermined coefficients in the general form of the MTL equation solution that was obtained in Step 2. The excitation for the MTL will have several forms. Independent lumped sources within the termination networks are one method of exciting the line. These sources are intended to be coupled to the endpoint of that line. However, the electromagnetic fields associated with the current and voltage on that line interact with neighboring lines inducing signals at those endpoints. This coupling is unintentional and is referred to as crosstalk. Another method of exciting a line is with an incident electromagnetic field as with a radio signal or a lightning pulse. This form of excitation produces sources that are distributed along the line and will also induce unintentional signals at the line endpoints that may cause Interference. Lumped sources can occur at discrete points along the line as with the direct attachment of a lightning stroke. The effect of incident fields either distributed along the line or at discrete points will be included in the MTL equations. This type of excitation wilt be deferred to Chapter 7. Lumped sources in the termination networks will constitute the primary excitations up to that point, and their effect will be included in the terminal network characterizations. In order to obtain the complete solution for the line voltages and currents, each of the above three steps must be performed and generally in the above order. Throughout our discussions this sequence of solution steps must be kept in mind and no steps can be omitted. It is as important to be able to determine the per-unit-length parameters for the particular line as it is to obtain the general form of the solution of the MTL equations! Electromagnetic fields are, in reality, distributed continuously throughout space. If a structure's largest dimension is electrically small, is., much less than a wavelength, we can approximately lump the EM effects into circuit elements as in lumped-circuit theory and define alternative variables of interest such as voltages and currents. The transmission-line formulation views the line as a distributed-parameter structure along the structure axis and thereby extends the lumped-circuit analysis techniques to structures that are electrically large in this dimension. However, the cross-sectional dimensions, e.g., conductor separations, must be electrically small in order for the analysis to yield valid results. The fundamental assumption for all transmission-line formulations and analyses, whether it be for a two-conductor line or a MTL, is that the field structure surrounding the conductors obeys a Transverse ElectroMagnetic or TEM structure. A TEM field structure is one in which the electric and magnetic fields in the space surrounding the line conductors are transverse or perpendicular to the line axis which will be chosen to be the z axis of a rectangular coordinate system. The waves on such lines are said to propagate in the TEM mode. Transmission-linestructureshaving electrically large cross-sectional dimensions have, in addition to the TEM mode of propagation, other higher-order modes of propagation [17-19). An analysis of these structures using the transmissionline equation formulation would then only predict the TEM mode component
4
INTROOUCrlON
and not represent a complete analysis. Other aspects, such as imperfect line conductors, also may invalidate the TEM mode transmission-line equation description. In addition, an assumption that is inherent in the MTL equation formulation is that the sum of the line currents at any cross section of the line is zero, In this sense we say that one of the conductors, the reference conductor, is the return for the other n currents. Even though the line cross section is electrically small, it may not be true that the currents sum to zero at any cross section; there may be other currents in existence on the line conductors [20-231. Presence of nearby conductors or other metallic structures which are not included in the formulation may cause these additional currents [24]. Asymmetries in the physical terminal excitation such as offset source positions (which are implicitly ignored in the terminal representation) can also create these non-TEM currents [24]. It is important to understand these restrictions on the applicability of the representation and the validity of the results obtained from it, and those aspects will also be discussed in this text. Although there is a voluminous base of references for this topic, important ones will be referenced, where appropriate, by [XI. These are grouped into two categories-those by the author (grouped by category) and other references. References consisting of publications on this topic by the author are listed at the end of the text and are grouped by category. Additional references will be listed at the end of each chapter. Limited numbers of problems are given at the end of each chapter to provide the reader with exercises for illumination of the important points and techniques. It is important to remind the reader that the sole purpose of this text is to present a complete and concise description of methods for solving the MTL equations that describe u MTL under the assumption of the TEM mode of propagation. Therefore we will derive and solve only the MTL equations. A complete solution of the MTL structure which does not presuppose only the TEM mode can be obtained with so-called full-wave solutions of Maxwell’s equations [17-19]. Generally these techniques require numerical methods for their solution. Our goal will be to examine methods for solving the MTL partial differential equations. So the effects of non-TEM field structures will not be considered. However, for parallel lines wherein the cross-sectional dimensions are much less than a wavelength, the solution of the MTL equations gives the significant contribution to the fields and resulting terminal voltages and currents. This is referred to as the quasi-TEM approximation and is an implicit assumption throughout this text. 1.1 EXAMPLES
OF MULTICONDUCTOR TRANSMISSION-LINE STRUCTURES
There are a number of examples of wave-guiding structures that may be viewed as “transmission lines.” Figure 1.1 shows examples of (n + 1)-conductor wire-type lines consisting of parallel wires. Throughout this text we will refer to conductors that have circular cylindrical cross sections as being wires. Figure
EXAMPLES OF MULTICONDUCTOR TRANSMISSION-LINE STRUCTURES
5
"t
I
"t
($jjj) Shield
+
FIGURE 1.1 Multiconductor lines in homogeneous media: (a) (n l)-wire line, (b) n wires above a ground plane, (c) n wires within a cylindrical shield.
l.l(a) shows an example of (n -t 1) wires. Typical examples of such lines are flatpack or ribbon cables used to interconnect electronic systems. Normally these wires are surrounded by circular cylindrical dielectric insulations. However, these insulations are omitted from this figure, and, in some cases, may be ignored in the analysis of such lines. Figure l.l(b) shows n wires above an infinite, perfectly conducting ground plane. Typical examples are cables which have a metallic structure as a return or high-voltage power distribution lines. In the case of high-voltage distribution lines, the return path is earth. Figure l.l(c) shows n wires within an overall, cylindrical shield. Shields are often placed around cables in order to prevent or reduce the coupling of electromagnetic fields to the cable from adjacent cables (crosstalk) or from distant sources such as radar transmitters or radio and television stations. The wires in each of these structures are shown as being of ungorm cross section along their length and parallel to each other (as well as the ground plane in Fig. l.l(b) and the shield axis in Fig. l.l(b)), Such lines are said to be uniform lines. Nonuniform lines in which either the conductors are not of uniform cross section along their length or are not parallel arise from either nonintentional or intentional reasons. For example, the conductors of a high-voltage power distribution line, because of their weight, sag and are not parallel to the ground. Tapered lines are intentionally designed to give certain desirable characteristics in microwave filters. The lines in Fig. 1.1 are said to be immersed in a homogeneous medium
6
INTRODUCTION
Multiconductorlines in inhomogeneous media, n lands on a printed circuit board (PCB):(a) n lands with a ground plane as reference, (b) (n + 1) lands.
ACURE 1.2
(logically free space since any dielectric insulations are not shown or ignored). There exist many useful transmission-line structures wherein the dielectric surrounding the conductors cannot be similarly ignored. Figure 1.2 shows examples of these. Figure 1.2(a) shows a structure having n conductors of rectangular cross section or lands supported on a dielectricsubstrate. A perfectly conducting, infinite ground plane covers the lower surface of the substrate. This is referred to in microwave literature as a coupled microstrip and is used to construct microwave filters. Figure 1.2(b) shows a similar structure where the ground plane is replaced by another land of rectangular cross section. This type of structure is common on printed circuit boards (PCBs)in modern electronic circuits. This type of structure is used to construct busses that carry digital data or control signals. The structures in Fig. 1.1 are, by implication, immersed in a homogeneous medium. Therefore the velocity of propagation of the waves is equal to that of the medium in which it is immersed or v = l/& where p is the permeability of the surrounding medium and 8 is the permittivity of the surrounding medium. F/m. For free space these become p, = 4n x IO-' H/m and e, w (1/36x) x For the structures shown in Fig. 1.2 which are immersed in an inhomogeneous medium (the fields exist partly in free space and partly in the substrate), there are n waves or modes whose velocities are, in general, different. This complicates the analysis of such structures as we will see.
PROPERTIES OF THE TRANSVERSE ELECTROMAGNETIC ( T E N MODE OF PROPAGATION
7
J
Y
Illustration of the electromagnetic field structure of the transverse electromagnetic (TEM)mode of propagation. FIGURE 1.3
1.2 PROPERTIES OF THE TRANSVERSE ELECTROMAGNETIC (TEM) MODE OF PROPAGATION
As mentioned previously, the fundamental assumption in any transmissionline formulation is that the electric field intensity vector, a ( x , y , z, t), and the magnetic field intensity vector, $(x, y, z, t), satisfy the transverse electromagnetic (TEM) field structure, i.e., they lie in a plane (the x-y plane) transverse or perpendicular to the line axis (the z axis). Therefore it is appropriate to examine the general properties of this TEM mode of propagation or field structure. Consider a rectangular coordinate system shown in Fig. 1.3 illustrating a propagating TEM wave in which the field vectors are assumed to lie in a plane (the x-y plane) that is transverse to the direction of propagation (the z axis). These field vectors are denoted with a t subscript to denote transoerse. It is assumed that the medium is homogeneous, linear and isotropic and is characterized by the scalar parameters of permittiuity, 8, permeability, p, and conductiulty, 0. Maxwell's equations become CA.1)
(Ma) (l.lb) The del operator, V, can be broken into two components, one component, V,, in the z direction and one component, V,, in the transverse plane as
v = v, + v,
(1.2a)
8
INTRODUCTION
where
Vt = d,
a + d, a ax ay
(1.2b)
V, = d,
a az
(1.2c)
where d,, d,, d, are unit vectors pointing in the appropriate directions. Separating (1.1) by equating those components in the z direction and in the transverse plane gives
a4 d, x -=
.
a, x
az
- p - a% at
(1.3a)
a% = u 4 + e a4 az
at
(1.3b) (1.3~) (1.3d)
Equations (1,3c) and (1,3d) are identical to those for staticjields. This shows that the electric and magnetic jields of a TEM field distribution satisfy a static distribution in the trunsuerse plane. Because of (1.3~)and (1.3d), we may define each of the transverse field vectors as the gradients of some auxiliary scalar fields or potential functions, 4 and +, such as CA.11 (1.4a) (1.4b) The scalar coefficients, g(z, t) and j ( z , t), are to be determined. Gauss' laws become CA.1) vt*#= 0 (1Sa)
VI.* = 0
(1.5b)
Applying (13) to (1.4) gives (1.6a) (1.6b) Equations (1.6) show that the auxiliary scalar potential functions satisfy Laplace's equation in any transverse plane as they do for static fields. This permits the unique definition of voltage between two points in a transverse plane
PROPERTIES OF THE TRANSVERSE ELECTROMAGNETIC (TEM) MODE OF PROPAGATION
9
as the line integral of the transverse electric field between those two points: (1.7a) This is a case where we may uniquely define voltage between two points for nonstatic time variation. Similarly, equation (1.6b) shows that we may uniquely define current in the z direction as the line integral of the transverse magnetic field around any closed contour lying solely in the transverse plane: (1.7b) Ordinarily the line integral in (1.7b) contains conduction current, d, and any source current, Js, as well as displacement current, e(&?/&), due to the time rate-of-change of the electric field penetrating the surface bounded by the contour. Since the electric field is confined to the transverse plane and therefore has no z component, the conduction current, uC?, and displacement, e(aC?/at), penetrating the transverse contour, c,, are zero thereby giving the current definition solely as source currents, such as may exist on the surfaces of conductors that penetrate the surface of this contour, as is the case for static fields. Once again, this permits a unique definition of current for nonstatic variation of the field vectors in a fashion similar to the static or dc case. Now suppose we take the cross product of the z-directed unit vector with (1.3a) and (1.3b). This gives (Ma) d, x d, x
a% = a(d, x 4) + E 82
(1.8b)
However, (1.9a)
(1.9b) as illustrated in Fig. 1.4. Therefore, equations (1.8) become (1.10a) (1.1 Ob)
10
INTRODUCTlON
c
FIGURE 1.4 Illustration of the identity 4 x dJ x e$
- -4.
Taking the partial derivative of both sides of (1.10) with respect to substituting (1.3) gives
2
and
(1.lla) (1.11 b)
Now let us consider the case where the medium is lossless, Le., u = 0. In this case equations (1.11) reduce to (1.12a) (1.12b) The solutions to these second-order differential equations are CA.1J &x, y, 2, t) =
t
;)
- + I-(*, y, t + ;)
*(x,y,z,t)=2+
=
1 a+(,,y, t - ;) rl
;
a-(,y, t +
(1.13a) (1.13b)
;)
PROPERTIES OF THE TRANSVERSE ELECTROMAGNETIC (TEN MODE OF PROPAGATION
11
where the intrinsic impedance of the medium is F
(I. 13c) and the velocity of propagation is V=-
1
(1.13d)
JG
The function J+(x, y, t - z/v) represents a forward-traueling waue since as t progresses z must increase to keep the argument constant and track corresponding points on the waveform. Similarly, the function f - ( x , y, t z/v) represents a backward-truueling waue, a wave traveling in the - z direction. Consequently we may indicate the vector relation between the electric and magnetic fields as
+
(1.14)
with the sign depending on whether we are considering the backward- or forward-traveling wave component. Equations (1.3a) and (1.3b) show that 4 and *are orthogonal so that (1.14) applies (with a different intrinsic impedance) even if the medium is lossy. If the time variation of the field vectors is sinusoidal, we use phasor notation CA.21:
Y,2, t ) = aa{$(x, y, zWoP‘) * ( x , y, z, t ) = a.(fi,(x, y , z)e@‘) &x,
Replacing time derivatives with differential equations:
a/& =jw in (1.12)
(1.1 Sa)
(1.15b)
gives the phasorform of the
(1.16a) (1.16b) The solutions to these equations become CA.11
12
INTRODUCnON
where
and the phase constant is denoted as
a=qlG
(1.17e)
The time-domain expressions are obtained by multiplying (1.17a) and (1.17b) by eloorand taking the real part of the result CA.1). For example, the x components of the transverse field vectors are:
where the x components of the complex components of E'* are denoted as E,f = E&&,f. If we now consider adding conductive losses to the medium, u # 0, this adds a transverse conductive current term, A -- a$, to Ampere's law, equation (Llb). The second-order differential equations become as shown in (1.11). In the case of sinusoidal excitation we obtain d2$ = y2Z, dz2
(1.19a)
d2f?I -= dz2
(1.19b)
Y
2f?I
where the propagation constant is (1.1912)
Y=J-
=u+jP
The phasor solutions in (1.17) for the lossless medium case become
E,(x, y, z) = Z+(x, y)e-aze-Jflz + E-(x, y)eaxeJflz 1 I?,(x, y, z) = - E+(x, y)e-'ze-'flz tt
- -1 P ( x , y)eazeJJz tt
( 1.20a)
(1.20b)
DERIVATION OF THE TRANSMISSION-LINE EQUATIONS FOR TWO-CONDUCTOR LINES
13
where the intrinsic impedance now becomes (1.204
Thus, in addition to a phase shift represented by e*'a', the waves suffer an attenuation represented by e*"', We find these properties of the TEM mode of propagation arising throughout our examination of MTLs in various guises. 1.3 DERIVATION OF THE TRANSMISSION-LINE EQUATIONS FOR TWO-CONDUCTOR LINES
The transmission-line equations are usually derived from a representation of the line as lumped circuit elements distributed along the line. While this gives the desired equations, a number of subtle aspects are obscured. In this section, we will derive the transmission-line equations for a general two-conductor line by three methods: 1. From the integral forms of Maxwell's equations. 2. From the diferential forms of Maxwell's equations. 3. From the usual distributed parameter, per-unit-length equivalent circuit. 1.3,1 Derivation from the Integral Form of Maxwell's Equations
Consider a two-conductor transmission line shown in Fig. 1.5(a). We assume that: 1. The conductors are parallel to each other and the z axis.
2. The conductors are perfect conductors. 3. The conductors have uniform cross sections along the line axis.
Because of the first and third properties this is said to be a uniform line. The medium surrounding the conductors may be lossy which is represented by a nonzero conductivity, u, and is homogeneous in u, s, and p. Maxwell's equations in integral form are CA.11 (1 2 1 a)
(1.21b)
14
INTRODUCIION
J
Y
Illustration of (a) the current and voltage and (b) the TEM fields for a two-conductor line FIGURE 1.5
Equation (1.21a) is referred to as Faraday's law, and equation (1.21b) is referred to as Ampere's law. Open surface s is enclosed by the closed contyr c and the directions are related by the right-hand rule CA.11. The quantity f is a current density in A/m a_"d contains conduction current, = ud', as well as any source as f = icurrent, A. We will assume the TEM field structure about the conductors in any cross-sectional plane as indicated in Fig. l.S(b). If we choose the contour in (1.21) to lie solely in the cross-sectional plane, cXrrand the surface enclosed to be a flat surface in the transverse plane, sxy, then (1.21) becomes
2'
(1.22a) =O
&@
DERIVATION OF THE TRANSMISSION-LINE EQUATIONS FOR TWO-CONDUCTOR LINES
+ c”
I
(b)
(8)
FIGURE 1.6
15
Definitions of (a) voltage and (b) current for a two-conductor line.
Observe that the right-hand side of Faraday’s law, (1.22a), is zero because there are, by the TEM assumption, no z-directed fields so that Hz= 0. Similarly, by the TEM assumption, (9. = 0, and there is no z-directed conduction or displacement current, only z-directed source currents, &. Thus Ampere’s law, (1.22b), simplifies as shown, Observe that equations (1.22) are identical tu those for static (dc) time variation.Therefore, we may uniquely define voltage between the two conductors, independent of path, so long as we take the path to lie in a transverse plane:
V(Z,t ) = -
s,’ 4.d7
(1 -23)
Figure 1.6(a) illustrates this point. We can choose either contour c1 or c2 for the definition of voltage. Since the conductors are perfect conductors, their surfaces are equipotential surfaces so that the contours can terminate at different points on them. Furthermore, by the TEM assumption, there is no component of the magnetic field penetrating the surface bounded by the contour enclosed by these two paths and the conductor surfaces which makes the voltage definition in (1.23) unique. Similarly, (1.22b) allows the unique definition of current as illustrated in Fig, 1.6(b). Choosing a closed contour in the transverse plane encircling one of the conductors gives the current on that conductor as (1‘24)
16
INTRODUCTION
x
L
FIGURE 1.7 Contours for the derivation of the first transmission-line equation: (a) longitudinal plane, (b) transverse plane.
This is unique because there is no z-directed electric field, 8' = 0, so that no conduction or displacement current penetrates the flat surface enclosed by this contour. The current so defined by (1.24) lies solely on the surface of the perfect conductor. If we enclose both conductors with the contour, it can be shown, because of (1.22b), that the net current is zero, i.e., the current at any cross section on the lower conductor is equal and opposite to the current on the upper conductor. (See Problem 1.3 at the end of the chapter.) We now turn to the derivation of the transmission-line equations in terms of the voltage and current defined above. Consider Fig. 1.7(a) where we have chosen an open surface, s, of uniform cross section in the z direction around which we integrate Faraday's law. The unit normal to this surface lies in the x-y (transverse) plane and is denoted by cf,. Integrating Faraday's law around
DERIVATION OF THE TRANSMISSION-LINE EQUATIONS FOR TWO-CONDUCTOR LINES
17
this contour gives
Observe that the second and fourth integrals on the left-hand side are zero since these are along the surfaces of the perfectly conducting conductors. Also note that the negative sign usually present on the right-hand side of Faraday's law is absent here. This is because of the direction chosen for the line integral, the choice of direction for the unit normal vector, d,, and the right-hand rule. Defining the voltages between the two conductors as in (1.23) gives (1.26a)
V(Z,t ) = -
fa' Jo
&x, y, Z , t ) d f
(1.26b)
Therefore, (1.25) becomes
d - V(Z,t ) + V(z + b z , t ) = p dt
Jl
@*d,ds
(1.27)
Rewriting this gives (1.28) Taking the limit as Az 3 0 gives (1.29) The right-hand side of (1.29) can be interpreted as an inductance of the loop formed between the two conductors. In order to do this, consider Fig. 1.7(b). The current, I(z, t), is again defined by (1.30)
18
INTRODUCTION
Therefore, the inductance for a Az section is (1.31)
A per-unit-length inductance, 1, can be defined at any cross section (since the line is uniform) as
1 - lim AS+O
L -
Az
(1.32)
This, combined with (1.29) gives the first transmlssion-line equation:
(1.33) We now turn our attention to the derivation of the second and remaining transmission-line equation. Recall the continuity equation which states that the net outpow of current from some closed surface equals the time rate of decrease of the charge enclosed by that surJace:
fi
kj*d3= - d p dv dt d' -- - -dt Qcnc
(1.34)
Enclose each conductor with a closed surface, J, of length Az just OR' the surface of the conductor as shown in Fig. 1.8(a). Integrating the continuity equation over this closed surface gives (1.35)
DERIVATION OF THE TRANSMISSION-LINE EQUATIONS FOR TWO-CONDUCTOR LINES
19
I
FIGURE 1.8 Contours for the derivation of the second transmission-line equation: (a) longitudinal plane, (b) transverse plane.
The portion of this closed surface over the ends is denoted by & whereas the portion of the surface over the sides is denoted by JoeThe terms in (1.35) become
ll,9
dg = I(z
+ Az, 1 ) - I @ , t )
( 1.36a)
(1.36b) The right-hand side of (1.35) can be defined in terms of a per-unit-length capacitance. The total charge enclosed by the surface is, according to Gauss' law, CA.11 (1.33
20
INTRODUCTION
The capacitance between the conductors for a Az section of the line is (1.38)
and the per-unit-length capacitance is c = lim AE-O
C -
(1.39)
AZ
Substituting (1.37) and observing Fig. 1.8(b) gives
(I .40)
Similarly, a conductance between the two conductors for a length of Az may be defined as
(L41) This leads, from (1.36b), to the definition of a per-unit-length conductance as g = lim
Az
$
AZ+O
d'
G -
(1.42)
$.b,d7
-1 8 * d f
Substituting (1.36a), (1.40) and (1.42) into (1.35), dividing both sides by Az, and taking the limit as Az -+ 0 gives the second and last transmission-line equation: (1.43)
Equations (1.33) and (1.43) are referred to as the transmission-line equations and represent a coupled set of first-order, partial differential equations in the line voltage, V(z,t), and line current, I(z, t). Solution of these equations will be one of our goals.
DERIVATION OF THE TRANSMISSION-LINE EQUATIONS FOR TWO-CONDUCTOR LINES
FIGURE 1.9
1.3.2
21
Illustration of the derivation of certain vector identities.
Derivation from the Differential Form of Maxwell’s Equations
We now obtain the transmission-line equations from the differential forms of Maxwell’s equations. We showed previously in (Moa) and (l.10b) that, for the TEM mode of propagation, the transverse field vectors satisfy (1.44a) d% = -a(d, az
x
8) - E
(1.44b)
Performing the line integral of both sides of (1.44a) between points a and a’ on the conductors along a path in the transverse plane and recalling the definition of voltage given in (1.26b) yields (1.45)
From Fig. 1.9 we see the following identities:
(a, x &*df= -.$*(a, df a, = a, x -
dl
x df)
(1.46)
(1.47)
Substituting (1.46) and (1.47) into (1.45) and recalling the definition of per-unit-length inductance given in (1.32) yields the first transmission-line equation given in (1.33). The second transmission-line equation is obtained from (1.3b). PerForming the line integral over both sides between points a and a’ on the two conductors
22
INTRODUCTION
vields
Using the identities in (1.46) and (1.47) in the definition of inductance in (1.32) and observing the definition of voltage given in (1.26b) yields
a a m , t) -i I(2, t ) = -aY(z, t ) - 8 -
cc a2
at
(1.49)
Rewriting gives (1.50)
We shall prove the following important identity relating the per-unit-length parameters, g, c, and 1 for a homogeneous surrounding medium as is assumed here: (1.5 1a) (1.5lb) Substituting these int (1.50) gives the second tra smission-line equation given in (1.43). 1.3.3 Derivation from the Perunit-length Equivalent Circuit
The previous two derivations of the transmission-line equations were rigorous and illustrated many important concepts and restrictions on the formulation. In this section we will show the usual derivation from a distributed-parameter, lumped circuit. The concept stems from the fact that lumped-circuit concepts are only valid for structures whose largest dimension is electrically small, i.e., much less than a wavelength, at the frequency of excitation. If a structural dimension is electrically large, we may break it into the union of electrically small substructures and can then represent each substructure with a lumped circuit model. In order to apply this to a transmission line, consider breaking it into small, Az length subsections as illustrated in Fig. 1.10. The per-unit-length inductance, I, derived previously represents the magnetic flux passing between the conductors due to the current on those conductors. We may lump this in each Az subsection by multiplying the per-unit-length parameter by Az. Since the line is assumed to be a uniform one, this can be done for all such subsections as shown in Fig. 1.10. Similarly, the per-unit-length capacitance, c, represents the displacement current flowing between the two conductors and can be similarly lumped in each subsection. The per-unit-length conductance, g,
23
DERIVATION OF THE TRANSMISSION-LINE EQUATIONS FOR TWO-CONDUCTOR LINES
IAz
rAz
== CAS -
+ Az, I )
-e---
f-’Azl
-sA-
V(z
z
+ Az
FIGURE 1.10 The per-unit-length model for use in deriving the transmission-line
equations.
represents the transverse conduction current flowing between the two conductors and can be lumped in a similar fashion. The previous derivations assumed that the two conductors are perfect conductors. Small conductor losses can be handled in this equivalent circuit in an approximate manner by including the per-unit-length resistance, r, (the total for both conductors) in series with the inductance element. The validity of this approximation will be discussed in a later section. From the per-unit-length equivalent circuit shown in Fig. 1.10, we obtain
V(z t Az, t ) - V(z,t ) = -rAz I(z, t ) - 1Az ”(”
(1.52a)
at
Similarly, we obtain
Dividing (1.52a) by Az and taking the limit as Az + 0 gives the first transmissionline equation: lim Ar*O
v(Z
+ A Z , t ) - v(Z,t ) am t ) Az
= 3 :
82
-rZ(z,t)-1-
at
t,
(1S3)
The second transmission-line equation can be derived from (1.52b) in a similar manner. However, before taking the limit as Az + 0, we should substitute the result for V(z + Az) from (1.52a) into (1.52b) giving I(z
+ Az, t ) - I(z, t ) = -gY(z, t) - c am t ) Az at
(1.54)
24
INTROUUCTION
Taking the limit of (1.54) as Az 4 0 gives the second transmission-lineequation: (1.55)
7.3.4
Pmpedm of the Perhielength Parameten
The per-unit-length parameters of inductance, 1, conductance, g, and capacitance, c, share important properties with each other for a homogeneous surrounding medium. These are: IC = pe (1.56a)
$1 = Occ
(1.56b)
In this section we will prove these important identities.
In order to prove (1.56a) we multiply the definitions of I and c given in (1.32) and (1.40), respectively:
(1.57)
From Fig. 1.7(b) and Fig. 1.8(b) we have the following identities: (1.58a) (1.58b) so that
(1.59)
We showed previously (see equation (1.14)) that (1.60a)
(1.60b)
DERIVATION OF THE TRANSMISSION-LINE EQUATIONS FOR TWO-CONDUCTOR LINES
Therefore [A. 1J
g * ( d , x d!) = -q(d, x,,?@*(d, x d!) = -&-dl
and, similarly,
g*(d,x dl) Substituting (1.61) into (1.59) yields
-
1 - #*dl tt
25
(1.61 a)
(1.61b)
(1.62) JC
Jo
The proof of the identity in (1.56b) follows an identical pattern using the expression for g given in (1.42). A simpler method of proving these identities is from the general, second-order relations for the fields of a general TEM mode given in (1.11): (1.63a) (1.63b) Performing the line integral between two points, a and u', on the conductors in a transverse plane on both sides of (1.63a) and recalling the definition of voltage given in (1.26b) yields (1.Ma) Similarly, performing the contour integral around the top conductor in a transverse plane on both sides of (1.63b) and recalling the definition of current from (1.30) yields (1.64b) If we differentiate the first transmission-line equation in (1.33) with respect to
z, differentiate the second transmission-line equation in (1.43) with respect to
26
INTRODUCnON
t and substitute we obtain a 2 V(2,t ) -r
a2
azqz, t)
-e a22
9
a V(2,t ) + IC azV(2, t ) '
7
t)
at2
q z , t) + -
91 - IC at
(1.65a) (1.65b)
at2
where the second equation was obtained by reversing the process. Comparing (1.65) to (1.64) we identify the two important identities given in (1.56). As mentioned previously, one must be able to determine the per-unit-length parameters for a given cross-sectional line configuration as well as be able to solve the transmission-line equations. All structural differences between classes of lines are contained in the per-unit-length parameters and nowhere else. The above identities show that we only need to obtain one of the three per-unitlength parameters, g, 1, or c. The transverse electric and magnetic fields satisfy Laplace's equation in any transverse plane (see equations (1.6)) so that determination of each of the per-unit-length parameters is simply a static field problem in the transverse plane. Numerous static-field-solution algorithms and computer codes can then be applied to this subproblem even though the eventual use of the parameters is in describing a problem whose voltages and currents vary with time! 1.4
CLASSIFICATION OF TRANSMISSION UNES
One of the primary tasks in obtaining the complete solutions for the voltage and current of a transmission line is the general solution of the transmission-line equations (Step 2). The type of line being considered significantly affects this solution. We are familiar with the difficulties in the solution of various ordinary differential equations encountered in the analysis of lumped circuits. Although the equations to be solved for lumped systems are ordinary differential equations (there is only one independent variable, time t ) and are somewhat simpler to solve than the transmission-line equations which are partial differential equations (since the voltage and current are functions of two independent variables, time t and position along the line, z), the type of circuit strongly affects the solution difficulty. For example, if any of the circuit elements are functions of time (a time-varying circuit), then the coefficients of the ordinary derivatives will be functions of the independent variable, t. These equations, although linear, are said to be nonconstant coeficient ordinary diflerential equations which are considerably more difficult to solve than constant coefficient ones cA.41. Suppose one or more of the circuit elements are nonlinear, Le., the element voltage has a nonlinear relation to its current. In this case, the circuit differentialequations become nonlinear ordinary differential equations which are equally difficult to solve cA.41. So the class of lumped circuit being considered drastically affects the difficulty of solution of the governing differential equations.
CLASSIFICATION OF TRANSMlSSlON LINES X
FIGURE 1.11
I
27
I
Illustration of a nonuniform line caused by variations in the conductor
cross section,
Solution of the transmission-line partial differential equations has similar parallels. We have been implicitly assuming that the per-unit-length parameters are independent of time, t, and position along the line, z. The per-unit-length parameters contain all the cross-sectional structural dimensions of the line. If the cross-sectional dimensions of the line vary along the line axis, then the per-unit-length parameters will be functions of the position variable, z. This makes the resulting transmission-line equations very difficult to solve. Such transmission-line structures are said to be nonuniform lines. This includes both the cross-sectional dimensions of the line conductors as well as the crosssectional dimensions of any inhomogeneous surrounding medium. If the cross-sectional dimensions of both the line conductors and the surrounding, perhaps inhomogeneous, medium are constant along the line axis, the line is said to be a uniform line whose resulting differential equations are simple to solve. An example of a nonuniform (in conductor cross section) line is shown in Fig. 1.11. Figure l.ll(a) shows the view along the line axis, while Fig. l.ll(b) shows the view in cross section. Because the conductor cross sections are different at z1 and z2, the per-unit-length parameters will be functions of one of the independent variables, in this case, position z. This type of structure occurs frequently on printed circuit boards (PCBs).A common way of handlihg
28
INlRODUCnON
I!
FIGURE 1.12 Illustration of a nonuniform line caused by variations in the surrounding
medium cross section.
this is to divide the line into three uniform sections, analyze each separately and cascade the results. Figure 1.12 shows a nonuniform line where the nonuniformity is introduced by the inhomogeneous medium. A wire is surrounded by dielectric insulation. Along the two end segments the medium is inhomogeneous since in one part of the region the fields exist in the dielectric insulation, cl, p,, and in the other they exist in free space, E,, 1.1,. In the miadle region, the dielectric insulation is also inhomogeneous consisting of regions containing e,, p,, E ~ p,, and E,, p,. However, because of this change in the properties of the surrounding medium from one section, zl, to the next, z2, the total line is a nonuniform one and the resulting per-unit-length parameters will be functions of z.The resulting transmission line equations for Figs. 1.11 and 1.12 are difficult to solve because of the nonuniformity of the line. Again, a common way of solving this type of problem is to partition the line into a cascade of uniform subsections. All of the previous derivations include losses in the medium through a per-unit-length conductance parameter, g. This loss does not invalidate the TEM field structure assumption. Most of the previous derivations assumed perfect conductors. In the derivation of the transmission-line equations from the distributed-parameter,lumped equivalent circuit shown in Fig. 1.10, we allowed
CLASSIFICATION OF TRANSMISSION LINES
FIGURE 1.13
29
Illustration of the effect of conductor losses in creating non-TEM field
structures.
the possibility of the line conductors being impefect conductors with small losses through the per-unit-length resistance parameter, r. Unlike losses in the surrounding medium, lossy conductors implicitly invalidate the TEM field structure assumption. Figure 1.13 shows why this is the case. The line current flowing through the imperfect line conductor generates a nonzero electric field along the conductor surface, &, t ) = rf(z,t), which is directed in the z direction violating the basic assumption of the TEM field structure in the surrounding medium. The total electric field is the sum of the transverse component and this z-directed component. However, if the conductor losses are small, this resulting field structure is almost TEM. This is referred to as the quasi-TEM assumption and, although the transmission-line equations are no longer valid, they are nevertheless assumed to represent the situation for small losses through the inclusion of the per-unit-length resistance parameter, r. An inhomogeneous surrounding medium, although nonuniform, also invalidates the basic assumption of a TEM field structure. The reason this is true is that a TEM field structure must have one and only one velocity of propagation of the waves in the medium. However, this cannot be the case for an inhomogeneous medium. If one portion of the inhomogeneous medium is characterized by 81, fi0 and the other is characterized by ea, po, the velocities of TEM waves in these regions will be v , = I/* and v2 = 1/= which will be different. Nevertheless, the transmission-line equations are usually solved in spite of this observation and assumed to represent the situation so long as these velocities (and corresponding e, and ez) are not substantially different. This is again referred to as the quusi-TEM assumption. A common way of characterizing this situation of an inhomogeneous medium is to obtain an eflectiue dielectric constant, e’ CA.31. This effective dielectric constant is defined such that if the line conductors are immersed in a homogeneous dielectric having this d, the velocities of ,propagation and all other attributes of the solutions for the original inhomogeneous medium problem and this one will be the same. In summary, the TEM jeid structure and mode of propagation charactetization
30
INTRODUCTION
of a transmission line is only valid for lines consisting of pedect conductors and surrounded by a homogeneous medium. Note that this medium may be lossy and not violate the TEM assumption so long as it is homogeneous (in 0, e, and p). Violations of these assumptions (perfect conductors and a homogeneous medium surrounding the conductors) are considered under the quasi-TEM assumption so long as they are not extreme [17,18]. 1.5 RESTRICTIONS ON THE APPUCABILITY EQUATION FORMULATION
OF THE TRANSMISSION-LINE
There are a number of additional, implicit assumptionsin the TEM, transmissionline-equation characterization. It was pointed out in the derivation from the distributed-parameter,lumped circuit of Fig. 1.10 that distributing the lumped elements along the line and allowing the section length to go to zero, limA,+oAz, means that line lengths that are electrically long, in., much greater than a wavelength A, are properly handled with this lumped-circuit characterization. However, nothing was said about the ability of this lumped-circuit model to adequately characterize structures whose cross-sectional dimensions, e.g., conductor separations, are electrically large. Structures whose cross-sectional dimensions are electrically large at the frequency of excitation will have, in addition to the TEM field structure and mode of propagation, other higherorder TE and TM field structures and modes of propagation simultaneously with the TEM mode [A.1,17-19], Therefore, the solution of the transmissionline equations does not give the complete solution in the range of frequencies where these non-TEM modes coexist on the line. A comparison of the predictions of the TEM transmission-line-equationresults with the results of a numerical code (which does not presuppose existence of only the TEM mode) for a two-wire line showed differences beginning with frequencies where the wire separations were as small as 12/40 CH.21. Analytical solution of Maxwell’s equations in order to consider the total effect of all modes is usually a formidable task. There are certain structures where an analytical solution is feasible and the next two sections consider these. 1.5.1
Higherorder Modes
In the following two subsections we analytically solve Maxwell’s equations for two closed structures to obtain the complete eolution and demonstrate that the TEM formulation is complete up to some frequency where the conductor separations are some significant fraction of a wavelength above which higherorder modes begin to propagate. 1.5.7.7 The lnffnik ParalleWak Transmission Line Consider ‘the Infinite, parallel-plate transmission line shown in Fig. 1.14. The two perfectly conducting plates lie in the y-z plane and are located at x = 0 and x = a. We will obtain
RESTRICTIONS ON APPLICABILITY OF TRANSMISSION-LINE EQUATION FORMULATION
31
Y
FIGURE 1.14 The infinite, parallel-plate waveguide for demonstrating the effect of cross-sectional dimensions on higher-order modes.
the complete solutions for the fields in the space between the two plates which is assumed to be homogeneous and characterized by 8 and p. Maxwell's equations for sinusoidal excitation become (1.66a)
(1.66b) Expanding these and noting that the plates are infinite in extent in the y direction so that a/ay = 0 gives CA.11
(1.67)
32
INTRODUCnON
In addition, we have the wave equations [A.lJ:
(1.68)
Expanding these and recalling that the plates are infinite in the y dimension so that a/ay = 0 gives [A. 1J
1
(1.69)
Let us now look €orwaves propagating in the +z direction. To do so we assume that separation of variables is valid where we separate the dependence on x, y and on z as Z(x, y, z ) = 3(x,
y)e-yg
(1.70)
where y is the propagation constant (to be determined). Substituting this into the above equation yields
Y E ; = -jopH: -YE:
aE: -ax
-J(wH;
aEf I,- j w H : ax yH;=jweEL
-yHL
(1.71)
an: -JoeE; -ax aIi; ax
e
joeE:
and
(1.72)
RESTRICTIONS ON APPLICABILITY OF TRANSMISSION-LINE EQUATION FORMULATION
33
The equations in (1.71) can be manipulated to yield CA.11
(1.73)
h2 = y2
+ o'pe
Observe that E'(x, y ) and W(X,y ) are functions of only x since there can be no variation in the y direction due to the infinite extent of the plates in this direction and also the z variation has been assumed, Thus the partial derivatives in (1,71), (1.72), and (1.73) can be replaced by ordinary derivatives. We now investigate the various modes of propagation. The Transverse Electric (TEI Mode (E, = 0) The transverse electric (TE)mode of propagation assumes that the electric field is confined to the transverse, x-y plane so that E, = 0. Therefore, from (1.73) we see that E: = Hi = 0. The wave equations in (1.72) reduce to
d2Et + h'Eb dx2
=0
(1.74)
whose general solution is
E ; = CIsin(hx)
+ C2cos(hx)
(1.75)
The boundary conditions are that the electric field tangent to the surfaces of the plates are zero:
Ey = O I x = o , x = L
(1.76)
which, when applied to (1.75), yields C2= 0 and ha = mn for rn = 0, 42, 3,. Thus, the solution becomes
..
(1.77)
34
INTRODUCTION
From (1.71) we obtain
I
(1.78)
and (1.79)
Since
h - - mn
(1.80)
a
the propagation constant becomes (1.81)
For the lowest-order mode, m = 0, all field components vanish. The next higher-order mode is the TE1 mode for m = 1 whose nonzero components are Ep and H,. The Transverse Magnetic (TM) Mode (HZ = 0) The transverse magnetic (TM) mode has the magnetic field confinedto the transverse, x-y plane so that H, = 0. Carrying through a development similar to the above for this mode gives the nonzero field vectors as
0 8
E, =j h D2sinrf)e-yz WB
.
i
(1.82)
for n = 0, 1, 2,. . The propagation constant is again given by (1.81) with m replaced by n. In contrast to the TE modes, the lowest-order TM mode is the TMo mode for n = 0. For this case the propagation constant reduces to the familiar Y =jw&
=jP
(1.83)
RESTRICTIONS ON APPLICABILITY OF TRANSMISSION-LINE EQUATION FORMULATION
35
and the field vectors in (1.82) reduce to
Hy= D2e-j@
I (1.84)
However, this is the TEM mode! Therefore, the lowest-order TM mode, TM,, is equivalent to the TEM mode and the TE, mode is nonexistent. We must then ascertain when the next higher-order modes begin to propagate thus adding to the total picture. The propagation constant in (1.81) must be imaginary, or at least have a nonzero imaginary part. Clearly for rn = n -- 0, we have the propagation constant of a plane wave: y =j m f i =]a. For higher-order modes to propagate, we require from (1.81) that oz/e 2 h2 giving 1 nn u2--
(1.85)
& a
The cutoff frequency for the lowest-order TEM mode, TM,, is clearly dc. The cutoff frequency of the next higher-order modes, TEI and TM1, are from (1.85) STBI.TMI =
1 n 2 n f ia
(1.86)
V
=-
2a
In terms of wavelength, A =
V
7’
we find that the TEM ntode will be the only
possible mode so long as the plate separation, a, is less than one-half of one wavelength, i,e., A (1.87) a s 2
This illustrates that so long as the cross-sectional dimensions ofthe line are electrically small, only the TEM mode can propagate! This is illustrated in Fig. 1.15.
36
INTRODUCTION
I
a = bl2
0
9
FIGURE 1.15 Illustration of the dependence of higher-order modes on cross-sectional electrical dimensions.
FIGURE 1.16 The coaxial cable for illustratingthe dependence of higher-order modes on cross-sectional electrical dimensions.
I.J.I.2 The Coaxial Transmission line Another closed system transmission line which is capable of supporting the TEM mode is the coaxial transmission line shown in Fig. 1.16. The general solution to Maxwell’s equations for the fields and modes in the space between the inner wire and the outer shield was solved in [l]. Clearly this structure can support the TEM mode with a cutoff frequency of dc. The higher-order TE and TM modes have the following cutoff properties. The lowest order TE mode is cutoff for frequencies such that the average circumference between the conductors is approximately less than one-half of one wavelength, Le.,
2x(a
+ 6 ) s It
(1.88)
Similarly, the lowest order TM mode is cutoff for frequencies such that the diflerence between the two conductor radii is approximately less than one-half of one wavelength, i.e.,
(6
- a) s A
(189)
These results again support the notion that the TEM mode will be the only mode of propagation in closed systems so long as the conductor separation is electrically small.
RESTRICTIONS ON APPLICABILITY OF TRANSMISSION-LINE EQUATION FORMULATION
37
1.5.1.3 fivo-wlre Llnes The two previous transmission-line structures are closed systems. For open systems such as the two-wire line, the issue of higher-order modes is not so clear-cut. Numerical analysis of a two-wire line given in CH.21 showed that the predictions of the transmission-line formulation for the two-wire tine begin to deviate from the complete solution when the cross-sectional dimensions such as wire separation are no longer electrically small. This supports our intuition. The problem was investigated in more detail in C19J where these notions are confirmed. Also certain “leaky modes” are capable of propagating with no clearly defined cutoff frequency. Thus, the TEM mode formulation and the resulting transmission-line equation representation for two-wire lines will be reasonably adequate so long as the wire separations are electrically small. Ordinarily, this is satisfied for practical transmission-line structures.
1.5.2
Transmission-Line Currents vs. Antenna Currents
There is one remaining restriction on the completeness of the TEM mode, transmission-line representation that needs to be discussed. It can be shown that under the TEM, transmission-line formulation for a two-conductor uniform line, the currents so determined on the two conductors at any cross section must be equal in magnitude and oppositely directed. Thus, the total current at any cross section is zero, This is the origin of the reference to the term that one of the conductors serves as a “return” for the current on the other conductor. Unless there is adherence to the following concepts, this will not be the case. Consider the pair of parallel wires shown in Fig. 1.17(a) supporting on their surfaces, at the same cross section, currents II and 12. In general, we may decompose, or represent, these as a linear combination of two other currents. The so-called differential-mode currents, f D , are equal in magnitude at a cross section and are oppositely directed as shown in Fig. 1.17(b). These correspond to the TEM mode, transmission-line currents that will be predicted by the transmission-line model. The other currents are the so-called common-mode currents, IC, which are equal in magnitude at a cross section but are directed in the same direction as shown in Fig. l.l7(c). These are sometimes referred to as “antenna-mode” currents [ZO, 21). This decomposition can be obtained by writing, from Fig. 1.17,
In matrix form, these can be written as (1.91)
38
INTRODUCTION
-------,-,-------
fc
-
LC
-------
I -
(4 FIGURE 1.17 Illustration of the decomposition of total currents into differential-mode (transmission-line-mode) and common-mode (antenna-mode)components. Equation (1.91)represents a nonsingular transformation between the two sets of currents since the transformation matrix is nonsingular. Therefore, its inverse can be taken and the transformation reversed to yield
(1.92) This gives
(1.93)
Ordinarily, the common-mode currents are much smaller in magnitude than the differential-mode currents so they do not substantially affect the results of an analysis of currents and voltages of a transmission line. However, the common-mode currents are significant, even though they are smaller in magnitude than the differential-mode currents, in the case of radiated emissions from this two-wire line. This is because the radiated electric fields from the differential-mode currents tend to subtract but those from the common-mode currents tend to add. Thus, a “small” common-mode current can give the same order of magnitude of radiated emission as a much larger differential-mode current. This was confirmed for cables and PCBs in CA.3, 22, 231. The significant point here is that if one bases a prediction of the radiuted emissions from a two-conductor line on the currents obtained from a transmission-line-
RESTRICTIONS ON APPLlCABlLlTY OF TRANSMISSION-LINE EQUATION FORMULATION
39
FIGURE 1,78 Decomposition of total currents of a three-conductor line into differentialmode and common-mode components.
equation analysis, the predicted emissions will generally lie far below those of the (unpredicted) emissions due to the common-mode currents. The commonmode currents can be ignored in a near-field, transmission-line analysis such as in determining crosstalk. This decomposition can be extended to lines consisting of more than two conductors. Consider the three-conductor line shown in Fig. 1.18. There are three currents to be decomposed, fI,f,, and I,. So we are free to redefine them in terms of three other currents such as is shown in Fig. 1.18 as fD1, fD2,and IC. We have chosen two of the currents, ID, and ID2 to be defined in the same fashion as the TEM mode currents in that they return through the lower conductor. The remainder current, I,., is the same in magnitude and direction of all three conductors. The transformation becomes
(1.94)
Inverting this transformation gives
[]; =;[
-; ; 2 -1
-1
I,
-1][12]
13
(1.95)
from which the decomposition currents can be obtained. There are a number of ways that these non-TEM mode currents can be created on a transmission line. Figure 1.19 illustrates one of these. It is important to remember that the TEM mode, transmission-line-equationformulation only characterizesthe line and assumes that the two (or more) conductors of the line continue indefinitely along the z axis, The field analysis does not inherently consider the field effects of the eventual terminations for a finitelength line. This problem was investigated in [24]. It was found that asymmetries
40
INTRODUCTION
FIGURE 1.19
Illustration of an asymmetry that creates common-mode currents.
as well as the presence of nearby metallic obstacles create these “nonideal” currents. For example, consider the two-wire line shown in Fig. 1.19 which is driven by a voltage source at the left end and terminated in a short circuit at the right end. This was analyzed using a numerical solution of Maxwell’s equations commonly referred to as a method of moments (MOM). This analysis gives the complete solution for the currents without presupposing the existence of only the TEM mode. It was found that if the voltage source was situated and modeled as being centered in the left segment on the centerline, then I , = -II; in other words, the currents on the wires are only differential-mode currents. However, if the voltage source was placed asymmetrically to the centerline such as shown and the resulting currents decomposed as in (1.93), common-mode currents appeared. This asymmetrical placement of the source, which is not explicitly considered in the transmission-line-equationformulation, was apparently the source. The important point here is that the TEM mode, transmission-line-equation formulation that we will consider in this text only predicts the differentialmode currents. If the line cross section is electrically small and one is interested only in predicting the currents and voltages on the line for the purposes of predicting signal distortion and crosstalk (the primary goal of this text), this prediction will be reasonably accurate. On the other hand, if one is interested in predicting the radiated electric field from this line, then the predictions of that field using only the currents predicted by the transmission-line-equation formulation will most likely be inadequate since the contributions due to the common-mode currents are typically the dominant contributors to radiated emissions [22,23]. REFERENCES
[l] [2] [3]
S. Ramo, J.R. Whinnery, and T.VanDuzer, Fields and Waves in Communication Electronics, 2d ed., John Wiley & Sons, NY, 1984. R.B. Adler, L.J. Chu, and R.M, Fano, Electromagnetic Energy Z’kansmission and Radiation, John Wiley, NY, 1963. S. Frankel. Multiconductor Z’kansmission Line Analysis, Artech House, Dedham, Massachusetts, 1977.
REFERENCES
[4]
[5) [6] [7]
[SI [9] [lo] [l 11 [123 [131
[14] [IS] [161
[I71 [IS]
[19]
[20]
E211 [22]
[23]
[24]
41
H. Uchida, Fundamentals of Coupled Lines and Multiwire Antennas, Sasaki Publishing Co., Sendai, Japan, 1967. S. Hayashi, Surges on Trunsmission Systems, Denki-Shoin, Kyoto, Japan, 1955. W.C. Johnson, Transmission Lines and Networks, McGraw-Hill, NY, 1950. L.V. Bewley, Traveling Waves on l’kansmission Systems, 2d ed., John Wiley, NY, 1951. P.I. Kuznetsov and R.L. Stratonovich, The Propagation of Electromagnetic Waves in Multiconductor Transmission Lines, Macmillan, NY, 1964. P.C. Magnuson, Transmission Lines and Wave Propagation, Allyn and Bacon, Newton, MA, 1970. R.E. Collin, Field Theory ofGuided Waves, 2d ed., IEEE Press, NY, 1991. J. Zaborsky and J. W. Rittenhouse, Electric Power Transmission, Ronald Press, NY, 1954. R.E. Matick, Transmission Lines for Digital arid Communication Networks, McGraw-Hill, NY, 1969. L. Young (ed.), Parallel Coupled Lines and Directional Couplers, Artech House, Dedham, Massachusetts, 1972. T. Itoh (ed.), Planar Transmission Line Structures, IEEE Press, NY, 1987. W.T. Weeks, “Multiconductor Transmission Line Theory in the TEM Approximation,” IBM J. Research and Development, pp. 604-611, November 1972. K.D. Marx, “Propagation Modes, Equivalent Circuits and Characteristic Terminations for Multiconductor Transmission Lines with Inhomogeneous Dielectrics,” I E E E Trans. on Microwave Theory and Techniques, MTT-21, 450-457 (1973). A.F. dos Santos and J.P. Figanier, “The Method of Series Expansion in the Frequency Domain Applied to Multiconductor Transmission Lines,” IEEE Trans. on Microwave Theory and Techniques, MTT-23,753-756 (1975). I.V. Lindell, “On the Quasi-TEM Modes in Inhomogeneous Multiconductor Transmission Lines,” IEEE Trans. on Microwave Theory and Techniques, M”29,812-817 (1981). Y. Leviatan and A.T. Adams, “The Response of a Two-Wire Transmission Line to Incident Field and Voltage Excitation, Including the Eflects of Higher Order Modes,” IEEE l’kans. on Antennas and Propagation, AP-30,998-1003 (1982). K.S.H.Lee, “Two Parallel Terminated Conductors in External Fields,” IEEE Trans. on Electromagnetic Compatibllity, EMC-20, 288-295 (1978). S.Frankel, “Forcing Functions for Externally Excited Transmission Lines,” IEEE Trans. on Electromagnetic Compatibility, EMC-22,p. 210 (1980). C.R. Paul and D.R. Bush, “Radiated Emissions from Common-Mode Currents,” Proceedings 1987 IEEE International Symposium on Electromagnetic Compatibility, Atlanta, GA, September 1987. C.R. Paul, “A Comparison of the Contribution of Common-Modeand DifferentialMode Currents in Radiated Emissions,” IEEE Trans. on Electromagnetic Compatibility, EMC-31, pp. 189-193 (1989). K.B. Hardin, “Decomposition of Radiating Structures to Directly Predict Asymmetric-Mode Radiation,” PhD Dissertation, University of Kentucky, 1991.
42
INTRODUCTION
PROBLEMS
1.1 Two, perfectly-conducting, circular plates are separated a distance d as shown in Fig. P1.1.The plates have very large radii with respect to d9
t‘ FIGURE P1.1
(ideally infinite), so that, in zylindrical coordinates,_we may assume a “TEM-mode” field structure, 4 ( p , t ) = 4 ( p , t)d,, and H(p,t ) = H4(p9t)d+. Define voltage and current as V(p, 0 = - U P 9 t ) d I ( P , t ) = 2ZPfl&(P,
0
Show, from Maxwell’s equations in cylindrical coordinates, that V and I satisfy the transmission-line equations:
where I and c arc static parameters defined by:
PROBLEMS
43
Would it be appropriate to classify this as a nonunfform line? Could the mode of propagation to which these equations apply be classified as a “TEM mode”? 1.2 The infinite, biconical transmission line consists of two, perfectly conducting cones of half angle 8, as shown in Fig. P1.2.Solve Maxwell’s equations
FIGURE P1.2
in spherical coordinates for this structure assuming that &r, 8) = d$(r* 8)h, and S ( r , 0) = X4(r, 8)d4. Show that the following definitions of voltage and current are unique:
V(r, t ) = J:-eh
$r dB
where V and I satisfy the following “transmission-line equations”:
44
INTRODUCTION
Show that
e=
VE
In( cot
$)
Would this be classified as a ungorm or nonuniform line? Would it be appropriate to classify the propagation mode as TEM?
(d)
FIGURE Pl.5
PROBLEMS
45
1.3 Show that, assuming a TEM field structure, the currents on the two conductors in Fig. 1.5 are equal in magnitude and oppositely directed at any cross section. 1.4 Show that, assuming a TEM field structure, the charge per unit length on one conductor in Fig. 1.5 is equal in magnitude and opposite in sign to
the charge per unit length on the other conductor at any cross section. 1.5 Derive the transmission-line equations from each of the circuits in Fig. P1.S in the limit as Az 3 0. Observe that the total inductance (capacitance)
in each structure is lAz (cAz). This shows that the structure of the per-unit-length equivalent circuit is not important in obtaining the transmission line equations from it so long as the total per-unit-length inductance and capacitance is contained in the structure and we let Az.+ 0.
CHAPTER TWO
The Multiconductor TransmissionLine Equations
The previous chapter discussed the general properties of all transmission-lineequation characterizations. The TEM field structure and associated mode of propagation is the fundamental, underlying assumption in the representation of a transmission line structure with the transmission-line equations. In this chapter we will extend those notions to multiconductor transmission lines or MTLs consisting of (n 1) conductors. In general, we will restrict the class of lines to those that are uniform lines consisting of (n + 1) conductors of uniform cross section that are parallel to each other. However, the conductors as well as the surrounding medium may be lossless or lossy. Lossless conductors are perfect conductors, while lossless media are media with zero conductivity, u = 0. The surrounding medium may be homogeneous or inhomogeneous.The development and derivation of the MTL equations parallel the developments for twoconductor lines considered in the previous chapter. In fact, the developed MTL equations have, using matrix notation, aform identical to those equations. There are some new concepts concerning the important per-unit-length parameters which contain the cross-sectional dimensions of the particular line.
+
2.1
DERIVATION FROM THE INTEGRAL FORM OF MAXWELL’S EQUATIONS
+
Figure 2.1 shows the general (n 1)-conductorline to be considered. It consists of n conductors and a reference conductor (denoted as the zeroth conductor) to which the n line voltages will be referenced. This choice of the reference conductor is not unique. Recall Faraday’s law in integral form:
46
DERIVATION FROM THE INTEGRAL FORM OF MAXWELL'S EQUATIONS
47
I
Reference conductor
r
I
Z4AZ
Y
FIGURE 2.1 Definition of the contour for derivation of the first MTL equation.
Applying this to the contour cl which encloses surface sI shown between the reference conductor and the i-th conductor and encircles it in the clockwise direction gives
where denotes the transverse electric field (in the x-y cross-sectional plane) and 4 denotes the longitudinal or z-directed electric field (along the surfaces of the conductors). Observe, once again, that because of the choice of the direction of the contour, the direction of d,, and the right-hand rule, the minus sign on the right-hand side of Faraday's law is absent in (2.2). Once again, because of
48
THE MULTICONDUCTOR TRANSMISSION-LINE EQUATIONS
the assumption of a TEM field structure, we may uniquely define voltage between the i-th conductor and the reference conductor (positive on the i-th conductor) as
,@I
t) = -
[
0'
(2.3a)
&x, y, Z, t).d7
J O
t;(z
+ Az, t) = -
Jbb'
&x, y, z
+ Az, t ) df
(2.3b)
The integrals along the surfaces of the conductors are zero if the conductors are considered to be perfect conductors. It was pointed out that the TEM mode cannot exist if the conductors are not perfect conductors. This is because a component of electric field will be directed in the z direction due to the voltage drop along the conductors. However, small losses can be accommodated as an approximation under the quasi-?'EM mode assumption. To allow for imperfect conductors, we define the per-unit-length conductor resistance, rCl/m. Thus
-[
b'
-[
b'
4*df=
&dz
=:
-rtAzll(z,t)
(2.4a)
where, along the toe of i-th conductor, 4 = drdrand df = dzhz, and along the bottom conductor, 4 = -8, h, and d f = -dzd,. The current is uniquely defined, because of the assumption of a TEM field structure, as
and contour 6, is a contour just off the surface of and encircling the i-th conductor in the transverse plane as shown in Fig. 2.2. Because of this definition of current and the TEM field structure assumption it can be shown, as was the case for two conductor lines, that the sum ofthe currents on all (n + 1) conductors in the z direction at any cross section is zero. This is the basis for saying that the currents of the n conductors return through the reference conductor. Substituting (2.3) to (2.5) into (2.2) yields
- K(z, t ) + tiAzlt(z, t) + &(z + dz,t ) + r0 Az
t) &-I
I,**
d = p ;il
d, ds
(2.6)
DERIVATION FROM THE INTEGRAL FORM OF MAXWELL’S EQUATIONS
49
x I
c,
4I
____)
Y
Illustration of the definitions of magnetic flux through a circuit for derivation of the per-unit-length inductances.
FIGURE 2.2
Dividing both sides by Az, and rearranging gives
&(z
+ Az, t ) - &(z, t ) = -roll Az
- ro12
- e . . -
+ rJI, - - - r
(to
*
0 4
(2.7)
1 d
Az dt Before taking the limit as Az 3 0, let us make some observations similar to the case of two-conductor lines. Clearly, the total magnetic flux penetrating the surface sIin Fig. 2.1 will be a linear combination of the fluxes due to the currents on the conductors. Consider a cross-sectional view of the line looking in the direction ojincreasing z shown in Fig. 2.2. The currents on the n conductors are implicitly defined in the positive z direction according to (2.5) since contour 6, is defined to be clockwise looking in the direction of increasing z. Therefore the magnetic fluxes due to the currents on the n conductors will also be in the clockwise direction looking in the direction of increasing z. The total magnetic flux, $,, penetrating the surface si between the reference conductor and the ith conductor is therefore deJined to be in this clockwise direction when looking in the direction ofincreasing z as shown in Fig. 2.2. Therefore, this total magnetic
50
THE MULTICONDUCTOR TRANSMISSION-LINE EQUATIONS
flux penetrating surface si can be written as
Taking the limit of (2.7) as Az -+ 0 and substituting (2.8) yields
This first MTL equation can be written in a compact form using matrix notation as
a V(Z,t ) = -RI(z, 82
t)
a I(z, t ) -Lat
(2.10)
where the voltage and current vectors are defined as
(2.11a)
(2.1lb)
The per-unit-length inductance matrix is defined from (2.8) as Y = LI
(2.12a)
where Y is an n x 1 vector containing the total magnetic flux per unit length, I(lr, penetrating the i-th circuit which is defined between the Gth conductor and
DERIVATION FROM THE INTEGRAL FORM OF MAXWELL'S EQUATIONS
51
the reference conductor:
(2.12b)
and the per-unit-length inductance matrix, L, contains the individual per-unitlength self-inductances, Ill, of the circuits and the per-unit-length mutual inductances between the circuits, l , , as
(2.12c)
Similarly, from (2.9) we define the per-unit-length resistunc- matrix as
(2.13)
Observe that this first transmission-line equation given in (2.10) is identical in form to the scalar first transmission-line equation for a two-conductor line. Consider placing a closed surface d around the i-th conductor as shown in Fig. 2.3. The portion of the surface over the end caps is denoted as J,, while the portion over the sides is denoted as 6 . Recall the continuity equation or equation of conservation of charge: (2.14)
Over the end caps we have
/l,9
*
d$ = l,(z
+ Az,t ) - I&,
t)
(2.15)
52
THE MULTICONDUCTOR TRANSMISSION-LINE EQUATIONS
Reference conductor
i Z
I
i
+ AS I
t
FIGURE 23 Definition of the surface for derivation of the second MTL equation.
Over the sides of the surface, there are two currents: conduction current, -- 04,and displacement current,&, = &(a&/&), where the surrounding homogeneous medium is characterized by conductivity, Q, and permittivity, &, These notions can be extended to an inhomogeneous medium surrounding the conductors in a similar but approximate manner. This is an approximation since an inhomogenous medium, uniform along the line or not, invalidates the TEM field structure assumption which requires that all waves propagate with the same velocity, that being the phase velocity of a plane wave in that medium. A portion of the left-hand side of (2.14) contains the transverse conduction current flowing between the conductors: (2.16)
This can again be considered by defining per-unit-length conductances, glj S/m, between each pair of conductors as the ratio of conduction current flowing
DERIVATION FROM THE INTEGRAL FORM OF MAXWELL'S EQUATIONS
53
>a
\
-
FIGURE 2.4 Illustration of the definitions used in computing the per-unit-length capacitances.
between the two conductors in the transverse plane to the voltage between the two conductors. (See Fig. 2.4.) Therefore,
c Jim AZ+O
_f_
Az
I I j o 4 . d i= gil(q - V,) +
+ Bit& + . + gin(y- V,) = -grl W , t ) - gI2 W, t ) - . . + b - Ba w , 4 - - Bln K(Z, t )
(2.17)
a
*
*
1
Similarly, the charge enclosed by the surface (residing on the conductor surface) is, by Gauss' law, Qe"0
,J
=e
(2.18)
4*di
The charge per unit of line length can be defined in terms of the per-unit-length capacitances, cU, between each pair of conductors as
e lim -!-. A Z + O Az
[lo4 - d
= cl,( V;
- 4) + - - + cI1V; + + cia( V; - V,) a
0
= -c,1V,(z,t)-***+
c c&y(z,t)-***n
&-1
Cln
(2.19)
v,@* 0
54
THE MULTICONDUCTOR TRANSMISSION-LINE EQUATIONS
FIGURE 2.5 Two-dimensional illustration of the per-unit-length conductances and capacitances as an aid in the determination of the entries in G and C.
These concepts are illustrated in cross section in Fig. 2.5. Substituting (2.15), (2.16), and (2.18) into (2.14), and dividing both sides by Az gives
Taking the limit as Az + 0 and substituting (2.17) and (2.19) yields
Equations (2.21) can be placed in compact form with matrix notation giving
a I(z, t ) = -GV(Z, t ) - C a V(Z, t ) 82
8t
(2.22)
where V and I are defined by (2.11). The per-unit-length conductance matrix, G, represents the conduction currentPowing between the conductors in the transverse
DERlVATtON FROM THE INTEGRAL FORM OF MAXWELL'S EQUATIONS
55
plane and is defined from (2.21) as
I
n
(2.23)
The per-unit-length capacitance matrix, C, represents the displacement current jowing between the conductors in the transverse plane and is defined from (2.21) as
(2.24)
Again observe that (2.22) is the matrix counterpart to the scalar second transmission-line equation for two-conductor lines. If we denote the total charge on the i-th conductor per unit of line length as qi, then the fundamental definition of C which is the dual to (2.12) is
Q=CV
(2.25a)
where
(2.25b)
and V is given by (2,lla). Similarly, the fundamental definition of G is I, = GV, where I, is the transverse conduction current between the conductors. The above per-unit-length parameter matrices once again contain all the cross-sectional dimension information that distinguishes one MTL structure from another. Although these were shown a's not being symmetric, it is logical
56
THE MULTICONDUCTOR TRANSMISSION-LINE EQUATIONS
FIGURE 2.6 The per-unit-length MTL model for derivation of the
MTL equations.
to expect that they are. This will be proven for isotropic surrounding media-the medium may be inhomogeneous.
2.2
DERIVATION FROM THE PER-UNIT-LENGTH EQUIVALENT CIRCUIT
As a final and alternative method we derive the MTL equations from the per-unit-length equivalent circuit shown in Fig. 2.6. Writing Kirchhoffs voltage law around the i-th circuit consisting of the i-th conductor and the reference conductor yields
Dividing both sides by Az and taking the limit as Az -+ 0 once again yields the
DERIVATION FROM THE PER-UNIT-LENGTH EQUIVALENT CIRCUIT
57
first transmission-line equation given in (2.9) with the collection for all I given in matrix form in (2.10). Similarly, the second MTL equation can be obtained by applying KirchhofT's current law to the i-th conductor in the per-unit-length equivalent circuit in Fig. 2.6 to yield
Dividing both sides by Az, taking the limit Az + 0, and collecting terms once again yields the second transmission-line equation given in (2.21) with the collection for all i given in matrix form in (2.22). Strictly speaking, the voltages in (2.26b) are at z + Az so that (2.26a) should be substituted before taking the limit. However, as was shown for two-conductor lines in the previous chapter, this yields the same result as when we take the limit Az -+ 0 in (2.26b) directly. 2-3 SUMMARY OF THE MTL EQUATIONS
In summary, the MTL equations are given by the collection
a V(Z,t ) = -RI(z, -
t)
82
a
- I(z, t ) = -GV(z, az
d -L I(z, t ) at
t)
(2.2 7a)
a V(z, t ) -c at
(2.27b)
The structures of the per-unit-length resistance matrix, R, in (2.13), inductance matrix, L,in (2,12), conductance matrix, G,in (2.23), and capacitance matrix, C, in (2.24) are very important as are the definitions of the per-unit-length entries in those matrices. The precise definitions of these elements are rather intuitive and lead to many ways of computing them for a particular MTL type. These computational methods will be considered in detail in Chapter 3. The important properties of the per-unit-length parameter matrices will be obtained in the next section. Again, these bear striking parallels to their scalar counterparts for the two-conductor line considered in the previous chapter. The MTL equations in (2.27) are a set of 2n, coupled,first-order, partial diyerentiul equations. They may be put in a more compact form as
d[v(i,t)]=-[ ][ ]-[ a2
I(Z, t )
0 R G 0
I-[]
a
V(z, t )
0 L
I(z, t )
C 0 at
V(z, t ) I(z, t )
(2.28)
58
THE MULTICONDUCTOR TRANSMISSION-LINE EQUATIONS
We will find this first-order form to be especially helpful when we set out to solve them in later chapters. If the conductors are perfect conductors, R = 0, whereas if the surrounding medium is lossless (a = 0), G = 0. The line is said to be lossless if both the conductors and the medium are lossless in which case the MTL equations simplify to (2.29) The first-order, coupled forms in (2.27) can be placed in the form of second-order, uncoupled equations by differentiating (2.27a) with respect to z and differentiating (2.27b) with respect to t to yield
a2 V(Z, t ) = -R a I(z, t ) - L -I(z, t ) azz az at a2
82
a2
---I(& az at
a
t ) = -c- V ( z , t )- c - V ( z , t ) at at2 a2
(2.30a) (2.30b)
Substituting (2.30b) and (2.27b) into (2.30a) and reversing the process yields the uncoupled, second-order equations: a2 a a2 V(z,t ) = (RG)V(z, t ) + (RC + LG) - V(z, t ) + LC - V(z, t ) az2 at at2 a2 I(z, t ) = (GR)I(z, t ) + (GL + CR) a I(z, t ) + CL a2 I(z, t )
at
a22
at2
(2,31a) (2.31b)
Observe that the various matrix products in (2.31) do not generally commute so that the proper order of multiplication must be observed.
2.4
PROPERTIES OF THE PER-UNIT-LENGTH PARAMETER MATRICES 1, C,
G
In the previous chapter we showed that, for a two-conductor line immersed in a homogeneous medium characterized by permeability, p, conductivity, a, and permittivity, e, the per-unit-length inductance, I, conductance, g, and capacitance, c, are related by IC = pe and ig =pa. For the case of a MTL consisting of (n 1) conductors immersed in a homogeneous medium characterized by permeability,p, conductivity, a, and permittivity, e, the per-unit-length parameter matrices are similarly related by
+
- -
LC = CL = Wl,
LG
GL p l ,
(2.32a) (2.32b)
PROPERTIES OF THE PER-UNIT-LENGTH PARAMETER MATRICES l,C, C
59
where the n x n identity matrix is defined as having unity entries on the main diagonal and zeros elsewhere:
'.'
1 0
0
1
0 0 0 0
.. .. .. . .
e
"*
.
0 0 0 0
e
*.*
*..
.
*
.
*
(2.33)
1 0
0 1.
Other important properties such as our logical assumption that these per-unitlength matrices are symmetric will also be shown. Recall from Chapter 1 that the transverse electric and magnetic fields of the TEM field structure satisfy the following differential equations (see equations ( 1.11)):
(2.34a) (2.34b) Define voltage and current in the usual fashion as integrals in the transverse plane (see Fig. 2.2) as I-
(2.35a) (2.35b) Applying (2,35)to (2.34) yields
a2 K(2, t ) = pa a c;(z, t ) + ps a2 Q(2, t ) az2
at
a2 a -I,(& t ) = pa - I&,
dz2
at
at2
t)
a2 + ps I,(2, t ) at2
(2.36a) (2.36b)
Collecting equations (2.36)for all conductors in matrix form yields (2.37a) (2.37b)
60
THE MULTICONDUCTOR TRANSMISSION-LINE EQUATIONS
Comparing (2.37) to (2.31) with R = 0 gives the identities in (2.32). Because of the identities in (2.32)) valid only for a homogeneous medium, we need to determine only one of the per-unit-length parameter matrices since (2.32) can be written, for example, as L = peC" (2.38a) U
G=-C e
(2.38b)
C = peL"
(2.38~)
The identities in (2.32) are valid only for a homogeneous surrounding medium as is the assumption of a TEM field structure and the resulting MTL equations. We will often extend the MTL equation representation, in an approximate manner, to include inhomogeneous media as well as imperfect conductors under the quusi-TEM assumption. Even in the case of an inhomogeneous medium, the per-unit-length parameter matrices, L, C, and G, have several important properties. The primary ones are that they are symmetric and positive-dejnite matrices. The proof that C, L, and G are symmetric matrices (regardless of whether the surrounding medium is homogeneous or inhomogeneous) can be accomplished from energy considerations [A. 1,1]. As an illustration, we will prove that C is symmetric; the proof that L and G are symmetric follows in a similar fashion. The basic relation for C is given in (2.24) and (2.25). Suppose we invert this relation to give (2.39) If we can prove that pu = pji then it follows that cfj= cJr.Suppose all conductors except the i-th and j-th are connected to the reference conductor (grounded) and all conductors are initially uncharged. Suppose we start charging the i-th conductor to a final per-unit-length charge of qr. Charging the i-th conductor to an incremental charge q results in a voltage of the conductor, from (2.39), of 6 = pllq. The incremental energy required to do this is d W = dq. The total energy required to place the charge qr on the i-th conductor is W = p' p l l q dq = pfrq:/2. Now if we charge thej-th conductor to an incremental charge of q in the presence of the charged i-th conductor, the voltage of the J-th conductor is 4 = plrqr + p f f qand the incremental energy required is d W = ( p f i q r+ pf,q) dq. The total energy required to charge thej-th conductor to a charge of qf becomes W = d W dq = pflqiqj+ pf,qf/2. Thus the total energy required to charge conductor i to qr and conductor j to q, is
PROPERTIES OF THE PER-UNIT-LENGTH PARAMETER MATRICES,.I C, C
61
If we reverse this process charging conductor j to qj and then charging conductor i to qr we obtain
Since the total energies must be the same regardless of the sequence in which the conductors are charged, we see, by comparing these two energy expressions, that P l j = Pjl Therefore, it follows that cIJ
= cJI
and therefore the capacitance matrix, C, is symmetric. Proof that L and G are also symmetric follows in a like fashion. Recall that this proof of symmetry relied on energy considerations and therefore is valid for inhomogeneous media. We next set out to prove that L,C,and G are positive deJnite. The energy stored in the electric field per unit of line length is (2.40)
where the transpose of a matrix M is denoted by M'.The vector Q contains the per-unit-lengthcharges on the conductors and is given in (2.25b). Substituting the relation for C in (2.25a), Q = CV, into (2.40) gives w, = - V'CV > 0
(2.41)
where we have used the matrix property that (MN)' = N'M'. This total energy stored in the electric field must be positive and nonzero for all choices of the voltages (positive or negative). Thus we say that C is positive definite if
V'CV > 0
(2.42)
for all possible values of the entries in V. It turns out that this implies that all of the eigenvalues of C must be positiue; a property we will find very useful in our later developments. Proof that L and G are also positive definite follows in a similar fashion. Finally we point out that all ofthe per-unit-length parameter matrices can be obtained >om capacitance calculations with and without the dielectric remoued.
62
M E MULTICONDUCTOR TRANSMISSION-LINE EQUATIONS
Designate the capacitance matrix with the surrounding medium (homogeneous or inhomogeneous) removed and replaced by free space having permeability 8, and permeability po as C,. Since inductance depends on permeability of the surrounding medium and the permeability of dielectrics is typically that of free space, po, the inductance matrix, L,can be obtained from C, (using the relations for a homogeneous medium (in this case, free space) given in (2.32)) as:
L = poC,eoC,-'
(2.43)
Therefore, L and C can be computed using only a capacitance calculation. This observation will be useful when we consider computing these parameters for inhomogeneous media in the next chapter.
REFERENCE [l]
R. Plonsey and R.E. Collin, Principles and Applications of Electromagnetic Fields, 2d ed., McGraw-Hill, NY, 1982.
PROBLEMS
2.1 Derive the MTL equations for the per-unit-length equivalent circuit of the
four-conductor line shown in Fig. P2.1.
FIGURE P2.1
PROBLEMS
2.2
63
A four-conductor line immersed in free space has the following per-unitlength inductance matrix:
Determine the per-unit-length capacitance matrix. If the surrounding S/m, determine the medium is homogeneous with conductivity cr = per-unit-length conductance matrix.
2.3 Derive the uncoupled, second-order MTL equations in (2.31). 2.4 Show that the criterion for positive definitenessof a real, symmetric matrix is that its eigenvalues are all positive and nonzero. (Hint: Transform the matrix to another equivalent one with a transformation matrix that diagonalizes it as T”MT = A where A is diagonal with its eigenvalues on the main diagonal. It is always possible to diagonalize any real, symmetric matrix such that T” = T‘ where T‘ is the transpose of T.) Show that the per-unit-length inductance matrix in Problem 2.2 is positive definite. 2.5 A matrix with the structure of G in (2.23) or C in (2.24) whose off-diagonal
terms are negative and the sum of the elements in a row or column are positive is said to be hyperdominant. Show that a hyperdominant matrix is always positive definite.
CHAPTER THREE
The PerlUnit-Length Parameters
The per-unit-length parameter matrices of inductance, L, capacitance, C, resistance, R,and conductance, C,are essential ingredients in the determination of the MTL voltages and currents from the MTL equations. It is important to recall that, under the fundamental TEM field structure assumption, the per-unit-length parameters of inductance, capacitance, and conductance are determined as a static (dc) solution to Laplace's equation, e.g., V 2 ~ ( xy ), = 0, in the two-dimensional cross-sectional (x, y ) plane of the line. Therefore the entries in L,C, and G are governed by the fields external to the line conductors and are determined as static field solutions in the transverse plane for perfect conductors. The entries in the per-unit-length resistance matrix, R,are governed by the fields interior to the conductors for imperfect conductors. In the case of perfect line conductors, R = 0. Technically, the fields exterior and interior to imperfect conductors interact so that the entries in R cannot be independently determined as the resistances of the isolated conductors. For typical line dimensions and frequencies of excitation the entries in R can be determined as the resistances of the isolated conductors to a reasonable degree of approximation. Cases where this interaction cannot be ignored will be discussed as needed. The purpose of this chapter is to investigate methods, analytical and numerical, for determining these per-unit-length parameters. The ease with which we can determine these for a particular MTL cross-sectional structure depends on the properties of the MTL. For example, we will find that for wire conductors (circular cylindrical cross sections) that are relatively widely spaced and immersed in a homogeneous surrounding medium, some simple, closedform analytical expressions can be obtained for the per-unit-length parameters. If the wires are closely spaced and/or the medium is inhomogeneous, one must typically resort to approximate numerical methods to obtain the per-unitlength parameters. Conductors of rectangular cross section such as are found on printed circuit boards (PCBs) also typically require approximate numerical methods for their determination regardless of whether the medium is 64
DEFINITIONS OF THE PER-UNIT-LENGTH PARAMETER MATRICES 1, C,
C
65
homogeneous or inhomogeneous (as is typically the case with PCB's). There exist some analytical solutions for the per-unit-length parameters for a twoconductor line having conductors of rectangular cross section but these are often quite involved. These analytical solution techniques generally attempt to transform the desired problem to a simpler problem using a transformation of coordinate variables, e.g., the Schwarz-Christoffel transformation. It is important to keep in mind that efficiency of solution of this step is critically determined by the class of MTL being considered. 3.1
DEFINITIONS OF THE PER-UNIT-LENGTH PARAMETER MATRICES 1, C, C
We first review the fundamental definitions, obtained in the previous chapter, of the per-unit-length parameter matrices of inductance, L,capacitance, C, and conductance, G.Recall that these per-unit-length parameters are determined as static field solutions in the transverse plane for perfect line conductors. There are numerous methods, analytic and numerical, for this static two-dimensional problem. The entries in the per-unit-length resistance matrix, R,will be obtained in Section 3.6. Again we will restrict our discussions to ungorm lines. 3.1.1
The Per-Unit-Length Inductance Matrix, L
The entries in the per-unit-length inductance matrix, L, relate the total magnetic flux penetrating the i-th circuit, per unit of line length, to all the line currents
producing it as
Y = LI
(3.la)
or, in expanded form,
(3.lb)
If we interpret the above relations in a manner similar to the n-port parameters CA.23, we obtain the following relations for the entries in L: (3.2a) (3.2b) Thus we can compute these inductances by placing a current on one conductor
66
THE PER-UNIT-LENGTH PARAMETERS
'L Y
I Y
Illustration of the definitions of flux through a circuit for determination of the per-unit-length inductances: (a) self-inductances, 11(, and (b) mutual inductances, l,j FIGURE 3.1
(and returning it on the reference conductor), setting the currents on all other conductors to zero and determining the magnetic flux, per unit of line length, penetrating the other circuit. The definition of the i-th circuit is critically important to obtaining the correct value and sign of these elements. This important concept is illustrated in Fig. 3.1. The i-th circuit is the surface between the reference conductor and the f-th conductor (which is of arbitrary shape but is uniform along the line). This surface shape may be a flat surface or some other shape so long as this shape is uniform along the line. The magnetic flux per-unit-length penetrating this surface (circuit) is defined as being in the
DEFINITIONS OF THE PER-UNIT-LENGTH PARAMETER MATRICES 1, C, G
FIGURE 3.2
67
Illustrations of the derivationof the per-unit-lengthinductancesfor a ribbon
cable structure.
clockwise direction around the i-th conductor when looking in the direction of increasing z. In other words, the flux direction, through su$ace si is the direction magnetic flux would be generated by the current of the i-th conductor. Figure 3.l(a) shows the calculation of Ill, and Fig. 3.l(b) shows the calculation of I , . In order to illustrate this important concept further, consider a threeconductor line consisting of three wires lying in a plane where the middle wire is chosen, arbitrarily, as the reference conductor as shown in Fig. 3.2(a). The surfaces and individual configurations for computing I , 1, Iz2, and Il2 are shown in the remaining figures. Observe that the surface for $* is between conductor number 2 and the reference conductor but observe the desired direction of this flux; it is chosen with respect to the magnetic flux that would be produced by current I , on conductor number 2. So, theflux directionfor the i-th circuit is defned by the direction of the current on the i-th conductor and the right-hand rule when looking in the direction of increasing z.
68
THE PER-UNIT-LENGTH PARAMETERS
FIGURE 3.3 Illustration of the contours and flux directions for the general derivation of the per-unit-length inductances.
This computation requires that the magnetic flux penetrating the surface by the desired current (and having the current return on the reference conductor) be computed. There are many ways of accomplishing this task, In some cases an analytical solution (exact or approximate) can be obtained, whereas in other cases numerical approximation techniques must be used. In either case, the precise definition of each of these liJ computations is, from Chapter 1, (3.3)
--
f Jwf
which is illustrated in Fig. 3.3. Although the above method is valid regardless of whether the medium is homogeneous or inhomogeneous in cc, if the medium is homogeneous in y (as are typical dielectric media), then L can be obtained from C using the fundamental relationship derived previously
L = flee-'
(3.4)
DEFINITIONS OF THE PER-UNIT-LENGTH PARAMETER MATRICES L, C, G
3.1.2
69
The Per-Unit-Length Capacitance Matrix, C
The entries in the per-unit-length capacitance matrix, C, relate the total charge on the i-th conductor per unit ofline length to all ofthe line ooltages producing it as (3.5a)
Q=CV or, in expanded form,
...
-c12
c n
C2k
*..
k- 1
(3.5b)
...
-C2r
If we denote the entries in C as [CIf,!these can be obtained by interpreting (3.5) as an n-port relation and applying the usual constraints of setting all voltages except the]-th voltage, 5, to zero and determining the charge, ( I l , on the i-th conductor (and -qf on the reference conductor) to give [C],,.The particular form lor the entries in C in (3.5b) can be readily seen by placing the per-unit-length capacitances between the conductors and writing the usual node-voltage equations of lumped-circuit theory CA.21. A simpler form for obtaining the elements of C is obtained by inverting (3.5a) as V-PQ (3.6a) or, in expanded form,
where
c
p-1
I
(3.W
The entries in P are referred to as the coeflctents ofpotential. Once the entries in P are obtained, C is obtained via (3.6~).The coefficients of potential are obtained from (3.6b) as (3.7a) (3.7b)
70
THE PER-UNIFLENCTH PARAMETERS
+ VI
FIGURE 3.4 Illustrations of the determination of the per-unit-length coefficients of potential: (a) self terms, p l l , and (b) mutual terms, pu.
These relationships show that to determine pu we place charge qJ on conductor j with no charge on the other conductors (but -qj on the reference conductor) and determine the resulting voltage G; on conductor i (between it and the reference conductor with the voltage positive at the t-th conductor). These concepts are illustrated in Fig. 3.4. Once P is obtained in this fashion, C is obtained as the inverse of P as shown in (3.6~).It is important to point out that the self-capacitancebetween the i-th conductor and the reference conductor, cff,Is not simply the entry in the i-th row and i-th column ofC. Observe the form of the entries in C given in (3.5b). The off-diagonal entries are the negatives of
DEFINITIONS OF THE PER-UNIT-LENGTH PARAMETER MATRICES I., C, G
71
FIGURE 3.5 Illustration of the general definitions of contours for the determination of charge and voltage in the determination of the per-unit-length coefficients of potential.
the mutual capacitances between the pairs of conductors whereas the maindiagonal entries are the sum of the self-capacitance and the mutual capacitances in that row (or column). Therefore, to obtain the self-capacitance, q,,we sum the entries in the i-th row (or column) of C. This is again a static problem in the two-dimensional transverse plane. There are many ways to compute the pu from the n-port definitions in (3.7). However, the fundamental definition was derived in Chapter 1 and is
as illustrated in Fig. 3.5. This method is valid regardless of whether the surrounding medium is homogeneous or inhomogeneous in E. If the surrounding medium is homogeneous in E, we can alternatively obtain C from L using the fundamental relationship
C = PEL-'
(3.9)
72
THE PER-UNIT-LENGTH PARAMETERS
3.1.3
The PerUnit-Length Conductance Matrix, C
The per-unit-length conductance matrix, G, relates the total transuerse conduction current passing between :he conductors per unit of line length to all the line voltages producing it as
I, = GY
(3.1 Ua)
or, in expanded form,
Again, the particular forms of the entries in G in (3.10b) can be readily seen by placing the per-unit-length conductances between the conductors and writing the usual node-voltage equations of lumped-circuit theory CA.2). Once again, the entries in G can be determined as several subproblems by interpreting (3.10b) as an n port. For example, to determine the entry in G in the i-th row and j-th column (which, according to (3.10b), is not g,,) we could enforce a voltage between thej-th conductor and the reference conductor, &, = 4, with all other conductor voltages set to zero, V, = 0, and determine the transverse current, I,,, flowing between 5+? = ''= the I-th conductor and the reference conductor. Denoting each of the entries in G as [GI,we have
-
= . . a =
(3.11)
as illustrated in Fig. 3.6.
