ÆTHERFORCE THEOEY AND CALCULATION OF TRANSIENT ELECTRIC PHENOMENA AND OSCILLATIONS BY CHARLES. PROTEUS STEINMETZ THIRD EDITION RTCVISED AND ENLARGED THIRD IMPRESSION McGRAW-HILL BOOK COMPANY, ING. NEW YORK: 370 SEVENTH AVENUE LONDON: & 8 BOUVEBIE ST., E. C. 4 1920 ÆTHERFORCE 3'/a7 COPYRIGHT, 1920, BY THE MCGRAW-HILL BOOK COMPANY, INC. COPYRIGHT, 1909, BY THE McGiiAw PUBLISHING COMPANY. FEINTED IN THE UNITED STATES OF AMEBICA. f', LIBRARY THE MAPLE PRESS - YORK PA ÆTHERFORCE DEDICATED TO TUB MWM.OHY OK MY FRIEND AND TEACHER HUDOLtf EKJKEMEYER ÆTHERFORCE ÆTHERFORCE PREFACE TO THE THIRD EDITION SINCE the appearance of the, first edition, ten years ago, the study of transients has been greatly extended and the term "transient" has become fully established in electrical literature. As the result of the increasing importance of the subject and our increasing knowledge, a large part of this book had practically to be rewritten, with the addition of inuch new material, espe- cially in Sections III and IV. In Section III, the chapters on "Final Velocity of the Electric Field" and on "High-frequency Conductors" have been re- written and extended. As Section V, an entirely new section has been added, com- prising six new chapters. The effect of the finite velocity of the electric field, that is, the electric radiation in creating energy components of inductance and of capacity and thereby effective series and shunt resistances is more fully discussed. These components may assume formid- able values at such high frequencies as are not infrequent in transmission circuits, and thereby dominate the phenomena. These energy components and the equations of the unequal current distribution in the conductor are then applied to a fuller discussion of high-frequency conduction. In Section IV, a chapter has been added discussing the relation of the common types of currents: direct current, alternating current, etc., to the general equations of the electric circuit. A discussion is also given of the interesting case of a direct current with distributed leakage, as such gives phenomena analogous to wave propagation, such as reflection, etc., which are usually familiar only with alternating or oscillating currents. A new chapter is devoted to impulse currents, as a class of non-periodic but transient currents reciprocal to the periodic but permanent alternating currents. Hitherto in theoretical investigations of transients, the circuit constants r L C and g have been assumed as constant. This, however, disagrees with experience at very high frequencies ÆTHERFORCE viii PREFACE or steep wave fronts, thereby limiting the usefulness of the theoretical investigation, and makes the calculation of many im- portant phenomena, such as the determination of the danger zone of steep wave fronts, the conditions of circuit design limit- ing the danger zone, etc., impossible. The study of these phenomena has been undertaken and four additional chapters devoted to the change of circuit constants with the frequency, the increase of attenuation constant resulting therefrom, and the degeneration, that is rounding off of complex waves, the flattening of wave fronts with the time and distance of travel, etc., added. The method of symbolic representation has been changed from the time diagram to the crank diagram, in accordance with the international convention, and in conformity with the other books; numerous errors of the previous edition corrected, etc. CHARLES P. STEINMETZ. Jan., 1920. ÆTHERFORCE PREFACE TO THE FIRST EDITION THE following work owes its origin to a course of instruction given during the last few years to the senior claas in electrical engineering at Union University and represents the work of a number of years. It comprises the investigation of phenomena which heretofore have rarely been dealt with in text-books but have now become of such importance that a knowledge of them is essential for every electrical engineer, as they include sonic? of the most important problems which electrical engineering will have to solve in the near future to maintain its thus far unbroken progress. A few of these transient phenomena were observed and experi- mentally investigated in the early clays of electrical engineering for instance, the building up of the voltage of direct-current generators from the remanent magnetism. Others, such a,s the investigation of the rapidity of the response of a compound generator or a booster to a change of load, have become of impor- tance with the stricter requirements now made on electric totems Iransient phenomena which were of such abort duration and' small magnitude as to be negligible with the small apparatus of former days have become of serious, importance in the, hu, generators and high power systems of to-day, as the discharge of generator fields, the starting currents of transformers the short circuit currents of alternators, etc. Especially is this , t tht ocl asses of phenomena closely related to " IK ÆTHERFORCE x PREFACE and others, dealing with the fairly high frequency of sound waves. Especially lightning and all the kindred high voltage and high frequency phenomena in electric systems have become of great and still rapidly increasing importance, due to- the great increase in extent and in power of the modern electric systems, to the interdependence of all the electric power users in a large territory, and to the destructive capabilities resulting from such disturbances. Where hundreds of miles of high and medium potential circuits, overhead lines and underground cables, are interconnected, the phenomena of distributed capacity, the effects of charging currents of lines and cables, have become such as to require careful study. Thus phenomena which once were of scientific interest only, as the unequal current distribu- tion in conductors carrying alternating currents, the finite velocity of propagation of the electric field, etc., now require careful study by the electrical engineer, who meets them in the rail return of the single-phase railway, in the effective impedance interposed to the lightning discharge on which the safety of the entire system depends, etc. The characteristic of all these phenomena is that they are transient functions of the independent variable, time or distance, that is, decrease with increasing value of the independent variable, gradually or in an oscillatory manner, to zero at infinity, while the functions representing the steady flow of electric energy are constants or periodic functions. While thus the phenomena of alternating currents are repre- sented by the periodic function, the sine wave and its higher harmonics or overtones, most of the transient phenomena lead to