TRANSMISSION LINES IN DIGITAL AND ANALOG ELECTRONIC SYSTEMS

TRANSMISSION LINES IN DIGITAL AND ANALOG ELECTRONIC SYSTEMS

TRANSMISSION LINES IN DIGITAL AND ANALOG ELECTRONIC SYSTEMS Signal Integrity and Crosstalk CLAYTON R. PAUL Department of Electrical and Computer Engineering Mercer University Macon, Georgia and Emeritus Professor of Electrical Engineering University of Kentucky Lexington, Kentucky

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This book is dedicated to the humane and compassionate treatment of animals and my beloved pets: Patsy, Dusty, Megan, Tinker, Bunny, Winston, Sweetheart, Lady, Tigger, Beaver, Ditso, Buru, Old Dog, Zip, Tara, Timothy, Kiko, Valerie, Red, Sunny, Johnny, Millie, Molly, Angel, Autumn, and Shabby. Those readers who are interested in the humane and compassionate treatment of animals are encouraged to donate to The Clayton and Carol Paul Fund for Animal Welfare c/o the Community Foundation of Central Georgia 277 MLK, Jr. Blvd Suite 303 Macon, GA 31202 The primary and only objective of this Fund is to provide monetary grants to (1) animal humane societies (2) animal shelters (3) animal adoption agencies (4) low-cost spay-neuter clinics (5) individual wildlife rehabilitators (6) as well as other organizations devoted to animal welfare in order to allow these volunteer organizations to use their enormous enthusiasm, drive and willingness to reduce animal suffering and homelessness through the monetary maintenance of their organizations where little or no monetary funds existed previously.

CONTENTS Preface xi 1 Basic Skills and Concepts Having Application to Transmission Lines 1 1.1 Units and Unit Conversion 3 1.2 Waves, Time Delay, Phase Shift, Wavelength, and Electrical Dimensions 6 1.3 The Time Domain vs. the Frequency Domain 11 1.3.1 Spectra of Digital Signals 12 1.3.2 Bandwidth of Digital Signals 17 1.3.3 Computing the Time-Domain Response of Transmission Lines Having Linear Terminations Using Fourier Methods and Superposition 27 1.4 The Basic Transmission-Line Problem 31 1.4.1 Two-Conductor Transmission Lines and Signal Integrity 32 1.4.2 Multiconductor Transmission Lines and Crosstalk 41 Problems 46 vii

viii CONTENTS PART I TWO-CONDUCTOR LINES AND SIGNAL INTEGRITY 49 2 Time-Domain Analysis of Two-Conductor Lines 51 2.1 The Transverse Electromagnetic (TEM) Mode of Propagation and the Transmission-Line Equations 52 2.2 The Per-Unit-Length Parameters 56 2.2.1 Wire-Type Lines 57 2.2.2 Lines of Rectangular Cross Section 68 2.3 The General Solutions for the Line Voltage and Current 71 2.4 Wave Tracing and Reflection Coefficients 74 2.5 The SPICE (PSPICE) Exact Transmission-Line Model 84 2.6 Lumped-Circuit Approximate Models of the Line 91 2.7 Effects of Reactive Terminations on Terminal Waveforms 92 2.7.1 Effect of Capacitive Terminations 92 2.7.2 Effect of Inductive Terminations 94 2.8 Matching Schemes for Signal Integrity 96 2.9 Bandwidth and Signal Integrity: When Does the Line Not Matter? 104 2.10 Effect of Line Discontinuities 105 2.11 Driving Multiple Lines 111 Problems 113 3 Frequency-Domain Analysis of Two-Conductor Lines 121 3.1 The Transmission-Line Equations for Sinusoidal Steady-State Excitation of the Line 122 3.2 The General Solution for the Terminal Voltages and Currents 123 3.3 The Voltage Reflection Coefficient and Input Impedance to the Line 123 3.4 The Solution for the Terminal Voltages and Currents 125 3.5 The SPICE Solution 128 3.6 Voltage and Current as a Function of Position on the Line 130 3.7 Matching and VSWR 133 3.8 Power Flow on the Line 134 3.9 Alternative Forms of the Results 137 3.10 The Smith Chart 138 3.11 Effects of Line Losses 147

