Orchard Publications, Fremont, California www.orchardpublications.com num=[0 1100 0]; den=[1 1100 10^5]; w=logspace(0,5,100); bode(num,den,w); grid Steven T. Karris Circuit Analysis II with MATLAB® Applications Circuit Analysis II with MATLAB® Applications Students and working professionals will find Circuit Analysis II with MATLAB® Applications to be a con- cise and easy-to-learn text. It provides complete, clear, and detailed explanations of advanced electri- cal engineering concepts illustrated with numerous practical examples. This text includes the following chapters and appendices: • Second Order Circuits • Resonance • Elementary Signals • The Laplace Transformation • The Inverse Laplace Transformation • Circuit Analysis with Laplace Transforms • Frequency Response and Bode Plots • Self and Mutual Inductances - Transformers • One and Two Port Networks • Three Phase Systems • Introduction to MATLAB • Differential Equations • Matrices and Determinants • Constructing Semilog Plots with Microsoft Excel • Scaling Each chapter contains numerous practical applications supplemented with detailed instructions for using MATLAB to obtain quick and accurate answers. Steven T. Karris is the president and founder of Orchard Publications. He earned a bachelors degree in electrical engineering at Christian Brothers University, Memphis, Tennessee, a masters degree in electrical engineering at Florida Institute of Technology, Melbourne, Florida, and has done post-master work at the latter. He is a registered professional engineer in California and Florida. He has over 30 years of professional engineering experience in industry. In addition, he has over 25 years of teaching experience that he acquired at several educational institutions as an adjunct professor. He is currently with UC Berkeley Extension. Orchard Publications, Fremont, California Visit us on the Internet www.orchardpublications.com or email us: info@orchardpublications.com ISBN 0-9709511-9-1 $39.95 U.S.A. Circuit Analysis II with MATLAB® Applications Steven T. Karris Orchard Publications www.orchardpublications.com Circuit Analysis II with MATLAB® Applications Copyright  2003 Orchard Publications. All rights reserved. Printed in Canada. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. Direct all inquiries to Orchard Publications, 39510 Paseo Padre Parkway, Fremont, California 94538, U.S.A. URL: http://www.orchardpublications.com Product and corporate names are trademarks or registered trademarks of the MathWorks, Inc., and Microsoft Corporation. They are used only for identification and explanation, without intent to infringe. Library of Congress Cataloging-in-Publication Data Library of Congress Control Number: 2003094467 Copyright Number TX-745-064 ISBN 0-9709511-9-1 Disclaimer The publisher has used his best effort to prepare this text. However, the publisher and author makes no warranty of any kind, expressed or implied with regard to the accuracy, completeness, and computer codes contained in this book, and shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the performance or use of these programs. This book was created electronically using Adobe Framemaker. Circuit Analysis II with MATLAB Applications Orchard Publications Preface This text is written for use in a second course in circuit analysis. The reader of this book should have the traditional undergraduate knowledge of an introductory circuit analysis material such as Circuit Analysis I with MATLAB® Applications by this author. Another prerequisite would be knowledge of differential equations, and in most cases, engineering students at this level have taken all required mathematics courses. It encompasses a spectrum of subjects ranging from the most abstract to the most practical, and the material can be covered in one semester or two quarters. Appendix B serves as a review of differential equations with emphasis on engineering related topics and it is recommended for readers who may need a review of this subject. There are several textbooks on the subject that have been used for years. The material of this book is not new, and this author claims no originality of its content. This book was written