Conversely we could invert the relationship in (3.10) as for the case of the capacitance matrix and determine the entries in that matrix. If the medium is homogeneous in a, we can obtain G from either L or C using the fundamental
DEFINITIONS OF THE PER-UNIT-LENGTH PARAMETER MATRICES I, C,
FIGURE 3.6
G
73
Contours for the determination of the per-unit-length conductances.
relationships obtained for a homogeneous medium: C = paL”
(3.12)
U
=-C 6
This result, although proven in Chapter 2, is apparent since the transverse conduction current density is related to the transverse electric field as 5, = d?, and the per-unit-length capacitance matrix, C, is governed solely by the transverse electric field. There are two different mechanisms that introduce losses in the medium. The first is through a nonzero conductivity and the other is through polarization loss [A. 11. Both mechanisms are implicitly included in the conductivity parameter. The incorporation of losses in the media for inhomogeneous media will be examined in Section 3.6.1 and is relatively straightforward as a modification of the determination of C for the inhomogeneous medium. If the surrounding medium, homogeneous or inhomogeneous, is lossless, i.e., the conductivities are zero, then G = 0. 3.1.4
The Generalized Capacitance Matrix, V
The above definitions as well as the derivation of the MTL equations assume that we (arbitrarily) select one of the (n + 1) conductors as the reference conductor to which all the n voltages, 6,are referenced. Once the reference conductor is chosen, all the per-unit-length parameter matrices must be computed for that choice consistently. Although the choice of reference conductor is arbitrary, choosing one of the (n + 1) conductors over another as reference may facilitate the computation of the per-unit-length parameters. For
74
THE PER-UNIT-LENGTH PARAMETERS
example, if n of the conductors are wires and the remaining conductor is an infinite, perfectly conducting plane, we will show that choice of the plane as the reference conductor simplifies the calculation of the per-unit-length parameters. However, choice of this plane as reference is not mandatory; we could choose instead one of the n wires as reference conductor. In this section we describe a technique for computing a certain per-uni t-length parameter matrix, the generalized capacitance matrix, '8, without regard to choice of reference conductor. Once this is computed, the other per-unit-length parameter matrices. L,C,and G, can be easily computed from it for a particular choice of reference conductor. The MTL voltages, 6,arc defined to be between each conductor and the reference conductor. We may also define the potentials, t#+, of each of the (n 1) conductors with respect to some reference point or line that is parallel to the z axis. The total charge per unit of line length, qr, of each of the (n -f 1) conductors can be related to their potentials, 4r,for f = 0, 1, 2,. . ,n with the (N 1) x (n 1) generalized capacitance matrix, dp, as
+
+
.
+
Q=W@
(3,13a)
or, in expanded form, as
+
+
Observe that dp is (n 1) x (n 1) whereas the previous per-unit-length parameter matrices, L, C,and G, are n x n. Also the generalized capacitance matrix, like the transmission-linecapacitance matrix, is symmetric, i.e., VI,= W,, for similar reasons. It can be shown that for a charge-neutral system as is the case for the MTL, the reference potential terms for the choice of reference point for these potentials, r$,, vanishes as the reference point recedes to infinity so that the choice of reference point does not affect the determination of the transmission-line-capacitancematrix, C, from the generalized capacitance matrix CB.1, C.51. Suppose that (8 has been computed and we select a reference conductor. Without loss of generality let us select the reference conductor as the zeroth conductor. In order to obtain the n x n capacitance matrix, C, from W, define the MTL line voltages, with respect to this zeroth reference conductor, as
.
6 * $1 - 4 0
(3.14)
for i = 1, 2,. , ,n. We assume that the entire system of ( n + 1) conductors is charge neutral, ie., qo + q1 + q2 + qn = 0. Therefore the charge (per unit
.- +
DEFlNlTlONS OF THE PER-UNIT-LENGTH PARAMETER MATRICES,.I C, C
75
of line length) on the zeroth conductor can be written in terms of the charges on the other n conductors as 40
=
-&i- I 9&
(3.15)
Denote the entries in the i-th row and 1-th. column of the per-unit-length capacitance matrix, with the zeroth conductor chosen as reference conductor, as C,,: 41
3
(3.16)
ctk and C,, = -ctJ. Comparing (3.16) to (3.5b) we observe that C,, = Substituting (3.14) and (3.15) into (3.13) and expanding gives
Adding all equations in (3.17) gives
or
(3.18b)
Substituting (3.18b) into the last n equations in (3.17) yields the entries in the
76
THE PER-UNIT-LENGTH PARAMETERS
per-unit-length capacitance matrix, C, given in (3.16) as CC.41
(3.19)
The first summation in the numerator of (3.19) is the sum of all the elements in the I-th row of%', whereas the second summation in the numerator of (3.19) is the sum ofall the elements in thej-th column of V. The denominator summation, Gg, is the sum of all the elements in Q. In the case of two conductors, the result in (3.19) gives the per-unit-length capacitance between the two conductors and reduces to (3.20) The generalized capacitance matrix, like the transmission-line capacitance, is symmetric so that VOl = Vlo, Eliminating the potential reference node (or line) and observing that capacitors in series (parallel) combine like resistors in parallel (series) one can directly obtain the result in (3.20) from the equivalent circuit of Fig. 3.7(a). Therefore we can obtain the per-unit-length generalized capacitance matrix, W,choose a reference conductor, and then easily compute C for that choice of reference conductor from W using the relation in (3.19). If the surrounding medium is inhomogeneousin e, we similarly compute the generalized capacitance matrix with the dielectric removed (replaced with free space). V,,, and from that compute the per-unit-length capacitance matrix with the dielectric remoued, C,, with the above method. Once C, is computed in this fashion, we may then compute L = ~ L E , , C ; ~ .
Potential reference point (line)
(4 FIGURE 3.7 Illustrationof (a) the meaning of the per-unit-lengthgeneralized capacitance matrix for a two-conductor line and (b) the elimination of the reference line to yield the
capacitance between the conductors.
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77
MULTICONDUCTOR LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
3.2
Conductors having cross sections that are circular cylindrical are referred to as wires. These types of conductors are frequently found in cables that interconnect electronic circuitry and form an important class of MTL's CA.31. These are some of the few conductor types for which simple closed-form equations for the per-unit-length parameters can be obtained. 3.2.1
Fundamental Subproblems for Wires
In order to determine simple relations for the per-unit-length parameters of wires, we need to discuss the following important subproblems cA.3, B.1,1,2]. 3.2.1.1 Magnetic Flux Due to a Filament of Current Consider a wire carrying a current I that is uniformly distributed ovecits cross section as shown in Fig, 3.8. The transverse magnetic field intensity, 8,is directed in the circumferential direction by symmetry. Enclosing the wire and current by a cylinder of radius r and applying Ampere's law for the TEM field CA.11:
(3.21)
gives
q = -I
21cr
FIGURE 3.8
Magnetic field intensity within and about a current-carryingwire.
(3.22)
78
THE PER-UNIT-LEm PARAMETERS
Ii
lm
v
FlCURE3.9 Illustration of the calculation of magnetic flux through a surface via a
simpler problem.
This result is due to the observations that: 1.
2 is tangent to df and therefore the dot product can be removed from
2.
4 is constant around the contour of radius r and so may be removed
(3.21) and the vectors r e p l a d with their magnitudes.
from the integral.
Now consider determiningthe magnetic flux from this current that penetrates a surface s that is parallel to the wire and of uniform cross section along the wire length as shown in Fig. 3.9(a). The edges of the surface are at distances R, and R2 from the wire. Next consider this problem in cross section as shown in Fig. 3.9( b). Consider the closed, wedge-shaped surface consisting of the original surface along with surfaces sl and s2. Surface s1 is the flat surface extending radially along R 1 to a radius of R2,and surface s2 is a cylindrical surface of constant radius R2 that joins surfaces s and sl. Gauss' law provides that there are no (known) isolated sources of the magnetic field. Thus the total magnetic flux, t,b,, through a closed surface must be zero CA.11: (3.23)
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
79
where 2 is the magneticflux density vector and is related foclinear,homogeneous, isotropic media to the magnetic field intensity vector, X , as W = p&'. From (3.23), the total magnetic flux through the wedge-shaped surface of Fig. 3.9 is the sum of the fluxes through the original surface, s, and surfaces sl and s2 since no flux is directed through the end caps because of the transverse nature of the magnetic field. But the flux through sl is also zero because the magnetic field is tangent to it. Thus the total magnetic flux penetrating the original surface s is the same as the flux penetrating surface s1 as shown in Fig. 3.9(c). But the problem of Fig. 3.9(c) is simpler than the original problem because the magnetic field is orthogonal to the surface. Thus the total magnetic flux through either surface is
JI,,, = =
935
$*d3
(3.24)
8,
g,*d3
where we have assumed that Rz > R, to give the indicated direction of The magnetic flux per unit of line length is
e,,,.
(3.25) We will find the results of this subproblem to be of considerable utility in our future developments. The above derivation has been made with two important assumptions: 1. The wire is infinitely long. 2. The current is uniformly distributed over the wire cross section or
symmetrical about its axis. The first assumption allows us to assume that the magnetic field is invariant along the direction of the wire axis. The second assumption means that we may replace the wire with ajlumentary current at its axis on which all the current I is concentrated, and implicitly assumes that there are no closely spaced currents to disturb this symmetry. The effects of this infinitely long filament of current on the flux penetrating the above surface will be unchanged.
80
THE PER-UNIT-LENGTH PARAMETERS
FIGURE 3.10 Illustration of the electric field about a charge-carrying wire. 3.2.1.3 VoltageDue to a Filament of Charge We next consider the dual problem of determining the voltage between two points due to a filament of charge. Consider the very long wire carrying a charge per unit of length q C/m as shown in Fig. 3.1qa). We assume that either the charge is uniformly distributed around the periphery of the wire or concentrated as a filament of charge. In this case, the electric field intensity, 3, will be radially directed in a direction transverse to the wire. Gauss' law provides that the total electric flux penetrating a closed surface is equal to the net positive charge enclosed by that surface CA.11: nn
(3.26)
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
81
where 5 is the electric flux density vector. For linear, homogeneo_us an! isotropic media, this is related to the electric field intensity vector by 9 = E&. Consider enclosing the charge-carrying wire with a cylinder of radius r as shown in Fig. 3.10(b). The electric field is obtained by applying (3.26) to that closed surface to yield (3.27) The dot product may be removed and the vectors replaced with their magnitudes since the electric field is orthogonal to the sides of the surface and no electric field is directed through the end caps of the surface. Furthermore, the electric field may be removed from the integral since it is constant in value over the sides of the surface. This gives a simple expression for the electric field away from the filament:
4 V/m 4 =-
(3.28)
2n~r
Next consider the problem of determining the voltage between two points away from the filament as shown in Fig. 3.1 l(a). The points are at radii R, and R , from the filament. The voltage is defined as (uniquely because of the transverse nature of the electric field) (3.29) A simpler problem is illustrated in Fig. 3.11(b). Contour c, extends along a radial line from the end of R 1 to a distance R,, and contour c2 is of constant radius R , and extends from that end to the beginning of the original contour. Thus, since (3.29) is independent of the path taken between the two points, we
(a)
(b)
Illustration of the calculation of voltage of a charge-carrying wire directly and via a simpler problem. FIGURE 3.11
82
THE PER-UNIT-LENGTH PARAMETERS
may alternatively compute (3.30)
This result was made possible by the observation that over c2 the electric field is orthogonal to the contour so that the second integral is zero, and over cl the electric field is tangent to the contour. We will also find this result to be of considerable utility in our future work. Once again, this simple result implicitly assumes that: 1. The filament is infinitely long. 2. The charge is uniformly distributed around its periphery.
The first assumption ensures that the electric field will not vary along the line length. The second assumption means that we may replace the wire with a filament of charge, and implicitly assumes that there are no closely spaced charge distributions to disturb this symmetry. 3.21.3 The Method of /mages The last principle that we will employ is the method of images. Consider a point charge Q situated a height h above an infinite, perfectly conducting plane as shown in Fig. 3.12(a). We can replace the infinite plane with an equal but negative charge -Q at a distance h below the previous location of the surface of the plane and the resulting fields will be identical in the space above the plane’s surface CA.11. The negative charge is said to be the image of the positive charge. We can similarly image currents. Consider a current I parallel to and at a height h above an infinite, perfectly conducting plane shown in Fig. 3.12(b). If the plane is replaced with an equal but oppositely directed current at a distance h below the position of the plane surface, all the fields above the plane’s surface will be identical in both problems CA.1). This can be conveniently remembered using the following mnemonic device. Consider the current I as producing positive charge at one endpoint and negative at the other (denoted by circles with enclosed polarity signs). Imaging these “point charges” in the way described gives the correct image current direction. A vertically directed current is similarly imaged as shown in Fig. 3.12(c). Current directions which are neither vertical nor horizontal can be resolved into their horizontal and vertical components and the above results used to give the correct image current distribution CA.13.
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
83
3
t
8
FIGURE 3.12 Illustration of the method of images for (a) a charge above an infinite, perfectly conducting plane, (b) and (c) its extension to current images.
3.2.2
Exact Solutions for Two-Conductor Wire liner
There exist few transmission-line structures for which the per-unit-length parameters can be determined exactly. The class of two-conductor lines of circular cylindrical cross section in u homogeneous medium considered in this section represents' a significant portion of such lines. 3.2.2.1 Two Wires Consider the case of two wires of radii rWl and rw2 as shown in Fig. 3.13(a). Let us assume that the currents are uniformly distributed around the wire peripheries as shown in Fig. 3.13(b). Using the above result for the magnetic flux from a filament of current, we obtain the total flux passing
84
THE PER-UNIT-LENGTH PARAMETERS
H .
s
rl 1
I
(c)
FIGURE 3.13 Illustration of (a) a two-wip line and calculation of (t the per-unit-length inductance and (c) the per-unit-length capacitance for widely separated wires.
between the two wires as
(3.31)
This result assumes that the current is uniformly distributed around each wire periphery. This will not be the case if the wires are closely separated since one current will interact and cause a nonuniform distribution of the other current (this is referred to as proximity effect). In order to make this result valid, we must require that the wires be widely separated. The necessary ratio of separation to wire radius to make this valid will be investigated when we
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
85
determine the exact solution. Therefore, because of the necessity to have the wires widely separated, the separation must be much larger than either of the wire radii so that (3.31) simplifies to
(3.32)
zP In(-)
S2
2x
rwirw2
H/m
In the practical case of the wire radii being equal, rwl = rw2 = rw,this reduces to (3.33) The per-unit-length capacitance will be similarly determined in an approximate manner. Consider the two wires carrying charge uniformly distributed around each wire periphery as shown in Fig. 3.13(c). The voltage between the wires can be similarly obtained using (3.30) as
(3.34)
and we have used the necessary requirement that the wires be widely separated. The per-unit-length capacitance is c=-4
V
(3.35)
For equal wire radii this simplifies to
We now turn to an exact derivation of these results. The essence of the method is to concentrate the total per-unit-length charge on each wire, q, on
86
THE PER-UNIT-LENGTH PARAMETERS
P
‘
FIGURE 3.14 two wires.
X
Illustration of the calculation of the exact per-unit-length capacitance of
filaments separated a distance d as shown in Fig. 3.14. Then find the equipotential contours about these filaments and locate the actual wires on these contours that correspond to the voltages of the wires. This gives the equivalent spacing between the two wires, s, that carry the same charge per unit of line length. The voltage at a point P shown in Fig. 3.14 due to the filaments of charge, one carrying q and the other carrying -4, with respect to the origin, x = 0 and y = 0, can be found using the previous basic subproblem as (3.37)
Thus points on equipotential contours are such that the ratio (3.38)
=K is constant where K is some constant. Substituting the equations for R + and R’: (3.39a) R + = J(x d/2)’ yz
+
R - = J(x
- d/2)’
+ + y2
(3.39b)
LINES HAVING CONDUCTORS OF CIRCULAR CYllNORlCAL CROSS SECTION
87
gives (3.40) Writing (3.40) in the form of the equation for a circle of radius r that is centered = 0: (x y2 = r2 (3.41a) gives h = -d- K 2 + 1 (3.41b) 2K2-1 at x = h, y
+
Kd K Z- 1
r=-
(3.41~)
The constant K can be eliminated by taking the difference of the squares of (3.41b) and (3.41~)to give (3.42) The value of potential for each of these equipotential surfaces can be found by solving (3.41) for K to give (3.43)
Substituting this into (3.38) and solving for the voltage gives (3.44) Recall that this is the voltage of the point with respect to the origin ofthe coordinate system which is located midway between the two wires. Thus the
voltage between the two wires that are separated by distance s is (3.45)
88
THE PER-UNIT-LENGTH PARAMETERS
and the per-unit-length capacitance becomes c = -4
(3.46)
V
This can be defined in terms of the inverse hyperbolic cosine as cosh”(x) = ln(x
+ =/,)
(3.47)
to give (3.48)
If the wire radii are not equal, the corresponding exact result for the per-unit-length capacitance is derived in [3] and becomes (3.49)
The exact result in (3.46) or (3.48) simplifies for widely spaced wires. For example, suppose s >> rw. The exact result in (3.46) reduces to the approximate result derived earlier and given in (3.36). The error for a ratio s/rw = 5 is only 2.7%. A ratio of separation to wire radius of 4 would mean that another wire of the same radius would just fit between the two original wires. For this very small separation, the error between the approximate expression (3.36) and the exact expression (3.46)is only 5.3%! So the wide-separation approximation given in (3.36) is quite adequate for practical wire separations. The per-unit-length inductance can be obtained from this result, assuming the surrounding medium is homogeneous in E and p as 1 = pw-1
=! 7t!cosh-l(L) 2rW H/m
(3.50)
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
89
1 0 1 i. T' T; 0 (b)
FIGURE 3.15 Determination of the per-unit-length capacitance of one wire above an infinite plane via the method of images.
Similarly, if the medium is lossy and homogeneous in u we can obtain U
g3-c
(3.51)
E
This latter result for the per-unit-length conductance is not particularly realistic since the only reasonably infinite, homogeneous medium that can exist around the two wires is free space which has u = 0. 3.2.2.2 One Wire Above an Inunite, Pehctly Conducting Plane Next consider the case of one wire at a height h above and parallel to an infinite, perfectly conducting plane (sometimes referred to as a ground plane) as shown in Fig. 3.15(a). By the method of images we may replace the plane with its image located at an equal distance h below the position of the plane as shown in Fig.
90
THE PER-UNIT-LENGTH PARAMETERS
3.15(b). The desired capacitance is between each wire and the position of the plane. But, since capacitances in series add like resistors in parallel, we see that this problem can be related to the problem of the previous section as: CIWOwtre
--
wire above
round
2
(3.52)
Therefore the capacitance of one wire above an infinite, perfectly conducting plane becomes, substituting h = s/2 in (3.48), C=
2ns cosh -
t)
F/m
(3.53)
or, approximately, for h >> r,: (3.54)
Similarly, the inductance can be obtained from this result as
1 = pec-1
(3.55)
3.2.2.3 The Coaxial Cabie Consider the coaxial cable shown in Fig. 3.16 consisting of a wire of radius rw within and centered on the axis of a shield of' inner radius rJ. The medium between the wire and the shield is assumed to be homogeneous. The case of an inhomogeneous medium can be solved so long as the inhomogeneity exists in annulae symmetric about the shield axis. (See the end-of-chapter Problem 3.1.) If we place a total charge q per unit of line length on the inner conductor, a negative charge of equal magnitude will be induced on the interior of the shield. Observe that by symmetry, the charge distributions will be uniformly distributed around the conductor peripheries regardless of the conductor separations. We earlier obtained the result for the electric field due to the charge, q, per unit of line length on the inner wire as
4=-
'
2ner
b,
r, s r s rJ
(3.56)
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
91
(4
FIGURE 3.16
(4 Calculation of the per-unit-length parameters far a coaxial cable: (a) the
general structure, (b) capacitance, (c) inductance, and (d) conductance. Observe that the field will be directed in the radial (transverse) direction. The voltage between the two conductors can be obtained as (3.57)
The capacitance per unit of line length is therefore c=-4
V
(3.58)
92
THE PER-UNIT-LENGTH PARAMETERS
Observe that, because of symmetry, the charge distributions will be uniform around the conductor peripheries regardless of conductor separation so that this result is exact. The per-unit-length inductance can be derived directly or by using 1 = jlsc-' (3.59) =! !In(5) H/m 2n rw
Similarly, the per-unit-length conductance is (3.60)
The per-unit-length inductance can be derived directly. Consider placing rl flat surface of length Az between the inner and outer conductors as shown in Fig. 3.16(c), The desired magnetic flux passes through this surface. Clearly the flux will be in the circumferential direction since, due to symmetry, the current will be uniformly distributed around the periphery of the inner wire and the inside of the shield. Thus we may use the fundamental result derived earlier to give I = - rl, (3.6 1)
I
The per-unit-length conductance given in (3.60) can similarly be obtained directly from Fig. 3.16(d) by finding the ratio of the transverse current to the voltage: g=
V
(3.62)
2na
3.2,3 Wideseparation Approximations for Wires in Homogeneous Media
The above results for two-conductor lines in a homogeneous medium are exact. For similar lines consisting of more than two conductors, exact closed-form
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
93
solutions cannot be obtained, in general. However, if the wires are relatively widely spaced, we can obtain some simple but approximate closed-form solutions using the fundamental subproblems derived in Section 3.2.1 CB.11. These results assume that the currents and charges are uniformly distributed around the wire peripheries which implicitly assumes that the wires are widely spaced. As we saw in the case of two-wire lines, the requirement of widely spaced wires is not overly restrictive. The following wide-separation approximations for wires are implemented in the FORTRAN program WIDESEP.FOR described in Appendix A.
+
1) wires in a homogeneous 3.2.3.1 fn + 1) Wires Consider the case of (n medium as shown in Fig. 3.17(a). The entries in the per-unit-length inductance are defined in (3.2). If the wires are widely separated, we can use the fundamental subproblems derived in Section 3.2.1 to give these entries. The self-inductance is obtained from Fig. 3.17(b) as (3.63a)
(3.63b)
The entries in the per-unit-length capacitance and conductance matrices can be obtained from this result as
c = p&L-’ U
G--C
(3.64) (3.65)
E
= afiL-’ 3.2.3.2 n Wires Above an Infinite, perfectlyConductingPlane Consider the case of n wires above and parallel to an infinite, perfectly conducting plane shown
94
THE PER-UNIT-LENGTH PARAMETERS
FIGURE 3.17 lllustration of the calculation of per-unit-length inductances using the wide-separation approximations for (n + 1) wires: (a) the cross-sectional structure, (b) self-inductance,and (c) mutual inductance.
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
95
FIGURE 3.18 Illustration of the calculation of per-unit-length inductances using the wide-separation approximations for n wires above a ground plane.
in Fig. 3.18. Replacing the plane with the image currents and using the fundamental result derived in Section 3.2.1 yields (3.66a)
(3.66b)
The entries in the per-unit-length capacitance and conductance matrices can then be found from these results using (3.64) and (3.65).
96
THE PER-UNIT-LENGTH PARAMETERS
FfCURE 3.19 Illustration of the calculation of per-unit-length inductances using the wide-separation approximations for n wires within a cylindrical shield: (a) the crosssectional structure, (b) replacement with images.
3.2.3.3 n Wires Within a Perfectly Conducting Shield Consider n wires of radii rwl within a perfectly conducting, circular cylindrical shield shown in Fig. 3.19(a). The interior radius of the shield is denoted by r, and the distances of the wires from the shield axis are denoted by dl while the angular separations are denoted by e,,. The perfectly conducting shield may be replaced by image currents located at radial distances from the shield center of ri/d, as shown in Fig. 3.19(b) [2,3]. The directions of the desired magnetic fluxes are as shown. Assuming the wires are widely separated from each other and the shield, we may assume that the currents are uniformly distributed around the wire and shield peripheries. Thus we may use the basic results of Section 3.2.1 to
LINES HAVING CONOUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
97
give
(3.67a)
(3.67b)
The entries in the per-unit-length capacitance and conductance matrices can then be found from these results using (3.64) and (3.65). 3.2.4
Numerical Methods for the General Case
The above results assumed that the medium surrounding the wires is homogeneous. For two wires or one wire above a ground plane, closed-form results have not been obtained for an inhomogeneous surrounding medium. For the coaxial cable, exact results can be obtained for an inhomogeneous medium so long as it is symmetric with respect to the shield axis. (See Problem 3.1 at the end of the chapter.) In the case of wire lines consisting of more than two conductors,exact results cannot be obtained even for a homogeneous surrounding medium, and wide-separation approximations must be used. There are some special cases for infinite structures of wires for which exact solutions can be obtained but these are not realistic for MTL applications [Z, 3). In this section we will discuss a numerical approximation technique which can be used to obtain accurate results for multiwire lines for an inhomogeneous surrounding medium as well as closely spaced conductors. The importance of considering wires that are immersed in inhomogeneous media stems from the practical requirement that circular cylindrical dielectric insulations must surround wire conductors to prevent shorting of the conductors. Closely spaced wires occur in many practical applications. An example is the use of ribbon cables wherein a group of dielectric-insulated wires are maintained in close proximity in a plane. The following method for dealing with these types of problems is described in [C.l-C.7). Consider an (n + 1)-wire line in a homogeneous medium. If the wires are closely spaced, proximity effect will cause the charge and current distributions around the wire peripheries to be nonuniform. With the exception of the
98
THE PER-UNIT-LENGTH PARAMETERS
two-wire line, this variation was ignored and assumed to be uniform around the wire peripheries. In the case of two wires that are closely spaced, the charge and current distributions will tend to concentrate on the adjacent surfaces (proximity effect). To model this effect, we could assume a form of the charge/current distribution around the i-th wire periphery in the form of a Fourier series in the peripheral angle, e,, such as (3.68a)
where
fu(O,) = cos(ke,), sin(k8,) k = 1,.
..,q
(3.68b) (3.68~)
and the (4 + 1) expansion coeficients, are determined to satisfy the boundary condition that the potential at points on each conductor due to all charge distributions equals the potential of that conductor. The charge distribution in (3.68a) has dimensions of C/mz since it gives the distribution around the wire periphery per unit of line length. The total charge on the i-th conductor per unit of line length is obtained by integrating (3.68a) around the wire periphery to yield Zn
4i = Jol-o
PFwl
del
(3.69)
= 21tr,,alO
This simple result is due to the fact that j&, cos(k0,) dei = ji;-o sin(k0,) de, = 0. We now determine the potential at an arbitrary point in the transverse plane at a position r, 8 from each of these charge distributions, i.e., q5,(rP,Op), as illustrated in Fig. 3.20(a). This can best be obtained by assuming the charge distribution around the periphery of the conductor is composed of filaments of charge, q, each of whose amplitudes are weighted by the particular distribution, Le., 1, cos(ke,), sin(k0,). Then we use the previous result given in equation (3.30) for the voltage between two points. With reference to Fig. 3.2qb) we obtain (3.70) It was shown in [C.5] that the potential of the reference point, t$o(ro, eo), may be omitted tfthe system of conductors Is electrically neutral, Le., the net charge per unit of line length is zero. Since this is satisfied for our MTL systems, we will henceforth omit the reference potential term. Thus the differential
LINES HAVING CONDUCTORS OF CIRCULAR CYLiNDRICAL CROSS SECTION
99
Potential reference point
FIGURE 3.20 Determination of the potential of a charge-carrying wire having various circumferential distributions: (a) definition of the problem and (b) replacement of the charge with weighted filaments of charge.
contribution to the potential due to a filamentary component of the charge distribution is d4,(rp,0,) =
4 - 2ne In(sp)
(3.71)
where the weighted charge distributions are given by (3.72) and the assumed charge distributions are
+
with N, 1 = 1
+ A, + B,. The distance from
the filament to the point is
100
THE PER-UNIT-LENCTH PARAMETERS
(according to the law of cosines) given by
Substituting these into (3.71) and integrating around the conductor periphery gives the total contribution to the potential due to the charge distributions:
where
Each of the integrals in (3.75a) can be evaluated in closed form giving [BJ]
Therefore, the contributions to the potential from each of these charge distributions are given in Table 3.1.
TABLE 3.1 Potentlal Due to Sinusoidal Charge Expansions Charge distribution 1
cos(m0) sin(m0)
Contribution to the potential 4(rP,e),
LINES HAVING CONDUCTORS
OF CIRCULAR CYLINDRICAL CROSS SECTION
101
FIGURE 3.21 Determination of the total potential at a point due to all chatge disttibu-
tions. Satisfaction of the boundary conditions is obtained if we choose a total of (3.77)
points on the wires at which we enforce the potential of the wire due to all the charge distributions on this conductor and all of the other conductors as illustrated in Fig. 3.21. This leads to a set of N simultaneous equations which must be solved for the expansion coefficients as
@=DA
(3.78a)
or, in expanded form,
..
...
...
(3.78b)
102
THE PER-UNIT-LENGTH PARAMETERS
The vector of potentials at the matchpoints on the i-th conductor is denoted as
a),=
[j
(3.78~)
and the vector of expansion coefficients of the charge distribution on the i-th conductor is denoted as
(3.78d)
Inverting (3.78b) gives
A = D'W
(3.79a)
or, in expanded form,
(3.79b)
The generalized capacitance matrix, 9,described in Section 3.1.4 can be obtained from (3.79), using (3.69), as [C.l-C.7] (3.80) row
This simple result is due to the fact that, according to (3.69).we only need to
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
103
determine a,,, and from (3.79),
(3.81)
or, in expanded form,
where [B,,],,,, denotes the entry in row m and column n of B,,. 3.2.4.1 Applications to Inhomogeneous Dielectric Media This method can be extended to handle inhomogeneous media such as circular cylindrical dielectric insulations around the wires by imposing the additional boucdary condition that the normal components of the electric flux density vector, 9,be continuous across the free-space-dielectric and dielectric-conductor interfaces [C. 1-C.7). To illustrate that application, we need to discuss the concepts of free charge and bound charge. Dielectric media consist of microscopic dipoles of bound charge. In polar dielectrics, such as water, the centers of positive and negative electric charge are separated slightly to give microscopic dipoles as illustrated in Fig. 3.22(a). However, any microscopic volume is electrically neutral. In nonpolar dielectrics, application of an external electric field causes the charges to separate to give these infinitesimal dipoles. I n either case, application of an external electric field causes these microscopic dipoles of bound charge to align with the field as illustrated in Fig. 3.22(b). If a slab of dielectric is immersed in an electric field,
FIGURE 3.22 Illustration of the effects of bound (polarization)charge: (a) microscopic dipoles, no external field (b) alignment of the dipoles with an applied electric field, and (c) creation of a bound surface charge.
104
THE PER-UNIT-LENGTH PARAMETERS
a bound charge density will appear on the surfaces of the dielectric as illustrated in Fig. 3.22(c). For high-frequency variation of the electric field, the charge dipoles cannot align instantaneously with the changes in direction of the electric field but lag behind it. This gives rise to a polarization loss which gives the same result as conduction loss due to a nonzero conductivity of the dielectric (usually small) cA.11. To account for both of these losses it is usually the practice to define an eflectiue conductiuity of the dielectric that includes both these losses in the following manner cA.11. The sum of conductive and displacement currents in Ampere's law for sinusoidal variation of the electric field is (a +joe)z. To account for polarization loss, the permittivity is written as the sum of a real and an imaginary part as e = 8'- je". Substituting gives a + j o e = (a we") +jwe' so that the efectiue conductivity is creff = (a oe"). Thus in any of our uses of a we intend that to mean the effective conductivity which includes both conductive and polarization losses. Charge consists of two types: free charge is that which is free to move and bound charge is the charge appearing on the surfaces of dielectrics in response to an applied electric field as shown in Fig. 3.22(c) which is not free to move. The lines of electric field intensity, 8, begin and end on-both free charge and bound charge, whereas the lines of electric flux density, B, begin and end only on free charge CA.11. At the interface between two dielectric surfaces the boundary condition is that the normal components of the electricjlux density vector, 9,must be continuous, i.e., g1,, = E ~ C ~=~ ,e282,, , = B2,,.A simple way of handling inhomogeneous dielectric media is to replace the dielectrics with free space having bound charge at the interface CC.1-C.7). At places where the dielectric is adjacent to a perfect conductor, we have both free charge and bound charge and the free charge density on the surface of the conductor is equal to the component of the electric flux density vector that is normal to the conductor surface, a = 9"C/m2, CA.1). Of course, the component of the electric field intensity vector that is tangent to a boundary is continuous across the boundary for an interface between two dielectrics,&t = c%;2, and is zero at the surface of a perfect conductor. In order to adapt the above numerical method to wires that have circular, cylindrical dielectric insulations, we describe the charge (bound plus free) around the wire periphery as a Fourier series in the peripheral angle, 0, as in (3.68):
+
+
(3.83)
A,
B,
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
PlI
105
-
FIGURE 3.23 The general problem of the determination of the potential of a dielectricinsulated wire due to free and bound charge distributions at the two interfaces.
where the expansion or basis functions are again
In (3.83), pfr denotes thefree charge distribution on the I-th conductor periphery and P l b denotes the bound charge distribution on the dielectric periphery facing the conductor as shown in Fig. 3.23. At the dielectric periphery facing the free-space region, there is only bound charge, which we similarly expand in a Fourier series in peripheral angle as
(3.85)
In (3.83) we have anticipated that the bound charge distribution around the conductor periphery will be opposite in sign to the bound charge distribution around the dielectric-free-space boundary. For each dielectric-insulated wire, there are a total of (4 1) unknown expansion coefflcients, a,,, for the free plus bound charge on the conductor peripheries and a total of (4 1) unknown expansion coefficients,blL,for the bound charge on the outer dielectric periphery. For a total of (n + 1) wires, this gives a total number of unknowns of N -Ii f where
+
+
n
N
(Nk k-0
+ 1)
(3.86a)
106
THE PER-UNIT-LENGTH PARAMETERS
fi = k=O
(fi& + 1)
=(n+l)+
f:
k=O
(3.86b)
f:
fik
k-0
unknowns. In order to enforce the boundary conditions we choose points on each conductor periphery at which to enforce the conductor potential, q5,, and points on each dielectric-free-space periphery at which to enforce the continuity of the normal components of the electric flux density vector due to all these charge distributions.This gives a set of N + fi simultaneousequations of a form similar to (3.78): (3.87)
+
(Nk 1) rows enforce the conductor potentials and the The first N = Epo second fi = Dl0(fik + 1) rows enforce the continuity of the normal components of the electric flux density vector across the dielectric-free-space (Nk+ 1) expansion coefficients interfaces. The vector A contains the N = of the free plus bound charge at the conductor peripheries, ark, and the vector d contains the fi = (Rk 1) expansion coefficients of the bound charge at the dielectric- free-space peripheries, a r k s The entries in (3.87) can be obtained by considering a cylindrical boundary of radius rb of infinite length shown in Fig. 3.24 which supports the charge distributions I, cos(mf&),sin(m0,) around its periphery. This is identical to the
+
zmo
FIGURE 3.24 The general problem of determining the potential inside and outside a charge distribution.
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
107
problem of free charge around a conductor periphery considered in the previous section but here the charge distribution can represent free or bound charge distributions. Proceeding as in the previous section by modeling the charge distributions as weighted filaments of charge gives the potential both inside and outside the boundary. Similarly, the electric field due to these charge distributions can be obtained from the gradient of these potential solutions CA.1): @rp, 0,) =
-w
(3.88)
Carrying out these operations, the potential and electric field at a point rb, 0, both inside and outside the charge distribution are given in Tables 3.2 and 3.3. Inverting equation (3.87) gives Boo
...
BlO
...
... ... ... Boo ... BNO
e . .
BON
. * a
BIN
..*
...
BNN
e . .
... ... ... . . . . . . . . . . . . ...
810
...
...
890
,..
...
@J
h N
h
N
...
...
... @N
I .
0 0 0,
(3.89)
108
THE PER-UNIT-LENGTH PARAMETERS
TABLE 3.2 Matchpoint Outside the Charge Distrlbution, rn 2
Contribution to the potential at P
Charge distribution
rb
Contribution to the electric field at P
1
rr+' cos(m0,) 2emr:
cos(mOb) sin(m0,)
rC+ sin(m0,) 2smr:
TABLE 3.3
Charge distribution 1
Matchpoint Inside the Charge Distribution, r,
Contribution to the potential at P
< rb
Contribution to the electric field at P
--rb In(rb)
0
E
cos(mOb) sin(mOb)
-
Recall that the charge at the conductor-dielectric interface consists of free charge plus bound charge, pv p#,. The entries in the generalized capacitance matrix relate the free charge on the conductors to the conductor potentials. Therefore, according to (3.83) and (3.85) we must add the total (bound) charge at the dielectric-free-space surface to the total (bound plus free) charge at the conductor-dielectric surface in order to obtain the total free charge on the conductor. Thus, in a fashion similar to the bare conductor case above, the entries in the generalized capacitance matrix can be obtained from (3.89) as w,j
=
I-
4) #o
a*.
-+I-
= 27rrwI flnt row
I
=+I+
I =*e*
+
-
(3.90)
= 4" 0
Blj 2arw1
6,
flrrt row
where ~filatrowBl, denotes the sum of the elements in the first row of the
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
109
--
d = 50 miti
r w = 7.5 mils (# 28 gauge, 7 x 36) I 10 miti e, 3.5 (PVC)
FJCURE 3.25
Dimensions of a five-wire ribbon cable for illustration of numerical results.
submatrix B,, of (3.89) relating the au coefficients of the bound plus free charge at the i-th conductor interface to the potential of thej-th conductor, $, and &rstrow fi,, denotes the sum of the elements in the first row of the submatrix fi,, of (3.89)relating the 6,kcoefficients of the bound charge at the dielectric-freespace interface for the i-th conductor to the potential of thej-th conductor, 4,. Details of this are described in [C.l-C.7] and numerical results are also presented. 3.2.5
Computed Results: Ribbon Cables
As an illustration of numerical results for the results of the previous sections, consider the five-wire ribbon cable shown in Fig. 3.25. The center-to-center separations of the wires are 50 mils (1 mil = 0.001 inch). The wires are identical and are composed of #28 gauge (7 x 36) stranded wires with radii of rw = 7.5 mils and polyvinyl chloride (PVC)insulations of thickness t = 10 mils and relative dielectric constant e, = 3.5. The generalized capacitance matrix was computed using the method of the previous section and ten Fourier coefficients around each wire surface (the constant term and nine cosine terms) and ten Fourier coefficients around each dielectric-free-space surface. The results are computed using the RIBBON.FOR computer program described in Appendix A which implements the method described in the previous section and are given in Table 3.4, The results computed using twenty Fourier coefficients around the wire and dielectric surfaces are virtually identical to those using ten coefficients indicating convergence of the solution. Alternatively, the results may be calculated with the GETCAP program described in CC.2, C.6, C.7). We have shown only the upper diagonal terms since, because of symmetry, the generalized capacitance matrix is symmetric. Choosing one of the outermost conductors as the reference conductor, the transmission-line-capacitance matrix is
110
M E PER-UNIT-LENGTH PARAMETERS
TABLE 3.4 Generalized Capacitances for the Five-Wire Ribbon Cable With and Without the Insulation Dielectric
Entry
With dielectric W/m)
-9.968 57 -2.575 14 - 1.555 14 - 1.780 84 23.549 2 -8.722 27 - 1.986 98 - 1.555 14 23.780 2 -8,722 27 -2.575 14 23.549 2 -9.968 58
-2.136 72 38.325 6 15.817 7 -2.109 30 1.671 38 38.541 2 15.817 7 2.899 41 38.325 5 17.497 8 26.775 8
-
-
38.152
- 15.974
CI4
-2.0343
e22
c23 c 24
e33
c34 c 4 4
Without dielectric
(PF/m)
CI I C12
ct 3
18.223 2
The Tranrmission-Une Capacitances for the FiveWire Ribbon Cable With and Without the Insulation Dielectrics
With dielectric Entry
(PF/m) 18.223 2
26.775 8
- 17.497 9 -2.899 39 - 1.671 39
TABLE 3.5
Without dielectric
-2.2829 38.401
- 15.974 -3.2263 38.152 - 17.861 26.01 7
(PW) 23.345
- 8.9057
-2.1907
- 1.9178 23.615
-8.9057 -2.9018 23.345 10.331 17.577
-
Effective dielectric
constant, E; 1.634 1.794 1.042 1.06 1 1.626 1.794 1.112 1.634 1.729 1.480
given in Table 3.5. Once again, only the upper diagonal elements are shown because of the symmetry of C. The rightmost column shows the egectiue dielectric constant which is the ratio of the per-unit-length capacitances with and without the dielectric.
LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION
111
TABLE 3.6 The Transmission-Line Inductances for the Five-Wite Ribbon Cable Computed Exactly and Using the Wide-Separation Approximations
Entry L, 1 LI 2
Ll 3 Ll4
4522 L23 L24
L3 3 L34 L44
Exact ()IH/m) 0.748 34 0.507 11 0.455 27 0.432 95 1,0132 0.7 19 84 0.645 69 1.1738 0.858 42 1.2914
Wide separation approx ()IH/m) 0.758 85 0.51805 0.460 52 0.436 96 1.036 1 0.737 78 0.656 68 1.198 3 0.87641 1.3134
Percent error 1.40 2.16 1.15 0.93 2.26 2.49 1.70 2.09 2.10 1.71
The per-unit-length inductance matrix, L, can be computed from the inverse of the capacitance matrix with the dielectric insulations removed, C,, as L = poeoC;'. Using the above computed results we obtain Table 3.6. The wideseparation approximations were computed from (3.63) using the FORTRAN program WIDESEP.FOR described in Appendix A as
112
THE PER-UNIT-LENGTH PARAMETERS
TABLE 3.7 'The Generalized Capacitances for the Threewire Ribbon Cable With and Without the Dielectric Insulations
With dielectric Entry Voo
Cg,, VOl VI 1 VI 1 Vll
Without dielectric (pF/m) 17.6900 10.5205 4.225 44 22.9694 10.5205 17.6901
(PF/m)
26.214 8
-18.0249 -5.033 25
-
37.8189
- 18.024 9
26.214 8
Observe that the wide-separation approximations are within some 2% of the exact results even though the ratio of adjacent wire spacing to wire radius is d/r, = 6.67.Consequently, the entries in the inductance matrix can be reliably computed with this less computationally expensive method. In order to gauge the effect of neighboring wires on these results and to obtain the capacitance and inductance matrices to be used in a later example, consider a three-wire ribbon cable. The wire separations, radii, insulation thicknesses and type are identical to the five-wire case. The exact generalized capacitance matrix is again computed with the RIBBON.FOR computer program described in Appendix A and the entries are given in Table 3.7.The per-unit-length transmission-line-capacitancematrices, C and Co,are given in Table 3.8. The per-unit-length inductances computed exactly as L = poeOC; and using the wide-separation approximations are given in Table 3.9. The wide-separation approximations are again computed from (3.63) as
I,, = li nI):( 112 = 2w InfA) rw lZ2 = - In Fo li
C:)
-
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
113
TABLE 3.8 The Transmission-Line Capacitancesfor the Three-Wire Ribbon Cable With and Without the Insulation Dielectrics
With dielectric Entry
c1I c12
Cl2
TABLE 3.9
(PFh)
- 18.716 37.432
24.982
Without dielectric (PFh) 22.494
- 11.247
16.581
Effective dielectric constant, e; 1.664 1.664 1.507
The Transmission-Line Inductances for the Three-Wire Ribbon Cable Computed Exactly and Using the Wide-Separation Approximations ~~
Entry LI I Ll2
L,,
Exact
Wide separation
(PH/m)
approx ( W m )
0.748 50 0.507 70 1.0154
0.758 85 0.51805 1.036 1
Percent error 1.38 2.04 2.04
Once again, the wide-separation approximations give results for the entries in the per-unit-length inductance matrix that are within some 2% of the exact values computed from L = pO8,C; Figure 3.26 illustrates the convergence of the method for the three-wire ribbon cable. The per-unit-length inductances and per-unit-length capacitances are plotted vs. the number of Fourier coefficients around the wire and dielectric boundaries in Fig. 3.26(a) and 3.26(b), respectively. Observe that the inductances converge to accurate values for only two Fourier coefficients, whereas the capacitances require of the order of three or four Fourier coefficients for convergence.
3.3 MULTICONDUCTOR LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
Determining the entries in the per-unit-length parameter matrices L, C, and C for conductors of rectangular cross section is the same as for conductors of circular cross section (wires)-the solution of Laplace’s or Poisson’s equation in the two-dimensional transverse plane, e.g.,
(3.91)
114
THE PER-UNIT-LENGTH PARAMETERS
.!
n
k
Y
Convergence of the pet-unit length parameters of the five-wire ribbon cable versus number of Fourier expansion coeficients: (a) inductances, (b) capacitances. FIGURE 3.26
There are various methods for solving this equation. Unless the problem boundaries fit some coordinate system, the usual solution methods determine an approximate solution using various numerical techniques that we will discuss in this section.
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
115
Y
(b)
FIGURE 3.27 Illustrations of the solution of Poisson’s equation: (a) in three dimensions and (b) in two dimensions.
3.3.1
Method of Moments (MOM) Technique,
Method of moments (MOM)techniques essentially solve integral equations where the unknown is in the integrand. An example is the integral form of Poisson’s equation CA.1):
(3.92)
where a charge distribution p is distributed throughout some volume v as illustrated in Fig. 3.27(a). Ordinarily we know or prescribe the potential at points in the region (for example, on perfectly conducting bodies) and wish to find the charge distribution that produces it. Thus we need to solve an integral equation for the integrand [A.1,4-7]. The problems of interest here are perfect conductors and/or dielectric bodies in the two-dimensional plane which are infinite in length (in the z direction) and have some unknown surface charge density, p(x, y ) C/mz, per unit of length in the z direction residing on their surfaces as illustrated in Fig. 3.27(b). In this case, the integral form of Poisson’s equation in (3.92) cannot be used to determine the potential distribution of this infinite length charge distribution since the structure extends to infinity in the z direction and we must find alternate methods. A very common way of doing this is to approximate the charge distribution around the two-dimensional conductor periphery as filaments of charge and use the basic problem of the potential of an infinitesimal line charge that was developed in Section 3,2.1.2. This forms the basis for numerical techniques that are used to analyze these two-dimensional structures of infinite length for determining the per-unit-length parameters c4-71. Thus
116
THE PER-UNIT-LENGTH PARAMETERS
we initially solve the problem of the potential of an infinitely long filament of charge carrying a per-unit-length charge (per unit length in the z direction in C/m2) which is uniformly distributed in the z direction. The potential at a point is the sum of the potentials of each line charge that makes up the desired charge distribution around the conductor periphery. The potential in this case is only meaningful with respect to the potential of a reference point in the twodimensional plane because the structure is infinite in length in the z direction. Again, as for the case of ribbon cables, it can be readily shown that we may omit the reference point and its potential so long as the system under consideration is charge neutral [CS, C.61. In order to illustrate the general method, consider a system of (n + 1) perfect conductors each having a prescribed potential 6,with i = 0, 1,. . ,n. In order to determine the potential distribution in the two-dimensional plane, we represent the per-unit-length charge distribution over the i-th conductor as a linear combination of NIbasisfunctions as in the case of ribbon cables considered earlier: (3.93) PI = a l l P l l + aL?P12 + a13P13 + ' * *
.
The Pik basis functions will be prescribed and the unknown coefficients ai&are to be determined to satisfy the boundary condition that the potential over the i-th conductor is The potential at a point due to this representation will be a linear combination of the charge expansion functions as
(3.94) Each coefficient, K l k , is determined as the contribution to the potential due to each basis function alone: Klk
= 6, 1
-
-
= 1, ail ,...,at& I,ma + I....,am, 0
(3.95)
As in the case of a ribbon cable considered previously, there are many possible forms for the expansion or basis functions, Entire domain expansions seek to represent the charge distribution over the conductor surface as functions each of which are nonzero over the entire contour of the surface in the same manner as a Fourier series represents a time-domain function using basis functionsdefined over the time interval encompassingone complete period. This was the technique used earlier to expand the charge distributions around the wire and dielectric insulation peripheries of ribbon cables. Subdomain expunsions seek to represent the charge distribution over discrete segments of the contour CC.1, C.31. Each of the expansion basis functions, P l k , is defined over the discrete segments of the contour, and is zero over the other segments. We will concentrate on the subdomain expansion method. There are many ways
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
r-------1
117
r-------1 am
a11
wlk
(b) FIGURE 3.28
Illustration of the pulse expansion ofa charge distribution on a flat strip.
of choosing the expansion functions over the segments. One of the simplest ways is to represent the charge distribution as a “staircase function” where the charge distribution is constant over the segments of the contour: Pu =
{;;;:
(3.96)
This is referred to as the pulse expansion method and the charge distribution is assumed constant over the segments Cik. Figure 3.28(a) illustrates this approximation of a charge distribution over the surface of an infinitesimally thin, perfectly conducting plate that extends to infinity in the z direction, Thus the charge distribution we are representing has variation in the x, y plane and is uniformly distributed in the z direction. The units of this charge distribution are therefore C/mz. Once the expansion functions are chosen we need to generate a set of linearly independent equations in terms of the expansion coeficients, afk,which can be solved for them thus generating via (3.94) an approximation to the charge distribution over that surface. The total charge (per unit length) in the z direction) in C/m is obtained by summing the charges of the subsections of that conductor: 41 =
Nc alk k= 1
Plk
dc
(3.97)
118
THE PER-UNIT-T-LENCTHPARAMETERS
In the case of the pulse expansion method, this simplifies to 41 =
NI
(3.98)
OC1kWlk
k=l
where w1k is the width of the k-th segment of the i-th conductor. From these results we can compute the per-unit-length capacitances. There are many ways of generating the required equations. One rather simple method is the method of point matching. For illustration consider a system of (n 1) conductors each having a prescribed potential of 4, for i = 0, n. We next enforce the potential of each conductor, r$l, due to all charge distributions in the system to be the potential of that conductor at the center of the subsection of the conductor. This is illustrated for the pulse expansion method in Fig. 3.28(b). A typical resulting equation is of the form
...,
+
Choosing a total of (3.100)
points on the conductors gives the following set of N equations in terms of the expansion coefficients: (3.101 a)
@=;A
or, in expanded form,
...
e . .
...
...
...
...
[i]
-
A0
A]
-
(3.101b)
An
The vector of potentials at the matchpoints on the i-th conductor is denoted as
@1=
(3.101c)
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
119
and the vector of expansion coefficients of the charge distribution on the i-th conductor is denoted as
(3.101d)
Inverting (3.101b) gives A = D-'@
(3.102a)
or, in expanded form,
(3.102b)
Once the expansion coefficients are obtained from (3.102), the total charge (per unit of length in the z direction in C/m) can be obtained from (3.97). The generalized capacitance matrix, %,' described in Section 3.1.4 can then be obtained. In the case of point matching and pulse expansion functions, as with flat conductors, the entries in the generalized capacitance matrix can be directly obtained from (3.102) as
where wIkis the width of the k-th subsection of the i-th conductor. If the widths of all the conductor segments are chosen to be w, then the elements of the generalized capacitance matrix simplify to WIJ =
BIJ
(3.103b)
120
THE PER-UNIT-LENGTH PARAMETERS
't
FIGURE 3.29 Calculation of the potential due to a constant charge distribution on a flat strip.
These simple results are due to the fact that a submatrix of (3.102b) is
(3.104a)
or, in expanded form, 'tk
{,zk
BiJ}#J
(3.104b)
Thus the basic subproblem is the integration in (3.95). In order to illustrate this, consider the infinitesimally thin conducting strip of width w and infinite length supporting a charge distribution p C/m2 that is constant along the strip cross section as shown in Fig. 3.29. If we treat this as an array of wire filaments each of which bears a charge per unit of filament length of p d x C/m, then we may determine the potential at a point as the sum of the potentials of these filaments again using the basic result for the potential of a filament given in
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
127
(3.30) or (3.71) [ 4 , 7 ] :
(3.105)
These integrals are evaluated using [8]. This may be simplified somewhat if we denote the distances from the edges of the strip as Rt and R' and the angles as 8' and 8- as shown in Fig. 3.29. In terms of these (3.105) becomes $(w,
xp,
Yp)=
[xP 2ns
h(z) - In(RtR-)
+,w - yp(flt - e-)
1
(3.106)
In the case where the field or observation point lies at the midpoint of the strip in question, the integral in (3.105) is singular but integrable. The result is [8]
(3.107) The electric field due to this charge distribution will be needed for problems that involve dielectric interfaces. The electric field can be computed from this result as
(3.108) where (3.109a)
Er=
-2RE [e' - e-)
(3.109b)
As an illustration of this method consider the rectangular conducting box shown in Fig. 3.30. The four walls are insulated from one another and are maintained at potentials of = 0, dz = lOV, $3 = 20V, 44 = 30V. Suppose
112
THE PER-UNIT-LENGTH PARAMETERS
61= 1ov
i
4.
0
FIGURE 3.30 A two-dimensional problem for demonstrationof the solution of Laplace's equation via the pulse expansion-point matching method. TABLE 3.10 Comparison of MOM and Exact Results for the Potential in Fig. 3.30
MOM
Exact
we divide the two vertical conductors into four segments each and the horizontal members into three segments each. Using pulse expansion functions for each segment and point matching gives fourteen equations in fourteen unknowns (the levels of the assumed constant charge distributions over each segment). Using the above results Table 3.10 gives the potentials at the six interior points. The exact results were obtained via a direct solution of Laplace's equation using separation of variables. In terms of the general parameters denoted in Fig. 3.30, the solution is [9]
+-nn sinh(nnu/6) sin(nnx'a) [ c ) ~ s i n h ( 7 ) +
&I
-
sinht!! (b }])y
-
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
123
where a = 3, b = 4, and 4, = 0, #2 3: lOV, t$3 = 20V, 4, 30V. The solution using finite difference and finite element methods will be given in a subsequent subsection. This method can be readily extended to systems that contain dielectric bodies as in the case of printed circuit boards by similarly expanding the bound charge on those surfaces as above. The solution technique follows that for the ribbon cable. The pulse expansion-point matching technique described above is particularly simple to implement in a digital computer program. Achieving convergence generally requires a rather large computational expense since the conductor subsections must be chosen sufficiently small to give an accurate representation of the charge distributions. This is particularly true in the case of lands on PCB’s where the charge distribution peaks at the edges of each land. There are other choices of expansion functions such as triangles or piecewise sinusoidal functions but the programming complexity also increases. Another way of improving convergence is to use another method of generating the required number of equations other than point matching. A reasonably simple but effective method is the Galerkin method. This is closely related to the Rayleigh-Ritz variational method of minimizing a functional [6]. Although the following explanation overlooks some of the finer points of the method, it illustrates the computational details. Consider Fig, 3.31(a) showing contours
wlk
L
(b)
FIGURE 3.31 Illustration of the general determination of potential via the Galerkin
method.
124
THE PER-UNIT-LENGTH PARAMETERS
on the i-th andj-th conductors of the system which is infinite in extent in the z direction. The potential at some point on thej-th conductor due to the charge expansion basis functions of the i-th conductor is a function of the distance rrj between the differential segments of each conductor: (3.1 10) Multiply this by the m-th basis function of the charge expansion of thej-th conductor and integrate over c j :
Likewise add in the contributions from the charge distributions of the other conductors to this equation to give (3.112) This method amounts to “weighting” the potential over the conductor rather than matching it at discrete points on the conductor. If pulse expansion functions are used, the method averages the potential over the conductor. Repeating this for the other expansion functions and all conductors gives the required number of equations to be solved for the charge expansion coefficients. In the case of pulse expansion functions and flat conductor segments of width wlk,this simplifies to N
NI K;kalk
wjk$J=
(3.113)
110 k = 1
Thus the form of the equations in (3.101b) is the same but the entries in D are changed and the entries in CP are multiplied by the segment widths, wjk. It is particularly interesting to observe that the wide-separation approximations developed for widely spaced wires in Section 3.2.3 can be shown to be equivalent to using the Galerkin method when only one expansion function (the constant one) is used for the charge distribution about each wire. This observation gives added credence to those seemingly crude approximations, 3.3.1.1 Applications to Printed Circuit Boards The above MOM method can be adapted to the computation of the per-unit-length capacitances of conductors
with rectangular cross sections as occur on printed circuit boards (PCB’s). Consider a typical PCB shown in Fig. 3.32(a) having infinitesimally thin conducting lands on the surface of a dielectric board of thickness t and relative dielectric constant of E,. The widths of the lands are denoted as wi and the
, ,
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION 5
WI
I
125
I
I
I
r0
(b)
FIGURE 3.32 A printed circuit board (PCB)for illustration of the determination of the per-unit-lengthcapacitances.
edge-to-edge separations are denoted as SfJe A direct approach would be to subsection each land into 4 segments of length wIL.The charge on each subsection could be represented using the pulse expansion method as being constant over that segment with unknown level of ai&.We could similarly subsection the surface of the dielectric and represent the bound charge on that surface with pulse expansions. A more direct way would be to imbed the dielectric in the basic Green'sfunction. We will choose to do this. Thus the problem becomes one of subsectioning the conductors immersed in free space as illustrated in Fig. 3.32(b). First we consider solving the problem with the board removed as in Fig. 3.32(b) then we will consider adding the board. Consider the subproblem of a strip of width w representing one of the subsections of a land shown in Fig. 3.33(a). We need to find the potential at a point a distance d from the strip center and in the plane of the strip. This basic subproblem was solved earlier and the results of (3.105) and (3.107)specialized to this case are
(3.1 14a)
= dselr(w)
W +[3(2D - 1)In(2D - 1) - t(2D + 1)In(2D + I)] 2n$
126
THE PER-UNIT-LENGTH PARAMETERS
4
-w,
X
T d (8)
y4
J
d
-
c-...lr
X
w/
WI
I X
(b)
FIGURE 333 Illustration of the determination of the potential due to a constant charge distribution on a flat strip via (a) point matching and (b) the Galerkin method.
The mutual result in (3.1 14b) is written in terms of the self term in (3.1 14a) and the ratio of the separation to subsection width:
D=-
d
(3.1 14c)
W
It is interesting to note that for the case of pulse expansions, the total per-unit-length charge on a strip is simply the strip width, Le., q
-
(1 C/m') x w = w
Thus if only one subdivision is used per land, the terms in (3.1 14a) and (3.1 14b) are the entries in the inuerse of the generalized capacitance matrix. In the case that the lands are divided into more than one subsection, these terms represent the entries in the inverse of a global generalized capacitance matrix as though each land consisted of several unconnected sublands. The Galerkin method given by (3.111) obtains these basic subproblems as illustrated in Fig. 3.33(b) and uses (3.114b): (3.115) continued
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
127
(3.1 16a)
(3.1 16b)
for the mutual terms. If we specialize this to strips of equal width, wt = w, = w, these results simplify to 1 (3.11 7a) W+,,,,(W) = -[3w2 - w2 In(w)J 2x8,
1 w ~ ( wd ,) = -[3w2 2RE0
+ d2 In@) - i(d - w ) In(d ~ - w) - 3(d + w ) In(d ~ + w)]
(3.117b) Dividing both sides by the common subsection width, w, these results can again be written in terms of the self term and the ratio of the subsection separation
128
THE PER-UNIT-LENGTH PARAMETERS
FIGURE 3.34 Illustration of the method of images for a point charge above an infinite dielectric half-space.
to width ratio, D, given in (3.114~)as (3.117c) @(w, d ) = r # r e l f ( ~ )
W + 2[D2 In@) - l(D - 1)2 In(D - 1) 7c.3,
(3.117d)
Next consider incorporating the dielectric board into these basic results. The first problem that needs to be solved is that of an infinite (in the z direction) line charge of q C/m situated a height h above the plane interface between two dielectric media as shown in Fig. 3.34. The upper half-space has free space permittivity e, and the lower half-space has permittivity e = ereo. This classic problem allows one to compute the potential in each region by images in the same fashion as though the lower region were a perfect conductor 13, lo]. The solution can be obtained by visualizing lines or tubes of electric flux, $, from the line charge. Electric flux lites through some open surface s are related to the electric flux density vector, 9 e$, as I) = J’j,9 . d & Consider one such flux line emanating from the line charge which is incident on the interface at some angle 8,.Some of this flux passes through the interface as (1 + k)@ while some is reflected at an angle 8, as -k$. Snell’s law shows that 8,= 8,. The boundary conditions at the interface require that the normal components of the electric flux density be continuous, i.e.,
-
I) sin 0, + k$ sin 8, = (1 + k)$ sin 0,
(3.118a)
Similarly, the tangential components of the electric field must be continuous
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
129
across the interface giving (3.1 18b) Recalling that 6, = 0, gives &=-
and
E,
e,
-1
+1
(3.1 19a)
a = ( l + & ) = - 2Er e, 1
+
(3.1 19b)
and (as will be needed later) (1
az - &2)= -
(3.1 19c)
8,
Thus the potential in the upper half-space ( y > 0) is as though it were due to the original charge, q, at the original height h and an image charge, -&q, with the dielectric removed and at a distance h below the interface: d)+(W) = --
4ne,
ln[x2 + ( y
-
kq In[x2 + ( y + h)'] + 4ne0 -
(3.120a)
The potential below the interface (y > 0) is due to a line charge (1 + k)q located a height h above the interface with the upper free-space region replaced by the dielectric: (3.120b) The problem now of interest is a line charge on the surface of a dielectric slab (the PCB)of relative permittivity and thickness t. First consider the more general problem of n line charge q located a height h above the dielectric slab. Using the above results we may construct the diagram of Fig. 3.35. These results follow similar lines as in optics. Observe that the potential in each of the three regions appear due to line charge images that produce the solid flux tubes in that region, and the appropriate dielectric constant to be used in the potential expression is that of the region. Now consider the problem at hand of a line charge on the surface of a dielectric slab of relative permittivity e, and thickness t as shown in Fig. 3.36(a). We wish to find the potential at a point on the board a distance d from the line charge. Specializing the results of Fig. 3.35 for h = 0 gives the images shown in Fig. 3.36(b). The potential then takes the form of a
130
THE PER-UNIT-LENGTH PARAMETERS
flCURE 3.35
Images of a point charge above a dielectric slab of finite thickness.
series:
This series converges rapidly since k < 1. Applying this result for a line charge to an infinitesimally thin strip on a dielectric slab by representing the charge distribution (pulse expansion function) as a set of 1 C/m line charges as shown in Fig. 3.37(a) requires performing the
LINES HAVING CONDUCTORS
k(t
- k')qO
80
-
k'( 1
OF RECTANGULAR CROSS SECTION
131
i2,
k2)q4
- k2)9 J' 0
FIGURE 3.36 Illustration of the replacement of a dielectric slab of finite thickness with images to be used in modeling a PCB.
following integral : (3.122)
The result is &,,lf(w)=
2118, e,
[w
1 a2 - w 1n@] + - k('"-l) 21280 8, n = i
(3.123a)
132
THE PER-UNIT-LENGTH PARAMETERS
FIGURE 3.37 Illustration of the determination of the per-unit-length capacitances for a PCB by (a) point matching and (b) the Galerkin method.
and
1 + (d - ;)ln(d - ):
+(w, d ) = 2ae0 e,E [ w
- (d
+ :)ln(d + :)]
(3.123b)
where a = 2nt. These results can be simplified and written in terms of the potentials with the dielectric removed as
4..,r(w) = &eIE(W) x
{i
-2m0$
ln[l
(3.123~)
pa-1)
et1
+ (4nq2] + 4nT tan"
-
( 4 9
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
133
and (3.123d)
x
{ -f(2D - l)ln(2D - 1) + i(2D + l)ln(2D + 1) + i(2D - 1) x InC(2D
+ -(4nT) 2
- 1)' + (4nr)'] - &2D + l)ln[(2D + 1)2 + (4nT)'I
[
tan-'
cnil)
- - tan-(%)])
where &,,(w) is the self term with the dielectric removed given in (3.1 14a), and $"(w, d ) is the mutual term with the dielectric removed given in (3.114b). The results have been written in terms of the ratios of subsection to subsection
width, D,and board thickness to land width, T: d D=-
(3.123e)
W
and
T = -t W
(3.123f)
Additionally the notation (3.1238) denotes the effective dielectric constant as the board thickness becomes infinite, t 4 00, such that it fills the lower half-space. This notion of an effective dielectric constant for this case is valid since half the electric field lines would exist in air and the other in the infinite half-space occupied by the board. Evidently the summation terms in (3.123~)and (3.123d) give the effect of the board. These results are used in the FORTRAN program PCB.FOR described in Appendix A to compute the entries in the per-unit-length capacitance matrix C of a PCB. The per-unit-length inductance matrix L is computed with the board removed from the basic relationship derived earlier:
L = po$C,-'
(3.124)
where Cois the per-unit-length capacitance matrix with the board removed. The Galerkin solution (specialized to equal width strips) is similarly obtained
134
THE PER-UNIT-LENGTH PARAMETERS
from Fig. 3.37(b), (3.111)yand (3.123b) as the basic integral l a 2ma E,
w ~ ( wd, ) = --
(3.125)
(3.126a)
and w ~ ( wd,) =
2
2x8,
[!2 -A(!- 2
Y:(
In(w) + w
(3.126b)
In(:)
1)-:('+
l>.In(:-
l)'ln(:+
I)]
w2 a2 +--2x80 E, I x - ln(u) + -[(->' 1 d - (;>']~n[-)' + 1 1 pn-1)
{i
n-
-![(! 4
- A[!( 4
2
w
-1 7w
(97"[(T7
+ 1 7-(
d/w
~
-1
+11
+
+
~ d/w ] ~1 n 1 [1
(
~
~
w continued
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
135
where, again, a = 2nt. Dividing the above by the common subsection width, w, these results can again be written in terms of the potential with the-board removed and the ratios of separation to width, D, given in (3.123e) and board thickness to width, T, given in (3.123f) as (3.1 264
and (3.126d)
- 1)' In(D - 1) + )(D + 1)' In(D + 1) + i [ D z - (2nT)']1n[D2 + (2nT)I - a[(D - 1)' - (2nT)']ln[(D - 1)' + (2t1T)~] - +[(D + - (2nT)2]ln[(D + + (2nT)']
x { - D z In@) t )(D
- ( D + l)tan-l(-)]} D + 1 2nT where +,",r(w) is the self term with the board removed given in (3.117~)and +"(w, d) is the mutual term with the board removed given by (3.1 17d). These
results are used to compute the capacitances of PCB's in the FORTRAN program PCBGAL.FOR described in Appendix A. 3.3.1.2 Computed Results: Printed Circuit Boards As an example consider the three-conductor PCB shown in Fig. 3.38 consisting of three conductors of equal width w and identical edge-to-edge separations s. The following computed results are obtained with the FORTRAN programs PCB.FOR (pulse expansion
136
THE PER-UNIT-LENGTH PARAMFTERS
FIGURE 3.38 A PCB consisting of identical conductors with identical separations for computation of numerical results.
functions and point matching) and PCBGAL.FOR (pulse expansion functions and Galerkin). The results will be computed for typical board parameters: e, = 4.7 (glass epoxy), t = 47 mils, w = s = 15 mils. Choosing the leftmost conductor as the reference conductor, Fig. 3.39(a) compares the elements of the per-unit-length transmission-line-inductancematrix, L, and Fig. 3.39(b) compares the elements of the per-unit-length transmission-line-capacitancematrix, C,for the two methods for various numbers of land subdivisions. The Galerkin method converges rather fast and much faster initially than point matching. Figure 3.40 compares the entries in C and C, for various numbers of land subdivisions. Figure 3.41(a) shows the ratios of the entries in C and C, for various board thickness using 50 divisions per land computed using the Galerkin method. Observe that these approach an effective dielectric constant that is the average of that of the board and free space: e,1 = -= 2.85
2
This would be the effective dielectric constant if the board occupied the infinite half-space since half the electric field lines would reside in free space and the other half in the board. Figure 3.41(b) shows these ratios as a function of the ratio of separation to land width using the Galerkin method and 50 divisions per land. Here we see that for wider separations, the effective dielectric constants are substantially lower than the average assuming the board was infinitely thick. For wide separations, more of the electric field lines exit the bottom of the board and are more important than for closely spaced lands. One might be tempted to obtain wide-separation approximations by approximating the charge as being uniformly distributed over each land, which essentially means using pulse expansions and only one division per land. Figure 3.42 shows the results for the inductances and capacitances versus the ratio of separation to land width. These inductances and capacitances give results that are within 10% for d/w > 5. However, as we have seen in Fig. 3.41(b), the finite thickness of the dielectric board has more of an effect in the case of wide separations so that an effective dielectric constant is not so easy to obtain. And finally we will compute the entries in C and L for a PCB that will be used in later crosstalk analyses: e, = 4.7 (glass epoxy), t = 47 mils, w = 15 mils, and s 3:45 mils. This separation is such that exactly three lands could be placed
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
1.3
---;
4 l1o. 12 : &-v
) .
( .
137
--C. LPMll
. # * . O . . . La11
.-.*-..LPMl2 ---e-. LO12
0.7 -
-
0.6 -
. l3-b*-6t.,
(E,
60 -
h
6
50
**.I.*
'
0.5L
-a-
-
4.7, width
-
.-. *....
----.--------.I
I
-
Point Matching va, Galerkin reparation 15 mils, thickness
-
4
LPM22 LG22
..'......---.-.-I-
I
1
-+-
.I
I
-
47 mils) 1
CPMll ..+ .o,,. *
I",
Y
CPMlZ COl2 -e- CPM22 .-..L..... c 0 2 2 ---E)--
201
0
'
I
5
I
10
I
15
,
1
20
25
I
38
Number of division, (b)
-
-
FiGURE 3.39 Illustration of the per-unit-length (a) inductances and (b) capacitances via
point matching and via the Galerkin method for various numbers of divisions of each land. E, = 4.7, width separation = 15 mils, thickness 47 mils.
138
THE PER-UNIT-LENGTH PARAMETERS
J Number of divisions FIGURE 3.40
The capacitances with and without the dielectric board versus the number
of divisions per land. e, = 4.7, width = separation = 15 mils, thickness = 47 mils.
between any two adjacent lands. The results for the. pulse expansion-point matching method for 50 divisions per land are (PCB,FOR) 1.105 13
C=[
-
40.599 1 20.299 6
]PH/m
0.690602
L=[ 0.690602 1.38120
-20.299
29.738 o
The results for the pulse expansion-Galerkin method for 50 divisions per land are (PCBGAL.FOR) 1.104 18
C=[
40.628 0 -20.3140
0,690094 1.380 19
-20.3 14 01 pF/m 29.7632
These compare with the results of a three-dimensional program for finite-width lands of length 10 inches (25.4 cm) and dividing those results by the length of the lands CA.31: 1.034 0 0.653 35 0.653 35 1.306 7 41.765 5 -20.882 7 C=[ -20.882 7 30.502 2
L=[
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
3 , 0 - ' ,
,
-
2.8
1
t
.
..,
r
,
.
' ,.......././-
2.6 2.4 -
/
i
2.2-
I
/'
'
**-.._..-----
*.--*
.I'
r
.
.
u
'
r
.
...................................
,e.-.--.
I
139
.
#
,,'
,'
,e'
I :
'
2.0-
1 :
1: 1: 1.6 1: 1.4
.........
1.8'
(e,
+ 1)12
C12lCO1 2
'
-//
1.2 '
1.0
3.0
'
$-*.
2.6-
-
2.2-
; 2.0 1.8
.
-
*-*.
*.
'
1
I
- .-.*-.-*.-
2.8
2.4
I
I
...
1
.
I
7
1
.
1
'
I
1
.
.
I
.
I
' .
.
I
,
.
1
I
.
.
-..* --.
-2-
-......... .--.*.* .-._ .................. .- ---.__.--.-_.5 -*.-: ......... ............""--.-..-..~
%.
-.-a*-
1*6;
1.4 1.2
(B, + 1)/2 ......... c1 llCOl1 ._._._..Cl2lCOl2
....... C22lC022
1 .o
,
,
. , .
,
.
-
FIGURE 3.41 Illustration of the effective dielectric constant of the PCB versus (a) board thickness (e, = 4.7, width 15 mils, separation = 15 mils); and (b) the ratio of land separation to width ratio (e, = 4.7, width = 15 mils, board thickness 47 mils). 5
This illustrates that the computations for infinite-length lands (two-dimensional) give adequate results for the per-unit-length parameters for finite-length lands if the lands are much longer than the widths and separations, i.e., the fringing of the electric field at the ends of the lands has negligible effect.
140
THE PER-UNIT-LENGTH PARAMETERS
3.0[,-.
(a,
.
(a,
.
-
-
ConvergonceVI. Land Separation 4.7, land width 15 mill, board thlcknesn
. . . . .
-
'
(
'
I
'
t
Convorpnco vi. Land Seprratbn 4.7, land wldtb 15 mlli, board thickneii I
'
I
.
I
.
I
-
.
I
.
-
47 mile) '
I
'
I
4
47 mlli)
1
.
I
'
-aCll(50divlirnd) * * * . o . *C . ll(1 divlhd)
-.+-
C22(SOdivlland)
-.c-
C12(SOdIvAand)
-..EF-. C22(1 dlv/land)
.- .P-....C12 (l*dlv/lyd)
1
3
s
1
9
11
12
14
16
18
20
dlw
(b)
RCURE 3.42 Illustration of the convergenceof(a) inductanceand (b) capacitance versus the ratio ofland separation to width. 9 L: 4.7, land width = 15 mils, board thickness = 47 mils. 3.3.2
Finite Difference Techniques
Recall that the entries in the per-unit-length inductance, capacitance and conductance matrices are determined as a static (dc) solution for the fields in the transverse (x-y) plane. Essentially, the transverse fields are such that the potential in the space surrounding the conductors, &c, y), again satisfies
LfNES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
141
FIGURE 3.43 The finite-differencegrid.
Laplace’s equation in the transverse plane: (3.127)
Finite difference techniques approximate these spatial partial derivatives in a discrete fashion in the space surrounding the conductors. For example, consider the closed region shown in Fig. 3.43. The space is gridded into cells of length and width h. First-order approximations to the first partial derivatives at the interior cell points a, b, c, d are [A.1,6] (3.128a) (3.128 b)
*I:;
-
=-
(3.128~)
42
Id
84 =-4 0 - 4 4 aY h
(3.128d)
142
THE PER-UNIT-LENGTH PARAMETERS
The second-order derivatives are similarly approximated using these results as: (3.129a)
(3.129b)
This amounts to a central diflerence expression for the partial derivatives [6]. Substituting these results into (3.127) gives a discrete approximation to Laplace’s equation: (3.130)
or
Thus the potential at a point is the average of the potentials of the surrounding 4 points in the mesh. Equation (3.131) is to be satisfied at all mesh points. Typically this is accomplished by prescribing the potentials of the mesh points on the conductor surfaces, initially prescribing zero potential to the interior points then recursively applying (3.131) at all the interior points until the change is less than some predetermined amount at which the iteration is terminated. An example of the application of the method is shown in Fig. 3.44 which was solved earlier using a MOM method and a direct solution of Laplace’s equation with those results given in Table 3.10. A rectangular box having the four conducting walls at different potentials is shown. The potentials of the interior mesh points obtained iteratively are shown. Observe that, even with this relatively course mesh, the potentials of the mesh points converge rather rapidly. This method could also be solved in a direct fashion rather than iteratively. Enforcement of the potential via (3.131) at the six interior mesh points gives
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
TABLE 3.11
Comparison of the Finite Difference and Exact Results for the Potential in Fig. 3.30
9, 4b 9c 4d cbe
cbr
143
1
Direct met hod
Iteration
16.44 21.66 14.10 20.19 9.77 14.99
16.4 1 21.63 14.07 20.16 9.76 14.98
4 -1
-1 4
-1
0
16.478 4 21.8499 14.1575 20.492 4 9.609 42 14.981 0
0 0 -1
0 -1
4 -1 0 -1 -1 4 0 -1 0 -1 0 4 -1 0 0 0 0 -1 -1
-1
Exact
0
(3.132)
30
Solving this gives the potentials of the interior mesh points. Table 3.11 compares the results of the direct method with the iteration of Fig. 3.44 and with exact results obtained from an analytical solution of Laplace’s equation given previously. The solution for the potentials via this method is only one part of the process of determining the per-unit-length generalized capacitance matrix. In order to compute the generalized capacitance matrix, we need to determine the total charge on the conductors (per unit of line length). To implement this calculation, recall Gauss’ Jaw: (3.133)
which provides that the charge enclosed by a surface is equal to the integral of the normal component of the electric flux density vector over that surface. For a linear, homogeneous, isotropic medium, 84 a, SB = E 8 = E 4
-
an
(3.134)
where io,, is the unit vector normal to the surface. In order to apply this to the problem of computing the charge on the surface of a conductor, consider Fig. 3.45.Applying (3.133) and (3.134) to a strip along the conductor surface between the conductor and the first row of mesh points just off the surface gives (3.135)
144
THE PER-UNIT-LENGTH PARAMETERS
10
30
FIGURE 3.44 Example for illustration of the finite difference method,
Conductor I
FIGURE 3.45
conductor.
Use of the finite difference method in determining surface charge on a
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
. FIGURE 3.46
hr
__ @4
--
149
hh
CP
Illustration of the implementation of the finite difference method for
multiple dielectrics.
Using the potentials computed at the interior mesh points, we can obtain the charge on the surface by applying (3.135) in discrete form as
where h, denotes the “vertical” length of the mesh perpendicular to the conductor surface, and hh denotes the “horizontal” length of the mesh parallel to the conductor surface. Once the charges on the conductors are obtained in this fashion, the generalized capacitance matrix can be formed and from it the per-unit-length capacitances can be obtained. Dielectric inhomogeneities can be handled in a similar fashion with this method. Gauss’ law requires that, in the absence of any free charge intentionally placed at an interface between two dielectrics, there can be no net charge on the surface. Consider the interface between two dielectrics shown in Fig. 3.46 where a mesh has been assigned at the boundary. Applying Gauss’ law to the surface surrounding the center point of the mesh shown as a dashed line gives (3.1 37)
146
THE PER-UNIT-LENCTH PARAMETERS
FIGURE 3.47 Illustration of (a) the finite element triangular element and (b) its use in representing two-dimensional problems.
Finite difference methods are particularly adapted to closed systems. For open systems where the space extends to infinity in all directions, a method must be employed to terminate the mesh. A rather simple but computationally intensive technique is to extend the mesh to a large but finite distance from the conductors and terminate it in zero potential, To check the sufficiency of this, extend the mesh slightly then recompute the results. 3.3.3
Finite Element Techniques
The third important numerical method for solving (approximately) Laplace’s equation is the finite element method or FEM. The FEM approximates the region and its potential distribution by dividing the region into subregions (finite elements) and representing the potential distribution over that subregion. We will restrict our discussion to the most commonly used finite element, the triangle surface shown in Fig. 3.47(a). Higher-order elements are discussed in [6, lo]. Figure 3.47(b) illustrates the approximation of a two-dimensional region using these triangular elements. The triangular element has three nodes at which the potentials are prescribed (known). The potential distribution over the element is a polynomial approximation in the x, y coordinates. +(x, y ) = a
+ bx f CY
(3.138)
Evaluating this equation at the three nodes gives
(3.139)
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
147
Solving this for the coefficients gives (3.140a) (3.140b) (3.140~) where (3.140d)
r, = (Yz - Y 3 )
r, = (Y3 - V I ) r, = (Yl - Yz)
1
(3.14Oe)
(3.1400
and the area of the element is 1 X I Y1 1 A = - 1 x2 Yz 2 1
x3
(3.141)
Y3
Observe the cyclic ordering of the subscripts in (3.140) as 1 -B 2 3 3. Given the node potentials, the potential at points on the element surface can be found from (3.138) using (3.140) and (3.141). The electric field over the surface of the element is constant: k!7= -V# (3.142)
Observe that when a surface is approximated by these triangular elements as in Fig.3.47(b), the potential is guaranteed to be continuous across the common boundaries between any adjacent elements.
148
THE PER-UNIT-LENGTH PARAMETERS
The key feature in insuring a solution of Laplace's equation by this method is that the solution is such that the total energy in thejield distribution in the region is a minimum [a, 101. The total energy in the system is the sum of the energies of the elements:
w=
5 W"'
(3.143)
Is1
The minimum energy requirement is that the derivatives with respect to all free nodes (nodes where the potential is unknown) are zero: (3.144) The energy of the i-th element is (3.145)
where 4f) is the potential of node k (k = 1,2,3) for the i-th element. This can be written in matrix notation as a quadraticform as (3.146a) where (3.146b)
and superscript t denotes transpose. The matrix
(3.146~)
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
149
3 FIGURE 3.48
An example to illustrate the use of the FEM.
is referred to as the local coejiclent matrix with entries
Now we must assemble the total energy expression for a system which is approximated by finite elements according to (3.143). In order to illustrate this consider the example of a region that is represented by three finite elements shown in Fig. 3.48. The node numbers inside the elements are the local node numbers as used above. The globar node numbers are shown uniquely for the five nodes. Nodes 1,2, and 3 arefree nodes whose potential is to be determined to satisfy Laplace's equation in the region. Nodes 4 and 5 are prescribed nodes whose potentials are fixed (known). The total energy expression can be written in terms of the global nodes as (3.147a)
where the vector of global node potentials is
(3.147b)
150
THE PER-UNIT-LENCTH PARAMETERS
and
cll
cl2
c13
c14
clS
c12
c22
c23
C24
c2S
(3.147~) c14
c24
c34
c44
c45
c25
c35
c45
c5S
This global coesfcient matrix, C, like the local coefficient matrix, can be shown to be symmetric by energy considerations CA.11. The entries in the global coefficient matrix can be assembled quite easily from the local coefficient matrix (whose entries are computed for the isolated elements independent of their future connection) with the following observation. For each element write the global coefficient matrix as
cy; c\y
Cyj 0 0 CyJ 0 0
0 0
0 0
cy4 cyj
cy4 cy4 (3.148a)
0 0 0 0 0 0 0 0 cyj
(3.148 b)
(3.148~)
Assembling these according to (3.147) gives the total energy:
w = W') + W2) + W3)
cY4
0
0
1
LINES HAVING CONDUCTORS OF RECTANGULAR CROSS SECTION
151
A very simple rule for assembling this global coefficient matrix can be developed from this example. The main diagonal terms are the sums of the main diagonal terms of the local coeficient matricesfor those elements that connect to this global node. The ofl-diagonal terms are the sums of the off-diagonal terms of the local coeficient matrices whose sides connect the two global nodes. Observe that the off-diagonal terms will have at most two entries, whereas the main diagonal terms will be the sum of a number of terms equal to the number of elements that share the global node. Now it remains to differentiate this result with respect to the free nodes to insure minimum energy of the system thereby satisfying Laplace's equation. To that end let us number the global nodes by numbering the free nodes first and then numbering the prescribed (fixed potential) nodes last. The above energy expression can then be written in partitioned form as (3. 150)
Differentiatingthis with respect to the free node potentials and setting the result to zero gives the final equations to be solved:
c//+/= - C/P+P
(3.15 1)
Solving this matrix equation for the free node potentials is referred to as the direct method of solving the FEM equations. Observe from the example of Fig. 3.48 that the global coefficient matrix, C, has a number of zero entries. This is quite evident since only those (two) elements that contain the nodes for this entry will contribute a nonzero term. Thus the coefficient matrices in (3.151) will be sparse. Although there exist sparse matrix solution routines which efficiently take advantage of this, FEM problems, particularly large ones, are generally solved more efficiently using the iterative method. This is similar to the iterative method for the finite difference technique in that the free nodes are initially prescribed some starting value (such as zero) and the new values of the free node potentials computed. The process continues until the change in the free node potentials between iteration steps is less than some value. Writing out (3.151) for the example of Fig. 3.48 gives
Writing each free potential in terms of the other free potentials and the
152
THE PER-UNIT-LENGTH PARAMETERS
prescribed potentials gives
where (3.153b)
+P
=
[3
(3.153c)
As a numerical illustration of the method consider the problem of a rectangular region of sides lengths 4 and 3 with prescribed potentials of OV, lOV, 20V, and 30V as shown in Fig. 3.30. This was solved earlier with the MOM technique and the finite difference technique. In order to compare this with the finite difference technique we will again choose six interior nodes and approximate the space between these nodes with twenty-four finite elements as shown in Fig. 3.49. Observe that the potentials at nodes 7, 10, 14, and 17 are not known since these are gaps between the adjacent conductors. We will prescribe these potentials as the average of the adjacent conductor potentials: 5V, 15V, 25V, and 15V. The solution for the six interior node potentials via the direct and the iterative method are given in Table 3.12. The iterative method converged after fifteen iterations. Some of the important features of the FEM are that it can handle irregular boundaries as well as inhomogeneous media. Inhomogeneous media are handled by insuring that the region is subdivided such that each finite element covers (approximately) a homogeneous subregion. The permittivities of the finite elements, 81, are contained in the C$, local node coefficients. TABLE 3.12 Comparison of the Finite Element and Exad Reaulta for the Potential in Fig. 3.30
41 42
43 44
45
46
Direct method
Iteration
Exact
16.438 9 21.656 3 14.099 4 20.186 3 9.772 26 14.989 6
16.438 9 21.656 3 14.0994 20.1863 9.772 24 14.989 6
16.478 4 21.8499 14.157 5 20.492 4 9.609 42 14.9810
MISCELLANEOUS ADDITIONAL TECHNIQUES
't 17
153
20 v
15
16
14
Y 3 4
10 v
30 V
+ X
ov FIGURE 3.49
x=3
Illustration of the FEM applied to a previously solved problem.
