Table Of ContentCircuit Analysis for
Power Engineering
Handbook
Circuit Analysis for
Power Engineering
Handbook
Arieh L. Shenkman
SPRINGER SCIENCE+BUSINESS MEDIA, B.V
A c.I.P. Catalogue record for this book is available from the Library of Congress
ISBN 978-1-4613-7418-3 ISBN978-1-4615-5299-4 (eBook)
DOI 10.1007/978-1-4615-5299-4
Printed an acid-free paper
All Rights Reserved
© 1998 Springer Science+Business Media Dordrecht
Originally published by K1uwer Academic Publishers in1998
Softcover reprint ofthe hardcover Ist edition 1998
No part of the material protected by this copyright notice may be reproduced or
uti1ized in any form ar by any means, electronic or mechanical,
including photocopying, recording, ar by any informati an storage and
retrieval system, without prior permission from the copyright owner.
To my wife Iris
Contents
1 Topological methods of circuit analysis
1.1 Introduction 1
1.2 The concept of a graph 1
1.3 Incidence, loop and mesh matrices 7
1.4 Generalized matrix approach in circuit analysis 14
1.5 Tellegen's theorem and conservation of energy 41
1.6 Dual graphs and dual circuits 44
1.7 Computer-aided analysis of very complex circuits 49
2 Sinusoidal steady-state analysis 57
2.1 Introduction 57
2.2 The phasor concept and complex representation of sinusoids 59
2.3 Phasor relationships for R, Land C elements; complex
impedance and admittance 73
2.4 Analysis of complex a.c. circuits 83
2.5 Resonance in a.c. circuits 132
2.6 Energy and power in a.c. circuits 159
2.7 Linear and circle diagrams 188
3 Magnetically Coupled Circuits 201
3.1 Introduction 201
3.2 Mutual inductance 202
3.3 Series and parallel connections of coupled elements 213
3.4 Energy storage and power transfer 232
3.5 Transformers 239
3.6 Resonance in coupled circuits 271
3.7 Circuits with more than two coupled elements 283
4 Three-phase systems 295
4.1 Introduction 295
4.2 Polyphase circuits 296
Vlll Contents
4.3 Three-phase generators 307
4.4 Three-phase connections 316
4.5 Power measurements in three-phase systems 349
4.6 Three-phase transformers 360
4.7 The rotating magnetic field 371
4.8 The principle of induction (asynchronous) and synchronous
machines 384
4.9 Symmetrical components 400
5 Non-sinusoidal behavior of electric circuits 447
5.1 Introduction 447
5.2 Fourier series 448
5.3 Circuit analysis for non-sinusoidal functions 477
5.4 Characteristics of non-sinusoidal functions 496
5.5 Power due to non-sinusoidal voltages and currents 501
5.6 Factors characterizing non-sinusoidal waves 512
5.7 Harmonics in three-phase systems 518
6 Transmission lines 527
6.1 Introduction 527
6.2 Transmission line (TL) parameters 527
6.3 Transmission line equations 530
6.4 Sinusoidal response of a transmission line 531
6.5 Waves in transmission lines 533
6.6 Solution of transmission line equations 538
6.7 Characteristic parameters of a transmission line 543
6.8 Some properties of transmission lines 548
6.9 A transmission line in various operating conditions 556
6.10 Equivalent circuit of a transmission line 566
6.11 Ladder network as a TL model 569
7 Transient analysis using Laplace transform techniques 573
7.1 Introduction 573
7.2 Definition of the Laplace transform 574
7.3 Laplace transform of some simple time functions 576
7.4 Basic theorems of the Laplace transform 578
7.5 Initial-value and final-value theorems 593
7.6 Convolution theorem 595
7.7 Inverse transform and partial-fraction expansions 600
7.8 Circuit analysis with the Laplace transform 608
8 Transient behavior of transmission lines (TL) 633
8.1 Introduction 633
Contents IX
8.2 The differential equations ofTL and their solution 633
8.3 Travelling-wave properties in a transmission line 636
8.4 Wave formations in TL at their connections 639
8.5 Wave reflections 648
8.6 Successive reflections of waves 660
8.7 Laplace transform analysis of transients in TL 668
8.8 Line with only LG or CR parameters 675
Appendix A 687
AppendixB 709
Index 723
Preface
The study of circuits is the foundation on which most other courses in the
electrical engineering curriculum are based. For this reason the first course
in circuit analysis must be appropriate to the succeeding specializations,
which may be classified into two groups. One is a specialization in electro
nics, microelectronics, communications, computers etc., or so-called low
current, low-voltage engineering. The other is in power electronics, power
systems, energy conversion devices etc., or so-called high-current, high
voltage engineering. It is evident that although there are many common
teaching topics in the basic course of circuit analysis, there are also certain
differences. Unfortunately most of the textbooks in this field are written
from the 'electronic engineer's viewpoint', i.e. with the emphasis on low
current systems.
This brought the author to the conclusion that there is a definite disad
vantage in not having a more appropriate book for the specializations in
high-current, high-voltage engineering. Thus the idea for this book came
into being.
The major feature distinguishing this book from others on circuit
analysis is in delivering the material with a very strong connection to the
specializations in the field of power systems, i.e. in high-current and high
voltage engineering. The author believes that this emphasis gives the
reader more opportunity for a better understanding and practice of the
material which is relevant for power system network analysis, and to
prepare students for their further specializations. This assertion is based
on the author's many years in engineering, and in teaching circuit analysis
to undergraduate and graduate students who have specialized in power
systems.