to fit the needs of the average student. Moreover, it is not restricted to computer oriented circuit analysis. While it is true that there is a great demand for electrical and computer engineers, especially in the internet field, the demand also exists for power engineers to work in electric utility companies, and facility engineers to work in the industrial areas. Chapter 1 is an introduction to second order circuits and it is essentially a sequel to first order circuits that were discussed in the last chapter of as Circuit Analysis I with MATLAB® Applications. Chapter 2 is devoted to resonance, and Chapter 3 presents practical methods of expressing signals in terms of the elementary functions, i.e., unit step, unit ramp, and unit impulse functions. Thus, any signal can be represented in the compex frequency domain using the Laplace transformation. Chapters 4 and 5 are introductions to the unilateral Laplace transform and Inverse Laplace transform respectively, while Chapter 6 presents several examples of analyzing electric circuits using Laplace transformation methods. Chapter 7 begins with the frequency response concept and Bode magnitude and frequency plots. Chapter 8 is devoted to transformers with an introduction to self and mutual inductances. Chapter 9 is an introduction to one- and two-terminal devices and presents several practical examples. Chapter 10 is an introduction to three-phase circuits. It is not necessary that the reader has previous knowledge of MATLAB®. The material of this text can be learned without MATLAB. However, this author highly recommends that the reader studies this material in conjunction with the inexpensive MATLAB Student Version package that is available at most college and university bookstores. Appendix A of this text provides a practical introduction to MATLAB. As shown on the front cover of this text the magnitude and phase plots can be easily obtained with a one line MATLAB code. Moreover, MATLAB will be invaluable in later studies such as the design of analog and digital filters. Circuit Analysis II with MATLAB Applications Orchard Publications As stated above, Appendix B is a review of differential equations. Appendix C is an introduction to matrices, Appendix D provides instructions on constructing semilog templates to be used with Bode plots, and Appendix E discusses scaling methods. In addition to numerous real-world examples, this text contains several exercises at the end of each chapter. Detailed solutions of all exercises are provided at the end of each chapter. The rationale is to encourage the reader to solve all exercises and check his effort for correct solutions and appropriate steps in obtaining the correct solution. And since this text was written to serve as a self-study or supplementary textbook, it provides the reader with a resource to test his knowledge. The author has accumulated many additional problems for homework assignment and these are available to those instructors who adopt this text either as primary or supplementary text, and prefer to assign problems without the solutions. He also has accumulated many sample exams. The author is indebted to the class of the Spring semester of 2001 at San Jose State University, San Jose, California, for providing several of the examples and exercises of this text. Like any other new book, this text may contain some grammar and typographical errors. Accordingly, all feedback for errors, advice, and comments will be most welcomed and greatly appreciated. Orchard Publications info@orchardpublications.com Circuit Analysis II with MATLAB Applications i Orchard Publications Contents Chapter 1 Second Order Circuits The Response of a Second Order Circuit 1-1 The Series RLC Circuit with DC Excitation 1-2 Response of Series RLC Circuits with DC Excitation 1-5 Response of Series RLC Circuits with AC Excitation 1-11 The Parallel GLC Circuit 1-14 Response of Parallel GLC Circuits with DC Excitation 1-16 Response of Parallel GLC Circuits with AC Excitation 1-26 Other Second Order Circuits 1-29 Summary 1-36 Exercises 1-38 Solutions to Exercises 1-40 Chapter 2 Resonance Series Resonance 2-1 Quality Factor Q 0s in Series Resonance 