Once again, as with the finite difference method, the finite element method is most suited to closed systems. For open systems whose boundaries extend to infinity, an infinite mesh method can be developed [6] or the mesh can be
extended sufficiently far from the main areas of interest and artificially terminated in zero potential thereby forming a closed system, albeit an artificial one. The FEM is a highly versatile method for solving Laplace's equation as well as other electromagnetic fields problems [6, lo], Other parameters of interest 'such as capacitances can be determined in the usual fashion by determining the resulting charge on the conductors as the normal component of the displacement vector just off the conductor surfaces. 3.4
MISCELLANEOUS ADDITIONAL TECHNIQUES
The previous sections have discussed methods for determining the per-unitlength capacitances (and implicitly inductances) via approximate methods. The only structures for which these parameters yielded exact solutions were for the cases : 1. Two wires in a homogeneous medium.
154
THE PER-UNIT-LENGTH PARAMETERS
2. One wire in a homogeneous medium above an infinite, perfectly conducting plane, 3. One wire within and located on the axis of an overall shield with the dielectric having symmetry about that axis. There exist similar closed-form solutions for infinite, periodic structures of wires [111. There also exist some analytical solutions for structures that consist of rectangular-cross-section conductors in inhomogeneous media (lands on PCB’s) [12]. But these are restricted to only two conductors. However, for the structures that exhibit crosstalk, there must exist more than two conductors and the number is finite. Furthermore, these structures typically are surrounded by an inhomogeneous medium. Thus the feasible way of determining the entries in the per-unit-length parameter matrices is through numerical methods discussed previously. Although not as useful for multiconductor lines as the previously discussed numerical methods, there are some additional solution techniques that are worth noting. These methods are the conformal mapping technique and the spectral-domaintechnique which will be briefly discussed in this section. 3.4.1
Conformal Mapping Techniques
Conformal mapping techniques seek to transform the desired two-dimensional geometry to another geometry which is easier to solve [ll-14). It is desired to obtain a transformation of variables that map the original x, y coordinates over to some other u, v coordinate system as
u = u(x, Y )
(3.154a)
v = v(x,y)
(3,154b)
The transformation is represented as
w =u +jv
= F(Z = x + j y )
(3.155)
The function F must be an analytic function of 2,d WfdZ = dF/dZ, in order that the capacitance of both structures is identical [13]. The function F will be guaranteed to be analytic if u and v satisfy the Cauchy-Riemann equations: (3.156a) (3.156b) Finding the appropriate transformation that simplifies the problem is, of course, the crucial issue with this method but it has been successfullyapplied to various PCB-type structures [12,14].
MISCELLANEOUS ADDITIONAL TECHNIQUES
3.4.2
155
Spectral-Domain Techniques
There are various versions of the so-called spectral-domain method. Our interest here is in the applications to the solution of the two-dimensional Laplace equation: a24 824 V2$(x, y ) = -+ -= 0 (3.1 57) axz ay2 Take the Fourier transform of 4 with respect to x or y. Typically we transform with respect to the variable that should not require imposition of a boundary condition. For example, if we label the axis parallel to a PCB as the x axis (along which the structure is infinite in length) and the axis perpendicular to the PCB as the y axis (along which boundary conditions will be imposed), we would transform with respect to x as (3.158) J
-OD
Laplace's partial differential equation when transformed becomes an ordinary di@erential equation : (3.159)
Which has the simple solution &I y, ) = Ae'"
+ Bepy
(3.160)
where A and E are, as yet, undetermined constants. Represent the charge distribution over the conductor surfaces as f ( x ) so that the total charge on a conductor is
Q=
J:",
S(x) dx
(3.161)
Applying the boundary conditions (in the y variable) to (3.160) to determine A and B, it can be shown, using Parseval's theorem, that the capacitance of a two-conductor system becomes simply [151
One advantage of this method is that the capacitance can be found from (3.162) without the need to determine the inverse Fourier transform of &/3,y). The other advantage is that this is a variational method so that a relatively crude
156
THE PER-UNIT-LENGTH PARAMETERS
Electrostatic isolation of two regions by a shield: (a) problem definition and (b) illustration of the resulting capacitances.
FIGURE 3.50
approximation to the charge distribution, f ( x ) , will yield very accurate results for the capacitance of the structure.
3.5
SHIELDED LINES
Some MTL’s have a subset of the conductors completely surrounded by a perfectly conducting shield as illustrated in Fig. 3.50(a). The shield separates k - 1 of the conductors electrostatically from the remaining set. The electric field lines of the conductors internal to the shield terminate on the interior of the shield, and the electric field lines of the conductors external to the shield terminate on the exterior of the shield. This shows that the mutual capacitances and self-capacitances between conductors interior and exterior to the shield will
INCORPORATION OF LOSSES; CALCULATION OF R,
4,
AND G
157
be zero as illustrated in Fig. 3.50(b) CL2'J. Therefore, the determination of the overall per-unit-length capacitance matrix is broken into two separate solutions and the overall capacitance matrix has the form
"1
c=["0 co
(3.163)
where CIis the (k - 1) x (k - 1) per-unit-length capacitance matrix of the system of k - 1 conductors within the shield, and Co is the (n - k + 1) x (n - k 1) per-unit-length capacitance matrix of the system of conductors external to the shield (including, in the case of a ground plane, the shield). If we assume that the shield is not ferromagnetic, ~r, = 1, then the per-unit-length inductance has inductances between all conductors of the system and is full:
+
(3.164)
If the two media within and without the shield are individually homogeneous, p and E,,, p, the various inductance and capacitance submatrices are related by clLi /%&rllk-l (3.165a) and COLO = P E o & r o L k + l (3.165b) E,,,
This simplifies the determination of the per-unit-length parameter submatrices and has been implemented in various computer codes. Shields inherently enclose electric fields and limit electrostatic coupling. They do not, however, inherently restrict or eliminate magnetostatic coupling unless some additional provision is made, as when the shield is grounded at both ends to allow a current to flow along the shield thereby generating a counteracting magnetic flux CA.31. These concepts are implemented in the FORTRAN program SHIELD for computing crosstalk between individually shielded cables that is described in [1.2,1.3]. 3.6 INCORPORATION OF LOSSES; CALCULATION OF R, 11, AND G
The remaining parameters, the entries in the per-unit-length resistance matrix, R, and the per-unit-length conductance matrix, C,provide the line loss. In addition, the currents of imperfect conductors do not flow solely on the conductor surfaces as with perfect conductors but are distributed over the conductor cross sections. This gives rise to a portion of the per-unit-length inductance matrix, the internal inductance, L,,due to magnetic flux internal to the conductors. This can be included in the total per-unit-length inductance
158
THE PER-UNIT-LENGTH PARAMETERS
+
matrix as L = L, Le where Le is the external inductance due to magnetic flux external to the conductors. We have previously assumed perfect conductors wherein L = Le. 3.6.1
Calculation of the PerUnit-Length Conductance Matrix, C
If the surrounding medium is homogeneous, G can be obtained from C or L as described earlier: G = - c = POL-: U
(3.1 66)
8
and L = Le is the external inductance matrix assuming perfect conductors. So the difficulty in obtaining G arises for inhomogeneous media. Actually, this turns out to be a simple modification of the calculation of the per-unit-length capacitance matrix, C (also computed assuming perfect conductors). Losses in the medium are due to: 1. Conductive losses due to the conductance parameter, a.
2. Polarization losses as described previously. Both of these losses can be represented by a complex permittivity. This can be readily seen by writing Ampere's law for sinusoidal excitation as [A.l]
v x d = (a +io(&- j & b ) ) z e(=.
+(eb
(3.167)
+;))E
=jlo(e' -j&'')B l?
where &b represents the polarhation loss due to bound charge and d represents the conduction current losses due to free charge in the dielectric. All of these parameters, 8, &b, a, are functions of frequency to varying degrees. Typically e is relatively independent of frequency up to frequencies in the low gigahertz range and ranges from 2e0 to 20e0. Ordinarily there is little free charge in typical dielectrics, and the loss is due to polarization loss which is normally significant only above the low gigahertz range of frequencies. In any event we may represent both loss mechanisms by using a complex permittivity J? = e' -je" where e' = 8
(3.168a) (3.168 b)
= E' tan 6
INCORPORATION OF LOSSES; CALCULATION OF R,
4,
AND C
159
The real and imaginary parts of the complex dielectric constant, E' and E", are not independent but are related by the Kramers-Kronig relations [16J The term tan 6 is referred to as the loss tangent and is tabulated for various materials and at various frequencies in numerous handbooks C17J The result in (3.167) shows that we can include losses in the medium by solvingfor the capacitance matrix for a medium having a complex permittivity, 12 = s'(1 -j tan S), and from that result directly obtain C and G. To show this write the per-unit-length admittance matrix as
P = G + jwC =j w < e
- c,
(3.169)
+jCl)
This shows that we could determine the capacitance matrix permittivity 8 = ~ ( 1 j tan 6) and obtain
using a complex (3.170a)
(3.170b)
This is a standard idea used for many years to include losses in media CA.13. It also applies to inhomogeneous media i f we use the complex permittivities of the various homogeneous regions in computing the complex capacitance matrix, CB.1, 18-21! It is a simple matter to modify the RIBBON.FOR, PCB.FOR, and PCBGAL.FOR codes described in Appendix A for computing the per-unitlength capacitance matrices of inhomogeneous, lossy media. Simply declare the permittivity (and all quantities involving it) to be complex, provide the loss tangent at thefiequency of interest as input and use a complex equation solver instead of a real equation solver. The generalized capacitance matrix will be complex from which can be determined in the usual fashion. Then C and G can be extracted from that result according to (3.170). Table 3.13 gives results for the three-wire ribbon cable considered previously, this time assuming a loss tangent at 100 MHz of tan S = 0.01. A value of loss tangent of tan S = 0.01 at 100 MHz is somewhat unrealistically large for typical insulation materials (polyvinyl chloride) at 100 MHz. However, the entries in C are still substantially less than those of OC at 100 MHz by over two orders of magnitude (a factor of the order of 300) so the dielectric loss is not significant. The entries in C are not substantially different for a loss tangent of tan 6 0.001 further indicating this is a low-loss situation. For typical dielectrics and dimensions of MTL's it is frequently possible to neglect G,i.e., set G = 0. In order 10 illustrate the reasonable nature of this
e,
-
160
THE PER-UNIFLENCTH PARAMETERS
TABLE 3.13
Transmiuion-llne Capacitance# and ConductancRibbon Cable (tan 8 5 0.01, 100 MHd
C,with loss
C, without loss
Entry ~
11 12 22
for the Three-Wire
(PW) -~
(PFb)
G(Wm)
37.432
64.763 -32.381 34.722
- 18.716
37.432
- 18.716
24.982
24.982
approximation and to bound the error incurred by neglecting G,suppose that the medium is homogeneous, The capacitance matrix for this lossless, homogeneous medium is related to a constant matrix that is dependent only on the cross-sectional dimensions as C = eK (3.171) The conductance matrix is similarly related to K for this lossy, homogeneous medium. Substituting the complex permittivity into (3.171) yields the per-unitlength admittance as P =jwC (3.172) = jws’(1 - j tan 6)K = we‘ tan 6K + j ~ e ’ K __*_._
c-c
G
C
from which we obtain
G = w tan 6C
(3.173)
If the loss tangent is constant, the entries in the per-unit-length conductance matrix, G,increase directly with frequency. The importance of G can be bounded by realizing that the entries in G will be added to the entries in j o C to give the total per-unit-length admittance matrix
P = G +jwC = o(tan S
+j l ) C
(3.174)
In the case of an inhomogeneous medium the entries in G can be no larger than (3.173) using the largest loss tangent for all the various media of the problem. Thus, for a lossy inhomogeneous medium, the entries in G will be negligible in comparison to the entries in wC and can therefore be neglected gthe largest loss tangent of all the various media is several orders of magnitude less than unity! This is typically (but not always) the case in practical situations. Molecular
INCORPORATION OF LOSSES; CALCULATION OF R, 11, AND G
161
resonances in the microwave range can lead to large loss tangents over certain frequency bands [21]. For example, for dielectrics used in typical MTL’s (ribbon cables, coupled microstrips, PCB’s, etc.), the loss tangent below the low gigahertz frequency range is of the order of lo-* [16,17]. For silicon substrates used in typical microstrip lines, the loss tangent is of the order of 2.5 x low4. Observe that the reciprocals of the entries in G represent resistances between each conductor and between each conductor and the reference conductor which carry the transverse conduction currents. The entries in G are of the order of S/m which represent transverse resistances of some 1 kQ to S/m to 10 kR between conductors. Use of values of entries in G considerably larger than this are not representative of useful transmission-line structures. Thus for typical MTL‘s and for frequencies up to the low gigahertz range, neglecting G, setting C = 0, typically does not appreciably affect the solutions for the line terminal voltages and currents. 3.6.2
Representation of Conductor losses
In contrast to losses in the surrounding medium, losses due to imperfect line conductors may be significant even for frequencies below the low gigahertz range. These losses are determined by the entries in the per-unit-length resistance matrix, R. At low frequencies, the entries in R are constant, whereas at the higher frequencies they typically vary as the square root of frequency, as a result of skin efjct. Also the imperfect conductors give rise to internal inductances which contribute to L,.These elements typically are constant at low frequencies and decrease as at the higher frequencies. Thus at the higher frequencies their inductive reactances, oL,,increase as whereas the externai inductive reactances, wL,, increase as f. The current density and the electric and magnetic fields in good conductors are governed by the dipusion equation. Ampere’s law relates the magnetic field to the sum of conduction current and displacement current as
a,
8
vx
fi,
--
I?=
+
aE
conduction
j
d
(3.175)
displacement
A good conductor is one in which the conduction current (which implicitly includes the polarization loss) greatly exceeds the displacement current which is satisfied for most metallic conductors and frequencies of reasonable interest. Thus Ampere’s law within the conductor becomes approximately VXA,-B-J
Similarly, Faraday’s law is
-.
v x E=
-jqd
(3.176a) (3.176b)
162
THE PER-UNIT-LENGTH PARAMETERS
FIGURE 3.51
Diffusion of currents and fields into a semi-infinite conductive half-space.
Taking the curl of Faraday's law gives CA.11
v x v x Lf=V(V*&V2e' = -jw/V x R
(3.1 77)
Substituting Gauss' law: gives
v*s,E= 0 v2e'
=jwpt7i
(3.178) (3.179)
Since the current density is related to the electric field in the conductor as we arrive at the dijiision equation:
j= tr$
3.6.2.1 Surface Impedance of Piane Conductors Consider a semi-infinite, conducting half-space with parameters 0, 6, p whose surface lies in the x-y plane as shown in Fig. 3.51 C16J. Assume the electric field and associated conduction current density are directed in the z direction. The diffusion equation in terms of this z-directed electric field becomes
d2& = jwpa& dx2
(3.181)
Similar Gquations govern the magnetic field, I?,,, and the current density, f,, The solutions are & goe-x/ae-IxIS (3.182a)
INCORPORATION OF LOSSES; CALCULATION OF R, I.,,AND C
163
where the familiar skin depth is (3.183) and $, go,f , are the appropriate quantities at the surface. This result shows that the fields and current density decay rapidly in the conductor and are essentially confined to layers at the surface of thickness equal to a few skin depths. The surJuce impedance can be defined as the ratio of the z-directed electric field at the surface and the total current density: (3.184) The total current in the conductor can be obtained by integrating the current density given in (3.182~)throughout the conductor:
(3.1 85)
Substituting into (3.184) gives (4 = v&,) (3.186) where
R, =
i= tra
a/square
(3.187a)
This surface impedance is the impedance of an area of the surface of unit width and unit length so that we use the term ohms/square. The term R, is referred to as the surface resistance, and the term L, is referred to as the internal whereas the inductance. Observe that the surface resistance increases as internal inductance decreases as so that the internal inductive reactance increases as Observe also that the surface resistance in (3.187a) could have been more easily calculated by assuming the current density to be constant in a thickness equal to one skin depth of the surface and zero elsewhere.
fi.
fi
8,
164
THE PER-UNIT-LENGTH PARAMETERS
Equivalently, the surface impedance can be written as
(3.188)
Essentially we could have obtained the same result from the intrinsic impedance
of the conductor CA.1):
+/&
(3.189)
by observing that within the conductor the conduction current dominates the displacement current, i.e., d >> 0 8 . 3.6.12 Resistance and JnternaJ Jnductance of Wires Next we consider the resistance and internal inductance of circular cylindrical conductors (wires). The resistance and internal inductance are again due to the wire conductance being finite. In the case of perfect conductors, the currents flow on the surfaces of the conductors. Resistance and internal inductance result from the current and magnetic flux internal to the imperfect conductors in a fashion similar to the case of the surface impedance of a plane conductor and can be computed in a straightforward fashion if we know the current distribution over the wire cross section. The internal inductance results from magnetic flux internal to the Conductor that links the current, whereas the external inductance results from the magnetic flux that is external to the conductors that penetrates the area between the conductor and the reference or other conductors as discussed previously. The determination of these parameters for wires is very straightforward fi we assume the current is symmetric about the axis ofthe wire [16,22]. At dc the current is uniformly distributed over the cross section, whereas at higher frequencies the current crowds to the surface, being concentrated in annuli of thickness of the order of a skin depth. This observation leads to a useful equivalent circuit representing this skin effect which attempts to mimic this phenomenon [23]. Essentially this assumption of current symmetry about the wire axis means that we,assume there are no nearby currents close enough to upset this symmetry (proximity effect) [24]. Neighboring conductors also affect this current distribution for conductors of rectangular cross section [25]. Determination of these parameters when other currents are close enough to upset this symmetry is considerably more difficult, and for typical wire radii and spacings the symmetrical current distribution assumption is adequate. For example, reference [16] gives the exact per-unit-length high-frequency resistance for a two-wire line in a homogeneous medium consisting of two identical wires
INCORPORATION OF LOSSES; CALCULATION OF
R, I, AND G
165
of radii rwseparated by s as
where
is the surface resistance of the conductor. For s = 4rwsuch that one wire exactly fits between the two wires the resistance is only 15% higher than that assuming a uniform current distribution. Internal to the conductor, the conduction current dominates the displacement current so that the diffusion equation again describes the conduction current distribution internal to the wire. Let us assume that this current density is z directed (along the wire axis) and is symmetric about the axis of the wire and write the diflusion equation in cylindrical coordinates. We orient the wire about the z axis of a cylindrical coordinate system. Because of the assumed symmetry, the current density is independent of z and q5 but is a function of the radius, r, from the wire axis so that the diffusion equation reduces to CA.11 d2fi + -1 dfi -- + k29, = 0 dr2 r dr
where
k2 = - j w p =
(3.190) (3.191)
2
-j-p
The solution to this equation is CA.1, 16, 221
+ +
ber(fir/d) j bei(,/%/d) (3.192) ber(firw/d) j bei(*rw/d) where ber(x) and bei(x) are the real and imaginary parts, respectively, of the Bessel function of the first kind of a complex argument [8, 16,211. The term f, is the current density at the outer radius of the wire, r = rw. It now remains to determine the total internal impedance (per unit length) of the wire. The total current in the wire can be found by integrating Ampere’s law around the wire surface (assuming that the displacement current within the wire is much less than the conduction current):
fi= f,
f = f &dl
(3.193)
L.
= 2nrWH#l,R,w
The magnetic field can be obtained from Faraday’s law multiplied by the wire
166
THE PER-UNIT-LENGTH PARAMETERS
conductivity:
v x P= -jcupuR
(3.194)
Substituting the result for curl in cylindrical coordinates CA.11 and recalling that the current density is z directed and dependent only on r and the magnetic field is 4 directed gives
Substituting this into (3.193) and using (3.192) gives the total current in terms of the current at the wire surface. The per-unit-length internal impedance becomes
where ber'(q)
r:
d ber(q) 4 d
bei'(q) = - bei(q) dq
and (3.196b) Writing this total internal impedance in terms of its real and imaginary parts as tint =r
+jwl,
(3.197)
INCORPORATION OF LOSSES; CALCULATION OF
R, I+,AND G
167
gives the conductor resistance and internal inductance as
-=-[
r rd,
q ber(q)bei'(q)
2
(bei'(q))2
- bei(q)ber'(q)
+ (ber'(q))2
1
(3.1 98)
(3.199) where
rdo=
1 n/m unr,
(3.200)
(3.201) = 0.5 x lo-' H/m
are the dc per-unit-length resistance and internal inductance, respectively, of the wire CA.11. Although the results in (3.198) and (3.199) are exact assuming a current distribution that is symmetric about the wire axis, they are somewhat complicated. Reasonable simplificationscan be obtained depending on whether the frequency is such that the wire radius is greater than or less than a skin depth. Figure 3.52 shows the ratio of the per-unit-length resistance and the dc resistance of (3.198) plotted as a function of the ratio of the wire radius and a skin depth, r,,,/d. Observe that the transition to frequency dependence commences around the point where the wire radius is two skin depths, rw 2s. Similarly, Fig. 3.53 shows the ratio of the per-unit-length internal inductance and the dc internal inductance of (3.199) plotted as a function of the ratio of wire radius frequency dependence commences at the to skin depth. The transition to same point as for the resistance. Consequently, the exact results can be approximated by rr- 1
-
3
fi
unr,
lI = f! = 0.5 x lo-' H/m 8n
r, < 26
(3.202a)
168
THE PER-UNIT-LENGTH PARAMETERS
FIGURE 3.52 Frequency dependence of the per-unit-length resistance of a wire as a function of the ratio of wire radius to skin depth.
Observe that the low-frequency resistance for rw < 26 in (3.202a) could have been computed by assuming the current is uniformly distributed over the cross section as illustrated in Fig. 3.5qa); this is the case at de. Similarly, the high-frequency resistance for rw > 26 in (3.202b) could have been computed by assuming the current is uniformly distributed over an annulus at the wire surface of thickness equal to one skin depth as illustrated in Fig. 3.54(b); this satisfies our intuition based on the plane conductor case. Considering the complexity of the exact results we will use these approximations in our future work. 3.6.2.3 Internal Impedance of Rectangular Cross Section Conductors Unlike wires, analytical solutions for the resistance and internal inductance of conductors of rectangular cross section are complicated by the fact that we do not know the current distribution over the cross section. The current distribution over the cross section of a rectangular conductor tends to be concentrated at the corners when the skin effect is well developed. Early works consisted of measured results for the skin effect [26-271. Wheeler developed a simple “incremental inductance rule” for computing the high-frequency impedance when the skin effect is well developed [28]. This rule continues to be used extensively. The direct solution for the resistance and internal inductance of conductors of rectangular cross section can be obtained using a variety of
INCORPORATION OF LOSSES; CALCULATION OF R, ,I AND C
OVo6
\
\
t
tt
0*04 0.02' 0.01
FIGURE 3.53
I
169
I
I
*
I I
l 1
l
r. 8
.n
1 I"
I
. a
#.,I
I
ma
1 .A,,
Frequency dependence of the per-unit-lengthinternal inductance of a wire
as a function of the ratio of wire radius to skin depth.
rwee26
(4 FIGURE
Illustration of the cross-sectional current distribution of a wire for ,-,low
frequencies and (b) high frequencies.
methods for solving the diffusion equation in the two-dimensional transverse plane for infinitely long conductors [28-37). For example, finite element methods are implemented in several commercial packages. Another common and less direct method is the perturbation technique [18]. A particularly simple (conceptually) numerical method for determining these quantities is described in [38). This technique determines the resistance and internal inductance for conductors of actual lengths and so includes end erects, whereas the two-dimensional methods do not. It uses the concepts of partial
170
THE PER-UNIT-LENGTH PARAMETERS
L
(b.1
(8)
FIGURE 3.55
Circuit representation of a rectangular bar.
inductance [A.3,39-41]. Figure 3.55(a) shows a bar of length L, width w, and thickness t. If we assume the current to be uniformly distributed over the cross section the total bar resistance can be easily determined as L R=-Q
awt
(3.203)
The partial inductance, L,, can be similarly determined for uniform current distribution over the cross section [39-41). This notion can be extended to bars that have nonuniform current distributions over their cross section by dividing the bar into N subbars of rectangular cross section over which we assume the current to be uniformly distributed but whose level is unknown as illustrated in Fig. 3.56(a). This essentially approximates the actual current distribution over the cross section as a step. The voltage across each subbar is (3.204)
where Lpfkis the mutual partial inductance between the i-th and k-th subbars which contains the relative locations of the subbars. Formulas for these mutual partial inductances between conductors of rectangular cross section having uniformly distributed currents over their cross section are also available in [39-411. Arranging (3.204) for all subbars gives
(3.205b)
INCORPORATION OF LOSSES; CALCULATION OF
+
R,
I.,, AND C
171
ke
(b)
FIGURE 3.56 Representation of (a) a rectangular bar having a nonuniform current distribution over the cross section in terms of subbars having constant current distribution and (b) the resulting circuit model.
for N subbars. The sum of the subbar currents equals the total current for the bar and the voltages across all subbars are equal to the voltage across the bar as illustrated in Fig. 3.56.These constraints can be imposed to give the total resistance and partial inductance of the overall bar by inverting ft in (3.205) to yield i = g-'&. The first constraint is imposed by summing the entries in the rows, and the second constant is imposed by summing the entries in the columns of that result to give the effective parameters of the complete bar including skin effect as
(3.206)
The resulting composite resistance, R ( j ) , includes the effect of nonuniform current distribution over the bar cross section. The imaginary part includes the
172
THE PER-UNIT-LENGTH PARAMETERS
internal inductance, L l ( f ) ,(due to flux internal to the conductor) which is also frequency dependent, This result includes the effect of flux external to the conductor through Le.The internal inductance decreases as due to less internal flux linkages for increasing frequency as the current crowds to the surface of the bar and is typically dominated by the external inductance. Proximity effects of nearby conductors on these skin effect parameters can also be included in the method. Consider a MTL consisting of n 1 conductors of rectangular cross section with the reference conductor numbered as the zeroth conductor. Each conductor is subsectioned into Nl subbars for i = 0, 1,. ,n. Writing (3.205) for the system gives NT = El': 3 equations:
8 +
..
Inverting this matrix gives
(3.208)
First the voltages of each subbar of a bar must be equal. To enforce this condition we sum the entries in the respective columns of the submatrices to yield
...
(3,209)
... where the vectors fir, are (3.2 10)
The sum of the subbar currents must equal the current of the bar. Implementing
INCORPORATION OF LOSSES; CALCULATION OF R,
4,
AND G
173
this in (3.209) yields pol
...
81
(3.21 1)
... where
8,= rows c 81,
(3.2 12)
The sum of the currents of the n conductors must equal that of the reference conductor n
& = - k-= E &
(3.2 13)
1
in order for the concept of partial inductance to make sense. The difference in the transmission line voltages at the two ends of the line are Afi = @ I -
bo
(3.214)
These concepts are virtually identical to the concepts of the generalized capacitance matrix obtained in Section 3.1.4. Therefore, by adapting the generalized capacitance relations in (3.19), we can obtain
(3.215)
where (3.216)
Inverting this n x n matrix gives
(3.2 17)
Dividing the d, entries by the segment length, L,gives an approximation to
174
THE PER-UNIT-LENGTH PARAMETERS
FIGURE 3.57 Reduction of the partial element model to the transmission-linemodel for a two-conductor line.
the per-unit-length impedance matrix of the line which contains resistance, internal inductance, and external inductance. This process is illustrated for a two-conductor line in Fig. 3.57. The inverse of (3.21 1) gives the resistances and partial inductances of the two conductors:
where we have used the necessary requirement that the net current at any cross section is zero. Writing the difference of the bar voltages gives
The reader can verify that this result, derived directly for the special case of a two-conductor line, is equivalent to using the general result in (3.216). Computations for various bar dimensions are given in [42]. The above method not only models the skin effect over the bar cross sections but also implicitly includes the proximity effect since the mutual partial inductances between each subbar within a bar and between those of the bars of the system are included.
INCORPORATION OF LOSSES; CALCULATION OF R, ,.I
4
W
AND G
175
c
w
4
(b)
t
c
-!%>>E)
w+t
FIGURE 3.58 Illustration of the current distribution over the cross section of a PCB land for low and high frequencies.
The above method is quite accurate if the number of subbars is chosen sufficiently large but it is computationally intensive. A simple method, analogous to that for a round wire, is tb approximate the resistance for the two cases where the bar dimensions are much less than or much greater than a skin depth. In the case where the bar dimensions are much less than a skin depth, the current is reasonably approximated as being uniform over the bar cross section as illustrated in Fig. 3.58(a). Thus the low-frequency, dc per-unit-length resistance can be simply calculated as 1 rdc = -Q/m uwt
(3.220)
For higher frequencies where the bar dimensions are much greater than a skin depth, i.e., the skin effect is well developed, we assume that the current is uniformly distributed over strips of thickness 6 and zero elsewhere as illustrated in Fig. 3.58(b). As will be shown subsequently, the current peaks at the corners of the bar when the skin effect is well developed but this effect is ignored here.
176
THE PER-UNIT-LENCTH PARAMETERS
Rerirtance of Land
'''\05
4''
id
* I " '
2' 3' ' '
-
""'107
2'
3' ' ' " * "
ld
2'
3'
'"'
to9
Frequency (Hr)
FIGURE 3.59 Results of the computation of the per-unit-length resistance of a land via the method of subbars. Thickness 1.4 mils, width .= 15 mils.
This gives the high-frequency per-unit-length resistance as rhf
1
=
a(26t
+ 26w)
1 2aqw
a.
(3.221)
+ t)
which increases as This high-frequency result is less than the actual value because of the peaking of the current at the bar corners at high frequencies. The two regions join at
wt 2 (w t)
SI--
1
t N-
-2
+
(3.222)
w >> t
as shown in Fig. 3.58(c). For bars having high aspect ratios, w >> t, the two curves join where the thickness (the smaller dimension) equals two skin depths, which satisfies our intuition. Results given in [43,44] indicate the sufficiency of the above approximation. Figure 3.59 illustrates the comparison of this approximation to the method of subbars described previously for various divisions of the bar (w/NW, t/NT). The resistance of the bar is the real part of 2 computed as in (3.206). The bar is typical of a "land" on the surface of a
INCORPORATION OF LOSSES; CALCULATION OF R,
4,
AND G
177
printed circuit board and has t = 1.4 mils and w = 15 mils. The results were computed for bar lengths of 1 inch and 10 inches, and the per-unit-length resistance obtained by dividing the total resistance by the bar length was virtually identical for the two lengths. The break frequency in (3.222) occurs around 16.514 MHz where the average dimension wt/(w t ) in (3.222) is two skin depths. The subbar method indicates that the true resistance in the high-frequency region is some 50% higher than that predicted by the approxiThis error is evidently due to the mate method but nevertheless varies as omission of the peaking of the current at the corners in the approximation. The approximate method could be modified by using a lower break frequency. Choosing the break frequency where the average dimension is equal to only 1.36 gives a result that exactly matches the high-frequency result. However this better approximation cannot be assumed to hold for other dimensions or for the case where nearby lands alter the current distribution so that the break frequency will be chosen as in (3.222). Considering the considerable computational effort involved in a numerical solution (a large number of simultaneous equations (3.205) for one bar or (3.207) for several bars must be inverted at eachfrequency of interest) we will choose to use the simple approximate method in computing the skin-effect per-unit-length resistances for rectangular-crosssection lands in our future computed results. Figure 3.60 shows the normalized current distribution over the land cross section computed for a land of total length 10 inches, using nine divisions along the thickness, and ninety-nine divisions along the width. These plots confirm that for frequencies where the land dimensions are much less than a skin depth, the current is uniformly distributed over the cross section; whereas for frequencies where the skin effect is well developed, the current distribution peaks at the land corners. It is reasonable to assume that the bar internal inductance similarly varies above as the dc value below the same break frequency and decreases as this. So a simple approximate method for its determination will similarly suffice. For typical line dimensions this is dominated by the external inductance due to magnetic flux external to the wires computed for perfect conductors. Again, the inductive reactance due to the high-frequency internal inductance varies as the square root of frequency, wl, N yet the inductive reactance due to the external inductance varies directly with frequency as ole= f.
+
a.
fi
a,
3.6.2.4 Approxjmate Representation of Conductor Internal Impedances in the Frequency Domain The total per-unit-length internal impedance of a conductor
in the frequency domain has a real part due to the conductor resistance and an imaginary part due to the internal inductive reactance: (3.223)
We will assume that the high-frequency resistance varies as
f i for f >f , and
178
THE PER-UNIT-LENGTH PARAMETERS
FIGURE 3.60 Illustration of the cross-sectional current distribution of a
(a) 100 kHz, (b) 10 MHz.
PCB land for
INCORPORATION OF LOSSES; CALCULATION OF R, I.,, AND C
FIGURE 3.60
Continued.
179
(c) 100 MHz. Thickness = 1.4 mils, width = 15 mils.
is constant at the dc value below this:
(3.224) Let us also assume that the high-frequency internal inductive reactance equals and also transitions to the dc the high-frequency resistance, q h , = internal inductive reactance, d i , d o , at h. Therefore the dc internal inductance can be written as II,do = rd,/o,. The equality of high-frequency resistance and internal inductive reactance is demonstrated for all conductor cross sections by Wheeler’s “incremental inductance rule” [28 3. Also we showed previously that the resistance and internal inductance of solid wires transition at precisely the same frequency. With these approximations the conductor internal impedance can be approximated as (3.225)
Thus in this approximation one need only know the dc per-unit-length resistance and the frequency that it transitions to the high-frequency
fi
180
THE PER-UNIT-LENGTH PARAMETERS
frequency dependence due to the skin effect. Of course this approximation will be in error at the transition frequency (see Figs. 3.52 and 3.53 for wires) but the required input data for the frequency-domain computer programs that use these will be minimized. Representation of this frequency dependence of the conductor internal impedances in the time domain will be investigated in Chapter 5 when we seek to determine the time-domain solution of the MTL equations. We will represent the conductor internal impedances as
z,(s) = A
+B
4
(3.226)
where the Laplace transform variable is denoted as s. That this is a reasonable representation of the conductor impedances can be demonstrated by substituting s .ojo giving (3.227) zI(o) = A B J / 7 ;
+ = A + BJ;;JT(I+ j )
Thus we may interpret, in this approximation, (3.228a) (3.228b)
REFERENCES
[l] [2] [3] [4], [S]
[a] [7] [8] [9]
W.B. Boast, Vector Fields, Harper & Row, N.Y, 1964. E. Weber, Electromagnetic Ffelds, Vol. I, John Wiley, NY, 1950. W.R. Smythe, Static and Dynamic Electricity, 3d ed., McGraw-Hill, NY, 1968. A.T.Adams, Electromagnetfcsfor Engfneers, Ronald Press, NY, 1971. R.E Harrington, Ffeld Computatfon by Moment Methods, Macmillan, NY, 1968. M.N.O.Sadiku, Numerical Techniques In Electromagnetics, CRC Press, Boca Raton, FL, 1992. W.T. Weeks, “Calculation of Coefficients of Capacitance of Multiconductor Transmission Lines in the Presence of a Dielectric Interface,” IEEE Trans. on Microwave Theory and Techniques, MTT-18, 35-43 (1970). H.B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 4th ed., 1961. M. Javid and P.M. Brown, Field Analysis and Electromagnetics, McGraw-Hill, NY, 1963.
REFERENCES
[lo] [ 1 11
[12] [13J [143
[Is] [16) [17]
1183 [19]
[20] [21 J
[22] [23J [24] [25]
[26] [27] E281 [29)
[30]
181
P.P. Sylvester and R.L.Ferrari, Finite Elements for Electrical Engineers, 2d ed., Cambridge University Press, NY, 1990. S. Frankel, Multiconductor Transmission Line Analysis, Artech House, Dedham, MA, 1977. K.C. Gupta, R. Garg, and I.J. Bahl, Microstrip Lines and Slotllnes, Artech House, Dedham, MA, 1979. R.E. Collin, Field Theory of Guided Waues, 2d ed., IEEE Press, NY, 1991. H.A. Wheeler, “Transmission-Line Properties of Parallel Strips Separated by a Dielectric Sheet,” IEEE Trans. on Microwave Theory and Techniques, MTT-13, 172-185 (1965). E. Yamashita, “Variational Methods for the Analysis of Microstrip-Like Transmission Lines,” IEEE Trans. on Microwave Theory and Techniques, MTT-16, 529-535 (1968). S. Ramo, J.R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 2d ed., John Wiley, NY, 1984. A.R. von Hippel, Dielectric Materials and Applications, MIT Press and Wiley, NY, 1954. R.F. Harrington and C. Wei, “Losses in Multiconductor Transmission Lines in Multilayered Dielectric Media,” IEEE Trans. on Microwave Theory and Techniques, MTT-32, 705-710 (1984). J. Venkataraman, et al,, “Analysis of Arbitrarily Oriented Microstrip Transmission Lines in Arbitrarily Shaped Dielectric Media over a Finite Ground Plane,” IEEE Trans. on Microwave Theory and Techniques, MTT-33, 952-959 (1985). M.V.Schneider, “Dielectric Losses in Integrated Circuits,” Bell Sysrem Technical Journal, 48, 2325-2332 (1969). R.E. Matick, TransmissionLinesfor Digital and CommunicationNefworks,McGrawHill, NY, 1969. W.C. Johnson, Transmission Lines and Networks, McGraw-Hill, NY, 1950. C. Yen, Z. Fazarinc and R.L. Wheeler, “Time-Domain Skin-EtTect Model for Transient Analysis of Lossy Transmission Lines,” Proc. IEEE, 70, 750-757 (1982). V. Belevitch, “Theory of the Proximity Effect in Multiwire Cables,” Philips Research Reports, 32, Part I , 16-43 (1977); 32, Part II,96-177 (1977). M.E. Hellman and 1. Palocz, “The Effect of Neighboring Conductors on the Currents and Fields in Plane Parallel Transmission Lines,” IEEE Trans. on Micrownue Theory and Techniques, MTT-17, 254-258 (1969). J.D. Cockcroft, “Skin Effect in Rectangular Conductors at High Frequencies,” Proc. Roy. Soc., 122,533-542 (1929). S.J. Haefner, “Alternating Current Resistance of Rectangular Conductors,” Proc. IRE, 25,434-447 (1937). H.A. Wheeler, “Formulas for the Skin-Effect,” Proc. IRE, 30, 412-424 (1942). P. Silvester, “Modal Theory of Skin Effect in Flat Conductors,” Proc. IEEE, 54, 1147-1151 (1966). P. Silvester, “The Accurate Calculation of Skin Effect of Complicated Shape,” IEEE Trans. on Power Apparatus and Systems, PAS-87,735-742 (1968).
THE PER-UNIT-LENGTH PARAMETERS
MJ.Tsuk and J.A. Kong, “A Hybrid Method for the Calculation of the Resistance and Inductance of Transmission Lines with Arbitrary Cross Sections,” IEEE Zkans. on Microwave Theory and Techniques, MIT-39, 1338-1347 (1991). F,Olyslager, N. Fache, and D. De Zutter, “A Fast and Accurate Line Parameter Calculation of General Multiconductor Transmission Lines in Multilayered Media,“ IEEE Runs. on Microwave Theory and Techniques, MTT-39, 901-909 (1991).
P. Waldow and I. Wolff, “The Skin-Effect at High Frequencies,” IEEE Trans. on Microwave Theory and Techniques, M’IT-33, 1076-1081 (1985). H. Lee and T. Itoh, “Phenomenological Loss Equivalence Method for Planar Quasi-TEM Transmission Lines with a Thin Normal Conductor or Superconductor,” IEEE Trans. on Microwave Theory and Techniques, 37, 1904-1909 (1989).
G.I.Costache, “Finite Element Method Applied to Skin-Effect Problems in Strip Transmission Lines,” IEEE Trans. on Microwave Theory and Techniques, 35, 1009-1013 (1987).
T. Itoh (ed.), Planar 7kansmission Line Structures, IEEE Press, NY, 1987. E.L. Barsotti, E.F. Kuester, and J.M. Dunn,” “A Simple Method to Account for Edge Shape in the Conductor Loss in Microstrip,” IEEE Trans. on Microwave Theory and Techniques, M”-39, 98-105 (1991). W.T.Weeks, L.L. Wu,M.F. McAlister, and A. Singh, “Resistive and Inductive Skin Effect in Rectangular Conductors,” IBM J. Research and Development, 23, 652-660 (1979).
A.E. Ruehli, “Inductance Calculations in a Complex Integrated Circuit Environment,” IBM J. Research and Deuelopment, 16,470-481 (1972). F.W. Grover, Inductance Calculations, Dover Publications, NY, 1946. C. Hoer and C. Love, “Exact Inductance Equations for Rectangular Conductors with Applications to More Complicated Geometries,” J . Res. Nat. Bureau o j Standurds-C. Eng. Instrum., 69C, 127-137 (1965). A.W. Barr, “Calculation of Frequency-Dependent Impedance for Conductors of Rectangular Cross Section,” AMP J. Technology, 1, 91-100 (1991). T.V. Dinh, B. Cabon, and J. Chilo, “New Skin-Effect Circuit,” Electronics Letters, 26, 1582-1584 (1990). A. Deutsch et al., “High-speed Signal Propagation on Lossy Transmission Lines,” IBM J. Research and Development, 34,601-615 (1990).
PROBLEMS
3.1
A coaxial cable with a symmetrically inhomogeneous interior medium is shown in Fig. P3.1.Determine the per-unit-length capacitance, inductance, and conductance for this transmission line.
3.2
Consider a pair of parallel #28 gauge (r, = 7.5 mils, 1 mil = 0.001 inch) wires separated a distance of 50 mils as in a ribbon cable. Determine the per-unit-length capacitance, c, and inductance, 1, for this transmission
PROBLEMS
183
FIGURE P3.1
line if the surrounding medium is free space. The wires are stranded with seven strands of # 36 gauge (r, = 2.5 mils) solid wires. Considering these seven wire strands to be connected electrically in parallel, compute the total wire resistance at low and high frequencies and the frequency where r, = 26. Repeat this calculation for the internal inductance. Show that the internal inductance is smaller than the loop inductance, I, c< 1. 3.3
Consider two #28 gauge solid wires (r, = 6.3 mils) separated a distance of 50 mils. Compute the per-unit-length capacitance, c, and inductance, 1, by wide-separation approximations and using exact results.
3.4
Repeat Problem 3.3 for one #28 gauge solid wire at a height of 1 cm above a ground plane.
3.5
Determine the per-unit-length capacitance and inductance for the RG58U coaxial cable which has rw = 16 mils, r, 58 rnits. The interior dielectric is polyethylene with e, = 2.3. If the shield has a thickness of 5 mils, determine the per-unit-length resistance of the interior wire and the shield and show that the resistance of the interior wire is the dominant resistance. Determine the per-unit-length capacitance and inductance of the threewire transmission line in Fig. P3.6 on p. 184 by using wide-separation approximations. 5
3.6
3.7
Repeat Problem 3.6 for two wires above ground shown in Fig. P3.7 on p. 184.
3.8
Repeat Problem 3.6 for the coaxial line in Fig. P3.8 on p. 184.
3.9
Prove the relation between the entries in the generalized capacitance matrix, W, and the entries in the transmission-line-capacitance matrix, C, given in (3.19).
3.10 Verify the results given in Tables 3.2 and 3.3. 3,ll Verify the results given in equations (3.105) to (3.109).
184
THE PER-UNIT-LENGTH PARAMETERS
.i
I
1 cm
All wires U28 gauge (r. = 6.3mils)
FIGURE P3.6
-
All wires #20 p ~ g (eI w 16 mils) FIGURE P3.7
\
\
FIGURE P3.8
PROBLEMS
185
FIGURE P3.12
3.12 A “trough transmission line” is shown in Figure P3.12.Determine the
per-unit-length capacitance by using a moment method and the results in (3.105)and (3.107). Confirm your result by a direct solution of Laplace’s equation. (See Chapter 10,pp. 588-590 of CA.11.) 3.13 Solve Problem 3.12 by using the finite difference method. 3.14 Solve Problem 3.12 by using the finite element method. 3.15 Determine the per-unit-length parameters with and without dielectric insulations for a three-wire ribbon cable with d = 50 mils, rw = 18 mils
(120 gauge), and insulation thickness t = 7 mils and e, = 4 using the program RIBBON.FOR. Plot these for various numbers of expansion coefficients.
3.16 Derive the Galerkin results for equal-width lands given in (3.117). 3.17 Determine the per-unit-length parameters for a PCB consisting of three identical lands with w = s I:8 mils and a silicon substrate with 10 mil thickness and e, = 12 using the programs PCB.FOR and PCBGAL.FOR. 3.18 Show that the capacitance matrix is related as in (3.171)to a matrix that
is independent of the properties of the surrounding homogeneous medium. 3.19 Investigate the solution of the diffusion equation in cylindrical coordin-
ates given in (3.190). 3.20
Solve the diffusion equation numerically using the finite difference method for a rectangular bar of width w and thickness t.
3.21 Verify the approximate relations for internal impedance of a conductor
in the frequency domain given in (3.225).
CHAPTER FOUR
Frequency-Domain Analysis
Having determined the entries in the per-unit-length parameter matrices of inductance, L, capacitance, C, conductance, C, and resistance, R, for the particular line cross-sectional dimensionsas in Chapter 3, we now embark upon the solution of the resulting MTL equations. In this chapter, we consider the frequency-domin solution of the MTL equations where the excitation sources are sinusoids which have been applied a sufficient length of time so that the line voltages and currents are in steady state. In the next chapter, we consider the time-domain solution of the MTL equations wherein the sources (and, consequently, the time variations of the line voltages and currents) may have arbitrary time variation. The time-domain solution will be the sum of the transient and steady-s tate responses.
4.1
THE MTL EQUATIONS FOR SINUSOIDAL STEADY-STATE EXCITATION
We assume that the time variation of the sources is sinusoidal and the line is in steady state. Therefore the line voltages and currents are also sinusoidal having a magnitude and a phase angle. Thus we denote the line voltages and line currents in their phasor form CA.21: (4.la) (4.1b) where &{ * } denotes the real part of the enclosed complex quantity, and the phasor voltages and currents have a magnitude and phase angle as (4.2a) 186
THE MTL EQUATIONS FOR SINUSOIDAL STEADY-STATE EXCITATION
187
(4.2b)
We will denote all complex (phasor) quantities with over the quantity. The radian frequency of excitation (as well as the radian frequency of the resulting line voltages and currents) is denoted by w where o = 2nf and f is the cyclic frequency of excitation. Applying (4,l) to (4.2) gives the resulting tlme-domain forms as CA.1, A.23 A
(4.3a) (4.3b) The time-domain MTL equations are given, in matrix form, in equation (2.27):
a V(Z,t ) -RI(z, 3 :
az
a
I(z, t ) az
a
t ) - L - I(z, t ) at
a
-GV(z, t ) - C - V(Z,t ) at
(4.4a) (4.4b)
For sinusoidal variation of the sources and line voltages and currents, the time variation is assumed to be ej@' as in (4.1) so that derivatives with re7pect to time, t, in (4.4) are replaced by jw. Substituting the phasor forms fortthe line voltages and currents given in (4.1) into (4.4) gives the MTL equations for sinusoidal, steady-state excitation as d - V(2) = -2i(z) dz
(4.5a)
d i(z) = -PV(z)
(4Sb)
dz
where the per-unit-length impedance matrix, 2,and admittance matrix, 9, are given by ~=R+JOL (4.6a)
9 = G +joC
(4.6b)
In taking the time derivatives to produce (4.9, we have assumed that the per-unit-length parameter matrices, R, L,G, and C are independent of time, t, i.e., the cross-sectional dimensions and surrounding media properties do not change with time. This is a local assumption but should be explicitly stated. The resulting equations in (4.5) to be solved are a set of coupled$rst-order, ordinary diflerential equations with complex coeficients, They can be put in a
188
FREQUENCY-DOMAIN ANALYSIS
more compact matrix form as d dz
- B(z) = AB(z)
(4.7a)
where (4.7b)
A=[
O
-?
-7 0
(4.7c)
Observe that for an (n + 1)-conductorline, o ( z ) and f(z) are n x 1 and contain the phasor line voltages and zurrcnts, respectively, and 2 and P are n x n. Therefore %(z) is 2n x 1 and A is 2n x 2n. Our task in this chapter will be to solve (4.7) and incorporate the terminal constraints. (The terminal constraints contain the lumped voltage and current source excitations and the load impedances.) Equations (4.7), being a set of coupled, first-order, differential equations, are similar in form to the state-variable equations found in the analysis of lumped systems wherein the independent variable is time, t, CA.21, whereas in (4.7) the independent variable is position along the line, z. Because of the direct similarity between the frequency-domain MTL equations and the state-variable equations, we will adapt the known solution properties for the state-variable equations directly to the solution of the frequency-domain MTL equations by making the simple analogy of time, t, in the state-variable solutions to position along the line, z,in the frequency-domain MTL equation solutions. This simple observation will obviate the necessity to obtain redundant solutions and will illuminate a number of interesting properties of the phasor MTL solution which are drawn by direct parallel from the state-variable solution. Alternatively, the coupled, first-order phasor MTL equations in (4.5) can be placed in the form of uncoupled, second-order, ordinary di$erential equations by differentiating both with respect to line position, z, and substituting the first-order equations given in (4.5) as df V(z)
= 2PV(z)
(4.8a)
da f(z) = P&f(z)
(4.8b)
dz'
dz'
Ordinarily, the per-unit-length parameter matrices 2 and
9 do not commute
so that the proper order of multiplication in (4.8) must be observed. In
differentiating (4.5) with respect to line position, z, we have assumed that the per-unit-length parameter matrices, R,L,G,and C, are independent of z. Thus we have assumed that the cross-sectional line dimensions and surrounding
SOLUTIONS FOR TWO-CONDUCTOR LINES
-
i(Z)
a
+
r L
+
189
-
media properties are constant along the line or, in other words, the line is a ungorrn line. Both the first-order, coupled forms of the MTL equations given in (4.5) or equivalently in (4.7) as well as the second-order, uncoupled forms given in (4.8) will be useful in obtaining the final solution. 4.2
SOLUTIONS FOR TWO-CONDUCTOR LINES
In this section we will summarize the well-known solutions for a two-conductor line shown in Fig. 4.1 CA.1, A.3). This wit1 be useful in the MTL solution since there are numerous analogies and parallels to this solution that appear in matrix form in the MTL solution. For a two-conductor line, n = 1, the per-unit-length parameter matrices become scalars, Y, I, g, and c, and the uncoupled second-order equations in (4.8) become d2 (4.9a) - P(2) = 9 2 P(z) dZ2
d f ( z ) = p2fiz) dz2
(4.9b)
where the propagation constant, 9, is (4.10)
190
FREQUENCY-DOMAIN ANALYSIS
where a is the attenuation constant whose units are nepers/m and /3 is the phase constant whose units are radians/m. The general form of the solution to these equations is CA.1, A.3)
F(z) = P+e+ + f(z) = f+e'9* + f-eP'
?+
=
z
(4.11a) (4.11b)
F-
e-9z
- ZC e9z -2-
The terms ?+, F-,I+,and r" are complex-valued, undetermined constants which will be determined when we incorporate the terminal conditions at the two ends of the line. The quantity gCis the characteristic impedance of the line and is given, in terms of the per-unit-length parameters, as (4.12)
(r +jol)
=G Substituting (4.10) and (4.12) into (4.11) gives (4.13a)
m-
-
V+e-~:e-np~+eg-e+) V - eu:eHpr-ez+s-) ZC ZC
where the magnitudes and phases of the undetermined constants are noted by
F+ = v + e +
(4.14a)
v- = v-4-
(4.14b)
The time-domain expressions are obtained from (4.1) as
- pz + 0') + V-e"' cos(wt + /3z + e-) (4.15a) V+ VI(z, t ) = -e-= cos(ot - /3z - 0, i-6 ' ) - -e"' cos(ot + pz - OZ + e-)
V(z, t ) = V+e-"cos(ot ZC
ZC
(4.15b)
SOLUTIONS FOR TWO-CONDUCTOR LINES
191
+
These expressions are the sums offorward-trawling waues, traveling in the z direction: v+(z, t) = V+e'"'cos(wt - pz 0') (4.16a)
+
I+(Z, t) = -e-"' COS(^^ - pz - ez + e+)
V+
(4.16b)
zc
and backward-trawling waoes, traveling in the -z direction:
+ + e-) cos(wt + pz - e, + e-)
Y-(z, t ) = V-eazCOS(WCpz
'V
I-(z, t ) = -ear zc
as
+ V-(z, t) I(z, t) = I + @ , t) - I - @ , t )
V(Z, c) = V'(2,
t)
(4.1 7a)
(4.17b) (4.18a) (4.18b)
That these are traveling in the +z and -z directions can be seen from the observation that as time t progresses, z must either increase or decrease in order to keep the arguments of the cosine terms constant in order to track corresponding points on the waveforms CA.13. The terms efar represent attenuation of the amplitudes of the waves. The velocity of these waves is CA.1): w
(4.19a)
Or-
B
If the line is lossless (perfect conductors and lossless medium) then the velocity of these waves is the velocity of TEM waves in the surrounding (assumed homogeneous) medium: 1
1)Z-Z-
1
fifi
r=g=O
(4.19b)
The ratios of backward-traveling and forward-traveling voltage waves at any point on the line is referred to as the reflection coefictent at that point. Taking the ratios of these in phasor form from (4.11) gives CA.1) (4.20a)
192
FREQUENCY-DOMAIN ANALYSIS
i! e)
/(O)
+
+ Transmission line as a 2 port
FIGURE 4.2
q(v9
Illustration of viewing a two-conductorline as a two port in the frequency
domain.
The reflection coefficients at two points on the line, z2, zl, are related as CA.11
Consider the two-conductor line shown in Fig. 4.1. The total line length is The solutions given in (4.1 1) contain undetermined constants, denoted by 9. 8’ and P-.These can be eliminated by putting the solution in the form of the chain parameter matrix as (4.21) This representation relates the line voltages at one end of the line, z = 2,to the line voltages and currents at the other end of the line, z = 0. In fact, the chain parameter matrix can be used to relate the voltage and current at any point on the line, z, to those at z = 0 by replacing Y with z in (4.21) and the results that follow. Similarly, the chain parameter matrix can be used to relate the line voltages and currents at two interior points on the line, z1 and z2 with z2 2 zl.by replacing Y with z2 and 0 with zi in (4.21) and the results that follow. The name, chain parameter matrix, is derived from the observation that the overall chain parameter matrix of several such lines in cascade is the product (in the appropriate order) of the chain parameter matrices of the individual lines in the chain. In fact, this observation provides an approximate method of modeling nonuniform lines such as twisted pairs of wires as a sequence or cascade of uniform lines CG.1-G.101. The chain parameter matrix can be viewed as a way of characterizing the line at its end points as a two port as illustrated in Fig. 4.2 CA.21. Evaluating the general solution in (4.11) at z = Y and at z 3:0 gives
SOLUTIONS FOR TWO-CONDUCTOR LINES
193
(4.22b)
Solving these gives the chain parameter matrix as
b = [ dll
[
421
=
-
dI2
]
422
cosh(f.9)
(4.23)
-& sinh(p9)
sinh(f.9)
cosh(f.9)
1
where the hyperbolic cosine and sine are cosh(9.9) = sinh(p9) =
e99 + e-99
2 e99 - e-99
2
(4.24a) (4.24b)
Now that the general form of the solution has been obtained, we incorporate the terminal constraints in order to evaluate the undetermined constants in that general solution. Reconsider the two-conductor line shown in Fig, 4.1. The line is terminated at the load end, z = 9,with a load impedance 2'. At the source end, z = 0, an independent voltage source, & = V,F,and source impedance, $, terminate the line. Thus the terminal Constraints are: V(0) =
- 2Sf(O)
Q.9)= 2 L f i . 9 )
(4.25a) (4.25b)
Substituting these constraints into the chain parameter form of the solution gives the explicit form of the solution for the line voltages and currents at any position along the line as CA.1, A.31 (4.26a) (4.26b)
194
FREQUENCY-DOMAIN ANALYSIS
where the rejection coeflcients of the source (PS)and the load (f,,)are given by (4.27a) (4.27b) The input impedance at any point along the line can be obtained as the ratio of the line voltage and current at that point as (4.28) = z, A
gL + ZCtanh(f(9 - z)) Zc + ZLtanh(j(9 - 2))
If the line is matched at the load, Le., 2' = &, then the reflection coefficient at the load is zero, PL = 0, and these relations simplify to
(4.29b)
Z&) = zc= gL
z, = z,
A
A
-
(4.29~)
The net flow of aoerage power in the +z direction is CA.1, A.21 P,&) = +B.9{V(z)f*(z)}
w
(4.30)
where A* denotes the conjugate of the complex quantity CA.21. If the line is matched at the load, the reflection coefficient is zero and (4.30) shows that all the power is traveling in the +z direction, i.e., there is no power reflected at the load and hence traveling in the - 2 direction. 4.3
GENERAL SOLUTION FOR AN (n
+ 1bCONDUCTOR LINE
In the previous section we discussed the well-known solution for a twoconductor line. In this section we begin our study of the solutions for an
GENERAL SOLUTION FOR AN (n
+ 1kCONDUCTOR LINE
195
(n + 1)-conductor line or MTL. In many cases, the results and properties of the solution for a two-conductor line carry over, with matrix notation, to a MTL.
4.3.1
Analogy of the MTL Equations to the State-Variable Equations
Transmission lines are distributed-parameter systems. If the electrical dimensions of a structure are small, it can be approximately modeled as a lumped-parameter system. The independent variables for a distributed-parameter system are the spatial dimensions,x, y, z,and time, t. In the case of a lumped-parameter system, the quantities of interest are lumped rather than distributed throughout space so that they depend only on time, t. Lumped-parametersystems are characterized by ordinary differential equations, whereas distributed-parameter systems such as transmission lines are characterized by partial differential equations as we have seen. If our interest is only in sinusoidal, steady-state behavior of the MTL, the use of phasor quantities removes the time dependence. In the case of a transmission line the only spatial parameter is the line axis, z, and the partial differential equations become ordinary differential equations with complexvalued coeficients as is illustrated in equations (4.9, (4.7), and (4.8). So we may make a direct analogy between the sinusoidal, steady-state transmission-line equations and those of a lumped-parameter system by viewing the spatial parameter, z, in the distributed-parameter system phasor equations as the equivalent of time, t, in the lumped-parameter system-governing equations. This important observation will allow a considerable simplification of the necessary work to obtain the solution to the phasor, MTL equations. It will also allow considerable insight into the properties of that solution since we may draw, by analogy, from the abundance of known properties of the solution for the lumped-parameter system. Consider a lumped-parametersystem. One way of representingsuch a system is via the n coupled, ordinary differential equations in state-uariuble form as CA.2, 1, 2, 3)
d - X(t) = AX(t) + BW(t)
dt
(4.31a)
where
(4.31b)
1%
FREQUENCY-DOMAIN ANALYSIS
... a l l
e . .
a,,
*
*
.
.* ..
... anI
(4.31c)
.
(4.31d)
The matrices A and B are assumed to be independent of the independent variable, t, in which case the system is said to be stationary, i.e., its parameters do not vary with time. This property is analogous to a uniform transmission line where d in (4.7) is assumed independent of z, Le., the line cross-sectional dimensions and media properties are constant along the line. The w,(t) are viewed as the p inputs to the system, and the xr(t) are viewed as the n state variables of the system. Any output of the system can be represented as a linear Combination of the state variables and the system inputs CA.2, B.11. In order to determine the response of this system to the (presumably known) inputs, we must prescribe the initial conditions on the state variables at some initial time to, X(to). As with any other set of ordinary differential equations, the total solution is the sum of the zero input response, with W(t) = 0, and the zero initial scute response with X(to) = 0. Let us begin this discussion of the solution to the state variable equations by considering ajirst-order, lumped system, n = 1, whose state-variable equations become the scalar equations d x(t) = ax(t) + bw(t)
(4.32)
dt
with a prescribed initial state, x(to),The solution of the homogeneous equations, w(t) = 0, is CA.21 xh(t)= ea('-'O)x(to) (4.33) = 40
- to)x(to)
The notation 4(t) = earis referred to as the state-transition function. Recall that the exponential is defined as the infinite series e " ' = 1 + -ta + - at 22 + - a 3t 3+ I! 21 3!
e * *
(4.34)
GENERAL SOLUTION FOR AN h
+ IKONDUCTOR LINE
197
The solution to the original equation in (4.32) is referred to as the particular solution. It can be obtained from the above homogeneous solution via the method ofoariation of parameters [B. 13 by replacing the initial state, x(to), with an undetermined constant that is a function oft, x,(t) = f?'k(t)
(4.35)
Substituting this into the original equation, (4.32), gives
+
d ae"'k(t) e"' - k(t) = ue"k(t) + bw(t) dt
(4.36)
Equation (4.36) becomes
d k(t) = e-"'bw(t)
dt
(4.37)
which has the solution CA.2, B.1) k(t) =
1:
e-drbw(t)ds
(4.38)
Substituting this into (4.35) gives the particular solution as (4.39)
Combining this with the homogeneous solution gives the total solution as (4.40)
The homogeneous solution, x,(t), is referred to as the zero input solution, whereas the particular solution, x,(t), is referred to as the zero state solution. Given the input, w(t), the key to obtaining the total response is obtaining the state-transition function &t) = e"', or exponential e"'. But this is simple for a first-order system. Before we extend these results to the general n-th order system, it is worthwhile to examine some important properties of the state-transition
198
FREQUENCY-DOMAIN ANALYSIS
function, 4(t) = e"! Perhaps the most important property is
Substituting t = to into the total solution in (4.40) gives x ( t ) = x(to). This also follows from the infinite-seriesdefinition of the exponential given in (4.34). The name state-transitionfunction is used for (b(t) = earsince it shows how the initial state, x(to), transitions to the final state, x(t). The second property is that in order to obtain the inverse of the state-transition function, we need only substitute --t for t : 4-1(t) = e-"' (4.41 b) = ( b( - t )
This property is rather obvious since we may obtain from the homogeneous solution in (4.33)
- to)x(t) = 4 4 0 - t)x(t)
x(to) = p ( t
(4.42)
which simply amounts to a reversal in time. We will find these important properties and the form of the general solution to the state-variable solution in (4.40) to carry over to the n-th order system considered next. Now consider the general n-th order lumped system characterized by (4.31). If we carry through the above development for the first-order system in like fashion we obtain the general solution as CA.2, B.1, 1, 2, 31 X(t) = @(t - to)X(to)
+
I
@(c io
- z)BW(r) dr
(4.43)
Given the vector of p inpnts, W(t), and the vector of initial states, X(to), equation (4.43) allows a straightforward determination of the states at some future time, X(t). The n x n state-transition matrix, @(t), has the same important properties as the first-order system: @(O) = 1, (4.44a)
a)-'@)= @ ( - t )
(4.44b)
and @(t) = eAr
1,
t2 t3 + -I!tA + -A2 + -A3 + 2! 3!
(4.44c) * * e
where the n x n identity matrix has ones on the main diagonal and zeros
GENERAL SOLUTION FOR AN h i l)-CONDUCTOR LINE
elsewhere: 1
0
0
1
.
I"=
[:0
...
1
0 e . .
0
199
(4.44d)
1
Now consider the phasor, transmission-line equations for an (n + 1)conductor line given in (4.7): d B(z) = AB(z) dz
(4.7a)
where (4.7b)
A=[
O
-9
-7 0
(4.7c)
Comparing these to the lumped-parameter state-variable equations given in (4.31) with W ( t ) = 0 shows that the general solution for the line voltages and currents are, by direct analogy,
where the &, are I I x n. If w,e choose z2 = 9 and z, = 0 we essentially obtain the chain parameter matrix CP for the overall line as (4.45b)
Because of the direct analogy between the state-variable equations for a lumped system and the phasor MTL equations we can immediately observe some important properties of the chain parameter matrix from comparison to (4.44):
b(0)= 12"
W(9)= &(--9)
(4.46a) (4.46b)
200
FREQUENCY-DOMAIN ANALYSIS
(4.46~) 3i
2z2p + -YA 33 + . . . +-
9 11
12, +--A
2!
3!
Once again, the property of the inverse of the chain parameter matrix given in (4.46b) is logical to expect since the inverse of (4.45b) yields (4.47)
This follows as simple reversal of the line axis scale (replacing with - z ) similar to the reversalin time for the state-transition matrix of lumped systems and the line is reciprocal (assuming the surrounding medium is linear and isotropic). We will find these properties to be important in obtaining insight into the interpretation of the MTL equation solution. 4.3.2
Decoupling the MTL Equations by Similarity Transformations
The essential task in solving the phasor MTL equations is to determine the chain parameter matrix 6(9), One obvious way of doing this is to use the matrix infinite series form given in (4.46~).Substituting the form of A given in (4.7~)gives d j l l ( 9 ) = 1,
9 Z A A +ZY + -[ZY]2 + 2! 41 9 4
A *
*
(4.48a)
612(9) = - 5 - -[2P]$ - -[29]22 + * .
(4.48b)
9 Y3 6)21(9) = - - 9 - -[YZ,P - y'[P2]2P + . . .
(4.48~)
3!
l!
5!
A -
I!
dj22(9) = 1,
5!
3!
pAA + -YZ + -[YZ12 9 4
2!
4!
A
A
+ * * a
(4.48d)
In theory, one ,could perform, using a digital computer, the various products of the per-unit-length parameter matrices and sum the terms in (4.48) for a sufficient number of terms to achieve convergence and truncate the series thereafter. However, a more practical, closed-form result can be obtained using the following idea,
GENERAL SOLUTION FOR AN (n
+ I)-CONDUCTOR LINE
201
The method of using a similarity trangormation is perhaps the most frequently used technique for determining the chain parameter matrix CB.1, 5-10]. We will find this to be of equal use in the time-domain solution in the next chapter. Define a change ofvariables as V(z) = ?vftm(z)
(4.49a)
f(z) = T,i,(Z)
(4.49b)
The n x n complex matrices 'fVand ?, are said to be similarity transformations between the actual phasor line voltages and currents, ft and ?, and the mode voltages and currents, ftm and im.In order for this to be valid, these n x n transformation matrices must be nonsingular, Le., ?;I and must exist where we denote the inverse of an n x n matrix M as M-I, in order to go between both sets of variables. Substituting these into the phasor MTL equations in (4.7) gives (4.50)
If we can obtain a *v and a ?, such that 9;
'29,and ?i1Qfv are diagonal as
(4.51b)
then the phasor MTL equations are uncoupled as
(4.52)
202
FREQUENCY-DOMAIN ANALYSIS
if we can find two n x n matrices TVand TI which simultaneously diagonalize both per-unit-length parameter matrices, 2 and 9, then the solution essentially reduces to the solution of n coupled, first-order diferential equations as in the case of two-conductor lines. But the solution of the n first-order differential equations in (4.52) was obtained earlier in the analysis of two-conductor lines. Therefore obtaining a similarity transformation that simultaneously diagonalizes both per-unit-length parameter matrices essentially solves the problem of the solution to the n coupled MTL equations! We will use this technique of decoupling the MTL equations on numerous occasions. The essential question becomes: When can we find a similarity transformation that diagonalizes a matrix? Before we address that question, let us examine the application of the similarity transformation to the uncoupled, second-order MTL equations given in (4.8): d2 V(2) dz2
= ZPV(2)
d2 f(z) = PZt(Z) dz2
Substituting the similarity transformations given in (4,49) gives d2 0 ( 2 ) = T;'2PT"Vm(Z) dz2 =
(4.8a) (4.8b)
(4.53a)
Ti%?&- 1PTv9m(z)
=2
d2 f,,,(z) = T;'YZT,t,,,(z) dz2 L I A A
(4.53b)
Recall that 2 and 9 are symmetric, i.e., 2'= 2 and 9' = 9, where the transpose o f a matrix M is denoted by M'.Since the transpose of the product of two matrices is the product of the transposes of the matrices in reverse order, we see that
= yz
GENERAL SOLUTION FOR AN (n
+ 1kCONDUCTOR UNE
203
where we have used the assumption that z and y are diagonal so that their product can be reversed. Comparing this to (4.53b) we observe that
Therefore it suffices to diagonalize the product 92 or the product 29.Let us arbitrarily choose to decouple (4.53b) as dZ im(z) = ?-'P2Tfm(z)
dzz
where
+[y
(4.56a)
= f"&>
? = ?,
(4.56b)
and 9' is a diagonal matrix as
9:
0
0
. e .
Pj:
...
001
0
(4.56~)
9,'
The general solution to these uncoupled equations is
i,,,(z) = e+I: - e9li;
(4.57)
where the matrix exponentials are defined as
(4.58a)
and the vectors of undetermined constants are
(4.58b)
204
FREQUENCY-DOMAIN ANALYSIS
The actual currents are obtained by multiplying these mode currents by the transformation matrix, 9, = 9,to give
Similarly, the uncoupled second-orderdifferentialequation in terms of the mode voltages is dZ V,(z) = T; '2PT"Vrn(z) (4.60) dz2 = P$P(P)- 'V,(z) = pV,(z)
with the general solution
The actual voltages can be obtained by multiplying this result by the trans= to give formation, $, = @;
The undetermined constants in these results are related. To determine this relation, substitute (4.59) into the second MTL equation, given in (4.5b) to give
where we have defined the characteristic impedance matrix as (4.64)
This can be placed in another form. From (4.56) we have (4.65)
Thus (4.66)
GENERAL SOLUTION FOR AN (n
I
Transmission
i
+ 1kCONDUCTOR LINE
205
-i"(99 +
line as a 2 n port
FIGURE 4.3
Illustration of viewing
frequency domain.
an (n
+ 1)-conductor line as
a 2n port
in the
Therefore, the characteristic impedance matrix can be written as
We will find this seemingly arbitrary definition of the characteristic impedance matrix to have physical significance in terms of backward- and forwardtraveling waves on the line in the following sections. 4.3.3
Characterizing the line as a 2n Port with the Chain Parameter Matrix
The phasor voltages and currents at the two ends of the line can be related with the chain parameter matrix as in (4.45b): (4.68)
This corresponds to viewing the (n + 1)-conductor line as a 2n port as illustrated in Fig. 4.3.The essential task in solving the phasor MTL equations is to determine the entries in the n x n submatrices, &, This section is devoted to that task. The general solutions of the phasor MTL equations are given, via similarity
206
FREQUENCY-DOMAIN ANALYSIS
transformations, in (4.59) and (4.63):
Evaluating these at z = 0 and z = 9 and eliminating parameter matrix submatrices as CB.13:
f$
gives the chain
(4.70~)
and qc= 2;'. As a check on this result, observe that the identity in (4.46a), 6(0)= l,, is satisfied. 4.3.4
Properties of the Chain Parameter Matrix
In this section we will define certain matrix analogies to the two-conductor solution CB.1, B.41. Although these will place the results in a form directly analogous to the two-conductor case, their use in numerical computation is limited. First let us define the square root of a matrix. In scalar algebra, the square root is defined as any quantity which when multiplied by itselfgives the original = a. The square root of a matrix can be similarly defined quantity, i.e., as a matrix which when multiplied by itself gives the original matrix, Le., = M.Recall the basic diagonalization in (4.65):
&&
,/%a
From this we may define the square root of the matrix product as (4.72)
GENERAL SOLUTION FOR AN (n
+ l)-CONDUCTOR LINE
207
This can be verified by taking the product and using (4.71): (4.73)
(4.74) as multiplication by itself shows. Similarly, we can define
m=9 - 1 f l P
f l as (4.75)
as a multiplication by itself shows. Therefore, the characteristic impedance matrix in (4.64) or (4.67) can be written, symbolically, as (4.76)
Observe that this result reduces to the scalar characteristic impedance for two-conductor lines. Additional symbolic definitions can be obtained for direct analogy to the two-conductor case by defining the matrix hyperbolic functions. First define the matrix exponentials as
(4.77b) In terms of these matrix exponentials, we may define the matrix hyperbolic functions as
208
FREQUENCY-DOMAIN ANALYSIS
(4.78b)
In terms of these symbolic definitions, the chain parameter matrix submatrices can be written, symbolically, as
1(9) =cosh(JB9)
(4.79a)
= P-'c o s h ( J E 9 ) P
aI2(9) = -2, s i n h ( m 9 )
(4.79b)
= -sinh(@9)2,
-
(4.79c)
&(9) =cosh(JE0)
(4.79d)
@21(9) = 2 ; sinh(m9) =
-s i n h ( m 9 ) Z ;
= Pcosh(JB9)P-l
Observe that these reduce to the scalar results obtained for the two-conductor line in (4.23). The final chain parameter identity has to do with the inverse of the chain parameter matrix given in (4.46b). Multiplying the chain parameter matrix by its inverse and using the identity for the inverse given in (4.47b) gives 6(9)&-1(0) = lzn
-
(4.80a)
= 6(Y)6( 0 )
Substituting the form of the chain parameter matrix gives
Multiplying this out gives the following identities for. the chain parameter submatrices:
GENERAL SOLUTION FOR AN In
+ 1CCONDUCTOR LINE
209
From the series expansions of the chain parameter submatrices in (4.48) we see (4.82a) (4.82b) (4.82~) (4.82d)
(4.83a) (4.83b) (4.83~) (4.83d) (4.83e) The last identity follows from the series expansions in (4.48) and the fact that and 9 are symmetric. These identities have proven of considerable value in reducing large matrix expressions that result from the solution of the MTL equations [B.I, B.4J. 4.3.5
Incorporating the Terminal Conditions
The general solutions to the phasor MTL equations given in (4.69) involve 2n undetermined constants in the n x 1 vectors 1; and I;. Therefore we need 2n additional constraint equations in order to evaluate these. These additional constraint equations are provided by the terminal conditions at z 3:0 and z = 9 illustrated in Fig. 4.4. The driving sources and load impedances are contained in these terminal networks that are attached to the two ends of the line. The terminal constraint network at z = 0 shown in Fig. 4.4(a) provides n equations relating the n phasor voltages v(0) and n phasor currents l(0). The terminal constraint network at z = 9 shown in Fig. 4,4(b) provides n equations relating the n phasor voltages and n phasor currents @'). Alternatively, the chain parameter matrix given in (4.68) relates the phasor The chain parameter matrix does not explicitly voltages at z = 0 and at z = 9. determine these voltages and currents. Essentially then we still need 2n relations to explicitly determine the terminal voltages and currents from the chain parameter matrix relation. These again will be provided by the terminal
v(9)
210
FREQUENCY-DOMAIN ANALYSIS
hY)
+
-
I
I
I
,
t ~ y ) j,(r/r) a :
-
+
I
I I I
- -f,(Ip)
Terminal constraint network atz-
V
constraints. The purpose of this section is to incorporate these terminal constraints to explicitly determine the terminal voltages and currents and complete this final but important last step in the solution. 4.3.5.1 The Generalized Thivenin Equivaient There are many ways of relating the voltages and currents at the terminals of an n port. If the network is linear, this relationship will be a linear combination of the port voltages and currents. One obvious way is to generalize the Th6venin equivalent representation of a 1- port as cA.2) (4.84a) V(0) qs- 2,f(O)
-
V(P) = VL f 2,i(P)
(4.84b)
The n x 1 vectors frs and V, contain the effects of the independent voltage and current sources in the termination networks at z = 0 and z = 9, respectively. The n x n matrices, 2, and 2, contain the effects of the impedances and any respectively. controlledsources in the terminal networks at z = 0 and z = 9,
GENERAL SOLUTION FOR AN (n
0s"
FIGURE 4.5
+ 1KONDUCTOR LINE
211
P t
The generalized Thhenin representation of a termination with no cross
coupling.
In general, the impedance matrices, &s or &,, arefull, i.e., there is cruss coupling between all ports of a network. However, there may be terminal-network configurations wherein these impedance matrices are diagonal and the only coupling occurs along the MTL. Figure 4.5 shows such a case wherein each line at z = 0 is terminated directly to the chosen reference conductor with an impedance and a voltage source. In this case, the matrices in (4.84a) become
(4.85a)
(4.85b)
The genera! forms of the solutions of the MTL equations for the line voltages and currents were obtained in (4.59) and (4.63) as (4.86s)
212
FREQUENCY-DOMAIN ANALYSIS
(4.86b)
where the characteristic impedance matrix is defined as (4.86~)
In order to solve for the 2n undetermined constants in 1; and I;, we evaluate (4.86) at z = 0 and at z = 9 and substitute into the generalized Th6venin equivalent characterizations given in (4.84) to yield
Writing this in matrix form gives
Once this set of 2n simultaneous equations is solved for 9; and the line voltages and currents are obtained at any z along the line by substitution into (4.86). An alternative method for incorporating the terminal conditions is to substitute the generalized ThCvenin equivalent characterizations in (4.84) into the chain parameter matrix characterization given in (4.68):
to yield [Bel]
(612 - 6,,gS- 2,,622 + ~ , 6 2 1 ~ s ) ~=(SL 0 ) - (611 - 2L621)?s(4.90a) i(9)= $21vs+ ($z2 - 6212s)i(o) (4.90b) Equations (4.90a) are a set if n simultaneous, algebraic equations which can be solved for the n terminal currents at z = 0, I(0). Numerous Gauss-eliminationtype subroutines for digital computers are available to solve these equations [A,2, 1.1). Once these are solved, the n terminal currents at z = 9,1(9),can
GENERAL SOLUTION FOR AN (n
FIGURE 4.6
+ 1KONDUCTOR LINE
213
The generalized Norton representation of a termination with no cross
coupling.
be obtained from (4.90b). The 2n terminal voltages, v(0) and obtained from the terminal relations in (4.84).
v(9),can be
4.3.5.2 The Generalized Norfon Equivalent The generalized Th6venin equivalent in the previous subsection is only one way of relating the terminal voltages and currents of a linear n port. An alternative representation is the generalized Norton equivalent wherein the voltages and currents are related by
i(0) = is - 9&0)
i(9)= - i L
+ P'V(3)
(4.91a) (4.91b)
The n x 1 vectors is and f,, again contain the effects of the independent voltage and current sources in the termination networks at z -- 0 and z = 9,respectively. The n x n matrices, ?, and 9, again contain the effects of the impedances and any controlled sources in the terminal networks at z = 0 and z = 9,respectively. Again, the admittance matrices, 9, or QL, may befull, i.e., there is cross coupling between all ports of a terminal network. However, there may be terminal network configurations wherein these admittance matrices are diagonal and the only coupling occurs along the MTL. Figure 4.6 shows such a case wherein each line is terminated at z = 0 directly to the chosen reference conductor with an admittance in parallel with a current source. In this case, the matrices in (4.91a) become
(4.92a)
214
FREQUENCY-DOMAIN ANALYSIS
1
0
(4.92b)
::: :::
".
The 2n undetermined constants, 1 : and I;,in the general solution given in (4.86) are once again found by evaluating these solutions at z = 0 and at z = 9 and substituting into the generalized Norton equivalent characterizations given in (4.91) to yield
Writing this in matrix form gives
Once this set of 2n simultaneous equations are solved for 1; and I;, the line voltages and currents are obtained at any z along the line by substitution into (4.86). An alternative method of incorporating the terminal conditions is to again substitute the generalized Norton equivalent terminal relations given in (4.91) into the chain parameter representation given in (4.89) to yield
Equations (4.95a) are once again a set of n simultaneous, algebraic equations which can be solved for the n terminal voltages at z = 0, V(0). Once these are solved, the n terminal voltajes at z = 9, ?(2), can be obtained from (4.95b). The 2n terminal currents, I(0) and i(2),can be obtained from the terminal relations in (4.91). 43.5.3 Mixed Representations There are numerous cases where both terminations cannot be represented as generalized Th6venin equivalents or as generalized Norton equivalents CF.8, G.41. For example, suppose some of the conductors are terminated at z = 0 to the reference conductor in short circuits. In this case, the generalized Norton equivalent in (4.91a) does not exist for this termination since the termination admittance is infinite. However, the generalized Thhvenin
GENERAL SOLUTION FOR AN (n
+ I)-CONDUCTOR LINE
115
equivalent in (4,84a) does exist since the short circuit is equivalent to a load impedance of 0 which is a legitimate entry in Shielded wires in which the shield (one of the MTL conductors) is "grounded" to the reference conductor represent such a case. Conversely, one of the conductors may be unterminated, Le., there is an open circuit between that conductor and the reference conductor. In this case we must use the generalized Norton equivalent (the termination has 0 admittance) since the generalized Th6venin equivalent does not exist (the termination has infinite impedance). An example of this is commonly found in balanced wire lines such as twisted pairs where neither wire is connected to the reference conductor CG.1-G.101. This calls for a mixed representation of the terminal networks wherein one is represented with a generalized Thkvenin equivalent whereas the other is represented with a generalized Norton equivalent. We now obtain the equations to be solved for these mixed representations. Using (4.84a) and (4.91b) yields
or, via the chain parameter matrix,
Similarly, using (4.91a) and (4.84b) yields
or, using the chain parameter matrix,
The above mixed representation can characterize termination networks wherein short-circuit terminations exist within one termination network and open-circuit terminations exist within the other termination network. Terminal networks wherein both short-circuit and open-circuit terminations exist within the same network can be handled with a more general formuIation such as
P,V(O) + ZJ(0) = €$ PL9(U)+ 2J(U)= P L
(4.1OOa) (4.100b)
216
FREQUENCY-DOMAIN ANALYSIS
--
For example, (4.100a) can be written in partitioned form as
pl' i7"I[ A
yz,
y22
9s
] + [-2
O,(O)
92(0)
212][p
l2
A
&(O)
]
=
[!'I + ["'I
(4.101)
z21 2 2 2 2(0) Is2 vsz p -
m
2s
V(0)
@S
Suppose there is no cross coupling within this termination network with the first set of terminals characterized as Norton equivalents as in Fig. 4.6 and the last set of terminals characterized as Thbvenin equivalents as in Fig. 4.5. The partitioned general form becomes
-- -- El 1,;[
[
Qll
,
0
9s
0
Vl(0)
ll[tZ(OJ
+
[:
;&oJ
=
+
(4.102)
A . -
2,
V(0)
-do)
MI
PS
where 9, and 2,, are diagonal. If any of the admittances are zero we set the appropriate entry in P,, to zero, whereas if any of the impedances are zero we set the appropriate entry in gZ2 to zero. The more general terminal constraints accommodated by (4.100) can also be incorporated into the MTL descriptions given by either (4.69) or the chain parameter representation given by (4.68) by similarly partitioning those MTL descriptions to yield a set of 2n or n simultaneous equations to be solved for the phasor terminal line voltages or currents. The result is somewhat more complicated than a single Th6venin or Norton representation and will be considered in more detail in Chapter 8. 4.3.6
Approximating Nonuniform Lines
As discussed previously, nonuniform lines are lines whose cross-sectional dimensions (conductors and media) vary along the line axis CB.1, 13, 141. For these types of lines, the per-unit-length parameter matrices will be functions of z, Le., R(z), L(z), G(z), and C(z). In this case the MTL differential equations become nonconstant-coe~cientdi@rentinl equations. Although they remain linear (if the surrounding medium is linear), they are as difficult to solve as nonlinear differential equations. A simple but approximate way of solving the MTL equations for a nonuniform MTL is to approximate it as a discretely uniform MTL. To do this we break the line into a cascade of sections each of which can be modeled approximately as a uniform line characterized by a chain parameter matrix as illustrated in Fig. 4.7. The overall chain parameter matrix of the entire line can be obtained as the product (in the appropriate order) of the chain parameter matrices of the individual uniform sections as
&(L?) = &(AzN) x h=l
* *
x
&h(AZh)
x
* *
x &l(Azl)
(4.103)
GENERAL SOLUTION FOR AN (n
+ I)-CONDUCTOR LINE
217
Representation of a line as a cascade of uniform sections each of which is represented by its chain parameter matrix. FIGURE 4.7
Observe the important order of multiplication of the individual chain parameter matrices. This is a result of the definitions of the chain parameter matrices as
Many nonuniform MTL’s can be approximately modeled in this fashion. Once the overall chain parameter matrix of the entire line is obtained as in (4.103), the terminal constraints at the ends of the line may be incorporated as in the previous section then the model solved for terminal voltages and currents. Voltages and currents at interior points can also be determined from these terminal solutions by using the individual chain parameter matrices of the uniform sections. For example, the voltage at the right port of the second subsection can be obtained from the terminal voltages and currents as (4.105) One such application is the analysis of MTL‘s consisting of twisted pairs of wires CG.1-G.101. Consider the case of two twisted wires shown in Fig. 4.8(a). This can be approximated as a sequence of abrupt loops in cascade as illustrated in Fig. 4.8(b). The chain parameter matrices of the uniform sections are then multiplied together along with an interchange of the voltages and currents at the junctions. If the lengths of the sections are assumed to be identical, then the overall chain parameter matrix is the N-th power of the chain parameter matrix of each section which can be computed quite efficiently using, for example, the Cayley-Hamilton theorem for powers of a matrix CA.2, 1, 2, 31.
