Agnes Scott College

Edith Clarke

Circuit Analysis of A-C Power
John Wiley & Sons, 1941 (Volume I), 1950 (Volume II)

cover page

Preface (Excerpts, Volume I – Symmetrical and Related Components)

This book is a compilation of notes and lectures given over a period of years to members of the Central Station Engineering Department of the General Electric Company in Schenectady, New York. Beginning in 1928, the notes were revised and extended for new groups of men entering the department, practical problems in power system performance with numerical solutions being added from time to time and they were presented by operating engineers. As the notes were helpful to members of the department and others receiving the, it was suggested that they be put in book form. In 1932, with Professor H. W. Bibber as co-author, a book on symmetrical components was undertaken. Parts of that unfinished book are included in Chapters I-IV of this one.

In answer to the repeated request that the methods of symmetrical and related components be presented very simply, the methods of solving unbalanced power system problems by means of components are analyzed and discussed in detail. The book has been divided into two volumes. Volume I deals largely with the determination of currents and voltages of fundamental frequency in poor systems during unbalanced conditions by means of symmetrical and related components. included in this volume are the electrical characteristics of overhead transmission circuits and information and data on transformers and synchronous machines which permit them to be represented by equivalent circuits in the solution of practical problems. Volume II will give additional characteristics of synchronous machines, equivalent circuits for types of transformers not included in Volume I, characteristics of insulated cables, induction machines, and other electrical equipment encountered in a-c power systems. Overvoltages from various causes and the effects of saturation in transformers and of amortisseur windings in synchronous machines will also be included in Volume II. In both volumes special attention is given to equivalent circuits and the solutions of practical problems.

Introduction (Excerpts, Volume I)

The problems of the power transmission engineer at any given time may be divided roughly into three classes:

  1. Problems which can be solved analytically by well-known methods in general use. The methods are satisfactory, because it is thought that all the factors influencing the problem are understood and can be evaluated, and the time required is not considered unduly long.
  2. Problems which can be solved analytically and the various factors evaluated, but the time and labor required are excessive.
  3. Problems for which there is no known analytic method of evaluating all the factors involved. This is not intended to imply that, for a given problem with all conditions specified, the engineer given sufficient time cannot provide a workable solution; but rather that, the effect of the various influences not being thoroughly understood, a different and independent problem is encountered with each change in given conditions.


The purpose of this book is to help the power transmission engineer solve some of his problems. Since it is expected that many of these problems will deal with systems during unbalanced conditions, where the use of symmetrical components and their related components will materially aid him, the greater part of the book is devoted to these components and their applications. But as he will also be expected to determine system conditions during normal operation, tables and charts are given to assist him in the solution of such problems.

Except for an occasional integral or differential equation, introduced for a better understanding of the fundamental principles involved, a knowledge of the elementary principles of alternating currents, algebra, plane geometry, trigonometry, and familiarity with electrical equipment are the only prerequisites for an understanding of this book.

Contents (Volume I)

  1. Definitions and Fundamental Concepts
  2. Symmetrical Components—Basic Equations for Three-Phase Systems
  3. Short Circuits on Systems with One Power Source
  4. Unsymmetrical Faults on Normally Balanced Three-Phase Systems
  5. Two Component Networks for Three-Phase Systems
  6. Transmission Circuits with Distributed Constants
  7. Simultaneous Faults on Symmetrical Three-Phase Systems—Analysis by the Method of Symmetrical Components
  8. Unsymmetrical Three-Phase Circuits—Analysis by the Method of Symmetrical Components
  9. Polyphase Systems of More than Three Phases, Single-Phase and Two-Phase Systems
  10. Alpha, Beta, and Zero Components of Three-Phase Systems
  11. Impedances of Overhead Transmission Lines
  12. Capacitances of Overhead Transmission Lines
    Appendix A. Determinants
    Appendix B. Tables and Charts for Overhead Transmission Circuits


Preface (Excerpts, Volume II)

Circuit Analysis of A-C Power Systems, Volume II, is a continuation of Volume I. In it, as in Volume I, circuits are analyzed by means of components. Basic equations, relating phase quantities and their symmetrical components and phase quantities and their αβ0 components, derived and applied in Volume I, are tabulated for ready reference in Chapter I of Volume II.

Insulated cables, various types of transformers and autotransformers, synchronous machines, and induction motors are discussed in detail in Volume II, and their electrical characteristics under normal and abnormal operating conditions determined. Overhead transmission lines, treated in Volume I, are not discussed in Volume II. Curves and charts are given for determining skin effect and proximity effect in circuits of non-magnetic solid and tubular conductors. The effects of open conductors in three-phase circuits supplying ungrounded transformer banks are discussed, and charts are given to show the conditions under which high overvoltage or phase reversal of induction motors or both may occur. Methods are given for determining the impedances seen from relays during power swings, with and without faults.


In Volume II, as in Volume I, the endeavor has been to present methods of procedure in determining the performance of a-c power systems under normal and abnormal operating conditions. Special attention is also given to the development of equivalent circuits for use in the component networks. Owing to space limitation in a book of this size, all types of equipment and all possible abnormal operating conditions have not been included. it is hoped, however, that the methods of analysis given here can be applied by the operating engineer to other types of equipment and to other abnormal operating conditions which may occur on his system.


Although the greater part of Volume II was written while the author was a member of the Central Station Engineering Department of the General Electric company, much remained to be done after she joined the Electrical Engineering Department of The University of Texas. The delay in production of Volume II is due in part to the change in point of view from that of an engineer in industry to that of a teacher, and to the attempt to make Volume II a textbook for seniors and graduate students as well as a reference for power system engineers.

Contents (Volume II)

  1. Introduction and Summary of Equations
  2. Impedances of Electric Circuits
  3. Electrical Characteristics of Insulated Cables
  4. Transformers and Autotransformers
  5. Transformers in System Studies
  6. Induction Machines
  7. Synchronous Machines
  8. αβ0 Components in Synchronous-Machine Analysis
  9. System Protection — Relays

    Appendix A. Tables
    Appendix B. Development of Equations — Rotating Machines
    Appendix C. Reciprocals of Equations for Circles and Straight Lines