GIFT OF Author THE INDUCTION MOTOR AND OTHER ALTERNATING CURRENT MOTORS (Frontispiece) THE INDUCTION MOTOR AND OTHER ALTERNATING" CURRENT MOTORS THEIR THEORY AND PRINCIPLES OF DESIGN 1 t B. A. BEHREND FELLOW, AND PAST SENIOR VICE PRESIDENT, AMERICAN INSTITUTE OF ELECTRICAL ENGINEERS i FELLOW, AMERICAN ACADEMY OF ARTS 4 SCIENCES; FELLOW, AMERICAN ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE; MEMBER, AMERICAN SOCIETY OF CIVIL ENGI- NEERS AND AMERICAN SOCIETY OF MECHANICAL ENGINEERS, ETC. SECOND EDITION REVISED AND ENLARGED SECOND IMPRESSION McGRAW-HILL BOOK COMPANY, INC. NEW YORK: 370 SEVENTH AVENUE LONDON : 6 & 8 BOUVERIE ST., E. C. 4 1921 COPYRIGHT, 1921, BY THE MCGRAW-HILL BOOK COMPANY, INC. PRINTED IN THE UNITED STATES OF AMERICA THE MAPLE FXtKSS Y O K K FA Go THE GREAT PIONEERS WHO HAVE BEEN MY FRIENDS NIKOLA TESLA, GISBERT KAPP, ANDRE BLONDEL, C. E. L. BROWN THIS BOOK IS AFFECTIONATELY INSCRIBED /f -I A O " Ignorance more frequently begets confidence than does knowledge." CHARLES DARWIN, "The Descent of Man," p. 3. "It is particularly interesting to note how many theorems, even among those not ordinarily attacked without the help of the Differential Calculus, have here been found to yield easily to geometrical methods of the most elementary character. "Simplification of modes of proof is not merely an indication of advance in our knowledge of a subject, but is also the surest guarantee of readiness for farther progress." LORD KELVIN AND PETER GUTHRIE TAIT, "Elements of Natural Philosophy," p. v. "The simplicity with which complicated mechanical interactions may be thus traced out geometrically to their results appears truly remarkable." SIR GEORGE HOWARD DARWIN, "On Tidal Friction," in "Treatise on Natural Philosophy." By KELVIN AND TAIT, p. 509. " the absence of analytical difficulties allows attention to be more easily concentrated on the physical aspects of the question, and thus gives the student a more vivid idea and a more manageable grasp of the subject than he would be likely to attain if he merely regarded electrical phenomena through a cloud of analytical symbols." SIR JOSEPH JOHN THOMSON, "Elements of the Mathematical Theory of Electricity and Magnetism," p. vi. "It is remarkable that such elementary cases of Newton's dynamics should require abstruse considerations for their explanation. But it is far worse in the more modern dynamics, with ignoration of coordinates, and modified Lagrangean functions. Dynamics as visible to the naked eye seems to disappear altogether sometimes, leaving nothing but complicated algebra." OLIVER HEAVISIDE, "Electromagnetic Theory," vol. iii, p. 401. "Let them make the effort to express these ideas in appropriate words without the aid of symbols, and if they succeed they will not only lay us laymen under a lasting obligation, but, we venture to say, they will find themselves very much enlightened during the process, and will even be doubtful whether the ideas as expressed in symbols had ever quite found their way out of the equations into their minds." "The Scientific Papers" of JAMES CLERK MAXWELL, vol. ii, p. 328. vii "The scientific career of Rankine was marked by the gradual develop- ment of a singular power of bringing the most difficult investigations within the range of elementary methods." "The Scientific Papers" of JAMES CLERK MAXWELL, vol. ii, p. 663. "Lagrange came to grief over the small conical oscillations of the spherical pendulum, yet he could have saved himself and detected his error but for the self-imposed restraint of excluding the diagram from his " Mecanique analytique." So it is curious to find the same fashion coming again in the modern school of pure analytical treatment, of doing away with an appeal to the visual sense of a geometrical figure." SIE GEORGE GREENHILL, Nature, April 17, 1919. viii PREFACE TO SECOND EDITION As indicated in the preface to the first edition of this little book, it owed its origin to a series of lectures delivered at the University of Wisconsin in January, 1900. These lectures were published in the Electrical World and appeared in 1901 in book form entitled "The Induction Motor." The book was translated into several languages among others into French and German. The American edition was soon exhausted and repeatedly attempts were made by myself and by assistants and associates of mine to revise it, but though several agreements were entered into between the publishers and Mr. A. B. Field and myself for a