218
FREQUENCY-DOMAIN ANALYSIS k
(b)
Approximate representation of a twisted-pair of wires as a cascade of uniform sections. FICURE4.8
’
Terminal
network
\
Shield’
’
Terminal network
I
I
2-0
I
2 - 9
/
2
Representation of a shielded line having pigtails as a cascade of three uniform sections.
FICURE4.9
Another application is shown in Fig. 4.9. Shielded cables frequently have exposed sections at the ends to facilitate connection of the shield to the terminal networks [FA-F.8). The shield is connected to the terminations via a “pigtail” wire over the exposed sections. The overall chain parameter matrix can then be obtained as the product of the chain parameter matrices for the pigtail and spR and the chain parameter matrix of the shielded sections of lengths 9pL section of length Psas
Once this overall chain parameter matrix is obtained, the terminal constraints are incorporated in the usual fashion in order to solve for the terminal voltages and currents.
SOLUTION FOR LINE CATEGORIES
4.4
219
SOLUTION FOR LINE CATEGORIES
One of the primary problems in this solution process is the determination of the chain parameter matrix, 6.The solution process for determining the submatricesdescribed previously assumes that one can find an n x n, nonsingular transformation matrix, ?, which diagonalizes the product of per-unit-length parameter matrices, 92,as = 91
f--'P$?
(4.107a)
where f a is diagonal as
(4.107b)
This is a classic problem in matrix analysis CA.4, 1-41, The n values,$, are said to be the eigenvalues of the matrix 92. Premultiplying (4.107a) by T gives
92T - 992 = 0 Let us denote the n x 1 columns of? as '
'
0
TI
(4.108)
where
?J
* *
(4.109)
Substituting (4.109) and (4.107b) into (4.108) and expanding the result gives n sets of simultaneous equations as
(92- ffl")?, = 0) 1
(4.1 10)
The columns of f-, ?,, are said to be the eigenvectors of the matrix 9%CA.41. Thus the question becomes whether we can find n linearly independent eigenvectors ofpi?, which will diagonalize it as in (4.107). Equations (4.1 10) are a homogeneous set of linear, algebraic equations. As such, they have: 1. The unique trivial solution TI= 0. 2. An infinite number of solutions for the I?; CA.41.
220
FREQUENCY-DOMAIN ANALYSIS
Clearly we want to determine the nontrivial solution which will exist only if the determinant of the coeficient matrix is zero; i.e.,
(92- #inl = 0
(4.111)
We now set out to investigate when this is possible and t o w to compute it. There are a number of known cases of n x n matrices, M , whose diagonalization is assured. These are CA.4, 31: 1. All eigenvalues of
h are distinct.
fi is real, and symmetric. fi is complex but normal, i.e., AM'* = M1*hwhere we denote the transpose of a matrix by t and its conjujate by *. 4. h is complex and Hermitian, i.e., 6l = MI*. For normal or Hermitian h,the transformation matrix can be found such that $-' = (f")*. For a real, symmetric Mythe transformation matrix can be found 2. 3.
such that T-' =TI.For other types of matrices, we are not assured that a nonsingular transformation can be found that diagonalizes it. The matrix product to be diagonalized is expanded as
92
-
(G + jwC)(R GR +@CR
+jwL) +jwGL - o'CL
(4.1 12)
There exist digital computer subroutines that find the eigenvalues and eigenvectors of a general complex matrix. These can be used to attempt to diagonalize 92.However, because the number of conductors, n, of the MTL can be quite large, it is important to investigate the conditions under which we can obtain an eficient and numerically stable diagonalization. The following sections address that point. 4.4.1
Perfect Conductors in Homogeneous Media
Consider the case of perfect conductors for which R = 0. The matrix product becomes (4.1 13) 9%= (G + jwC)(jwL)
=@GL
- o'CL
If the surrounding medium is homogeneous with parameters o, we have the important identities:
6,
and p, then
CL = LC = pc1,
(4.1 14a)
GL = LG = pol,,
(4.1 14b)
SOLUTION FOR LINE CATEGORIES
and
221
% is already diagonal. In this case we may choose
9s 1,
(4.1 15a)
and all the eigenvalues are identical giving the propagation constants as
f=Jm
(4.115b)
=a+jP
In this case, the chain parameter submatrices in (4.70) become
61,
cosh(Q9)1,
(4.116a)
&12
= -sinh(99)&
(4.1 16b)
$21
= -sinh(Q9)2;'
(4.116~)
$22
= cosh(fS)l,
(4.116d)
3 :
where A
jw
2cS-L
(4.1 17a)
9
(4.1 17b)
In the medium, in addition to being homogeneous, is also lossless, u = 0, the propagation constant becomes
9=jwfi
(4.118)
so that the attenuation constant is zero, a = 0, and the phase constant is /? = w f i . The velocity of propagation becomes
u=-
0
P
(4.119)
The chain parameter submatrices simplify to (4.120a) (4.120b)
222
FREQUENCY-DOMAIN ANALYSIS
-jsin(@.EP>2;1
$21
3 :
$22
= cos(pa)l,
= -j u sin(p9)C
(4,120c) (4.120d)
where the characteristic impedance becomes real given by (4.12 la) (4.12 1b)
This case of perfect conductors in a homogeneous medium (lossless or lossy) forms the purest form of TEM waves on the line, In the following sections we investigate the quasi-TEM mode of propagation wherein the conductors can be lossy and/or the medium may be inhomogeneous. 4.4.2
Loay Conductom in Homogentour Mtdia
Consider the case where we permit imperfect c?nductors, R # 0,but assume a homogeneous medium. The matrix product PZ in (4.1 12) becomes, using the identities for a homogeneous medium given in (4.1 14).
P2 e: GR + fuCR + ( CR
j ~ -~w'PE)~,, a
(4.122)
+ (jopa - w2ps)ln
where we have substituted the identity U G=C
(4.123)
E
and have neglected the internal inductance of the wires, L,= 0, for reasons discussed in Chapter 3. From (4.122) we need only diagonalize CR as
SOLUTION FOR LINE CATEGORIES
The eigenvalues of
223
92 become (4.125)
The key problem here is finding the transformation matrix which diagonalizes CR as in (4.124). Thus we need to diagonalize the product of two matrices. A numerically stable transformation can be found to accomplish this in the following manner. Recall that C is real, symmetric, and positive dejnlte and R is real symmetric. First consider diagonalizing C. Since C is real and symmetric, one can find a real, orthogonal tran$ormation U which diagonalizes C where U-' = U' CB.1, 31: U'CU = B2 (4.126)
Since C is real, symmetric, and positive definite, its eigenvalues 0; are all real, nonzero, and positive CB.11. Therefore the square roots of these, 0, = ,/@,will be real, noitzero numbers. Next form the real, symmetric matrix product
-
e-wcue-'= i, Now form
(4.127)
e-'u'cRue= e - ~ u ~ c u e - ~ ~ ~ ~ u (4.128) e 1"
= OU'RUO
The matrix OU'RU8 is real and symmetric so it can also be diagonalized with a real, orthogonal transformation, S,as S'[BU'RUBJS = S'[O-'U'CRUBJS = A2
(4.129)
This result shows that the desired transformation is
T = UBS
(4.130)
The inverse of T is T-' = S'8-'Ut = TIC-'. There are numerous digital computer subroutines that implement the diagonalization of the product of two
224
FREQUENCY-DOMAIN ANALYSIS
real, symmetric matrices, one of which is positive definite in the above fashion [1.1]. The entries in the per-unit-length resistance matrix, R, are functions of the square root of frequency, at high frequencies due to the skin effect. This does not pose any problems in the phasor solution since we simply evaluate these at the frequency of interest and perform the above computation at that frequency. Reevaluate R at the next frequency and perform the above computation at that frequency. Reevaluate R at the next frequency and perform the above computations for that frequency and so forth. We will find in the next poses some significant problems in chapter that this dependence of R on the time-domain analysis of MTL's.
a,
fi
4.4.3
Perfect Conductors in Inhomogeneous Media
We next turn our attention to the diagonalization of the matrix product $2 where we assume perfect conductors, R = 0, and lossless media, G = 0. The surrounding, lossless medium may be inhomogeneous in which case we no longer have the fundamental identity in (4.1 14a), Le., CL # p l , , In fact, the product of the per-unit-length inductance and capacitances matrices is generally not diagonal. The matrix product to be diagonalized becomes
92 = -02CL
(4.131)
Once again, C and L are real, symmetric and C is positive definite. Therefore we can diagonalize CL as in the previous section with orthogonal transformations as (4.132)
The transformation matrix, 'f', which accomplishes this is real and given by
T=UM
(4.133a)
where U and S are obtained from
U'CU = e2
(4.133b)
SOLUTION FOR LINE CATEGORIES
o
e;
Lo
225
-. o
s y e u w e ) s = sye- 1u'cLue)s =: A'
e:J e : (4.133~)
The inverse of T is T-' = S'0-'Ut = T'C''. The desired eigenvalues of 9%are
.p: = -a%: 4.4.4
(4.134)
The General Case: lossy Conductors in lossy Inhomogeneous Media
In the most general case wherein the line is lossy, R # 0, G # 0, and the medium is inhomogeneous so that LC # l/u21,, we have no other recourse but to seek a fieqrtertcy-dependent transformation, T(o), such that
Although we are not assured of a numerically stable diagonalization, a more important computational problem here is that the transformation matrix, T(w), is frequency dependent and must be recomputed at each frequency! Thus an eigenvector-eigenvalue subroutine for complex matrices must be called repeatedly at each frequency which can be quite time-consuming if the responses at a large number of frequencies are desired. In order to provide for this general case, a general frequency-domain FORTRAN program, MTL.FOR, which determines T(w) via (4.135) and incorporates the generalized Thevenin equivalent terminal representations in (4.84) is described in Appendix A. Other FORTRAN codes that efficiently implement the considerations in Sections 4.4.1,4.4.2, or 4.4.3 which make assumptions about the losses of the line and/or the homogeneity of the surrounding medium are described in [I.lJ. 4.4.5
Cyclic Symmetric Structures
The MTL structures considered in Sections 4.4.1, 4.4.2, and 4.4.3 are such that the matrix product, 92,can always be diagonalized with a numerically efficient and stable similarity transformation, T, which is frequency independent. Not all structures can be diagonalized in this fashion. One can try to diagonalize Pi! with a digital computer subroutine that determines the eigenvalues and eigenvectors of a general, complex matrix such as 9%but we are not assured
226
FREQUENCY-DOMAIN ANALYSIS
that the eigenvectors will be linearly independent. Furthermore, the transformation matrix, T(o),will befiequency dependent as was demonstrated in the previous section. This section discusses MTL's which have certain structural symmetry so that a numericall stable (and trivial) transformation 9 can always be found which diagonalizes 2.Furthermore this transformation isfrequency independent regardless of whether the line is lossy or the medium is inhomogeneous, i.e., the general case. Consider structures composed of n identical conductors and a reference conductor wherein the n conductors have structural symmetry with respect to the reference conductor so that the per-unit-length impedance and admittance matrices have the following structural symmetry CB.1, 113:
4
$3
(4.136a)
(4.136b)
Examples of structures which result in these types of per-unit-length parameter matrices are shown in Fig. 4.10. Observe that in order for the main diagonal terms to be equal, the conductors and surrounding media (which may be inhomogeneous) must also exhibit symmetry. For example, if the n conductors are dielectric-insulated wires, the dielectric insulations of the n wires must have identical 8 and thicknesses. The reference conductor need not share this property. A general cyclic symmetric matrix fi has the entries given by (4.137a) where (4.137b) (4,137~)
+
and indices greater than n or less than 1 are defined by the convention: n j =j and n + i = i EB.11. Because of this special structure of the per-unit-length
SOLUTION FOR LINE CATEGORIES
227
/ -4
I
/
/
/
FIGURE 4,lO Cyclic symmetric structures which are diagonalizable by a frequencyindependent transformation.
and we are guaranteed matrices, they are normal matrices, &(2')*= (&')*2, that each can be diagonalized as CB.1, 1-41
(4.138a) (4.138b) &here the n x n matrices 9; and ff are diagonal CB.1). In fact, the transformation
228
FREQUENCY-DOMAIN ANALYSIS
is trivial to obtain CB.11: (4.139) and $-1
Similarly, the eigenvalues of mined as CB.11
= @)*
(4.140)
92 (the propagation constants) are easily deter-
(4.141) As an illustration of these results, consider a four-conductor (n = 3) line with a cyclic symmetric structure so that
(4.142a)
(4.142b)
The transformation matrix is
(4.143a)
(4.143b)
and the propagation constants are
SOLUTION FOR LINE CATEGORIES
229
There are a number of cases where a MTL can be approximated as a cyclic symmetric structure. A common case is a three-phase, high-voltage power transmission line consisting of three wires above earth. In order to reduce interference to neighboring telephone lines, the three conductors are transposed at regular intervals. As an approximation, we may assume that each of the three (identical) wires occupy, at regular intervals, each of the three possible positions along the line (all of which are at the same height above earth and produce identical separation distances between adjacent wires). With this assumption, the per-unit-length matrices, 2 and 9, take on a cyclic symmetric structure:
(4.145a)
(4.145b)
LP
P)
PJ
The transformation matrix is given by (4.143) and the propagation constants simplify to
9: = (2 + 22')(P + 2P) 9; = (2 - 2/)(P - P) p i = (2- 2/)(F- P)
(4.146a) (4.146b) (4.146~)
Two of the propagation constants, j$ and f3, are equal and these are associated with the aerial mode of propagation. The third propagation constant, 9,, is associated with the ground mode of propagation. This transformation is referred to in the power transmission literature as the method ofsymmetrical components. Such lines are said to be balanced. In the case of unbalanced lines where, for example, one phase may be shorted to ground, this transformation does not apply. Other approximations of MTL's as cyclic symmetric structures are useful. Cable harnesses carrying tightly packed, insulated wires have been assumed to be cyclic symmetric structures on the notion that all wires occupy at some point along the line all possible positions. This leads to a cyclic symmetric structure of the n x n per-unit-length impedance and admittance matrices that is similar to the special case of transposed power distribution lines shown in (4.145). Other common cases are the cyclic symmetric, three-conductor lines shown in Fig. 4.11. Two, identical, dielectric-insulated wires are suspended at equal heights above a ground plane as shown in Fig. 4.11(a). The per-unit-length
230
FREQUENCY-DOMAIN ANALYSIS
d
k
W
I
i
w
FIGURE 4.11 Certain three-conductors structures that are cyclic summetric: (a) two
identical wires at identical heights above a ground plane, (b) two identical lands of a coupled microstrip configuration.
impedance and admittance matrices become (4.147a) (4.147b) The transformation matrix and propagation constants simplify to (4.148a)
p: = (2+ 2#)( P+ 9) 9j = (2 - 2?(P- P)
(4.148b) (4.148~)
This transbrmation is referred to in the microwave literature as the euen-odd mode transformation and has been applied to the symmetrical, coupled microstrip line shown in Fig. 4.1 l(b).
231
LUMPED-CIRCUIT ITERATIVE APPROXIMATE CHARACTERIZATIONS
r.
kY
-e
'I""
( -e p
4".
Y
@
tu
I I
-
I
N'
I
o I
I I
I
I
I
f
-
FIGURE 4.12
Lumped-circuit iterative approximate structures: (a) lumped 7,(b) lumped
1.
4.5
LUMPED-CIRCUIT ITERATIVE APPROXIMATE CHARACTERIZATIONS
Lumped-circuit notions apply to circuits whose largest dimension is electrically small, Le., <
232
FREQUENCY-DOMAIN ANALYSIS
0 0
@
0
0 @ (4 FIGURE 4.12 (Continued)
Lumped-circuit iterative approximate structures: (c) lumped
PI, and (d) lumped T.
that the total parameter is the per-unit-length multiplied by the section length,
9/N.These structures are named for the symbols their structures represent:
r,
1, A, T. The chain parameter matrices of these structures can be derived in a straightforward fashion as CB.1J (4.149a)
(4.149b)
LUMPED-CIRCUIT ITERATIVE APPROXIMATE CHARACTERIZATIONS
233
This overall chain parameter matrix of a line that is represented as a cascade of N such lumped sections is
6
(4.150)
&kr,n,T)
Once this overall chain parameter matrix is obtained, the terminal conditions are incorporated as described in Section 4.3.5 to give the terminal voltages and currents of the MTL. Lumped-circuit analysis programs such as SPICE can be used to analyze the resulting lumped circuit as an alternative to obtaining the overall chain parameter matrix via (4.150) and then incorporating the terminal conditions. Nonlinear terminations such as transistors and diodes can be readily incorporated into the terminations since these lumped-circuit programs include sophisticated models for them. We will find this notion of the lumped iterative model to be a useful approximation in the time-domain analysis of the MTL in the next chapter since these lumped-circuit programs can be used to perform the time-domain analysis with a simple change in a control statement. However, there are additional considerations in the use of this approximate model in the time-domain analysis of MTL‘s. It is interesting to compare the chain parameter matrices for the above lumped-circuit structures to the exact chain-parameter matrix given in series form in (4.48) for the entire line of length 9:
9P -LZ3 s5 -[92]9- [92]2P l! 31 5!
1.
t
(4.1 51) * *
Comparing these submatrices to the submatrices in the chain parameter
234
FREQUENCY-DOMAIN ANALYSIS
matrices for the lumped-iterative structures given in (4.149) we observe that the lumped a and lumped T structures give a better representation for N = 1, i.e., representing the total line with only one lumped section, than the lumped 1 and lumped structures. Representing the line with smaller sections and representingeach section by one of the lumped iterative models gives the overall chain parameter matrix as the power of the lumped iterative chain parameter matrix as shown in (4.150). We are assuming that this would give a better approximation to (4.151). As a rudimentary investigation of this premise, let us represent the line with two lumped 1sections shown in Fig. 4.12(a) each of length 2 / 2 . The overall chain parameter matrix is the chain parameter matrix in (4.149a) with N replaced by 2 and raised to the second power:
r
-UP - 8 LP219)
a"
{
1,
g2 +34 92 + 16 [92]')
(4.152)
Comparing this to the exact chain parameter matrix in (4.151) we do observe some convergence although the quantitative aspects of convergence are not clear. 4.6
ALTERNATIVE 2wPORT CHARACTERIZATIONS
The chain parameter matrix is not the only way of relating the voltages and currents of the MTL viewed as a 2n port. Other obvious ways are the impedance parameters CA.21: (4.153) and the admittance.parameters-CA.21; (4.154) The currents at both ends, i(0) and -i(2),are defined as being directed into the 2n port in accordance with the usual convention. These alternative representations can be obtained from the chain parameter submatrices. For example, the impedance parameter submatrices are defined, from (4.153), by setting currents equal to zero as
LUMPED-CIRCUIT ITERATIVE APPROXfMATE CHARACTERIZATIONS
V(0) =
- $1 2 W)I~(0)=0
VW) = t 2 l f ( o ) I i ( 9 ) - 0
235
(4.155b) (4.15%)
V(0) = ~ 1 1 4 ~ ) 1 ? ( 1 i . , 4
(4.155d)
From the chain parameter matrix, setting i(0) = 0, we obtain
V ( 8 = $1 I V W f(9) = 6,,9(0) from which we obtain
(4.156a) (4.156b)
t2,= -6 I 1 6-1 21 != - & - I21 &
(4.157a)
22
2,, = -6;;
(4.157b)
Similarly setting i(9)= 0 in the chain parameter matrix gives
V ( 9 ) = &llv(o)+ 612~(o)
(4.158a)
0 = 4,,0(0)-1- 6 2 2 1 ( 0 )
(4.158b)
from which we obtain
2,, = 6,, - d$l&;:622 = -6;; 2,,= -6-16 21 22 611$ - I21
(4.159a) (4.159b)
3:-
We have used the matrix chain parameter identities given in (4.83) to give equivalent forms of these imeedance parameters which demonstrate that the line is reciprocal, i.e., gIl= ZZ2and g12 = The admittance parameters can also be derived from the chain parameters in like fashion or by realizing that the admittance parameter matrix is the inverse of the impedance parameter matrix. This yields
PI, = 9 2 , = - 9 1A-16 2 11 = PI, = P,, = 6;;
22 6-1 I2
(4.160a) (4.160b)
Although these appear to be viable alternatives to the chain parameter matrix representation, they have some unique problems. The primary problem is that the impedance and admittance parameter matrices do not exist for certain frequencies where the line length is some multiple of a half-wavelength, i.e., Y = kA/2. For example, for the case of a lossless line, R = 0, G = 0, in a
236
FREQUENCY-DOMAIN ANALYSIS
homogeneous medium, LC e: pel,, the impedance parameters are obtained froin the chain parameter matrices given in (4.120) as (4.161a) I it,,= 2,,= -6;: = -1 2,
sin(p.9')
(4.161b)
Recall that p = 2n/A so that the denominators of these expressions, sin(p.9') = sin(2&"/l), are zero for frequencies where the line length is some multiple of a half-wavelength. Similar remarks apply to the admittance parameter matrix in (4.154). The chain parameter matrix exists and is nonsingular for all frequencies and all line configurations as was shown by analogy to the state-variable formulation in Section 4.3. 4.7
POWER AND THE REFLECTION COEFFICIENT MATRIX
As a final analogy to the two-conductor line let us define the reflection coeBcient matrix, f(z), and investigate the flow of power on the line. The phasor voltages and currents are written in the form of forward-traveling waves, ?+(z) and It@),and backward-traveling waves, ?-(z) and f-(z), from (4.59) and (4.63) as
where
?(z) = O+(z) + V-(z) f(z) = f+(z) I-(z)
(4.162b)
8 + ( z )= 2c9e-9zf:
(4.163a)
?-(z> = 2c+?e9zi;;;
(4.163b)
-
(4.162a)
and (4.164a) (4.164b) Let us define the reflection coefficient matrix in a logical manner relating the reflected or backward-traveling voltage waves to the incident or forwardtraveling voltage waves at any point on the line as
V ( z ) = f(Z)?+(Z) Substituting (4.163) gives
(4.165)
POWER AND THE REFLECTION COEFFICIENT MATRIX
237
where the load reflection coefficient matrix is defined as f, = P ( 9 ) . From this relation, the reflection coefficient matrix at any point on the line can be related to the load reflection coefficient matrix which we will show can be explicitly calculated knowing the termination impedance matrix and the characteristic impedance matrix. Thus, the voltage and current vectors can be written as
The input impedance matrix at any point on the line relates the voltages and currents at that point as
Substituting (4.167) yields
Similarly, the reflection coefficient matrix can be written in terms of the input impedance matrix at a point on the line from (4.169) as
If the line is termined at z = 9 as (4.171) then the reflection coefficient matrix at the load is
These formulae reduce to the corresponding scalar results for a two-conductor line. From this last result we observe that in order to eliminate all reflections at the load, the line must be terminated in its characteristic impedance matrix, Le., 2, = 2,. This is the meaning of a matched line in the MTL case. It is not sufficient to simply place impedances only between each line and the reference conductor. Impedances will need to be placed between all pairs of the n lines since the characteristic impedance matrix is, in general, full. The total average power transmitted on the MTL in the +z direction A
238
FREQUENCY-DOMAIN ANALYSIS
(4.173)
where * again denotes the conjugate of the complex-valuedquantity, Substituting the voltages and currents in terms of forward- and backward-traveling waves as in (4.162) gives pa, &&{Q+'f+* + V-'f+*
-
?+If-*
+ Q-'f-*}
(4.174)
The first term, ?+'f+*, gives the average power carried by the forward-traveling waves, and the last term, v-'f-*, gives the average'Fower carried b i the backward-traveling waves. The middle two terms, V ' I+* and ?+'f- , are cross-coupling terms between the waves. Suppose the line is matched at its load, i.e., gL = gC,So that the load reflection coefficient matrix is zero, i.e., f, = 0. Equation (4.166) shows, as expected, that the reflection coefficient matrix is zero at all points on the line, i.e., f(z) = 0. Thus there are only forward-traveling waves on the line, and there is no power flow in the -2 direction. These properties are, of course, also directly analogous to the scalar, two-conductor line. The use of matrix notation allows a straightforward adaptation of the scalar results to the MTL case although there are some peculiarities unique to the MTL.case. For example, equation (4.172) reduces to the familiar twoconductor case where the termination and characteristic impedance matrices become scalars. 4.8
COMPUTED RESULTS
In this section we will show some computed and experimental results that demonstrate the prediction methods of this chapter. The frequency-domain prediction model is implemented in the computer program MTLFOR described in Appendix A. This program determines the 2n undetermined constants in the general form of solution in (4.86):1.; The terminal configurations for both structures are shown in Fig. 4.13. These are characterized as a generalized Thtvenin equivalent as in (4.84) where
COMPUTED RESULTS
*
--
P
239
c
--
0 0 0
-
+ +I
(P)
-
-
son
so n
FIGURE 4.13 A three-conductor line for illustrating numerical results.
Two configurations of a three-conductor line ( n = 2) are considered: a threewire ribbon cable and a three-conductor printed circuit board. Experimentally determined frequency responses will be compared to the predictions of the MTL model as well as those of the lumped-pi iterative approximation. 4.8,l Ribbon Cables
The cross section of the three-wire ribbon cable is shown in Fig. 4.14. The total line length is 9 = 2 m. The per-unit-length parameters for this configuration were computed using the computer program RIIEiBON.FOR described in Appendix A and are given in Chapter 3: [,748 50 0.507 7 1 PH/m 0.5077 1.0154
[
37.432 - 18.7 16
- 18.7161 pF,m 24.982
The experimental results are compared to the predictions of the MTL model using MTL.FOR, with and without losses, over the frequency range of 1 kHz to 100 MHz in Fig. 4.15. Observe that below 100 kHz, losses in the line conductors are important and cannot be ignored. The dc resistance was
240
FREQUENCY-DOMAIN ANALYSIS
d
d
@@@ ill
@ d-50milr
0
8,
0I
-
-
7.5 mils (+2S gauge stranded 7x36) = 10 mils E, 3.5 (PVC) ru I
FIGURE 4.14 Dimensions of a three-conductor ribbon cable for illustrating numerical
results.
Frequency (Hz)
G FIGURE 4.15 Comparison of the frequency response of the near-end crosstalk of the
ribbon cable of Fig. 4.14 determined experimentally and via the MTL model with and without losses: (a) magnitude.
computed using (3.200) for one of the #36 gauge strands (tw= 2.5 mils) and dividing this result by the number of strands (seven) to give 0.19444 Q/m. The skin effect was included by determining the frequency where the radius of one of the #36 gauge strands equal two skin depths, rw = 28 = 2/,/=, according
COMPUTED RESULTS
Near-End Crosstalk Voltage (Ribbon cable) .v..,. ... ' ....' ' " ' ' 1
80 -
I
..'..a
' " 1 9 1 - 1
I
I
241
I * .
60 40
-
-a
v
-60 -80
---
.........
10'
104
10'
106
10
10'
Frequency (Hz) (b) FIGURE 4.15
=
(Continued) (b) phase.
r(f)=
to Fig. 3.52 (S, 4.332 MHz) and taking the resistance vary as rdom above this. Both skin effect resistance and internal inductance are Both the magnitude and the phase are well included according to
to (3.225).
predicted. Observe that the line is one wavelength (ignoring the dielectric insulation) at 150 MHz. So the line is electrically short below, say, 15 MHz. Observe that the magnitude of the crosstalk for the lossless case (and a significant portion of the lossy case) increases directly with frequency, i.e., 20 dB/decade. We will find this to be a general result in Chapter 6, Figure 4.16 shows the predictions of the lumped-pi (n) approximate model of Fig. 4.12(c) using one and two pi sections to represent the entire line. The wire resistances and internal inductances are assumed to be the dc values over the entire frequency range for these lumped-pi models since the skin effect dependency is difficult to model in the lumped-circuit program SPICE which was used to solve the resulting circuit. Both one and two pi sections give virtually identical predictions to the exact MTL model for frequencies below which the line is one-tenth of one wavelength long. Observe that two pi sections do not substantially improve the accuracy of the predicted frequency range even though the circuit complexity is double that for one pi section.
(a)
4.8.2
Printed Circuit Boards
The next configuration is a three-conductor printed circuit board whose cross section is shown in Fig. 4.17. The total line length is Y = 10 inches = 0.254 m.
242
FREQUENCY-DOMAIN ANALYSIS
Now-End Crosstalk Voltrpo
(Ribbon rrbk) I
1
I
T l l l t l l ,
I 1 1 1 1 1 1 1
I 1 1 1 1 1 1 1 ,
I
I , , ,
I*
Frequency (Hz) (a)
L 10’
0
f
i
10‘
0
~
~
~
o
~
1a
n ~ - a * ~ ,~ * 1 1 ~
106
10’
~I1
I
I
O
~
C ~*
-
,
.
107
8
~ a t o s t a t J~
1 0’
Frequency (Hz) (b)
Comparison of the frequency response of the near-end crosstalk of the ribbon cable of Fig. 4.14 determined via the MTL model with losses and via the lumped-pi model using one and two sections to represent the line: (a) magnitude, (b) phase. FIGURE 4.16
COMPUTED RESULTS
FIGURE 4.17
243
Dimensions of a three-conductor PCB for illustrating numerical results. Near-End Crosstalk Voltage (Printed circuit board)
Frequency (Hr) (0)
Comparison of the frequency response of the near-end crosstalk of the PCB of Fig. 4.17 determined experimentally and via the MTL model with and without losses: (a) magnitude. FIGURE 4.18
The per-unit-length parameters were computed using the computer program PCBGAL.FOR described in Appendix A and are given in Chapter 3: 1.104 18 0.690094 0.690094 1.38019 40.6280 -20.3140
-20.3140]
pF,m
29.7632
The experimentally obtained results are compared to the predictions of the MTL model using MTLFOR with and without losses over the frequency range of 10 kHz to 1 GHz in Fig. 4.18. The conductor resistances and internal
244
FREQUENCY-DOMAIN ANALYSIS Near-End Crosatalk Voltage (Printed circuit board)
....
L,
an
.zI
-
80
1 ___
'I....
'... I
,
,
,
,
,I
,
I
,
I
\ I
r
.........
,,,,,
It
81
Experlmental
MTL model (loray) MTL model (loailem)
Frequency (Hr) (b)
FIGURE 4.18
(Continued) (b) phase.
inductances were computed in a similar fashion to the ribbon cable. The dc resistance is rdo= l/wta = 1.291 n/m where w is the land width (w = 15 mils) and t is the land thickness for 1 ounce copper (t = 1.38 mils). The frequency where this transitions to a behavior was approximated as being where the land thickness equals two skin depths: t = 26 or f, = 14.218 MHz. Observe that the frequency where the conductor losses become important is of the order of 100 kHz. Both the magnitude and the phase are well predicted. The line is one wavelength (ignoring the board dielectric) at 1.18 GHz. Thus the line can be considered to be electrically short for frequencies below some 100MHz. Again note that the magnitude of the frequency response increases directly with frequency, 20 dB/decade, where the line is electrically short for the lossless case (and a significant portion of the lossy case). Figure 4.19 shows the predictions of the lumped-pi approximate model of Fig. 4.12(c) using one and two pi sections to represent the entire line. The conductor resistances and internal inductances are again assumed to be the dc values over the entire frequency dependency is range for these lumped-pi models since the skin effect diffcult to model in the lumped-circuit program SPICE which was used to solve the resulting circuit. Both one and two pi sections give virtually identical predictions to the exact MTL model for frequencies below which the line is one-tenth of one wavelength long. Observe that, as in the case of the ribbon cable, two pi sections do not substantially improve the accuracy of the predicted frequency range even though the cirkuit complexity is double that for one pi section.
fi
(a)
COMPUTED RESULTS
245
Near-End Crosstalk Voltage (Printed circuit board)
5a'
10
Y
ou
'a s c)
10'
Frequency (HI) (n)
Near-End Croiitalk Voltage (Printed circuit board) 80
-
I
I
.
1 1 1 1 1 1 1
1 1 . 1 1 1 1
I
A
-8
<
.. ..
-20-
-60
MTL model (lossy). 1 Pi section (lossy) 2 Pi sections (lossy) I
104
.
1o5
I
I
. . a 1 1 1 1
9
I 1 d 1 1 . 1 1
1 0'
107
I
, 1 1 1 1 . 1 1
10"
I
I
a,,-
1 09
Frequency (Hs) (b)
Comparison of the frequency response of the near-end crosstalk of the MTL model with losses and via the lumped-pi model using one and two sections to represent the line: (a) magnitude, (b) phase. FIGURE 4.19
PCB of Fig. 4.17 determined via the
246
FREQUENCY-DOMAIN ANALYSIS
REFERENCES [13 K. Ogata, State Space Analysis of Control Systems, Prentice-Hall, Englewood Cliffs, NJ, 1967. [2] C.T. Chen, Linear System rireory and Design, PIolt, Rinehart and Winston, NY, 1984. [3] F.E. Hohn, Elementary Matrix Algebra, 2d ed., Macmillan, NY, 1964. [4] A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, NY, 1965. [5] L.A. Pipes, “Matrix Theory of Multiconductor Transmission Lines,” Phil. Mag., 24,97-113 (1937). [6] S.O. Rice, “Steady State Solutions of Transmission Line Equations,” Bell System Technical Journal, XX, 131-178 (1941). [7] L.A. Pipes, “Steady-State Analysis of Multiconductor Transmission Lines,” J . A&. PhyS., 12, 782-789 (1941). [8] L.A. Pipes, “Direct Computation of Transmission Matrices of Electrical Transmission Lines,” J . Franklin Institute, Part I, 281, 275-292 (1966); Part 11, 281, 387-405 (1966). [9] D.E. Hedman, “Propagation on Overhead Transmission Lines I-Theory of Model Analysis, and II-Earth-Conduction Effects and Practical Results,” Trans. IEEE, PAS-84,200-211 (1965). [lo] L.M. Wedepohl, “Applications of Matrix Methods to the Solution of Traveling Wave Phenomena in Polyphase Systems,” Proc. IEE, 110,2200-2212 (1963). [ll] D.F. Strawe, “Analysis of Uniform Symmetric Transmission Lines,” The Boeing Company, Boeing Document D2-19734-1, August 1971. [12] W.I. Bowman and J.M. McNamee, “Development of Equivalent Pi and T Matrix Circuits for Long Untransposed Transmission Lines,” IEEE ?’runs. on Power Apparatus and Systems, 625-632 (1964). [13] E.N. Protonotarios and 0. Wing, “Analysis and Intrinsic Properties of the General Nonuniform Transmission Line,” IEEE ?’runs. on Microwave Theory and Techniques, Mn‘-15, 142-150 (1967). [14] E.C. Bertnolli, “Analysis of the N-Wire Exponential Line,” Proc. IEEE, 55, 1225 (1967).
PROBLEMS
4.1
Determine the velocity of propagation and characteristic impedance for the following transmission lines: 1. I = 0.25 pH/m, c = 100 pF/m. 2. Coaxial cable; c = 50 pF/m, e, = 2.1. 3. Two bare #28 gauge solid wires (r, = 6.3 mils) separated by 100 mils. 4. One bare # 16 gauge solid wire (r, = 25.4 mils) 1/4 inch above a ground plane.
PROBLEMS
4.2
247
Consider a lossless transmission line operated in the sinusoidal steady state. For the following problem specifications determine: 1. The line length as a fraction of a wavelength. 2. The input impedance to the line. 3. The time-domain voltage at the line input and at the load. 4. The average power delivered to the load.
The specifications are: = 1 m , f = 262.5MHz, & = SOQ i? = (30 - j 2 0 0 ) Q 2, = (100 +J50)Q, v = 300m/ps, and = l O E V. = - j 3 0 Q & = 500Q Y = 36 m, f = 28 MHz, & = l50Q (b) v = 300 m/ps, and & = 100/0" V. (c) Y = 2 m, f = 175 MHz, Zc = 100 Q, & = (200 j30) R, 2, = 50 a, v = 200 m/ps, and = 1 O R V.
(a)
8
-
4.3
Verify that (4.11) are solutions to (4.9).
4.4
Verify (4.14).
4.5
Verify (4.26).
4.6
Verify (4.28). Suppose the line is lossless and its length is one-quarter of one wavelength, Y = A/4. If it is terminated in a short circuit, ZL= 0, determine the input impedance to the line. Repeat this for an open-circuit termination, 2' = 00.
4.7
Verify (4.30).
4.8
Solve the first-order, ordinary differential equation given by (4.32) with a = -2, b = 3, and ~ ( t = ) u(t) where u(t) is the unit step function, ' ~ ( t )= 0, t .c 0,u(t) = 1, t > 0, and an initial state of x(0) = 0.
4.9
Solve the second-order,ordinary differentialequation given by (4.3 1) with
A=[-:
and ~
( t=) ~ ( t ) .
-3
248
FREQUENCY-DOMAIN ANALYSIS
4.10 Determine the state-transition matrices, U)(t), for the following:
4.1 1 Verify that (4.44~)satisfies the homogeneous n-th order state variable equations in (4.3 1). 4.12
Verify the chain parameter matrix entries given in (4.48).
4.13 Verify (4.57) and (4.63). 4.14
Verify the chain parameter submatrices in (4.70).
4.15
Verify (4,74), (4.79, and (4.76).
4.16 Determine a 2 x 2 matrix,
T, which will diagonalize the matrices in
Problem 4.10. 4.17 Consider a ribbon cable consisting of three #28 gauge wires (r, = 7.5 mils) lying in a plane with adjacent separations of 50 mils. Assume the
wires are lossless and immersed in free space. Determine the characteristic impedance matrix if one of the outer wires is chosen as the reference conductor. Calculate the chain parameter matrix for this line. 4.18 Determine the generalized Tlihenin equivalent representation, V(0) = V, - Z,I(O), of the source termination network shown in Fig. P4.18,
I
2-0
1
2
=8
c z
FIGURE P4.18
4.19
Repeat Problem 4.18 for the source termination network of Fig. P4.19.
PROBLEMS
249
FIGURE P4.19
4.20 Characterize the source and load termination networks shown in Fig. P4.20.
Rs
FIGURE P4.20
4.21 Derive the relations given in (4.90), (4.95), (4.97), and (4.99). 4.22 Verify the relations in (4.116) and (4.120). 4.23 Show that a 2 x 2 real, symmetric matrix, M, can be diagonalized with the orthogonal transformation:
T-[
cos e
sin 9
-sin
e
1
cos 9
where tan 29 =
2MI 2
MI, - M22
250
FREQUENCY-DOMAIN ANALYSIS
Show that the eigenvalues are A: = Mll cos2 0 t 2M,,cos 0 sin 0 + M,, sin2 8
A,2 = M~~sin2 e - 2M12COS e sin e
+ M2, cos20
4.24
Diagonalize the following product, CL,where
4.25
Diagonalize the following matrix which has a cyclic symmetric structure:
; : :]
4 3 2 3
M-[:
3 2 3 4
Show by direct calculation that the you obtain does in fact diagonalize Verify from this that the eigenvalues are this matrix and that ?-l = ft*. as expected according to (4.141). 4.26
Verify the relations in (4.146) for the 3 x 3 cyclic symmetric matrices given in (4.145).
4.27
Verify the chain parameter matrices for the lumped-circuit iterative structures given in (4.149).
4.28
For the ribbon cable of Problem 4.17, construct lumped-n and lumped-T
1 cm
I
FIGURE P4.30.
PROBLEMS
251
iterative circuits for a line length of 5 m where the line is represented as one section, i.e., N = 1. 4.29 Derive the relations for the admittance parameter submatrices in (4.160).
Evaluate these for a lossless line in a homogeneous medium to obtain the duals to (4.161). 4.30
Consider the MTL shown in Fig. P4.30 consisting of two #20 gauge, solid wires (r, = 16 mils) above an infinite, perfectly conducting ground plane. Determine the per-unit-length inductance and capacitance matrices and the characteristic impedance matrix assuming the line to be lossless. Determine the structure and values for a termination network that will match the line. For a total line length of 5 m and the termination structure of Fig. 4.13 plot the crosstalk magnitude and phase from 100 kHz to 100 MHz. Compare these to the predictions of the lumped-n approximation. Show your results for the lossless and lossy cases.
CHAPTER FIVE
Time-Domain Analysis
The previous chapter dealt with the solution of the MTL equations for the case of sinusoidal steady-state excitation of the line; that is, the sources are sinusoids at a single frequency and are assumed to have been applied for a sufficiently long time such that all transients have decayed to zero leaving only the steady-state solution. In this chapter we will examine the total solution of the MTL equations for general time variation of the sources. This solution will include both the transient and the steady-state components of the solution and will apply for arbitrary, time-domain excitation signals. It is important to understand the relative solution difficulty for the various classes of uniform lines at the outset. In the case of the frequency-domain solution, we have seen in the previous chapter that the solution of the MTL equations is a straightforward process whether the line is lossless or lossy! We will see in this chapter that the time-domain solution of the MTL equations for a lossless line is also a straightforward computational process! Conductors exhibit a frequency-dependent resistance and internal inductance due to the which is difficult to characterize skin effect.This dependence on frequency is in the time domain. Incorporation of these losses into the frequency-domain solution is trivial; compute the resistance and internal inductance at the frequency of interest, include them as constants in the MTL equations and solve them. For the next frequency of interest, recompute the resistance and internal inductance for this frequency and repeat the solution. Typically the loss introduced by a nonzero conductivity of the surrounding media is not as significant as the loss introduced by imperfect conductors although it can be easily incorporated into the frequency-domain solution. Although the inclusion of these skin-effect losses does not significantly complicate the frequencydomain solution, it adds considerable complications to the general, timedomain solution of the MTL equations. The additional complication in the time-domain solution over the frequencydomain solution for lossy lines is the decoupling of the MTL equations. In the frequency-domain solution, we determine an n x n transformation matrix,
fi
252
253
TWO-CONDUCTOR LOSSLESS LINES
9, that simultaneously diagonalizes two complex-valued matrices, 2(w) = R(o)+ JwL, and q(o)= C(w) + jwC although we can frequently ignore the
losses in the medium contained in C.In the case of the time-domain solution for lossless lines, in order to decouple the MTL equations we must determine an n x n transformation matrix, T,that simultaneously diagonalizes only two matrices, L and C.Because L and C are real, symmetric, and positive definite, this is always possible as we showed in the previous chapter. In the case of the time-domain solution for lossy lines, in order to decouple the MTL equaations we must determine an n x n transformation matrix, T, that simultaneously diagonalizes three matrices, R, L,and C.In general, this is not possible! So the decoupling of the MTL equations for the time-domain analysis of lossy tines is not a viable technique. We begin the discussion with a review of the solution for two-conductor, lossless lines. This serves to introduce the notion of traveling waves on the line and also introduces an important solution technique for numerical computation: the method of characteristics or Branin's method. Next we discuss the solution of the MTL equations for lossless lines. And finally, we tackle the difficult problem of the time-domain solution of the MTL equations for lossy lines.
5.1 TWO-CONDUCTOR LOSSLESS LINES
The scalar transmission-line equations for two-conductor lossless lines are (5.la) (5.lb) Differentiating one equation with respect to z and the other with respect to t and substituting yields the uncoupled, second-order differential equations (5.2a) (5.2b) The solutions to these equations are CA.13 V(Z,t ) = v + ( t
-
;) + v - ( t + ;)
(5.3a)
254
TIME-DOMAIN ANALYSIS
I(z, t ) = P ( t
- ;) + r ( t + ;)
(5.3b)
where the characteristic impedance is (5.4)
= ul 1
a-
vc and the velocity of propagation of the forward-traveling waves, V+, and backward-traveling waves, V', is
The functions of t and z, Vt(t, z ) and V-(t, z), are as yet unknown but have time and position related only us t f z/v. As an alternative method of obtaining this general solution that will prove useful in our later results, let us obtain the Laplace transform of the transmissionline equations CA.21. Denoting the Laplace transform of the voltage and current with respect to the time variable as V(z,s) and I(z, s) where s is the Laplace transform variable, the transmission-line equations in (5.1) become d V(2,s) = -sll(z, s) dz d f(2, s) = - S C V ( Z , s)
dz
(5.6a) (5.6b)
The uncoupled second-order equations in (5.2) become d2 V(2,s) = s21cV(z,s) dz2
(5.7a)
d2 Z(Z, s) = s2cN(z, s) dz2
(5.7b)
TWO-CONDUCTOR LOSSLESS LINES
255
The general solutions to these equations are (5.8a) (5.8b)
This is equivalent to the spectral-domain method of solving Laplace's equation discussed in Chapter 3 wherein the transform with respect to one of the independent variables converts the partial differential equations into ordinary differential equations. The inverse Laplace transform of these results can be easily obtained by recognizihg the fundamental time-delay result rA.2):
Applying this to (5.8) with a corresponding to z/u again gives the general form of the solution in (5.3). The Laplace-transformed chain parameter matrix obtained in the previous chapter can be obtained from the frequency-domain result derived there and given, for a lossless line, by (4.23)with f =#I:
where p = O/Y. The chain parameters relate the phasor voltages and currents at the two ends of the line. The Laplace-transformed result can be obtained by simply replacing ja, with s and assuming the line is lnithffy relaxed so that V(z, t ) = I(z, t ) = 0 for all 0 5 z 5 Y and t s 0. Recalling the definitions of cosine and sine and replacingjo with s in those relations: cos(B.9) = j sin(p9) =
gives
e loDS'/u
-
2
e JCuS?/u
-
+ e [email protected]/v
-
2
+ e JT
c+
2
enT
e-]w.rPu 0
- e-#T 2
(5.lla) (5.11b)
256
TIME-DOMAIN ANALYSIS
where we have defined the one-way defay of the line as (5.13)
The time-domain result can be obtained by applying the basic time-delay Laplace transform pair given in (5.9) to (5.12) to give
+ T ) + iV(0, t - T )- $ZcI(O, t + T )+ $Zcl(O,t - Z‘) (5.14a) = iCV(0, t + T ) - ZcI(0, t + T)] + $[V(O,t - T)+ ZcI(0, t - r)] 1 1 1 1 I ( 9 , t ) - --- V(0,t + r ) +-- V(0,t - T ) + (5.14b) t i - T) 2 ZC 2 ZC
V ( 9 ,t ) = iV(0, t
+ 3 40,t - T ) - 2-V(0,t + T ) +I(O, t + T ) + - - V(0,t -
=’[ 2
1 :[5t,
zc
T ) +I(O, t
-T)
1
This latter result will prove useful in subsequent work. It relates the voltage and current at z = 9,V ( 9 ,t), and I ( 9 , t), to those values at z = 0, V(0,t), and I(0, t), that are either delayed in time by one time delay, V(0,t - T)and I(0,t Z‘),or are adoanced in time by one time delay, V(0,t + T)and f(0, t + T). These can be placed in another form by multiplying (5.14b) by Z, and adding and subtracting (5.14a) and (5.14b) to give
-
V ( Y ,t )
+
ZCZ(9,
t) =
+
V(0,t - T ) ZCZ(0,
V ( 9 ,t ) - Z c I ( 9 , t ) = V(0,t
t
- T)
+ T ) - Z,Z(O, t + T )
(5.15a) (5.15b)
These can also be time shifted to yield
+ Z c f ( 9 , t ) = V(0,t - T ) + Z,I(O, t - T ) V ( 9 , t - T ) - ZcZ(9, t - T ) = V(0,t ) - ZcZ(0, t ) V ( 9 ,t )
(5.16a) (5.16b)
Equations (5.16) will form the basis of a very effective method of solution of the MTL equations for lossless lines that is referred to as the method of characteristics or Branin’s method, It is important to point out that all of the above transform solution methods and results can be directly translated to the solution of the MTL equations for a uniform, multiconductor, lossless transmission consisting of any number of conductors. So we see why the analysis of lossless MTL’s will be a straightforward task.
TWO-CONDUCTOR LOSSLESS LINES
FIGURE 5.1
5.1.1
257
A two-conductor line in the time domain.
Graphical Solutions
We will again consider lines of total length 9.Consider the two-conductor line shown in Fig. 5.1 where we assume for the moment resistive loads, Rs and RL. The forward- and backward-traveling waves are related at the load, z = 9,by the load reflection coeficient as V-(t + P/u) rL= V’(t -9/u)
(5.17)
- zc R L + zc
=-R L
Therefore the reflected waveform at the load can be found from the incident wave using the load reflection coefficient as (5.18)
The reflection coefficient so defined applies to voltage waves only. A current reflection coefficient applying to the current waves can be similarly obtained and is the negative of the voltage reflection coefficient (due to the negative sign in (5.3b)) CA.1) r(t
);
+
=
);
-
(5.19)
The reflection at the load is illustrated in Fig. 5.2. The reflection process can be viewed as a mirror that produces, as a reflected V-,a replica of V’ that is “flipped around,” and all points on the V - waveform are the corresponding points on the Vt waveform multiplied by r,. Note that the total voltage at
258
TIME-DOMAIN ANALYSIS
2
t
-9
2
#
U \ \
z
=9
+ z
FIGURE 5.2 Illustration of reflections at a termination of a mismatched line.
-
the load, V ( 9 ,t), is the sum of the individual waves present at the load at a particular time as shown by (5.3a). Now let us consider the portion of the line at the source, z 0, shown in Fig. 5.3. When we initially connect the source to the line, we reason that a forward-traveling wave will be propagated down the line. We would not expect a backward-traveling wave to appear on the line until this initial forwardtraveling wave has reached the load, a time delay of T = 9 / v , since the incident wave will not have arrived to produce this reflected wave. The portion of the incident wave that is reflected at the load will require an additional time T to move back to the source. Therefore, for 0 g t s 2 9 / u = 2T, no backwardtraveling waves will appear at z = 0, and for any time less than 2T the total voltage and current at z = 0 will consist only of forward-traveling waves, V+
TWO-CONDUCTOR LOSSLESS LINES
259
FIGURE 5.3 Characterization of the initially transmitted pulae.
and I
'. Therefore, from (5.3) V(0, t ) =
;) - ;)
v+(t-
I(0, t ) = I+(C
(5.20a) (5.20b)
Since the ratio of total voltage and total current on the line is Zc for 0 S t s ZT, the line appears to have an input resistance of 2, over this time interval. Thus the forward-travelingvoltage and current waves that are initially launched are related to the source voltage by
(5.21a) (5.21b)
260
TIME-DOMAIN ANALYSIS
This initially launched wave has the same shape as the source voltage, &(t), (but is reversed; see Fig. 5.2). The initially launched wave travels toward the load requiring a time T = S/L, for the leading edge of the pulse to reach the load. When the pulse reaches the load, a reflected pulse is initiated, as shown in Fig. 5.2. This reflected pulse requires an additional time T = .V/u for its leading edge to reach the source. At the source we can obtain a voltage reflection coefficient
Rs - zc rs = Rs + z c
(5.22)
as the ratio of the incoming incident wave (which is the reflected wave at the load) and the reflected portion of this incoming wave (which is sent back toward the load). A forward-traveling wave is therefore initiated at the source in the same fashion as at the load. This forward-traveling wave has the same shape as the incoming backward-traveling wave (which is the original pulse sent out by the source and reflected at the load), but corresponding points on the incoming wave are reduced by r,. This process of repeated reflections continues as re-reflections at the source and the load. At any time, the total voltage (current) at any point on the line is the sum of all the individual voltage (current) waves existing on the line at that point and time, as shown by (5.3). As an example, consider the transmission line shown in Fig. 5.4(a). At t = 0 a 30V battery with zero source resistance is attached to the line, which has a total length of 14 = 400 m, a velocity of propagation of o = 200 m/ps and a characteristic impedance of 2, = 50 Q. The line is terminated at the load in a 100SI resistor so that the load reflection coefficient is 100 - 50 r, = 100 + 50
-4 and the source reflection coefficient is
0 - 50 r, = 0+ 50 = -1
The one-way transit time is T = 9 / v = 2ps, At t = 0 a 30 V pulse is sent down the line, and the line voltage is zero prior to the arrival of the pulse and 30 V after the pulse has passed. At t = 2 ps the pulse arrives at the load, and a backward-traveling pulse of magnitude 30r, = 1OV is sent back toward the source. When this reflected pulse arrives at the source, a pulse of magnitude r, of the incoming pulse or TsrL3O= - 10 V is sent back toward the load. This
261
TWO-CONDUCTOR LOSSLESS LINES
4.
1
t
4 I
2-0
z = 200 m
(b)
(e)
t- 1 p
I
-
z
-
10 v
400m
z-600m
z
6.5 p8
FIGURE 5.4 An example illustrating computation of the line voltage at various points on the line.
pulse travels to the load, at which time a reflected pulse of r, of this incoming pulse or TLTsrL30= -3.33 V is sent back toward the source. At each point on the line the total line voltage is the sum of the waves present on the line at that point. The previous example has illustrated the process of sketching the line voltage at various points along the line and at discrete times. Generally we are only
262
TIME-DOMAIN ANALYSIS
interested in the voltage at the source and load ends of the line, V(0,t ) and V ( 9 ,t), as continuous functions of time, In order to illustrate this process, let us reconsider the previous example and sketch the voltage at the line output, z = 9, as a function of time, as illustrated in Fig. 5.5. At t 3:0 a 30 V pulse is sent out by the source, The leading edge of this pulse arrives at the load at t = 2 ps. At this time a pulse of rL30 = 10 V is sent back toward the source. This 10 V pulse arrives at the source at t = 4 ps, and a pulse of rsrL30-- - 10 V is returned to the load. This pulse arrives at the load at t = 6 ps, and a pulse of T,,TJL30 = -3.33 V is sent back toward the source. The contributions of these waves at z = 9 are shown in Fig. 5.5(b) as dashed lines, and the total voltage is shown as a solid line. Note that the load voltage oscillates during the transient time interval about 30V, but asymptotically converges to the expected steady-state value of 30 V. If we had attached an oscilloscope across the load to display this voltage as a function of time, and the time scale were set to 1 ms per division, it would appear that the load voltage immediately assumed a value of 30V. We would see the picture of Fig. 5.5(b) only if the time scale of the oscilloscope were sufficiently reduced to, say, 1 ps per division. In order to sketch the load current I(.Y,t), we could divide the previously sketched load voltage by RL.We could also sketch this directly by using current reflection coefficients rs= 1 and rL= -3 and an initial current pulse of 30 V/Zc 3:0.6 A. The current at the input to the line is sketched in this fashion in Fig. 5.5(c). Observe that the current oscillates about an expected steady-state value of 30 VIR, = 0.3 A. There are a number of other graphical methods which are equivalent to this graphical method. One of the more popular ones is referred to as the so-called lattice diagram CA.1). This is illustrated in Fig. 5.6(a). Position along the line is plotted along the upper horizontal axis and time is plotted vertically. The initial waveform applied to the line is Z,/(Z, + R,) G(t)as shown in Fig. 5.6(b). The lattice diagram simply tracks a particular point on this applied waveform, K at t', as it travels back and forth along the line. An exact solution for this general result can be obtained for any &(t) waveform by carrying through the above graphical method to yield (5.23a)
This succinct form of the solution gives the time-domain voltages explicitly as
263
TWO-CONDUCTOR LOSSLESS LINES
0.3334 A 0.2 A
0.3001 A
0.0667 A c
t(rd -0.2 A
FIGURE 5.5 An example illustrating computation of the load voltage and source current as a function of time.
264
TIME-DOMAIN ANALYSIS
(b)
FIGURE 5.6
The lattice diagram.
scaled versions of the input signal waveform, &(t), which are delayed by multiples of the line one-way delay, T.To compute the total solution waveforms, V(0,t ) and V ( 9 ,t), one simply draws the scaled and delayed &(t) and adds the waveforms at corresponding time points according to (5.23). Alternatively, this general result can be proven from an earlier frequencydomain exact solution given in (4.26a) specialized to a lossless line, Le., p =j p * jufv: (5.24) where we have substituted the one-way delay, T = 9 / v . In terms of the Laplace transform variable s this becomes (5.25a)
(5.25b)
265
TWO-CONDUCTOR LOSSLESS LINES
Multiplying both sides by the common denominator and using (5.9) gives
In order to examine the impact of this result we introduce the time-shljl or diflerence operator, D, as D*"f(t) = f ( t f m T )
(5.27)
In other words, the difference operator operates on a function of time to shift it ahead or backward in time. Substituting this result into (5.26) gives [I [i
+ r,D-2]~(t)
(5.28a)
- rSrL~-zjv(~, t) = " [(l + T,)D"JV,(t)
(5.28b)
- r,r,~-~]v(o, t) = zc [l zc + Rs zc
+ Rs
Multiplying through by D' and rearranging into the form of a transfer function gives (5.29a)
(5.29b)
which match the results in (5.23). Thus, for a two-conductor lossless line having resistive loads, one can immediately sketch the terminal voltage waveforms as scaled and delayed versions of the source voltage waveform, G(t), with the result given in (5.23) or equivalently in (5.29).
266
TIME-DOMAIN ANALYSIS
5.1.2
The Method of Characteristics (Branin's Method)
The previous section has demonstrated graphical methods for sketching the time-domain solution of the transmission-line equations for linear, resistive loads. It is frequently desirable to have a numerical method that is suitable for a digital computer and will handle nonlinear as well as dynamic loads. The following method is referred to as the method of characteristics. The numerical implementation is attributed to Branin and was originally described in [l]. It is only valid for lossless lines. The method of characteristics seeks to transform the partial differential equations of the transmission line into ordinary differential equations. To this end we define the characteristic curves in the z, t plane as (5.30a) (5.30b) The differential changes in the line voltage and current are
av(z,t, dz dV(z, t ) = -
aZ
')dt +at
(5.3 1a) (5.31b)
Substituting the transmission-lineequations in (5.1) into (5.31) gives (5.32a) (5.3 2b) Along the forward characteristicdefined by (5.30a), dz = 1/& dt, these become (5.33a) (5.33b) Similarly,along the backward characteristicdefined by (5.30b), dz =
- I/&
dt,
TWO-CONDUCTOR LOSSLESS UNES
267
‘t
z5Y
2 3 0
FIGURE 5,7
2
Illustration of characteristic curves for the method of characteristics.
these become (5.34a) (5.34b) Multiplying (5.33b) by the characteristic impedance, Z, = equations gives dV(2, t ) t 2, dl(z, t ) = 0
a,
and adding the (5.35a)
Similarly, multiplying (5.348) by the characteristic impedance, Z, = ,/@, and subtracting the equations gives dV(2, t ) - 2 , df(Z, t )
0
(5.358)
Equation (5.35a) holds along the characteristic curve defined by (5.30a) with phase velocity u = 1/&, while (5.35b) holds along the characteristic curve defined by (5.30b) with the same phase velocity. This is illustrated in Fig. 5.7.
268
TIME-DOMAIN ANALYSIS
These are directly integrable showing that the difference between two voltages at two points on a given characteristic is related to the difference between two currents on the same characteristic. Therefore, from Fig. 5.7 we may obtain [ V ( s ,t ) - V(0,t
- T)]= - Z c [ I ( 9 , t ) - I(0, t - T)]
(5.36a)
where the one-way delay is T = 9 / v . This result was derived earlier using the Laplace transform and given in (5.16). As an alternative derivation, we simply manipulate the solutions of the transmission-line equations given in (5.3). Rewrite these as V(2,t ) =
;)
v+(t- + V - ( t +
;)
( - -:> - v-( + 3-
Z,I(Z,t) = y+ t
t
(5.37a) (5.37b)
Evaluating these at the source end, z = 0, and at the load end, z = 9,gives V(0,t ) = V + ( t )+
v-(t)
(5.38a)
ZcI(0, t ) = V+(t)- V ( t )
and
V ( 9 ,t ) =
Z"(t
(5.38b)
- T)+ V-(t + T )
zcr(9,t ) = v+(t- T ) - v-(t+ T )
(5.39a) (5.39b)
where the one-way delay for the line is T = 9 / v . Adding and subtracting (5.38) and (5.39) gives V(0,t ) + ZcI(0, t ) = 2 V + ( t ) (5.40a) V(0,t)
- ZCI(0, t ) = 2V-(t)
+ Z C Z ( 9 , t ) = 2V+(t - T ) ~(9 t) , zcr(9, t) = 2 ~ + T 7) V ( 9 ,t )
(5.40b) (5.40~) (5.40d)
Shifting both (5.40a) and (5.40d) ahead in time by subtracting T from t along with a rearrangement of the equations gives
V(0,t ) = ZcI(0, t ) + 2V-(t) V ( 9 ,t ) = - Z c I ( 9 , t )
+ 2 V ( t - T)
V(O,t-r)+Z~I(O,t-T)~2V+(t-r) V ( 3 ,t
- r ) - Z , I ( 9 , t - r ) = 2V'(t)
(5.4 la)
(5.41b) (5.4 1c) (5.4 1d)
TWO-CONDUCTOR LOSSLESS LNES
E,@, I
- T)
T
V(gI - T)- Z c l ( g t - T)
Et(0,t
- 7')
269
T V ( O , t - 7') t Zc/(O,t-
7')
FIGURE 5.8 A time-domain equivalent circuit of a two-conductor line in terms of time-delayed controlled sources obtained from the method of characteristics (Branin's method) as implemented in SPICE.
Substituting (5.41d) into (5.41a) gives where
w,t ) = ZcI(0, t ) + E , ( 9 , t - r ) &(Y,t
- T ) = V ( 9 ,t - T) - ZcI(Y, t - T )
(5.42a) (5.42b)
= 2V-(t)
Similarly, substituting (5.40~)into (5.40b) gives t) =
where E& t
- T)
- Z c I ( 9 , t ) + E& V(0, t - T )
= 2V+(t
- T)
(5.43a)
+ ZcI(0, t - T )
(5.43b)
- T)
t
Equations (5.42) and (5.43) suggest the equivalent circuit of the total line shown in Fig. 5.8. The controlled source E,(O, t - T) is produced by the voltage and current at the input to the line at a time equal to a one-way transit delay earlier than the present time. Similarly, the controlled source E,@', t - 7') is produced by the voltage and current at the line output at a time equal to a one-way transit delay earlier than the present time. The equivalent circuit shown in Fig, 5.8 is an exact solution of the transmissionline equationsfor a lossless, two-conductor, uniform trartsmission line. The circuit
270
TIME-DOMAIN ANALYSIS
analysis program SPICE contains this exact model among its list of available circuit element models that the user may call cA.21. The model is the TXXX element, where XXX is the model number chosen by the user. SPICE uses controlled sources having time delay to construct the equivalent circuit of Fig. 5.8. The user need only input the characteristic impedance of the line 2, (SPICE refers to this parameter as ZO) and the one-way transit delay T (SPICE refers to this parameter as TD). Thus SPICE will produce exact solutions of the transmission-line equations. Furthermore, nonlinear terminations such as diodes and transistors as well as dynamic terminations such as capacitors and inductors are easily handled with the SPICE code whereas a graphical solution or the hand solution of the equivalent circuit of Fig, 5.8 for these types of loads would be quite difficult. This author highly recommends the use of SPICE for the incorporation of two-conductor transmission-line effects into any analysis of an electronic circuit. It is simple and straightforward to incorporate the transmission-line effects in any time-domain analysis of an electronic circuit, and, more importantly, models of the complicated, but typical, nonlinear loads such as diodes and transistors as well as inductors and capacitors already exist in the code and can be called on by the user rather than the user needing to develop models for these loads. As an example of the use of SPICE to model two-conductor, lossless transmission lines in the time domain, consider the time-domain analysis of the circuit of Fig. 5.5. The 30 V source is modeled with the PWL (piecewise linear) model as transitioning from O V to 30V in 0.1 ps and remaining there throughout the analysis time interval of 20 ps. The SPICE program is FIGURE 5 . 5 VS 1 0 PWL(0 0 .1U 30 20U 30) T 1 020 20~50 TD=2U R L 2 0 100 TRAN . 1 U 2 0 U P R I N T TRAN V(2) I(VS) . P L O T TRAN V(2) I(V9) END
.
. .
The results for the load voltage are plotted using the .PROBE option of the personal computer version, PSPICE, [A.2] in Fig. 5.9(a) and the input current to the line is plotted in Fig. 5.9(b). Comparing these with the hand-calculated results shown in Fig. 5.5 shows exact agreement. 5.1.3
The Bergeron Diagram
The following graphical method was originally developed for analyzing transients in hydraulic systems by L. Bergeron in 1949 and has been adapted to transmission lines [2-41. It can be easily proven using the equivalent circuit shown in Fig. 5.8 that was obtained from the method of characteristics. The
TWO-CONDUCTOR LOSSLESS LINES
40
-
30
-
20
10
I
7
271
r
-
0,
I
I
I
I
b
loot 00
5
10
15
20
Time (pa) (b)
FIGURE 5.9 Results of the exact SPICE model for the problem of Fig. 5.5.
advantage of the method is that it readily handles nonlinear resistive loads on the line, such as diodes, but the disadvantage is that it is only valid for step-function excitation and resistive loads. Consider the lossless, two-conductor line shown in Fig. 5.1qa) having resistive source and load impedances and driven by a step voltage source: G(t) = &u(t). Substituting the equivalent circuit from Fig. 5.8 gives the circuit of Fig. 5.10(b). In order to simplify the notation we designate the line voltages and currents at the input and output of the line as V(0, t ) = Kn(t),I(0, t ) = I&), V(U,t ) = <,, (c), and f ( Yt,) = fOu,(t). Let us choose to implement Branin's method via this equivalent circuit for discrete times that are multiples ofthe one-way line delay T. Thus we solve the circuit of Fig. 5.10(b) at t = 0, T, ZT, 3 T , . , , Observe that the controlled source outputs are related to the ooltage and current at the opposite end of the line and at one time delay earlier.
.
272
TIME-DOMAIN ANALYSIS
-7
E,(r
--
I
- r ) vw,(r- T ) - zciw,(r- T ) - r ) VI.(# - T ) + zcc(t- T)
E,O
(b)
FIGURE 5.10 Illustration of the proof of the validity of the Bergeron diagram method: (a) the structure and (b) the equivalent circuit using Branin's method.
First consider the circuit at t = O+, that is immediately after t = 0, as shown
in Fig. 5.11. The controlled source outputs depend on the line voltages and 0 - T = - T which are of course zero because we assume the currents at t = '
line is initially relaxed. The solution for V;,(O+) and I,"(O+) are obtained from the simultaneous solution of the two equations written for the left circuit (at 2 = 0): (a) Kn(O+) &I,n(O+) (5.44a) (b) Kn(0')
3
V, - &Iin(O+)
(5.44b)
Equation (5.44b) can be rewritten as
Equations (5.44) are plotted in Fig. 5.1 1 and their intersection gives the solution Observe that equation (5.44a) has slope l/Zc and passes for V;,(O+) and IIn(O+), through the origin, whereas equation (5.44b) has slope -1/& and passes through V = 4.
TWO-CONDUCTOR LOSSLESS LINES
\
273
/
FIGURE 5.11 The Bergeron diagram at t = 0’.
Next we obtain the solution at t =: T from the circuit of Fig, 5.12. The solutions for Y,,,(T) and loul(T)are obtained from the simultaneous solution of the equations written for the right circuit (z = 3): (5.45a) (5.45b) Equation (5.45b) can be rewritten as
Equations (5.46) are plotted in Fig. 5.12 and their intersection gives the solutions for KU,(T)and &,,{T). Observe that equation (5.45a) has slope 1/RL and passes through the origin, whereas equation (5.45b) has slope l/Zc and passes through the first solution point. Thus the solution proceeds along line (b). Next we obtain the solution at t -- 2T from the circuit of Fig. 5.13. The solutions for &,(2T)and 1,,(2T) are obtained from the simultaneous solution
-
274
TIME-DOMAIN ANALYSIS
FIGURE 5.12
The Bergeron diagram at t = T.
of the equations written for the left circuit (z = 0):
Equations (5.46) can be rewritten as
Equations (5.46) are plotted in Fig. 5.13 and their intersection gives the solutions for Kn(2T) and Iin(2T), Observe that equation (5.46a) has slope -l/Rs and passes through V - V,, whereas equation (5.46b) has slope l/Zc and passes through the second solution point. Thus the solution proceeds along line (b). The solution continues in like fashion by alternating between the left and right circuits at time increments of T.This produces the time-domain solution valid at each end at times separated by 2T as shown in Fig. 5.14. The value of
MULTICONDUCTOR LOSSLESS LINES
FIGURE 5.13
The Betgeron diagram at t
-
275
2T.
this method lies in its ability to handle nonlinear resistive loads such as diodes as illustrated in Fig. 5.15. Numerous electronic design manuals use this method
for these purposes. 5.2
MULTICONDUCTOR LOSSLESS LINES
In this section we examine the time-domain solution of the MTL equations for a lossless line:
a I(2, t ) a V(2, t ) = -L -
az
a
-I(z,t)= az
at
a
-c-V(2,t) at
(5.47a)
(5.47b)
276
TIME-DOMAIN ANALYSIS
f I
I T
I 2T
I 31'
I
c
4T
I
w T
2T
3T
4T
I
FlCURE5.14 Illustration of the line input and output voltages computed via the Bergeron diagram.
I
Diode
FIGURE 5.15 Use of the Bergeron diagram to compute time-domain responses of the line for nonlinear, resistive loads.
MULTICONDUCTOR LOSSLESS LINES
277
The uncoupled, second-order equations are
a2
-V(z, t ) = LC -V(Z, t ) aZ2 at2
(5.48a)
a2 I(z, t ) = CL a2 I(z, t )
(5.48b)
aZ2
a2
at2
Note that the order of multiplication of L and C in the second-order equations in (5.48) must be strictly observed. The objective in this section is to determine the general solution of these equations and the incorporation of the terminal conditions. 5.2.1
Decouplhg the MTL Equations
The primary technique used to determine the general form of the solution of the MTL equations for sinusoidal,steady-state excitation in the previous chapter was to decouple them with a similarity transformation, In the case of the general time-domain solution of lossless lines, the technique of decoupling the MTL equations in (5.47) or (5.48) via a similarity transformation is again a useful technique that can always be used to find the solution. As in the previous chapter we define the similarity transformations to mode voltages and currents as: (5.49a) (5.49b) Substituting these into (5.47) give
a
- V,(Z, t ) = -T;'LT, aZ
a
- I,(z, t ) = -Ti'CTY
az
a
I (Z t ) at '
(5.50a)
a Vm(z,t ) at
(5.50b)
or for the uncoupled second-order equations in (5.48)
a2
-Vm(z,t ) = T i 'LCTy -V,(Z, t ) 822 at2
(5.5 1a)
-Im(z,t ) = T[ 'CLT, -I (z t ) az2 at2 '
(5.51b)
a2
a2
a2
If we can choose a T, and a T, such that (5.50) or (5.51) are uncoupled then
278
TIME-DOMAIN ANALYSIS
we have the general form of the mode solutions as those of two-conductor lines of the previous section. This is always guaranteed for lossless lines as was shown in the previous chapter for sinusoidal excitation. For example, consider the coupled first-order equations in (5.50). Suppose we can find a TYand a T, such that they simultaneously diagonalize both L and C as = L,,,
T;'LT,
(5.52a)
TT'CT, = C,
(5.52b)
where L, and C,,, are diagonal as 0
...
ImZ
01
...
0
(5.52~)
lmn
(5.52d)
Then the mode equations in (5.50) become
a
a
-V,(z,
t) =
t)
(5.53a)
a &(z, aZ
t ) = -Cm - V,(Z, t) at
(5.53b)
az
-L, - 1 at
a
(2
'
or
(5.54)
MULTICONDUCTOR LOSSLESS LINES
279
These are the same equations as for uncoupled, two-conductor lines having characteristic impedances of (5.55)
and velocities of propagation of urn, =
1 -
(5.56)
Therefore the solutions for these mode voltages and currents have the same general form as for the two-conductor line. The actual voltages and currents can be obtained from the forms of the mode voltage and current solutions via (5.49). We now address the determination of the diagonalizing similarity transformation for several classes of lines. 5.2.1.1 Lossless lines in Homo@neous Media
For this class of line we have the
important identity
LC = CL = psl.
(5.57)
where the surrounding homogeneous medium is characterized by permittivity, and permeability, p, The similarity transformations that simultaneously diagonalize L and C as in (5.52) can be found in the following manner as shown in the previous chapter. Because L is real symmetric, a real orthogonal transformation T can be found such that E,
TILT = L,,,
(5.58a)
where
(5.58b)
and the inverse of T is its transpose [SI: (5.58~) Similarly, with the aid of the identity in (5.57) expressed as
c = flcL”
(5.59)
280
TIME-DOMAIN ANALYSIS
we may form
= T'CT
T"C(T')-'
(5.60)
= pT'L"T
Comparing (5.58) and (5.60) to (5.52) shows that the transformations can be defined as TI= T (5.61a)
T, = T
(5.61b)
T;' = T;'
r
T'
(5.6 1c)
Therefore the mode characteristic impedances in (5.55) are
and all modes have the same velocity of propagation: 1 vmi
=
(5.63)
7F
For the special case of a three-conductor line, n = 2, the mode transformation is simple: cos8 -sin 8 (5.64a) sin8 cos 6 where
1
tan 28 =
2112 4 1
- I22
(5.64b)
For n 2 3 a numerical computer subroutine implementing, for example, the Jacobi method must be used to obtain the orthogonal transformation [SI. Appendix A describes a FORTRAN subroutine, JACOBISUB supplied with this text which accomplishes this reduction. 5.2.7.2 Lossless Lines in InhomogeneousMedia For this we no longer have the identity in (5.57). However, we showed in Chapter 4 that because L and C are real, symmetric, and positive definite we can find a transformation that simultaneously diagonalizes these matrices in the following manner, First we find an orthogonal transformation that diagonalizes C as
u'cu = e 2
(5.65a)
281
MULTICONDUCTOR LOSSLESS LINES
where
(5.65b)
and (5.65~) Since C is positive definite, we can obtain its square root, 0, which is real and nonsingular and form the product BU'LUB. Since this is real and symmetric, we can diagonalize it with another orthogonal transformation as (5.66a)
(5.66b)
and
s-'
= S'
(5.66~)
Define the matrix T as
T = U0S
(5.67)
The columns of T can be normalized to a Euclidean length of unity as
T",,, = Ta
(5.68)
where a is the n x n diagonal matrix with entries (5.69a)
all = 0
(5.69b)
The mode transformations in (5.49) that simultaneously diagonalize L and C
282
TIME-DOMAIN ANALYSIS
as in (5.52) can then be defined as (5.70a) (5.70b) Also (5.71a) (5.71 b)
Substituting (5.70) and (5.71) into (5.52) gives (5.72a) (5.72b)
Comparing (5.72) and (5.52) shows that
L, C,
= aAZa
(5.73a)
=a-2
(5.73b)
Since a and A are diagonal matrices, the mode characteristic impedances and velocities of propagation are given by (5.74)
and Urn( =
1 -
JLG
(5.75)
One can show that Zc = ~l)'lZc,,,T;l. The desired mode transformation
MULTICONDUCTOR LOSSLESS LINES
283
matrices are (5.76a) (5.76b)
The above diagonalization is implemented in the digital computer FORTRAN subroutine DIAG.SUB described in Appendix A. This subroutine calls the FORTRAN subroutine JACOBLSUB to compute the two orthogonal transformations required by DIAGSUB. 5.2.1.3 Incorporating the Terminal Conditions via the SPICE Program Since we have uncoupled the equations via the mode transformation in the preceding sections, the mode voltages and currents are essentially associated with n uncoupled two-conductor transmission lines as is illustrated in (5.54) whose general solutions are known. Again each general mode solution contains two undetermined constants so that there are a total of 2n undetermined constants. It remains to incorporate the terminal conditions at the two ends of the line in order to evaluate these 2n undetermined constants. There are a number of ways of doing this. The simplest and most useful way is to utilize the exact time-domain model that exists in the SPICE code as shown in Fig. 5.8 CA.2, A.3, IO]. But each of these models only relates the n mode voltages and currents at the two ends of the line. In order to relate these mode quantities to the actual voltages and currents we implement the mode transformations given in (5.49). These transformations can be implemented in the SPICE program through the use of controlled sources as illustrated in Fig. 5.16. Writing out the transformations in (5.49) gives
and q,,,respectively. Inverting where we denote the entries in Tvand T, as Tv,,
284
TIME-DOMAIN ANALYSIS
FIGURE 5.16 Illustration of the implementation of the mode transformations using controlled sources.
(5.77b) gives . a .
(5.7 7c)
...
I . .
where we denote the entries in TF1as T;). The transformations in (5.77a) and (5.77~)can be implemented in SPICE using the controlled source representation illustrated in Fig. 5.16. Zero-volt voltage sources are placed in each input to sample the current Il(z,t)for use in the controlled sources representing the transformation in (5.77~). The interior two-conductor mode lines having characteristic impedance Zcmrand time delay T, = 9'/urn, are simulated with the existing two-conductor SPICE model as TXXX il 12 20 = ZCmi TD=Ti
as shown in Fig. 5.17. The advantages of this method of implementing the terminal conditions is that the model for the line is independent of the terminations, and any of the available device models in SPICE such as resistors, capacitors, inductors as well as the nonlinear models such as diodes and transistors can be called. The user need not redevelop the mathematical models of those devices. The overall model of the line can be implemented as a subcircuit model in SPICE and the
MULTICONDUCTOR LOSSLESS LINES
FIGURE 5.17
285
Characterization of the uncoupled modes as two-conductor lines with
Branin's method.
appropriate line terminations attached to the ports of this subcircuit. Thus the solution is an exact one within the time-step discretization in the SPICE solution. As an example of the implementation of this valuable technique, consider the example of three rectangular-cross-sectionconductors (lands) on the surface of a printed circuit board (PCB)shown in Fig. 5.18 that has been considered previously. This problem is that of an inhomogeneous medium and we shall assume a lossless medium and lossless conductors so that the results of Section 5.2.1.2will apply. The per-unit-length capacitance matrix was computed using the numerical technique described in Chapter 3 via the PCBGAL.FOR program discussed in Appendix B:
40.6280 -20.3140
-20.3140
]
29.7632
pF/m
The per-unit-length inductance matrix was computed from the capacitance matrix with the dielectric board removed, C,, as
1.10418 0.6900941CIH/m = [0.690094 1.38019 The similarity transformations become
[
0*51
1.118 T' = - 1.234 x 10-5 1.0
and
1.118 -1.234 x
Ti1= Tb = [o,5
1.o
lo-']
286
TIME-DOMAIN ANALYSIS
Rs'50n
10 inches = 0.254 m-
-P=
(c) FIGURE 5.18 An example to illustrate the modeling of a MTL via Branin's method.
The mode characteristic impedances and propagation velocities are 109.354 SZ vml = 1.80065 x lo8 m/s Zcm, 265.325 SZ vmz = 1.92236 x lo8 m/s Zcml
MULTICONDUCTOR LOSSLESS LINES
287
zc-,= 109.354 n
TI
@
FIGURE 5.19
-
1.410605 nr
@
The SPICE equivalent circuit for the structure of Fig. 5.18.
This gives the mode circuit one way time delays as
T,= 1.410605 ns T, = 1.321 295 ns The SPICE program (implemented on the personal computer version, PSPICE) is obtained from the circuit of Fig. 5.19 as SPICE MTL MODEL VS 1 0 PULSE(0 1 0 6.25N 6.25N 43.75N 100N) RS 1 2 50 v1 2 3 RL 7 0 50 V3 7 6 RNE 13 0 50 V2 13 12 RPE 8 0 50 V4 8 9 EC1 3 0 POLY(2) (4,O) (11,O) 0 1.118 0.5 EC2 12 0 POLY(2) ( 4 , O ) (11,O) 0 -1.234E-6 1.0 EC3 6 0 POLY(2) ( 5 , O ) (10,O) 0 1.118,0.5 EC4 9 0 POLY(2) (5,O) ( l 0 , O ) 0 -1.2343-6 1.0 PC1 0 4 POLY(2) V1 V2 0 1.118 -1,234E-5 FC2 0 11 POLY(2) V1 V 2 0 0.5 1.0
288
TIME-DOMAIN ANALYSIS
FC3 0 5 POLY(&) V3 V 4 0 1.118 -1.234E-5 FC4 0 10 POLY(2) v3 v4 0 0.5 1.0 T1 4 0 5 0 20 = 109.354 TD = 1.410605N T2 11 0 10 0 ZO = 265.325 TD = 1.321295N .TRAN .1N 20N 0 .05N .PRINT TRAN V(2) V(7) V(13) V(8) .PLOT TRAN V(2) V(7) V(13) V(8) END
.
Comparisons with experimentally obtained data will be shown in Section 5.2.6.2.
Another important advantage of the SPICE implementation of the solution is that thefrequency-domain or sinusoidal steady-state phasor solution considered in Chapter 4 can also be obtained from this program with only a slight change in the control statements. These are to redefine the voltage source as VS 1 0 AC 1 and redefine the control, print and plot statements as
.AC DEC 50 1OK lOOOME0 (which solves the frequency-domain circuit from 10 kHz to 1 GHz in steps of 50 per decade) and
.
PRINT AC VM(2) VM( 7 ) VM( 13) VM(8) .PLOT AC VM(2) VM(7) VM(l3) VM(8) The remaining statements in the original time-domain program above are unchanged. The above method is implemented in the SPICEMTL.FOR digital computer program described in Appendix A. A SPICE subcircuit model is generated by the program which characterizes the MTL at its terminals. 5.2.2 Extension of Branin's Method to lossless Multiconductor lines in Homogeneous Media
Branin's method was developed for two-conductor lossless lines. It can be extended in the following manner to lossless MTL's in homogeneous media. The following extension of that method cannot be readily extended to handle inhomogeneous media. Recall thefrequency-domain chain parameter matrix as (5.78)
289
MULTICONDUCTOR LOSSLESS LINES
where the submatrices are given in (4.120) for a lossless line in a homogeneous medium as 6,1 = cos(pu)l, (5.79a)
t,, = - j
sin(pY)zc
(5.79b) (5.79c) (5.79d)
The characteristic impedance matrix is defined as
2, = UL 2;' = uc
(5.80a) (5.80b)
and the velocity of propagation is again defined by 0=-
1
(5.81)
JG
The Laplace transform of the corresponding time-domain result can be obtained from this by substituting the Laplace transform variable s for jo, assuming the line to be initially relaxed, V(z, t ) = I(z, t) = 0 for all 0 5 z s 9 and t 5 0, and recalling the trigonometric expansions given in (5.1 1) as (5.82a)
-(
esT
I(9, s) =
- e-sT
+(
eaT + e-rT
)Z;'V(O,
s)
)I(O, s)
(5.82b)
where the line one-way delay is again defined as
T 3 -Y v
(5.83)
Multiplying (5.82b) by Zc and adding and subtracting the equations gives V ( 9 , s)
+ ZcI(u, s) = e-$V(O, s) + e-aTZcI(O,s)
V ( 9 , s) - ZCI(9,s) = eaTV(O,s) - earZcI(O,s)
(5.84a) (5.84b)
Observing once again the delay transform pair given in (5.9) gives the
290
TIME-DOMAIN ANALYSIS
time-domain forms as
+ ZcI(9, t ) = V(0, t - T ) + ZCI(O,t - T) V ( 9 , t ) - ZcI(9, t ) = V(0, t + T ) - ZCI(0, t + 7‘) V ( 9 ,t )
(5.85a) (5.85b)
Time shifting (5.85b) and rearranging gives V(0, t ) - ZcI(0, t ) = V ( 9 , t
- T ) - ZcI(9, t - T )
(5.8%)
Define the vectors E,(t
- 7‘)
3
Er(t -. T )
- T ) + ZcI(0, t - 7‘) V ( 9 , t - T ) - ZcI(9, t - T) V(0, t
(5.86a) (5.86b)
The subscripts r and Idenote “reflected” and “incident” respectively. Equations (5.8 5) become V ( 9 , t ) ZcI(9, t ) = E,(t - 7‘) (5.87a)
+
V(0, t ) - ZcI(0, t ) = Er(t - T )
(5.87b)
Equations (5.86) time shifted become
E,(t)
V(0, t )
+ ZcI(0, t)
E,(t) = V ( 9 , t ) - ZcI(9, t )
(5.88a) (5.88 b)
Substituting (5.87) into (5.88) gives the terminal voltages in terms of the E vectors as V(Os t ) 3 W t ) + 3Wt - T ) (5.89a) V ( 9 , t ) = fEr(t)
+ fE,(t - T)
(5.89b)
Describe the resfstioe terminations as generalized Thbvenin equivalents:
- R,I(O, t ) V ( 9 , t ) = VL(t) + RLI(9, t) V(0, t ) = Vs(t)
(5.90a) (5.90b)
Substituting these terminal relations into (5.88) and using (5.87) gives
E,@) = 2MSVdt) + rsE,(t - T ) Er(t) = ~ M L V L (+~r) ~ E , (-. t T)
(5.91a) (5.91b)
where we have defined, in the fashion for two-conductor lines, the reflection
MULTICONDUCTOR LOSSLESS LINES
291
coefficient matrices as (5.92a) (5.92b) and the voltage division coefficients are defined as (5.93a) (5.93b) The scheme is to discretize the time axis into N computation steps in each one-way time delay T as At = T/N. Then form the four n x N arrays EPLD, EfLD,EYew,EYeW, The columns of these are associated with the time increment and the rows are associated with the particular conductor of the line. To begin the recursion algorithm we assume an initially relaxed line and fill EPLDand EpLDwith zero entries. Then compute the entries in EfJewand ETEwaccording to (5.91) and compute the terminal voltages from (5.89). Once all entries are filled for all time increments for 0 5 t s T, write the entries in EYew and ,YEW into EPLDand EYLDrespectively and repeat the calculations for the times in the next interval T 5 t 5 2T. This is repeated to solve for the terminal voltages in blocks of time of length equal to the one-way line delay, ?i This extension of Branin's method to MTLs is implemented in the FORTRAN program BRANINOFORdescribed in Appendix A. The method implicitly assumes that all modes propagate with the same velocity so that it does not appear feasible to extend it to lines in inhomogeneous media. The recursive method can be solved in series form to give
These reduce to the exact scalar results for a two-conductor line given in (5,23) but the order of multiplication of the matrices must be preserved here.
