Circuit Analysis I with MATLAB Computing and Simulink/SimPowerSystems Modeling

678 Pages·2009·4.813 MB·English

by  Steven Karris

Circuit Analysis II
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Steven T. Karris
Orchard Publications
www.orchardpublications.com
Circuit Analysis II

with MATLAB® Computing and
Simulink® / SimPowerSystems®
Modeling
Steven T. Karris
Orchard Publications, Fremont, California
www.orchardpublications.com
Circuit Analysis II with MATLAB® Computing and Simulink® / SimPowerSystems® Modeling
Copyright  2009 Orchard Publications. All rights reserved. Printed in USA. 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: 2009930247
ISBN10: 1934404201
ISBN13: 9781934404209
TX 5745064
Disclaimer
The author has made every effort to make this text as complete and accurate as possible, but no warranty is implied.
The author and publisher shall have neither liability nor responsibility to any person or entity with respect to any loss
or damages arising from the information contained in this text.
This book was created electronically using Adobe Framemaker.
Preface
This text is written for use in a second course in circuit analysis. 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.The reader of this book should have the traditional undergraduate
knowledge of an introductory circuit analysis material such as Circuit Analysis I with
MATLAB®Computing and Simulink®/ SimPowerSystems®Modeling, ISBN 978-1-934404-17-1.
Another prerequisite would be a basic knowledge of differential equations, and in most cases,
engineering students at this level have taken all required mathematics courses. Appendix H 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 discussed in the last chapter of Circuit Analysis I with MATLAB®Computing and
Simulink®/ SimPowerSystems®Modeling, ISBN 978-1-934404-17-1. 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. Accordingly, any
signal can be represented in the complex 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 is an introduction to state space and state
equations. Chapter 8 begins with the frequency response concept and Bode magnitude and
frequency plots. Chapter 9 is devoted to transformers with an introduction to self and mutual
inductances. Chapter 10 is an introduction to one- and two-terminal devices and presents several
practical examples. Chapters 11 and 12 are introductions 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, Appendix B is an introduction to Simulink, and Appendix C
introduces SimPowerSystems. The pages where MATLAB scripts, Simulink / SimPowerSystems
models appear are indicated in the Table of Contents.
The author highly recommends that the reader studies this material in conjunction with the
inexpensive Student Versions of The MathWorks™ Inc., the developers of these outstanding
products, available from:
The MathWorks, Inc.
3 Apple Hill Drive
Natick, MA, 01760
Phone: [phone],
www.mathworks.com
[email protected].
Appendix D is a review of complex numbers, Appendix E is an introduction to matrices,
Appendix F discusses scaling methods, Appendix G introduces the per unit system used
extensively in power systems and in SimPwerSystems examples and demos. As stated above,
Appendix H is a review of differential equations. Appendix I provides instructions for
constructing semilog templates to be used with Bode plots.
In addition to numerous 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 is indebted to several readers who have brought some errors to our attention.
Additional feedback with other errors, advice, and comments will be most welcomed and
greatly appreciated.
Orchard Publications
39510 Paseo Padre Parkway
Suite 315
Fremont, California 94538
www.orchardpublications.com
[email protected]
Table of Contents
1
Second Order Circuits 11
1.1 Response of a Second Order Circuit....................................................................11
1.2 Series RLC Circuit with DC Excitation...............................................................12
1.2.1 Response of Series RLC Circuits with DC Excitation...............................13
1.2.2 Response of Series RLC Circuits with AC Excitation.............................111
1.3 Parallel RLC Circuit...........................................................................................115
1.3.1 Response of Parallel RLC Circuits with DC Excitation..........................117
1.3.2 Response of Parallel RLC Circuits with AC Excitation..........................126
1.4 Other Second Order Circuits.............................................................................130
1.5 Summary.............................................................................................................136
1.6 Exercises..............................................................................................................138
1.7 Solutions to EndofChapter Exercises.............................................................140
MATLAB Computing: Pages 16, 17, 19, 113, 119, 1through 123,
125, 126, 128, 129, 132 through 134, 142,
144, 145
Simulink/SimPowerSystems Models: Pages 110, 114, 129, 153
2
Resonance 21
2.1 Series Resonance..................................................................................................21
2.2 Quality Factor Q in Series Resonance..............................................................24
0s
2.3 Parallel Resonance...............................................................................................26
2.4 Quality Factor Q in Parallel Resonance...........................................................29
0P
2.5 General Definition of Q.......................................................................................29
2.6 Energy in L and C at Resonance........................................................................210
2.7 Half-Power Frequencies  Bandwidth...............................................................211
2.8 A Practical Parallel Resonant Circuit................................................................216
2.9 Radio and Television Receivers.........................................................................218
2.10 Summary............................................................................................................221
2.11 Exercises.............................................................................................................223
2.12 Solutions to EndofChapter Exercises.............................................................225
MATLAB Computing: Pages 25, 26, 225, 227, 230, 231
Simulink / SimPowerSystems models: Pages 215, 216
3
Elementary Signals 31
3.1 Signals Described in Math Form...........................................................................31
Circuit Analysis II with MATLAB  Computing and Simulink / SimPowerSystems Modeling i
Copyright © Orchard Publications
3.2 The Unit Step Function........................................................................................32
3.3 The Unit Ramp Function.....................................................................................39
3.4 The Delta Function............................................................................................311
3.4.1 The Sampling Property of the Delta Function..........................................311