For this purpose the emphasis and a great amount of material are dedi
cated to a.c. circuit analysis, inducing three-phase circuits, using sinusoidal
steady-state phasor techniques. Power systems are based on employing
three-phase networks, including three-phase generators, three-phase trans
mission lines and three-phase utilities. Thus the study of three-phase
systems has been enlarged and also expanded for a thorough analysis of
unbalanced systems with their treatment, using the method of sequence
components.
xii Preface
The concept of a rotating magnetic field is comprehensively covered,
and on that basis the introductory study of a.c. machines, asynchronous
and synchronous, is presented. In coupled circuit analysis the focus of atten
tion is on the study of power transformers. The study of three-phase trans
formers is also covered in detail.
The problem of high harmonics in power systems is of great importance,
especially when using modern power-electronic equipment. A full chapter
on the non-sinusoidal behavior of a.c. circuits is included in the book.
As mentioned above, transmission lines are one of the most important
parts of power systems. So a detailed analysis of three-phase transmission
lines in their steady state and also their transient behavior is also given.
Another distinguishing feature of the book is the differentiation between
the steady state and transient behavior of circuits. Such a division is based
on the concept that steady-state behavior is normal and transients charac
terize the faults. Transient analysis is given by using Laplace transform
techniques, which are the most appropriate for the transient analysis of
power system circuits.
The presentation of the course material is geared to readers who are
being exposed to (a) the basic concept of electric circuits based on their
earlier study of physics and/or introductory courses in circuit analysis, and
(b) basic mathematics, including differentiation and integration techni
ques.
However, Chapter 1 is dedicated to a generalized introductory scope of
circuit theory using matrices and computers. It covers the systematic formu
lation of circuit equations based on Kirchhoff's two laws, and using elemen
tary graph theory and matrix algebra. The important solution methods for
these equations, such as Cramer's rule, Gauss elimination and using an
inverse matrix, are discussed here. The development of algorithms for
computer-aided solutions is also given.
Chapter 2 covers the a.c. analysis with a strong emphasis on phasor
methods based on using complex numbers. Although rather lengthy, the
topics of superposition, Thevenin and Norton equivalents, and node
voltage and mesh-current methods of analysis are all included in this
chapter in order to demonstrate their application to the phasor concept of
a.c. circuit analysis. The resonance in a.c. circuits, power relations, power
measurements, and using linear and circle diagrams for a.c. circuit analysis
are also presented.
Chapter 3 is concerned with magnetically coupled circuits. Attention is
focused on developing a circuit model of a real iron-core transformer and
analysing resonance in coupled circuits. The generalized solution of a large
system with many coupled elements is also given.
Chapter 4 covers all the most important topics of three-phase circuit
systems: different kinds of Y and .1 connections in balanced three-phase
circuits; the detailed analysis of unbalanced three-phase circuits; three
phase transformers and rotating magnetic fields. The principles of induction
Preface Xlll
(asynchronous) and synchronous machines are discussed, and a non
symmetrical component technique is used for analysing different non
symmetrical faults in power systems.
Chapter 5 discusses the non-sinusoidal behavior of a.c. circuits. The
Fourier series method, introduced in general in this chapter, is used for non
sinusoidal circuit analysis. The power relation and different characteristics
of non-sinusoidal waves are also considered.
Chapter 6 is dedicated to three-phase transmission lines in their steady
state behavior. The transmission line equations are developed and their
solution for various operating conditions of transmission lines are given.
The most important properties of transmission lines and their equivalent
circuits are discussed.
Chapter 7 discusses the Laplace transform and its application to the tran
sient analysis of circuits, with an emphasis on power system circuits.
Starting with the mathematical introduction to Laplace transform techni
ques, the different engineering applications of this method are covered.
Finally, in Chapter 8, the transient behavior of transmission lines (TL)
is presented. Believing that this topic is very important for power engi
neers, the author has included this material to provide the reader with the
basic knowledge of transients in TL. For this purpose the solution of a
transmission line (as a network with distributed parameters) differential
equation is given, and the method of travelling waves is introduced.
Different engineering approaches using this method are discussed. A tran
sient analysis of TL is extended to the solution of the transient behavior of
a ground rod (which is part of the lightning protection of a transmission
line) when affected by a stroke of lightning, and the transients in under
ground cables.
Numerous examples, most of which are relevant to power engineering,
accompany the theoretical and explanatory treatment of the material,
making it more clearly understood. Although the level of the book is high,
the reader can follow the material without any special difficulties because it
is presented in sequence from simple to more complicated. Also, for the
reader's convenience, two mathematical appendixes dedicated to matrix
theory and complex numbers are included.
I hope that this textbook will be helpful to all readers specializing in
power system engineering, and of value in assisting professors and lecturers
in the educational process.
I wish to acknowledge useful discussions with Professor Moses Zarudi
and his significant contributions to the material in Chapters 2 and 3. It is
also a pleasure to acknowledge the assistance of Dr. Izida Tchernina in
providing solutions to the computerized examples.
I am grateful to David Hatter, the commissioning editor, for his support
and helpful suggestions in the difficult compromise between the abundance
of material originally presented and the assigned limits as to the volume of
the published text.