292
TIME-DOMAIN ANALYSIS
FIGURE 5.20 Illustration of the time-domain to frequency-domain transformation method of computing time-domain responses of linear MTLs: (a) a single-input, single-output linear system in the time domain; (b) a single-input, single-output linear system in the frequency domain; and (c) representation of a MTL as a single-input, single-output linear system.
5.2.3
Time-Domain to Frequency-Domain Transformations
Perhaps the most common technique for determining the time-domain response of an MTL is known as the time-domain to frequency-domain transformation. This is a straightforward adaptation of a common analysis technique for lumped, linear circuits and systems CA.2). Consider the single-input, singleoutput lumped, linear system shown in Fig. 5.2qa). The input to the system is denoted as ~ ( t ) the , output is denoted by y(t), and the unit impulse response (x(t) = S(t), y(t) = h(t)) is denoted by h(t). The independent variable for the lumped system is time, denoted by t. In the time domain, the response is obtained via the convolution integral CA.21:
(5.95a)
(5.95b)
MULTICONDUCTOR LOSSLESS LINES
293
In the frequency domain this translates to
where &jw) is the Fourier transform of h(t) and is called the tranqer function of the system as illustrated in Fig. 5.20(b) CA.21. The frequency-domain transfer function, fi(jw),can be easily obtained by applying unit-magnitude sinusoids to the input and computing the response using the usual phasar computational methods with the frequency of those sinusoids varied over the desired frequency range as illustrated in Fig. 5.20(b). Once l?(jo)is obtained in this manner, the time-domain impulse response is obtained as the inverse Fourier transform of
fi( j w ) :
h(t) = . F - ' { f i ( j ~ ) }
(5.97)
The method is a very intuitive one. Any time-domain waveform can be decomposed into its sinusoidal components with the Fourier series if it is periodic and the Fourier transform if it is not periodic CA.23. A periodic waveform with period T has these frequency components appearing at discrete frequencies that are multiples of the basic repetition frequency. If the waveform is nonperiodic, these frequency components appear as a continuum. Nevertheless, the time-domain to frequency-domain method for computing the time-domain response can be described quite simply in the following manner. Decompose the input signal via Fourier methods into its constituent sinusoidal components (magnitude and phase). Pass each component through the system and sum in time the time-domain responses at the output to each of these components by using phasor methods. The magnitudes of the output sinusoidal components are the products of the magnitudes of the input sinusoidal components multiplied by the magnitude of &lo) at that frequency. The phases of the output sinusoidal components are the sums of the phases of the input sinusoidal components and the phases of f i ( j w ) at that frequency. Therefore
(5.98b) Clearly a major restriction on the method is that the system be linear since we are implicitly applying the principle of superposition. In the case of a periodic waveform with period,'Z these frequency components appear at discrete frequencies that are multiples of the basic repetition frequency, w, = 27th = 27t/T, and x ( t ) can be represented as the sum of these time-domain sinusoidal components with the Fourier series as CA.21 x ( t ) = cg
+ n = 1 c, cos(nwot + /h,) "
(5.99)
294
TIME-DOMAIN ANALYSIS
where we have truncated the series to contain only NH harmonics. In the case of a periodic pulse train having trapezoidal pulses of peak magnitude X,duty cycle D = z/T where z is the pulse width (between 50% points), and equal rise/fall times of T,.,the items in (5.99) become CA.31:
cg = X D c,, = 2XD
(5.1 OOa)
sin(nnD) sin(nlcf,r,.) nnD nnL7,
&,, = -nn(D +AT,)
(5.100b) (5.1OOc)
For this case of a periodic waveform, we compute the frequency-domain transfer function at each of the NH harmonic frequencies with the methods of Chapter 4 (including skin-effect losses if we choose):
Then multiply each appropriate magnitude and add the angles to give the time domain output as:
fi
In this way we can include dependent skin-effect losses which are difficult to characterize directly in the time domain as we will discuss in Section 5.3 but again this supposes u linear system, This method is implemented in the FORTRAN program TIMEFREQ.FOR described in Appendix A. Actually the periodic waveform results can be used for a nonperiodic waveform if we choose the pulse waveshape over a period to be that of the desired pulse and also choose a repetition frequency low enough that the response reaches its steady-state oalue before the onset of the next pulse. This technique is equivalent to avoiding “aliasing” in the application of the fast Fourier transform (FFT). Consider applying these concepts to a MTL. Suppose we apply a voltage source, G(t),to the i-th conductor at z = 0 and desire the time-domain response of the j-th line voltage say, at the far end of the line z = 9, Q(9, t). In order to view this problem as a single-input, single-output system we imbed the MTL along with the terminations into a two port as illustrated in Fig. 5.20(c) and extract the input to the system, G(t),and the output of the system, b(9,t). The frequency-domain transfer function between these two ports, l?,,(ja),can be computed by the phasor methods of Chapter 4 wherein &(r) is a sinusoid, G(t) = sin(ot). Once the frequency-domain transfer function is obtained in this fashion, the time-domain response can be determined for any time variation of &(r) using the above summation of the responses to its sinusoidal components in (5.102) which amounts to the convolution integral of (5.95). An important
MULTICONDUCTOR LOSSLESS LINES
295
advantage is that this method can directly handle frequency-dependent losses such as skin-effect resistance of conductors. Another advantage is that only phasor computational methods are required; there is no need to numerically integrate the time-domain MTL equations. A major disadvantage is that a linear MTL is assumed, Le., the line parameters and the terminations are assumed linear. For example, corona breakdown of the surrounding medium as well as nonlinear loads such as diodes and transistors cannot be handled with this method since it implicitly assumes a linear system. Another disadvantage is that the input signal will consist of a wide spectrum so that accuracy depends on computing the frequency-domain transfer function for what could be a large number of frequencies. Nevertheless, the method is simple to implement for determining the time-domain response of a MTL having linear terminations. 5.2.4
lumped-Circuit lteralive Approximate Characterizations
Lumped-circuit iterative characterizations were discussed in Chapter 4 for approximately characterizing the MTL in the frequency domain. The same circuits can be directly used to characterize the MTL in the time domain but there is an added degree of approximation inherent in their use for time-domain calculations over and above the approximations for the frequency domain. The reader is referred to the structures of these lumped-circuit iterative models, the lumped 1, r, n, and T models shown in Fig. 4.12. These structures were developed under the assumption that the MTL was electrically short, 9 << A, at the fiequency under tnuestigation. In the application of these approximate structures to time-domain calculations, we must recognize that a time-domain input signal contains, in theory, an infinite range of spectral components. So if these structures are used to characterize the time-domain response, those spectral components of the input that are below the frequency where the MTL becomes electrically long will be processed correctly and the higher-frequency components will not be processed correctly. Typically the higher-frequency spectral components contribute to the fine detail of the input (and output) waveshape such as rise and fall times. Incorrectly processing the higherfrequency components will contribute to the “rounding” of the sharp pulse edges. Nevertheless, within this approximation, the lumped-circuit iterative models are simple to use since they may be readily incorporated into any of the various lumped-circuit CAD programs such as SPICE, and the user need only compute the per-unit-length parameters. The FORTRAN program SPICELPLFOR described in Appendix A generates a SPICE subcircuit model for a lumped-pi model of a MTL. 5.2.5
Finite Difference-Time Domain (FDTD) Methods
A common way of approximately determining the time-domain response of a MTL is the use of finite difference-time-domain methods or FDTD [ll-l3J. The derivatives in the MTL equations are discretized and approximated with
2%
TIME-DOMAIN ANALYSIS
various finite differences. In this method the position variable, z, is discretized as Az and the time variable, t, is discretized as At. There are many ways of approximating the derivatives, 3/82 and a/& in those equations [SI. We used a particularly simple discretization in the finite difference method of solving Laplace's equation in two dimensions in Chapter 3. Consider a real function of one variable, f ( t ) . Expanding this in a Taylor series in a neighborhood of a desired point gives f(t
+ h) = f ( t ) + hf'(t) + h2 f " ( t ) + 31 h3 f " ( t ) +
* * *
(5.103)
where the primes denote the various derivatives with respect to t of the function. Solving this for the first derivative gives f ' ( t ) 8 f0 + h, - f ( 0 h
h2 f'"(t) - . . - h2 f " ( t ) - 6
*
(5.104)
Thus the first derivative is approximated as (5.105)
where O(h) denotes that this approximation is valid within an error of order h. So the first derivative may be approximated with the first-order forward direrence : (5.106)
This amounts to approximating the derivative of f ( t ) using its slope about the region in advance of the desired point. Similarly, the derivative can be approximated as its slope about the region at the desired point by expanding the function f(t - h) with a Taylor series to give
This gives the first-order backward dtrerence accurate to order h: (5,108)
Similar approximations for the higher-order derivatives can be found in like
MULTICONDUCTOR LOSSLESS LINES
297
fashion. The second forward difference is
f "(t) 2
f(t
+ 2h) - 2f(t + h) + f(t)
(5.109)
h2
and the second backward difference is f"(t)z
f(t)
- 2 f ( t - h) + f(t - 2h)
(5.1 10)
h'
both of which are also accurate to within order h. More accurate approximations known as central digerences can be found by expanding f(t + h) and f ( t - h) in Taylor series as
f(t
- h)
h' h3 f ( t ) - hf'(t) + 21 f " ( t ) - 5 f"'(t)
+
* *
(5.1 Ilb)
Subtracting the equations and solving gives the Erst-order central direrence (5.112)
which is accurate to order h'. Similarly, the second-order central dverence is (5.1 13)
which is accurate to order h'. The discretization of Laplace's equation in the finite difference method of Chapter 3 amounts to a second-order centra1 difference approximation to Laplace's equation. The stability of the solution of Laplace's equation resulting from that discretization is assured, unlike the stability of the discretizationsof the transmission-line equations which we now investigate. Consider the discretization of the transmission-line equations for a twoconductor line. In order to give a general result and to effectively incorporate the terminal conditions we will state these for a lossy line having incident field excitation. This will be covered more completely in Chapter 7 but for the moment, the incident field gives rise to distributed voltage and current sources,
298
vo
TIME-DOMAIN ANALYSIS
Io
-
+ A212
0
v,
v3 V N O Z
INDZ vNDZ+I I N D Z + l
0 VNOZ+2
I
z
ZPO
-9
Illustration of the discretization of a two-conductor line for implementation of the finite difference-time-domain (FDTD) method.
FIGURE 5.21
Vp and IF as (5.114a) (5.1 14b)
We divide the line into NDZ sections each of length Az as shown in Fig. 5.21. Similarly, we divide the total solution time into segments of length At, In order to insure stability of the discretization and to insure second-order accuracy we interlace the NDZ 1 voltage points, vi, 6,. , VNDz,VNDz+l,and the NDZ current points, I,, I,, , ,INDz, as shown in Fig. 5.21. Each voltage and adjacent current solution point is separated by A2/2. In addition, the time points must also be interlaced and each voltage time point and adjacent current time point are separated by At/2 as illustrated in Fig. 5.22. The second-order, central difference approximations to (5.1 14) become
+
v;:: - vi+' + I Az
..
..
I;+W
- I;+w
At
1;+312
+r
+ 1;+112 ,
2
- v;:312 + v;:112 -
2
(5.115a)
(The superscript n should not be confused with the number of conductors of a general MTL.)The incident field source b k is evaluated at the current location I,, and Irk is evaluated at the voltage location V,. Solving these gives the recursion relations:
MULTfCONOUCTOR LOSSLESS LINES
299
"I
I
i (& -#)AS
(&
- I)Az (k AI
_____L
&A2
I
FIGURE 5.22 Illustration of the interlacing of the discrete voltages and currents in position and time to insure stability in the FDTD method.
These are solved in a "bootstrapping" fashion. First the voltages along the line are solved for a fixed time from (5.116b) in terms of the previous solutions and then the currents are solved for from (5.116a) in terms of these and previous values. The solution starts with an initially relaxed line having zero voltage and current values at all points along the line. Next consider incorporating the terminal conditions. Referring to Fig. 5.21, represent these as Norton equivalents where we allow for lumped sources and characterized loads at z = 0, characterized by I, = G/R, and R,, and at z = 9, by IL = V,/R, and RL.We will show that the exact way of incorporating these terminal constraints is to substitute into (5.116b) for k = 1: io = 0
(5.1 17a)
(5.1 17b)
v, I,, = Rs AZ
(5.1 17c)
300
TIME-DOMAIN ANALYSIS
Similarly we impose the terminal constraints at z (5.116b) for k = NDZ 1:
+
h Z + 1
=0
8"-
-
Y by substituting into
(5.lf8a) 1
RL
AZ
(5118b) (5.1 lac)
+
We will also show that setting lo = INDz+l = 0 k = 1 and k = NDZ 1 in (5.116b) requires that we replace cAz with cAzj2 in only those two equations. Equation (5.116b) for all other k, k = 2, 3 , . . , ,NDZ, must use CAZ.In this section we will consider lossless lines so we set
r=Gk=O
(5.119a)
in (5.116a) for all k and set
..
in (5.1 16b) for k = 2, 3 , . ,NDZ. This gives the final difference equations to be solved. Equation (5.116b) for k = 1: (5.120a)
Equation (5.116b) for k = 2, 3 , . , ,,N D Z :
MULTICONDUCTOR LOSSLESS LINES
Equation (5.116b) for k = NDZ
+ 1:
c Az
Equation (5.116a) for k
301
(5.120~)
e:
1, 2,.
. . ,NDZ:
The voltages and currents are solved by iterating k for a fixed time (solving first for the voltages and then for the currents) and then iterating time. The initial conditions of zero voltage and current are used to start the iteration. The conditions for this set of recursion relations to be stable is the Courant condition : AZ
At S -
(5.121)
U
which amounts to the condition that the time step must be no greater than the propagation time over each cell. The Az discretization is chosen sufficiently small such that each Az section is electrically small at the significant spectral components of the source voltages, &(t) and Q(t). It is not obvious that the method of incorporating the terminal constraints by using the distributed conductance, g, and distributed induced field source, IF, in (5.1 15b) or (5.116b) and making the correspondences as in (5.117) and (5.118) properly incorporates these lumped loads. Nor is it obvious that we must use cAz/2 in the end sections and CAZin the interior sections in (5.1 16b). We will now prove this and, in addition, will show that if we choose the time and position discretizations such that At =
Az V
(5.122)
(which is sometimes referred to as the "magic time step"), the above difference equations will yield the exact solution with no approximation error! In order
302
TIME-DOMAIN ANALYSIS
Ik t = O
El
--
A2
VB(l
Vt(r E, = vt(r ~2
FIGURE 5.23
vB(r
4 I
-*I - Zcrl(1- $1 - b + z c ~ -tI ) - 5 - zcIc(r- $1 - b + zcil(r- $1
Use of Branin’s equivalent circuit to prove the exactness of the FDTD
voltage recursion relation at the source.
to show this let us model the source end of the line for the first section of length Az using the exact time-delay model of Branin’s method as shown in Fig. 5.23. The voltages and current to be solved for in the finite difference solution are 6,I,, &. The currents and voltage I,,, V,, I , are auxiliary variables and are
not solved for in the finite differencemethod. Each half of length A2/2 is modeled with its own exact solution with reference to Fig. 5.8 with time delay Of 7/2 where ?=-
Az
(5.123)
V
In order to simplify the derivation, we again use the digerence operator: D*”f(t) = f(t f mz)
(5.124)
The resulting equations relating all voltages and currents in Fig. 5.23 are
The objective here is to eliminate I,,, V,, IC from these equations and write the
MULTICONDUCTOR LOSSLESS LINES
303
result in terms of V,, 11, V,. The terminal constraints at z = 0 are (5.126) Substituting (5.126) into (5.125a) and (5.125b) gives (5.127a)
Operating on (5.127a) with D and (5.127b) with D112 and substituting gives (1
+2 ) D h
=(1
-
2)
6 - 22cD'/211 + 2, ( D b + G)
(5.128)
RS
Rewriting (multiply both sides by Rs/2Z,) gives
where we have written this result in terms of the source reflection coefficient,
rS.In order to show that (5.129) is equivalent to (5.120a) for the magic time
step of (5.122), we substitute (5.122) along with the fundamental relation bet ween the per-uni t-length capacitance and the characteristic impedance, uc = Z;', into (5.120a) which gives (5.129). This shows that (5.120a) is the proper finite difference relation for the end section and that for this section we must use cAz/2 rather than CAZ! Similarly we may draw the exact circuit for the last Az section as shown in Fig. 5.24. The equations are
The objective is to incorporate the lumped terminal conditions at z = 2 ' and to eliminate I,,, G , I , to give an equation in VNDz,I N D Z ,VNDztI which, for the
304
TIME-DOMAIN ANALYSIS
&
I
-2
- I --
2
At
k
I
FIGURE 5.24 Use of Branin's equivalent circuit to prove the exactness of the FDTD
voltage recursion relation at the load.
I
vc I
---
I AZ
I
(Y)A* El E1
E3 4
-
vk(r
- 2: -
I (k+)AZ
zClB(r
- 3)- $1
V A 0 - 1 + ZcL-,(t Vc(t -f' Z c W v&(r 1 + zcls(t
-
-
- $1 - 3)
FlGURE 5.25 Use of Branin's equivalent circuit to prove the exactness of the FDTD voltage recursion relation at an intermediate point on the line.
magic time step of (5.122) is equivalent to (5.120~).Again this can be readily done as above. The remaining tasks are to show that (5.120b) and (5.120d) are correct for the magic time step, First we show that (5.120b) is correct by representing a Az section containing 4,f, with the exact Branin's model as shown in Fig. 5.25. The equations from that model are = Z&,
+ D-I'*(& - Z&)
(5.13 1a)
MULTICONDUCTOR LOSSLESS LINES
305
FIGURE 5.26 Use of Branin’s equivalent circuit to prove the exactness of the FDTD current recursion relation at an intermediatepoint on the line.
Operating on (5.131a) with Dill and (5.131b) with D and substituting gives DV,
V, + 2Z~D’121k-1- Z,-(DIB + I B )
(5.132)
Similarly, operating on (5.131d) with D - 1 / 2and substituting into (5.131~)gives D G = V, - 2ZcD’”Ik
+ ZJDIB + 1,)
(5.133)
Adding (5.132) and (5,133) gives DV, = V, + ZcD’12(Ik - I&-’)
(5.134)
JI we SumtitUte tne magic time step In (>.ILL)along wirn uc = 2 ;’ into (5.120b) we obtain (5.134) demonstrating their equivalence. Similarly we can demonstrate the correctness of (5.12Od) from the exact circuit model of Fig. 5.26 whose equations are
306
TIME-DOMAIN ANALYSIS
I 2-0
At
I-
At
I
FIGURE 5.27 Alternative derivation of the FDTD recursion relations using a lumped-pi representation of each section.
The objective is to eliminate I,, V,, ICfrom these equations and to show that they are equivalent to (5.120d) for the magic time step of (5.122). Subtracting (5.135a) from (5.135d) and operating on the result with D gives
Operating on (5.135b) and (5.13%) with D3I2and subtracting gives
Substituting (5.137) into (5.136) gives
Substituting the magic time step of (5.122) along with the fundamental relation 2,s ul into (5.120d) we obtain (5.138) demonstrating their equivalence. The above has demonstrated that the finite difference recursion relations given in (5.120) are an exact representation of a two-conductor, lossless line for the magic time step of (5.122). It is possible to provide an alternative, intuitive derivation of those relations in the following manner. A common way of approximating transmission lines, as discussed before, is with the lumpedcircuit, iterative approximations. One such representation is the lumped-pi model illustrated in Fig,4.12(c). Let us use this model and divide the line into Az segments and represent the per-unit-length distributed parameters of inductance and capacitance by lumped elements as shown in Fig. 5.27. The division points are chosen at the above finite difference solution voltages, VI, V,, , , , bDz, V,,,,. Each Az segment is represented by its inductance, Mz, and the capacitance is split and placed at the ends of each section as (c/Z)Az. This illustrates why (c/2)Az is used in the equations for the end segments as pointed out above. The finite difference current solution points, ZI,Z2,. , ,INDz,
.
.
MULTICONDUCTOR LOSSLESS LINES
307
are through the inductors at the center of the segment. Observing the interlacing of the solution time points according to Fig. 5.22, one can derive equations (5.120) directly from this circuit. This shows that, although (5.120) was derived for resistive terminations, it can be extended to include dynamic terminations so long as the derivatives in those relations are approximated according to Fig, 5.22. Similarly, line losses and incident field effects can be incorporated using Fig. 5.27 and Fig. 5.22 as a guide for their discretization. As an example, consider the two-conductor lossless line considered earlier and shown in Fig. 5.5. The exact solution for the load voltage, V ( 9 ,t), is sketched by hand in Fig. 5.5(b) and the SPICE results (using a 30V source voltage with a 0.1 ps rise time) are sketched in Fig. 5.9(a). In the following computed results we will designate Y
AZ = NDZ At =
final solution time NDT
The Courant condition given in (5.121) for stability of the FDTD solution given in (5.120) translates to NDT 2NDZ
v x final solution time
9
The magic time step in (5.122) occurs for an equality in this expression. Figure 5.28(a) shows the FDTD predictions for zero risetime and N D Z = 200 or Az = 2 m and the magic time step of At = Az/u = 10ns. For a final solution time of 20 ps, this gives 2000 time steps or N D T = 2000. The results are exactly those for the hand calculations. If we reduce the time step below that of the magic time step using 4000 tjme steps or At = 5 ns we observe considerable “ringing” on the leading edge of each transition, The FDTD solution for NDZ = 1 (dividing the line into only one section) and the “magic time step” of NDT = 10 or At = 2 ps is denoted as X and gives the exact solution even for this course gridding of the line! Figure 5.28(b) shows the FDTD predictions for a 0.1 ps risetime and N D Z = 200 or Az = 2 m and the magic time step of At = Az/u = 10 ns. Again, the results are exactly those for the SPICE or hand calculations. If we reduce the time step below that of the magic time step using 4000 time steps or At = 5 ns we observe less “ringing” on the leading edge of each transition than for the zero-risetime solution. The spectrum of this 0.1 ps risetime pulse rolls off at -4OdBldecade above 1/m,= 3.18 MHz CA.3). So the section length, Az, should be electrically short above this, requiring Az < 6.29 m. Thus, if we do not wish to use the magic time step, we must further reduce Az to eliminate this ringing.
308
TIME-DOMAIN ANALYSIS
-
Finite Difference -Time Domain (risetime 0 ps)
il
?.
Time (pa)
(4
-
Flnlte Difference -Tlme Domrln (rlwtimo 0. I p)
10
0
1
2
...... ...... .
1
4
-
NDZ 200, NDT = 2000 NDZ = 200, NDT 9 4000 NDZ = 1, NDT = 10
.
6
8
10
12
14
16
18
20
Time (ps) (b)
FIGURE 5.28 Illustration of the computation of the load voltage of Fig. 5.5 using the
FDTD method and comparison of the convergence to the choice of spatial and temporal discretization:(a) step response, (b) risetime of 0.1 ps.
MULTICONDUCTOR LOSSLESS LINES
309
Therefore we have demonstrated that the finite difference-time domain recursion relations of (5.120) provide the exact solution of the transmission-line equations for lossless, two-conductor lines with lumped loads for the magic time step of At = Az/u! It is a simple matter to extend these results to multiconductor lines using matrix notation. Assume that the multiconductor line has lumped terminal source and load, represented as generalized Thhenin equivalents: (5,139a) V(0, t ) = Vs(t) - R,qI(O,t) (5.139b) V ( 9 , t) = VL(t) RLI(9, t )
+
The voltages and currents are again interlaced in position and time as in Figs 5.21 and 5.22. The resulting FDTD recursion relations are V"+' 1 =
(g
RsC + I)-'
(5.140a)
(5.140b) (5.140~)
I;+3/2
= I;+'/'
- - At ~-l(vn+l AZ
k+1
- Vi+')
k = 1,. , ,N D Z
(5.14Od)
These equations are implemented in the FORTRAN program FINDIF.FOR described in Appendix A. 5.2.6
Computed Results
In this sectiqn we will give some predicted and experimentally obtained results for the ribbon cable and printed circuit board configurations investigated earlier. The terminal configurations for both structures are shown in Fig. 5.29(a). A 50 SZ time-domain source produces an open-circuit voltage, G(t), shown in Fig. 5.29(b) consisting of a trapezoidal periodic pulse train with a 50%duty cycle and equal rise and fall times, T, = zf, with various values. We will show results for rise/fall times that are of the order of the one-way time delay, T = S / u , and greater. The level of the pulse will be V, = 1 V in all cases.
310
TIME-DOMAIN ANALYSIS
three-conductor line for illustrating the predictions of various models: (a) terminal representations and (b) representation of the open-circuit voltage waveform flGWIE 5.29 A
of the source.
These are characterized as generalized Thbvenin equivalents:
where
Two configurations of a three-conductor line (n = 2) are considered: a threewire ribbon cable and a three-conductor printed circuit board. Experimentally
MULTICONDUCTOR LOSSLESS LINES
311
Near-End Cromtrlk Voltage (Ribbon -.-. cable)
0
50
1so
100
200
Time (ni) FIGURE 5.30 Comparison of the time-domainresponse of the near-end crosstalk voltage of the ribbon cable of Fig. 4.14 determined via the SPICE model, the lumped-pi model, and Branin’s method implemented directly for a pusle risetime of 20 ns.
determined responses for the neur-end ooltuge of line # 1, c(0, t), will be compared to the predictions of the SPICE model (SPICEMTLFOR), one lumped-pi section (SPICELPLFOR), Branin’s method (BRANIN.FOR), the time-domain to frequency-domain method (TIMEFREQ.FOR), and the finite difference-time domain method (FINDIF.FOR). The names in parentheses denote the appropriate FORTRAN program described in Appendix A which implements that method. 5.2.6.1 Ribbon Cable The cross section of the three-wire ribbon cable is shown in Fig. 4.14. The pulse risetime will be z, = 20 ns. The line length is 9 = 2 m so that the one-way delay (ignoring the dielectric insulations) is T = 6.67 ns. The per-unit-length parameters for this configuration were computed using the computer program RIBBON,FOR described in Appendix A and are given in Chapter 3: L - p 7 4 8 50 0.507 71 PH/m 0.507 7 1.015 4
C=[
34.432 - 18.716
- 18.716]
pF,m
24,982
Figure 5.30shows a comparison of the predictions of the SPICE model, one lumped-pi section, and Branin’s method for a risetime of r, = 2011s. The
312
TIME-DOMAIN ANALYSIS
FIGURE 5.31 Experimentally determined crosstalk for the ribbon cable and a pulse risetime of 20 ns.
experimentally determined response is shown in Fig. 5.3 1. The experimental results give a peak voltage at 20 ns of some 110 mV. The SPICE model predicts 113 mV and the lumped-pi model predicts 109.1 mV. Branin’s method predicts a value of 104 mV. Both the SPICE model and Branin’s method are exact (for a lossless line as assumed here) and should show the same values. The fact that they do not is due to the fact that the per-unit-length parameters of the SPICE model include the dielectric inhomogeneity whereas Branin’s method can only be used for a homogeneous medium. The inductance matrices for both cases are, of course, identical since the dielectric does not affect this parameter. Using the capacitance matrix with the dielectric removed, Coycomputed in Chapter 3:
-
22.494 11.247
- 11.247 16.581
along with L in the SPICE model gives the identical comparison shown in Fig. 5.32 which confirms this hypothesis. Figure 5.33 shows the predictions of the time-domain to frequency-domain method using TIMEFREQ.FOR and a frequency-domain transfer function (computed using the FORTRAN program MTLFOR without considering losses) computed for 100, 50, 20, and 10 harmonics of the fundamental frequency, f‘ -- 1 MHz. (Actually it doesn’t matter what repetition frequency or duty cycle or fall time is used so long as the response to the rising edge, which is shown in the figure, has died out
313
MULTICONDUCTOR LOSSLESS UNES
Near-End Croirtalk Voltage (Ribbon cable)
- SPICE
I
r
I
Branin's method model (homogeneous medium)
1
I so
200
Time (nr)
Comparison of the predictions of the SPICE model (for a homogeneous medium) and Branin's method implemented directly for the ribbon cable showing the effect of the wire insulation dielectric. Pulse risetime = 20 ns. FIGURE 9.32
Near-End Crosstalk Voltage (Ribbon cable)
Time (ns)
FIGURE 5,33 Comparison of the predictions of the SPICE model and the time-domain to frequency-domain transformation method for the ribbon cable. Pulse risetime = 20 ne.
314
TIME-DOMAIN ANALYSIS
Near-End Crosrtalk Voltage (Ribbon cable)
0
50
100
'
150
Time (ns)
FIGURE 5.34 Comparison of the predictions of the SPICE model and the FDTD method for the ribbon cable.
sufficiently.) The predictions using 100 and 50 harmonics are virtually identical to the exact SPICE results. The predictions using 20 and 10 harmonics are not so good. This is because the spectrum of the pulse begins to roll off at a rate of -40 dB/decade above a frequency off = l/m, = 15.9 MHz CA.3). Thus for a fundamental frequency of 1 MHz, the twentieth harmonic is at 20 MHz and the frequency response at the frequencies above this apparently contribute significantly to the response. Figure 5.34 shows the predictions of the finite difference-time domain model using FINDIF.FOR versus the exact SPICE results. Two discretizations are shown. The mode velocities computed with SPICEMTL,FOR are u1 = 2.323 97 x lo8 m/s and v2 = 2.51064 x lo8m/s. Since the spectrum of the pulse roils off at -40 dB/decade above f = l/m, = 15.9 MHz we choose Az = 1 m from the smaller mode velocity giving NDZ = 2 so that each section is electrically short above around 25 MHz. To use the magic time step, we choose, for a final solution time of 200 ns, 50 time steps corresponding to the largest mode velocity, u2. This will insure that we are in the stable range of the Courant condition for the smaller mode velocity, ul. The comparison with the SPICE results (which are exact) is excellent. Even though we reduce the time step below the magic time step the predictions are still quite good. In Chapter 4 it was pointed out that in order to match a MTL consisting of more than two conductors (n > l), it is not sufficient to terminate each line to the reference conductor with a single resistance. In order to match the line, the termination impedance matrix in the generalized Th6venin characterization
MULTICONDUCTOR LOSSLESS LINES
Zcm
-
315
ZlU
Illustration of matching of a MTL for (a) completely matched lines and (b) partially matched lines. FIGURE 5.35
of the termination must equal the characteristic impedance matrix, i.e., R, = Z, and/or R,.= Zr..In order to demonstrate this let us terminate this ribbon cable with a matched termination, ie., R, = Z,, as shown in Fig. 5.35(a). The characteristic impedance matrix is
z, = U,L =
G::
= c224.395 152'205'3 152.205 304.409
*
where we. will ignore the dielectric insulations and assume a homogeneous medium (free space) with 0, = 2.997925 x lo8 m/s. In order to simulate this termination we insert the resistors as shown in Fig. 5.35(a). (See Problem 4.19.) We will use the SPICE model with L and C, given above. Figure 5.36(a) shows the results of the simulation for the above 1 MHz trapezoidal pulse train with z, = 20 ns. Observe that the line is evidently matched with this termination. The question of whether one can partially match the line by placing termination
316
TIME-DOMAIN ANALYSIS
Time (ns) (8)
1.0
-
0.9 0.8
c 7w.z
-
1
/ ; 1 : / ; 1 : / ;
0,lc
0.6-
54 0.5
5 0.4
-
z
0.2
,/
/
; : I
0.0 LL!.. -*. -0.11
El: .-.--.- - - .VNE VFE
...,....*
/ ;
- 1/
O*I-/
--------
z.7-=-=s-.Jw-
1;
I-
$0.3-
;
-a-
....*,
-.-
e...
-..,...,.. -.-
*'..
",L,ltl,".~."
I
-.-.-
.-I-.--.--I-.I-.--.--.-.-
MULTICONDUCTOR LOSSLESS
LINES
317
resistors Z,,, and Z,,, only between each line and the reference conductor as in Fig. 5.35(b) is investigated next. The results of the simulation of Fig. 5.35(b) are shown in Fig. 5.36(b). Observe that this line is clearly not matched but the results do converge to a steady state quite rapidly. So, ideally, to eliminate all reflections on a MTL it must be terminated in its characteristic impedance matrix, Z,. However, because there is cross coupling in Zc due to Z,,, # 0, this gives a nonzero crosstalk level of about 100 mV, as shown in Fig. 5.36(a), that is not present in results for the partially matched line shown in Fig. 5.36(b). This cross coupling essentially acts like common-impedance coupling through the nonzero Z,,, resistance in Fig. 5.35(a). However, this is the price we pay for completely matching the line. 5.2.6.2 Printed Circuit Board The remaining structure to be examined is the three-conductor printed circuit board considered previously and shown in cross section in Fig. 4.17. The above methods will be used to predict the response for a 1 MHz trapezoidal pulse train shown in Fig. 5.29 having a rise/fall time of z, = 6.25 ns. The line length is 9 = 0,254 m = 10 inches so that the one-way delay (ignoring the board) is T -- 0.847 ns. The per-unit-length parameters for this configuration were computed using the computer program PCBGALFOR described in Appendix A and are given in Chapter 3:
]
1.104 18 0.690 094 PH/m 0,690094 1,38019 40.628 0 c = [-20.314 0
-20.3140 29.763 2
A comparison of the predictions of Branin’s method, the SPICE model, and one pi section are shown in Fig. 5.37. The predictions of the exact SPICE model and one lumped-pi section are again quite close. Once again, Branin’s method shows some error since it effectively ignores the board dielectric and assumes a homogeneous medium (in this case free space). The experimental results are shown in Fig. 5.38. The peak measured voltage is 94mV compared with a prediction of the SPICE model of 95 mV. Figure 5.39 compares the predictions of the SPICE model and the time-domain to frequency-domain method computed using the code TIMEFREQ.FOR. For the risetime of 6.26ns the spectrum of the pulse begins to roll off at a rate of -40 dB/decade above a frequency of J = l/m, = 51 MHz so we should expect to require more than 50 harmonics to achieve convergenceas is evident in the plot. Figure 5.40 compares the SPICE predictions to those of the finite difference-time domain model computed using the code FINDJF.FOR. The mode velocities are u1 = 1.80065 x lo8 m/s and 02 = 1.92236 x lo8 m/s. The Az discretization is again chosen to make each section electrically short at l/nz, = 51 MHz giving, using the smaller velocity ut, Az < 0.353 m. So we choose NDZ = 2. Using the larger mode velocity, 02, gives the magic time step of NDT = 60. The comparison
318
TIME-DOMAIN ANALYSIS
Near-End Crosstalk Voltage (Printed circuit board)
Time (na)
RGURE S.37 Comparison of the time-domain response of the near-end crosstalk of the printed circuit board of Fig. 4.17 determined via the SPICE model, the lumped-pi model, and Branin’s method implemented directly for a pulse risetime of 6.25 ns.
FIGURE 5.38 Experimentally determined crosstalk for the printed circuit board and a pulse risetime of 6.25 ns.
MULTICONDUCTOR LOSSLESS LINES
319
Near-End Crosstalk Voltare
SPICE model (lossless) --TDFD (lossless, N = 300) -.- -.- TDFD (lossless N 150)
.-----TDFD (lossless N
. .. .
--
50) TDFD (lossless N = 25)
Time (ns)
FIGURE 5.39 Comparison of the predictions of the SPICE model and the time-domain to frequency-domain transformation method for the printed circuit board. Pulse risetime = 6.25 ns.
Near-End Crosstalk Voltage (Printed circuit board) 100
I
- -
SPICE model ---. Finite Finite difference (NDZ = 2, NDT difference (NDZ 2. NDT (lossle88)
.-I-.-
30
0
60) 600)
40
Time (na)
FIGURE 5.40
Comparison or the predictions of the SPICE model and the FDTD method
for the printed circuit board. Pulse risetime 5 6.25 ns.
320
TIME-DOMAIN ANALYSIS
between the exact solution and the FDTD solution for the magic time step is excellent. Observe that even when the time discretization is reduced from that of the magic time step, N D Z = 2, NDT = 600, the correlation remains excellent. This again demonstrates that even in the case of MTL's in inhomogeneous media where the mode velocities are different, and the discretization can be chosen equal to the magic time step for only one of those velocities, the FDTD method can still give adequate results even though the other mode velocities do not satisfy the magic time step condition. 5.3
INCORPORATION
OF LOSSES
Losses arise from either the nonzero conductivity and polarization loss of the surrounding medium or from imperfect conductors. Of the two mechanisms, the loss introduced by imperfect conductors is usually more significant than the loss due to the medium for typical transmission-line structures. It is for this reason that the surrounding medium is often assumed to be lossless, i.e., set G = 0 in the MTL equations. The resistance due to imperfect conductors is represented in the per-unit-length resistance matrix, R. The per-unit-length inductance matrix, L,can be separated into a portion, L,,due to magnetic flux internal to the conductors and a portion, Le,due to magnetic flux external to the conductors as L = L, Lo.In the case of perfect conductors, the current flows on the conductor surfaces and the internal inductance is zero: L = Le, In Chapter 3 we discussed the per-unit-length resistance and internal inductance for conductors. At the lower frequencies where the skin depth, 6 -- 1-,/ is much larger than the conductor cross-sectional dimensions, the resistance and internal inductance are constant and equal the dc values because the current may be assumed to be uniformly distributed over the cross section. At the higher frequencies where the skin depth is much smaller than the conductor cross section the resistance increases as and the internal inductance decreases as because the current crowds to the outer edges of the conductor cross section and can be represented, as an approximation, to be uniformly distributed over a strip at the surface of thickness equal to one skin depth and zero elsewhere. This is exact for a conductor of circular cross section (a wire) in the absence of neighboring conductors, but for a conductor of rectangular cross section, such as a PCB land, the current crowds to the corners so that this underestimates the resistance as we saw in Chapter 3. Thus the entries in R are constant at lower frequencies and increase as at the higher frequencies. The entries in the per-unit-lengthinductive reactance matrix, oL,are the sum of the internal inductive reactance matrix, oL,,and the external inductive reactance matrix, oL,. At the lower frequencies, the internal inductance is constant, and at the higher frequencies it decreases as so that the entries at the higher in the internal inductive reactance, 04,increase as frequencies.
+
fi
fl
fi
fi
INCORPORATION OF LOSSES
321
In the frequency domain, the MTL equations can be written as
d O(Z) = - [ZI(O) +fwL]i(z)
(5.141a)
dz
d - I(z) =
dz
- [G+ ~oC]V(Z)
(5.141 b)
where L represents the external inductance, and the internal inductance is included in &,(w)= R +joiL,. To obtain the time-domain results we represent the MTL equations with the Laplace transform as
d I(z, s) = - [G + sC]V(z,
dz
s)
(5.142b)
A common way of approximating the internal impedance term with the Laplace transform variable is as C14, 151
That this represents a reasonable approximation to the skin effect behavior can be seen if we substitute s =j w to yield &(w) = A
+ Bfi
(5.144)
Thus A represents the dc per-unit-length resistance matrix. The component B & a ( 1 + j) represents the high-frequency per-unit-length resistance matrix as well as the high-frequency per-unit-length internal inductive reactance matrix. This assumes that the high-frequency resistance and internal inductive reactance are equal, which can be shown to be true for all conductor cross sections using Wheeler’s “incremental inductance rule” (see reference [28] in Chapter 3). Although it may appear that we may include the dc internal inductance in the inductance matrix, L, this is not possible for the following reason. Above a frequencyf, the internal inductance decreases as If we simply added the dc internal inductance to L, this would dominate the and we would high-frequency internal inductance (which decreases as effectively be ignoring this behavior at the high frequencies, which would affect the early-time response and artificially change the time delay of the line. The approximation in (5.143) stems from the observation that for low frequencies,
a.
8)
322
TIME-DOMAIN ANALYSIS
where the skin effect is not well developed, the dc values dominate the high-frequency values, where the skin effect is well developed, and vice-versa, This is shown for wires in Figs 3.52 and 3.53, and for conductors of rectangular cross section in Fig. 3.58, Consequently the per-unit-length impedance of the conductor is modeled as
tr(w)= A
+B
f i
The inaccuracy in this approximation occurs where the impedance transitions from the low-frequency value to the high-frequency value at f,. In order to obtain the quantities A and B for typical conductors, consider a wire or radius rw. From (3.202) we observe that
The resistance of the reference conductor would be similarly added to all entries of A. Similarly, for a conductor of rectangular cross section having width w and thickness t, the approximations given in (3.220) and (3.221) yield
There is one final observation apparent in (5.142). The product of two Laplace-transformed variables translates in the time domain to the convolution of their time-domain representations CA.21:
Therefore, (5.142) translates in the time domain to
a V(2, t ) = -Z,(t) * I(2, t ) - L a I(2, t ) az at
(5.146a)
323
INCORPORATION OF LOSSES
a I(z, t ) = -CV(2, t ) - c -& a V(2, t ) -
(5.146b)
az
This shows that the high-frequency skin-effect resistance and internal inductance that vary as require a convolution of the time-domain representation, Z,(t), and the line currents as was pointed out in [16]. The explicit representation of (5.146a) can be obtained in the following way. Expanding (5.142a) using the approximation given in (5.143) yields
a
d - V(Z,S) = -[A dz
+ B&]I(z,
= -AI(z,
S)
S)
I;[
- sLI(z,
(5.147)
S)
- B - sI(z, S) - sLI(z, S)
& = s/&
The skin-effect term is placed in the form of since the inverse transform of l/& is [16,17] 1
1
to simplify the result
1
(5.148)
3-73 Therefore the result simplifies to
a
a
az
at
- V(z, t ) = - Z,(t) * I(z, t ) - L - I(2, t ) = -AI(z, t ) - -
J' .B [ ro '
p
(5.149)
ia
3
I@, t - t) dt]
a I(z, t ) - Lat
Observe that this result shows that the convolution integral requires knowledge of the entire past history of (the derivative of) the current. Of course there are various approximate ways of including skin-effect losses that avoid this operation; these we will also investigate. 5.3.1
Two-Conductor Lossy Liner
In the case of two-conductor lines the MTL equations and the results in (5.141) to (5.149) reduce to scalars and the transmission-line equations become:
d V(z,S) = - [ A
ds
+ B& + sL]I(z, S)
(5.150a)
d I(z, S) = - [G
+ sC] V(Z,S)
(5.1 Sob)
ds
324
TIME-DOMAIN ANALYSIS
Many of the MTL results are extensions of the results obtained for twoconductor lines which we now investigate. 5.2 1.1 1umped-Circuit Appmximate Characterizations Perhaps the simplest way of including losses is via an approximate lumped-circuit model. The obvious structural choices are the lumped-pi or lumped-T structures investigated previously. The advantage of these representations is that simple lumped-circuit analysis programs can be used to solve for the line voltages either in the frequency domain or the time domain. The primary difficulty with these approximations is that they do not correctly process certain of the highfrequency spectral components of the input signal because their validity is based on the assumption that they are electrically short at all frequencies of interest. Figure 5.41(a) shows a lumped-pi representation of a section of the line. Typically this type of representation requires a large number of such sections to represent the line for input signals having significant high-frequency spectral content. Implementation of this lumped-circuit approximation is straightforward if we omit consideration of the skin-effect behavior of the line resistance (and internal inductance) and only include the dc resistance and internal inductance which can be represented as constant elements. A recurring problem is how we shall represent the Jf skin-effect behavior in the time domain which is required in these lumped-circuit representations. A novel method of doing this is to simulate the physical process that occurs in the development of the skin effect [lS]. For a conductor of circular cross section the current resides in annuli and is more strongly concentrated in the outer annuli as frequency is increased. Each of these annuli can be represented by a resistor and an inductor. The circuit representation of this process shown in Fig. 5.41(b) serves to simulate this process, At dc all resistors are in parallel giving the dc resistance. As frequency increases, each branch is successively removed from the circuit generating a frequency dependence. Of course, a larger number of branches is required to extend this Jf behavior to higher frequencies. Another representation that is particularly useful for conductors of rectangular cross section is to simulate the representation of the conductor as numerous bars in parallel as shown in Fig. 3.56. The current over each subbar is assumed constant but of unknown value and the dc resistances and partial inductances of the individual subbars are used to give the circuit shown in Fig. 5.41(c); this was implemented in [19]. Because of the interaction between the subbars via the mutual partial inductances, the currents in the subbars adjust to simulate the actual skin effect that occurs. The FORTRAN code SPICELPLFOR described in Appendix A for lossless lines generates a lumped-pi SPICE subcircuit model. This can be then modified to include losses by adding either dc resistances or one of the skin-effect simulations in Fig. 5.41(b) or (c). In this way only the line is modeled and nonlinear loads can be handled in the CAD code in which this model is imbedded. This is a simple approximation but suffers from the lengthy
fi
fi
INCORPORATION OF LOSSES
e
**
325
4
rAr
(a
fi
FIGURE 5.41 Representation of skin-effect conductor internal impedance: (a) the lumped-pi model, (b) approximation of the internal impedance as a frequency-selectivo network, and (c) approximation of the internal impedance via the method of coupled subconductors.
computation time which a sufficiently large circuit model requires to model the very-high-frequency spectral components of the input signal. 5.3.1.2 Time-Domain to Freguency-Domain Transformations Once again, a very straightforward way of including losses, particularly f i skin-effect losses, that are difficult to model directly in the time domain, is the time-domain to frequency-domain transformation. The frequency-domain transfer function (magnitude and phase) between the input and the desired output is obtained with the frequency-domain methods of Chapter 4. Observe that all of the line terminations are imbedded in this transfer function. Thus if any of the
326
TIME-DOMAIN ANALYSIS
terminations are nonlinear, the overall system is nonlinear and the notion of a frequency-domain transfer function is not useful. Skin-effect dependence causes no complications in the frequency-domain computation of the transfer function. The Fourier transform or the Fourier series of the input signal is obtained, and its individual spectral components are passed through the frequency-domain transfer function to yield the spectral components of the output. The time-domain output is the inverse Fourier transform of this spectrum. This straightforward technique is implemented in the FORTRAN code TIMEFREQ,FOR described in Appendix A which uses the frequencydomain transfer function computed with MTL.FOR. This method requires that the frequency-domain transfer function be computed at a sufficient number of frequencies of the input spectrum. It is a very simple and often-used method of indirectly determining the time-domain response but suffers from the basic restriction that a linear line and terminations are required since superposition was implicitly used.
fi
5.3.1.3 Finite Difimnce-Time Domain (FDTD) Methods The next rather obvious method is the finite difference-time domain (FDTD) method described earIier for lossless lines. This can be straightforwardly adapted to handle losses including skin-effect dependence [20,21]. The FDTD discretization of the transmission-line equations are given earlier in (5.1 15). The only change in the discretization of these equations is in the voltage change equation (5.115a) which requires the addition of the discretized convolution term shown in (5.149). The term containing the constant, A, representing the conductor dc resistance can be represented in the usual fashion as the average of the currents about the present cellular point as shown in (5.115a). The convolution term can be discretized in the following fashion. Divide the time axis into At segments. Kunz showed that the discrete convolution can be approximated in the following manner assuming that the function F(t) may be approximated as being constant over the At segments [21]:
fi
J: $F(t - r ) dr s
&
-F((n + 1)At - r ) dr ( m t 1)AI
FR+1-m
g
fmAt
m=O
z f i
(5.151a)
1
-dr
&
F"+1-mz0(rn) m-0
where (5.15 1b) Adapting this result to the discretization of the transmission-line equation gives,
INCORPORATION OF LOSSES
327
with reference to Fig. 5.22, (5.152a)
Only the first equation has changed from the previous discretization. Solving (5.152a) gives the recursion relation for this equation:
(5.153)
or (5.154a)
for k = 1 , . . , ,NDZ where
(5.154b) This replaces equation (5.120d) of the lossless case. Unfortunately, this requires storage of all past values of the currents. A clever solution to this problem is presented in [21). Prony's method can be used to
328
TIME-DOMAIN ANALYSIS
TABLE 5.1
Coefficients of the Prony Approximation of &(m)
i
-
0.79098180E 1 0.11543423E0 0.134353 80E0 0.218 70422E0 0.98229667E - 1 0.51360484EO -0.20962898EO 0.119 74447E1 0.11225491E- 1 -0.744 252 55EO
1 2 3 4 5 6 7 8 9 10
-0.1 14844 27E - 2 -0.138 18329E - 1 -0.540 375 96E - 1 -0.142 1649480 -0.301 284 37E0 -0.561 421 85EO -0.971 171 26EO -0.163 38433E1 -0.28951329El -0.504 10969E1
approximate Zo(m)as (5.155)
Kunz showed that a reasonable approximation can be obtained by using ten terms where the coefficients are given in Table 5.1 [21]. Substituting (5.155) into (5.154) yields (5.156a)
for k = 1,.
..,NDZ where
The Yy functions are updated via (5.156b) before evaluating (5.156a). Thus only one additional past value of current needs to be retained. In the case of a multiconductor line a virtually identical development
INCORPORATION OF LOSSES
329
provides (5.157a)
for k = 1,. . . ,NDZ where
(5.157~) This replaces equation (5.1 40d) of the lossless-line development. All other equations in (5.140) remain unchanged. This is implemented in the FORTRAN code FDTDLOSS.FOR described in Appendix A. In order to compare the predictions of these models we will investigate a high-loss, two-conductor transmission line shown in Fig. 5.42. The dimensions are typical of thin-film circuits. Two conductors of rectangular cross section of width 20 pm and thickness 10 pm are separated by 20 jtm and placed on one side of a silicon substrate (E, = 12) of thickness 100 pm. The total line length is 20 cm and is terminated at the near and far ends in 50 R resistors. The source is a ramp fiinction rising to a level of 1 V with a risetime of 50ps. The per-unit-length inductance and capacitance were computed using PCBGALFOR giving I = 0.805 969 pH/m and c = 88.248 8 pF/m. This gives a velocity of propagation in the lossless case of o = 1.185 73 x 10" m/s and a one-way time delay of T = 1.68672 ns which gives an effective dielectric constant of si = 6.3925 and a characteristic impedance of 2, = 95.566 R. The per-unit-length dc resistance is computed as A = rd, = l/(awt) = 86.207 R/m. The break frequency where this transitions to the high-frequency resistance that varies as Jf is computed from (3.222)as f , = 393.06 MHz. The factor B is computed as described previously to be B = 1/2(t + w ) = r d c / a = 2.45323 x Figure 5.43(a) shows the comparison of the SPICE (lossless) predictions, the time-domain to frequency-domain transformation (TDFD), and the finite difference-time domain (FDTD) results for the near-end and far-end voltages. The time-domain to frequency-domain transformation modeled the source as a 10 MHz periodic trapezoidal waveform with 50% duty cycle and rise/fall times of 50 ps, and the code TIMEFREQ.FOR was used which uses the frequency-
330
TIME-DOMAIN ANALYSIS
FIGURE 5.42 A lossy printed circuit board for illustration of numerical results: (a) line dimensions and terminations, (b) cross-sectional dimensions, and (c) representation of the open-circuit source-voltago waveform.
domain transfer function obtained from MTL.FOR. The high-frequency spectrum of this waveform rolls off at -40 dB/decade above l/m, = 6.3662GHz so 2000 harmonics were used in the computation of the frequency response via the M"L.FOR code (giving an upper limit of 20 GHz). The spatial discretization for the FDTD results was chosen so that each cell was A/lO at twice this break frequency giving NDZ = 215. The magic time step of At = Az/v for this spatial discretization and a total solution time of 10 ns is NDT = 1275. The effect of line loss is evident and significant, but the time-domain to frequencydomain transformation method and the FDTD method give different results. This is due to the fact that the FDTD method simply adds the low-frequency dc resistance to the high-frequency impedance and models the total conductor impedances as
INCORPORATION OF LOSSES
LOWYPCB (riretime = 50pr)
0 . q . .
1
.
1
.
.
.
.
,
.
.
,
..................................................................... ---------)
=P 7
0.5.
0
9
PY z
0.4
.............
------------{,-.-z*-,--,-*-*-#<.....................................................................
-
,
1
I
0.3 0.2
.
331
;
I..>'
I
-
SPICE (lossless) --- TDFD NH
0.1 -
-
(IOSSY, 9 2000) 215, NDT
.............. FDTD (lorey, NDZ
0.0
I
,
.
I
.
I
.
.
.
I
.
-
1275) .
.
Time (ne) (0)
0.01 0
A.
I
2
.
.
I
I
.
.
,
6
4
.
.
I
8
.
.
'J
10
Time (ne) (b)
FIGURE 5.43 Comparison of the source and load voltages predicted by the SPICE (lossless) model, the time-domain to frequency-domain transformation (TDFD), and the finite difference-time-domain (FDTD) methods for the lossy PCB of Fig. 5.42: (a) complete models and (b) adding the dc and high-frequency impedance in the TDFD method to compare with the FDTD method.
332
TIME-DOMAIN ANALYSIS
The time-domain to frequency-domain method uses the frequency-domain transfer function computed with MTLFOR which uses a frequency-selective model of the conductor impedances:
At the break frequency, f , = 393.06 MHz, the FDTD results are a factor of 2 larger than those of the frequency-selectiveTDFD method. The true value lies between these two values (see Figs. 3.52 and 3.53.) This break frequency lies in the -20 dB/decade region of the spectrum which begins at ~/ZT= 6.366 MHz and ends at l/m, = 6.366 GHz CA.31. Thus we should expect a difference between the two model predictions. To verify that this is the case, we recompute the time-domain to frequency-domain transformation results wherein the low-frequency and high-frequency conductor impedances are added together in the frequency-domain transfer function as in the FDTD code rather than using a frequency-selective computation as described above. Figure 5.43(b) shows the results of this computation for the near-end and far-end voltages. These results confirm the hypothesis and show that the FDTD code gives virtually identical results to the time-domain to frequency-domain transformation method when the same equations are being solved. In the applications of the FDTD technique to electromagnetic scattering problems, a surface-impedance boundary condition is frequently used. This condition essentially has a frequency dependence and requires no constant resistance [la, 213. Numerous solutions of the transmission-line equations for lossy lines include only the high-frequency, impedance of the line conductors and ignore their dc, constant resistance. We have included this in the FDTD solution via the A term. The question is whether the dc resistance can be neglected and only the high-frequency impedance used. Figure 5.44(a) shows the results of including only the dc resistance or only the high-frequency impedance of the conductors. Including only the dc resistance gives the correct late-time dc level of the responses but the early-time results are not well predicted at the beginning of the transitions. Including only the high-frequency impedance of the conductors does not predict the late-time dc levels. So it appears that both the dc and the high-frequency impedances must be included in the formulation. Figure 5.44(b) shows the comparison between the TDFD and FDTD models using only the dc resistance or only the highfrequency resistance. This again confirms the importance of including the dc resistance and additionally shows that the TDFD and FDTD models give virtually identical results when the same representations of the conductor impedances are used. The slight difference when zi = q h f is probably due to approximation of the high-frequency convolution term in the FDTD code.
fi
fi
fi
fi
fi
O
,
S
I
.
.
2
0
,
.
-
INCORPORATION OF LOSSES
Loiry PCB (riretlme SOpr) .
,
*
4
.
6
,
.
.
,
.
.
333
I
10
8
Time (ns)
(4
Loary PCB
i
.. . ....
0.2 0.I
0
2
6
4
SPICE (lossless) TDFD (lossy, ZI.IIdc) TDFD (lossy, 21 = 4 h l ) FDTD (lossy, 21 = rdc) FDTD (lossy, 21 ~th,) 8
10
Time (ns) (b)
FIGURE 5.44 Comparisons as in Fig. 5.43 with (a) the TDFD method using only the dc resistance and only the high-frequency impedance and (b) both methods using only the dc resistance or only the high-frequencyimpedance.
334
TIME-DOMAIN ANALYSIS
53.1.4 Direct Solution via Inversion ofthe Laplace Transform The next solution method is the direct solution of the transmission-line equations given in (5.150). The propagation constant and characteristic impedance are
y(s) = J ( A
+ B& + sL)(G + sC)
(5.158) (5.159)
We will omit the conductance term and assume small losses. The propagation constant and characteristic impedance require the square root of quantities involving the Laplace transform variable. This can be approximated by expanding the square root in a Taylor series and using the first two terms: A
+ BJS SL
w
(5.160)
a(+ +) 1
A 2sL +
which is valid for IA + B$l < lsLl which implies high frequencies and/or small losses. Under this approximation, the propagation constant and characteristic impedance become (5.161)
s A B =-+-+-4 2z0 22, 0,
(5.162)
= z o +Avo - + - Bo, 2s
2&
where the characteristic impedance and velocity of propagation for the lossless line are denoted by (5.163) vo =
1 -
fl
(5.164)
335
INCORPORATION OF LOSSES
The general solution to the transmission-line equations is of the form (5.165a) (5.165b)
The direct solution and inversion of these expressions for general sources and loads is quite complicated. We will show the result for a unit-step-function source, &(t) = Y , U ( t )
Y,
(5.166)
S
zero source impedance, Z,(s) = 0, and either an infinite or a matched line,
Z,(s) = 2,-(s). In this case the general solution in (5.165) becomes
The term e - A / z Z represents ~r simple attenuation of the waveform amplitude, whereas the term represents time delay:
where the time delay of the lossless line is denoted as T, = z/v,,. The inverse Laplace transform of these results becomes [17] (5.169a)
336
TIME-DOMAIN ANALYSIS
and erfc denotes the tabulated complementary error function:
- -J"e-rsdr
erfc(x) = 1
6
(5.170)
0
2
m
3-
e"'dr
6.
The representation of Z,(s)in (5.143)characterizes the initial or early-time response when A = 0 and characterizes the final or late-time response when B = 0. So we would expect that ignoring the high-frequency resistance and internal inductance, B = 0, would yield early-time responses that are not realistic. This can be confirmed by setting B = 0 in (5.143) yielding
(5.171) Z(S) = Z"J1
+A
(5.172)
The current expression for the step response of a matched line becomes
qz, s) =
v,
e-~ l l v o ~ J - z
(5.173)
The inverse Laplace transform is E171 I(z, t ) =
5 e - ( A / 2 L ) f ~ o ( -A 2 0
and I&)
JiT-;-j.f),(t
-
2L
(5.174)
is the Bessel function of the first kind [17,22]: Io(x)= 1
x2 x4 X6 +-+-+-+,.* 2 ' 2242 224262
(5.175)
Consider a time immediately after the time delay to a point z, t = c+.The
INCORPORATION OF LOSSES
337
result in (5.174) becomes
since, according to (5.175), I O ( ( A / 2 L ) J ! ) NN 1. This result predicts a stepfunction change in the line voltage at these points on the line but that is not realistic; the current should not change instantaneously. So evidently the high-frequency skin-effect impedance must be included in the time-domain result, B # 0, in order to generate realistic early-time responses. Other responses such as the impulse response
can be similarly obtained either by direct inversion of the Laplace transform or by differentiation of the step function result. For example, using the approximations in (5.161) and (5.162), the impulse response becomes
~(z, s) = V ,e
(4%)~ e -( W 2 Z e ) J ; z e -W
u d
(5.178)
whose inverse Laplace transform is [17]
The step response or the response to any other form of &(t) can be obtained by convolution using this impulse response of the line CA.21. Clearly, even for this simple case of a two-conductor line, the incorporation of losses is quite complicated and so are the results. Furthermore this direct solution has required the assumption of a matched or infinitely long line in order to provide simple transforms which are invertible. Thus we expect considerable solution difficulties for more practical cases of mismatched lines with nonlinear loads. The inversion of the Laplace transform result was also investigated in [23-25). It is also possible to invert the Laplace transform numerically using the fast inversion of the Laplace transform (FILT) [26] sometimes known as the numerical inversion of the Laplace transform (NILT) [27,28]. An extensive investigation of the properties of the Laplace transform of the response of a two-conductor lossy line is contained in [29]. 5.3.1.5 Time-Domain Characterization of the Line as a Two Port The final solution method is to characterize the line as a two port representation in the time domain. Sources and loads (including nonlinear ones) can then be attached and the solution obtained. The advantage of this type of characterization is that the terminations are external to it and can be nonlinear.
338
TIME-DOMAIN ANALYSIS
The lumped-circuit iterative' models such as the lumped-pi or lumped-T structures discussed previously are common examples of this notion. However, these models suffer from the requirement that many such cascaded sections that are electrically short at the highest frequency of interest are needed in order to correctly process the high-frequency spectral content of the input signal. An alternative to the requirement for a large number of cascaded sections is the following [30]. The chain parameter matrix of a lossy two-conductor line of length 9 is
Similarly, the admittance parameter representation is
where the admittance parameters can be obtained from the chain parameter matrix as (5.18 1b)
(5.1 8 1c)
This admittance representation can be synthesized as shown in Fig. 5,45(a). The task now becomes the synthesis of the networks that may be used to represent the elements (5.182a)
(5.182b) First let us consider the frequency domain and ignore losses so that 7.9 = j o 9 / v =j p 9 =j2n9/A and 2, = The admittances of the elements
fi.
INCORPORATION OF LOSSES
339
FIGURE 5.45 Synthesis of equivalent circuits to represent the line’s admittance parameters: (a) the admittance parameter reprsentation and (b) series representation of the individual admittance parameters for a lossless line.
become
&(jo)- YM(jo)=j
(5.183a) 4
(2. f) 1
= -j
sin
(5.1 83b)
These trigonometric functions may be expanded in series form valid for 9 < 4 2 using [22) tanx 3: x
2 +-x3 +-xs 3! 15
1 -=-
7 + -x + -x3
1 sinx x
6
360
+ . . e
+ .*
I
340
TIME-DOMAIN ANALYSIS
This gives
[ (
>'+-(->'+...I
SY - YM(jcu)=j u - Y 1 + -1 -
yS(ju)
2
3 A
2 15
SY
1
(5.184a)
These admittance functions can be synthesized as cascades of admittances as illustrated in Fig. 5.45(b). Clearly, using the first terms gives the usual lumped-pi model that we have used previously. Although these expansions are only valid for lossless lines and line lengths less than one-half of one wavelength there are other expansions of the trigonometric functions that extend this region of validity and, in addition, accommodate lossy lines [30]. A number of other methods also seek to obtain expansions using infinite-ladder-type networks to represent the appropriate impedance/admittance functions [31]. Another frequently-used model is the extension to lossy lines of the method of characteristics, Branin's method, discussed previously for lossless lines [27,28,32-34). These methods frequently assume only dc resistance and ignore dependence. The chain parameter matrix in (5.180) is inverted skin-effect to yield
fi
Z,(s) =
/-+ g
(5.185b) sc
(5.185~)
These equations can be converted into the following equations which are similar to the method of characteristics for lossless lines:
- Z,(s)I(O, s) = e-'(*)[ V ( 9 ,s) - Zc(s)I(Y,s)] V ( 9 ,s) + Zc(s)I(Y,s) = e-'(')[V(O, s) + Z,(s)I(O, s)] V(0, s)
(5.186a) (5.186b)
These relations may be represented as shown in Fig. 5.46 where
- E,@)] = e-e(r)[2V(0, s) - E&)]
Eo(s) = e-e(8)[2V(Y,s)
(5.187a)
E&)
(5.187b)
INCORPORATION OF LOSSES
-
I
341
-
I
FIGURE 5.46 Representation of a lossy, two-conductorline with the generalized method of characteristics.
ZC(4
FIGURE 5.47
(b)
Ladder representations of the characteristicimpedances of a lossy line.
Therefore, the lossy line can be simulated as in Fig. 5.46 if we can synthesize appropriate representations for ZJs) and e-e(s! For example, the characteristic impedance for lossy lines, Z,(s) c: J(r sl)/(g sc), can be expanded in the form of a ladder network consisting of an infinite cascade of symmetrical T networks consisting of series and parallel impedances shown in Fig. 5.47 using the Pad6 approximation method [27,32]. Recall that this expansion is valid for constant r and g, i.e., the skin effect is ignored. Chang showed that a small number of these structures (approximately four) will simulate the characteristic impedance quite well. The network shown in Fig. 5.47(a) is used if the characteristic impedance is “capacitive” meaning c > gl/r, and the network shown in Fig. 5.47(b) is used if the characteristic impedance is “inductive” meaning 1 > rc/g. The elements are given in Table 5.2 [32]. A useful way of interpreting the exponential propagation function is by determining the unit-impulse response that it represents and using conoolution in conjunction with (5.187). The impulse response of the propagation function is
+
~ ( = ~e - e m1
+
e - J i ; i ~ J b ’+ (re + W I C ) ~ + (rg/lc)
(5.188)
342
TIME-DOMAIN ANALYSIS
TABLE 5.2
Capacitive Z&)
Elements of the Circuits of Fig. 5.47
p = rc/gl> 1
Inductive Z&)
p = rc/gl < 1
The inverse Laplace transform is [27] (5.189a) where a=-
rc
+ gl
(5.189b)
- rcl
(5.189~)
21c
b = -Ik3
21c
T = .9/0is the total one-way line delay, and II is the modified Bessel function of the first kind. A similar direct convolution is obtained in [35]. A first-order approximation to the propagation function can also be obtained using the small loss approximation of the first two terms in a Taylor series expansion given in (5.161): e -@(a) e- A Y / 2 5 e - ( B U / 2 Z , ) 4 e - ~ T (5.190)
-
whose inverse Laplace transform is [I71
In either case the outputs of the controlled sources in Fig. 5.46 are obtained from (5.187) by convolution using these impulse responses as
INCORPORATlON OF LOSSES
5.3.2
343
Multiconductor lines
Many of the techniques for two-conductor lines are adaptable to MTL's. The prominent ones are the lumped-circuit iterative approximate structures such as the lumped-pi and lumped-T structures, the time-domain to frequency-domain transformation, and the finite difference-time domain (FDTD) technique (equations (5.140) with equation (5.157) replacing (5.14Od)). The FORTRAN programs SPICELPLFOR, TIMEFREQ.FOR and FDTDLOSS.FOR are written to handle these cases for MTL's. With the exception of these techniques, the time-domain solution of the MTL equations for lossy lines is generally a formidable task. In this section we discuss some of the remaining methods for lossy MTL's. The MTL equations can be easily decoupled via similarity transformations in the case of lossless lines as demonstrated previously. In the general case of lossy lines this is not possible except for certain special case8 that exhibit structural symmetry. In attempting to decouple the MTL equations for lossy lines we again convert to modal quantities with similarity transformations: (5.193a) (5.193b) where we have denoted the line voltages and currents as their Laplacetransformed variables to simplify the equations with respect to the time variable. Substituting these into the complete MTL equations yields
d dz
- Vm(z,s) = T;'R(s)T,I,(z,
s) + sT;'(L,(s)
+ L,)T,I,,,(z, s)
(5.194a) (5.194b)
Observe that the resistance matrix, R(s), and the internal inductance matrix, L,(s), are each shown as being functions of s which will be the case unless skin-effect losses are ignored and only dc resistance and internal inductance are considered. Similarly, the conductance matrix, G(s), is, in general, a function of s. The intention is to find the n x n transformation matrices TYand TI such that these modal equations are uncoupled. If this were possible, we could simulate the modes with uncoupled, two-conductor lossy modal lines using the previous results for two-conductor lines from Fig. 5.46. The transformations from mode to actual line currents and voltages via (5.193) could be implemented in the same manner as for lossless lines using controlled sources as illustrated in Fig. 5.48. Recall also that R and a portion of L, the internal inductance contribution, contain frequency-dependent parameters corresponding to the internal impedances of the conductors. So, in general the transformation matrices will be functions of s. However, it is desirable that the transformations
344
TIME-DOMAIN ANALYSIS
FIGURE 5.48 Representation of a lossy MTL via the generalized method of characteristics in terms of uncoupled modes and lossy two-conductor mode lines.
be independent of s so that the controlled-source parameters are constants. Controlled sources that are functions of s which represent the general case were implemented in [31]. This general goal of decoupling the MTL equations is often referred to as the generalized method of characteristics [36-42) although it has not been fully realized because of the inability to completely decouple the lossy-line MTL equations. The procedure for accomplishing thb is more easily observed by writing the MTL equations as uncoupled, second-order equations as shown in equation (2.31):
d'
7VJZy dz
d' -I dzz
T;'[RG]T,V,(z,
3)
S)
+s2T; '[LC)T,V,(z,
( Z S)
'
5
+ sT;'[RC + LG]TyV,(z,
(5.195a)
s)
T,"[GRJT~I,(z, S) + sT,"[GL
+s2T; '[CL]T,I,(z,
S)
+ CR]T~I,,,(Z,S)
(5.195b)
s)
Because the per-unit-length parameter matrices are symmetric it suffices to
INCORPORATION OF LOSSES
345
decouple (5.195b) and let -T = TI e
(5.196a)
T;’
(5.196b)
So we must find an n x n matrix T which simultaneously diagonalizes the following three matrices:
T-’(GR)T = A I
T-’(GL
+ CR)T = A,
T- ‘(CL)T = A,
(5,197a) (5.197b) (5.197~)
We observed previously that it is possible to obtain a transformation T that will diagonalize the product of two real, symmetric, positive definite matrices as are C and L. Clearly it is too much to expect to be able to find one transformation T that will simultaneously diagonalize all three matrices in (5.197) with one transformation and, furthermore, to have that transformation be independent of s. In order to show the special cases which can be decoupled with one constant transformation, T, suppose that we assume a perfect medium, G = 0. Now we must diagonalize two matrices: T”(CR)T = A2
(5.198a)
A3
(5.198b)
T”(CL)T
3 :
There are certain special cases where a constant transformation, T, can be found that simultaneously diagonalizes both matrices in (5.198). These are: 1. (n 2. (n
+ 1) identical conductors ignoring loss in the reference conductor. + 1) identical conductors in homogeneous media.
The key here is the observation that for n identical conductors, the per-unitlength resistance matrix is of the form
(5.199)
= rl,
+ roU,
where 1, is the n x n identity matrix with ones on the main diagonal and zeros elsewhere and U, is the n x n unit matrix with ones in all positions. In the first
346
TIME-DOMAIN ANALYSIS
case, R = rl, and
[r(s) + sl,(s)]T”(C)T = A2 T-’(CL)T = AS
(5.2Wa) (5.200b)
where we have extracted the internal inductances so that L = Le. By the procedure given in Section 5.2.1.2 this is always possible (see (5.72)). In the next case we have the fundamental identity for a homogeneous medium, CL = 1/u21,, so that [r(s) sll(s)]T”[C(l, U,)]T = A2 (5.201a)
+
+
1 1 U 2 T-’[l,]T = U 2 1, =
(5.20 1b)
But C and (1, + U,) are both real, symmetric and positive definite so that diagonalization of (5.201a) is assured. The above has demonstrated those special cases that can be decoupled. There are other apparent cases such as the assumption that only nearest-neighbor coupling occurs so that C and L are tridiagonal Toeplitz matrices [42]. But this requires identical conductors and neglecting loss in the reference conductor also in order to work so we may as well diagonalize the full C and L as in (5.200) and not assume only nearestneighbor coupling. When neither of the above special cases of identical conductors exist, the MTL equations in terms of modes in (5.194) cannot be decoupled. However, they can be placed in the form of ordinary differential equations (which are coupled) and numerically integrated. For example, suppose we assume R and G are constant (omit skin-effect losses) and neglect the internal inductance, Ll(s) = 0. If we choose a modal transformation that diagonalizes the inductance and capacitance matrices, then the modal equations become
where
T;’L,T, = L,
(diagonal)
(5.203a)
T i ‘CTv = C,
(diagonal)
(5.203b)
Ti ‘RT,= R,
(not diagonal)
(5.203~)
T;’C;Tv = G,
(not diagonal)
(5.203d)
Note that only L and C can be diagonalized so that the modal equations in
INCORPORATION OF LOSSES
347
(5.202) are not uncoupled. Tripathi in [36] places these in the form of ordinary differential equations via the generalized method of characteristics in the following manner. Define the modal characteristics as
+ d t at-aV ,
(5.204a)
a I, + dt at
(5.204b)
a
dV, Z d Z - V , az
a
dl, = dZ - I , a2
where dz is a diagonal matrix containing the derivatives of the n axes of the z-t plane defining the characteristic curves. Substituting (5.202) into (5.204) yields dtl,
dV, dV,
- dz(L,C,)”2
H
=0
(5.205a)
+ Zc, d I , + dz(ZcmG,V, + RmIm) 0 dtl, + dz(L,C,,,)”’ = 0
(5.205b)
0
(5.205d)
- Zcmd l , + dz( -Z,-,G,V, + R,I,,,)
=i
(5.205~)
where 1, is the n x n identity matrix and Zcm is the n x n diagonal modal on the characteristic impedance matrix with entries [Zc,,Irr = main diagonal. Equation (5.205a) defines the n families of characteristic curves along which (5.205b) holds, and (5.20%) defines the n families of characteristic curves along which (5.205d) holds. The ordinary differential equations in (5.205) are solved numerically by approximating the time variables as At and approximating the derivatives as first-order differences. The method was extended to include frequency-dependent skin-effect losses of the conductors and frequency-dependent losses in the medium in [37]. This technique follows the usual method of characteristics for two-conductor lines with the added complexity of coupled equations. Another interesting technique that shows promise is the asymptotic waveform evaluation (AWE)method [43,44]. It seeks to expand the Laplace-transformed voltages and currents in power series in the Laplace transform variable s or moments. By matching matrix coefficients of corresponding powers of s the MTL equations are transformed into a series of differential equations each of which can be diagonalized, and the various voltages and currents in the series expansions are solved for recursively. The method is currently restricted to constant R, i,e., the skin effect is not included.
Jm
5.3.3
Computed Results
The computed results are for the ribbon cable (shown in cross section in Fig. 4.14) and the PCB (shown in cross section in Fig. 4.17) with 50 R loads shown in Fig. 5.29. We will show the predictions of the lossless SPICE model, the
348
TIME-DOMAIN ANALYSIS
Near-End Croritalk Voltage (Ribbon cable, rigetime = 20 ns)
Time (n8)
FIGURE 5.49 Comparison of the time-domain response of the near-end crosstalk of the ribbon cable of Fig. 4.14 determined via the SPICE model (lossless) and the lumped-pi model (dc losses included) using one and two sections to represent the line for a pulse risetime of 20 ns.
lumped-pi approximate model, the time-domain to frequency-domain transformation method (TDFD), and the finite difference-time domain (FDTD) methods for the lossy lines.
5.3.3.1 Ribbon Cable Figure 5.49 shows the comparison of the SPICE (lossless) model and the predictions of the lumped-pi iterative model using one and two sections for the 2 m ribbon cable and a rise/fall time of the pulse of 2011s. Figure 5.50 shows the comparison of the SPICE (lossless) model, the time-domain to frequency-domain transformation (using 100 harmonics of the 1 MHz pulse train), and the FDTD model (using 2 divisions of the line and 50 divisions of the time axis). All models give approximately the same results, and the losses appear to be inconsequential for this configuration and rise/fall time. Figure 5.51 shows the comparison of the lossless SPICE model, the time-domain to frequency-domaintransformation (using 1000 harmonics of the 1 MHz pulse train) and the FDTD model (using 25 divisions of the line and 750 divisions of the time axis) for a pulse risetime of 1 ns. The losses give a dc offset during the period when the pulse is in its 1 V stable state which is referred to as common-impedance coupling and is due to the resistance of the reference conductor.
INCORPORATION OF LOSSES
-
349
Near-End Crorrtalk Voltage (Ribbon cable, rlretlmr 20 nr)
0
50
150
100
Time (na)
Comparison of the time-domain respqnse of the near-end crosstalk of the ribbon cable of Fig. 4.14 determined via the SPICE model (lossless), the time-domain to frequency-domain method (TDFD) (with skin-effect losses), and the finite differencetime-domain (FDTD) method (with skin-effect losses) f0r.a pulse risetime of 20 ns. FIGURE 5.50
Near-End Cronrtalk Voltase (Ribbon cable, rirstims = 1 na)
TDPD (loary, NH
-
1OOO)
T i m e (ns)
FIGURE 5.51 Comparison of the time-domain response of the near-end crosstalk of the
ribbon cable of Fig. 4.14 determined via the SPICE model (lossless), the time-domain to frequency-domain (TDFD) method (with skin-effect losses), and the finite differencetime-domain method (FDTD) (with skin-effect losses) for a pulse risetime of 1 ns.
350
TIME-DOMAIN ANALYSIS
-
Near-End Crosrrtalk Voltage (Printed circuit board, risetime 6.25 ns)
Time (no)
FIGURE 5.52 Comparison of the time-domain response of the near-end crosstalk of the printed circuit board of Fig. 4.17 determined via the SPICE model (lossless) and the lumped-pi model (dc losses included) using only one and two sections to represent the line for a pulse risetime of 6.25 ns.
100
c$ -4
Near-End Crosstalk Voltage (Printed circuit board, risetime = 6.25 ns)
Y
,"
50
E
0
0
10
20
30
Time (ns)
FIGURE 5.53 Comparison of the time-domain response of the near-end crosstalk of the printed circuit board of Fig. 4.17 determined via the SPICE model (lossless), the timedomain to frequency-domain (TDFD) method (with skin-effect losses), and the finite difference-time-domain method (FDTD)(with skin-effect losses) for a pulse risetime of 6.25 ns.
I
REFERENCES
-
351
Near-End Crosatalk Voltage (Printed circuit board, risetime 50 pa)
r
9
--
I
II .........
I
1
I
5
10
SPICE model (lossless) TDFD (loasy, NH 1OOO) FDTD (lo#sy, NDZ 88, NDT
20
-
2000)
25
1s Time (no)
FIGURE 5.54 Comparison of the time-domain response of the near-end crosstalk of the printed circuit board of Fig. 4.17 determined via the SPICE model (lossless), the time-domain to frequency-domain (TDFD) method (with skin-effect losses), and the finite difference-time-domain method (FDTD) (with skin-effat losses) for a pulse risetime of 50 ps.
5.3.3.2 Printed Circuit Board Figure 5.52 shows the comparison of the SPICE (lossless) model and the predictions of the lumped-pi iterative model using one and two sections for the 25.4 cm PCB and a pulse rise/fall time of 6.25 ns. Figure 5.53 shows the comparison of the SPICE (lossless) model, the time-domain to frequency-domain transformation (using 100 harmonics of the 10 MHz pulse train), and the FDTD model (using 2 divisions of the line and 60 divisions of the time axis). All models again give approximately the same results. Figure 5.54 shows the comparisons of the lossless SPICE model, the time-domain to frequency-domain transformation (using lo00 harmonics of the 10 MHz pulse train), and the FDTD model (using 88 divisions of the line and 2000 divisions of the time axis) for a pulse rise-time of 50 ps. Again the losses are not significant for this configuration except for the slight dc offset caused by commonimpedance coupling. REFERENCES
[l]
F.H. Branin, Jr., “TransientAnalysis of Lossless Transmission Lines,” Proc. IEEE, 55,2012-2013, 1967.
[2]
L. Bergeron, Water Hammer in Hydraulics and Wave Surges in Electrfcity, John Wiley, NY, 1961.
352
[3]
[4] [5] [6] [73 [S]
[9] [lo] [ll] [12) [131
[14] [l5]
El61 1171 [I81 [19] [20] [Zl]
TIME-DOMAIN ANALYSIS
W. Frey and P. Althammer, “The Calculation of Electromagnetic Transients on Lines by Means of a Digital Computer,” The Brown Boueri Reuiew, 48, 344-355 (1961). R.S. Singleton, “No Need to Juggle Equations to Find Reflection-Just Draw Three Lines,” Electronics, 41, 93-99 (1968). A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, NY,1965. H. Amemiya, “Time Domain Analysis of Multiple Parallel Transmission Lines,” RCA Review, 28,241-276 (1967). EY. Chang, “Transient Analysis of Lossless Coupled Transmission Lines in Inhomogeneous Dielectric Media,” IEEE Trans. on Microwave Theory and Techniques, MTT-18,616-626 (1970). C.W. Ho,“Theory and Computer-aided Analysis of Lossless Transmission Lines,” IBM J. Research and Development, 17, 249-255 (1973). K.D. Marx, “Propagation Modes, Equivalent Circuits, and Characteristic Terminations for Multiconductor Transmission Lines with Inhomogeneous Dielectrics,” IEEE Trans. on Microwave Theory and Techniques, MTT-21,450-457 (1973). V.K. Tripathi and J.B. Rettig, “A SPICE Model for Multiple Coupled Microstrips and Other Transmission Lines,” IEEE Trans. on Microwave Theory and Techniques, MTT-33, 1513-1518 (1985). A.K. Agrawal, H.J.Price, and S,H. Gurbaxani, “Transient Response of Multiconductor Transmission Lines Excited by a Nonuniform Electromagnetic Field,” IEEE Trans. on Electromagnetic Compatibility, EMC-22, 119-129 (1980). D.F. Higgins, “Calculating Transmission Line Transients on Personal Computers,” Proc. IEEE International Symposium on Electromagnetic Compatibility, August 25-27, 1987, Atlanta, GA. A.R. Djordjevic, T.S. Sarkar, and R.F. Harrington, “Time-Domain Response of Multiconductor Transmission Lines,” Proc. IEEE, 75, 743-764 (1987). R.L. Wiggington and N.S. Nahman, “Transient Analysis of Coaxial Cables Considering Skin Effect,” Proc. IRE, 45, 166-174 (1957). N.S. Nahman and D.R. Holt, “Transient Analysis of Coaxial Cables Using the IEEE Trans. on Circuit Theory, 19, Skin Effect Approximation A B&“ 443-451 (1972). EM.Tesche, “On the Inclusion of Losses in Time-Domain Solutions of Electromagnetic Interaction Problems,” lEEE Trans. on Electromagnetic Compatibility, 32, 1-4 (1990). R.V. Churchill, Operational Mathematics, 2d ed., McGraw-Hill, NY, 1958. C. Yen, 2.Fazarinc, and R.L. Wheeler, “Time-Domain Skin-Effect Model for Transient Analysis of Lossy Transmission Lines,” Proc. IEEE, 70,750-757 (1982). T. Vu Dinh, B. Cabon, and J. Chilo, “Time Domain Analysis of Skin Effect on Lossy Interconnections,” Electronics Lett., 26, 2057-2058 (1990). E.S.Mok and 0.1. Costache, “Skin-Effect Considerations on Transient Response of Transmission Line Excited by an Electromagnetic Pulse,” IEEE Trans. on Electromagnetic Compatibility, 34, 320-329 (1992). K.S. Kunz and RJ. Luebbers, The Finite Difference Time Domain Method in Electromagnetlcs, CRC Press, Boca Raton, FL, 1993.