3.4.2 The Sifting Property of the Delta Function..............................................312
3.5 Higher Order Delta Functions............................................................................313
3.6 Summary.............................................................................................................319
3.7 Exercises..............................................................................................................320
3.8 Solutions to EndofChapter Exercises..............................................................321
Simulink model: Pages 37, 38
4
The Laplace Transformation 41
4.1 Definition of the Laplace Transformation..............................................................41
4.2 Properties and Theorems of the Laplace Transform...............................................42
4.2.1 Linearity Property........................................................................................42
4.2.2 Time Shifting Property.................................................................................43
4.2.3 Frequency Shifting Property........................................................................43
4.2.4 Scaling Property...........................................................................................44
4.2.5 Differentiation in Time Domain Property..................................................44
4.2.6 Differentiation in Complex Frequency Domain Property...........................45
4.2.7 Integration in Time Domain Property........................................................46
4.2.8 Integration in Complex Frequency Domain Property................................47
4.2.9 Time Periodicity Property...........................................................................48
4.2.10 Initial Value Theorem.................................................................................49
4.2.11 Final Value Theorem................................................................................410
4.2.12 Convolution in Time Domain Property....................................................411
4.2.13 Convolution in Complex Frequency Domain Property............................411
4.3 Laplace Transform of Common Functions of Time..............................................412
4.3.1 Laplace Transform of the Unit Step Function u t.................................412
0
4.3.2 Laplace Transform of the Ramp Function u t .......................................412
1
4.3.3 Laplace Transform of tnu t ....................................................................414
0
4.3.4 Laplace Transform of the Delta Function t .........................................417
4.3.5 Laplace Transform of the Delayed Delta Function t–a ......................417
4.3.6 Laplace Transform of e–atu t ..................................................................418
0
4.3.7 Laplace Transform of tne–atu t ...............................................................418
0
4.3.8 Laplace Transform of sint u t .................................................................419
0
4.3.9 Laplace Transform of cost u t ................................................................419
0
4.3.10 Laplace Transform of e–atsint u t.........................................................420
0
4.3.11 Laplace Transform of e–atcost u t ........................................................420
0
4.4 Laplace Transform of Common Waveforms.........................................................421
ii Circuit Analysis II with MATLAB  Computing and Simulink / SimPowerSystems Modeling
Copyright © Orchard Publications
4.4.1 Laplace Transform of a Pulse.......................................................................422
4.4.2 Laplace Transform of a Linear Segment......................................................422
4.4.3 Laplace Transform of a Triangular Waveform.............................................423
4.4.4 Laplace Transform of a Rectangular Periodic Waveform............................424
4.4.5 Laplace Transform of a HalfRectified Sine Waveform..............................425
4.5 Using MATLAB for Finding the Laplace Transforms of Time Functions.............426
4.6 Summary.................................................................................................................427
4.7 Exercises.................................................................................................................430
Laplace Transform of a Sawtooth Periodic Waveform.......................................431
Laplace Transform of a FullRectified Sine Waveform......................................431
4.8 Solutions to EndofChapter Exercises.................................................................432
MATLAB Computing: Page 4-37
Simulink Model: Page 4-38
5
The Inverse Laplace Transformation 51
5.1 The Inverse Laplace Transform Integral................................................................51
5.2 Partial Fraction Expansion.....................................................................................51
5.2.1 Distinct Poles...............................................................................................52
5.2.2 Complex Poles..............................................................................................55
5.2.3 Multiple (Repeated) Poles............................................................................58
5.3 Case where F(s) is Improper Rational Function...................................................513
5.4 Alternate Method of Partial Fraction Expansion.................................................514
5.5 Summary...............................................................................................................518
5.6 Exercises...............................................................................................................519
5.7 Solutions to EndofChapter Exercises...............................................................520
MATLAB Computing: Pages 53 through 56, 58, 510
512 through 514, 520
6
Circuit Analysis with Laplace Transforms 61
6.1 Circuit Transformation from Time to Complex Frequency..................................61
6.1.1 Resistive Network Transformation.............................................................61
6.1.2 Inductive Network Transformation............................................................61
6.1.3 Capacitive Network Transformation..........................................................62
6.2 Complex Impedance Z(s).....................................................................................611
6.3 Complex Admittance Y(s)...................................................................................613
6.4 Transfer Functions...............................................................................................616
6.5 Using the Simulink Transfer Fcn Block...............................................................620
6.6 Summary..............................................................................................................623
6.7 Exercises...............................................................................................................624
Circuit Analysis II with MATLAB  Computing and Simulink / SimPowerSystems Modeling iii
