Analysis of Multiconductor Transmission Lines (Wiley Series in Microwave & Optical Engineering, 28)

WlLEY SERIES IN MICROWAVE AND OPTICAL ENGINEERING KAI CHANC, Editor Texas A & M University INTRODUCTION TO ELECTROMAGN...

Author: Clayton R. Paul

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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

A NOTE TO THE READER

This book has been electronically reproduced Erom digital infommtion stored at John Wiley k Sons,Inc. We are pleased that the use of this new technology will enable us to keep works of enduing scholarly value in print as long as there is a reasonable demand for them. The content of this book is identical to previous printings.

This text is printed on acid-free paper. Copyright Q 1994 by John Wiley & Sons, Inc, All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008.

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

&@


+ 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


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)


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.


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


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)


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)


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)


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.


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


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


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


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


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


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


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


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


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


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


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)


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


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


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.

LINES HAVING CONDUCTORS OF CIRCULAR CYLINDRICAL CROSS SECTION

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)


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


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)


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


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.


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


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


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


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


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)


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


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)


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


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


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


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.


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


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


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)


..

...

...

(3.78b)

102


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)


(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


103

determine a,,, and from (3.79),

(3.81)


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


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,


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


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.


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


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


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.


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


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


.!

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.


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


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


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


...

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)


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)


(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


'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


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


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

-


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


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


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


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


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


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


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


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


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


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


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


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


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=[


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


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 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


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


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


. 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


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)


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 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~)


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


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


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


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


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


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].


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


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


+

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


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


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


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


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)


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


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


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)


+

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


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


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


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


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


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


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


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).


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


.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


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


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)


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


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


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


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


+ 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%


... 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


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


(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,


+ 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


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


+ 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


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)


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


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)


+ 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


(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


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,


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


(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


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


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


+ 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


--

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)


+ 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


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


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)


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


-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


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


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)


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


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


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


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


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


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., = 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


(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


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


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

- 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 ) = '

. -

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


7.6


7.7


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


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


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


-

.

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%


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


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


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


..

(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


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


I5

I

1

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I I

40 -

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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

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