2-4 Parallel Resonance 2-6 Quality Factor Q 0p in Parallel Resonance 2-9 General Definition of Q 2-10 Energy in L and C at Resonance 2-11 Half-Power Frequencies - Bandwidth 2-12 A Practical Parallel Resonant Circuit 2-16 Radio and Television Receivers 2-17 Summary 2-20 Exercises 2-22 Solutions to Exercises 2-24 Chapter 3 Elementary Signals Signals Described in Math Form 3-1 The Unit Step Function 3-2 The Unit Ramp Function 3-10 ii Circuit Analysis II with MATLAB Applications Orchard Publications The Delta Function 3-12 Sampling Property of the Delta Function 3-12 Sifting Property of the Delta Function 3-13 Higher Order Delta Functions 3-15 Summary 3-19 Exercises 3-20 Solutions to Exercises 3-21 Chapter 4 The Laplace Transformation Definition of the Laplace Transformation 4-1 Properties of the Laplace Transform 4-2 The Laplace Transform of Common Functions of Time 4-12 The Laplace Transform of Common Waveforms 4-23 Summary 4-29 Exercises 4-34 Solutions to Exercises 4-37 Chapter 5 The Inverse Laplace Transformation The Inverse Laplace Transform Integral 5-1 Partial Fraction Expansion 5-1 Case for m t n 5-13 Alternate Method of Partial Fraction Expansion 5-15 Summary 5-18 Exercises 5-20 Solutions to Exercises 5-22 Chapter 6 Circuit Analysis with Laplace Transforms Circuit Transformation from Time to Complex Frequency 6-1 Complex Impedance Z(s) 6-8 Complex Admittance Y(s) 6-10 Transfer Functions 6-13 Summary 6-17 Exercises 6-18 Solutions to Exercises 6-21 Circuit Analysis II with MATLAB Applications iii Orchard Publications Chapter 7 Frequency Response and Bode Plots Decibel 7-1 Bandwidth and Frequency Response 7-3 Octave and Decade 7-4 Bode Plot Scales and Asymptotic Approximations 7-5 Construction of Bode Plots when the Zeros and Poles are Real 7-6 Construction of Bode Plots when the Zeros and Poles are Complex 7-12 Corrected Amplitude Plots 7-25 Summary 7-36 Exercises 7-38 Solutions to Exercises 7-39 Chapter 8 Self and Mutual Inductances - Transformers Self-Inductance 8-1 The Nature of Inductance 8-1 Lenz’s Law 8-3 Mutually Coupled Coils 8-3 Establishing Polarity Markings 8-11 Energy Stored in a Pair of Mutually Coupled Inductors 8-14 Circuits with Linear Transformers 8-20 Reflected Impedance in Transformers 8-25 The Ideal Transformer 8-28 Impedance Matching 8-32 A Simplified Transformer Equivalent Circuit 8-33 Thevenin Equivalent Circuit 8-34 Summary 8-38 Exercises 8-42 Solutions to Exercises 8-44 Chapter 9 One- and Two-port Networks Introduction and Definitions 9-1 One-port Driving-point and Transfer Admittances 9-2 One-port Driving-point and Transfer Impedances 9-7 Two-Port Networks 9-12 iv Circuit Analysis II with MATLAB Applications Orchard Publications The y Parameters 9-12 The z parameters 9-19 The h Parameters 9-24 The g Parameters 9-29 Reciprocal Two-Port Networks 9-34 Summary 9-38 Exercises 9-43 Solutions to Exercises 9-45 Chapter 10 Three-Phase Systems Advantages of Three-Phase Systems 10-1 Three-Phase Connections 10-1 Transformer Connections in Three-Phase Systems 10-4 Line-to-Line and Line-to-Neutral Voltages and Currents 10-5 Equivalent Y and ' Loads 10-10 Computation by Reduction to Single Phase 10-20 Three-Phase Power 10-21 Instantaneous Power in Three-Phase Systems 10-23 Measuring Three-Phase Power 10-27 Summary 10-30 Exercises 10-32 Solutions to Exercises 10-33 Appendix A Introduction to MATLAB® MATLAB® and Simulink® A-1 Command Window A-1 Roots of Polynomials A-3 Polynomial Construction from Known Roots A-4 Evaluation of a Polynomial at Specified Values A-6 Rational Polynomials A-8 Using MATLAB to Make Plots A-10 Subplots A-19 Multiplication, Division and Exponentiation A-20 Script and Function Files A-26 Display Formats A-31 [...]