+
REFERENCES
353
[22] H.B. Dwight, Tables oflntegrals and Other Mathematical Data, 4th ed.,Macmillan, NY, 1961. [23] K.R. Shah and Y. Yavin, “Equivalent Representation of Lossy Transmission Lines-Part I” and “Equivalent Representation of Lossy Transmission LinesPart 11: Skin Effect Consideration,” Proc. IEEE, 1258-1261 (1971). [24] M.S. Lin, A.H. Engvik, and 5 8 . Loos, “Measurements of Transient Response on Lossy Microstrips with Small Dimensions,” IEEE Trans. on Circuits and Systems, 37, 1383-1393 (1990). [25] M. Cases and D.M. Quinn, “Transient Response of Uniformly Distributed RLC Transmission Lines,” IEEE Trans. on Circuits and Systems, 27, 200-207
(1980). [26] T. Komuro, “Time-Domain Analysis of Lossy Terminations with Arbitrary Terminal Networks,” I E E E Trans.on Circuits and Systems, 38,1160-1 164 (1991). E271 E.C. Chang and S. Kang, “Computationally Efficient Simulation of a Lossy [28]
[29]
[30]
[31 J
[32] [33]
[34] [35’J
Transmission Line with Skin Effect by Using Numerical Inversion of Laplace Transform,” IEEE Trans. on Circuits and Sysrems, 39, 861-868 (1992). F.Y. Chang, “Transient Simulation of Nonuniform Coupled Lossy Transmission Lines Characterized with Frequency-Dependent Parameters Part I!: DiscreteTime Analysis,” IEEE Trans. on Circuirs and Systems, 39,907-927 (1992). M.S.Ghausi and J.J. Kelly, Introduction to Distrlbuted-Parameter Networks, Holt, Rinehart and Winston, NY, 1968. L. Monroe and C.R. Paul, “Lumped Circuit Modeling of Transmission Lines,” Proc. 1985 IEEE International Symposium on Electromagnetic Compatibility, Wakefield, MA, August 1985, pp. 282-286. (See also L. Monroe, “Modeling of Transmission Lines: A New Iterative, Lumped-Circuit Model,” MSEE Thesis, University of Kentucky, May 1985.) V.K, Tripathi and A. Hill, “Equivalent Circuit Modeling of Losses and Dispersion in Single and Coupled Lines for Microwave and Millimeter-Wave Integrated Circuits,” IEEE Trans. on Circuits and Systems, 36, 256-262 (1988). F.Y. Chang, “Waveform Relaxation Analysis of RLCG Transmission Lines,” IEEE Trans. on Circuits and Systems, 37, 1394-1415 (1990). F.Y. Chang, “Waveform Relaxation Analysis of Nonuniform Lossy Transmission Lines Characterized with Frequency-Dependent Parameters,” I E E E Trans. on Circuits and Systems, 38, 1484-1500 (1991). J.I. Alonso, J.B. Borja, and F.Perez, “A Universal Model for Lossy and Dispersive Transmission Lines for Time Domain CAD of Circuits,” I E E E Z’hns. on Microwave Theory and Techniques,40,938-946 (1992). P.S. Yeung, “Lossy Transmission Lines: Time Domain Formulation and Simulation Model,” IEEE Trans. on Microwave Theory and Techniques,41, 1275-1279
(1993). [36] N. Orhanovic, P.Wang, and V.K.Tripathi, “Generalized Method of Characteristics for Time Domain Simulation of Multiconductor Lossy Transmission Lines,” Proc. IEEE Symposium on Circuits and Systems, May 1990. [37] V.K. Tripathi and N. Orhanovic, “Time-Domain Characterization and Analysis of Dispersive Dissipative Interconnects,” IEEE Trans. on Circuits and Systems, 39,938-945 (1992).
TIME-DOMAIN ANALYSIS
J. Mao and 2.Li, “Analysis of the Time Response of Multiconductor Transmission Lines with Frequency-Dependent Losses by the Method of Convolution Characteristics,” IEEE Trans.on Microwave Theory and Techniques,40,637-644 (1992). F.-Y.Chang, “The Generalized Method of Characteristicsfor Waveform Relaxation Analysis of Lossy Coupled Transmission Lines,” IEEE Trans. on Microwuoe Theory and Techniques,37,2028-2038 (1989). A.J. Gruodis, “Transient Analysis of Uniform Resistive Transmission Lines in a Homogeneous Medium,”IBM J. Research and Developmemt,23,675-681 (1979). A.J. Gruodis and C.S. Chang, “Coupled Lossy Transmission Line Characterization and Simulation,”IBM J . Research and Deuelopment, 25, 25-41 (1981). D.S.Gao, A.T. Yang, and S.M. Kang, “Modeling and Simulation of Interconnection Delays and Crosstalks in High-speed Integrated Circuits,” IEEE Trans. on Circuits and Systems, 37, 1-8 (1990). T.K. Tang, M.S.Nakhla, and R. Griffith, “Analysis of Lossy Multiconductor Transmission Lines Using the Asymptotic Waveform Evaluation Technique,” IEEE Trans. on M1crowaue Theory and Techniques,39,2107-2116 (1991). J.E. Bracken, V. Raghavan, and R.A. Rohrer, “Interconnect Simulation with Asymptotic Waveform Evaluation (AWE),”IEEE Trans. on Circuits and Systems, 39,869-878 (1992).
PROBLEMS
5.1
Show, by direct substitution, that (5.3) satisfy the transmission-line equations given in (5.2).
5.2
Consider a lossless two-conductor line that has R, = 300 Q, RL = 60 R, 2 , = 100 Q, v = 200 m/ps, 9 = 200 m, and &(t) = 400u(t) where ~ ( tis) the unit-step function. Sketch V(0,t), I(0, t), V ( 9 , t), I ( 9 , t ) for 0 5 t i; lops. Do the results converge to the expected steady-state values?
5.3
Confirm your results using SPICE. Repeat Problem 5.2 for R L = 0 (short-circuit load).
5.4
Repeat Problem 5.2 for RL = a0 (open-circuit load).
5.5
A time-domain reflectometer (TDR) is an instrument used to determine properties of transmission lines. In particular, it can be used to detect the locations of imperfections such as breaks in the line. The instrument launches a pulse down the line then records the transit time for that pulse to be reflected at some discontinuity and to return to the line input. Suppose a TDR having a source impedance of 50 SZ is attached to a 50 Q coaxial cable having some unknown length and load resistance. The dielectric of the cable is Teflon (e, = 2.1). The open-circuit voltage of the TDR is a pulse of duration 10 ps. If the recorded voltage at the input of the TDR is as shown in Fig. P5.5,determine the length of the cable and the unknown load resistance. Confirm your results using SPICE.
"'"1 ,
PROBLEMS
355
120v,
100 v
FIGURE P5.5
5.6
A 12 V battery (R, = 0) is attached to an unknown length of transmission line that is terminated in a resistance. If the current to that fine for 6 ps is as shown in Fig. PJ.6, determine the line characteristic resistance and the unknown load resistance. Confirm your results using SPICE.
- 10 mA
I
FIGURE P5.6
5.7
Digital data pulses should ideally consist of rectangular pulses. Actual data, however, have a trapezoidal shape with certain rise/fall times. Matching the data transmission line eliminates reflections and potential logic errors arising from these reflections. However, matching cannot always be accomplished. In order to investigate this problem, consider a line having R , = 0 and R, = 00. Assume that the source voltage Q(t)is a ramp waveform given by
&(t)
=
{-
for
t
s F',
r:
where
t,
is the risetime of the pulse. Sketch the load voltage for line
356
TIME-DOMAIN ANALYSIS
lengths having one-way transit times T such that: 1. T, = &T
2. T, = 2T 3. T, = 3T 4. z, = 4T. This example shows that in order to avoid problems resulting from mismatch, one should choose line lengths short enough for T, to be >> T for the desired data. Confirm your results using SPICE.
FIGURE P5.8
5.8
Consider the three-conductor line shown in Fig. P5.8. The driven or generator conductor is a 120gauge solid wire (radius 16 mils) and the pickup or receptor wire is a 128 gauge solid wire (radius 6.3 mils). The wires are separated by l.Scm and are suspended above an infinite, perfectly conducting ground plane at heights of 2 cm and 1 cm as shown.
PROBLEMS
357
The lines are terminated as shown and driven by a ramp voltage source. Neglect all losses, assume a homogeneous medium and compute and plot and I$&), versus time the near-end and farcnd coupled voltages, I$,&) by the following methods: 1. SPICE model
2. Branin’s method 3. The time-domain to frequency-domain transformation method 4. The finite difference method. 5.9
Verify, by long division, the results given in (5.29).
5.10
Use the Bergeron diagram method to verify the results of Problem 5.2.
5.1 1 Diagonalize the following inductance matrix for a homogeneous medium via the method of Section 5.2.1.1: 5 1
L = C 1 3] )IH/m 5.12 Diagonalize the product CL using the method of Section 5.2.1.2 using L given in Problem 5.11 and
.=[
-5lo -5]pF/m 15
5.13 A three-conductor line in an inhomogeneous medium is characterized with C and L of Problem 5.12. The total length is 2 m and is terminated in 1 kn resistors and driven by a 50 Q 1 V pulse source having a 10 ns
rise time. Solve for the terminal crosstalk voltages using: 1. SPICE 2. Branin’s method 3. The time-domain to frequency-domain transformation method 4. A lumped-pi circuit 5. The FDTD method. 5.14
Verify the relations in (5.94).
5.15 Derive the Fourier series coefficients for a trapezoidal waveform given in (5,100). 5.16
Derive the FDTD results given in (5.120).
5.17 Verify the relations given in (5.151),
358
TIME-DOMAIN ANALY SI5
5.18 Derive the recursion relation given in (5.156). 5.19 Plot the time-domain step response for the lossy PCB shown in Fig. 5.42
for a zero source impedance and a matched line using the low-loss result in (5.169). 5.20 Determine the impulse response for Problem 5.19 using (5.179). 5.21 Examine how you would implement the time-domain result shown in Fig. 5.46 using the results of Table 5.2 and (5.189). 5.22 Determine the time-domain crosstalk for a three-conductor PCB having w = s = 8 mils and a board having a thickness of 10 mils and e, = 12. The line length is 15cm and the terminations are 1 kf2 resistors. The source is a 1 V trapezoidal pulse train shown in Fig. 5.29 having a 50 fl source resistance and 50 ps rise/fall times. Compare the lossy and lossless cases using: 1. The SPICE model 2. The time-domain to frequency-domain transformation 3. A lumped-pi structure 4. The FDTD model.
CHAPTER SIX
Literal (Symbolic) Solutions for Three-Conductor lines
The previous chapters have considered the solution of the MTL equations for a general (n + 1)-conductor line. In general, this solution process must be accomplished with digital computer programs; i.e., a numerical result is obtained. Although exact, this numerical process does not reveal the general behavior of the solution. In other words, the only information we obtain is the solution for the specific set of input data, e.g., line length, terminal impedance levels, source voltages, frequency, etc. In order to understand the general behavior of the solution, it would be helpful to have a literal solution for the induced crosstalk voltages in terms of the symbols for the line length, terminal impedances, per-unit-length capacitances and inductances, the source voltage, etc. From such a result we could observe how changes in some or all of these parameters would affect the solution. This advantage is similar to a transfer function which is useful in the design and analysis of electric circuits and automatic control systems CA.21. In order to obtain this same insight from the numerical solution we would need to perform a large set of computations with these parameters being varied over their range of anticipated values. Such transmission-lineliteral transfer functions for the prediction of crosstalk have been derived in the past for use in the frequency-domain analysis of microwave circuits [l-41 or for time-domain analysis of crosstalk in digital circuits [S-ll]. However, all of these methods makc one or more of the following assumptions about the line in order to simplify the derivation: 1. The line is a three-conductor h e , Le., n = 2, with two signal conductors and a reference conductor. 2. The line is symmetric, i.e., the two signal conductors are identical in cross-sectional shape and are separated from the reference conductor by identical distances. 359
360
LITERAL (SYMBOLIC3 SOLUTIONS FOR THREE-CONDUCTOR LINES
3. The line is weakly coupled, i.e., the effect of the induced signals in the receiving circuit on the driven circuit is neglected (widely separated lines tend to satisfy this in an approximate fashion the wider the separation), 4. Both lines are matched at both ends, Le., the line is terminated at all four ports in the line characteristic impedances. 5. The line is lossless, i.e., the conductors are perfect conductors and the surrounding medium is lossless. 6. The medium is homogeneous.
The obvious reason why these assumptions are used is to simplify the difficult manipulation of the symbols that are involved in the literal solution. The assumption of a symmetric line and the subsequent literal solution is referred to in the microwaves literature as the even-odd mode solution. However, numerous applications are not symmetric nor perfectly matched. The purpose of this chapter is to derive the literal or symbolic solution of the MTL equations for a three-conductor line and to incorporate the terminal impedance constraints into this solution to yield explicit equations for the crosstalk. Both the frequency-domain and the time-domain solutions will be obtained. In addition, the derivation will not presume a symmetric line nor will it presume a matched line. We presume that the entries in the per-unit-length parameter matrices, L and C, are known. The idea is to simply proceed through the usual solution steps that would be involved in a numerical solution but instead to use symbols for all quantities rather than numbers, It is important to remind the reader of the steps to be taken. First we solve for the general solution in terms of 2n (4 in this case) undetermined constants and then we incorporate the terminal constraints in order to evaluate these undetermined constants. For a three-conductor line the last step, incorporation of the terminal conditions, involves the simultaneous solution of symbolic equations with, for example, Cramer’s rule, and is quite difficult. It therefore does not appear to be feasible to extend this to lines consisting of more than three conductors. Even for a three-conductor line, the solution effort is so great that we must make some other simplifying assumptions. The primary simplification is to assume a homogeneous surrounding medium so that we may take advantage of the important identity for a homogeneous medium, LC = pel2. This identity essentially reduces the number of symbols and allows the consolidation of certain other groups of the per-unit-length symbols. To further aid in simplification we first assume a lossless line; perfect conductors in a lossless, homogeneous medium. We then extend this result in an approximate fashion to consider imperfect conductors. Besides providing considerable insight into the effect of each parameter on the solution, this literal solution also provides verification of some long-held intuitive notions. The first is that for an electrically short, weakly coupled line and a sufficiently small frequency, the total crosstalk can be written as the sum of two contributions. One contribution is due to the mutual inductance between
LITERAL (SYMBOLIC) SOLUTIONS FOR THREE-CONDUCTOR LINES
361
i
- eI
I
I , 1
FIGURE 6.1 The three-conductor MTL (a) line dimensions and terminal characterization and (b) the per-unit-length equivalent circuit.
the two circuits and the other is due to the mutual capacitance between the two circuits. This is the basis for the widely used, inductiue-cupucitiue coupling approximation. From the literal solution we cannot only verify this concept but can also determine the specific restrictions on its applicability. We will also obtain the literal solution for time-domain analysis. From this solution we can immediately see the appropriate restrictions on the applicability of various approximate techniques that are outlined in various handbooks. Although these approximate techniques are simpler than a full solution of the MTL equations, it is important to know the limitations on their applicability. The general statement of the problem is illustrated in Fig. 6.l(a). The line consists of three perfect conductors immersed in a lossless, homogeneous medium characterized by permittivity e and permeability p. The generator circuit is composed of a generator conductor with the reference conductor. It
362
LITERAL (SYMBOLIC3 SOLUTIONS FOR THREE-CONDUCTOR LINES
is driven at the left end with an open-circuit source voltage, &(t), and source resistance, R,, and is terminated at the right end in a load resistance, R,. The receptor circuit is composed of a receptor conductor and the reference conductor. It is terminated at the left or “near end” in a resistance, RNb,and at the right or “far end” in a resistance, RFB.Although resistive terminations are used in the following developments the frequency-domain phasor crosstalk results apply to complex-valued terminal impedances. The per-unit-length equivalent circuit is shown in Fig. 6.l(b). From this, the MTL equations can be derived in the usual manner and become
a
a
aZ
at
- V ( z , t ) = -L--I(z,t)
(6.la)
a I(z,t ) = - c a V(z, t ) -
(6.lb)
az
at
where (6.2a) (6.2b) (6.2~) (6.2d) Subscript G denotes quantities associated with the generator circuit, whereas subscript R denotes quantities associated with the receptor circuit. Because of the assumption of a homogeneous surrounding medium, we have the important identity 1
LC = p&l,= - 1, V2
(6.3)
where u = 1/fi is the velocity of propagation of the waves in the medium. This identity gives the following relations between the per-unit-length parameters: (6.4a) MCO + cm) = IR(CR + cm) (6.4b) lm(cO + cm) = lRCm Im(CR
+ cm) e 1Gcm
(6.4~)
The terminal conditions are written in the form of generalized ThCvenin
FREQUENCY-DOMAIN SOLUTION
363
equivalent characterizations as V(0, t ) = Vs(t)- RSI(0, t )
(6.5a) (6.5b)
where (6.6a)
1
(6.6b) (6.6~)
The objective is to obtain equations for the near-end and far-end crosstalk voltages; = V'(0, t ) and ?$&) = h(9,t). 6.1
FREQUENCY-DOMAIN SOLUTION
The MTL equations for sinusoidal steady-state excitation become d V(2) dz
= -jOL@)
d - it@) = -jOCV(Z) dz
(6.7a) (6.7b)
where the phasor line voltages and currents are (6.8a) (6.8b) The phasor generalized ThCvenin equivalent characterization of the terminations becomes (6.9a) q(0)= Vs - RJ(0) (6.9b)
where
os=[!]
(6.1 Oa)
364
LITERAL (SYMBOLIC) SOLUTlONS FOR THREE-CONDUCTOR LINES
(6.10b) (6.10~) The objective is to obtain equations for the phasor near-end and far-end crosstalk voltages; 3 : &(O) and &E = PR(9). The chain parameter matrix was derived for this case in Chapter 4 and becomes (6.1 I) The chain parameter submatrices simplify for this case of perfect conductors in a lossless, homogeneous medium to (6.12a) (6.12b) (6.12~) (6.12d) where
c = cos(pa)
(6.13a) (6.13b)
The velocity of propagation is (6.14a) and the phase constant is
p 3 -0
(6.14b)
V
The terminal characterization in (6.9) is substituted into the chain parameter matrix in (6.11) to give
FREQUENCY-DOMAIN SOLUTION
365
Substituting the chain parameter submatrices given in (6.12) into (6.15) gives
cos(fiY)(Rs + RJ
cos(fiS?)(R,
+sin(pa) (R, joCYR, +joL) w
1
f(0)
1 -
+ RL) + sin(pa) (R, jwCSRL + jwL @') fiz
(6.16a)
Vs
(6.16b)
Once these are solved, the near-end and far-end crosstalk voltages are obtained from the second entries in these solution vectors as FNE= FR(0)= - R N E f R ( O ) and = VR(U) = RFEfR(9). Equations (6.16) were solved in [B.SJ via Cramer's rule in literal form to yield the following exact literal solution for the crosstalk voltages:
cE
(6.17a) (6.17b)
+
Den = Cz+ (jo)ZSzzorRP joCS(t,
+ TR)
(6.17~)
The various quantities in these equations are [B.5] (6.18a) (6.18b) where the inductlve-coupling coefficients are (6.19a) (6.19b) and the capacitive-coupling coefficients are (6.20a) (6.20b)
366
LITERAL ISYMBOLIQ SOLUTIONS FOR THREE-CONDUCTOR LINES
The remaining quantities are defined in the following way. The coefficient KNE is defined by (6.21)
The coupling coefficient between the two circuits is defined by (6.22)
and the circuit characteristic impedances are defined by
,z,
= VIff
4-
(6.23a) (6.23b)
The line one-way delay is denoted by (6.24) The relationships of the termination impedances to the characteristic impedances are important parameters. In order to highlight this dependency, the various ratios of termination impedance to characteristic impedance are defined by
(6.25)
In terms of these ratios, the factor P in Den becomes
Observe that P 3: 1 if the line is weakly coupled, k cc 1, and/or the lines are matched at opposite ends, as, = aLR= 1, or ctSR = aL, = 1. The circuit time
FREQUENCY-DOMAIN SOLUTION
367
constants are logically defined as
(6.27a)
(6.27b)
Observe that a line time constant is equal to the line one-way delay if the lines are weakly coupled, k << 1, and that line is matched at one end. In other words, tl = T if k << 1 and asr = 1 or aLf= 1. 6.1.1
Inductive and Capacitive Coupling
The above results are an exact literal solution for the problem. No assumptions about symmetry or matched loads are used. Therefore they cover a wider class of problems than have been considered in the past. Although they have been simplified by defining certain terms, they can be simplified further if we make the following assumptions. First let us assume that the line is electrically short ut thefrequency of ititerest, i.e., 9 << A. In this case the terms C and S simplify to
c = cOs(p9)
(6.28a)
(6.28b)
Further let us assume that the line is weakly coupled, Le., k << 1 . Under these assumptions, there exists a sufficiently small frequency such that the exact results in (6.17) simplify to
368
LITERAL (SYMBOLIC3 SOLUTIONS FOR THREE-CONDUCTOR LINES
FIGURE 6.2
The frequency-domain inductive-capacitivelow-frequency coupling model.
These low-frequency results can be computed from the equivalent circuit of Fig. 6.2. The terms depending on the per-unit-length mutual inductance 1, are referred to as the inductive coupling contributions, whereas the terms depending on the per-unit-length mutual capacitance c, are referred to as the capacitive coupling contributions. Observe that the low-frequency,weakly coupled approximate results in (6.29) show that the crosstalk varies directly with frequency or 20 dB/decade and the total coupling can be written as the sum of inductive-coupling and capacitivecoupling components as
RNE
= RNE
+ RFE
1
l m 9RS
+ RL
(6.3 1a) (6.31b)
FREQUENCY-DOMAIN SOLUTION
369
f
(4 FIGURE 6.3 Illustration of the dominance of inductive (capacitive) coupling for (a) low- and (b) high-impedance terminations,
(6.31~) Depending on the levels of the load impedances, the inductive-coupling contribution may dominate the capacitive-coupling contribution or vice versa. This is illustrated in Fig. 6.3, If the termination impedances are much smaller than the line characteristic impedances, Le., low-impedance loads, then inductive coupling dominates. On the other hand, capacitive coupling dominates in the case of high-impedance loads, Although this approximation is valid only for weakly coupled lines and for a sufficiently small frequency where the line is electrically short, this separation of the total coupling into an inductive-coupling and a capacitive-coupling component provides considerable understanding of crosstalk phenomena. In particular, it readily explains how shields and/or twisted pairs of wires may or may not reduce crosstalk cA.31. 6.1.2
Common-Impedance Coupling
The above derivation assumes that all three conductors are peqect conductors and the surrounding homogeneous medium is lossless. Losses can be ignored
370
LITERAL (SYMBOLIC) SOLUTIONS FOR THREE-CONDUCTOR LINES
Reference conductor
(4
/
//
/
\ \Common impedance Electromagnetic coupling contribution coupling contribution assuming perfect conductors L
Frequency
I
(b) FIGURE 6.4
Illustration of common-impedance coupling due to a lossy reference
conductor. in many practical problems. However, there is a potentially significant contribution to crosstalk via imperfect conductors that occurs at the lower frequencies. This is referred to as common-impedance coupling and is contributed by the impedance of the reference conductor. Figure 6.4 illustrates the problem. As the frequency of excitation is lowered, the crosstalk decreases directly with frequency. At some lower frequency, this contribution due to the electric and magnetic field interaction between the two circuits is dominated by the common-impedance coupling component. At a sufficiently low frequency, the current in the generator circuit, returns predominantly in the reference conductor and can be computed from
rG,
1 fG
Rs + R L
(6.32)
If we lump the per-unit-length resistance of the reference conductor, r, as a total
TIME-DOMAIN SOLUTION
371
resistance, R = r Y , then a voltage drop of
is developed across the reference conductor. This is voltage-divided across the termination resistors of the receptor circuit to give
P$&= -
R~~
+ RFE
r g
RS
' P , + RL
(6.34a)
(6.34b)
h.4:;
In an approximate sense, we may simply combine these contributions with the inductive-capacitive coupling contributions in (6.30) to give the total as A
vNE
A
&E
z
Pi%D + Pi;' + Pi;
= jwMA%D& + jwM$f'&
":p:
+ P$iP+ @&
= jwM;iD
+Mii&
& + ] w M $ ~P, + M $ i
(6.35a) (6.35b)
This approximate inclusion of the impedance of the reference conductor at low frequencies was verified in CB.141 by deriving the exact chain parameter
matrix with the per-unit-length resistance of the reference conductor included, 6.2
TIME-DOMAIN SOLUTION
To obtain the exact time-domain solution we will assume &(t) = 0 for t S 0 and the line is initially relaxed: V(z, 1) = I(z, c) = 0 for all 0 s z s 49 and t b; 0 CB.151. In this case the Laplace transform variable s can be substituted for jo in the above frequency-domain exact solution. Substituting j w + s in (6.13) gives ):ntis j w s =j w
049
=+
e~T
- e-sT 2T
(6.36a)
372
LITERAL (SYMBOLIC) SOLUTIONS FOR THREE-CONDUCTOR LINES eaT + e-aT
c -- cost;)
r+
2
(6.36b)
where the line one-way time delay is 9 T=V
(6.37)
Substituting these along with j w + s into (6.17) gives the exact Laplacetransformed time-domain solution:
(6.38b)
(6.39b) (6.39~) where &(s) is the Laplace transform of &(t). We now take the inverse Laplace transform of these results. Rewrite (6.38a) and (6.38b) as (1
+
+ be-4;T)VN,(s)
(6.40a)
m
(1
T + ue-2'T + be-4JT)VF&)= 2 M,&"T - e - 3 a T ) ~ ( ~(6.40b) ) X
In taking the inverse Laplace transform of this result we recall the simple time-delay transform pair: F(t f n T ) o efrrTF(s)
(6.41)
373
TIME-DOMAIN SOLUTION
Therefore the exact time-domain expressions become
&E(t)
= -aVFE(t
- 2 T ) - b?$E(t - 4T) + 2-X MNE[&(t - T ) - &(c - 3T)] T
(6.42b) These equations can be solved recursively for the crosstalk voltages realizing that &(t) = VNE(t)= Vfs(t) = 0 for t s 0. The solutions are in terms of values of the source voltage at the present time, &(t); at various time delays prior to the present time, &(t - T), &(t - 2T), &(t - 379, and &(t - 4T); as well as prior solutions at various time delays prior to the present time, V,&(t 2T), &E(t - 4T), GE(t - 2T), and &E(t - 4T).
-
6.2.1
Explicit Solution
Although the solution in (6.42) is exact, it requires knowledge of the solution at previous times that are multiples of the line one-way delay, T. Thus a recursive solution is required. In order to obtain a solution that depends only on &(t) we introduce the time-shiji or dlgerence operator, D, where DmF(t)3 F(t + rnT) D-’”F(t) F(t - m T )
(6.43a) (6.43b)
Using the time-shift operator in (6.40) gives (1
+ aD-2 + b D ’ 4 ) V ~ ~ ( t ) =
(6.44a)
+ &E) - 2KNED-’ +
T [(MNB X
(&E
M N E ) D - ~&(t) ]
m
Multiplying the equations by powers of D gives the transfer functions in terms of the time-shift operator, D, as
374
LilERAL (SYMBOLIC) SOLUTIONS FOR THREE-CONDUCTOR LINES
(6.45b) These expressions can be inverted by carrying out the long division of the following basic problem: 1 aoD'4 alD'6 + a 2 D - 8 a,D"O + a,D'12 (6.46a) D4+aDZ+b +a,D"* + a6D"6 + * * * where a. = 1 a1 = - a a2 = a2 b a3 = -a3 + 2ab (6.46b) u4 = a4 3a2b + b2 a, -- -a5 + 4a3b - 3ab2 a6 = a6 5a4b + 6a2b2 b3
-
+
+
-
-
Substituting the results in (6.46) into (6.45) gives the time-domain crosstalk voltages in terms of V,(t) delayed by multiples of the one-way line delay, T,as
+ N2 &(t - 27') i-N4 Y,(t - 47') + NS &(t - 6 T ) + F1 G(t - T ) + F3 &(t - 3 T ) + F5 Y,(t - 5T) +F7 V,(t - 77') +
V,,(t) = No $(t)
(6.47a)
* * *
&j(t)
* * *
where the constants are T
(6.47b)
TIME-DOMAIN SOLUTION
375
and m
m
m
(6.48b)
These final expressions give the explicit relationships for the crosstalk voltages as linear combinations of the source voltage delayed in time by various multiples of the line one-way delay, T. 6.2.2
Weakly Coupled Lines
The above time-domain solutions are exact but somewhat complicated. Quite often the lines can be considered to be weakly coupled, k << 1. This assumption simplifies the above exact results. For a weakly coupled line, the factor P in (6.26) approximates to unity, P r 1, and the characteristic impedances approximate to ZcG2 V l G , ZcR 3 V I R . The quantities in (6.39) approximate to
(6.49b) (6.49~) Substituting the time constants in terms of the ratios of the termination resistances to the line characteristic impedances given in (6.27) gives
376
LITERAL (SYMBOLIC) SOLUTIONS FOR THREE-CONDUCTOR LINES
where the reflection coefficients are logically defined as (6.51a) (6.51b) (6.51~) (6.51d) If we define the quantities
KG
~ S G ~ L G
(6.52a)
KR
~ S R ~ L R
(6.52b)
then the quantities in (6.46b)become
(6.53)
Suppose that the line is weakly coupled and the generator circuit is matched at one end, r,, 3 0 or r,, = 0. In this case KG 3:0 so that uo = 1, ul = K R = r S R r L R , a2 = Ki = C R E R , a3 = Ki = . ,R& cI Alternatively, suppose that the line is weakly coupled and the receptor circuit is matched at one end, rsR = 0 or r L R = 0. In this case, K R = 0 so that a, = 1, a1 = KG = rsGrLG, a2 = Kg = l&GO, a3 = Ki = l'&rjLG,. . For either of these cases the results in (6.47) simplify somewhat. Now suppose that the line is weakly coupled and that one of the ends of each line is matched; rsG = 0 or rLG = 0 and rsR e: 0 or r L R = 0. In this case, To = T and T R = T so that X = 4 P . The quantities K G and K R in (6.52) are zero so that a1 = a j = = 0. The near-end and far-end crosstalk expressions
.. .
..
0
e
-
TIME-DOMAIN SOLUTION
377
(6.54a)
+[&(t) 4T KNE
[I$(t
bE(t) =
- 2&(t - 2 T ) + &(t
- 7') - &(t
- 4T)]
- 3T)J
(6.54b)
If the line happens to be matched at the load end of the generator circuit, = 0, then K N E = MNE and (6.54) simplify to
r,,
VNE(c)= !?!!E[&(t) 2T
VF&) = 5 [Vs(r 2T
- G(t - 2T)]
- T ) - b(t - 3T)]
(6.55a)
(6.55b)
There exist numerous electronic design handbooks and other publications that contain time-domain crosstalk prediction equations for three-conductor lines [S-111. However, as pointed out previously, these invariably make the following assumptions: 1. The line is weakly coupled. 2. All ports are matched: r,, = r,, = r,, = r,, = 0. 3. Both circuits have identical cross sections, e.g., two identical wires at the
same height about a ground plane. These are very special restrictions that are generally not fulfilled in practical cases. Nevertheless, for a line that satisfies the above ideal conditions, the previously derived exact crosstalk results reduce to (6.56a)
v,&)
=
-(-lrn - C m z , ) c v d ' - T ) - &(t - 3T)] 8T Zc
(636b)
-0
These results are equivalent to results derived intuitively in CS-7J. Observe that the far-end crosstalk is zero for this completely matched line in a homogeneous medium.
378
LITERAL (SYMBOLIQ SOLUTIONS FOR THREE-CONDUCTOR LINES
6.2.3
Inductive and Capacitive Coupling
In the frequency-domain solution we observed that for a weakly coupled, electrically short line and for a sufficiently small frequency, the near-end and far-end crosstalk voltages reduce to &(fa)
(6.57a)
GE(jw) 2j a M p s &W
(6.57b)
&E(.b)
.faMNE
The time-domain results can be obtained from these by substituting d ja*dt
(6.58)
to give
(6.59b) For trapezoidal pulses representing, perhaps, digital clock or data signals, these results give crosstalk pulses occurring during the transitions of %(t) with the levels of those pulses dependent on the slope or “slew rate”of V,(t)as illustrated in Fig. 6.5. Substituting the definitions of MNEand MpEin terms of inductiveand capacitive-coupling contributions as given in (6.18) and (6.19) shows that this approximated time-domain crosstalk result can be computed from the equivalent circuit shown in Fig. 6.6. Therefore, the time-domain crosstalk is the sum of an inductive-coupling component and a capacitive-coupling component. This result can also be seen from the approximate result derived for a completely matched, weakly coupled line given in (6.55): (6.60a) (6.60b) If the one-way delay is sufficiently small compared to the rise/fall times of the source voltage, Le., T,, 7f >> T (6.61) the derivatives of the source voltage are reasonably approximated by (6.62a)
TIME-DOMAIN SOLUTION
379
FIGURE 6.5 Illustration of near-end and far-end time-domain crosstalk predicted by the inductive-capacitivecoupling model.
FIGURE 6.6
The time-domain inductive-capacitive coupling model.
380
LITERAL (SYMBOLIC) SOLUTIONS FOR THREE-CONDUCTOR LINES
FIGURE 6.7
The frequency-domain representation of a periodic, trapezoidal pulse train.
(6.62b) and the results in (6.60) are identical to those in (6.59). The question of how much greater than the line one-way delay must be than the rise/fall times of the pulse in order to satisfy (6.61) can be answered in the following way. Consider V,(t) as being a periodic train of trapezoidal pulses with period T (repetition frequency f = 1/T) as illustrated in Fig. 6.7(a). The pulses have amplitude V,, rise/fall times of rr, .tf and pulse width T between the 50% points. This is a common approximation to digital signals such as clock or data CA.31. Suppose that the tisetime andfalltime are identical, i.e., 7, = Tf. In this case, the signal can be represented as a Fourier series as CA.3) (6.63)
TIME-DOMAIN SOLUTION
381
FIGURE 6.8 The frequency-domain transfer function predicted by the inductive-capacitive coupling model.
where t
c,=2%-T
sin(nm/T) sin(nnt,/T) nnt/T nq/T
4,, = -nn(+)
(6.64a) (6.64b)
It is possible to obtain bounds on these magnitudes as shown in Fig. 6.7(b) CA.31. These bounds consist of three line segments: one with slope 0 dB/decade out to a frequency of l/m, one with slope -20dB/decade from this point out to a frequency of 1/nt,, and'one with slope -4OdB/decade thereafter. The high-frequency spectral content of the pulse is therefore primarily governed by the pulse rise/falI time. Consider the basic assumption of the low-frequency model as shown in Fig. 6.8: the frequency-domain crosstalk increases linearly with frequency (20 dB/decade) up to some frequency,&, where the line becomes electrically long. Combining the pulse spectrum in Fig. 6.7 and the crosstalk transfer function in Fig. 6.8 gives the resulting spectrum of the crosstalk pulse shown in Fig. 6.9. Observe in Fig. 6.9 that the high-frequency spectrum of the crosstalk pulse rolls off at -2OdB/deade. Let us choose the upper limit of validity for the low-frequency model, f , , sufficiently greater than the beginning of this slope, l/m,, in order to assure that the higher frequency components
382
-----
LITERAL (SYMBOLIC) SOLUTIONS FOR THREE-CONDUCTOR LINES
Input pulse spectrum
T
I1
I
L IIT
1
=
I I
r I I
@
'
f. I
I
I I I
'
I I
I
I
I
f
!
FIGURE 6.9 Combination of Figs. 6.7 and 6.8 to yield the frequency spectrum of the crosstalk voltages.
above f,, which are not properly processed by the low-frequency model, will not be significant: (6.65)
Substituting f , in terms of some fraction of line length, say, one-tenth: (6.66)
gives 5' ,
.
10 9 >> -- = 3.18T Itv
(6.67)
Therefore the pulse rise/fall times must be larger than the line one-way delay. Typically choosing 7, > 10T provides sufficient accuracy. 6.2.4
Common-Impedance Coupling
The effect of the impedance of the reference conductor can be handled in an approximate manner at the lower frequencies of the pulse by simply adding the
COMPUTED RESULTS
383
common-impedancecoupling term to the inductive-capacitivecontributions in (6.59): (6.68)
where the M's are as given previously. The effect of common-impedance coupling is to add a scaled replica of b(t) to the crosstalk resulting from inductive and capacitive coupling. 6.3 COMPUTED RESULTS
In this section we will give some computed results comparing the predictions of the SPICE MTL (exact) model, the lumped-pi model, and the inductivecapacitive coupling model developed in this chapter for three-conductor lines that were examined in the previous two chapters. In all models the conductors are considered to be lossless. For both structures, the SPICE subcircuit models were computed with the SPICEMTLFOR, SPICELPLFOR and SPICELC,FOR computer programs described in Appendix A and combined into one SPICE program for ease of plotting the results. The nodes at the input to the generator lines are designated as S1, S2, and S3, whereas the nodes at the near end of the receptor circuit are designated as NE1 (SPICE), NE2 (lumped-pi),and NE3 (inductive=capacitivecoupling model). The termination impedances are all 50 f l resistive, i.e., R, = RL = R,, = R F E 3: 50 Q. Predictions of the exact time-domain solution of (6.42) are compared to those of the explicit solution of (6.47) using 7 terms in [B.15]. This reference also gives comparisons with the finite d!,fference-time domain model and the time-domain to frequency-domain model. The exact frequency-domain result of (6.17) has been verified on numerous occasions and will therefore not be used here. 6.3.1
A Three-Wire Ribbon Cable
The first configuration is a three-wire ribbon cable considered previously. The wires are #28 gauge stranded (7 x 36) and are separated by 50 mils. One of the outer wires is the reference conductor and the line is of total length 2 m as shown in Figs 4.13 and 4.14. The per-unit-length parameters were computed in Chapter 3 with the RIBBON.FOR computer program described in Appendix A. Figure 6.10 compares the frequency-domain predictions for all three models over the frequency range of 1 kHz to 100MHz. The line is one wavelength (ignoring the dielectric insulations) at 150 MHz so we may consider it to be electrically short for frequencies below, say, 15 MHz. The lumped-pi model gives good predictions below 10 MHz, whereas the inductive-capacitivecoupling
I
-2a
,/-q /
.......................................................................................
.
*
I
.
.
*
. * ,
*
.................................................... ."
c
v
d
a. .
. .. . . . .
-40
0
0
z
-60 -RO
I
*
............
.,. .......................... I 04
............
IO'
. . . .
..
. . .
....*............... I@
. . . . ...............
16'
I
FIGURE 6.10 Illustration of the frequency response of the ribbon cable of Fig. 4.14 via
-
the SPICE model, one lumped-pi section, and the inductive-capacitive coupling model. (a) Magnitude: 0 VDB(NE1) VDB(Sl), VDB(NE2) VDB(S2), 0 VDB(NE3) VDB(S3). (b) Phase: 0 VP(NE1) VP(Sl), H VP(NE2) VP(S2), 0 VP(NE3) VP(S3).
-
-
-
-
COMPUTED RESULTS
(8)
385
Time (ar)
FIGURE 6.11 Illustration of the time-domain response of the ribbon cable of Fig. 4.14 via the SPICE model, one lumped-pi section, and the inductive-capacitivecoupling model for a rise/fitll time of (a), 60 ns.
model gives good predictions below 1 MNz. This shows that the simple inductive-capacitive coupling model can give adequate predictions for a significant frequency range so long as its basic limitations are observed. Figure 6.11 shows the correlation between the three models for the time domain. The source, &(t), is a 1 MHz trapezoidal pulse train with 50% duty cycle. The trapezoidal pulses have 1 V magnitude with various rise times. The one-way time delay for the line (ignoring the dielectric insulations) is T = U/u = 6.67ns so we should not expect the inductive-capacitive coupling model to give adequate predictions for risetimes less than, say, 60 ne. Figure 6.11(a) shows the predictions for t, = 60 ns. The predictions of the lumped-pi model compare well with those of the SPICE model. Figure 6,11(b) shows the predictions for t, = 120 ns, and Fig. 6.1l(c) shows the predictions for T, = 240 ns. Again, the predictions of the exact SPICE model and the lumped-pi model are excellent. For this latter risetime of 240ns, there is good agreement between the inductive-capacitivecoupling model's prediction of the peak crosstalk and t, = 36T. Figure 6.12 shows oscilloscope photographs of the experimental results. For
(b)
T h o (nr)
FIGURE 6.11 (Continued) (b) 120 ns, and (c) 240 ns. 0 V(NEl),
V(NE2), 0 V(NE3).
COMPUTED RESULTS
387
FIGURE 6.12 The experimentally determined time-domain response of the ribbon cable
of Fig. 4.14 for a rise/fall time of (a) 60 ns, (b) 120 ns.
388
LITERAL (SYMBOLIC) SOLUTIONS FOR THREE-CONDUCTOR LINES
FIGURE 6.12 (Continued) (c) 240 ns.
z, = 60 ns in Fig. 6.12(a) the measured peak {oltage is 80 mV compared with a predicted value of 80 mV. For z, = 120 ns in Fig. 6.12(b) the measured peak voltage is 50 mV compared with a predicted value of 45 mV, and for 5, = 240 ns in Fig. 6.12(c) the measured peak is 26mV compared with a predicted value of 23 mV. Observe in the measured results that, while the pulse is in its quiescent state of 1 V, the crosstalk does not go to zero as predicted by the lossless model but appears to asymptotically approach a value of the order of 2 5 mV. This is a result of common-impedance coupling. The per-unit-length dc resistance of one strand is obtained by dividing the dc resistance of one of the #36 gauge strands by 7 (the number of strands in parallel to give r 9 = 0.19444 Q/m. Substituting this into (6.34a) gives a common-impedance coupling level of 1.94 mV. 6.3.2
A ThncConductor Printed Circuit Board
The next configuration is a threeconductor printed circuit board also considered previously and shown in Fig. 4.17. The conductors (lands) are 15 mils in width and have thicknesses of 1.38 mils (1 ounce copper). They are on one side of a 47 mil thick glass epoxy board and have edge-to-edge separations of 45 mils. One of the outer lands is the reference conductor and the line is of total length 10 inches = 0.254 m. The per-unit-length parameters were computed in Chapter 3 with the PCBGAL.FOR computer program described in Appendix A. Figure 6.13 compares the frequency-domain predictions for all three models over the
COMPUTED RESULTS
389
(
-9 d S
-ZG
I
:-4a a
-60
-SO
-T8 Y00
4
. ...
IO‘
._..._...____.. _..,. . * ..................... I 0’
.......*.....‘...................... IO6
.
............................*...................._._....... I
*
10’
Id
I@
Frequency (Hz)
(b)
FIGURE 6.13 Illustration of the frequency response of the printed circuit board of Fig, 4.17 via the SPICE model, one lumped-pi section, and the inductive-capacitive coupling model. (a) Magnitude: 0 VDB(NE1) VDB(Sl), I VDB(NE2) VDB(SZ), 0 VDB(NE3) VDB(S3), (b) Phase: 0 VP(NE1) VP(Sl), D VP(NE2) - VP(S2), 0 VP(NE3) - VP(S3).
-
-
-
-
390
LITERAL (SYMBOLIQ SOLUTIONS FOR THREE-CONDUCTOR LINES
FIGURE 6.14 Illustration of the time-domain response of the printed circuit board of
Fig. 4.17 via the SPICE model, one lumped-pi section, and the inductive-capacitive coupling model for a rise/fall time of (a) 6.25 ns, (b) 20 ns.
COMPUTED RESULTS
391
FIGURE 6.14 (Confinued) (c) 60 ns.
frequency range of 10 kHz to 1 GHz. The line is one wavelength (ignoring the dielectric insulations) at 1.18 GHz so we may consider it to be electrically short for frequencies below, say, 100 MHz. The lumped-pi model gives good predictions below 100 MHz, whereas the inductive-capacitive coupling model gives good predictions below 10 MHz. This again shows that the simple inductivecapacitive coupling model can give adequate predictions for a significant frequency range so long as its basic limitations are observed. Figure 6.14 shows the correlation between the three models for the time domain. The source, V,(t),is again a 1 V, 1 MHz trapezoidal pulse train with various risetimes. The one-way time delay for the line is (again ignoring the board dielectric) T = .V/u = 0.85 ns so we should not expect the inductivecapacitive coupling model to give adequate predictions for risetimes less than, say, 10 ns. Figure 6.14(a) shows the predictions for r, = 6.25 ns. The predictions of the lumped-pi model compare well with those of the SPICE model. Figure 6.14(b) shows the predictions for tr = 20 ns, and Fig. 6.14(c) shows the predictions for r, = 60 ns, Again the predictions of the SPICE model and the lumped-pi model are virtually identical. For this latter risetime of 60ns, the inductive-capacitive coupling model gives excellent prediction of the peak crosstalk magnitude and T, = 70T.
392
LITERAL (SYMBOLIC) SOLUTIONS FOR THREE-CONDUCTOR LINES
FIGURE 6.15 The experimentallydeterminedtime-domain response of the printed circuit board of Fig. 4.17 for a rise/fall time of (a) 6.25 ns, (b) 20 ns.
REFERENCES
393
FIGURE 6.15 (Continued) (c) 60 ns.
Figure 6.15 shows oscilloscope photographs of the experimental results. For t, = 6.25 ns in Fig. 6.1J(a) the measured peak voltage is 94 mV compared with a predicted value of 95 mV. For 7, = 20 ns in Fig. 6.15(b) the measured peak voltage is 46 mV compared with a predicted value of 46 mV, and for r, = 60 ns in Fig. 6.15(c) the measured peak is 17.5 mV compared with a predicted value of 15.8 mV. Observe in Fig. 6.15(c) that, again while the pulse is in its quiescent state of 1 V, the crosstalk does not go to zero as predicted by the lossless model but appears to asymptotically approach a value of the order of 2.5 mV. This shows that common-impedance coupling can be significant even for short conductors. The dc resistance of one land is rY = 0.3279 n/m. Substituting this into (6.34a) gives a common-impedance coupling level of 1.64 mV.
REFERENCES
[l]
[Z]
L. Young, Parallel Coupled Lines and Directional Couplers, Artech House, Dedham, MA., 1972. V.K. Tripathi, “Asymmetric Coupled Transmission Lines in an Inhomogeneous Medium,” IEEE Trans. on Microwave Theory and Techniques, MTT-23,731-739 (1975).
[3]
R. Speciale, “Even- and Odd-Mode Waves for Nonsymmetrical Coupled Lines in Nonhomogeneous Media,” IEEE Trans.on Microwave Theory and Techniques, MTT-23, 897-908 (1975).
394
LITERAL (SYMBOLIC) SOLUTIONS FOR THREE-CONDUCTOR LINES
c41 J.C. Isaacs, Jr., and N.A. Strakhov, “Crosstalk in Uniformly Coupled Lossy
c51 E61
c71 E81 c91
[lo1 c111
Transmission Lines,” Bell System Technical Journal, 52, 101-1 15 (1973). W.R. Blood, Jr., MECL System Design Handbook, Motorola Semiconductor Products, 4th ed., 1988. A. Feller, H.R. Kaupp, and J.J. Digiacoma, “Crosstalk and Reflections in High-speed Digital Systems,” Proc. Fall Joint Computer Coflerence, 1965, pp. 511-525. J.A. DeFalco, “PredictingCrosstalk in Digital Systems,” Computer Design, 69-75, (1973). D.B. Jarvis, “The Effects of Interconnections on High-speed Logic Circuits,” IEEE nuns. on Electronfc Computers,” EC-12,476-487 (1963). I. Catt, “Crosstalk (Noise) in Digital Systems,” IEEE l).ansactlons on Electronic Computers, EC-16,743-763 (1967). A.J. Rainal, ”Transmission Properties of Various Styles of Printed Wiring Boards,” Bell System Technical Journal, 58,995-1025 (1979). H.You and M.Soma, “Crosstalk Analysis of InterconnectionLines and Packages on Circufts and Systems, 37, in High-Sped Integrated Circuits,” IEEE 1019-1026 (1990).
PROBLEMS 6.1 Consider a problem consisting of two wires above an infinite, perfectly conducting plane. The wires are #28 gauge stranded wires (7 x 36) both at heights of 1 cm above the ground plane and separated by 100 mils (2.54 mm). Ignore the dielectric insulations. The terminations are Rs = 50 Q, R, = 1 kR, RNE= 300 n, RFE= 100 R. If the total line length is 2 m, compute the near-end and far-end frequency-domain crosstalk (magnitude and angle) using (6.17) from 1 kHz to 100 MHz. Determine the coupling coefficient, the near-end and far-end crosstalk coefficients, M&%!’’FE and M$;A,P,,, and the frequency where the line is 0.11 long.
6.2 For the structure of Problem 6.1, repeat the crosstalk calculations using the low-frequency, inductive-capacitive coupling model in (6.29).
6.3 For the structure of Problem 6.1, assume that G(t) is a 1 MHz, periodic train of trapezoidal pulses with equal rise/fall times of 100 ns and a level of 1 V. Compute the time-domain near-end and far-end crosstalk using the iterative solution given in (6.42). Repeat these calculations for rise/fall times of 50 ns and 10 ns. 6.4
Repeat Problem 6.3 using the explicit solution given in (6.47).
6.5
Repeat Problem 6.3 using the low-frequency, inductive-capacitivecoupling approximation given in (6.59).
CHAPTER SEVEN
Incident-Field Excitation of the Line
The previous chapters have been devoted to the analysis of MTL’s that are excited or driven by lumped sources in the termination networks. In this chapter we will examine the response of a MTL wherein the sources are electromagnetic fields incident on the line. These incident fields may be in the form of uniform plane waves such as are generated by distant transmitting antennas or they may be nonuniform fields such as are generated by nearby radiating structures. We will find that we may incorporate the effects of these incident fields by including distributed sources along the line. Once the MTL equations are derived from the resulting per-unit-length equivalent circuit, we concentrate on their solution. 7.1 DERIVATION OF THE MTL EQUATIONS FOR INCIDENT-FIELD EXCITATION
Incorporation of the effects of incident electromagnetic fields into the transmission-line equations was considered for two-conductor lines in C1-51. This was later extended to multiconductor lines in [6-81 and CH.3-H.51. A digital computer program has been developed for the multiconductor case [HA, H.61. The transmission-line formulation has been compared to the full-wave MOM solution in CH.2, H.73 and to experimental results in CH.81. A literal solution for the two-wire case was obtained in CH.91. The formulation has also been adapted to twisted pairs [9] and to shielded cables [IO]. The derivation of the MTL equations follows essentially the same pattern as in Chapter 2. The MTL equations which we will obtain will be solved and the termination networks incorporated into that general solution in the usual fashion in order to determine the line currents at the ends of the line. The currents that are modeled with these MTL equations are, once again, diflereential-mode or transmission-line currents in that the sum of the currents directed in the z direction on all (n + 1) conductors is zero at every line cross section. In other 395
3%
INCIDENT-FIELD EXCITATION OF THE LINE
words, the differential-mode currents of n of the conductors “return” on the reference conductor. In addition to these differential-mode currents, there can exist certain common-mode or antenna-mode currents that are not modeled by the MTL equations as discussed in Chapter 1 [3-5, 11-13]. So at points along the line there will be a combination of both currents only one component of which (the differential-mode) will be modeled by the MTL equations that we will develop. If the MTL cross section is electrically small at the frequency of interest, the total current at the terminations is that predicted by the M T L equations, This is because for a‘ line with electrically small cross-sectional dimensions, we can surround each termination with a closed surface (approximately a “node” in lumped-circuit analysis terminology), and Kirchhoff‘s current law shows that the total current entering this closed surface must be zero. Therefore at the terminations of a line having electrically small crosssectional dimensions, the total current at the terminations must sum to zero (differential-mode or transmission-line current) so that the common-mode current not predicted by the M T L equations must go to zero at the terminations and is therefore of no importance in predicting the terminal responses of a MTL [13]. This also applies to crosstalk on MTL’s that have electrically small cross-sectional dimensions. Common-mode currents are important in modeling radiated emissions or for determining currents at some intermediate point along the line such as a folded dipole antenna C13). Therefore the MTL equations we will obtain provide the complete prediction of the terminal response of a MTL which satisfies the necessary requirement of electrically small crosssectional dimensions. Consider an (n 1)-conductor, uniform MTL where the conductors are parallel to the z axis as shown in Fig. 7.1. In order to derive the first MTL equation we again integrate Faraday’s law around the contour C,between the reference conductor and the i-th conductor enclosing surface S,in the clockwise direction as shown in Fig. 7.1:
+
or
where 4 is the transverse electric field in the x-y cross-sectional plane and is the longitudinal or z-directed electric field along the surfaces of the conductors. Observe that again the negative sign is absent from the right-hand side of Faraday’s law because of the choice of the direction of the contour and the normal to the enclosed surface. At this point it is necessary to distinguish between incident and scattered field quantities. The incident field is that produced by the distant or nearby source in the absence of the line conductors. The scattered fields are produced by the currents and charges that are induced
DERIVATION OF THE MTL EQUATIONS FOR INCIDENT-FIELD EXCITATION
(
.
397
@ (]
;+k(24;
Reference conductor
.
AZ
*
J Y
FIGURE 7.1 Definition of the contour for the derivation of the first MTL equation for incident-field illumination.
on the line conductors [SI. The total field is the sum of a scattered and an incident component as
where superscript i denotes incident and superscript s denotes scattered. There are some important assumptions that need to be stated. First we assume that the currents on the conductors are z directed. Therefore the scattered magne$ fields will lie entirely in the transverse plane. Since there is no z-directed 43' field, Faraday's law shows that we may uniquely define the scarrered voltage
398
INCIDENT-FIELD EXCITATION OF THE UNE
between the i-th conductor and the reference conductor independent of path in the transverse plane as (7.4a) Ja
(7.4b) Second, because of the transverse nature of the scattered magnetic field, the scattered magnetic field can be related to the currents that produced it with the usual per-unit-length inductances as
Let us represent the imperfect conductors with per-unit-length resistances, rl. Although these will logically be functions of frequency due to the skin effect, let us for the moment assume they are constant with the assurance that the frequency-domain result will handle this dependence. The total longitudinal fields are related to the currents on the conductors as
[4 * d f =[dz
dz = r,Azll(z, t )
(7.6a)
Substituting (7.3)) (7.4)) (7.5)) and (7.6)into (7.2)) dividing by Az and taking the limit as Az -t 0 yields
a Vf(z,t ) + [ro aZ
*
rl I- ro
(7.7)
DERIVATION OF THE MTL EQUATIONS FOR INCIDENT-FIELD EXCITATION
399
Repeating this for the other conductors and arranging in matrix form gives
r
1
This can be written in an alternative form by recognizing that the transverse and longitudinal electric fields and the normal magnetic fields are related by Faraday's law. This can be shown by writing Faraday's law around the contour in terms of the incident fields, given in (7.2), dividing both sides of the result by Az, and taking the limit as Az -P 0 to give
$ldf.dl+
f
df.d,Sv = b:(i-th conductor, z, t )
(7.9)
- 8; (reference conductor, z, t ) where bl(m-th conductor, z) is the longitudinal or z-directed incident electric field along the position of the m-th conductor with it remoued. Substituting (7.9) into the right-hand side of (7.8) gives the first MTL equation:
a az
-V ( Z t ),
a I(z, + RI(z, t ) + L at
r
t)
(7.10)
1
I:(i-th conductor, z,t ) - &(reference conductor, z, t )
Observe that the voltages in this expression are the scattered voltages and not the total voltages.
400
INCIDENT-FIELD EXCITATION OF THE LINE
z
+ As
c Z
FIGURE 7.2 Definition of the surface for the derivation of the second MTL equation for incident-field illumination.
The second MTL equation can be derived as in Chapter 2 by enclosing the i-th conductor with a closed surface as illustrated in Fig. 7.2 and applying the continuity equation:
a
g*dS= - - Q,,, at
(7.11)
Over the end caps we have (7.12)
Although there are some subtleties involved [6,8] we again define the per-unit-length conductance and capacitance matrices in terms of the scattered voltages as
DERIVATION OF THE MTL EQUATIONS FOR INCIDENT-FIELD EXCITATION
-gfl
* *
f
k=I
g,k
*
401
1
(7.13) where I,,(z, t ) is the transverse conduction current between the i-th conductor and ail other conductors, and
lim AS+O
-Az
k- 1
Dividing both sides of (7.11) by Az, taking the limit as Az + 0,and substituting (7.12),(7.13),and (7.14)yields the second MTL equation in matrix form as
a
- I(2, t ) az
a vyz, t ) = 0 + cvyz, t ) + c at
(7.15)
The above results are in terms of the scattered uoltages. They can be placed in terms of the total voltages by writing
&(z, t ) = Vf(z,t ) + V;(Z,t ) = V:(Z,t ) -
$:.dl
(7.16)
Substituting this into (7.8) and (7.15) gives the MTL equations in terms of the
a
- I(z, t) aZ
a V(2, t ) = -G + GV(2, t ) + c at (7.17b)
402
INCIDENT-FIELD EXCITATION OF
THE LINE
7.1.1 Equivalence of Source Representations
Evidently (7.17) shows that incident electromagnetic fields modify the MTL equations by adding sources to the usual homogeneous MTL equations. The MTL equations in terms of total voltages given in (7.17) can be written as
a V(Z, t ) + RI(z, t ) + L a I(Z, t ) = VF(Z,t ) 82
(7.18a)
at
a I ( Z , t ) + Gv(Z,t ) + c a v(Z,t ) = I F ( Z , t ) az
(7.18b)
at
where
r
i
i (7.19a)
Thus the sources are the component of the incident magnetic field normal to the i-th circuit as shown in (7.19a) and the component of the incident electric field transoerse to the i-th circuit as shown in (7.19b). Recall that the i-th circuit is the surface bounded by the i-th conductor and the reference conductor between z and z + Az. These MTL equations can be derived from the per-unit-length equivalent circuit shown in Fig. 7.3. This is the same as the previous circuits except that distributed sources are added to incorporate the incident-field source functions given in (7.19). Observe that the right-hand side of (7.18a) depends on the incident magnetic field, whereas the right-hand side of (7.18b) depends on the incident electric field. These distributed “sources” can both be written in terms of only the incident electricjield by using Faraday’s law in (7.9) to replace the incident magnetic field in V,(Z, t ) to give
vF(z, t ) =
I -
L
8f.d+ (A(i-th conductor, z , t )
- &’;(reference
conductor, z, t ) }
1
1
(7.19~)
DERIVATION OF THE MTL EQUATIONS FOR INCIDENT-FIELD EXCITATION /&
403
+ Ar, I )
mo
FIGURE 7.3 The per-unit-length equivalent circuit for a MTL with incident-field illumination.
On the other hand, in terms of the scattered voltages, the MTL equations become
a V'(Z, t ) + RI(z, t ) + L a I(z, t ) = V;(Z, aZ
a
82 J(2, t ) + GV'(2, t ) where
at
t)
(7.20a)
+ c Bt-a vyz, t ) = 0
(7.20b)
I
Bi(i-th conductor, 2, t) - bf(reference conductor, z, t )
(7.21)
This latter form in terms of scattered voltages shows that the source term for the MTL equations is solely the difference in the incident electric field that is in the longitudinal direction and along the positions of the conductors (with them removed). It was shown above that these two forms are completely equivalent (although one of them uses the total voltages and the other uses the
404
INCIDENT-FIELD EXCITATION OF THE UNE
scattered voltages). That these are equivalent is intuitive since Faraday's law essentially relates the net circulation of the electric field around a contour to the magnetic flux penetrating that contour. There are some computational advantages to using (7.20) instead of (7.18) and converting the scattered voltages back to total voltages via (7.16): &(z, t ) = Vi(&t )
- :J t f* d l
(7.16)
However, there is a significant difference between the two forms when incorporating the terminal conditions. For example, suppose the terminal constraints are resistive with no lumped sources and are in the form of generalized ThCvenin equivalents (incorporation of the terminal constraints will be addressed in more detail in Section 7.2.2): V(0, t ) -RSI(0, t ) (7.22a) V ( 9 ,t ) = RLI(9, t )
(7.22b)
These can be used directly with the form of the MTL equations in (7.18) which are in terms of the total ooltages. On the other hand, suppose we wish to use the form of the MTL equations in (7.20) which are in terms of scattered uoltages. In this case, the appropriate terminal constraints must be in terms of the scattered voltages. To obtain these we substitute (7.16) into (7.22a) and (7.22b) to yield
(7.22d)
Therefore when we use the formulation in terms of scattered voltages, the driving sources in the MTL equations are simply the longitudinal incident electric fields but the integrals of the transverse incident electric field are nevertheless included in the results since they appear as sources in the terminations as shown by (7.22). This important subtlety is frequently overlooked and leads to incorrect results for the terminal currents and voltages as well as controversy over the proper form of the incident-field-excitation sources. When this is observed, equivalent results for the terminal voltages and currents will
FREQUENCY-DOMAIN SOLUTIONS
405
be obtained with either of the forms of the incident field excitation: (7.18) or (7.20). Throughout this chapter we will use the form in terms of the total voltage given in (7.18) with the source vectors given in (7.19).
7.2
FREQUENCY-DOMAIN SOLUTIONS
In the frequency domain the phasor MTL equations become (7.23a) (7.23b) where
$=R+joL
(7.24a)
P = G +jwc
(7.24b)
and
(7.24~)
Alternatively, the forcing function OF@)can be written in terms of the incident electric field as in (7.19~)as
O,(Z)
=
i
-
1' + &.dl
(Ei(i-th conductor, z )
I
- &(reference conductor, z)}
(7.24e)
Once again these can be written in state-variable form as a coupled set of
406
INCIDENT-RELD EXCITATION OF THE LINE
first-order, ordinary differential equations in matrix form as (7.25)
There are considerable advantages in writing the phasor MTL equations in this form as we show in the next section. In fact, drawing from the wealth of properties of the analogous solution for lumped, linear systems, we can immediately write the solution to these phasor MTL equations. 7.2.1
Solution of the MTL Equations
The solution to the phasor MTL equations given in (7.25) can again be immediately obtained by observing that they are directly analogous to the state-variable equations for lumped systems discussed in Chapter 4 CA.2, B.1,
H.31: d X(t) = AX(t) + BW(t) dt
(7.26)
The solution to these state-variable equations was discussed in Chapter 4 and becomes X(t) = @(t
- to)X(to)+
s’
@(c
io
- .r)BW(.r)d.r
(7.27a)
where the state-transition matrix is
Therefore, the solution to the phasor MTL equations in (7.25) becomes, by direct analogy,
where d(2) =
E];
and the chain parameter matrix is defined as
(7.28b)
FREQUENCY-DOMAIN SOLUTIONS
407
The n x n submatrices of the chain parameter matrix, 6&), are given in (4.70):
where T"YZT = f2, Observe in (7.28a) that the incident fields add a convolution term to the usual chain parameter relation. Writing (7.28a) out for a MTL of total length 9 gives
where the total source voltages &,(.5?)'
and &-,,,(Y) are, according to (7.28a),
According to (7.29a) the equivalent circuit of the complete line can be viewed and fpTI(9) as an unexcited line in series with the equivalent sources &.,,,(9) located at z = Y as illustrated in Fig. 7.4. 7.2.1.1 Simplified Forms of the Excitations The total excitation source vectors and qFT(9'), in (7.29b) and (7.29~)can be due to the incident field, fF,(Y) simplified using the definitions of the phasor distributed sources, &(z) and v,,(z), given in (7.24d) and (7.24e) and the properties of the chain parameter submatrices obtained in Section 4.3.3. The total forcing function, qFT(9), given in (7.29b) becomes, by substituting I&) from (7.24d) and OF@)from (7.24e),
408
INCIDENT-FIELD EXCITATION OF THE LINE
FIGURE 74 Illustration of the representation of a MTL with incident-field illumination as a 2n port having lumped sources that represent the effects of the incident field.
1
= ~ 0 9 @ 1 ~&i-th ( z - conductor, ~)[ 7)
(7.30)
]dz
- &(reference conductor, T)
Using the chain rule for differentiation:
FREQUENCY-DOMAIN SOLUTIONS
a a?
- [A(r)B(r)J
= A(r)
409
(7.31)
we obtain
dr
(7.32)
We now show that the last term in (7.32) is identically zero, i.e.,
This can be easily shown using the chain parameter submatrix definitions given in (4.70):
Thus the final result is
ds (7.35)
410
INCIDENT-FIELD EXCITATION OF THE LINE
where we have used the fact that &,l(0)= 1, where 1, is the n x n identity matrix. In a similar fashion we can show that
dT
(7.36)
where we have used the property that &zl(0)= 0 (see (4.70~)).These results depend , on a show that the total forcing functions, V,,,(9) and &,(9) convolution of one of the chain parameter submatrices and the longitudinal incident electric field along the positions of the conductors (with them removed) as well as the product of that chain parameter submatrix and the integrals of the transverse incident electric field (with the conductors removed) which are evaluated at the left end of the line (z = 0) and at the right end of the line (z = 9)These . forms are simpler to evaluate than those in (7.29). The important simplifications of the total forcing functions, qP,(9)and &.,(9 given ), in (7.35)and (7.36)can also be derived in a different fashion if we are careful to recall the distinction between total and scattered voltages given in (7.16). The solution for the form of the MTL equations in terms of scattered voltages given in (7.20) can be similarly obtained using the state-variable equation result given in (7.28). In that result the forcing functions become v,(z) and $,,(T) e-&(r) = 0 and the line voltage is the scattered voltage, v(z) br p(z).Writing out the solution in (7.29) and substituting the relation between scattered and total voltages in (7.16) yields the simplified forms of the total forcing functions given in (7.35)and (7.36).
e(?)
7.2.2
Incorporation of the Terminal Conditions
Now that the general solution of the phasor MTL equations has been obtained
in terms of the chain parameter matrix and the incident-field forcing functions,
FREQUENCY-DOMAIN SOLUTIONS
411
we next incorporate the terminal conditions to arrive at an explicit solution for the phasor line voltages and currents. Consider Fig. 7.4. The voltages and currents at the right end of the unexcited line are denoted as V ' ( 9 ) and P ( 9 ) . These are related to the actual desired voltages and currents, V ( 9 )and @'), as
(7.37a) (7.37b) Therefore, we can incorporate the efsects of incident fields into the solution by replacing V ( 9 ) and i(9)in the equations for the terminal responses without incident field illumination given in Section 4.3.5 with q(9)and i(9)- ipr(9)! This is further confirmed by rewriting the general solution given in (7.29a) in terms of the chain parameter matrix with sources. Rewriting this in the form of the chain parameter relation without incfdent-field illumination gives
vFT(9)
Consider the terminal conditions written in the form of generalized ThCvenin equivalents as (7.39a) V(o) = Os- 2,jiCo)
V ( 9 ) = VL + 2,i(U)
(7.39b)
Making the substitutions in (7.37) gives
Observe that (7.41) shows we could modify (4.90) for no incident-field illumination by replacing VL with VL- V F r ( 9 )+ gLipT(W)and f(9) with i(S)- ip,(S)to yield
412
INCIDENT-FIELD EXCITATION OF THE LINE
Similarly, the alternative result given in (4.88) becomes
(7.43) where the characteristic impedance matrix is & = Z+f?-'T-' = ?-'+f?T-'. The solutions for the terminal voltages so obtained will be
The terminal constraints for the generalized Norton equivalent representation: (7.45a) l(0) = Is - 9,V(O)
F ( 9 ) = -l L
+ 9LV(9)
(7.45b)
become, substituting (7.37),
f(0) I= fs - P,V(O)
(7.46a)
W )= [-L- ld9) + 9,0,,(9)] + PLV'(9)
(7.46b)
This shows that we can modify the equations developed for the generalized Norton equivalent in (4.95) by replacing the source lLwith f L lFT(9) fi,VF,(9) and V ( 9 ) with V ( 9 ) - V F T ( 9 ) :
+
FREQUENCY-DOMAIN SOLUTIONS
413
The solutions for the terminal voltages so obtained will again be given by (7.44). The results of the mixed termination representation given in (4.96) to (4.99) are similarly obtained as
(7.50a)
~ 2 . 2 , 1Lossless Lines in Homogeneous Media The above final equations for determining the terminal currents and voltages of the line are considerably simplified if we assume lossless lines in a homogeneous medium. For this case, the per-unit-Iengt h parameter ma trices satidy
R=O
(7.53a)
G=O
(7.53b)
LC = CL = P61,
(7S3C)
414
INCIDENT-FIELD EXCWATION OF THE LINE
The chain parameters were derived in Chapter 4 and simplify to
611 = cOs(pa)l, &12
= -10 sin(jW)L
(7.54a) (7.54b)
= - j sin(pp)&
621 = -ju
sin(p9)C
(7.54c)
= - j sin(@g)2; 1 $22
= cos(p9)1,
(7.54d)
where the characteristic impedance matrix becomes real given by
2, = oL 2;1= uc
(7.55a)
(7.55b)
and the phase velocity of propagation in the surrounding homogeneous medium is V=-
1
J;1;;
(7.56)
The final equations to be solved for the terminal voltages simplify considerably for this assumption. For example, consider the generalized Th6venin equivalent characterization of the terminations given in (7.42). Substituting the simplified forms of the total forcing functions, 8,#) and f,,,(9), given in (7.35) and (7.36) into (7.42) gives CH.41:
(7.57a)
FREQUENCY-DOMAIN SOLUTIONS
415
and
f ( 9 ) = -ju
sin(/39)CqS + [cos(/3.V)tR + j u sin(/?.V)C&$(O)
(7.57b)
I
&(i-th conductor, z) - &reference conductor, t) dz
The generalized Norton equivalent representation in (7.47) as well as the mixed representations in (750) and (7.52) simplify in a similar fashion.
7.2.3
lumped-Circuit Iterative Approximate Characterizations
If the line is electrically short at the frequency of the incident field, then the usual lumped-circuit,iterative approximate circuits are adequate to characterize the line. For example, the per-unit-length equivalent circuit in Fig. 7.3 as well as the MTL equations show that the Lumped 1, II, and T circuits in Fig. 4.12 can be modified to incorporate the effects of incident fields by simply multiplying the per-unit-length phasor voltage and current sources created by the incident field by the total line length, thus inserting the phasor voltage sources, GI.%’,in series with each line self-inductance, and inserting the phasor current sources, fF,9,in parallel with each line self-capacitance. Since the line must be electrically short for the lumped-circuit approximation to be valid, and the cross-sectional dimensions must be electrically small for the TEM assumption to be valid, the phasor fields do not vary significantly over the line. Thus, the per-unit-length phasor voltage and current sources, &(z) in (7.24d) and in (7.24e), can be approximately evaluated at any convenient location on the line, say, at the center of the line. These are then multiplied by the total line length to give the lumped sources in the model. Lumped-pi and lumped-T circuits are illustrated in Fig. 7.5.
r,
v&)
416
INCIDENT-FIELD EXCITATION OF THE LINE I
Adaptation of the lumped-pi and lumped-Tapproximate equivalent circuits for incident-field illumination. FIGURE 7.5
7.2.4
Uniform PlaneWave Excitation of the line
A significant set of problems to which these results apply is the case of illumination of the line by a untform plane wave from some distant source [A.l]. The radiated fields in the far field of a radiating structure are spherical waves which locally resemble uniform plane waves. This assumption also simplifies the evaluation of the sources in the above results. In order to characterize the frequency-domain response of the line let us describe the incident uniform plane wave angle of incidence and polarization with respect to a spherical coordinate system as illustrated in Fig, 7.6. The
FREQUENCY-DOMAIN SOLUTIONS
417
FIGURE 7.6 Definitions of the parameters characterizing the incident-field as a uniform plane wave.
propagation vector of the wave is incident on the origin of the coordinate system (the phase reference for the wave) at angles 8, from the x axis and & from the projection,onto the y-z plane from the y axis as shown in Fig. 7.6(a). The polarization of the electric field vector is described in terms of the relation to the unit vectors in the spherical coordinate system, de and d,, as illustrated in Fig. 7.6(b). In terms of these, the general expression for the phasor electric field vector can be written as CA.11
where the components of the incident electric field vector along the x , y, and z axes of the rectangular coordinate system describing the line are CA.1J e, = sin 8 , sin 8,
ey = -sin 8, cos 9, cos 4, e, = -sin 8, cos 8,
and
sin 4,
- cos 0,
sin 4,
+ cos 9, cos 4,
I
(7.59a)
(7.59b)
418
INCIDENT-FIELD EXCITATION OF THE LINE
The components of the phase constant along those coordinate .axes are
e,
p, = -!cos
PI= -/3
sin 8, cos 4,
p, = - B sin e, sin cPp
I
(7.60)
The phase constant is related to the frequency and properties of the medium as (7.61)
where u, = 1/& is the phase velocity in free space and the mtdium is characterized by permeability, p = popr, and permittivity, 8 E,.!?,, E, is the complex amplitude of the sinusoidal wave. Although not needed, the magnetic field intensity vector of the incident wave is related to the electric field intensity vector by 1 ;t A I = - d p x E‘ (7.62) 5:
tl
where a, is the unit vector in the direction of propagation, This gives
where the components of the incident magnetic field vector along the x, y, and z axes of the rectangular coordinate system describing the line are CA.11
h, = -cos 0, sin 0, h, = cos OE cos 8, cos t$p - sin 0, sin 4, h, = COS 0, cos e, sin 4,
and again
hf
1 1
(7.64a)
+ sin 19, COS 4,
+ h; + h: = 1
(7.64b)
The intrinsic impedance of the medium is (7.65)
vF,(.9)
The phasor sources due to this incident plane wave, given in (7.35) andip#’) given in (7.36), can be easily evaluated in terms of the above results.
FREQUENCY-DOMAIN SOLUTIONS
"t Xk
, A
419
----
around plane
(b) FlGURE7.7 Derivation of the contributions to the equivalent sources due to the transverse component of the incident electric field for (a) an (n 1)-wire line, and (b) n wires above a ground plane.
+
If the reference conductor is placed at the origin of the coordinate system, x = 0, y = 0, as shown in Fig. 7.7(a), the transverse field contributions can be written in terms of the cross-sectional coordinates of the k-th conductor (xk,yk) using the general form of the incident field in (7.58) as
(7.66a)
420
INCIDENT-FIELD EXCITATION OF THE LINE
,/m
where dk = is the straight-line distance between the reference conductor and the k-th conductor in the transverse plane and (7.66b) Similarly, the contributions due to the longitudinal field are (7.67)
Substituting the expressions for the chain parameter submatrices given in (4.70):
into the expressions for ?,#)'
given in (7.35) and I&.?)
given in (7.36)yields
FREQUENCY-DOMAIN SOLUTIONS
421
Observe that the evaluation of these forcing functions requires that we evaluate the following n x 1 vectors: (e9W-r) f e-9(9-9$-1*
(7.71)
and (7.72)
The entries in these vectors become, in terms of the results in (7.66) and (7.67),
where $k is given by (7.66b) and
These results can be extended to' the case where the reference conductor is an infinite, perfectly conductingground plane illustrated in Fig. 7.7(b). The total incident field is the sum of the incident field (with the ground plane and the other conductors removed) and the reflected field. Snell's law shows that the angles of incidence and reflection are the same CA.1). Similarly, continuity of
422
INCIDENT-FIELD EXCITATION OF M E LINE
the tangential electric fields at the surface of the ground plane gives constraints on the y and z components of the electric field. Thus the incident and reflected (at the ground plane) fields are given by [A.l, H.11
Thus the total fields are (7.78a) (7,78b) (7.78~) Thus the entries in the vectors in (7.71) and (7.72) become
and
where (7.80b) and (7.81)
These results are substituted into the equations for the terminal voltages for the generalized ThCvenin equivalent characterization of the terminal networks given in (7.43) and (7.44) and implemented in the FORTRAN program 1NCIDENT.FOR that is described in Appendix A.
FREQUENCY-DOMAIN SOLUTIONS
7.2.5
423
Two-Conductor lines
The previous results for a MTL are, of necessity, couched in matrix notation and are somewhat complex. In the case of a two-conductor line where n = 1 as illustrated in Fig. 7.8, the above results simplify considerably. The chain parameter submatrices for a two-conductor line become scalars even for the most general case of a lossy line in an inhomogeneous medium, equation (4.79): (7.82a) (7.82b) (7.82~) (7.82d) where the characteristic impedance is (7.83a) and the propagation constant is (7.83b)
In order to simplify the notation, let us place both conductors in the x-z plane parallel to the z axis as shown in Fig. 7.8. The reference conductor is placed at x = 0, and the other conductor is placed at x = d. Thus the conductors are
Definition of the line parameters for the incident-line field illumination of two-conductor.
FIGURE 7.8
a
424
INCIDENT-FIELD EXCITATlON OF THE LINE
separated by d. Also, since the terminations are implicitly assumed to be linear, we will omit any lumped sources in them, Le., & = = 0. The contributions to the terminal currents due to these lumped sources in the termination networks can be obtained with the results of Chapter 4 by superposition. The total forcing functions due to the incident field given in (7.35) and (7.36) reduce to
e
&,(9) =
r
9
cosh(p(9
Jo
- [s,” &(x,
- t))[&(d, r ) - &O,
9)d x ]
(7.84)
r)] dz
+ cosh(f9)[-; &(x,
0) d x ]
and
GT(9)= -Jog sinh(p(9 - r))S;’[&(d,
r ) - eL(0, r)] dr
(7.85)
--~inh(PJ)i;l[[~~ &(x, 0) d x ] For & = = 0, the solution for the terminal current at the source end, z = 0, for the Thevenin equivalent representation of the terminations given in (7.42a) reduces to (7.86)
[ - B [6E!&, + [ 5
= -x1 ‘ D o
cosh(P(9 - r)) + sinh(P(9 - 7)) -][i?!(d, 2L ZC
t)
- &(O,
t)]
dr
14) d x ]
cosh(P9)
s,”
+ sinh(99) $][ ZC
&(x, 0) d x ]
The terminal current at z = 9 can be easily obtained from this result with the following observation. First reverse the z coordinate and add the line length, 9, to it, i.e., replace z with 9 - z. This serves to reverse the roles of the terminations. Therefore, the terminal current at z = 9 can be obtained by replacing z with 9 - z, multiplying the equation by a-minus sign (to reverse the directions of the currents), and replacing Sswith ZL and vice versa. This yields
f ( 9= )
a1
FREQUENCY-DOMAIN SOLUTIONS
9
[cosh(fz)
0
-
+
B
[cosh(f9)
;[1.'
+ sinh(9.r) $][&d,
$][s,"
2,
+ sinh(f9)
7) - &O,
r)] d7
425
(7.87)
&(x, 9) dx]
ZC
B x x , 0) d x ]
The denominator in both expressions, 6, is
6 = cosh(fY)(&
+ &) + sinh(99)
(7.88)
These results agree with those obtained by Smith [SI. 1.2.5.1 Uniform Plane-Wave Exciration of the Line In the case of uniform plane-wave excitation of the line with the incident electric field described by (7.58) to (7.61), the terminal current at z = 0 becomes
(7.89)
x
{
-j&e, s,'[cosh(~(Y
- 7 ) ) + sinh(f(9 - 7))
Similarly, the terminal current at z = Y becomes
(7.90)
426
INCIDENT-FIELD EXCITATION OF THE LINE
for uniform plane-wave excitation of two-conductor lines, are somewhat complicated. In this section we will specialize those results to three cases. These are illustrated in Fig. 7.9. The first case, illustrated in Fig. 7.9(a), is referred to as endfire excitation and has 8, = go", 4, = -go", and 8, -- 90". Thus the wave is propagating in the + z direction with the electric field polarized in the +x direction. For this incidence and polarization, e, = 1 1.2.5.2 Special Cases The above results, although exact and general
(7.9 la) e, = 0
"t "t
Three important cases of wave incidence: (a) endfire, (b) sidefire, and (c) broadside.
FIGURE 7.9
FREQUENCY-DOMAIN SOLUTIONS
427
and the components of the phase constant along those coordinate axes are
(7.91b)
as expected. The complete expression for the electric field vector is, from (7.58),
Substituting these into (7.89) and (7.90) and evaluating gives f(0) = f(9) =
dE0
[cosh(f9')
''
+ sinh(99) z - cos(P9) +j sin(p9)
(7.93)
(cosh(p9) - J sin(B9))
(7.94)
dE, [1 - (cosh(99) + sinh(99) D
1 1
In the case of a lossless, homogeneous medium,
P =jP
(7.95a) (7.95b)
these simplify to (7.96)
The next case, illustrated in Fig. 7.9(b), is referred to aB sidefire excitation and has 8, = 180", 4, = 0", and 8, = 0". Thus the wave is propagating in the + x direction with the electric field polarized in the +z direction. For this incidence and polarization, e, = 0 e,, = 0 e, = 1
(7.98a)
428
INCIDENT-FIELD EXCITATION OF THE LINE
and the components of the phase constant along those coordinate axes are
(7.98b)
as expected. The complete expression for the electric field vector is, from (7.58),
Substituting these into (7.89) and (7.90) and evaluating gives
-
r(0) = -12 80
?D
f(9) = -j2
sin
(cosh(p9) - 1) + sinh(99)
1
80 e-J@*/*sin(T)[sinh(f9)
+ (cosh(f9) - 1)
fD
"I
?r
(7.100) (7.101)
ZC
In the case of a lossless, homogeneous medium, these simplify to
The final case, illustrated in Fig. 7.9(c), is referred to as broadside excitation and has 8, = go", 4, = O", and = 90". Thus the wave is propagating in the - y direction with the electric field polarized in the +x direction. For this incidence and polarization, e,, = 0
e, = 0
(7.104a)
FREQUENCY-DOMAIN SOLUTIONS
429
and the components of the phase constant along those coordinate axes are
(7.104b)
as expected. The complete expression for the electric field vector is, from (7.58), ( Y = 0)
(7.105)
Substituting these into (7.89) and (7.90) along with y = 0 and evaluating gives
D D
cosh(99) - 1 1 - cosh(99)
+ sinh(fg)
-x-
i.]