second edition, other and more urgent demands upon our time prevented the completion of the work. Once more then, twenty years after its first appearance, this little book addresses itself to the engineering public. The first edition contained at the time almost entirely new matter and almost all of this originated with the author. Tested though it was by the most careful laboratory work, yet a certain diffidence prevented the author from pressing his claims to recognition. Rarely perhaps has any early work become so absorbed into the texture of thought of engineers as the substance of this little book. The kindly words of my friend Dr. Addams S. McAllister in a presentation copy of his own excellent treatise on " A Iternating Current Motors," in which he says that to the present author "all writers on induction motors, and all students of induction motor phenomena, are indebted for the first presentation of the conception of the phenomena now considered modern," I would not here repeat though I treasure them very highly were it not for the fact that as a perusal of the introduction may indicate my work is constantly quoted as done by others and these quota- tions are as a dispassionate analysis indicates not in accord- ance with the plain facts. The circle diagram has become indispensable to the engineer. Its first demonstration and proof were developed by me in 1895. In its present form it is used exactly as given by me in "The Induc- tion Motor," New York, 1901. The idea of the leakage coefficient ix X PREFACE TO SECOND EDITION and its characteristics have been found correct and have been universally adopted. The conception of the single-phase motor with the primary exciting belt resolved into two component motors simulated by two poly-phase motors in series with opposite torque, which conception I worked out quantitatively, has recently been commended as the best method for students in a paper by Mr. B. G. Lamme* (A. I. E. E., April 1918). Yet, here as elsewhere, "A prophet is not without honor, save in his own country, and in his own house." I think the explanation for this must be found in the tendency of mankind to prefer to give recognition to those remote from us rather than to associates or acquaintances. It is easier, for instance, to name some one whom the students do not know and will not come in contact with, as the originator of a certain theory, than a man whom they are likely to meet in their professional relations. Mankind, and especially professional mankind, is chary of praise of its fellow-workers. An interesting example of this is furnished in the theory of the regulation of alternators. Numerous references in American textbooks are made to "Po tier's Method" for determining the regulation of alternators. Now, I believe those who call a certain method by this name can never have referred to A. Po tier's paper "Sur la Reaction d'Induit des Alternateurs," p. 133. L' Eclairage Electrique, 28th July, 1900. Prof. A. Potier was a great savant and a gentleman. His paper abounds in carefully selected references. He claimed no new method. He stated that Mr. Kapp many years before (about 1893 and since) used a method for the determination of the regulation of alternators in which he resolves the total effect of the armature currents into an "armature reaction" component and a " self- induction" component, forming a right angle triangle in the regulation curve in the case of zero power factor He then refers to copious data published by me in the E. T. Z., in L' Eclairage Electrique, in the Electrical World, and other data which I sent him at his request privately, pointing out the corroboration of Kapp's method and indicating and this is the only new point in the paper that this method therefore implies identity of the zero power factor regulation curves as given by Dr. Behn- Eschenburg and myself for years, with the saturation curve, a? * See also F. W. Alexanderson, Transactions A. I. E. E., 1918, Part I, p. 691, 692, 693. PREFACE TO SECOND EDITION xi the two have been proved by my tests to be equidistant and displaced from each other. After this interesting theoretical remark, he completes his paper by citing Kapp and giving the method of determining the regulation, upon my experimental data of which he bases his theoretical conclusion. Therefore, the method of two components is to be designated by no single name ; while the intrinsic importance of zero power factor regula- tion was urged continually* since 