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6.8 Solutions to EndofChapter Exercises...............................................................627
MATLAB Computing: Pages 66, 68, 615, 619 through 621,
629 through 6-32, 637
Simulink / SimPowerSystems models: Pages 68 through 611, 620 through 622
7
State Variables and State Equations 71
7.1 Expressing Differential Equations in State Equation Form...................................71
7.2 Solution of Single State Equations........................................................................76
7.3 The State Transition Matrix.................................................................................78
7.4 Computation of the State Transition Matrix......................................................710
7.4.1 Distinct Eigenvalues (Real of Complex)...................................................711
7.4.2 Multiple (Repeated) Eigenvalues..............................................................715
7.5 Eigenvectors.........................................................................................................718
7.6 Circuit Analysis with State Variables..................................................................722
7.7 Relationship between State Equations and Laplace Transform..........................729
7.8 Summary..............................................................................................................737
7.9 Exercises..............................................................................................................740
7.10 Solutions to EndofChapter Exercises..............................................................742
MATLAB Computing: Pages 74, 76, 78, 712, 713, 715, 717, 721
730, 744, 745, 746, 748, 750
Simulink models: Pages 79, 710
8
Frequency Response and Bode Plots 81
8.1 Decibel Defined ....................................................................................................81
8.2 Bandwidth and Frequency Response.....................................................................83
8.3 Octave and Decade...............................................................................................84
8.4 Bode Plot Scales and Asymptotic Approximations...............................................85
8.5 Construction of Bode Plots when the Zeros and Poles are Real...........................86
8.6 Construction of Bode Plots when the Zeros and Poles are Complex..................812
8.7 Corrected Amplitude Plots..................................................................................824
8.8 Summary..............................................................................................................835
8.9 Exercises..............................................................................................................837
8.10 Solutions to EndofChapter Exercises..............................................................838
MATLAB Computing: Pages 819, 820, 822, 823, 833, 840, 843, 845
9
Self and Mutual Inductances  Transformers 91
9.1 SelfInductance.......................................................................................................91
iv Circuit Analysis II with MATLAB  Computing and Simulink / SimPowerSystems Modeling
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9.2 The Nature of Inductance.....................................................................................91
9.3 Lenz’s Law..............................................................................................................93
9.4 Mutually Coupled Coils.........................................................................................93
9.5 Establishing Polarity Markings............................................................................911
9.6 Energy Stored in a Pair of Mutually Coupled Inductors.....................................914
9.7 Circuits with Linear Transformers.......................................................................919
9.8 Reflected Impedance in Transformers.................................................................924
9.9 The Ideal Transformer.........................................................................................927
9.10 Impedance Matching...........................................................................................930
9.11 Simplified Transformer Equivalent Circuit.........................................................931
9.12 Thevenin Equivalent Circuit...............................................................................932
9.13 Autotransformer..................................................................................................936
9.14 Transformers with Multiple Secondary Windings...............................................937
9.15 Transformer Tests................................................................................................937
9.16 Efficiency..............................................................................................................942
9.17 Voltage Regulation..............................................................................................946
9.18 Transformer Modeling with Simulink / SimPowerSystems.................................949
9.19 Summary..............................................................................................................957
9.20 Exercises...............................................................................................................962
9.21 Solutions to EndofChapter Exercises..............................................................965
MATLAB Computing: Page 913, 914, 922, 944
Simulink / SimPowerSystems model: Page 949 through 956
10
One and TwoPort Networks 101
10.1 Introduction and Definitions...............................................................................101
10.2 One-Port Driving-Point and Transfer Admittances...........................................102
10.3 One-Port Driving-Point and Transfer Impedances.............................................107
10.4 Two-Port Networks...........................................................................................1011
10.4.1 The y Parameters...................................................................................1011
10.4.2 The z parameters...................................................................................1017
10.4.3 The h Parameters..................................................................................1022
10.4.4 The g Parameters...................................................................................1026
10.5 Reciprocal Two-Port Networks.........................................................................1031
10.6 Summary............................................................................................................1035
10.7 Exercises.............................................................................................................1040
10.8 Solutions to EndofChapter Exercises............................................................1042
MATLAB Computing: Page 1049
Simulink / SimPowerSystems model: Page 1050
Circuit Analysis II with MATLAB  Computing and Simulink / SimPowerSystems Modeling v
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