(7.106)
ZC
- sinh(92’) -r
(7.107)
In the case of a lossless, homogeneous medium, these simplify to (7.108)
(7.109) 7.2.5.3 One Conductor Above a Ground Plane As a final application of the results for uniform plane-wave illumination of a two-conductor line we consider the case of one conductor at a height h above an infinite, perfectly conducting ground plane as illustrated in Fig. 7.10(a). In a fashion similar to the multiconductor case considered previously, this problem can be replaced with an equivalent problem by replacing the ground plane with images as illustrated in Fig. 7.1qb). The image field can be viewed as the wave reflected by the ground plane, and the total incident field is the sum of the original field and the image or reflected field. Again, Snell’s law provides that the angle of incidence and the angle of reflection are equal CA.1, HA]. Thus the fields can be written as
A t the position of the ground plane, y = 0, continuity of the tangential electric
430
INCIDENT-FIELD EXCITATION OF THE LINE
3
^t
FIGURE 7.10 Replacement of the ground plane with images for incident-field illumination.
field requires that e: = -et
I
-e,
(7.1 1 la)
Thus the components of total field (incident plus reflected) can be written as (the y component is not needed)
Equations (7.86) to (7.88) become (y = 0)
FREQUENCY-DOMAIN SOLUTIONS
431
(7.113) x
{
-jPxe,
f ( 9= ) 2h
s,"
[cosh(9(9
- t)) + sinh(f(9 - t))
Bo sin( P, h)
(7.114)
5-5-
Comparing (7.1 13) and (7.1 14) to (7.89) and (7.90) we see that the results for the case of a ground plane can be obtained from the case without a ground and replacing d with 2h: d 2h. We will plane by removing the factor find this general rule to be helpful in later results. Of course, the characteristic impedance, Z,, in these expressions must be that of one conductor above a , of two conductors separated by twice ground plane which is one-half the 2 the height above ground (the image problem). Again, these expressions are complicated. However, if we restrict our consideration to the special cases shown in Fig. 7.9, the results simplify. For the endfire excitation, 8, = go", 4, = -go", and OE = 90" we obtain e, = 1, p, = P, and all others zero. The total field vector components become (7.1 15a) (7.115b) This compares to the results without the ground plane given in (7.92): (7.1 16a) (7.1 16b) Therefore we double (7.93) and (7.94) (replacing d = 2h) which gives
+ sinh(f2') z "- cos(/39) + j sin(p9)
I
D
f(9)= 2hEo
T
D
[1
(cosh(f9)
+ sinh(f9)
(7.1 17)
1
(cos(P9) - j sin(p9)) (7.118)
432
INCIDENT-FIELD EXCITATION OF THE LINE
For a lossless line in a homogeneous medium, these simplify to (7.1 19)
For the sidefire excitation (propagating from above the ground plane in the P 0", 4, = O", and 8, = 0". For this case, we obtain e, = 1, = -8, and all others zero, The total field vector components become
-x direction), 8,
P,
(7.12 1a) (7.121b) This compares to the results without the ground plane given in (7.99): (7.122a) (7.122b) Substituting (7.122) into (7.1 13) and (7.1 14) gives (cosh(f9) - 1) + sinh(f9)
1
f(3) =j2hp$
90
m[sinh(f9) Bh
(7.123)
+ (cosh(f9) - 1)
For a lossless line in a homogeneous medium, these simplify to
4, = 0" and t& = 90". For this case, For the Broudsfde excitation, 8, = !No, we obtain e, = 1,3/, 3: -Band all others zero. The total field vector components become gpi I 0 (7.127a) &a'
Y=
24
(7.127b)
433
FREQUENCY-DOMAIN SOLUTIONS
This compares to the results without the ground plane given in (7.105):
E* = 0 Ex= Eo
(7.128a) (7.1 28b)
Therefore we simply double the results in (7.106) to (7.109) and replace d = 211 to give
- 1 t sinh(99)
-r
1 - cosh(f9) - sinh(9.Y)
f(9) =
(7.129)
(7.130)
and for a lossless, homogeneous medium, (7.131)
D
1 - cos(p9)
-j sin(p9) -
(7.132)
Z2.5.4 Elecfrkally Short Lines The previous results are exact within the TEM mode assumption, If the line length, 9,is substantially less than a wavelength at the frequency of interest, i.e., .Y < 1 = u/l; we can approximate the line with lumped-circuit iterative structures such as the lumped-pi or lumped-T structures containing independent sources representing the effects of the incident field. Lumped-circuit codes such as SPICE can be used to solve the resulting structures. If the line length is significantly less than a wavelength, i.e., 9 << 1 = u / j , a very simple but approximate model can be further developed. This model is simply the lumped-circuit model wherein the line inductances and capacitances are neglected. So long as the terminal impedances, gs and gL,are not significantly different from the line characteristic impedance,&,this very simple model can give adequate results with a minimum of computational effort cA.31. This model is illustrated in Fig. 7.11, It is obtained from the per-unit-length equivalent circuit in Fig. 7.3 by omitting the per-unit-length inductance, I, and per-unit-length capacitance, e, and lumping the distributed sources representing the effect of the incident field, @(z)Az =* & 9 / 2 ) 9 and &(z)Az &(.Y/2)9. Since the line is assumed to be very short, electrically, we can evaluate these sources at any convenient point on the line and have arbitrarily chosen to evaluate them at a midpoint of the line. Substituting the forms of these sources
434
INCIDENT-FIELD EXCITATION OF
"t
FIGURE 7.11
ME
LINE
"( Q)Y
The low-frequency equivalent circuit for incident-field illumination.
given in (7.24) yields (7.133a)
(7.133b)
The alternative form of (7.133a) is obtained by writing the normal magnetic field in terms of the transverse and longitudinal electric fields via (7.9) as in (7.24e). The terminal voltages can be easily obtained from the model of Fig. 7.11 using voltage division and current division as
For the case of uniform plane-wave excitation, these results can be explicitly obtained by substituting (7.58) (with y = 0) into (7.133) to yield
FREQUENCY-DOMAIN SOLUTIONS
435
(7.135b)
where the area of the loop formed by the conductors of the line is denoted as
A=d9
(7.13%)
In the case of one conductor at a height h above an infinite ground plane illustrated in Fig. 7.10, using the forms of the total field given in (7.112) gives
(7.136b)
where the loop area becomes A = hlp. 7.2.6
Computed Results
In this section we will give some computed results and compare them to results obtained by alternative methods. This will illustrate the accuracy of the formulation. 7.2.6.1 Comparison with Predictions of the Method of Moments Codes First we consider a lossless, two-wire transmission line with uniform plane-wave excitation CH.21. The wires have radii of 1 mm, separations of 1 cm and total length of 1 m. The terminations at each end are identical resistors, Ss= 2‘ R, with three sets of values: R = 552.2262R (matched loads), R = 5OR (low-impedance
-
436
INCIDENT-FIELD EXCITATION OF THE LINE
loads), R = 10 kn (high-impedance loads). Three polarizations of the incident uniform plane wave are illustrated in Fig. 7.9. The endfire case has the wave propagating along the line ( 2 ) axis with the electric_field vector lying in the plane of the line and polarized in the x directions: %‘ 3: le-jfl‘d,. The sidejre case has the uniform plane wave propagating in the plane of the wires in the x direction perpendicular to them with the electric field vector polarized parallel to the wires in the z direction, 8‘= le-JflXh,.The broadside case has the uniform plane wave propagating perpendicular to the plane of the line in the - y direction with the electric field vector polarized in the plane of the wires in the x direction, = lejflyd,. Results for only certain polarizations, angles of incidence, and termination impedance values will be shown here. The reader is referred to CH.2) for more extensive numerical comparisons. The numerical code used to give the baseline results is a method of moments (MOM) code developed at Ohio State University by J.H. Richmond [14, 151. It will be referred to as OSMOM and uses piecewise sinusoidal expansion functions and a Galerkin technique. These features give this code exceptional accuracy. It is important to point out that the MOM method is a direct implementation of Maxwell’s equations and, within numerical error, should provide “exact” solutions to compare against the predictions of the MTL model. Figure 7.12 shows the results for the endfire case and low-impedance loads (R -- 50 n).Observe that the line is one wavelength long at 300 MHz so we should expect to see nulls in the frequency response at multiples of 150 MHz. Figure 7.12(a) shows the magnitude of the frequency response, whereas Fig. 7.12(b) shows the phase. Excellent correlation with the MOM results (OSMOM) are observed. The approximate, low-frequency model in (7.134) (using (7.135) with e, = 1, e, = 0, /?, = 0, /?, = @)gives a value of 2.28 x at f = lo’ = 10 MHz. This compares with the exact value of approximately 1.5 x lo-’. The error here is incurred for the following reason. Observe that the sources in (7.135) vary with frequency as& = OF. Therefore, the model is only valid if the magnitude of the frequency response varies linearly with frequency, 20 dB/decade, and the phase is f 90”.This is only satisfied for frequenciesbelow the lowest plotted frequency. Figure 7.13 shows the results for the sidefire excitation and matched loads (R = 552.2262 n),whereas Fig. 7.14 shows the results for the broadside excitation and high-impedance loads (R = 10 kQ). Excellent predictions of the MTL model are obtained for these orientations of the incident wave. The approximate low-frequency model in (7.134) (using p, = 0) gives a value of 1.896 x low6at (7.135) with e, = 0, e, = 1, 6, = /?, f = lo7 3: 10MHz for the sidefire case of Fig. 7.13. This compares with the The accuracy here is better since the exact value of approximately 1.9 x frequency response clearly varies directly with frequency and the phase is -90” at the lowest frequency off = lo’ = 10 MHz. The approximate, low-frequency model in (7.134) (using (7.135) with e, = 1, e, = 0,?/, = p, = 0) gives a value at f = 10’ = 10 MHz for the broadside case of Fig. 7.14. This of 1.896 x compares with the exact value of approximately 9 x lo-’, Again, the accuracy
ei
FREQUENCY-DOMAIN SOLUTIONS
Excltation: endfire Loading: low impedance Transmission-line model
-
---
Line dimensions (m) 1.0 Length Separation 0.01 Radius 0.0001
OSMOM
0
I
1
10,-7
10'
437
I o9
108
Frequency (a)
Excitation: endfire Loading: low Impedance Transmission-line model
-1sol 107
-
1
10'
--
Line dlmentiom (m) -io Length Separation 0.01 Radiut 0.001 OSMOM 0
I
,
,
10'
Frequency
0) FIGURE 7.12 Comparison of the predictions of the transmission-line model and the method of moments for a two-wire line for endfire illumination:(a) magnitude, (b) phase.
438
INCIDENT-FIELD EXCITATION OF THE LINE
Excitation: sidefire Loading: matched Tfan#mhBiOn-l~nemodel 10-4
lo-'
I
107
,
I
,
--
Line dimensions (m) Length 1.0 Separation 0.01 Radius 0.001 OSMOM 0
-
I
,
I
,
1
1
lo'
1
1
1
109
Frequency (8)
Excitation: ridefire Loading: matched Transmission4lne model
-
--
Line dimenrions (m) Length 1.0 Separation 0.01 Radiur 0.001 OSMOM 0
FIGURE 7.13 Comparison of the predictions of the transmission-line model and the method of moments for a two-wire line for sidefire illumination:(a) magnitude,(b) phase.
FREQUENCY-DOMAIN SOLUTIONS
Excitation: broadaide Loading: high impedance Transmission-line model
-
-
Line dimeniiona (m) Length -1.0 Separation 0.01 Radius -0.001 OSMOM Q
1
I
10-7
439
0 I
I 07
I
r 1 I
IO’
/OB
Frequency (a)
Excitation: broadside Loading: high impedance Tranami8sion-linc model 1807
90
z
OSMOM
0
-
3 i*”
-
--
Line dimenrlonr (m) Length 1.0 Separation 0.01 Radiui -0.001
. O-
-90-
e
e
-1g01-
FIGURE 7.14 Comparison of the predictions of the transmission-line model and the method of moments for a two-wire line for broadside illumination: (a) magnitude, (b) phase.
440
INCIDENT-FIELD EXCITATION OF THE LINE
here is not as good as the sidefire case since the frequency response varies directly with frequency and the phase is -90" below the lowest frequency of f = lo' =r: 10 MHz. If the proper restrictions on the approximate, low-frequency model are observed, it can give reasonable estimates of the frequency response with minimal computational effort. For frequencies where the line is electrically long, we have no other recourse but to use the transmission-line model. 1.2.6.2 A Thtee-Win line in an Incident Unihrm Plane Wave This eTample consists of a three-wire lossless line of length 1 m as illustrated in Fig. 7.15.The three identical wires have radii of 1 mm and lie in the x-z plane. The adjacent wire separations are 1 cm, and impedances terminate each end in a, star = ZSz = 500 and ZLo = configuration. For the first case studied, Zs0= ZSl gLI= Z,, = 500 $2. The generalized Thtvenin equivalent representation becomes A
8(0)= -
1000 500 500 1000
&,
L
r.0
- 1.1
= r,a
1 Vlm
lmm
FIGURE 7.75 A three-win line for illustration of computed results.
FREQUENCY-DOMAIN SOLUTIONS
441
An incident field in the form of a uniform plane wave propagating in the + x direction with the electric field polarized in the +z direction excites the line. This problem was studied in [16], and two frequenciesof the incident field were investigated: /.?9 = 1.5 and P 9 = 3.0. For the first frequency, the line is approximately A/4 and for the second frequency the line is approximately A/2. This should provide a good test of the model since the response of MTL's tends to be the most sensitive around multiples of A/2. The results computed for the line currents at z = 0 by Harrison in [la] via a different technique are fo(0) = 1.7666
-5A
fl(0) = 9.076E - 8 A
I2(0)= 1.7678 - 5 A fO(0) = 5.454E - 5 A
-7A 5.461E - 5 A
fl(0) = 7.736E
I2(0)
The results computed from the distributed source MTL model described herein using the code 1NCIDENT.FOR described in Appendix A are l(0) = 2.295 26E - 71- 35.636' A = 1.76362E - 5 / - 109.721" A
/32' = 3.0
f,(O) = 8.000E - 7/-86.278' A fZ(0) 5.45725E - 5/- 170.97"A
'
which correspond well to Harrison's results. Another result obtained by Harrison in [16] was for a 10 m line with 280
= (50 -j 2 5 ) Q
Zsl = (loo +jloo) R Zs2
and
= (25 +j25) Q
gt0 = (50 + 125) Q ZL1= (100 -j50) n
2L2= (100 4 5 0 ) R
442
INCIDENT-FIELD EXCITATION OF M E LINE
The generalized Th6venin equivalent representation becomes
+
25 +J25
2s2 2SO Y
200 -j25
125 +I125
150 -j50
The results computed by Harrison are Io(0) = 6.255E
- 5 A, I o ( 9 ) = 3.650E - 5 A
1.065E - 5 A, I1(5?) = 1.220E - 5 A 12(0) 5.644E - 5 A, I z ( 9 ) F 2.784E - 5 A Il(0)
The results computed by the program WIRE described in CH.11 are
PA?'
= 1.5{:(O) 2(0)
= 1.066E - 5/-99.83' A, I I ( 9 ) 3 1.221E - 51158.65" A 5.6476 - 5/- 159.07' A, I z ( 9 ) 3:2.784E - 51- 148.26' A
The results computed from the distributed source MTL model described herein using the code 1NCIDENT.FOR described in Appendix A are
pu = lS(1Il(0) = 1.066E - 5/-99.83' A, 2(0)
-
I 1 ( 2 ) = 1.221E 5/158.65' A 5.647E - 51- 159.07' A, I,(.") = 2.784E - 5/- 148.26' A
which correspond well to Harrison's results. The last example is the case of two wires above an infinite ground plane illustrated in Fig. 7.16 which was considered in CH.1). Wire # 1 has a radius of 30 mils and is located at y = 0 and height above ground of 5 cm. Wire # 2 has a radius of 10 mils and is located at y = 4 cm and height above ground of 2 cm. The line length is 1 m and = 100a, gL1= 500 a, gs2= 500 a, and ZL2= lo00 Q. This results in a generalized Thbvenin equivalent characterization of the terminations as
FREQUENCY-DOMAIN SOLUTIONS
443
FIGURE 7.16 A three-conductor line consisting of two wires above a ground plane for illustration of computed results.
The first case has 'a 1 V/m uniform plane wave traveling parallel to the ground plane with the electric field perpendicular to the ground plane: 6, = 90", 6, = go", & = -90". The results of the code 1NCIDENT.FOR described in Appendix A at a frequency of 100 MHz are II(0) = 4.63789E - 4/-37.082' A, I I ( 9 ) = 1,1502E- 4/- 156.86"A
IZ(0)
6.601 77E - 5/-24.279" A, I Z ( 9 ) = 3.0208E - 6/70.1546" A
These are identical to those of the WIRE program described in CH.11. The next case has the 1 V/m uniform plane wave traveling in the --x direction (perpendicular to the ground plane) with the electric field in the + z direction
INCIDENT-FIELD EXCITATION OF THE LINE
444
= 0", & 0". The results of the code (parallel to the wires): 0, = O", 1NCIDENT.FOR described in Appendix A at a frequency of 100 MHz are 5:
II(0)
5.3161B - 4/-33.826" A, II(U)p 1.9875E - 4/-6.817" A
12(0)
8.391 8E
- 5/52.795" A, X z ( 9 ) = 4.633 88E - 5/35.7719'
A
These are again identical to those of the WIRE program described in CH.13. The frequency response for this case is plotted for wire 1 2 in Fig. 7.17. As before, these frequency-domain transfer functions can be used along with the TIMEFREQ.FOR program to compute the time-domain response for a plane wave having any waveshape. 7.3
TIME-DOMAIN SOLUTIONS
The time-domain solutibns can be obtained with techniques analogous to those for no incident field as discussed in Chapter 5. Virtually all of those techniques can be used for the incident-field case. We essentially need to solve the time-domain MTL equations given in (7.18) and (7.19) or the form given in (7.20) and (7.21) for some arbitrary time variation of the incident field. We will choose to solve the forms in (7.18) and (7.19) which are in terms of the total voltages and currents. The following sections will concentrate on the solution for lossless lines although applications to the lossy-line case will be considered. Thus the equations to be solved are given in (7.18) and (7.19) with R G = 0:
-
a V(z, t ) + L a I(2, t ) V&, 5
az
at
a I(2, t ) + c a V(2, t ) = I&, a2 at
t)
t)
(7.137a) (7.137b)
where (7.138a)
-
r
L
$fed1
+ {&;(i-th conductor, z, t )
- &;(reference conductor, z, t ) }
- - +, -
TIME-DOMAIN SOLUTIONS
fncldent Uniform Plane Wave
(ec o*, e,
00,
00)
10'
106
10'
10'
Frequency (Hz) (a)
100 h
8
--_. -.-.
*. .*
e 3
z
:-... ....-----..-...____.._._ ---....
I 4
0-
-100
-200,
, \
-
I
I
I
I
I 1 . 1 1
I
.,,
, ',, $I
8
'*
" 1 :
t
w
445
446
INCIDENT-FIELD EXCITATION OF THE LINE
(7.138b)
We first concentrate on the solution for two-conductor lines. The solution for the multiconductor case will be obtained in terms of this solution by diagonalizing the MTL equations in the usual fashion. 7.3.1 7.3.7.7
Twdondudor Lossless lines The General Solution vir the Method of Charactedstks The trans-
mission-line equations for a two-conductor, lossless line become
a a a V(Z,t ) + I - l(z, t ) = - 64?& az at at = cb&,
t)
a B&, - az -
a a a -I(z, t ) + c - V(z, t ) = -c - 8&
t)
t)
(7.139b)
8’(~, t ) 8 f ( d ,Z, t ) - 8:(0, Z, t )
(7.140a)
az
where
(7.139a)
t)
at
cb&,
t) =
at
Iod
Si(x, 2, t ) dx
(7.140b)
and the conductors lie in the x-z plane with the reference conductor located at x = 0 and the other conductor located at x = d as illustrated in Fig. 7.8.The method of characteristics was used to obtain the time-domain solution in Chapter 5 for the transmission-line equations that did not have forcing functions on the right-hand side. Here we extend that method to solve (7.139). A similar derivation was given in [17]. The total differentials are dV = E d z az
+ Eatd t
ai dl = c d z + -dt az
at
dgT = 3 d z + s d t az at
(7.14 1a)
(7.14 1b)
(7.141~)
447
TIME-DOMAIN SOLUTIONS
(7.141d) Substituting the transmission-line equations given in (7.139) into (7.141a) and (7.141b)yields dV = (-I
5)dz + E d t
ai + 8’at
dI=(-cKat
at
aZ
d Z + -ai dt
C -a’T)
at
at
(7.142a) (7.142b)
The characteristic curves are drawn as shown in Fig. 5.7. Along theforward characteristic, dz = u dt where the velocity of propagation along the line is o = l/& Substituting this into (7.142) and defining the characteristic impedance of the line as Z, = ul = l/uc yields dV = (-2, - + u8’ at a1
-u
(7.143a) (7.143b)
Adding these gives
d V + Z,dI = (08’ - u-
aZ
-at
(7.144)
Substituting (7.141c) yields dV + 2 , dI
8’ dz
- d8r
(7.145)
Rearranging and integrating along the forward characteristic from z = 0 to
z = 9 Rives
(7.146) or
where the one-way line delay is T = 9 / u . The result in the integral is due to the fact that along the forward characteristic, t and z are related as t = z/u C’.
+
448
INCIDENT-FIELD EXCITATION OF THE LINE
Similarly, along the backward characteristic, dz = - u dt. Substituting this into (7.142) yields dV = (Zc-ai dL 0 % + dt (7.148a) az at at
- +
E)
1 aV d I = ( -z-, at +-z,1 &at+ a 1 -)dt at
(7.148b)
Subtracting these gives (7.149) Substituting (7.141~)yields
d V - Zc dI
8 L dz
- d&r
(7.150)
Rearranging and integrating along the forward characteristic from z = 0 to z = 9 gives (7.151) or
The result in the integral is due to the fact that along the backward characteristic, t and z are related as t = -z/u + C ' . The results in (7.147) and (7.152) relate the voltages and currents at one end of the line to those at the other end delayed by the line one-way delay, T,as well as various values of the transverse and longitudinal incident electric fields. These are similar to the results derived in Chapter 5 for no incident field and will again allow us to construct a SPICE model of the line. 7.3.72 The General Solution via the Fquency Domain As before, a simple method of determining the timadomain solution is to transform the frequencydomain solution. The frequency-domain chain parameter matrix for a lossless line excited by an incident field is
fw)= &,(~)40)+ f(a= 6 2 , < 9 P ( O ) +
&2(9)f(O) 422<9)f(O)
+ P,?W) + c,(a
(7.153a) (7.153b)
TIME-DOMAIN SOLUTIONS
449
where the scalar chain parameters are given by
Moving the forcing functions produced by the incident field to the left-hand side of (7.153) gives
If we take the inverse Fourier transform of this result, we see that the incident-field-excited line shown in Fig. 7.18(a) can be characterized as the cascade of the unexcited line and independent sources representing the eflects of the incident field, VFT(U, t ) and I F T ( 9 , t), as shown in Fig. 7.18(6) where
Therefore, the basic problem is to determine the time-domain representations of the incident field forcing functions as in (7.156). The method of characteristics was used previously to derive relations between the voltages and currents at one end of the line in terms of the voltages and currents at the other end delayed by the one-way time delay and the longitudinal and transverse components of the incident electric field and are given in (7.147) and (7.152). We now prove, by transforming the above frequency-domain result to the time domain, that the correct derivation was obtained. Substituting the chain parameters given in (7.154) into (7.153) yields
(7.157b)
(We have multiplied the second equation by the characteristic impedance, Z,,
450
INUDENT-FIELD EXCITATION OF THE LINE
“ti
&(I)
V’@, r
. I
- T)- Zcf’(&? r - T)
Fdr)
-
V(0, I
- T)t
ZCf(0, r
- T)
(4 FIGURE 7.18 A generalized method of characteristics model of a two-conductor line
with incident field illumination: (a) the line, (b) lumping the effects of the incident field as sources, and (c) representing the unexcited portion with time-delay-controlledsources.
Recognizing the basic time delay: (7.159)
TIME-DOMAIN SOLUTIONS
451
where w
B--V
(7.160a)
and the line one-way delay is again 9
T=-
(7.160b)
V
these transform to the time domain as
The equivalence of these results to those obtained with the method of characteristics and given in (7.147) and (7.1 52) can be demonstrated by substituting the frequency-domain expressions for GT(9)given in (7.84) and fFT(9) given in (7.85) into (7.158) to yield
(7.162b)
Comparing these to (7.147) and (7.152) we observe the equivalence between the two results. We next derive an explicit result for the time-domain voltages at the ends = R L f ( 9 ) .Substituting of the line for resistive loads, p(0) = -R,f(O) and Q(9) these terminal constraints into (7.158) yields the terminal voltages as
452
INCIDENT-FIELD EXCITATION OF THE LINE
where the reflection coefficients are given by (7.164a) (7.164b) The time-domain solution is obtained from this result by recognizing the basic result given in (7.159) as
These are implicit relations in that the value of a solution variable, V(0,t), depends on the value two one-way line delays earlier, V(0,t - 27'). An explicit recursion relation can be obtained with the time-shift or diflerence operator, D, as was done in Chapter 6: D*'fS(t) = f(t f k T )
Thus the results in (7.165) become,
(7.166)
453
TIME-DOMAIN SOLUTIONS
where
a = r,r,
(7,168)
Carrying out the long division and substituting (7.166) gives RS V(0,t ) = ___ Rs + zc
(7.169a) {(r~ - l)Ch,(a,t - T ) + a h T ( 9 , t - 37') + a 2 b , ( 9 , t - 57') + - .I + (rL+ l ) Z C [ f P T ( st ,- T) + a f F T ( 9t, - 37") + a 2 f f T ( 9 t, - S T ) + - .]} (7.169b) + a VFd9, t - 2 T ) + a2V,,(9,t - 4T)+ . + Zc[fF,(S, t ) + a f F T ( st, - 2T)+ a 2 f F T ( st,- 4T)+ -1 - rs[V,,(3',t - 2T)+ aV,,(9, t - 47') x a2 V , T ( 9 ,t - 6T) + -3 + r S Z C C I F T W , - 279 + d F r ( 9 , t - 47') + a 2 f F r ( 9t, - 6T) + .I}
RL V ( 9 ,t) = { C bd9, t) RL
+ zc
*
a ]
e
* *
Thus the terminal voltages are weighted sums of the functions produced by the bT(3', t ) and 4,(9) e f F , ( 9 , t), delayed in time by incident field, multiples of the line one-way delay, T. Thus the basic problem here is again to determine the frequency-domain to time-domain transformations of the functions representing the effect of the incident field given in (7,156).Other series expansions for the time-domain solution are given in [S, 181.
cT(9)
7.3.1.3 Uniform Piane-Wave Excitation of the Line Although the above results are valid for any time form of the incident-field excitation, a useful form is that of a uniform plane wave. Consider a two-conductor line shown in Fig. 7.8 with line conductors located in the x-z plane at x = 0, y = 0 and x = d, y = 0 and extending from z = 0 to z = 9,The frequency-domain forcing functions are given in (7.84)and (7.85). We consider a uniform plane wave incident on the line whose frequency-domain representation is
&(x, y, z, o)= &(w)(e,d,
+ eydy+ e,c5,)e-'P"~e-'~~ye-'B~z (7.170)
where the components of the incident electric field vector along the x, y, and z axes of the rectangular coordinate system describing the line are e, = sin 0, sin 0,
- cos 0, sin dP 0, COS 0, sin 4, + COS OB COS 4,
ey = -sin 0, cos 0, cos 4, e, = -sin
I
(7.171a)
451
INCIDENT-FIELD EXCITATION OF THE LINE
The angles are with reference to Fig. 7.6. The components of the phase constant along those coordinate axes are
p, = - p COS e, py = - p sin 0, COS 4, 8, = -/I sin 0, sin 4,
I
(7.171b)
In the time domain, (7.170) translates to cA.1J
df(x,y, z,t )
-
(e$,
+ eydy+ e,d.)t&( t - - - - - -') vx
vy
(7.172)
vz
The time form of the electric field is denoted by do@)where do@) c+ ko(o), and the velocities of propagation along the axes are denoted by
(7.173) 0
V , P - S
p,
- sin 0, sin 4, V
The frequency-domain forcing functions, given in (7.84) and (7.85), become
(7.1 75)
TIME-DOMAIN SOLUTIONS
455
FIGURE 7.19 The pulse function representing the effect of propagation in the crosssectional plane.
The time-domain forms can be obtained by first noting that
(7.176) is equivalent, in the time domain, to the pulse function CA.2, A.31 0
Tx
t<--
2
(7.177a)
[o
f > T, 2
where
z=-d
(7,177b)
vx
is the transit delay from one conductor to the other as illustrated in Fig. 7.19. Therefore the time-domain forcing functions, G T ( 9 t, ) and I F T ( 9 , t) given in (7.174)and (7.175), are the convolution of the pulse function and the functions I
rf
V(t) = 2 / e x p o ( t
+ T - $)+ tifo(t - T-
2) -
%&(t
- T, -
:)]
(7.178)
456
INCIDENT-FIELD EXCITATION OF THE LINE
f(t) =
L{-e,[.(t 2ZC
tT-
+)-
b.(t
-T-
+)]
(7.1 79)
as
where the transit time along the line of the z component of the wave propagation vector is
c = Y-
(7.182)
0,
and the transit time of the waves on the line is
Y
T=-
(7.183)
0
The transit time of the wave in the cross-sectional plane:
d T, = -
(7.184)
0 ,
is normally small because of the requirement that the line cross-sectional dimensions be electrically small in order for the transmission-line model to be valid, so we can usually disregard this time delay. When v, = f v as when the wave is tra,veling solely in the f z direction,'one term in (7.178) and (7.179) appears to be undefined. However that term also contains u, = 00 so the term becomes zero.
TIME-DOMAIN SOLUTIONS
457
Observe that if the line cross section is sufficiently small over the spectrum of &(o)then the pulse function, p(t), approximates an impulse function: p ( t ) r d(t)
small T,
(7.185)
and the forcing functions are approximated by
(7.186) (7.187) If the propagation vector has no component in the x direction, flX= 0 or u, = 00, then (7.186)and (7.187)are exact. Again, in order for the transmissionline model to be valid, the line cross-sectional dimensions must be electrically small at all frequencies of interest, Thus, as a practical matter, we may assume that (7.186)and (7.187)are valid and may also omit the cross-sectional time delay, T, = d/u,, from the functions in (7.178) and (7.179). In the case of a conductor located at a height h above an infinite ground plane as illustrated in Fig. 7.10, the above functions are easily modified. The total incident electric field (the incident field plus the field reflected from the ground plane) is given in the frequency domain in (7.112). The frequency domain forcing functions given in (7.84) and (7.85) depend on the z and x components of the electric field, Substituting (7.112) (x = h and y 3:0) gives
[
= -J2Pxh -]&(w)eze-’”’.
(7.188a)
0
This compares to the case with no ground plane:
Similarly, (7.189a)
This compares to the case with no ground plane:
Therefore, to convert the previous frequency-domain results to the case of a
INCIDENT-FIELD EXCITATION OF THE LINE
458
ground plane, we simply remove the factor e-JPXdl2and replace 4 2 e h in those results. The frequency-domain forcing functions, given in (7.84) and (7.85), become
Therefore the time-domain forcing functions are again given by the convolutions in (7.180)and (7.181) with TJ2 removed and the pulse function, p(r), has 4 2 replaced with h: 4 2 cs h. The explicit, time-domain series solution given for resistive loads in (7.169) can be written in a more compact form for uniform plane-wave excitation as
(7.192a)
(7.192b)
* ( [ M + ( t )+ a M + ( t - 2T) + aZM+(t- 4T) + - r,[M-(t - 279 + aM-(t - 4T)+ aZM'(t - 67') + - .I} 1 9
3
where M + ( t ) = V(t) + Z,I(t)
(7.193a)
TIME-DOMAIN SOLUTIONS
-
M - - ( t )= V(t) Z,I(t)
459
(7.193b)
{ (- + -t> - ;))[6.”
r A e,
e#(
+
2
+ -
-
and the loop area of the line is denoted by
A=d9
(7.193~)
In the case of a ground plane, replace d o 2h in M*(t) and remove TJ2. 1.3.1.4 Electrically Short f ines Simple frequency-domain results were obtained in Section 7,2.5.4 for a line that is electrically very short, i.e., S << A = u/f, This simple but approximate model, illustrated in Fig. 7.11, can be used to obtain a simple approximate model for the time-domain results. Essentially we must require that the significant spectral components of the waveform such as that of a uniform plane wave, g0(t),lie below this frequency. In the time domain, this requires that, for example, for a trapezoidal waveform the pulse rise/fall times be much longer than the line’s one-way delay, i.e., r,., rf >> T. If this is the case, the frequency-domain forcing functions for a uniform plane-wave excitation given in (7.135) approximate to
(7.194a)
/ . ( : ) 2 ’
r -ce,A
Jo&(o)
(7.194b)
where the area of the loop formed by the line is denoted as A = d 9 . Recognizing that d jwe(7.195) dt gives the time-domain values of these sources as (7.196a) (7.196b)
460
INCIDENT-FIELD EXCITATION OF THE
LINE
For the case of a ground plane, replace the loop area with A = 2 h 9 , The time-domain terminal voltages of the line are obtained from the circuit of Fig. 7.11 as
V(0,t ) =
- Rs RS+ RL ,&:(
t)
+
$::;,IF($,
t)
(7.197a)
Again, if the limits on validity of this result are observed (rise/fall times much longer than the line one-way delay), reasonably accurate time-domain predictions can be obtained with minimal computational effort. 7.3.1.5 A SPICE Equivalent Ciwit In this section we will describe a simple SPICE model for computing the response of a lossless, two-conductor transmission line to an incident uniform plane wave. This will be extended to the MTL case in subsequent sections. The frequency-domain chain parameters are given in (7.153):
These illustrate that one model of the line consists of the unexcited line in series with a voltage and a current source representing the effects of the incident field as illustrated in Fig. 7.18(b). Thus one form of a possible SPICE model consists of the usual unexcited line alreadiy available in SPICE in cascade with the as illustrated time-domain sources, &(A?, t ) cs &(9)and IF#, t ) cs &,(9) in Fig. 7.18(c). We now examine this possible model. These time-domain sources are derived, for an incident uniform plane wave, as the convolution of the pulse function, p(t), and the functions V(t) and I ( t ) as shown in (7.180) and (7.181). It was pointed out there that so long as the line cross-sectional dimensions are significantly less than one wavelength at the significant frequencies of the spectrum of the incident waveform, dfO(t),the pulse function approximates an impulse function, and we obtain, with good approximation, (7.198a) (7.198 b) where V(t)and I(t) are given in (7.178) and (7.179). However, this implementation has a significant problem in that both of these require that we obtain &#) advanced in time as do(t+ T)! The other functions of 4(t)in V(t)and I ( t ) require only that we delay 4(t)85 8Jt T ) and 4(t - T,)which can be accomplished by simply applying do(t)to an ideal matched line having the appropriate time
-
TIME-DOMAIN SOLUTIONS
461
delay. The ideal delay line is readily available in SPICE but one to advance a function in time is not. So we next address a more viable model. Another possible structure for the SPICE model results from the form of the solution given in (7.161):
- T ) - Z,1(9, t - T ) ] + &(t) vy,t) + z c w ,t ) = [V(O,t - T ) + ZJ(0, t - 2.91+ V&) V(0,t ) - ZcI(0, T ) = [V ( 9 ,t
(7.199a) (7.199b)
where
W )= - c V F T ( 9 , t - T ) - Z&(9, s -[V(t - T ) - Z,Z(t
- 773
t
- 771
(7.200a)
(7.200b)
Observe that these sources only require that we obtain delayed versions of 8#). This suggests the usual model of an unexcited line containing time-delayed voltage sources. The problem is that we do not have access to internal nodes of the usual SPICE transmission model so that we may add the sources in (7.200) in series with the ones already present (see Fig. 5.8) according to (7.199). The remedy here is simply to build the model using ideal delay lines and controlled sources. Such a model is shown in Fig. 7.20.The external terminals of the line are denoted as 101 and 202 with the 0 node being common. The controlled sources in that model are obtained as E,(t) = V(2) - d[ex
- e*(
49 )][V(l00) h(T,+ T )
- V(6)]
(7.201a)
where the usual SPICE designations for the voltage of node n is denoted as V(n). The reader can observe that the circuit of Fig. 7.20 with the controlled source parameters given in (7.201) yields equations (7.199) and (7.200) and is therefore an exact representation. The characteristic impedances Zcl, Zcz,Zc3, Zc4, Z,, in the five auxiliary delay lines need not equal the characteristic impedances of the original line, Z,,but each line must be appropriately matched
462
INCIDENT-FIELD EXCITATION OF THE UNE
@I
-
a 'W I-1 A
@ FIGURE 7.20 The
I
SPICE model representing incident-field illumination of a two-
conductor line. to its chosen characteristic impedance to prevent reflections on that line and provide for an ideal delay. The velocities of propagation and line lengths (or equivalently the line time delays) must be as shown. If the incident wave has no z component of its propagation vector, Le., it is propagating solely in the
TIME-DOMAIN SOLUTIONS
463
x-y plane, then u, = 00 so that = 0 and the last two delay lines in Fig. 7.20 are removed. Similarly, if the wave is propagating solely in the z direction, T, = f T,then one term in (7.201) is removed since it also contains u, = co. This basic model will be extended to the MTL case in a subsequent section by decoupling the MTL equations. 1.3.7.6 Computed R e d s In order to illustrate the accuracy of the SPICE model and to provide an understanding of the relationship between the time-domain parameters of the incident field waveform, cB,(t), such as rise/fall time we will consider the example shown in Fig. 7.21.A wire of radius rw = 10 mils and length Y = 1 m is suspended at a height h = 2 cm above an infinite ground plane. The terminations are resistive with Rs = 500 Q and RL = lo00 Q. A uniform plane wave is incident from above (e, = 0, e, = 1, u, = -u, u, = 00) and has a 1 V/m amplitude and various risetimes. The line characteristic impedance is Z, = 303.347 82 n and the line one-way delay is T = 3.33564 ns. We will consider three cases for the risetime: T, = 50 ns, T, = 10 ns, T, = 1 ns. This will illustrate cases where the risetime is greater than or less than the line one-way delay.
A 2 cm
v
-------
ALE1 V/m
r,
(b)
t
FIGURE 7.21 Characterization of a two-conductor line for illustration of numerical results.
464
INCIDENT-FIELD EXCITATION OF THE LINE
We will compaic the predictions of four models: 1. The SPICE model described in the previous section and illustrated in
Fig. 7.20. 2. The series solution given by (7.192)using seven terms. 3. The time-domain to frequency-domain transformation method described in Chapter 5. 4. The finite difference-time domain (FDTD) method also described in Chapter 5. The FDTD method will be discussed in more detail for the case of incidentfield illumination in a subsequent section. For the SPICE model, the conductor separation, d, in (7.201) should be replaced by 2h since the problem is above a ground plane for reasons discussed previously. This also applies to the series solution parameters given in (7.193).Also because u, = 00, and u, = -u, the terms containing these in (7.201)are f 1, and thus lines 4 and 5 containing u, in Fig. 7.20 are removed from the model. The time-domain to frequency-domain transformation technique simply views the problem as a two port with r&(t) as the input and V(0,t) as the output. The incident waveform, &(t), is modeled as a 1 MHz, periodic trapezoidal waveform with equal rise and fall times. The frequency of this waveform is sufficiently long with respect to the transient behavior of the result that the response to the leading edge should be the same as to the actual waveform. The frequency-domain transfer function was computed at 500 harmonics of the basic repetition rate of 1 MHz using the frequency-domain code 1NCIDENT.FOR described in Appendix A. These were combined with the spectral amplitudes and phase angles of the periodic waveform as described in Chapter 5 using the code TIMEFREQ.FOR that was used earlier and is also described in Appendix A. In fact, this process is no different from the case where the excitation resides in a termination; the frequency response contains this information. The FDTD solution divides the line into NDZ discrete sections and the time variable into NDT divisions. The FDTD results were obtained with the FDTDINCFOR code described later. The transmission-line equations in (7.139)are discretized, and the solution is obtained recursively. Again the implementation of the FDTD method is virtually identical to the case of no incident-field excitation described in Chapter 5. Figure 7.22(a) shows the result for a risetime of rr = 50 ns. The SPICE model, the series result, and the time-domain to frequency-domain transformation method using 500 harmonics all give virtually identical results. The time-domain to frequency-domain transformation method using only 100 harmonics is not so accurate. As discussed in Chapter 5, the high;frequency behavior of this waveform rolls off at -40 dB/decade above l / k q = 6.37MHz. Although the final harmonic used, 100 MHz, seems well above this, we evidently require more harmonics to give adequate time-domain predictions with this method. The
465
TIME-DOMAIN SOLUTIONS
Incident Ulriform Plane Wave
1
-2
1
.
.
, ._ . -- -. (risetime . ?,,SO ns)
,
I
1
.
ViO) SPICE model ..... ... V(0) clerles aolution, 7 terms
-----
.----......
-
.-I-.-..-I-.-.-
1I
'
.
25
0
-
.
V(0) time-frequency, 500 terma V(0) time-frequency, 100 terms V(0) V(0) PDTD (NDZ = = 2, NDT 1000) 1000)
.
.
..
1
.
,I
..
.
.,
.
50
,I
..
..
,
IS
, '
100
Time (ns) (0)
Incident Uniform Plane Wave
.........
-----.-
-.-
"
~
V(0) SPICE model V(0) seriea rolulion, 7 terma V(0) time-frequency, SO? terms V(0) time-frequency, 100 terms V(0) FDTD (NDZ 5, NDT IOOO)
- -
I
-10
0
'
~
~
5
'
~
10
~
15
~
.
20
'
Time (nu)
'
'
25
~
'
'
30
r
3 I
.
35
'
~
~
40
.
'
.
(h)
FIGURE 7.22 Predictions of the time-domain near-end voltage of the line of Fig. 7.21
using the SPICE model, the series solution, the time-domain to frequency-domain transformation method, and the FDTD method for an incident uniform plane wave with a risetime of (a) 50 ns, (b) 10 ns, and (c) 1 ns.
'
'
"
466
INCIDENT-FIELD EXClTATlON OF THE LINE
-
Incident Uniform Plane Wave (risetime 1 no) 2
O
\
'
.
'
I
'
'
.
I
~
~
~
1
~
~
~
I
"
V(0) time-frequency, 500 terms
Time (nr)
(4 FIGURE 7.22 (contfnued)
FDTD results using NDZ = 2 and NDT = loo0 also give accurate predictions. This is to be expected since dividing the line into two sections means each section is one-tenth of a wavelength at 30MHz. Observe that the resulting waveform appears to be related to the derioatiue of ef"(t). In fact, this is the case for this sufficiently long risetime. One can compute from the results for an electrically short line given in (7.197) an expected level of V(0, t ) E -0.889 m V which Is exactly the level computed by the other exact methods. Here the risetime is more than 10T so the result is expected. Figure 7.22(b) shows the result for a risetime of 2, = 10 ns.Again all four methods (using 500 harmonics) give virtually identical results. However, the risetime is of the order of the line one-way delay so that the result no longer resembles the derivative of rfo(t) as expected. Figure 7.22(c) shows the result for a risetime oft, = 1 ns. The SPICE model and the series solution again give virtually identical results. The time-domain to frequency-domain transformation method even using 500 harmonics is beginning to show errors, Here the high-frequency spectrum of t&(t) rolls off at -40 dB/decade above 318 MHz so we would expect the results using only 500 harmonics to be inadequate. Similarly, the FDTD solution shows some ringing in its solution. 7.3.2
Multiconductor lines
We now apply many of the notions developed for two-conductor lines to the case of lossless MTL's. The primary method will again be to uncouple the MTL equations via similarity transformations. This yields n uncoupled two-conductor
~
l
TIME-DOMAIN SOLUTIONS
467
lines which can be modeled using all of the previous techniques. In particular, a simple SPICE model will be developed. In addition, there are several other direct methods that will be discussed: lumped-circuit iterative models, timedomain to frequency-domain transformations, and finite difference-time domain (FDTD) methods. 1.3.2.1 Decoupling the MTL Equations Diagonalization of the MTL equations as in Chapter 5 is a viable method for lossless lines. Consider the MTL equations for a lossless line:
a V(2, t ) = -L a I(2, t ) + V,(z, -
t)
(7.202a)
a I(2, t ) = -c a V(2, t ) + I&, -
t)
(7.202b)
at
a2
at
a2
Xn a fashion virtually identical to the technique of Chapter 5, define the transformation to mode quantities as (7.203a) (7.203 b) Substituting these into (7.202) gives (7,204a) (7.204b) where L,,,and C, are the n x n matrices
L, = TF'LT, C, = TF'CT"
(7.205a) (7.205b)
and the incident-field forcing functions for the modes become
If T, and T, can be chosen such that L,,, and C, are diagonal matrices, then the equations become uncoupled sets of two-conductor lines each with incidentfield excitation through elements of the vectors VpM(z,t ) and IF,@, t). This can
468
INCIDENT-FIELD EXCITATION OF THE LINE
always be done for lossless lines as was shown in Chapters 4 and 5. So the basic solution technique utilizes the solution for a field-illuminated, lossless two-conductor line which was obtained previously. As discussed in Section 5.2.1.2, a suitable transformation can @efound which accomplishes this. This transformation is obtained by first determining a real orthogonal transformation U that diagonalizes C as u t u =e2 (7.207) where O2 is a diagonal matrix and U' denotes the transpose of U. Since C is real and symmetric, this can always be done. Furthermore, since C is positive definite, all elements of O2 are real and positive so that we can form the square root of that matrix, 8, which will have real elements on its main diagonal and zeros elsewhere. Next find a real orthogonal transformation, S,such that s ' ( e u w e ) s = A?
(7.208)
where A2 is again a diagonal matrix with real positive elements on its diagonal. Define T = U8S (7.209) Normalizing the columns of T to unit length gives
where a is an n x n diagonal matrix. The above transformations can now be defined as T, = uesa (7.211) with the property that
TY= UB'lSa'l
(7.212)
T; = a2T; 'C
(7.213)
= T:
The modal per-unit-length parameter matrices given in (7.205) become (7.214a)
(7.214 b)
TIME-DOMAIN SOLUTIONS
469
Consequently the mode characteristic impedances and velocities of propagation become C z ~ m l i zcmi (7.21Sa)
=&
= a,2Al Urn{
3
1 -
Ji;nlcmr
(7.2 15 b)
1
a-
Ai
In order to obtain the time-domain solutions, we again transform the frequency-domain chain parameter matrix to the time domain. The frequencydomain chain parameter matrix relates the phasor voltages and currents at one end of the line to those at the other as
where the total forcing functions are given by
and the chain parameter submatrices are @ll(U) = )C-'T,(eJaA9 + e-JmAY)T~'C
(7.21Sa)
612(9) = -fC-'T,A(eJ"PA"- e-JcuAY)Ti'
(7.218b)
&,(U) = -iT,(e'oA9
(7.2 18c)
+
- e-JmA9)A-'Ti 'C
&(9)= iT,(eJaAg e-JmAg)T;'
(7.2 18d)
470
INCIDENT-FIELD EXCITATION OF THE LINE
where the modal chain parameter submatrices are (7.220a)
(7,2206)
(7.220~)
(7.220d)
and the total modal forcing functions due to the incident field become
We now obtain the time-domain version of the chain parameter representation. This will show how to construct a timadomain equivalent circuit that is implementable in the SPICE program. Substituting (7.220) into (7.219) yields 1.3.2.2 A SPICE Equivalent Circuit
TIME-DOMAIN SOLUTIONS
471
Adding and subtracting these yields ?,,,(O)
- ZC,,,f,,,(O)= e-'""g[vm(9)
- Zcmlm(9)]+ a0(9) (7.223a)
q,,,(U)+ Zcmi,,,(9)= e-'''A2[8m(0) t Z,f,,,(O)]
+ &,(9) (7.223b)
Recognizing the basic time-delay transformation:
these become, in the time domain,
where we denote the i-th entry in a vector V as [VI,, and the one-way time delay of the i-th modal line is denoted by (7.227) =A i 9
The additional sources are
Equations (7.226) suggest the usual SPICE equivalent circuit for the modes where we add to it the sources due to the incident field, [Eo(t)Irand [E9(t)],. The transformation back to the actual line voltages and currents is accomplished
472
INCIDENT-FIELD EXCITATION OF THE LINE
with controlled sources that represent the mode transformations according to (7.203) : CV(Z, t ) ~ i
{ ~ ~ v ~ i k ~ v r~n) (I zA,
(7.229a)
{[TI' 11ik~1(z, ~ > I J
(7.229b)
k= 1
cIrn(z, t11t = k= 1
as in Chapter 5. Thus the only difference between the SPICE model derived for the case of the excitation in the terminal networks and this case, wherein the excitation is via an incident field, is the implementation of the sources due to that field given in (7.221) and (7.226), Although the above is valid for any time-domain form of the incident field, we will now restrict our attention to uniform plane-wave excitation of the line. The modal forcing functions in (7.224) become
(7.230b)
where $, denotes the vector of contributions from the transverse electric field, and @' denotes the vector of contributions from the longitudinal electric field. If the reference conductor is placed at the origin of the coordinate system, x = 0, y 3 0, as shown in Fig. 7.7(a), the transverse field contributions can again be written in terms of the cross-sectional coordinates of the k-th conductor ( x k , yk) using the general form of the incident field in (7.58) as (7.231a)
is the straight-line distance between the reference where dk ,/= conductor and the k-th conductor in the transverse plane and 5
TIME-DOMAIN SOLUTIONS
473
Similarly, the contributions due to the longitudinal field are CeL(z)lk
= Eli(xk, yk, 2)
- $:(Os
(7.232)
0,Z)
Denoting the cross-sectional time delay as (7.233) and (7.234) along with the time delay of the modes:
T=E=A18
(7.235)
Vl
(7.230) becomes
(7.236a)
(7.236b)
I
Again, recognizing the basic time-delay transformation in (7.225), these sources transform to the time domain as
X
(T +
n
J
I
where pk(t) is the pulse function for the k-th conductor given by
P&(t) =
0
t<
--Txyk 2
1 Txyk
0
t>
Txyk 2 Txyk 2
(7.238)
--
as illustrated in Fig. 7.19 and * denotes the convolution operation, Again, let us assume that the line cross-sectional dimensions are electrically small at the significant frequencies of the waveform, (ao(t), so that pk(t) approximates an impulse: pk(t) d(t) (7.239) Similarly we will neglect the cross-sectional time delay in the expressions involving tifO(t),TXykY 0 for all k, so that the forcing functions simplify to
(7.240a) (7.240b) where the entries in the n x 1 vectors a. and ag are given by
Therefore, the modal expressions in (7.226) simplify to
TIME-DOMAIN SOLUTIONS
475
Equations (7.242) suggest the SPICE model shown in Fig. 7.23. The external Nodes 30i, 40i, 50i, and nodes are denoted as 1Oi (at z = 0) and 20i (at z = 9). 60i attach to controlled sources which implement the transformations between actual and mode quantities given in (7.229). The remainder of the network implements (7.242) for the i-th mode. The five delay lines simulating equation (7.242) for each mode need not be terminated in the mode characteristic impedances but can be terminated in any characteristic impedance in order to remove reflections and give the desired ideal delay of 80(t).If the wave has no propagation component in the t direction, = 0, then the last two lines are removed from the model. If the wave has a component in the + z direction, it may happen that 1; = T, in which case (7.242b) appears to be undefined. However, this term becomes
(G - T,)
d dt
= - f&(t
TI+ TI
- q)
(7.243)
so that the last delay line is replaced as shown in Fig. 7.24. Waves having a component in the - 2 direction, may result in the last two delay lines having negative delays which SPICE does not allow. Thus we must restrict this model to positive T,. For negative T, simply reverse the ends of the model. This model is implemented via a SPICE subcircuit generated by the FORTRAN program SPICEINC.FOR described in Appendix A. 1.3.2.3 Lumped-Circuit Iterative Approximate Characterizations The use of lumped-circuit approximations such as the lumped 1, n, and T circuits parallels all of the previous such uses. The lumped n and T circuits shown for the phasor, frequency-domain solution in Fig. 7.5 can be used directly for the time-domain solution by simply replacing the phasor sources, &,Sand with their time-domain equivalents, and I&'.
r,
7.3.2.4 Time-Domain to Frequency-Domain Transformations Perhaps the most straightforward method of obtaining the time-domain response is to view the problem as a two port with 4(t)as the input and the desired terminal response voltage (or current) as the output. The magnitude and phase of 4(t)can be obtained with the Fourier transform or can be approximated as a periodic
476
INCIDENT-FIELD EXCITATION OF THE UNE
I Transformation
'
Modit linea
I FIGURE 7.23
Tran8fknation
-
The SPICE model for a MTL with incident-field illumination.
TIME-DOMAIN SOLUTIONS
FIGURE 7,24
477
Replacement of one of the sources in Fig. 7.23 for the special case where I direction and the mode velocity equals that of the
the propagation is solely in the incident wave.
waveform with a sufficiently long period. The frequency-domain transfer function can be obtained with the previous methods such as the code 1NCIDENT.FOR described in Appendix A. The magnitudes of the spectral components of &&) are multiplied by the magnitudes of the transfer function and the phase angles of 8#) are added to the angles of the transfer function to produce the magnitude and phase of the response in the usual fashion. Then the inverse Fourier transform can be used to convert this back to the time domain. The method is straightforward and can handle skin-effect losses. However, it relies on superposition so that the two port must be linear. This again means that the line terminations as well as the surrounding medium must be linear. Therefore nonlinear loads as well as corona breakdown in lightning studies cannot be handled with this method.
fi
1.3.2.5 Finite Difference-Time Domain Methods Another straightforward way of solving the MTL equations is with the finite difference-time domain (FDTD) method described in Chapter 5, The FDTD discretization of the transmissionline equations with incident-field illumination was obtained in Chapter 5. Other FDTD derivations and discretizations are given in [8] and C19-211. The space and time derivatives are discretized as Az and At, respectively, giving a recursion relation in the form of a set of difference equations. The transmission-line equations in (7.18) become
a
- V(z, t )
8Z
a I(z, t ) = - a E,(z, t ) + E&, t ) + V;(z, + RI(z, t ) f L at az
t ) (7.244a)
478
INCIDENT-FIELD EXCITATION OF THE LINE
where the n x 1 vectors E T and E L are due to the transverse and longitudinal components of the incident electric field, respectively. The vectors V;(z, t ) and rjp(z, t ) as well as R and G are used to model lumped sources along the line such as at the terminations, as described in Chapter 5. These were given, in the frequency domain, in (7.231) and (7.232). Thus (removing sin &/$k and e''Pk)
This translates in the time domain to
(7.247a)
(7.247b)
Similarly, (7.248) The voltage and current solution points are interlaced one half-cell apart as shown in Figs. 5.21 and 5.22 for stability purposes. Similarly, the voltage and current solution times are also interlaced one half-time-step apart as shown in Fig. 5.22. Using this scheme, the difference equations become
479
TIME-DOMAIN SOLUTIONS
1
- (1;' 1'2
Az
-
1 +C(V;+' - Vi) + $G(Vi+' + Vi) At
(7.249b)
Rearranging these gives
(&L +
R)1i+3/2 =
(&L - f R)l;+ll2 - Az-(Vi:: 1
+
(6
AT
- AL)
c..(1"+312
-
(7.250b)
v,
The terminal conditions are incorporated as before. We characterize these as resistive in the form of generalized Thbvenin equivalents as V(0, t) = V,y - R,yI(O, t)
(7.251a)
+ RLl(9, t)
V ( 9 , t ) = V,,
(7.251 b)
First the line voltages are solved for from (7.250a). With reference to Fig. 5.21, we let I, = 0, G = (l/Az)R;', 1; = (l/Az)Rt1Vs along with replacing C with (1/2)C as before for k = 1 to yield V I" t l = & R ~ C + l ) - l { ( ~ : - R,C
)V; - 2Rs(I;*'/* - It+1'2) (7.252a)
-1
-RsCA,[80(t"+1) - 80(t")] + (V;+ + V;)} At AZ
l
\
480
INCIDENT-FIELD EXCITATION OF THE LINE
For k -- 2,.
. .,NDZ we obtain (7.252b)
', +
With reference to Fig. 5.21, we let INDZ+= 0, G = (l/Az)R; I; = (l/Az)R; 'V, along with replacing C with (1/2)C as before for k = NDZ 1 to yield (7.252~)
Once the voltages are determined along the line for a particular time step, the currents are obtained from (7.250b) for k = 1,. ,N D Z as
..
Once again, for stability, the position and time discretizations must satisfy the Courant condition : At S-
AZ
(7.253)
UIMAX
where v,,,,~ is the maximum of the mode velocities. This method is implemented in the FORTRAN program FDTDINCFOR that is described in Appendix A. 1.3.2.6 Computed Results The computed results compare the SPICE model, the time-domain to frequency-domain transformation, and the finite differencetime domain (FDTD) model. The structure is shown in Fig. 7.25 and consists of a 2 m length of ribbon cable. The wire radii are 7.5milsJ the dielectric
TIME-DOMAIN SOLUTIONS
481
(a)
"t
I*=
7.5 mila
t = 10 mils
33
f50 mila
t
FIGURE 7.25 A
three-wire line for illustrating numerical results.
thicknesses are 10 mils and the dielectric constant is 3.5. The wires have center-to-center separations of 50 mils. The middle wire is chosen as the reference conductor, and a uniform plana wave is incident in the z direction with the electric field polarized in the x direction. The line is terminated in 500 Q resistors giving R,=RS=[
]
500 0 0 500
The time form of the incident electric field, c&(t), is a ramp waveform rising to a level of 1 V/m with various risetimes T, = 100ns, 10ns, and 1 ns. It is important to note that the incident field should be computed as the field with the conductors removed. Thus, in computing this incident field we should
482
-
INCIDENT-FIELD EXCITATION OF THE LINE
Incident Uniform Plane Wave (risetime 100 ns)
---
......... 9-
Vl(0) SPICE model Vl(0) time-frequency (500 terms) VI(0) time-frequency (100 terms) Vl(0) PDTD (NDZ= 1, NDT = 2000)
D
Time (na) (a)
1 - 2 1 . . '
-
Incident Uniform Plane Wave (riaetime 10 ns) I
.
"
I
.
"
I
.
.
-
.
1
Vl(0) time-frequency (500 terma)
V1(0) time-frequency (100 termc) *-.-.-e-
Vl(0) FDTD (NDZ
5, NDT
1000))
Time'(nr)
e3 FIGURE 7.26 Predictions of the time-domain near-end voltage of the line of Fig. 7.25 using the SPICE model, the series solution, the time-domain to frequency-domain transformation method, and the FDTD method for an incident uniform plane wave with a risetime of (a) 100 ns, (b) 10 ns, and (c) 1IM.
TIME-DOMAIN SOLUTIONS
483
-
Incident Uniform Plane Wave (risetime 1 ns)
0
25
so
100
Time (nc) (C)
FIGURE 7.26 (continued)
leave the dielectric insulations in place. However, to simplify the problem we will assume the incident field to be that with the conductors and their insulations removed. Figure 7.26(a) shows the results for the 10011s rise time. All three models show excellent correlation. The time-domain to frequency-domain transformation simulated the B”,(t) waveform as a 1 MHz trapezoidal pulse with a SO% duty cycle and equal rise and fall times. The results were obtained using 100 harmonics and 500 harmonics. The FDTD results were obtained by discretizing the line into N D Z = 1 sections of length Az = 9 / N D Z = 2 m length and the time into NDT = 2000 time steps of At = final time/NDT = 0.1 ns. The spectrum of &#) rolls off at -40 dB/decade above l / q = 3.18 MHz. Each section is All0 at 15 MHz which is well above the point where the waveform spectrum begins to roll off at -40 dB/decade so the results are not surprising. The results for the 1011s risetime is shown in Fig. 7.26(b). For the 1Ons risetime, all three models give excellent correlation. The spectrum of 4(t)rolls off at -40 dB/ decade above 1/mr = 31.8 MHz so the time-domain to frequencydomain transformation method using only lo0 harmonics gives reasonably accurate results. The FDTD method uses NDZ = 5 and N D T = 1OOO. Each section is A/lO at 75 MHz which is well above the point where the waveform spectrum begins to roll off at -40dB/decade so the results are again not surprising. The results for the 1 ns risetime are shown in Fig. 7.26(c). The spectrum of 8#) rolls off at -40dB/decade above l / n ~ ,= 318 MHz so the time-domain to frequency-domaintransformation method using 500 harmonics gives reasonably accurate results. The FDTD method uses NDZ = 20 and
484
INCIDENT-FIELD EXCITATION OF THE LINE
..
Incident Uniform Plane Wave 900, er = 908, 4p = -900)
(ea
-..***.a*
Vl(0) lolry
Frequency (Hr) (1)
Incident Uniform Plane Wave . I90°, .$r = - 9 O O )
(eE7 90', er
90
4s A
V
ti $
0
V
3
4
-4s
-
90 103
104
lo"
10'
10'
10'
Frequency (Hr) (b)
FIGURE 7.27 Predictions of the frequency response of the near-end voltage of the line of Fig. 7.25 using the SPICE model, and the MTL model with and without losses: (a) magnitude, (b) phase. Incident uniform plane wave, Os = 90°, 8, = 90°, 4, = 90".
TIME-DOMAIN SOLUTIONS
1 . 2 . 7 . .
.,
1
9
I
1
.
,
. , . .
- Vl(0) time-frequency lorsles~(1000 terms)
......-+.Vl(0) time-frequency
1.0 8
485
lossy (1000 terms)
0.8 '
-8
'i 3
0.6
0.4
'
-
i
0.2 -
o . o o . .
"
'
'
*
/
'
*
> '
. -
Time (n8)
Predictions of the time-domain near-end voltage of the line of Fig. 7.25 using the time-domain to frequency-domain transformation method with and without losses for an incident uniform plane wave with risetime of 1 ns. FIGURE 7.28
NDT = 1OOO. Each section length, Az -- 0.1 m, is 2/10 at 300 MHz which is of the order of the point where the waveform spectrum begins to roll off at -40 dB/decade, so some ringing on that prediction is expected. The SPICE model is not restricted to the time domain and can be used to give frequency-domain results although restricted to lossless lines. These are shown in Fig. 7.27 from 1 kHz to 200 MHz. The results of the frequency-domain direct calculation with and without losses are obtained using the code 1NCIDENT.FOR. All three models give virtually identical results except around frequencies where the line is some multiple of a half-wavelength. The effect of losses on the time-domain results can be investigated by computing the results using the time-domain to frequency-domain method using the TIMEFREQFOR program described in Appendix A. The frequencydomain transfer lunction is computed with and without losses using the program 1NCIDENT.FOR. The time-domain results for 1 ns risetime are compared in Fig. 7.28 using lo00 harmonics of the 1 MHz waveform. The highest spectral component, 1OOO M H g is a factor of 3 higher than the point the spectrum rolls off at -40 dB/decade, l/m, = 318 MHz, so the excellent predictions are to be expected. The results with and without losses are virtually identical. This is not to say that losses are always unimportant, but for the particular load-impedance level, line dimensions, and spectral content of the waveform used here, they are apparently not significant.
486
INCIDENT-FIELD EXCITATION OF THE LINE
REFERENCES
c11 C.D. Taylor, R.S. Satterwhite,and C.W. Harrison, ”The Response ofa Terminated
Two-Wire Transmission Line Excited by a Nonuniform Electromagnetic Field,” IEEE 7’rans. on Antennas and Propugation, AP-13, 987-989 (1965). C2l A.A. Smith, Jr., “A More Convenient Form of the Equations for the Response of a Transmission Line Excited by Nonuniform Fields,” IEEE Duns. on Electromagnetic Comparibility, EMC-15, 151-152 (1973). c31 K.S.H. Lee, “Two Parallel Terminated Conductors in External Fields,” IEEE Trans. on Electromagnetic Compatibility, EMC-20, 288-295 (1978). S. Frankel, “Forcing Functions for Externally Excited Transmission Lines,” IEEE c41 7’rans. on Electromagnetic Compatibility, EMC-22, 210 (1980). c51 A.A. Smith, Coupling of External Electromagnetic Fields to Transmission Lines, 2d ed., Interference Control Technologies, 1987. E61 F.M. Tesche, T.K. Liu, S.K. Chang, and D.V. Giri, “Field Excitation of Multiconductor Transmission Lines,” Technical Report AFWL-TR-78-185, Air Force Weapons Lab, Albuquerque, NM, February 1979. ~ 7 1 G.W. Bechtold and D.J. Kozakofl, “Transmission Line Mode Response of a Multiconductor Cable in a Transient Electromagnetic Field,” IEEE Trans. on Electromagnetic Compatibility, EMC-12, 5-9 (1970). C81 A.K. Agrawal, H.J. Price, and S.H. Gurbaxani, “Transient Response of Multiconductor Transmission Lines Excited by a Nonunibrm Electromagnetic Field,” IEEE Trans. on Electromagnetic Compatibility, EMC-22, 119-129 (1980). c91 C.D. Taylor and J.P. Castillo, “On the Response of a Terminated Twisted-Wire Cable Excited by a Plane-Wave Electromagnetic Field,” IEEE 7’runs. on Electromagnetic Compatibility, EMC-22, 16-19 (1980). [lo] E.F. Vance, Coupling to Shielded Cables, John Wiley & Sons, NY, 1978. Ell] Y. Leviatan and A.T. Adams, “The Response of a Two-Wire Transmission Line to Incident Field and Voltage Excitation, Including the Effects of Higher Order Modes,” IEEE 7kans. on Antennas and Prppagatlon, AP-30,998-1003 (1982). [12] G.E. Bridges and L. Shafai, “Plane Wave Coupling to Multiple Conductor Transmission Lines Above a Lossy Earth,” IEEE Duns. on Electromagnetic Compatibility, 31, 21-33 (1989). [13] F.M.Tesche, “Plane Wave Coupling to Cables,” in Handbook of Electromagnetic Compatibility, Academic Press, San Diego, CA, 1994. [14] J.H. Richmond, “Radiation and Scattering by Thin-Wire Structures in the Complex Frequency Domain,” Interaction Note 202, Air Force Weapons Laboratory, Kirtland Air Force Base, Albuquerque, NM, May 1974. [15] J.H. Richmond, “Computer Program for Thin-Wire Structures in a Homogeneous Conducting Medium,” Technical Report, NASA CR-2399, National Aeronautics and Space Administration, Washington, DC, June 1974. [16] C.W. Harrison, “Generalized Theory of Impedance Loaded Multiconductor Transmission Lines in an Incident Field,” IEEE Trans. on Electromagnetic Compatibility, EMC-14, 56-63 (1972).
PROBLEMS
487
1171 F. Schlagenhauferand H.Singer, “Investigationsof Field-Excited Multiconductor Lines with Nonlinear Loads,” Proc, 1990 International Symposium on Electromagnetic Compatibility, August 1990, Washington, DC,pp, 95-99. [18] Y. Kami and R. Sato, “Transient Response of a Transmission Line Excited by
an ElectromagneticPulse,” IEEE Trans.on Electromagnetic Compatibility, EMC30,457-462 (1988). [19] D.E. Meriwether, “A Numerical Solution for the Response of a Strip Transmission
Line over a Ground Plane Excited by Ionizing Radiation,” IEEE 7 h s . on Nuclear Science, NS-18, 398-403 (1971). [20] D.F. Higgins, “Calculating Transmission Line Transients on Personal Computers,” IEEE International Symposium on Electromagnetic Compatibility, August 25-27, 1987, Atlanta, GA. [21] E.S.M. Mok, and G.I. Costache, “Skin-Effect Considerations on Transient Response of a Transmission Line Excited by an Electromagnetic Pulse,” IEEE Trans. on Hectromagnetic Compatibility, EMC-34, 320-329 (1992).
PROBLEMS
7.1
Derive the relation between the transverse and longitudinal electric fields and the normal magnetic field given in (7.9) from Faraday’s law.
7.2
Derive the result for the phasor chain parameter matrix given in (7.29).
7.3
Prove the relation given in (7.36).
7.4
Derive the relations given in (7.42) and (7.43).
7.5
Derive the relations given in (7.47) and (7.48).
7.6
Derive the relations given in (7.49) and (7.50).
7.7
Derive the relations given in (7.51) and (7.52).
7.8
Derive the result for a lossless line in a homogeneous medium given in (7.57).
7.9
Derive the relations given in (7.59).
7.10 Derive the relations given in (7.66) and (7.67). 7.11 Derive the relations given in (7.78). 7.12 Derive the relations given in (7.89) and (7.90). 7.13 Derive the relations given in (7.93) and (7.94). 7.14
Derive the relations given in (7.100) and (7.101).
7.15 Derive the relations given in (7.106) and (7,107). 7.16 Consider a two-wire line consisting of two #20 gauge (radius 16 mils) bare wires separated by 5 cm and total length 5 m. The loads are 50 R
488
INCIDENT-FIELD EXCITATION OF THE LINE
at z = 0 and 1OOOn at z 5 9. Sketch the frequency response of the voltage across each load from 1 kHz to 100 MHz for a 1 V/m incident field that has 8, = 60°, 8, = 120”,4, = -30”. Compare the exact results to the electrically short line model of Section 7.2.5.4 which predicts a 20 dB/decade variation for all frequencies. 7.17
Repeat Problem 7.16 replacing the reference wire with an infinite ground plane.
7.18
Repeat Problem 7.16 replacing the two wires with a PCB having two 10 mil lands of length 10 cm separated by 50 mils. The board is constructed of glass epoxy (E, = 5) and is 64 mils thick.
7.19 Derive the solution given in (7.163). 7.20 Derive the series solutions given in (7.169).
7.21 Derive the relations given in (7.200). 7.22 Derive the modal chain parameter matrices given in (7.220). 7.23 Derive the modal forcing functions given in (7.237)and (7.240). 7.24
Derive the functions given in (7.245)to (7.247).
7.25 Derive the difference equations given in (7.252). 7.26
Repeat Problem 7.16 but obtain the time-domain response using:
1. The SPICE model 2. The time-domain to frequency-domain model 3. The FDTD model The incident waveform is a periodic “sawtooth” wave rising from zero to its maximum in 7,= 200 ns and returning to zero in zf 50 ns. Compare these to the electrically short line model of Section 7.3.1.4. 7.27 Repeat Problem 7.26 replacing the reference wire with an infinite ground
plane. 7.28 Repeat Problem 7.26 replacing the two wires with a PCB having two 10 mil lands of length 10 cm separated by 50 mils. The board is constructed
of glass epoxy (e, = 5) and is 64 mils thick.
CHAPTER ElCHT
Transmission-Line Networks
The previous chapters of this text have considered the analysis of uniform transmission lines that have one important restriction: all (n 1 ) conductors are parallel to each other. Numerous practical configurations consist of interconnections of these types of lines as illustrated in Fig. 8.l(a). These practical configurations will be referred to as transmission-line networks. Lines may end in termination networks or may be interconnected by interconnection networks. Each transmission line of the network will be referred to as a cube after Cl-31, A convenient way of describing the overall network is with a graph as illustrated in Fig. 8.l(b) [l, 2,4]. The transmission lines are represented with single lines or branches of the graph. The termination networks are defined as a node having only one tube incident on it and are represented by rectangles. The interconnection networks are defined as a node having more than one tube incident on it and are rebresented bv-circles. The excitation for the network may be in the form of lumped sources in the termination or interconnection networks or it may be due to either distributed excitation from an incident electromagnetic field or a point excitation along the line as with the direct attachment of a lightning stroke. Point excitation of a tube as in the case of a direct attachment of a lightning stroke can be handled by characterizing the segments of the tube to the left and right of the excitation point with any of the following models and treating the point excitation as an interconnection network between these tube subsegments. Lumped sources in this interconnection network then represent this point excitation at the junction. Distributed excitation must be included in the overall characterization of the tube as described in Chapter 7, whereas lumped sources within the termination/interconnection networks are included in their description. The purpose of this chapter is to examine methods for characterizing these types of interconnected lines. Evidently, any method for characterizing this network seeks to characterize each tube in some fashion, as outlined in the previous chapters, and to interconnect these tubes by enforcing the constraints on the line voltages and
+
489
490
TRANSMISSION-LINE NETWORKS
c Termination network
#l
Termination network
r
,
-
I
#2
--
Interconnection network #4
Transmiidon line #1 (tube)
--
Termination network
c
#3
(4
(b)
FIGURE 8.1 Illustrationof a transmission-line network: (a) tube and network definitions, (b) representation with a graph.
currents via Kirchhoffs laws and the element characteristics within the termination/interconnection networks. One obvious representation method is to use SPICE subcircuit models for the tubes developed in Chapters 5 and 7 and interconnect and/or terminate the nodes of those subcircuit models in the resulting SPICE code. This method is very straightforward using the programs SPICEMTLFOR or SPICEINC.FOR described in Appendix A to provide the SPICE subcircuit models of each tube. The advantages of this method are that it is Straightforward to implement, and dynamic as well as nonlinear loading
INTRODUCTION
491
and elements within the termination/interconnection networks, such as diodes and transistors, are already available in the SPICE code and can be readily used to build the termination/interconnectionnetworks to complete the overall characterization. So a wide variety of practical terminations can be analyzed without the need for developing either the models of complicated elements or the numerical integration routines to give a time-domain analysis. The disadvantage of this method is that it is so far applicable only to lossless lines. An approximate method is to use lumped-circuit iterative models of each tube such as the lumped-pi or lumped-T models and use any lumped-circuit analysis program such as SPICE to analyze the resulting interconnection. Losses can be incorporated into this result, but the method is restricted to tubes that are electrically short. Time-domain results can again be reasonably approximated if the rise/fall times of the source waveforms are sufficiently longer than the tube one-way delays. It is also possible to construct an exact model of the line using the admittance or impedance parameter characterizations of each tube [4,5]. These methods have the advantages of simple construction of the overall equations which are to be solved to give the tube terminal voltages and currents. Losses such can be approximated as lumped ciras skin-effect losses which vary as cuits or analyzed directly by determining the frequency response of the network. With the exception of the SPICE subcircuit method, all methods ultimately must face the problem of the systematic interconnection of the tube models. Computer implementation of the interconnection of the tubes for a large network is not a simple task and must be designed so that a user can easily and unambiguously describe the interconnections to the resulting computer code. Of course all standard lumped-circuit analysis codes such as SPICE must address this problem of systematic and unambiguous implementation of the element interconnections via user input to the code, and the characterization of transmission-line networks is similar in that respect. Characterization of the tubes via the admittance parameters as in [4] was designed so that a systematic interconnection process will be effected. Another novel method is the use of the scattering parameters for the tubes [l, 21. This leads to the so-called BLT equations (apparently named for the authors). The implementation of the BLT equations directly in the time domain was described in [3]. All of the above methods must address both the frequency-domain as well as the time-domain analysis of the network. The time-domain analysis of the network can be obtained in the usual fashion using the timedomain to frequency-domain transformation discussed earlier wherein the source waveform is decomposed into its spectral components and each component is passed through the previously computed frequency-domain transfer function. The time-domain result is the inverse Fourier transform of this. As before, the time-domain to frequency-domain method can readily handle skin-effect losses that are difficult to characterize in the time domain, but it suffers from the fundamental restriction that the network must be linear, Le., the line and all terminations must be linear, since superposition is used.
z/s
492
TRANSMISSION-LINE NETWORKS
8.1 REPRESENTATION WITH THE SPICE MODEL
Perhaps one of the more straightforward methods of characterizing and analyzing the crosstalk on transmission-line networks is with the SPICE equivalent circuit developed in Chapter 5 or for incident field illumination in Chapter 7. Each tube is characterized by its SPICE subcircuit model generated with the programs SPICEMTL.FOR or SPICEINC.FOR described in Appendix A. These subcircuit models are then interconnected and the terminations added to produce the final SPICE model of the network. The method is straightforward using the above codes to generate the subcircuit models but is restricted to lossless tubes. Again, this method can handle, in a straightforward way, dynamic and/or nonlinear loads in the termination/interconnectionnetworks. In order to illustrate the methods of this chapter we will use the example shown in Fig. 8.2. The network consists of three tubes. Tube # 1 contains four wires, whereas tubes #2 and # 3 contains two wires. All tubes will consist of bare wires above a ground plane as illustrated in Fig. 8.3. Tube # 1 is of length 2 m and tubes # 2 and # 3 are of length 1 m. The cable is suspended 1 cm above an infinite, perfectly conducting ground plane, and the wires have radii of 7.5 mils. A source, b#), in network # 1 drives line # 1 of tube # 1. This source is in the form of a ramp waveform with a risetime of 1 ns as shown in Fig, 8.3(c). The tubes are terminated in various resistive terminations at termination networks # 1, 12,and #3. Interconnection network # 4 contains a variety of terminations, open circuit, short circuit, series impedance, shunt impedance and direct connection, to illustrate the versatility of the method. The desired output will be the voltage, Ku1(t),across the termination of wire # 1 of tube # 3 at termination network #3, The graph of this transmission-line network is shown in Fig. 8.l(b). Each termination or interconnection network has the number of that node included within the symbol. The number of each tube is noted on that branch of the graph. Figure 8.4 illustrates the resulting construction of the overall SPICE network with node numbering. The SPICE subcircuit models of the tubes are constructed using SPICEMTL.FOR, and the per-uni t-length parameters are computed using WIDESEP.FOR.
8.2
REPRESENTATION WITH LUMPED-CIRCUIT ITERATIVE MODELS
The next method is to approximately characterize each tube with a lumpedcircuit iterative structure such as a lumped-pi structure. These characterizations are obtained using the SPICELPLFOR code. The resulting overall SPICE model of the network is virtually identical to that of Fig. 8.4 with the only exception being that the subcircuit models of the tubes are lumped-pi structures. Figure 8.5 shows the comparison of the predictions of the output voltage, Ku1(t), obtained with the SPICE model of Fig. 8.4 and the lumped-pi structure using
REPRESENTATION WITH LUMPED-CIRCUIT ITERATIVE MODELS
493
i
Tube 2
Tu be 3
FIGURE 8.2 An example of numerical results.
8
transmission-line network to illustrate and compare
only one lumped-pi section to represent each tube. The correlation is obviously very poor due to the fact that the tube one-way delays are on the order of 10 ns which is not significantly smaller than the waveform rise time of 1 ns. Figure 8.6 shows this correlation for a risetime of 10011s which is much better.
494
TRANSMISSION-LINE NETWORKS
-
.
I.=
7.5 mils
d
50 mils
f;cm
t (4 Cross-sectional dimensions of the tubes of the transmission-line network of Fig, 8.2: (a) tuba 1, (b) tubes 2 and 3, (c) V, versus t.
FIGURE 8.3
8.3 REPRESENTATION VIA THE ADMITTANCE OR IMPEDANCE PARAMETERS
The use of the admittance parameters to characterize the tubes was described in [4]. This leads to a straightforward way of incorporating the termination and interconnection networks since we essentially need to add admittances in order to construct the admittance matrix of the overall network. The frequency-domain chain parameters of a uniform line are
+ o, f ( 9 ) = 6'210(0+)dj,,,i(O) + i,,
V ( 9 ) = 6ilv(o)+ 6&0)
v,,
(8. la)
(8.lb)
where and f,, are due to any incident field excitation of the line. The frequency-domain admittance parameters are derived in Chapter 4 from these
REPRESENTATION VIA THE ADMITTANCE OR IMPEDANCE PARAMETERS
495
FIGURE 8.4 Illustration of the SPICE model of the transmission-line network of Fig. 8.2.
(8.3b)
4%
TRANSMISSION-LINE NETWORKS
75 I
40
-
Network Roiponie (ritetime 1 ne) I
I
-
I
1
I
\
I I
\
S
$
a v
-#J
S-
I
I
!
CI
I I
\
I
4
-30-
Y
%
8
I
II l
i
\
-65-
I
/
1,
‘-1
- 100
I
/
/
SPICE model Lumped Pi model I
Time (ni) FIGURE 8.5 Comparison of crosstalk voltage at the termination of conductor # 1 of
tube # 3 for the transmission-line network of Fig. 8.2 for a risetime of 1 ns using the SPICE model and using one lumped-pi section to represent each tube.
Network Response (risetime = 100 ns)
15 10
S
5
-g
B
s
Y
.y
o
Y
E
u
-5
- 10 - 15 0
SPICE model Lumped Pi model 50
100
150
200
250
300
Time (ns)
FIGURE 8.6 The predictions of Fig. 8.5 for a risetime of 100 ns showing the adequacy of the lumped-pi representation.
REPRESENTATION VIA THE ADMlllANCE OR IMPEDANCE PARAMETERS
4!V
(8.3d) Observe that the currents are defined as directed into each end ofthe cube. The admittance parameters show that the tube is reciprical as it should be, The various parameters in these are as defined in Chapter 4 where the per-unitlength impedance and admittance parameters are diagonalized as
and the characteristic admittance matrix is
The only potential disadvantage to the admittance parameter description of the tubes is that the parameters do not exist for frequencies where the tube is some multiple of a half-wavelength. The tubes are characterized by the above admittance parameters with the following notation illustrated in Fig, 8.7(a). Consider the i-th tube connecting thej-th network and the k-th network at its end oints. Denote the vector of currents and voltages at the ends of the tube as f{, {, if,0: where the subscript denotes the tube and the superscript denotes the network at that end:
f
v:Er~ination/interconncctionnetwork
^termination/interconnection network Itube
The admittance parameters (8.2) and (8.3) become
The characterization of the termination and interconnection networks must be general enough to include open and short circuits as well as lumped source8 and impedances and direct connections within the termination/interconnection networks. As discussed in Chapter 4, a general way of characterizing these is in the form of a combination of generalized Thbvenin and generalized Norton equivalents [l-4, 6, 71. Consider characterizing the m-th interconnection network which has the i-th, j-th, and k-th tubes interconnected,by it as
498
TRANSMISSION-LINE NETWORKS
I
Tubei
I
0) FIGURE 8.7 Definitions of the tube voltages and currents for (a) an individual tube and (b) an interconnection network.
illustrated in Fig. 8.7(b). The tube voltages and currents can be interrelated as
The total number of equations in (8.7) equals the number of conductors incident at the termination/interconnection network (node). For the example of Fig. 8.2 this is 4 2 2 = 8. The fact this representation is completely general can be proven from the fact that it can be derived from a chain parameter representation of the ports of the network which always exits for any linear network. The representation in (8.7) has the sole purpose of enforcing Kirchhoffs voltage law (KVL), Kirchhoff's current law (KCL), and the element relations that are imposed by the particular interconnections within the interconnection network. Figure 8.8 illustrates some common examples. Figure 8.8(a) illustrates the k-th conductor of the i-th tube terminating in an open circuit within the m-th network. The constraint here is that the current is zero:
+ +
REPRESENTATION VtA THE ADMITTANCE OR IMPEDANCE PARAMETERS
499
(e)
FIGURE 8.8 Illustration of the determination of the network characterizations for (a) an open circuit, (b) a short circuit, (c) a direct connection, (d) a Thbvenin equivalent, and (e) a Norton equivalent.
Therefore a one appears in the column of @ ! corresponding to the current of that conductor of that tube in @'Figure . 8.8(b) illustrates the k-th conductor of the I-th tube terminating in a short circuit within the m-th network. The constraint here is that the voltage is zero:
500
TRANSMISSION-LINE NETWORKS
Therefore a one appears in the:olumn of q$corresponding to the voltage of that conductor of that tube in V$. Figure 8.8(c) illustrates a direct connection between the k-th conductor of tube f and the n-th conductor of tube j within the m-th network. The constraints here are that the voltage of the conductor of the i-th tube and the voltage of the conductor of thej-th tube are equal and the sum of the currents of the conductor of the i-th tube and the conductor of the j-th tube equals zero:
The first constraint is imposed by placing a one in the column of corresponding to the voltage of that conductor of that tube in q;l and by placing a negative one in the column of 27 corresponding to the voltage of that conductor of that tube in 97. The second constraint is imposed by placing a one in the column of 2;(corresponding to the current of that conductor of that tube in f;l and by placing a one in the column of 2,.corresponding to the voltage of that conductor of that tube in 17. Multiple connections of conductors can be similarly handled. For example, consider the case of three conductors, k of tube i, n of tube j , and p of tube l, connected at a common point within the network. KCL recyires that the sum of the currents at that interconnection equals zero: [fr]k + ~171.+ [Qlp= 0.Similarly, KVL requires that the differences of two of the three pairs of the voltages that are interconnected equal zero: [or]k - [07],,= 0,[V;l]& - [0r"3, = 0.Figure 8.8(d) illustrates a series connection of an impedance and a lumped voltage source. The constraints are that the currents are equal and the voltages are related by the element relations:
The first constraint is imposed by placing a one in the column of 2;l corresponding to the current of that conductor of that tube in f;l and by placing a one in the col%mn of 2,. corresponding to the voltage of that conductor of that tube in 17. The second constraint is imposed by placing a one in the column of Q$.corresponding to the voltage of that conductor of that tube in $;", by placing a negatiue one in the column of PT corresponding to the voltage of that conductor of that tube in 97, placing & in the column of 2$ corresponding to the current of that conductor of that tube in f$, and
REPRESENTATION VIA THE ADMWANCE OR IMPEDANCE PARAMETERS
507
placing & in in the row corresponding to the equation being written. Figure 8.8(e) illustrates a parallel connection of an admittance and a lumped current source. The constraints are that the voltages are equal and the currents are related by the element relations:
ern
The first constraint is imposed by placing a one in the column of P;I correspondingto the voltage of that conductor of that tube in 81"and by placing a negative one in the column of 97 corresponding to the voltage of that conductor of that tube in 07. The second constraint is imposed by placing a one in the column of @' corresponding to the current of that conductor of that tube in iy, by placing a one in the column of 27 corresponding to the current of that conductor of that tube in fy, placing & in the column of P;I correyonding to the voltage of that conductor of that tube in 8?,and placing & in Pmin the row corresponding to the equation being written. The final element of the process is the combination of the admittance parameters of the tubes and the constraint relations imposed by the termination/ interconnection networks. A simple example will illustrate that result. Consider the m-th network interconnecting tubes i,j, and k as shown in Fig. 8.7(b) where the i-th tube connects to termination network p at the other end, the]-th tube connects to termination network q at the other end, and the k-th tube connects to termination network r at the other end. The tube admittance characterizations at the m-th end are
The network characterization is given in (8.7). Substituting (8.13) gives
This provides a simple rule for constructing the overall admittancematrix which can be solved for the voltages at the ends of each tube:
502
TRANSMISSION-LINE NETWORKS
..
(8.15)
e
X
0, As an example, consider the transmission-line network in Fig. 8.2 with graph shown in Fig. 8.l(b). The admittance matrix becomes
(9: f 2t%d
ft:p,,
0
0
0
0
2$,,
cp:+ t f P , , )
0
(9: +
ws*> 0
2:9,,
0
0
2:9,,
0
0
(9: + 2,*:
(9: f 28983) W M 3 2:9,, (9: + 2:9s,,
Numbering each conductor of each tube as shown gives the following. First we examine termination network # 1. The constraints are
REPRESENTATION VIA THE ADMlllANCE OR IMPEDANCE PARAMETERS
503
= -~oo[~:J,
[q],= - 1op:-J4
Therefore
50
0
0
p; = 0 0 0 1
0
10
0 0.
The number of equations equals the number of conductors incident on this node: 4. Similarly, termination networks # 2 and # 3 are characterized by
and
giving 5k
0
0 50 and
1]2:-[ loo
1 0
0
i33=[;]
The number of equations equals the number of conductors incident on each node: 2. Interconnection network # 4 has the following constraints. KVL imposes
KCL imposes
504
TRANSMISSION-LINE NETWORKS
Observe that the total number of constraint equations for network 1 4 equals the total number of conductors incident on that node: 4 + 2 + 2 = 8. This requirement must always be met for any set of constraint equations for a termination/interconnectionnetwork. The matrices in (8.7) become 1 0 0 0
[OO 0 0 0
0 1 0 0
0
0 0 0
0 0 1 0
0
0 0 0
0
0 0 0
1
0 0 0
0
0 1 0
0
0 0 1
0 0 0 0
0
0 0 0
-1
0 0' 0 0 0 0 0 0
:I
0 O o0 o0 1 0 I
2: =
0
0 0
0 0 0 0
0 0 0 0
2; =
1
0
0
0
0 0
0
0
0 0-
0
1
0 0
0 0' 0 0 0 0 0 0 0 0
0 0
0 -1 -1
0
0 0 0 0
0
4
0 0
and
0
1
1
0
D
O
-
REPRESENTATION VIA THE ADMtllANCE OR IMPEDANCE PARAMETERS
505
Time (nn)
Comparison of crosstalk voltage at the termination of conductor # 1 of tube # 3 for the transmission-line network of Fig. 8.2 for a risetime of 1 ns using the SPICE model and the time-domain to frequency-domain transformation which utilizes the frequency-domaintransfer function. FIGURE 8.9
0
0 0
p4
0 3
0
The predictions of this model are compared to those of the SPICE model for a risetime of 1 ns in Figure 8.9. The predictions of the time-domain to frequency-domain transformation are obtained by first determining the frequency-domain transfer function with this model at the spectral harmonics of the input. The ramp waveform of Fig. 8.3(c) is modeled as a trapezoidal waveform with identical rise and fall times and a 1 MHz repetition rate. This is decomposed into its spectral components and combined with the frequencydomain transfer function computed with the above admittance parameter
506
TRANSMISSION-LINE NETWORKS
I5
I
1
I
I I
40 -
>
A
6
v
!
5-
B
a
Y
-30-
1
U
-65
-100.