a function which is the product of exponential and trigono- metric terms, and may be called an oscillating function, and its overtones or higher harmonics. A second variable, distance, also enters into many of these phenomena; and while the theory of alternating-current appara- tus and phenomena usually has to deal only with functions of one independent variable, time, which variable is eliminated by the introduction of the complex quantity, in this volume we have frequently to deal with functions of time and of distance., ÆTHERFORCE PREFACE xi We thus have to consider alternating functions and transient functions of time and of distance. The theory of alternating functions of time is given in " Theory and Calculation of Alternating Current Phenomena." Transient functions of time are studied in the first section of the present work, and in the second section are given periodic transient phenomena, which have become of industrial importance, for instance, in rectifiers, for circuit control, etc. The third section gives the theory of phenomena which are alternating in time and transient in distance, and the fourth and last section gives phenomena transient in time and in distance. To some extent this volume can thus be considered as a con- tinuation of "Theory and Calculation of Alternating Current Phenomena." In editing this work, I have been greatly assisted by Prof. 0. Ferguson, of Union University, who has carefully revised the manuscript, the equations and the numerical examples and checked the proofs, so that it is hoped that the errors in the work are reduced to a minimum. Great credit is clue to the publishers and their technical staff for their valuable assistance in editing the manuscript and for the representative form of the publication they have produced. CHARLES P. STEINMETZ. SCHENECTADY, December, 1908. ÆTHERFORCE [...]... Conditions for oscillations, and the possibility of excessive currents and voltages 13 14 of transient Example Example of the gradual and the an alternating current to 15 16 17 its oscillatory approach 17 of permanent value 20 Conditions for appearance of transient terms, and for Effect of capacity their harmlessness or danger Relations of transient terms and their character to the stored energy of the circuit... Effect of frequency on transient terms to be expected in transformers 181 103 Effect of magnetic stray field or leakage on transient starting current of transformer 182 104 Effect of the resistance, equations, and method of construction of transient current of transformer when 105 Construction of numerical examples, ISO excessive transient currents of transient 181 currents 185 starting by 188 table of. .. CHAPTER 1 I THE CONSTANTS OF THE ELECTRIC Flow of electric energy, the CIRCUIT electric field 3 and its components 2 The electromagnetic field, the electrostatic field and the power consumption, and their relation to current and 3 The electromagnetic energy, the electrostatic energy, and the power loss of the circuit, and their relations to the 4 Effect of conductor shape voltage and and material on circuit... Equations of quarter-wave and half-wave 66 Terminal conditions Distribution of current and voltage at start, and evaluation of the coefficients of the trigonometric series length, and angular measure of distance 554 equations of quarter-wave and half-wave 67 Final 550 oscillation 68 69 558 oscilla- 559 tion Numerical example of the discharge of a transmission line Numerical example of the discharge of a... character of the arc rectifier 255 Constant-potential and constant- current type 18 Mode 25(3 of operation of mercury arc rectifier: Angle of 258 over-lap 19 Constant-current 20 Theory and rectifier: Arrangement of apparatus calculation: Differential equations 21 Integral equations 22 Terminal conditions and 23 Calculation of numerical example final equations Performance curves and oscillograms Transient. .. 25 27 Equations of the equivalent sine waves of the mercury arc rectifier Numerical example 261 262 264 266 268 269 273 275 277 SECTION ^5) TRANSIENTS IN SPACE CHAPTER 283 INTRODUCTION I 1 Transient phenomena in space, as periodic functions of time and transient functions of distance, represented by transient functions of complex variables 283 2 Industrial importance of transient 284 phenomena in space... the finite velocity of the electric 394 industrial importance magnetic and dielectric field leading to energy components of inductance voltage and capacity current and of 69 Lag 70 Conditions under which this effect of the 395 thereby to effective resistances finite velocity is considerable and therefore of importance A 394 field is of Inductance of a Length Return Conductor, lo of an Infinitely radiation... resistance, inductance and capacity 5 6 The resistance of materials metals, electrolytes, insulators and pyroelectrolytes Inductance and the magnetic characteristics of materials Permeability and saturation, and its effect on the mag: netic field of the circuit 7 Capacity and the dielectric constant of materials The disruptive strength of materials, and its effect on the electrostatic field of the circuit... voltmeter The coefficients L and C, which are the proportionality factors of the magnetic and of the dielectric component of the electric field, are called the inductance and the capacity of the circuit, respectively As electric power P is resolved into the product of current i and voltage e, the power loss in the conductor, Pb therefore can also be resolved into a product of current i and voltage &i which... Discussion of reactance and frequency 90 Discussion of effective resistance 430 91 433 92 Discussion of size, shape and material of conductor, and 434 frequency 93 Discussion of size, shape and material on circuit constants 94 Instances, equations and tables 95 Discussion of tables 96 Continued 97 Conductor without return conductor 435 430 437 442 444 ÆTHE ORCE RF CONTENTS SECTION IV xxv TRANSIENTS IN TIME AND . transient functions of time and of distance. The theory of alternating functions of time is given in " Theory and Calculation of Alternating Current Phenomena. " Transient functions of time are. ÆTHERFORCE THEOEY AND CALCULATION OF TRANSIENT ELECTRIC PHENOMENA AND OSCILLATIONS BY CHARLES. PROTEUS STEINMETZ THIRD EDITION RTCVISED AND ENLARGED THIRD IMPRESSION McGRAW-HILL BOOK COMPANY, ING. NEW YORK: 370. periodic transient phenomena, which have become of industrial importance, for instance, in rectifiers, for circuit control, etc. The third section gives the theory of phenomena which are alternating in time and transient in distance, and the fourth and last section gives phenomena transient in time and in distance. To some extent