Circuit Analysis for Power Engineering Handbook

Circuit 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 8.2 The differential equations oftl and their solution 8.3 Travelling-wave properties in a transmission line 8.4 Wave formations in TL at their connections 8.5 Wave reflections 8.6 Successive reflections of waves 8.7 Laplace transform analysis of transients in TL 8.8 Line with only LG or CR parameters Appendix A AppendixB Index IX 633 636 639 648 660 668 675 687 709 723

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 electronics, microelectronics, communications, computers etc., or so-called lowcurrent, low-voltage engineering. The other is in power electronics, power systems, energy conversion devices etc., or so-called high-current, highvoltage 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 lowcurrent systems. This brought the author to the conclusion that there is a definite disadvantage 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 highvoltage 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 dedicated 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 transmission 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 attention is on the study of power transformers. The study of three-phase transformers 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 characterize 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 techniques. However, Chapter 1 is dedicated to a generalized introductory scope of circuit theory using matrices and computers. It covers the systematic formulation of circuit equations based on Kirchhoff's two laws, and using elementary 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 nodevoltage 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; threephase transformers and rotating magnetic fields. The principles of induction

Preface Xlll (asynchronous) and synchronous machines are discussed, and a nonsymmetrical component technique is used for analysing different nonsymmetrical 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 nonsinusoidal 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 steadystate 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 transient analysis of circuits, with an emphasis on power system circuits. Starting with the mathematical introduction to Laplace transform techniques, 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 engineers, 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 transient 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 underground 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.

XIV Preface Last but not least, my sincere appreciation goes to my wife, Iris, who prodigiously supported and aided me throughout the writing of this book. I am also extremely grateful for her patience and assistance in editing and typing in English.