CONTENTS ix 3.12 Lumped-Circuit Approximations for Electrically Short Lines 161 3.13 Construction of Microwave Circuit Components Using Transmission Lines 167 Problems 170 PART II THREE-CONDUCTOR LINES AND CROSSTALK 175 4 The Transmission-Line Equations for Three-Conductor Lines 177 4.1 The Transmission-Line Equations for Three-Conductor Lines 177 4.2 The Per-Unit-Length Parameters 184 4.2.1 Wide-Separation Approximations for Wires 185 4.2.2 Numerical Methods 196 Problems 205 5 Solution of the Transmission-Line Equations for Three-Conductor Lossless Lines 207 5.1 Decoupling the Transmission-Line Equations with Mode Transformations 208 5.2 The SPICE Subcircuit Model 210 5.3 Lumped-Circuit Approximate Models of the Line 227 5.4 The Inductive-Capacitive Coupling Approximate Model 232 Problems 236 6 Solution of the Transmission-Line Equations for Three-Conductor Lossy Lines 239 6.1 The Transmission-Line Equations for Three-Conductor Lossy Lines 240 6.2 Characterization of Conductor and Dielectric Losses 244 6.2.1 Conductor Losses and Skin Effect 244 6.2.2 Dielectric Losses 248 6.3 Solution of the Phasor (Frequency-Domain) Transmission-Line Equations for a Three-Conductor Lossy Line 251 6.4 Common-Impedance Coupling 260 6.5 The Time-Domain to Frequency-Domain Method 261 Problems 270 Appendix A Brief Tutorial on Using PSPICE 273 Index 295

PREFACE This book is intended as a textbook for a senior/first-year graduate-level course in transmission lines in electrical engineering (EE) and computer engineering (CpE) curricula. It has been class tested at the author s institution, Mercer University, and contains virtually all the material needed for a student to become competent in all aspects of transmission lines in today s highfrequency analog and high-speed digital world. The book is also essential for industry professionals as a compact review of transmission-line fundamentals. Until as recently as a decade ago, digital system clock speeds and data rates were in the hundreds of megahertz range. Prior to that time, the lands on printed circuit boards (PCBs) that interconnect the electronic modules had little or no impact on the proper functioning of those electronic circuits. Today, the clock and data speeds have moved into the low gigahertz range. As the demand for faster data processing continues to escalate, these speeds will no doubt continue to increase into the gigahertz frequency range. In addition, analog communication frequencies have also moved steadily into the gigahertz range and will no doubt continue to increase. Although the physical dimensions of these lands and the PCBs supporting them have not changed significantly over these intervening years, the spectral content of the signals they carry has increased significantly. Because of this the electrical dimensions (in wavelengths) of the lands have increased to the point where these interconnects have a significant effect on the signals they are carrying, so that just getting the systems to work properly has become a major design problem. This has generated a new design problem, referred to as signal integrity. Good signal integrity means that the interconnect conductors should not adversely xi

xii PREFACE affect the operation of the modules that the conductors interconnect. Prior to some 10 years ago, these interconnects could be modeled reliably with lumped-circuit models that are easily analyzed using Kirchhoff s voltage and current laws and other lumped-circuit analysis methods. Because these interconnects are becoming electrically long, lumped-circuit modeling of them is becoming inadequate and gives erroneous answers. Most interconnect conductors must now be treated as distributed-circuit transmission lines. In the last 30 years there have been dramatic changes in electrical technology, yet the length of the undergraduate curriculum has remained four years. Since the undergraduate curriculum is a zero-sum game, the introduction of courses necessitated by the advancements in technology, in particular digital technology, has caused many of the standard topics to disappear from the curriculum or be moved to senior technical electives which not all graduates take. The subject of transmission lines is an important example of this. Until a decade ago, the analysis of transmission lines was a standard topic in the EE and CpE undergraduate curricula. Today most of the undergraduate curricula contain a rather brief study of the analysis of transmission lines in a one-semester junior-level course on electromagnetics (often the only course on electromagnetics in the required curriculum). In some schools, this study of transmission lines is relegated to a senior technical elective or has disappeared from the curriculum altogether. This raises a serious problem in the preparation of EE and CpE undergraduates to be competent in the modern industrial world. For the reasons mentioned above, today s undergraduates lack the basic skills to design high-speed digital and high-frequency analog systems. It does little good to write sophisticated software if the hardware is unable to process the instructions. This problem will increase as the speeds and frequencies of these systems continue to increase, seemingly without bound. This book is meant to repair that basic deficiency. In Chapter 1, the fundamental concepts of waves, wavelength, time delay, and electrical dimensions are discussed. In addition, the bandwidth of digital signals and its relation to pulse rise and fall times is discussed. Preliminary discussions of signal integrity and crosstalk are also given. Part I contains two chapters covering two-conductor transmission lines and designing for signal integrity. Chapter 2 covers the time-domain analysis of those transmission lines. The transmission-line equations are derived and solved, and the important concept of characteristic impedance is covered. The important per-unit-length parameters of inductance and capacitance that distinguish one line from another are obtained for typical lines. The terminal voltages and currents of lines with various source waveforms and resistive terminations are computed by hand via wave tracing. This gives considerable insight into the general behavior of transmission lines in terms of forward- and