0
””
‘
- Tlmelfrequency (lossless), 1000 terms ----Time-frequency (IOIBY),1000 termr
30
,
1
60
120
90
150
Time (ns)
FIGURE 8.10 Comparison of crosstalk voltage at the termination of conductor # 1 of for the transmission-line network of Fig. 8.2 for a risetime of 1 ns using the time-domain to frequency-domain transformation with and without losses. tube # 3
model at 500 harmonics. The resulting spectral components of the output voltage are combined using TIMEFREQ.FOR giving &(t). The spectrum of this waveform rolls off at -40 dB/decade above ~ / R T = , 318.3 MHz so a final frequency of 500 MHz is marginally sufficient and some ringing appears on the waveform at the transitions. The frequency-dependent losses of the conductors are included in the transfer function and the results recomputed and shown in Fig. 8.10 using 1000 harmonics. This upper limit of 1000 MHz for the frequency-domain transfer function is a factor of 3 higher than the point where the spectrum rolls of at -40 dB/decade. This gives better predictions than the use of 500 harmonics. Observe that the wire losses appear to have a minor effect on the output voltage waveshape. Figure 8.11 shows the frequency response of the transfer function obtained with this method with and without losses. This further confirms that the wire losses have little effect in this problem. Time-domain results can be directly obtained using the admittance matrix characterization and convolution as described in [SI. The admittance parameters are not, of course, the only way of characterizing the tubes. The dual is the impedance parameter characterization:
+ os, + 2,(-ji<9)) +
V(0) = 2&0) t 2,(-1(9))
V ( 9 )= Z,l(O)
VSL
(8.1 6a)
(8.16b)
REPRESENTATION VIA THE ADMITTANCE OR IMPEDANCE PARAMETERS
t/
I
1
I
507
I
Frequency (Hr)
(4
Frequency (Hz) (b)
FIGURE 8.11 The frequency-domain crosstalk voltage I$.,at the termination of conductor # 1 of tube # 3 for the transmission-line network of Fig. 8.2 with and without losses: (a) magnitude, (b) phase.
506
TRANSMISSION-LINE NETWORKS
The chain parameters in (8.1) can be manipulated to yield
The overall matrix to be solved can be obtained by substituting (8.16) for the tubes connected to node m which is characterized by (8.7) to give
This result has the same form as (8.14) and (8.15) with the following substitutions:
9;.* 21"
(8.19a)
2p e.P1"
(8.19b)
Pis, s-&Si
(8.19~)
* 2M,
i;, e.v;, 81" fp
(8.1 9d)
(8.19e) (8.19f)
With the impedance parameters we solve for the currents incident on the nodes, 11".
8.4
REPRESENTATION WITH THE BLT EQUATIONS
An alternative to the above methods is the use of the scattering parameter representation of the tubes [l-31. Consider a tube having lumped excitation at some point, z = 7, along its length illustrated in Fig. 8.12. These lumped excitations may represent point sources such as the direct attachment of a lightning stroke or can be extended to include distributed incident-field excitation as we will show. The general frequency-domain solution of the MTL equations for this tube can be written as in Chapter 4 for the tube segments to the left and right of the sources as
REPRESENTATION WITH
THE
BLT EQUATIONS
503
FIGURE 8.12 Determination of the scattering parameters for a tube having lumped excitation at a point on the tube.
The subscripts on the undetermined-constant vectors, L and R, for z c z 4 9. represent the left and right segments, respectively. The characteristic impedance matrix is the inverse of the characteristic admittance matrix given in (8.5), and the propagation matrix is determined as in (8.4). Evaluating these at z = i yields
Adding and subtracting these yields
The currents at the line endpoints can be logically decomposed into incident and reflected waves by evaluating (8.20b) at z = 0 and (8.21b) at z = 9 as
lL= l(0) = fy, + 1i
IR = -1(9= )8k + &
A
(8.24a) (8.24b)
510
TRANSMISSION-LINE NETWORKS
where (8.25a) (8.25b) (8.2%)
(8.25d) where superscripts i and r denote incident and rejected, respectively. This is a sensible designation if we designate the incident wave as the portion incoming at the termination and the reflected wave as the portion outgoing from the junction and also observe that a component containing is traveling to the right and a component containing e?' is traveling to the left. To conform with the results of the previous section, the tube currents are directed into the tube at both ends. Combining (8.23) and (8.25) yields the reflected components in terms of the incident quantities as
E = -(?ePWl)tk - 3(fe9$-1)(IF + 2c1VF) % = -(fef9f- ')fi- @e9(9-')f- ')OF- 2,-'vF)
(8.26a)
(8.26b)
(There are sign differences between these results and those of [l-31 due to our choice of having the total currents directed into the tube at both ends.) In matrix notation these become
[PI]
h
3
where the tube propagation matrix
a[f7fk
+ f,
(8.27)
fi is (8.28a)
with entries (8.28b) The current source vector due to the lumped sources at z = 7 is (8.29a) In the case of distributed excitation such as is the case for incident field illumination of the tube, the source vector becomes simply
REPRESENTATION WITH THE BLT EQUATIONS
511
The total voltages and currents at the left end of the tube become, by evaluating (8.20) at z = 0 and substituting (8.25a) and (8.25b), 8(0) =
VL = 2&
i(0) = 1' = 1: -
+ 1:. 1
- ii)
(8.30a) (8.30b)
Similarly, the total voltages and currents at the right end of the tube become, by evaluating (8.21) at z = 9 and substituting (8.25~)and (8.25d),
v(9)
OR
2C(&
- fk)
-i(9)= PR = i;+ ik
(8.31a) (8.31b)
Observe that, as in the case of the admittance parameters, the total currents are defined as being directed into the tubes at both ends. The incident components are defined as being the components traveling out of the tubes or into the network attached to that end. The reflected components are similarly defined as being the components traveling into the tubes or out of the network attached to that end. Hence the origin of the names incident and rejected: with respect to the network or node at which the end of the tube is incident. These results can be derived from the chain parameter matrix representation of the line with incident-field illumination. Substituting the chain parameter matrix of the line given in (7.29) into (8.30) and (8.31) we obtain (8.26). This gives the current source vector in (8.29b) in terms of the total incident-field and O F T , given in (7.29) as vectors, iFT (8.32) We next form the junction scattering matrix for node rn, the reflected and incident current components at node m as
f:, = $,it,
&, which relates (8.33)
The number of equations here equals the total number of conductors incident on that node. First consider a termination node where only one tube is incident as shown in Fig. 8.13. Suppose the network contains no lumped sources and is characterized by a generalized Th6venin equivalent as
9, = -2J, =
-2,& + it,)
(8.34)
(Recall that the total currents are defined as being directed into the tube ends and are therefore directed out of the termination networks.) The relations for
512
TRANSMISSION-LINE NETWORKS
Definitions of incident and scattered waves for determining the scattering parameters of a termination network.
FIGURE 8.13
the voltage and current at the end of the tube that is incident on this node are (8.35) Solving (8.34) and (8.35) yields the current scattering matrix: (8.36) This has a direct parallel to the scalar current reflection coeficient for a two-conductor line CA.11. The next form of the scattering matrix is for an interconnection network wherein there are two or more tubes incident. In the admittance parameter representation of the previous section we characterized these in a general sense as a combination of generalized Thbvenin and generalized Norton equivalents so that series and parallel impedances and excitation sources could be included. The formulation of the BLT equations in [1-33 has the excitation sources along the tubes due to incident fields. We will derive the BLT equations with that assumption although we will later modify them for the more general case of lumped sources within the networks. Thus the interconnection networks will simply have: 1. Conductors directly connected. 2. Conductors terminated in short circuits.
3. Conductors terminated in open circuits as shown in Fig. 8.14. First consider the direction connection of several conductors as shown in Fig. 8.14(a). This is characterized as c,lm= 0 (8.37a)
c,v,
=0
(8.37b)
where &, contains the total currents of the conductors incident on the m-th contains the voltages of those conductors. The first relation in node and (8.37a) enforces KCL at the connection so that for each connection a one
v,,,
REPRESENTATION WITH THE BLT EQUATIONS
513
FIGURE 8.14 Interconnection networks: (a) direct connection, (b) open circuit, and (c) short circuit.
appears in the columns of C1corresponding to the currents of i,,,for the conductors that are connected. The second condition in (8.37b) enforces KVL at the connection so that for n conductors connected, there are (n - 1) equations enforcing - 6 = 0 for (n - 1) pairs of the voltages. Thus a one appears in the column of Cy corresponding to the voltage of the pair in 8, and a negative one appears in the column of Cycorresponding to the other voltage of the pair in qm.The second situation is an open circuit as illustrated in Fig. 8.14(b). Enforcing KCL requires that we place a one in the column of CI eorresponding to the current of !,, that is constrained to zero by the open circuit. The third constraint is a short circuit illustrated in Fig. 8.14(c). Enforcing KVL requires that we place a one in the column of Cv corresponding to the voltage of 0, that is constrained to zero by the short circuit. The sum of the row dimensions of C,and Cv must equal the total number of conductors incident on the node. The scattering matrix for this interconnection node can now be formulated. Decomposing the total currents and voltages into their incident and reflected components gives &
-cII: cv2,f:,
= C,C
(8.38a)
= c,2cmif,
(8.38b)
where kCm contains the characteristic impedance matrices of the tubes incident upon the node on the main diagonal and zeros elsewhere. Solving (8.38) yields the scattering matrix of the interconnection node as (8.39) Arranging the tube characterizations for all tubes in the network and the scattering matrix representations for all nodes in the network yields the overall characterization (8.40a) @ = f i T f k + f,T
I; = 3,f:
A
(8.40b)
514
TRANSMISSION-LINE NETWORKS
The overall tube propagation matrix, itT,is 2nT x 2nT where nT is the total number of conductors in the overall transmission-line network and has the individual tube characterization matrices given in (8.27) on the main diagonal and zeros elsewhere. Similarly the overall junction scattering matrix, ST,is 2n, x 2n, and contains the individual scattering matrices for the interconnection and termination networks on the main diagonal and zeros elsewhere. Combining (8.40) gives (&' = (8.41) which are referred to as the BLT (Baum, Liu, and Tesche) equations. The total currents can be obtained by writing
Solving (8.40) and (8.42) simultaneously gives the total currents as IT
5=
(1
+ &)(&- fiT)-'im
(8.43)
This formulation can be extended to include series and parallel impedances and lumped sources in the termination/interconnection networks. In order to provide that extension, consider the general characterization of the m-th termination/interconnection node illustrated in Fig. 8.7(b) and given in (8.7):
This can be written in matrix form as
The currents and voltages can be decomposed into incident and reflected components according to (8.31) as (8.45a) (8.45b)
and likewise for tubej and tube k. Substituting (8.45) into (8.44) yields
(8.46)
REPRESENTATION WITH THE BLT EQUATIONS
515
and the collection of characteristic impedance matrices of the tubes incident on the termination/interconnection network is
The current source vector is
For example, consider the case of a termination network (only one tube incident on the node) and a generalized Thtvenin representation of this network:
(8.48)
The above representation becomes
8,
= -(2,
+ 2c,)-'(2rn- 2c,)
Q = (2,+ 2c,)-lVSm
(8.49a) (8.49b)
Thus the scattering parameter representation in (8.33) has been modifled to include lumped sources in the termination/interconnection networks. In the case of an interconnection network containing only short circuits, open circuits, and/or direct connections, this representation becomes (8.50) and
[2F 27 ];[=.2
(8.51)
where Cv and C, were originally defined in (8.37). Substituting (8.50) and (8.51) into (8.47a) yields (8.39).
516
TRANSMISSION-LINE NElWORKS
The tube characterizations in (8.27) remain unchanged. Thus the overall representation of the network is of the form in (8.40): (8.52a) (8.52b) where STis 2nT x 2nT and contains the individual scatterin4 matrices given in (8.47a) on the main diagonal and zeros elsewhere, and IsT is 2nT x 1 and contains the in (8.47~).In a fashion similar to the earlier derivation we obtain the general form of the BLT equations for the total tube currents as
For example, consider the two-conductor line shown in Fig. 4.1 (without incident-field illumination). The total tube propagation matrix fiT is simply (8.28) since there is only one tube and becomes
Writing (8.44) at the source and at the load gives
Thus the scattering matrices at the source and load in (8.47a) become
where ps and f" are the voltage reflection coe@cients at the source and load, respectively. The current refection coeficients are the negatives of these [A. I]. Thus the overall scattering matrix is
REPRESENTATION WITH THE BLT EQUATIONS
517
Also the current source vectors in (8.47~)become
so that the overall current source vector becomes
Forming the BLT equation in (8.53) for the total currents yields
Recalling that fl = f(0) and f4 = -f(9) these results are identical to those given in (4.26b). The time-domain solution can be obtained from this formulation by first using the BLT equations to obtain the frequency-domain transfer function and then using the timedomain to frequency-domain transformation method as before. Again, that technique for obtaining the time-domain solution assumes linear loads and tubes since superposition is implicitly used. 8.5
DIRECT TIME-DOMAIN SOLUTIONS IN TERMS OF TRAVELING WAVES
The above solution methods concentrated on the frequency-domain solution of transmission-line networks. The time-domain solution can be obtained from this solution via the time-domain to frequency-domain technique, which is very straightforward and has been used on numerous occasions. Of course, the time-domain to frequency-domaintechnique requires linear terminations. In this final section we briefly investigate the direct time-domain solution via the traveling waves existing on the various tubes of the network. To begin the discussion consider a tandem connection of two-conductor lines having a discontinuity as shown in Fig. 8.15(a). Each line is characterized by a characteristic impedance, Zcr, and time delay, ?i.Lossless lines and resistive terminations are assumed to simplify the discussion. The frequency-domain
516
TRANSMISSION-LINE NETWORKS
FIGURE A15 Determination of the scattering parameters at the junction of two different
two-conductor lines: (a) line configuration, and (b) definition of incident and reflected voltage waves at the junction.
solutions on each line are of the form
in accordance with the general solution of the transmission-line equations for each tube. The discontinuity may be characterized as before with a form of voltage scattering parameter matrix that relates the incident and reflected voltage waves at that point as illustrated in Fig. 8.15(b). The incident waves are the portions of (8.54a) that are incoming at the junction, P:e-’flIa and p;eJflSa. The reflected waves are the portions of (8.54a) that are outgoing at the junction, Pid”4 and ple-Jfi2H.
-
(8.55)
vlncldenl
The r,,elements are the voltage rejection coeficients, and the Ti) elements are the voltage transmission coeflcients. Figure 8.16 illustrates the determination of these parameters for various discontinuities. Consider the direct connection shown in Fig. 8.16(a). The voltages and currents must be continuous. Equating
DIRECT TME-DOMAIN SOLUTIONS IN TERMS OF TRAVELING WAVES
519
(b) FIGURE 8.16
Illustration of the simplified determination of the scattering parameters
for (a) a direct connection, and (b) a resistive interconnection network.
equations (8.54) for the left and right sections and putting them in the form of (8.55) yields (8.56a) (8.56b) This result is directly analogous to the case of a uniform plane wave incident interface between two media CA.1). It is a sensible result for the following reason. In the time domain, the wave incident from line # 1 "sees" an impedance at the junction equal to the characteristic impedance of line # 2 since it has not arrived at the termination of line 6 2 and therefore line # 2 appears infinite in length. So the reflection coefficient can be calculated as though line # 1 is terminated in the characteristic impedance of line #2. The transmission coeficient is easily calculated from the reflection coefficients since the total voltage incident on the junction from line # 1 is the sum of the incident and reflected waves on that line which, because of the direct connection, must equal the transmitted voltage or (1 rll)= T2,. In a similar fashion, (1 rZ2)= TI2. The BLT equations can be easily formed for this network although their solution is best suited to computer implementation.The overall tube propagation matrix is at an
+
L o
+
520
TRANSMISSION-LINE NETWORKS
Writing (8.7) at the source, the junction and the load gives
where m designates the “middle node” (the junction). The junction scattering matrices are obtained from (8.47a) as
sm= -{[;
3+[’
0 -0 1 1 0 p
O]y{[; 3-[’ -‘IC“’ z,
z,
0
0
0
O]]
Thus the overall scattering matrix becomes
0
-rll
-T12
0
0
-T12
-rz2
0
L o
0
o
-r,J
These reflection and transmission coefficients are the negative of the voltage reflection and transmission coefficients in (8.56) because we are considering currents and because of the current directions at the junctions (into the tubes). Similarly, the total current source vector is
DIRECT TIME-DOMAIN SOLUTIONS IN TERMS OF TRAVELING WAVES
521
The BLT equations are then formed from (8.53) with f, = 0 (no incident-field illumination). Figure 8.16(b) shows a connection consisting of a resistive network. For the above reasons we may similarly determine (8.57a)
The transmission coefficients are determined by first obtaining the total voltage incident on the junction, (1 + rll)or (1 + r2Jand then using voltage division. In the time domain, the source initially “sees” a termination of Zcl so that the voltage wave sent out is, by voltage division, V(t) =
Rs + ZCI zcl
&(t)
(8.58)
This wave travels down line # 1 reaching the discontinuity at one time delay rllV(c - TI)is reflected and T2,V(t - TI) is transmitted across the junction. The reflected portion arrives at the source at 2T, where a portion of it, rsrllV(t - 2T1), is sent back towards the junction where a portion of it is reflected, ~ l l ~ sV(t ~l 3T1), l and a portion is transmitted, T,ITsT,, V(t - 3Ti), etc. Meanwhile, the portion of the original wave that was transmitted across the junction, Tzl V(t - q),arrives at the load where it is reflected as rLTZlV(t - Tl - T2).This is sent back to the junction where a V(t - 2T,), is reflected back toward the load and a portion of it, r22rLTZl portion, Tl2rLTZl V(t - TI - 2T2),is transmitted across the junction onto line # 1. This process of continued reflections and transmissions continues, and the total voltage at any point on either line is the sum of the total waves at that point and time. Clearly the line voltages will be linear combinations of the initial transmitted wave, V(t), delayed in time by various sums of multiples of the line delays, TI and T2.A lattice diagram can be constructed as described in Chapter 5 to aid in determining these total voltages but the process is clearly very complicated owing to the multitude of reflections and transmissions, If of TI where
522
TRANSMISSION-LINE NETWORKS
either line is terminated in its characteristic impedance, the summation is simplified considerably, and a series s o htion can be developed. But for completely mismatched lines, the summation is tedious. A symbolic solution can be obtained as in Chapters 5 and 6 if we use the time-shift or difference operator as before:
in the BLT equations of (8.53). Thus
The BLT equations in (8.53) with fTT= 0 (no incident-field illumination) can be written as
+
and [RT(D) 13” is simple to obtain. However, the literal solution of the BLT equations is very tedious even for this simple case. If we substitute a previously described SPICE model for each line, the summation of these waves is taken care of and nonresistive loads can be considered. A direct time-domain summation of the traveling waves was implemented using the scattering parameters of the termination/interconnectionnetworks in [3] but, once again, keeping track of all the incident/reflected waves on the line is a very tedious task because of the multiple reflections/transmissionsand the different mode velocities on each tube.
REFERENCES
[l] C.E. Baum, T.K. Liu, E M . Tesche, and S.K. Chang, “Numerical Results for Multiconductor Transmission-Line Networks,” Interaction Note 322, Air Force Weapons Laboratory, Albuquerque, NM, September 1977. [2] C.E. Baum, T.K. Liu, and F.M. Tesche, “On the Analysis of General Multiconductor Transmission-LineNetworks,” Interaction Note 350,Air Force Weapons Laboratory, Albuquerque, NM, November 1978. [3] A.K. Agrawal, H.K. Fowles, L.D.Scott, and S.H. Gurbaxani, “Application of Modal Analysis to the Transient Response of Multiconductor Transmission lines with Branches,” IEEE Zhans. on Electromagnetic Compatibility, EMC-21,256-262 (1979).
PROBLEMS
923
C.R.Paul, “Analysis of Electromagnetic Coupling in Branched Cables,” Proc. 1979 IEEE International Symposium on Electromagnetic Compatibility, San Diego, CA, August 1979. [SI J.L. Allen, “Time-Domain Analysis of Lumped-Distributed Networks,” IEEE Trans. on Microwave Theory and Techniques, MTT-27, 890-896 (1979). [6] F.M. Tesche, and T.K. Liu, “User Manual and Code Description for QV7TA: A [4]
General Multiconductor Transmission-Line Analysis Code,” Interaction Application Memo 26, LuTech, August 1978. [7] A.R. Djordjevic and T.K. Sarkar, “Analysis of Time Response of Lossy Multiconductor Transmission Line Networks,” IEEE Trans. on Microwave Theory and Techniques, MTT-35, 898-907 (1987).
PROBLEMS
8.1 8.2
Derive the admittance parameters (8,3). Derive the admittance parameter relation for a transmission-line network (8.14).
8.3
Derive the impedance parameters (8.17).
8.4
Derive the relations given in (8.26) and (8.27).
8.5
Derive the scattering parameter relation for a termination network (8.36).
8.6
Derive the scattering parameter relation for an interconnection network (8.39).
8.7
Derive the BLT equations given in (8.41) and in (8,43).
8,8
Derive the scattering parameter matrix and current source vector given for the general case in (8.47).
8.9
Derive the BLT equations for the general case given in (8.53).
8.10 For the tandem connection of two-conductor lines shown in Fig. 8.15
derive series expressions for the time-domain source and load voltages in terms of delayed source-voltage waveforms for a matched load on line # 2, i.e., R, = Z,, . 8.11 Consider the transmission-line network shown in Fig. P8.11.All conduc-
tors are #20 gauge wires (r, = 16 mils) at a height of 1 cm above a ground plane. Solve for the voltage ?&,(f) where &(t) is a 10MHz trapezoidal pulse train with rise/fall times of 1 ns and 50% duty cycle. Compute this using:
524
TRANSMISSION-LINE NETWORKS
B’
10
FIGURE P8.11
1. The SPICE model. 2. The lumped-pi model. 3. The admittance parameter model. 4. The impedance parameter model. 5. The BLT equations.
Publications by the Author Concerning Transmission Lines
A.
BOOKS
CA.1J Introduction 9 Electromagnetic Fields, 2d ed., McGraw-Hill, NY, 1987 (with S.A. Nasar). [A.2] Analysis o j Linear Circuits, McGraw-Hill, NY, 1989. [A.3] Introduction to Electromagnetic Compatibility, John Wiley Interscience, NY, 1992. CA.43 Essential Engineerfng Equations, CRC Press, Boston, MA, 1991 (with S A . Nasar).
B.
GENERAL
[B. 1J “Application of Multiconductor Transmission Line Theory to the Prediction of Cable Coupling-Vol. I-Multiconductor Transmission Line Theory,” Technical Rep&, Rome Air Development Center, Griffiss AFB, NY, RADCTR-76-101, April 1976. (A0258028) “Lumped Model Approximations of Transmission Lines: Erect of Load Impedances on Accuracy,” Technical Report, Rome Air Development Center, Griffiss AFB, NY, RADC-TR-82-286, Vol. IV E, August 1984 (with W.W. Everett, 111). “On Uniform Multimode Transmission Lines,” IEEE Trans. on Microwave Theory and Techniques, MTT-21, 556-558 (1973). “Useful Matrix Chain Parameter Identities for the Analysis of Multiconductor Transmission Lines,” IEEE Trans. on Microwave Theory and Techniques, MTT-23, 756-760 (1975). “Solution of the Transmission Line Equations for Three-Conductor Lines in Homogeneous Media,” IEEE Trans. on Electromagnetic Compatibility, EMG 20, 216-222 (1978). “Computation of Crosstalk in a Multiconductor Transmission Line,” IEEE Trans. on Electromagnetic Compatiblllty, EMC-23, 352-358 (1981). “On the Superposition of Inductive and Capacitive Coupling in Crosstalk Prediction Models,” IEEE Trans. on Electromagnetic Compatibility, EMC-24, 335-343 (1982). 525
526
PUBLICATIONS BY THE AUTHOR
CB.81 “Estimation of Crosstalk in Three-Conductor Transmission Lines,” IEEE Trans. on Electromagnetic Compatibility, EMC-26, 182-192 (1984). of Electromagnetic Coupling in Branched Cables,” Proc. 1979 IEEE ~ ~ - 9“Analysis 1 International Symposium on Electromagnetic Compatibility, San Diego, CA, October 1979. [B.lO] “Adequacy of Low-Frequency, Crosstalk Prediction Models,” Proc. 4th Symposium and Technical Exhibition on Electromagnetic Compatibility, Zurich, Switzerland, March 1981. CB.111 “Coupling to Transmission Lines: An Overview,” Proc. 1983 International Symposium and Technical Exhibition on Electromagnetic Compatibility, Zurich, Switzerland, March 1983. CB.121 “A Simple Technique for Estimating Crosstalk,” Proc. 1983 IEEE International Symposium on Electromagnetic Compatibility, Washington, DC, August 1983. CB.13) “Lumped Circuit Modeling of Transmission Lines,” Proc. 1985 IEEE International Symposium on Electromagnetic Compatibility, Wakefield, MA, August 1985 (with L. Monroe). CB.141 “Derivation of Common Impedance Coupling from the Transmission-Line Equations,” IEEE 7kans. on EIectromagnetic Compatibility, EMC-34, 315-319 (1992). CB.151 “Literal Solutions for Time-Domain Crosstalk on Lossless Transmission Lines,” IEEE Tkans. on Electromagnetic Compatibility,EMC-34,433444,1992,
C.
PER-UNIT-LENGTH PARAMETERS
“Computation of the Capacitance Matrix for Dielectric-Coated Wires,” Technical Report, Rome Air Development Center, Griffiss AFB, NY, RADCTR-74-59, March 1974, (with J.C. Clements). ”Applications of Multiconductor Transmission Line Theory to the Prediction of Cable Coupling-Vol. 11-Computation of the Capacitance Matrices for Ribbon Cables,” Technical Report, Rome Air Development Center, Griffiss AFB, NY,RADC-TR-76-101, April 1976, (with A.E. Feather). (A025029). “Two-Dimensional Systems of Dielectric-Coated,Cylindrical Conductors,” IEEE 7kans. on Electromagnetic Compatibility, EMC-17, 238-248 (1975) (with J.C. Clements and A.T. Adams). “Computation of the Transmission Line Inductance and Capacitance Matrices from the Generalized Capacitance Matrix,” IEEE Trans. on Electromagnetic Compatibility, EMC-18, 175-183 (1976) (with A.E. Feather). “Reference Potential Terms in Static Capacitance Calculations via the Method of Momen’ts,” IEEE Trans. on Electromagnetic Compatibility, EMC-20,267-269 (1978). “Application of Moment Methods to the Characterization of Ribbon Cables,” Computers and Electrical Engineering, 4, 173-184 (1977) (with A.E. Feather). “Application of Moment Methods to the Characterization of Ribbon Cables,” Proc. international Symposium on Innovative Numerical Analysis in Applied Enoineerina Science. Paris. France. Mav 1977 (with A.E. Feather).
PUBLICATIONS BY THE AUTHOR
527
CC.81 “Moment Method Calculation of the Per-Unit-Length Parameters of Cable Bundles,” Proc. 1994 IEEE International Symposium on Electromagnetic Compatibility, Chicago, IL, August 1994, (with J.S. Savage and W.T.Smith).
D. CABLE HARNESSES
[D. 11 “Applications of Multiconductor Transmission Line Theory to the Prediction of Cable Coupling-Vol. 111-Prediction of Crosstalk in Random Cable Bundles, Technical Report, Rome Air Development Center, Griffiss AFB, NY, RADCTR-76-101, February 1977. (A038316). CD.21 “Sensitivity of Multiconductor Cable Coupling to Parameter Variations,” Proc. 1974 IEEE International Symposium on Electromagnetic Compatibility, July 16-18, San Francisco, CA. CD.31 “Sensitivity ofCrosstalk to Variations in Wire Position in Cable Bundles,” Proc. 1987 International Symposium and Technical Exhibition on Electromagnetic Compatibility, Zurich, Switzerland. [D.4] “Sensitivity of Crosstalk to Variations in Wire Position in Cable Bundles,” Proc. IEEE Intertrational Symposium on Electromagnetic Compatibility, Atlanta, GA, September 1987.
E.
RIBBON CABLES
CE.11 “Applications of Multiconductor Transmission Line Theory to the Prediction of Cable Coupling-Vol. IV-Prediction of Crosstalk in Ribbon Cables,” Technical Report, Rome Air Development Center, Griffiss AFB, NY, RADCTR-76-101, February 1978. (A053548). CE.2) “Prediction of Crosstalk in Ribbon Cables: Comparison of Model Predictions and Experiment Results,” IEEE Trans. on Electromagnetic Compatibility, EMC20, 394-406 (1978). [E.3] “Prediction of Crosstalk in Ribbon Cables,” IEEE International Symposium on Electromagnetic Compatibility, Atlanta, GA, June 1978.
F.
SHIELDED WIRES
[F.11 “Applications of Multiconductor Transmission Line Theory to the Prediction of Cable Coupling-Vol. VIII-Prediction of Crosstalk Involving BraidedShield Cables,” Technical Report, Rome Air Development Center, Griffiss AFB, NY, RADC-TR-76-101, August 1980. CF.21 “Prediction of Crosstalk in Flatpack, Coaxial Cables,” Technical Report, Rome Air Development Center, Griffiss AFB, NY, RADC-TR-82-286, Vol. IV F, December, 1984 (with W.E. Beech). CF.31 “Effect of Pigtails on Crosstalk to Braided-Shield Cables,’’ IEEE Trans. on Electromagnetic Compatibility, EMC-22,161-172 (1980).
528
PUBLICATIONS BY THE AUTHOR
CF.41 “Transmission-Line Modeling of Shielded Wires for Crosstalk Prediction,” IEEE Transactions on Electromagnetic Compatibility, EMC-23,345-35 1 (1981). [FS] “Effect of Pigtails on Coupling to Shielded Wires,” Proc. IEEE International Symposium on Electromagnetic Compatibility, Sen Diego, CA, October 1979. CF.61 “Prediction of Crosstalk in Flatpack, Coaxial Cables,” Proc. 1984 IEEE International Symposium on Electromagnetic Compatibility, San Antonio, Texas, April 1984 (with W.E, Beech). CF.71 “Literal Solution of the Transmission-LineEquations for Shielded Wires,” Proc. 1990 IEEE International Symposium on Electromagnetic CompatibiNty, Washington, DC, August 1990 (with B.A. Bowles). CF.83 “Symbolic Solution of the Multiconductor Transmission-Line Equations for Lines Containing Shielded Wires,” IEEE Trans. on Electromagnetic Compatibility, EMC-33, 149-162 (1991) (with B.A. Bowles).
G. TWISTED PAIRS OF WIRES “Applications of Multiconductor Transmission Line Theory to the Prediction of Cable Coupling-Vol. V-Prediction of Crosstalk Involving Twisted Wire Pairs,” Technical Report, Rome Air Development Center, Griffiss AFB, NY, RADC-TR-76.101, February 1978. (A053559). “Crosstalk in Twisted-Wire Circuits,” Technical Report, Rome Air Development Center, Griffiss AFB, NY, RADC-TR-82-286, Vol. IV C, November 1982 (with M.B. Jolly). “Prediction of Crosstalk in Balanced, Twisted Pair Circuits,”Technical Report, Rome Air Development Center, Grifiss AFB, NY,RADC-TR-82-286,Vol. IV D, August 1984 (with D. Koopman). “Prediction of Crosstalk Involving Twisted Pairs of Wires, Part I, A Transmission Line Model for Twisted Wire Pairs,” IEEE Trans. on Electromagnetic Compatibility, EMC-21, 92-105 (1979) (with J.A. McKnight). “Prediction of Crosstalk Involving Twisted Pairs of Wires, Part 11, A Simplified, Low-Frequency Prediction Model,” IEEE nuns. on Electromagnetic Compatibility, EMC-21, 105-1 14 (1979) (with J.A. McKnight). “Sensitivity of Crosstalk in Twisted-Pair Circuits to Line Twist,” IEEE Trans. on Electromagnetic Compatibility, EMC-24, 359-364 (1982) (with M.Jolly). “Sensitivity of Coupling to Balanced, Twisted Pair Lines to Line Twist,” Proc. 1983 International Symposium and Technical Exhibition on Electromagnetic Compatibility, Zurich, Switzerland, March 1983 (with D. Koopman). “Prediction of Crosstalk in Twisted Pairs of Wires, A Simplified, LowFrequency Model,” Proc. IEEE International Symposium on Electromagnetic Compatibility, Atlanta, GA, June 1978, “Coupling to Twisted-Pair Transmission Lines,” Proc. 4th Symposium and Technical Exhibition on Electromagnetic Compatlbllity, Zurich, Switzerland, March 1981. “Crosstalk in Balanced, Twisted-Pair Circuits,” Proc. 1981 IEEE International Symposium on Electromagnetic Compatibility, Boulder, CO, August 1981 (with M.B. Jolly).
PUBLICATIONS BY THE AUTHOR
529
H. EFFECTS OF INCIDENT FIELDS
[H. 13 “Applications of Multiconductor Transmission Line Theory to the Prediction of Cable Coupling-Vol. VI-A Digital Computer Program for Determining Terminal Currents Induced in a Multiconductor Transmission Line by an Incident Electromagnetic Field,” Technical Report, Rome Air Development Center, Griffiss AFB, NY, RADC-TR-76-101, February 1978. (A053560). CH.2) “Coupling of Electromagnetic Fields onto Transmission Lines: A Comparison of the Transmission Line Model and the Method of Moments,” Technical Report, Rome Air Development Center, Griffiss AFB, NY, RADC-TR-82-286, Vol. IV A, November 1982 (with R.T. Abraham). EH.31 Efficient Numerical Computation of the Frequency Response of Cables Illuminated by an Electromagnetic Field,” IEEE Trans. on Microwave Theory and Techniques, MTI-22,454-457 (1974). [H.4] “Frequency Response of Multiconductor Transmission Lines Illuminated by an Incident Electromagnetic Field,” IEEE Trans. on Electromagnetic Compatibility, EMC-18, 183-190 (1976). CH.51 ‘‘Frequency Response of Multiconductor Transmission Lines Illlrminated by aa Incident Electromagnetic Field,” Proc. 1975 IEEE International Symposium on Electromagnetic Compatibility, San Antonio, TX, October 1975. CH.61 “WIRE, A Digital Computer Program for Determining Terminal Currents Induced on a Multiconductor Transmission Line by an Incident Electromagnetic Field,” Proc. 1978 National Aerospace and Electronics Conuention, Dayton, OH, May 1978. [H.7] “Coupling of Electromagnetic Fields to Transmission Lines,” Proc. 1981 IEEE International Symposium on Electromagnetk Compatibtlity, Boulder, CO, August 1981 (with R.T. Abraham). CH.81 “Coupling of Electromagnetic Fields to Transmission Lines,” Proc. 1982 IEEE International Symposium on Electromagnetic Compatibility, Santa Clara, CA, September 1982 (with D.F. Herrick). [H.9] “Bounds on Currents Induced in Transmission Lines by Incident Fields,” Proc. 1984 I E E E Southeascon, Louisville, K Y , April 1984 (with D.R. Bush). ‘I
1.
DIGITAL COMPUTER PROGRAMS
[I. 13 “Applications of Multiconductor Transmission Line Theory to the Prediction of Cable Coupling-Vol. VII-Digital Computer Programs for the Analysis of Multiconductor Transmission Lines,” Technical Report, Rome Air Development Center, Griffiss AFB, Rome NY, RADC-TR-76-101, July 1977. (A046662). [I.2] “SHIELD, A Digital Computer Program for Computing Crosstalk between Shielded Cables,” Technical Report, Rome Air Development Center, Griffiss AFB, NY, RADC-TR-82-286, Vol. 1V B, November 1982. [I.3] “SHIELD-A Digital Computer Program for the Prediction of Crosstalk to Shielded Cables,” Proc. 1983 International Symposium and Technical Exhibition 011Electromagnetic Compatibility, Zurich, Switzerland, March 1983.
530
WBLICATIONS BY THE AUTHOR
CI.41 “A Simple SPICE Model for Coupled Transmission Lines,” Proc. 1988 IEEE International Symposium on Electromagnetic Compatibility, Seattle, WA, September 1988. J.
PRINTED CIRCUIT BOARDS
[JJ] “Modeling Crosstalk on Printed Circuit Boards,” Technical Report, Rome Air Development Center, Griffiss AFB, NY,RADC-TR-85-107, July 1985 (with W.W. Everett, 111)). [J.2] “Modeling of Printed Circuit Boards for the Prediction of Crosstalk and Ground Drop,” 1BM J. Research and Development, 33, 33-50 (1989). cJ.31 “Printed Circuit Board EMC,” Proc. 1985 International Symposium and Technical Exhibition on E/ectromagnetic Compatibility, Zurich, Switzerland, March 1985. CJ.4) “Printed Circuit Board Crosstalk,” Proc. 1985 IEEE lnternaffonalSymposium on Electromagnetic Compatibility, Wakefield, MA, August 1985 (with W.W. Everett, 111). [J.5] “Modeling and Prediction of Ground Shift on Printed Circuit Boards,” Proc. 1986 IERE Symposium on Electromagnetic Compatibility, University of York, England, September 1986.
K. POWER TRANSMISSION LINES CK.11 “Solutions of the Transmission Line Equations for Lossy Conductors and Imperfect Earth,“ Proc. IEE (London), 122, 177-182 (1975). CK.21 “A Modal Decomposition for Power Transmission Lines with Imperfect Earth Return,” Proc. I E E (London), 124,647-648 (1977).
APPENDIX A
7
Description of Computer Software
Several FORTRAN computer programs that implement the methods described in this text are included on a diskette with the textbook. This appendix gives a description of those codes and their algorithms. The codes use certain arrays, matrices, and vectors in which to store information. These array dimensions are determined primarily by the total number of conductors of the transmission line and are stored in parameters at the beginning of the code. An example is the parameter MSIZE:
PARAMETER (MSIZE= 99) The user need only change these parameters in order for the program to handle larger numbers of conductors in the line. Each uncompiled code (source code) contains at the beginning a brief description of the code, the array dimensions, and the names and contents of the required input files and the name of the resulting output file. The programs are written in standard FORTRAN 77 language. A conscious attempt was made to use rudimentary FORTRAN programming commands so that the codes would be compilable with a wide variety of FORTRAN compilers on a large number of platforms. The codes fall into two distinct categories: those which compute the per-unit-length parameters of inductance and capacitance and those which solve the MTL equations and incorporate the terminal conditions. The codes that determine the per-unit-length parameters are WIDESEP.FOR which implements the wide-separation approximations for wires, RIBBON.FOR for considering ribbon cables, PCB.FOR and PCBGAL.FOR for considering lands on printed circuit boards, and MSTRP.FOR and MSTRPGALFOR for considering coupled microstrip structures. All of these codes require an input file which holds the two-dimensional, cross-sectional dimensions and material characteristics of the line. These files are called WIDESEPJN, RIBBONJN, PCBJN, and MSTRPJN, respectively, The output of these programs is the file PULDAT, 531
532
DESCRIPTION OF COMPUTER SOFTWARE
The upper triangle of the per-unit-length parameter inductance, L, and capacitance, C, matrices are stored by rows in this file with the inductance matrix given first followed by the capacitance matrix. The input data are printed at the end of these files but are not read by the programs that use these per-unit-length data. These and the remaining codes input the total number of conductors, (n + 1). The remaining codes solve the MTL equations and incorporate the terminal conditions to give the total solution for the line voltages and/or currents. These codes are further subdivided into those for the frequency-domain solution and those for the time-domain solution. Each of these codes require one or more input files that are described at the beginning of the code listings. For example, the code SPICEMTLFOR requires the input files SPICEMTLJN and PULDAT, The input file SPICEMTLJN contains structural information such as line length. The file PULDAT is the output of the codes WIDESEP.FOR, RIBBON.FOR, PCBGAL.FOR, PCB.FOR, MSTRP.FOR, or MSTRPGAL.FOR. The user may a1ternatively provide his/her per-uni tlength data from other sources. The output file of SPICEMTLFOR is simply SPICEMTLOUT. The output files generally contain either the magnitude and phase of the line voltages and/or currents at each frequency or the line voltages and/or currents at each time step. There is one exception to this: the programs SPICEMTLFOR, SPICELPLFOR, SPICELC.FOR, and SPICEINC.FOR provide as output a SPICE subcircuit that models the MTL on a port basis. Both the input data and the output data for each code appear as one item per line in the appropriate file. These data are in a form of free field format. The READ statements in the calling program are in an unformatted type:
Each data item appears on a separate line of the input file followed on that line with several blank spaces and an equals sign followed by descriptive remarks. The annotation (in the input and output files) of =***** simply describes the data entry preceeding the equals sign to the user and is not read by the calling program. In order for this to function properly, at least one blank space must appear between the data item and the equals sign. This is in the same format in both the input and output files and anticipates that the subsequent program that may use this output file has unformatted reads such as READ
@,*I. A.1 PROGRAMS FOR CALCULATION OF THE PER-UNIT-LENGTH PARAMETERS
In this section we describe six codes for the calculation of the entries in the per-unit-length inductance matrix, L, and capacitance matrix, C: WIDESEP.FOR which implements the wide-separation approximations for wires, RIBBON,FOR for ribbon cables, PCB.FOR and PCBGAL.FOR for
APPENDIX A
533
printed circuit boards, and MSTRP.FOR and MSTRPGAL.FOR for coupled microstrip structures. The output of these codes is the file PUhDAT that contains the upper triangles of these matrices with the inductance matrix given first followed by the capacitance matrix. A.1.1
Wide-Separation Approximations for Wires: WIDESEP.FOR
Required Input Files: WIDESEPJN File: PULDAT
Output
The code computes the entries in the per-unit-length inductance matrix, L, for three configurationsof widely spaced wires in a homogeneous medium described in Section 3.2.3: (a) (n 1) wires (3.63), (b) n wires above an infinite, perfectly conducting ground plane (3.66), and (c) n wires within an overall circular cylindrical shield (3.67). The per-unit-length capacitance matrix is obtained from C = p o p r ~ o ~ r Lwhere -l p, and s, are the relative permeability and permittivity, respectively, of the homogeneous medium surrounding the wires. These structures are depicted in Fig. A.1. The first data parameter in WIDESEPJN is the number of wires, n, (exclusive of the reference conductor).
+
(8)
FIGURE A.1 Cross-sectional configurationsof wire lines for the WIDESEP.FOR FORTRAN program input data: (a) (n + 1) wires, (b) n wires above a ground plane, and (c) n wires within a cylindrical shield.
534
DESCRIPTION OF COMPUTER SOFTWARE
(4 FIGURE A.1 (Continued)
The relative permittivity and relative permeability of the surrounding homogeneous medium are the next input parameters. The type of structure is input to WIDESEPJN by the reference conductor parameter: 1 = wire, 2 = ground plane, and 3 = overall shield. The remaining parameters describe the wire radii and their cross-sectional positions. The first case, n wires with another wire as the reference wire, is depicted in Fig. A.l(a). The reference wire is located at the origin of a rectangular coordinate system. There are five pieces of input data in WIDESEPJN.
APPENDIX A
535
1. The number of wires, n, (exclusive of the reference wire. 2. The reference conductor type (type = 1). 3. The relative permittivity and permeability of the surrounding homogeneous medium. 4. The radius of the reference wire (in mils). 5. The radii of the other wires (in mils) along with their x coordinates (in meters) and y coordinates (in meters).
The last group are input sequentially for each wire: rwl, xl,yl, rw2, x2, Yz,
*
9
rwn, Xn, Y n .
The second case, n wires above an infinite ground plane as the reference conductor, is depicted in Fig. A.l(b). The ground plane forms the y coordinate axis (x = 0) of a rectangular coordinate system. There are four pieces of input data in WIDESEPJN. 1. The number of wires, n. 2. The reference conductor type (type = 2). 3. The relative permittivity and permeability of the surrounding homogeneous medium. 4. The radii of the other wires (in mils) along with their x coordinates (height above ground) (in meters) and y coordinates (in meters).
The last group are again input sequentially for each wire: rwl, hl, yl, rw2,h2, . . . ,Y, h,, yn. (The radius of a reference wire is again input but not read by the code.) The last case, n wires within an overall shield of radius r,,, as the reference conductor, is depicted in Fig. A.I(c). There are five pieces of input data in WIDESEPJN. yz,
1. The number of wires, n. 2. The reference conductor type (type 3). 3. The relative permittivity and permeability of the surrounding homogeneous medium. 4. The (interior) radius of the shield (in mils). 5. The radii of the other wires (in mils) along with their distances from the shield center, dl, (in mils) and their angular locations, e,, (in degrees).
The last group are again input sequentially for each wire: rwl,dl, el, rw2, d2, 029 * * * r w n , dn, e n * The entries in the upper triangle of the per-unit-length inductance and capacitance matrices are output to PUL.DAT. The last items in PUL.DAT simply restate the input parameters to insure that the correct problem was solved. These are not read by the subsequent application program. 9
536
DESCRIPTION OF COMPUTER SOFTWARE
A.1.2
Ribbon Cables: RIBBON.FOR
Required Input Files: RIBBONJN Output File: PULDAT
Consider the general N-wire ribbon cable shown in Fig. A.2. The line consists of N identical, dielectric-insulated wires. The wire radii are denoted in the code as RW,the insulation thicknesses are denoted by TD, and the identical adjacent wire spacings are denoted as S. The relative permittivity of the insulation is denoted by ER in the code. These input data are contained in RIBBONJN. The results of the computation are contained in the output file PUL.DAT. The last items in PULDAT simply restate the input parameters to insure that the correct problem was solved. These are not read by the subsequent application program. The structure of the main program RIBBON.FOR is as follows. Utilizing Tables 3.2 and 3.3 we can write the matrix that satisfies the boundary conditions. 1. The potentials of a conductor at points on that conductor due to all
charge distributions in the system are set equal to the potential of that conductor. 2. The components of the electric flux density vector, 9 = e&, that are normal to the dielectric-free-space boundary at points on that boundary are continuous across the boundary. This corresponds to matrix equation (3.87) which becomes of the form
FIGURE A2 Cross-sectional definition of the ribbon cable parameters for the RIBBON.
FOR FORTRAN program input data.
APPENDIX A
Expanding these gives
537
AU iBu' = Q,
(A.2a)
CU
(A.2b)
+ Du' = 0
The first set in (A.2a) enforce the potentials on the conductors and the second set in (A.2b) enforce continuity of the normal components of the electric flux density vector across the dielectric-free-space boundary. We will assume t b t the number of Fourier expansion functions around the conductor-dielectric boundary and around the dielectric-free-space boundary are equal. The number of expansion coefficients around each of these two boundaries is denoted as NF. The (N x NF) x 1 vector of Fourier expansion coefficients for the free and bound charge around the conductor-dielectric boundary is denoted by u. Similarly, the (N x NF) x 1 vector of Fourier expansion coefficients for the bound charge around the dielectric-free-space boundary is denoted by u', The (N x NF) x 1 vector Q, contains the potentials of the conductors at the matchpoints on those conductors. The (N x NF) x 1 zero vector 0 results from the satisfaction of the continuity of the normal components of the electric flux density vector across the dielectric-free-space boundary. The submatrices A, B, C, D are (N x NF) x (N x NF). Because of the repetitive nature of the ribbon cable structure, these submatrices have a special form:
and B, C,and D have a similar structure. This simplifies the fill time for (A.1). In order to avoid a singular solution matrix in (AJ), the matchpoints are chosen from the following scheme CC.2, C.61.On each conductor-dielectric interface and each dielectric-free-space interface, the N F matchpoints are separated by the angle e=- 2n (A.4a) NF These are rotated by an angle
A=-
R
2NF
(A.4b)
as illustrated in Fig. A.3. The matrix equations in (A.1) or (A.2) are solved as (A.4a)
(A.4b)
538
DESCRIPTION OF COMPUTER SOFTWARE
313
+
FIGURE A.3 Illustration of the scheme for rotating the matchpoints of a ribbon cable to avoid singular solution matrices.
Once these are solved, the total free charge on each conductor and the elements of the generalized capacitance matrix are obtained according to (3.90). The transmission-line-capacitance matrix, C, is obtained from the generalized capacitance matrix according to the algorithm given in (3.19) with the chosen reference conductor IREF. The generalized capacitance matrix with the dielectric removed is computed by computing the free charge with the dielectric removed:
and the transmission-line-capacitance matrix, C,,, is computed for the chosen reference conductor IREF.The transmission-line-inductancematrix is computed according to
L = p,,e,,c,"
(A.6)
The upper triangle of C and L are printed out to the fife PUL.DAT with L printed first. Observe the numbering of the conductors illustrated in Fig. A.4. The conductors to the left of the reference conductor are numbered sequentially from left to right and the remaining conductors to the right of the reference conductor are also numbered sequentially. This numbering is critical to observe when one generates the entries of the matrices and vectors of the terminal characterization as with a generalized Thtvenin equivalent in the programs that use these data. The inversion of the above real-valued matrices is accompiished with a standard Gauss-Jordan subroutine, GAUSSJ which is supplied as a part of the code RIBBON.FOR, This subroutine was taken with permission from [11.
APPENDIX A
539
Refe;ence conductor
I
FIGURE A.4 Illustration of the numbering scheme for determining the transmission-line capacitance matrix from the generalized capacitance matrix for a ribbon cable.
A.1.3
Printed Circuit Boards: PCBJOR, PCBGALFOR
Required Input Files: PCBJN Output File: PULDAT
This code determines the entries in the per-unit-length inductance and capacitance matrices, L and C,for all N-land PCB illustrated in Fig. AS. The problem that is solved is a special case wherein all lands have the same width, W,and identical edge-to-edge separations, S.The board has thickness, T,and relative dielectric constant, E?. The lands are numbered from left to right as shown. The generalized capacitance matrix for the structure is first solved using the results for potential given in Section 3.3.1. One of the input data parameters provided in PCBJN is the number of the desired reference land as illustrated in Fig.A.6. The code then solves for the transmission-line-capacitance matrix from the generalized capacitance matrix based on the desired reference conductor. The
Cross-sectional definition of the printed circuit board parameter8 for the PCB.FOR and PCBGALFOR FORTRAN program input data.
FIGURE A S
540
MSCRlFTlON OF COMPUTER SOFTWARE
I
Reference land FIGURE A.6 Illustration of the numbering scheme for determining the transmission-linecapacitance matrix from the generalized capacitance matrix for a printed circuit board.
code first solves for the capacitance matrix with the board removed and replaced with free space, C,, from which the transmission-line-inductance matrix is obtained as L = c(,E,C;~.Then the capacitance matrix with the board present, C, is determined. The matrix given in (3.101) relating the charge distributions over the land subsections to the potentials of the lands is formed and inverted as in (3.102). From this result the generalized capacitance matrix is formed as in (3.103). Then the algorithm in (3.19) is used to obtain the entries in the transmissionline-capacitance matrices, C and C, for the chosen reference land. For both cases, the matrix equation to be solved has, because of the assumption of identical land widths and edge-to-edge spacing, the following structure:
where t denotes the transpose of the matrix, u1contains the unknown levels of the charge distributions (assumed constant) of each subsection of the i-th land, and Cp, contains the potentials of the subsections of the 1-th land. The upper triangle of the per-unit-length inductance matrix is printed to PUL.DAT followed by the upper triangle of the per-unit-length capacitance matrix. These data are used by all subsequent analysis codes. Additionally, the per-unit-length capacitance matrix with the board removed, C,, is printed to PUL.DAT but is not used by the analysis codes. And finally the problem parameters are printed to PUL.DAT so that the user can insure that the problem solved is as desired. The inversion of (A.7) is again performed using the Gauss-Jordan subroutine that was used in RIBBON.FOR. Additionally, the code PCBGAL.FOR is supplied. This code implements the pulse expansion-Galerkin method discussed in Section 3.3.1, whereas the code PCB.FOR implements the pulse expansion-point matching method. As was shown in Chapter 3, the Galerkin method gives better convergence for a small
APPENDIX A
541
number of land subdivisions, but both methods converge rapidly as the number of land subsections is increased. A.1.4
Coupled Microstrip Structures: MSTRP.FOR, MSTRPCALFOR
Required Input Files: MSTRPJN Output File: PULDAT
These codes determine the n x n per-unit-length transmission line capacitance and inductance matrices, C and L, consisting of n equal-width lands with identical separations on a dielectric substrate with a ground plane on the opposite side. This code is useful in simulating a printed circuit board that has one or more inner planes that are buried at various distances within the board. Essentially it simulates the surface of the PCB where the traces are located and the innerplane closest to that surface. Since we assume a board of infinite width the remaining innerplanes are of no consequence since they are isolated in this calculation from the surface lands by the first innerplane. This code also simulates many other microstrip structures that are used in microwave circuits. The construction of these codes requires two simple modifications of the previous codes PCB.FOR and PCBGAL.FOR. The first modification is that the generalized capacitance matrix computed in the codes PCRFOR and PCBGAL.FOR is the transmission-line capacitance matrix for the microstrip structures. This is because the voltages of the n lands of the microstrip structure are taken with respect to the ground plane. It can be shown that these voltages are essentially the absolute potentials computed in the generalized capacitance matrix CB.11. Therefore the first modification of the codes PCB.FOR and PCBGAL.FOR is to simply remove the computations that determine the transmission-linecapacitance matrices for the generalized capacitance matrices. The second modification of the codes PCB.FOR and PCBGAL.FOR is to determine the absolute potential of an infinitesimal line charge on the surface of a dielectric sheet having a ground plane on the opposite side. This basic subproblem was solved for the PCB having no ground plane in Chapter 3 by imaging across the two dielectric surfaces as shown in Figure 3.36(b) resulting in
#(d) = -- aq ln[d2J 4nsre,
--a’q
4n&re,7 w -
i
kf2n-1)ln[d2+ (2nt)’J
(3.121)
In the case of the microstrip structure we image the line charge across the top dielectric surface and across the ground plane and obtain in a similar fashion
Comparing (3.121) and (A.8) we see that the codes PCB.FOR and
542
DESCRIPTION OF COMPUTER SOFTWARE
PCBGALFOR can be simply modified to analyze the microstrip structures by replacing in all summations
Cm k(2n-1) n= 1
0
PCB.FOR, PCBGAL.FOR
a\ -1
-
(A.9)
*
MSTRP-FOR, MSTRPGAL.FOR
There is one additional difference. In the case of PCB.FOR and PCBGAL.FOR with the dielectric removed (replaced with free space) k = 0 and the summation is zero.In the case of microstrip structures, MSTRP.FOR and MSTRPGALFOR, the first term of the summation is nonzero for k = 0, i.e., with the dielectric removed. This term represents the image across the ground plane and is present even with the dielectric removed.
A.2
FREQUENCY-DOMAIN ANALYSIS
One code, MThFOR, serves to determine the frequencydomain response of a MTL.According to the input data, one can consider lossless or lossy lines as well as homogeneous media or inhomogeneous media. The lossy nature of the conducton is contained in M l L I N and the inhomogeneity of the surrounding medium is determined by the per-unit-length parameters contained in PUhDAT. A.2.1
General: MTLFOR
Required Input Files: MTLIN, PUL.DA1, FREQJN Output File: MTL.OUT
The program forms the per-unit-length impedance and admittance matrices:
8 = R(f) +jwL 9 =j w c
(A. 1Oa) (A,lOb)
where the entries in R, L, and C are given in (2.13), (2.12c), and (2.24). The entries in the upper diagonal of L and C are read from PULDAT. The entries in R are input data from MTLIN. These are given for each conductor as the dc resistance, rda, and the break frequency, f,, where the resistance transitions to a skin-effect frequency dependence as rhr = The per-unitlength internal inductances of the conductors are also included in OL = wL, oL, and are determined from these data according to the scheme described in Section 3.6.2,4 assuming the high-frequency resistance and internal
fi +
rdom.
APPENDIX A
543
inductive reactance are equal, rhf = and they both transition at the same frequency, f,. If no conductor loss is desired, set the dc resistance to zero, and if no skin-eflect dependence is desired set fo larger than the largest analysis frequency, The analysis frequencies are read from the file FREQJN. The code determines the eigenvalues, y, and eigenvectors,T, of YZ as in (4.56). The entries in the generalized Thbvenin equivalent characterizations of the terminations as in (4.84) are read from the file MTLJN. Incorporating these into the general solution in (4.86) gives the 2n x 2n matrix to solve, (4.88), for the 2n undetermined constants in the vectors 1 .: The eigenvectors and eigenvalues of YZ are determined using the International Mathematical and Statistical Libraries (IMSL) version 9.2 subroutine EIGCC, Equation (4.88) is solved using the IMSL subroutine LEQTIC. Both subroutines were taken with permission from [2].
A.3
TIME-DOMAIN ANALYSIS
Four time-domain analysis codes are included. These implement the timedomain to frequency-domain transformation (TIMEFREQ.FOR), Branin's method (for a homogeneous medium) extended to MTL's (BRANIN.FOR), the finite difference-time domain method for lossless lines (FINDIF.FOR), and the finite difference-time domain method for lossy lines (FDTDLOSSFOR).
A.3,1 Time-Domain to Frequency-Domain Transformation: TIMEFREQ.FOR Required Input Files: TIMEFREQJN, MTLFREQ.DAT Output File: TIMEFREQ.OUT
This code determines the time-domain response of the MTL using the time-domain to frequency-domain transformation described in Section 5.2.3. The input signal is assumed to be a periodic pulse train with trapezoidal pulses. The Fourier series of this input signal is given by (5.99) and the coefficients are given by (5.100). The input file TIMEFREQJN contains the number of harmonics used, the pulse level, the repetition frequency of the pulse train, the duty cycle, the pulse rise/fall time, and the final solution time. In addition, the dc level of the transfer function, H(O), is input and multiplied by the dc level of the input spectrum, co, to give the dc level of the output spectrum. The input file MTLFREQ.DAT contains the frequency-domain transfer function (magnitude and phase) at each of the desired harmonics of the input signal. These can be computed with the frequency-domain analysis code MTLFOR that was described in the previous section. The program combines these data to compute the time-domain response according to (5.102).
544
DESCRIPTION OF COMPUTER SOWARE
A3.2
Branin's Method Extended to Multiconductor Lines: BRANIN.FOR
Required Input Files: BRANINJN, VSVLIN, PULDAT Output File: BRANIN.OUT
This code implements the extension of Branin's method extended to MTL's as described in Section 5.2.2. Equations (5.91) are solved recursively. The source and load voltage waveforms in Vs(t) and V,(t) in the generalized Th6venin equivalent representation of the terminations in (5.90) are described in a piecewise-linear fashion in the input file VSVLIN. The resistive entries in the matrices Rs and R, in (5.90) are input data in the file BRANINJN. These are assumed to be symmetric and the upper triangle of these are input. The code uses the subroutine GAUSSJ, described earlier, to invert L It is implicit in this method that the surrounding medium is homogeneous (for which L and C were computed).
A.3.3
Finite Difference-Time Domain Method: FINDIF.FOR
Required Input Files: FINDIFJN, VSVL.IN, PULDAT Output File: FINDIF.OUT
This code implements the finite difference-time domain method described in Section 5.2.5. Equations (5.140) are solved recursively. The discretizations of position and time are contained in FINDIFJN. The source and load input voltages are described in a piecewise-linear manner in the input file VSVLJN. Subroutine GAUSSJ is used to invert L and C.
A.3.4
Finite DiHerence-Time Domain Method: FDTDLOSS.FOR
Required Input Files: FDTDLOSS.IN, VSVL.IN, PULDAT Output File: FDTDLOSS.OUT
This code implements the finite difference-time domain method for lossy lines described in Section 5.3.1.3. It is essentially identical to FINDIF.FOR with the addition of the discrete convolution for the dc and skin-effect losses (resistance and internal inductance) described in Section 5.3.1.3. Equations (5.140) are solved recursively with the exception that (5.157) replaces (5.14Od), The discretizations of position and time are contained in FDTDLOSS.IN along with the dc resistance and skin-effect transition frequencies of the conductors according to the scheme described in Section 3.6.2.4. The source and load input voltages are described in a piecewise-linear manner in the input file VSVLIN. Subroutine GAUSSJ is used to invert L and C.
APPENDIX A
A.4
545
SPICE/PSPICE SUBCIRCUIT GENERATION PROGRAMS
This section describes three FORTRAN codes that generate SPICE/PSPICE subcircuit models that implement the SPICE method (SPICEMTL.FOR), the lumped-pi model (SPICELPLFOR), and the low-frequency, inductive-capacitive model (SPICELC.FOR). A.4.1
General Solution, Lossless Lines: SPICEMTL.FOR
Required Input Files: SPICEMTL.IN, PULDAT Output File: SPICEMTL.OUT
This code implements the exact SPICE model described in Section 5.2.1.3. The line is assumed to be lossless but the surrounding medium may be inhomogeneous as determined by the per-unit-length parameters in PUL.DAT. The per-unit-length inductance and capacitance matrices are read from PUL,.DATand diagonalized as in Section 5.2.1.2 using the subroutines DIAG and JACOBI. A subcircuit model is developed with the node numbering as 101, 102,. . ,10nfor the nodes of conductors 1,2,. . ,n at z = 0 and 201, 202,. , ,20n for the nodes of conductors 1, 2,. , ,n at z = 9 as illustrated in Fig. A.7. This subcircuit model can then be incorporated into a SPICE/PSPICE program. The total number of conductors, n 1, and the total line length are specified in the input file SPICEMTLIN.Note that the resulting SPICE/PSPICE code can also handle frequency-domain solutions by replacing the .TRAN run instruction with the .AC instruction CA.21,
.
A.4.2
.
+
.
.
Lumped-pi Circuit, Lossless lines: SPICELPI.FOR
Required Input Files: SPICELPIJN, PULDAT Output File: SPICELPI.OUT
This code implements a SPICE subcircuit for the lumped-pi model of a MTL described in Sections 4.5 and 5.2.4. The line is again assumed to be lossless but the surrounding medium may be inhomogeneous as determined by the perunit-length parameters in PUL.DAT. The per-unit-length inductance and capacitance matrices are read from PUL.DAT. A subcircuit model is again developed with the node numbering as before as 101,102,. , 1On for the nodes of conductors 1, 2,. , . ,n at z = 0 and 201, 202,. , , ,20n for the nodes of conductors 1, 2,. . ,n at z = 2 as is illustrated in Fig. A.7. This subcircuit model can then be incorporated into a SPICE/PSPICE program. The total number of conductors, n + 1, and the total line length are specified in the input file SPICELPLIN.Conductor losses can be incorporated by adding additional nodes and the appropriate line resistances and internal inductances to the resulting subcircuit model.
.
..
546
DESCRIPTION OF COMPUTER SOFTWARE
Subcircuit model conductor MTL
2-0
I i z-*v-z
FIGURE A7 Terminal node numbering scheme for the SPICE subcircuit models generated by the FORTRAN programs SPICEMTLFOR and SPICELPI.FOR models.
FIGURE A8
The SPICE subcircuit model generated by the code SPICELCFOR.
APPENDIX A
0-
@
. 9 9 -
Generator conductor
. 1 1 1 . ) 1 .
Receptor conductor
-0
.-
2-0
FIGURE A.9
A.4.3
547
@ 2-P -2
The node numbering scheme for the SPICE subcircuit model generated by the code SPICELC.FOR.
Inductive-Capacitive Coupling Model: SPICELC.FOR
Required Input Files: SPICELCJN Output File: SPICELC.OUT
This code implements the low-frequency, inductive-capacitive coupling model described in Section 6.2.3 for a three-conductor line. The line is assumed to be lossless but the surrounding medium may be inhomogeneous as determined by the per-unit-length mutual inductance and mutual capacitance parameters in SPICELC.IN. This input file also specifies the total line length. Figure A.8 shows the SPICE model that is developed. A subcircuit model, shown in Fig. A.9, is developed with the node numbering as S, NE for the two nodes of conductors 1, 2, z = 0 and L, FE for the nodes of conductors 1, 2 at z = 9. This subcircuit model can then be incorporated into a SPICE/PSPICE program. A.5
A.5.1
INCIDENT FIELD EXCITATION Frequency-Domain Program: INCIDENT.FOR
Required Input Files: INCIDENT.IN, PULDAT, FREQJN Output File: INCIDENT.OUT
This code determines the frequency-domain response of a general MTL to a single=frequency,uniform plane wave. Two types of line structures are provided for: an (n + 1)-conductor MTL and n conductors above an infinite, perfectly conducting ground plane. The terminations are characterized by generalized
548
DESCRIPTION OF COMPUTER SOWARE
ThCvenin equivalents. Lumped voltage sources can also be included in these terminations. Equations (7.43) are solved for the vectors of undetermined constants in the general solution, and the terminal voltages are obtained from equations (7.44). The forcing functions due to a uniform plane-wave incident field, 9,,,(9') and f,,(9'), are determined from equations (7.69) to (7.76). Essentially, the program MTL.FOR described earlier was modified to include the incident-field sources. The code requires the input files INCIDENTJN, PULDAT, and FREQ.IN. The file PUL.DAT is as before and gives the usual per-unit-length parameters of the line. The file FREQJN is also the same as for the program MTL.FOR and gives the solution frequencies sequentially. The file INCIDENTJN describes (in this order):
+
1. The total number of conductors (n 1). 2. The total line length. 3. The line type (1 = no ground plane, 2 = ground plane). 4. The output conductor for the terminal solution voltages. 5. The incident uniform plane-wave description (go,OE, e,, 4, with reference to Fig. A.10). 6. The conductor cross-sectional coordinates (sequentially according to Fig. A.11). 7. The dc resistances and skin-effect break frequencies (for the reference conductor and sequentially for the other n conductors) as described for
MTLFOR. 8. The generalized Thbvenin equivalent characterizations for the terminations
as described for MThFOR. The output file, INCIDENT.OUT, gives the termination solution voltages for the chosen output conductor (with respect to the reference conductor) sequentially for each solution frequency in FREQJN. These can be used in the previously described code, TIMEFREQ,FOR, to compute the time-domain response to a uniform plane wave that has a periodic, trapezoidal waveform.
A.52
SPICE/PSPICE Subcircuit Model: SPICEINC.FOR
Required Input Files: SPICEINCJN, PULDAT Output File: SPICEINC.OUT
This code generates a SPICE/PSPICE subcircuit model for an (n + 1)conductor MTL excited by a uniform plane wave, The input file, SPICEINCJN, contains the total number of conductors, the line length, the line type (1 = no ground plane, 2 = ground plane), the incident wave polarization and propaga-
APPENDIX A
549
(b) FIGURE A.10 Definition of (a) the angles of incidence and (b) polarization of the electric field for an incident uniform plane wave for the codes INCIDENT.FOR, SPICEINC,FOR, and FDTDINC.FOR.
tion direction with reference to Fig. A.10 (eE,e, 4,), and the cross-sectional conductor locations with reference to Fig. A.11 (X(i), Y(i)).The cross-sectional coordinates are entered sequentially as the last items in this input file. The other input file is the usual PUL.DAT which contains the per-unit-length parameters. The output file, SPICEINCOUT, contains the SPICE subcircuit model. The external nodes are labeled as in SPICEMTL and are shown in Fig. A.12. The code implements the model shown in Fig. 7.23.The external nodes at z = 0 are denoted as 101, 102,. . ,ion and the external nodes at z = 9 are denoted as 201, 202,...,20n. One additional node, 100, is external to which the time waveform source of the incident wave, d#), is attached. The code is restricted to 4, positive, i.e., components of the propagation vector in the +z direction. For propagation directions giving a component in the - 2 direction, simply reverse the line.
.
550
DESCRIPTION OF COMPUTER SOWARE
"t
(b)
FJCURE A.11 Definition of the cross-sectional dimensions for input data to the codes INCIDENT.FOR, SPICEINC.FOR, and FDTDINCFOR for (a) (n + 1) wires and
(b) n wires above a ground plane.
A.5.3
Finite Difference-Time Domain (FDTD) Model: FDTDINCFOR
Required Input Files: FDTDINCJN, PULDAT, EOJN Output File: FDTDINC.OUT
This code implements the FDTD method contained in equations (7.252). The input file FDTDINCIN contains the total number of conductors, the line length, the line type (1 = no ground plane, 2 = ground plane), the incident wave polarization and propagation direction with reference to Fig. A.10 (OB, e,, +,), the cross-sectional conductor locations with reference to Fig. A.11 (XU), the number of line position cells, NDZ,the number of time discretizations,
vi)),
APPENDIX A
551
. . . . I .
0
Subcircuit model for (n + 1) conductor MTL with incident field illumination
i
I
I
z=O
I
2-9
5
FIGURE A.12 Node numbering for the SPICE subcircuit model generated by the code SPICEINCFOR.
NDT, the final solution time, the print solution index, the output conductor for the terminal voltages, and the line terminations in the form of a generalized Thevenin equivalent without sources (Rs,RL,V, = 0, V L = 0). The input file PUL.DAT contains the usual per-unit-length parameters. The input file, EOJN, contains a piecewise-linear specification of t&(t). The output file, FDTDINCOUT, contains the solution at the discrete time steps at the endpoints of the chosen output conductor. REFERENCES
[l J W.H.Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes: The Art of Scient@c Computing, Cambridge University Press, NY, 1989. [Z] Visual Numerics, Inc., 9990 Richmond Ave., Suite 400, Houston, TX 770424548,
Index
Index Terms
Links
A Admittance matrix: lossy, homogeneous medium
160
lossy, inhomogeneous medium
159
234
Admittance parameters: series expansions of
338
transmission-line networks
495
Aerial mode Ampère’s law
229 14
frequency domain
161
lossy medium
158
77
Asymptotic waveform expansion, for lossy MTLs Attenuation constant, two-conductor line
347 190
B Basis functions, for charge expansions
105
Bergeron diagram
270
Bessel function
165
Biconical transmission line
43
BLT equations: symbolic solution
522
116 336
165
Index Terms
Links
BLT equations: (Cont.) transmission-line networks
491
508
514
519 Bound charge, in inhomogeneous media
103
Branin’s method
256
extended to lossy lines
340
extended to MTLs
288
two-conductor line
266
302
Broadside excitation, incident field illumination
428
432
C Capacitance: coaxial cable
91
two-conductor line
19
two-wire line
85
wire above ground line
90
88
Capacitance matrix: definition of
55
homogeneous medium
62
MTL definition of
69
Capacitive coupling coefficients
365
Cayley–Hamilton theorem, for powers of a matrix
217
Central differences, finite difference method
142
Chain parameter matrix:
221
incident field illumination
406
lossy two-conductor line
423
469
436
516
Index Terms
Links
Chain parameter matrix: (Cont.) phasor MTL
199
phasor solution
206
properties of
206
series expansion
200
two-conductor line
192
Characteristic impedance: lossy two-conductor line
423
two-conductor line
190
Characteristic impedance matrix frequency domain
1
237
204
Characteristic impedances: of modes
279
three-conductor line
366
Circuit, for inductances
66
Classes of transmission lines
26
Coaxial cable
91
capacitance of
91
conductance of
92
inductance of
92
Coaxial line, higher-order modes of
36
Coefficients of potential
69
Common-impedance coupling: frequency-domain solution
371
time-domain solution
382
Common-mode currents: creation by asymmetries incident field illumination
40 396
280
282
Index Terms
Links
Common-mode currents: (Cont.) three-conductor line
39
two-conductor line
37
Complementary error function
336
Computed results, incident field illumination
435
463
480
52
72
Conductance: coaxial cable
92
two-conductor line
20
two-wire line
89
Conductance matrix: definition of
55
homogeneous medium
158
inhomogeneous medium
158
MTL definition of Conduction current
72 9
Conductive loss
158
Conductivity, effective
104
Conformal mapping method
154
Continuity equation incident field illumination time domain
18 400 51
Convolution, incident field illumination
455
Convolution integral
292
lossy line solution Coupled microstrip
474
323 6
Coupling coefficient, three-conductor line
366
Courant condition
301
Cramer’s rule
365
307
Index Terms Crosstalk definition of
Links 39
40
3
Cross-sectional dimensions: parallel-plate line
35
TEM restrictions
3
Current : antenna
37
common mode
37
definition of
9
differential mode
37
transmission line
37
two-conductor line
15
Cutoff frequency: coaxial line
36
parallel-plate line
35
Cyclic symmetric structures: propagation constants of
228
transformation matrix of
228
parameter matrices of
226
D Decoupling MTL equations: frequency domain
202
incident field illumination
467
time domain
277
Dielectric half space, potential of line charge
129
Dielectrics: nonpolar
103
220
Index Terms
Links
Dielectrics: (Cont.) polar Difference operator Differential equations, types of
103 265
302
374
26
Differential-mode currents: incident field illumination
395
three-conductor line
39
two-conductor line
37
Diffusion equation wires Displacement current
162 165 9
52
E Effective dielectric constant: printed circuit board
136
ribbon cable
110
two-conductor line
29
Eigenvalues, of MTL
219
Eigenvectors, of MTL
219
Electric field, sinusoidal charge expansion
108
Electric flux density vector
104
Electrical length, definition of
30
Electrically-short line, incident field illumination
433
Electrically-small cross section
396
Endfire excitation, incident field illumination
426
Entire domain expansion functions
116
Equipotential contours
86
459 431
436
452
Index Terms
Links
Even–odd mode transformation
230
Exponential, matrix
207
F Faraday’s law
14
frequency domain
161
incident field illumination
396
integral form
165
46
Finite difference method
140
Finite differences
296
Finite difference–time domain method incident field illumination lossy lines
295
477
326
Finite element method
146
FORTRAN
531
FORTRAN codes: BRANIN.FOR
291
BRANIN.OUT
544
DIAG.SUB
283
E0.IN
550
EIGCC
543
FDTDINC.FOR
464
FDTDINC.IN
550
FDTDINC.OUT
550
FDTDLOSS.FOR
329
FDTDLOSS.IN
544
FDTDLOSS.OUT
544
FINDIF.FOR
311
311
544
545
480
550
343
544
312
317
544
Index Terms
Links
FORTRAN codes: (Cont.) FINDIF.IN
544
FINDIF.OUT
544
FREQ.IN
542
GAUSSJ
538
INCIDENT.FOR
422
441
442
443
444
464
477
485
283
545
547
547 INCIDENT.IN
547
INCIDENT.OUT
547
JACOBI.SUB
280
LEQTIC
543
MSTRPGAL.FOR
531
532
541
MSTRP.FOR
531
532
541
MSTRP.IN
531
541
MTLFREQ.DAT
543
MTL.FOR
225
238
239
243
312
326
332
330
542 MTL.IN
542
MTL.OUT
542
PCBGAL.FOR
135
138
159
243
285
317
329
531
532
539
133
135
138
159
531
532
539
PCB.IN
531
539
PUL.DAT
531
532
PCB.FOR
533
536
Index Terms
Links
FORTRAN codes: (Cont.)
RIBBON.FOR
539
541
542
544
545
547
549
550
109
112
159
239
311
383
531
532
536 RIBBON.IN
531
536
SPICEINC.FOR
490
492
532
SPICEINC.IN
548
SPICEINC.OUT
548
SPICELC.FOR
383
532
547
SPICELC.IN
547
SPICELC.OUT
547
SPICELPI.FOR
295
311
324
343
383
492
532
545
490
SPICELPI.IN
545
SPICELPI.OUT
545
SPICEMTL.FOR
311
312
383
492
532
545
SPICEMTL.IN
532
545
SPICEMTL.OUT
532
545
TIMEFREQ.FOR
294
311
312
317
326
329
343
444
464
485
506
543
TIMEFREQ.IN
543
TIMEFREQ.OUT
543
VSVL.IN
544
Index Terms
Links
FORTRAN codes: (Cont.) WIDESEP.FOR WIDESEP.IN Fourier series, for charge distributions
93
111
532
533
531
533
98
Fourier transform
293
Free charge
103
Free nodes, finite element method
149
449
G Galerkin
436
Galerkin method: general description of
123
potential of land
134
wide-separation approximation
124
Gauss’ law finite difference method Gauss’ laws, time domain Generalized capacitance matrix
80 143 8 173
conversion to MTL
76
definition of
74
Fourier charge expansions
102
pulse expansions
119
two-conductor line
76
Generator circuit, three-conductor line
361
Global nodes, finite element method
149
Green’s function
125
Ground mode
229
145
492
531
Index Terms
Links
Ground plane, incident field illumination
457
H Hermitian matrix
220
Higher-order modes
3
coaxial line
36
parallel plate line
30
two-wire line
37
Homogeneous medium
68
incident field illumination parameter properties
71
73
429
435
413 22
Hyperbolic cosine, inverse
88
Hyperdominant matrix
63
I Identity matrix
59
Impedance, internal of wires
166
Impedance parameters
234
transmission-line networks IMSL
506 543
Incident field illumination: explicit solution
453
ground plane
421
implicit solution
452
sources
407
SPICE equivalent circuit
460
time-domain solution
448
uniform plane wave solution
455
470 458
76
Index Terms
Links
Incident fields
396
Incident waves
509
Incremental inductance rule
168
Inductance: coaxial cable partial
92 170
two-conductor line
18
two-wire line
85
wire above ground line
90
88
Inductance matrix: definition of
50
MTL definition of
65
printed circuit board
133
Inductive and capacitive coupling: frequency-domain solution
367
time-domain solution
378
Inductive coupling coefficients
365
Infinite parallel-plate line, modes of
30
Inhomogeneous media: bound charge of invalidation of the TEM mode
103 29
Interconnection network, transmission-line networks Interference
489 3
Internal impedance, representation of
177
Internal inductance
161
surface impedance
163
wires
164
179
Index Terms
Links
Internal inductance matrix
320
Intrinsic impedance
418
lossless media
11
lossy media
13
164
Iterative solution: finite difference method
142
finite element method
151
K Kirchhoff’s laws, transmission-line networks
498
L Lands, on PCBs Laplace transform
6 180
254
371 in lossy line solution Laplace’s equation
334 8
exact solution for trough
122
finite difference method
141
finite element method
146
minimum energy solution
148
spectral-domain method
155
transverse fields
113
transverse fields of MTL
64
two-conductor line
26
Leaky modes, two-wire line
37
Lightning
489
Local nodes, finite element method
149
297
508
289
321
Index Terms
Links
Loss: conduction
104
polarization
104
Loss tangent
159
Lossy conductors, invalidation of the TEM mode
29
Lossy MTLs, decoupled
343
Lumped system
26
Lumped-circuit iterative structures
231
incident field illumination
415
lossy lines
324
in time-domain solution
295
transmission-line networks
492
292 475
M Magic time step
301
Magnetic flux density vector
79
Matched line, illustration of
315
Maxwell’s equations: differential form
7
frequency domain
31
integral form
13
Method of characteristics
256
incident field illumination
446
for lossy MTLs
344
two-conductor line
266
Method of images dielectric half space
82 128
307
Index Terms Method of moments incident field illumination
Links 40
115
435
Mixed termination representation: incident field illumination
413
terminal constraints
214
transmission-line networks
497
Mode voltages and currents
277
frequency domain
201
Modes: incident field illumination of MTL
467 2
MTL equation phasor solution: mixed terminal representation
214
Norton equivalent
214
Thévenin equivalent
212
MTL equations
54
frequency domain
187
188
general phasor solution
199
204
incident field illumination
402
444
lossless time, time domain
275
lossless lines lossy lines
58 321
matrix form
57
second-order form
58
series solution time domain
291 50
MTL equations solution, plane wave illumination
475
57
446
Index Terms
Links
MTL solution, general process
2
MTLs, examples of
4
N n wires above ground, inductance matrix
95
n wires within a shield, inductance matrix
97
Nonuniform line: approximate representation of
216
definition of
27
examples of
27
Normal matrix
220
Norton equivalent: incident field illumination
412
terminal constraints
213
Numerical recipes n + 1 wires, inductance matrix
551 93
O Orthogonal transformation
223
P Parameter, in FORTRAN codes
531
Partial inductance
170
Permeability, of free space
6
Permittivity: complex or free space
158 6
Per-unit-length equivalent circuit: of MTL
56
227
Index Terms
Links
Per-unit-length equivalent circuit: (Cont.) two-conductor line
22
Per-unit-length parameter matrices: eigenvalues of
61
homogeneous medium
58
positive definiteness of
61
properties of
58
symmetry of
60
Per-unit-length parameters, two-conductor line Phase constant, two-conductor line
60
24 190
Phasor MTL equations, incident field illumination Phasors
405 186
Pigtails on shielded lines, approximate representation
218
Point matching
118
Poisson's equation
115
Polarization charge
103
Polarization loss
158
Positive definite matrix
61
Potential, sinusoidal charge expansion
100
Power flow
238
Prescribed nodes, finite element method
149
Printed circuit board
124
computed results
317
lossy line
350
computed results for
136
dimensions of
136
108
388
Index Terms
Links
Printed circuit board (Cont.) effective dielectric constant
136
example of
125
Galerkin method potential
127
inductance matrix
133
phasor computed results
241
potential of land
132
potential of line charge
130
pulse expansion potential
125
SPICE equivalent circuit
285
wide-separation approximation
136
Printed circuit board lands: computed results
176
internal inductance of
168
resistance of
168
Prony's method, internal impedance
327
Propagation constant: lossy two-conductor line
423
of the TEM mode
12
two-conductor line
189
Propagation matrix, tube
510
Pulse expansion: for charge distributions
117
potential of land
132
Q Quadratic form, finite element method
148
134
134 132
Index Terms Quasi-TEM assumption definition of
Links 4 29
R Radiated emissions, two-conductor line Receptor circuit, three-conductor line Reference conductor, of MTL Reflected waves
38 362 46 509
Reflection coefficient: current
512
two-conductor line
191
194
1
236
current
257
516
voltage
257
516
Resistance, wires
164
Reflection coefficient matrix
257
Reflection coefficients:
Resistance matrix, definition of
51
Ribbon cable
67
computed parameters of
109
computed results
311
computed results, lossy line
348
conductance matrix for
160
phasor computed results
239
dimensions of
109
S Scattered fields
396
Scattered voltage, definition
397
383
518
521
Index Terms
Links
Scattering matrix: current
512
junction
511
Scattering parameters transmission-line networks Separation of variables
491 508 32
Shielded MTLs, parameters of
157
Sidefire excitation, incident field illumination
427
432
436
Similarity transformation, phasor MTL equation solution Similarity transformations, for lossy lines
201 343
Sinusoidal charge expansions: electric field of
108
potential of
100
108
Skin depth
320
Skin effect
161
174
Snell's law
421
429
Spectral-domain method
155
SPICE
295
MTL model
283
two-conductor line model
270
SPICE equivalent circuit, incident field illumination
460
Square root, of matrix
206
State-transition matrix
198
State-variable equations
405
analogy to MTL equations
188
of lumped systems
195
470 406
324
Index Terms
Links
State-variable equations (Cont.) phasor MTL equations analogy
199
state-transition matrix
198
Subdomain expansion functions
116
Superposition
293
Surface impedance, conductive half space
162
Symmetric line
359
Symmetrical components
229
T TE mode, parallel-plate line
33
TEM mode, basic MTL assumption
3
TEM mode of propagation, properties of
7
4
Terminal constraints: incident field illumination
404
mixed representations
214
Norton equivalents
213
Thévenin equivalents
210
Termination network, transmission-line networks
489
Thévenin equivalent: incident field illumination
411
terminal constaints
210
Thin-film circuit
329
Three-conductor line
359
characterization
362
explicit solution
374
frequency-domain solution
365
410
Index Terms
Links
Three-conductor line (Cont.) recursive solution
373
Time-domain reflectometer
354
Time-domain to frequency-domain: incident field illumination
475
lossy line solution
325
transformation
292
Time-Shift operator
265
TM mode, parallel-plate line Transmission coefficients, voltage Transmission-line equations, two-conductor line
374
34 518
521
18
20
Transmission-line networks
489
Transposed line
229
Trapezoidal pulse train: Fourier series of
293
spectral bounds
381
380
Traveling waves: TEM mode
11
transmission-line networks
517
two-conductor line
191
Tube, transmission-line network
489
Twisted pair lines, approximate representation of Two port, two-conductor line as
217 192
Two-conductor line: characteristic impedance of
254
equivalent circuit
269
frequency-domain solution
190
257
452
Index Terms
Links
Two-conductor line: (Cont.) incident field illumination
423
446
input impedance of
194
lossless line equations
253
phasor voltage and current of
193
power flow on
194
propagation constant of
189
series solution
262
265
time-domain solution
190
256
uniform plane wave
425
Two-conductor lossy line: equations of
323
as a two port
337
Two-wire line: capacitance of
85
88
conductance of
89
inductance of
85
88
5
13
U Uniform line
189
Uniform plane wave: frequency-domain characterization
416
time-domain characterization
395
453
Velocities of propagation, of modes
279
280
Visual Numerics
551
V 282
Index Terms
Links
Voltage: definition of two-conductor line Voltages, of MTL
9 15 48
W Wave equations, frequency domain
32
Waves, traveling
11
Weakly-coupled line three-conductor line
360 366
Wide-separation approximations: n wires above ground
95
n wires within a shield
97
n + 1 wires
93
printed circuit board
136
ribbon cable
111
Wire: electric field of internal inductance of
81 164
magnetic field of
77
magnetic flux of
77
resistance of voltage of
164 81
Wire above ground line: capacitance of
90
inductance of
90
375