... '); disp(vC0); fprintf(' \n') Solution was entered as y0 = 115*exp (-2 00*t )-1 10*exp (-3 00*t) 1-7 Circuit Analysis II with MATLAB Applications Orchard Publications Chapter 1 Second Order Circuits 1st derivative of solution is y1 = -2 3000*exp (-2 00*t)+33000*exp (-3 00*t) 2nd derivative of solution is y2 = 4600000*exp (-2 00*t )-9 900000*exp (-3 00*t) Differential equation is satisfied since y = y2+y1+y0 = 0 1st... voltage v t for t 0 iG iS u0 t v t iC iL G L C Figure 1.10 Parallel RLC circuit For this circuit 1-1 4 Circuit Analysis II with MATLAB Applications Orchard Publications The Parallel GLC Circuit iG t + iL t + iC t = iS t or 1 Gv + -L t 0 dv v dt + I 0 + C - = i S dt t 0 By differentiation, 2 dv v di S dv- C - + G - + = -2 dt L dt dt t (1.40) 0 To find the forced response, we must first specify... derivative y1 = -1 94*exp (-4 *t)+696*exp (-1 6*t) 1-1 8 Circuit Analysis II with MATLAB Applications Orchard Publications Response of Parallel GLC Circuits with DC Excitation y2=diff(y0,2) % Compute and display second derivative y2 = 776*exp (-4 *t )-1 1136*exp (-1 6*t) y=y2/640+y1/32+y0/10 % Verify that (1.40) is satisfied y = 0 The plot is shown in Figure 1.12 where we have used the following MATLAB code: t=0:... derivative y1 = -8 *exp (-8 *t)*(5+5080*t)+5080*exp (-8 *t) y2=diff(y0,2) % Compute 2nd derivative y2 = 64*exp (-8 *t)*(5+5080*t )-8 1280*exp (-8 *t) y=y2/640+y1/40+y0/10 % Verify differential equation, see (1.40) y = 0 The plot is shown in Figure 1.14 where we have used the following MATLAB code: 1-2 2 Circuit Analysis II with MATLAB Applications Orchard Publications Response of Parallel GLC Circuits with DC Excitation... be expressed as v C t = - C i dt and as before, the circuit – can be represented by the integrodifferential equation di 1 - Ri + L + dt C 1-8 t i dt = 15 u 0 t (1.18) – Circuit Analysis II with MATLAB Applications Orchard Publications Response of Series RLC Circuits with DC Excitation and since dv C i = i C = C -dt we rewrite (1.18) as 2 dv C dv C RC - + LC - + v C = 15 u 0 t 2 dt dt... the derivative using MATLAB as follows: syms t k2; v0=exp( 8*t)*(5+k2*t); v1=diff(v0); % v1 is 1st derivative of v0 v1 = -8 *exp (-8 *t)*(5+k2*t)+exp (-8 *t)*k2 1-2 1 Circuit Analysis II with MATLAB Applications Orchard Publications Chapter 1 Second Order Circuits Then, dv - = – 8e –8t 5 + k 2 t + k 2 e –8t dt and dv dt = – 40 + k 2 (1.63) t=0 iC dv Also, i C = C - or dv = - and dt dt C dv dt... conditions * State variables and state equations are discussed in Signals and Systems with MATLAB Applications, ISBN 0970951 1-3 -2 by this author Circuit Analysis II with MATLAB Applications Orchard Publications 1-1 Chapter 1 Second Order Circuits 1.2 The Series RLC Circuit with DC Excitation Let us consider the series RLC circuit of Figure 1.1 where the initial conditions are i L 0 = I 0 , v C 0 = V 0 ,... = -6 600*exp (-3 00*t)+6900*exp (-2 00*t) y2=diff(y0,2) % The second derivative of y(t) y2 = 1980000*exp (-3 00*t )-1 380000*exp (-2 00*t) y=y2+500*y1+60000*y0 % Summation of y and its derivatives y = 900000 Using the expression for v C t we can find the current as dv C 100 i = i L = i C = C = -dt 6 1-1 0 10 –3 6900e – 200t – 6600 e – 300t = 115e – 200t – 110 e – 300t A (1.29) Circuit Analysis II with MATLAB. .. = 2.5 or 1-9 Circuit Analysis II with MATLAB Applications Orchard Publications Chapter 1 Second Order Circuits k 1 + k 2 = – 12.5 (1.24) Also, dv C i L = i C = C dt dv C i L dv C = - and -dt C dt t=0 iL 0 5 = - = = 300 100 C –3 10 6 (1.25) Next, we differentiate (1.23), we evaluate it at t = 0 and equate it with (1.25) Then, dv C dv C – 200 t – 300 t - = – 200k... E-9 Circuit Analysis II with MATLAB Applications Orchard Publications v NOTES vi Circuit Analysis II with MATLAB Applications Orchard Publications Chapter 1 Second Order Circuits his chapter discusses the natural, forced and total responses in circuits containing resistors, inductors and capacitors These circuits are characterized by linear second-order differential equations whose solutions . 8-4 4 Chapter 9 One- and Two-port Networks Introduction and Definitions 9-1 One-port Driving-point and Transfer Admittances 9-2 One-port Driving-point and Transfer Impedances 9-7 Two-Port Networks 9-1 2 iv. Applications Circuit Analysis II with MATLAB Applications Students and working professionals will find Circuit Analysis II with MATLAB Applications to be a con- cise and easy-to-learn text. It provides. 9-1 2 iv Circuit Analysis II with MATLAB Applications Orchard Publications The y Parameters 9-1 2 The z parameters 9-1 9 The h Parameters 9-2 4 The g Parameters 9-2 9 Reciprocal Two-Port Networks 9-3 4 Summary