Transmission Lines in Digital and Analog Electronic Systems: Signal Integrity and Crosstalk

By Clayton R. Paul

In the last 30 years there have been dramatic changes in electrical technology--yet the length of the undergraduate curriculum has remained four years.  Until some ten years 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. 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.

• Language: English
• Publisher: Wiley
• Release date: Jan 11, 2011
• ISBN: 9781118058244

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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

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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.

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 high-frequency 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 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 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 three-conductor 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 .

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 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

Chapter 1

Basic Skills and Concepts Having Application to Transmission Lines

We are rapidly moving into a digital era. Until as recently as some 10 years ago, clock speeds and data rates of digital systems were in the hundreds of megahertz (MHz) range, with rise and fall times of the pulses in the nanosecond ( ) range. Prior to that time, the lands (conductors of rectangular cross section) that interconnect the electronic modules on printed circuit boards (PCBs) had little or no impact on the proper functioning of those electronic circuits. Today, the clock and data speeds have moved rapidly into the low-gigahertz (GHz) range. The rise and fall times of those digital waveforms have decreased into the picosecond ( ) range. For example, a 1-GHz digital clock signal consists of trapezoidal-shaped pulses having rise and fall times on the order of 100 ps or less. A digital clock waveform is illustrated in Fig. 1.1.

Figure 1.1 Typical digital clock/data waveform.

The period of the periodic waveform T is the reciprocal of the clock fundamental frequency . The rise and fall times are denoted as and , respectively, and the pulse width (between 50% levels) is denoted as . Digital data waveforms are similar except that the period starts immediately after the previous pulse [i.e. ], and the occurrence of a pulse during the adjacent time intervals is random. As the frequencies of the clocks increase, the period T decreases and hence the rise and fall times of the pulses must be reduced commensurately in order that pulses resemble a trapezoidal shape rather than a sawtooth waveform, thereby giving adequate setup and hold time intervals. Reducing the pulse rise and fall times has had the consequence of increasing the spectral content of the waveshape according to the Fourier series of the waveform. Typically, this spectral content is significant up to the inverse of the rise and fall times, , as we will see.

For example, a 1-GHz digital clock signal having rise and fall times of 100 ps has significant spectral content at multiples (harmonics) of the basic clock frequency (1 GHz, 2 GHz, 3 GHz, ….) up to around 10 GHz. As the demand for faster data processing continues to escalate, these speeds will no doubt continue to increase into the gigahertz frequency range. The pulse rise and fall times will be reduced commensurately, thereby increasing the spectral content further into the gigahertz frequency range. This also applies to mixed-signal systems containing both digital and analog signals.

Although the physical lengths of the lands that interconnect the electronic modules on the PCBs have not changed significantly over these intervening years, their electrical lengths (in wavelengths) have increased because of the increased spectral content of the signals that the lands carry. Today these interconnects can 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. Remember that it does no good to write sophisticated software if the hardware cannot execute those instructions faithfully. This has generated a new design problem, referred to as signal integrity. Good signal integrity means that the interconnect conductors (the lands) should not adversely 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 lumped-circuit analysis methods, or could be ignored altogether. Because these interconnects are becoming electrically long, lumped-circuit modeling of them is becoming inadequate and gives erroneous answers. The interconnect conductors must now be treated as distributed-circuit transmission lines. The interaction of the electric and magnetic fields between two adjacent transmission lines also causes portions of the voltage and current waveforms on one line to appear inadvertently at the ends of the adjacent line, thereby creating potential interference problems in the electronic devices that the adjacent line interconnects. This is called crosstalk and is also rapidly becoming a significant problem in high-speed digital electronics.

This book is intended to be a thorough but concise description of the analysis of transmission lines with respect to signal integrity and crosstalk in modern high-speed digital and high-frequency analog systems. This chapter covers some important basic skills and concepts that facilitate an understanding of the behavior of those transmission lines as well as demonstrating when the interconnects need to be considered as transmission lines.

1.1 Units and Unit Conversion

The internationally accepted system of units is the International System, or SI system, where the primary units are the meter, kilogram, second, and ampere, thus termed the MKSA system. All quantities in any formula or law must be in these units. For example, Coulomb's law for the force between two point charges that are separated a distance R is

The SI units of the charges and are coulombs (C), and the distance between the two charges, R, is in meters (m). The SI units of force are newtons (N). The constant in the denominator is the permittivity of free space (essentially air):

We will see in the other electromagnetic laws a similar constant, known as the permeability of free space:

These two constants have the units of a capacitance in farads (F) per unit of length and an inductance in henrys (H) per unit of length. These important constants appear throughout the laws of electromagnetics and in transmission-line applications and should be committed to memory. The speed of light in a vacuum (and essentially in air) is

Throughout science we must deal with numbers spanning many orders of magnitude. Table 1.1