Classical Circuit Theory

Classical Circuit Theory Omar Wing Classical Circuit Theory Omar Wing Columbia University New York, NY USA Library of Congress Control Number: 2008931852 ISBN 978-0-387-09739-8 e-ISBN 978-0-387-09740-4 Printed on acid-free paper. 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. While the advice and information in this book are believed to be true and accurate at the date of going to press, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. 9 8 7 6 5 4 3 2 1 springer.com To all my students, worldwide Preface Classical circuit theory is a mathematical theory of linear, passive circuits, namely, circuits composed of resistors, capacitors and inductors. Like many a thing classical, it is old and enduring, structured and precise, simple and elegant. It is simple in that everything in it can be deduced from first principles based on a few physical laws. It is enduring in that the things we can say about linear, passive circuits are universally true, unchanging. No matter how complex a circuit may be, as long as it consists of these three kinds of elements, its behavior must be as prescribed by the theory. The theory tells us what circuits can and cannot do. As expected of any good theory, classical circuit theory is also useful. Its ulti- mate application is circuit design. The theory leads us to a design methodology that is systematic and precise. It is based on just two fundamental theorems: that the impedance function of a linear, passive circuit is a positive real function, and that the transfer function is a bounded real function, of a complex variable. In this book, we begin with basic principles of circuits, derive their analytic properties in both the time and frequency domains, and state and prove the two important theorems. We then develop an algorithmic method to design common and uncom- mon types of circuits, such as prototype filters, lumped delay lines, constant phase difference circuits, and delay equalizers. Along the way, we learn about the relation between gain and phase, linear and minimum phase functions, group delay, sensitivity functions, scattering matrix, synthesis of transfer functions, approximation of filter functions, all-pass circuits, and circuit design by optimization. The book is written as a text suitable for use by seniors or first year graduate students in a second course in circuit theory. It can be covered in one semester or one quarter at a brisk pace. Chapter Two, which is on fundamentals, and Chapter Three, which is on circuits in the time domain, can be omitted, if the principal aim of the course is circuit design. In this age of digital signal processing and analog electronic filters, one may ask why we want to study passive filter design. The reason is that the operating frequency of a digital filter is limited by how fast we can sample an input signal and by how fast we can process the signal samples digitally. While digital circuitry has vii viii Preface made steady improvement in its operating speed, it is still too slow for some circuits used in many wireless communication systems. As to analog electronic filters, scores of configurations have been proposed over the years. The conventional design is to realize a transfer function as a cascade of second order sections, each of which is implemented in a circuit of resistors, capacitors, integrators and summers. More recently, it is found that a better methodology is to first realize a passive filter as a lossless (inductor-capacitor) ladder terminated in resistors, and then to simulate the passive filter with integrators and summers. The new configuration has superior sensitivity properties. What all this means is that we need to know how passive filters are designed in the first place. Similarly, a microwave filter is usually first designed as a passive filter with inductors and capacitors. Then each inductor is replaced by a length of transmission line of certain characteristic impedance and each capacitor by another line of dif- ferent length and characteristic impedance. More commonly, the inductor-capacitor resonant sub-circuits in a passive filter are replaced by microwave resonators con- nected by short lengths of transmission lines. The basis of design is again classical circuit theory. Modern textbooks on analog electronic filters and microwave filters abound, but not those on classical circuit theory. It seems there is a need for a modern text on the mathematical foundations of passive circuit analysis and design, which are what classical circuit theory is all about. Classical circuit theory is old, but it has survived the test of time and it is still relevant today because it is basic. In writing this book, I have benefited from feedback from students at Columbia University, where the draft of the book had been classroom-tested three times. Their comments had been helpful and are hereby gratefully acknowledged. I also want to thank the Department of Electrical Engineering, Columbia Univer- sity, where I taught circuit theory for thirty-five years, for its continual support. In addition, I am grateful to the following institutions for having invited me to teach various aspects of circuit theory to a collective group of outstanding students. The institutions are: Chiao Tong University, Taiwan (1961); The Technical University of Denmark (1973); Indian Institute of Technology, Kanpur (1978); South China University of Technology (1979); Eindhoven University of Technology (1979-80); Shanghai Jiao Tong University (1982, 2000); East China University of Technology (1985, 1990); Beijing Post and Telecommunications University (1990); The Chinese University of Hong Kong (1991-98); Fudan University, Shanghai (2001, 2003); and Columbia Video Network (2005). A book is the work of its author. But to craft it into an endearing product requires the assistance of an experienced editorial and production staff. The staff at Springer played this role, to whom I am most grateful. Pomona, New York Omar Wing June 2008 Contents 1 Introduction ................................................... 1 1.1 A brief history . 1 1.2 What drives circuit theory? . 4 1.3 Scope of this book . 5 1.4 Mathematical programming tools . 6 1.5 Notable people in classical circuit theory . 7 2 Fundamentals ................................................. 11 2.1 Kirchhoff’s laws . 11 2.2 Linear and nonlinear elements . 12 2.3 Linear and nonlinear circuits . 14 2.4 Small-signal equivalent circuits . 15 2.5 Fundamental KVL equations . 16 2.5.1 Conventions . 17 2.5.2 KVL equations. 17 2.6 Fundamental KCL equations . 19 2.7 Tellegen’s theorem . 20 2.8 Energy in coupled inductors . 21 2.9 Passive circuits . 22 2.10 Modified node equations . 23 2.11 Numerical solution . 26 2.11.1 Backward Euler method . 27 2.11.2 Consistent initial conditions . 27 2.11.3 Verification of Tellegen’s theorem . 28 2.11.4 Remarks. 29 Problems . 29 3 Circuit Dynamics .............................................. 35 3.1 State equations . 35 3.1.1 A simple example . 35 3.1.2 Uniqueness of solution. 36 ix x Contents 3.1.3 Normal form . 37 3.2 Independent state variables. 38 3.2.1 Circuit with a capacitor loop . 38 3.2.2 Circuit with an inductor cutset . 39 3.3 Order of state equations . 40 3.3.1 Capacitor loops and inductor cut sets . 40 3.4 Formulation of state equations . 40 3.4.1 Circuits without capacitor loops or inductor cut sets . 41 3.4.2 Circuits with capacitor loops . 42 3.4.3 Circuits with inductor cut sets . 42 3.4.4 Remarks. 43 3.5 Solution of state equations . 44 3.5.1 Impulse response . 45 3.5.2 Examples . 46 3.6 Repeated eigenvalues . 48 3.7 Symbolic solution . 51 3.8 Numerical solution . 52 3.9 Analog computer simulation . 52 3.10 Exponential excitation . 53 Problems . 54 4 Properties in the Frequency Domain ............................. 59 4.1 Preliminaries . 59 4.2 Modified node equations . 60 4.3 Circuits with transconductances . 61 4.4 Reciprocity . 61 4.5 Impedance, admittance . 62 4.5.1 Poles and zeros . 63 4.5.2 Real and imaginary parts . 64 4.5.3 Impedance function from its real part . 64 4.6 Transfer function . 66 4.6.1 Frequency response . 67 4.6.2 Transfer function from its magnitude . 68 4.6.3 All-pass and minimum phase transfer functions . 69 4.6.4 Linear phase and group delay . 70 4.7 Relation between real and imaginary parts . 71 4.8 Gain and phase relation . 74 4.9 Sensitivity function . 76 4.9.1 Computation of sensitivity. ..

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