PREFACE xiii backward-traveling waves and their reflections. The SPICE computer program and its personal computer version, PSPICE, contain an exact model for a two-conductor lossless line and is discussed as a computational aid in solving for transmission-line terminal voltages and currents. SPICE is an important computational tool since it provides a determination of the terminal voltages and currents for practical linear and nonlinear terminations such as CMOS and bipolar devices, for which hand analysis is very formidable. Matching schemes for achieving signal integrity are covered, as are the effects of line discontinuities. Chapter 3 covers the corresponding analysis in the frequency domain. The important analog concepts of input impedance to the line, VSWR and the Smith chart (which provides considerable insight), are also discussed. The effect of line losses, including skin effect in the line conductors and dielectric losses in the surrounding dielectric, are becoming increasingly critical, and their detrimental effects are discussed. Part II repeats these topics for three-conductor lines in terms of the important detrimental effects of crosstalk between transmission lines. Crosstalk is becoming of paramount concern in the design of today s high-speed and high-frequency electronic systems. The transmission-line equations for threeconductor lossless lines are derived, and the important per-unit-length matrices of the inductance and capacitance of the lines are covered in Chapter 4. Numerical methods for computing the per-unit-length parameter matrices of inductance and capacitance are studied, and computer programs are given that compute these numerically for ribbon cables and various structures commonly found on PCBs. Chapter 5 covers the solution of three-conductor lossless lines via mode decoupling. A SPICE subcircuit model is determined via this decoupling and implemented in the computer program SPICEMTL.EXE. This program performs the tedious diagonalization of the per-unit-length parameter matrices and gives as output a SPICE subcircuit for modeling lossless coupled lines. As in the case of two-conductor lines, this allows the study of line responses not only for resistive loads but, more important, nonlinear and/or reactive loads such as CMOS and bipolar devices that are common line terminations in today s digital systems. How to incorporate the frequency-dependent losses of the line conductors and the surrounding dielectric into a solution for the crosstalk voltages is discussed in Chapter 6. The frequency-domain solution of the MTL equations is again given in terms of similarity transformations in the frequency domain. The time-domain solution for the crosstalk voltages is obtained in terms of the frequency-domain transfer function, which is obtained by superimposing the responses to the Fourier components of V S ðtþ. The appendix gives a brief tutorial of SPICE (PSPICE), which is used extensively throughout the book. Several computer programs used and described in this book for computing the per-unit-length parameter matrices

xiv PREFACE and a subcircuit model for three-conductor lines are contained in a CD that is included with the book along with two MATLAB programs for computing the Fourier components of a digital waveform. The CD also contains two versions of PSPICE. Each chapter concludes with numerous problems for the reader to practice his or her understanding of the material. The answers to those that are simply stated are given in brackets, [], at the end of the question. The answers to most of the other problems can be verified using PSPICE. In those cases, the hand calculations should be checked using PSPICE. If these disagree, there is an error in either (1) the hand calculation, (2) the PSPICE setup, or (3) both. In this case, the reader should determine the error so that both answers agree. Getting the hand calculations and those obtained with PSPICE to agree is a tremendously useful learning tool. This book grew out of the realization that most of today s EE and CpE graduates lack a critically important skill: the analysis of transmission lines. If we, as educators, are to prepare our graduates adequately for the increasingly difficult design problems of a high-speed digital world, it is imperative that we institute a dedicated course devoted to the analysis of transmission lines. This book is devoted to achieving that objective. CLAYTON R. PAUL Macon, Georgia