1896 by myself until its final adoption by the American Institute in 1914! As I once wrote to Mr. Oliver Heaviside, quoting Huxley, " Magna est veritas et praevalebit! Truth is great, certainly, but, considering her greatness, it is curious what a long time she is apt to take about prevailing." And to one with scholarly inclina- tions, eking out a livelihood by the practice of engineering, it is a matter of inward gratification to see one's work generally adopted " among the rubble of the foundations of later knowledge and forgotten." Remembering that I was twenty years old when I published the much referred to circle diagram I will say this to a young reader of the present generation by way of advice: Let him not trouble his head with recognition. "If truth does not prevail in his time, he will be all the better and the wiser for having tried to help her. And let him recollect that such great reward is full payment for all his labor and pains." (Huxley.) In the treatment of the theory, the diagram of fluxes as devel- oped in 1895 by A. Blondel and used ever since by me, has been continued throughout this book. Its simplicity and brilliant elegance is much to be preferred to the older method of Kapp's, so extensively adopted by Steinmetz and others. It is also greatly to be preferred to the " equivalent circuit" methods which are interesting as an exercise but somewhat artificial and removed from intimate contact with the physical phenomena. On account of their identity with Hopkinson's, I have adopted Blondel's stray coefficients, greater than one, which are the recip- rocals of my former ones, smaller than one, in order to establish uniformity of notation. It is opportune to say a few words on the subject of the absence of complex algebra in this little volume. I have given this question a great deal of thought. At first I intended to give in parallel chapters the results of the theory in complex algebraic * "The Experimental Basis for the Theory of the Regulation of Alterna- tors." By B. A. Behrend, Am. Inst. El. Engrs., May 19, 1903. xii PREFACE TO SECOND EDITION form. But I became discouraged in working out a number of problems. The algebraization, to borrow a term from Heaviside, is certainly cumbersome, and one may be happy indeed if one succeeds in avoiding algebraic or arithmetic errors. Page after page is covered with algebraic symbols at which the careful and conscientious calculator looks with much anxiety. It is indeed a beautiful method, this method of resolution of directed quantities into rectangular coordinates, but I doubt whether it is suitable for all types of engineering minds. Perhaps here as elsewhere, it is charitable to let men work out the methods best suited to themselves and not to press intolerance to the point of imposing one method upon all. This is particularly advisable as a graph- ical method can, and should, be checked by an algebraic process and then the graphical process is explanatory of, and elucidating, the physical process. I have therefore decided to omit the use of the symbolic method, and the reader should turn to other works if he desires algebraic treatment. It may be necessary to emphasize that the treatment of the phenomena loses nothing in accuracy or elegance by the adoption of graphic methods which have been used and advocated by Maxwell, Kelvin, Sir George Darwin, Sir. J. J. Thomson, and others. The squirrel cage motor with two secondaries with different resistances and leakages is here treated graphically, and so is the theory of concatenation. A chapter on speed regulation of induction motors is also added. On the subject of leakage in induction motors a great deal has been published, but I have found it inadvisable to embody much of it in this edition. Practical formulae and calculations based on them should be used sparingly, excepting in the workshop, and they should invariably be based on personal experience. One should not encourage begetting the formula habit. The theory of the single-phase induction motor has been given in two ways. First, as originally given by me in 1897 with the assumption of two rotating fields, and the equivalence of two rotating field motors in series; and secondly, as first given by Potier and Goerges and beautifully completed by Prof. Sumec with the use of the cross-magnetization as used and advocated in this country by A. S. McAllister. In the chapter on the poly-phase series motor, I have followed to some extent the brilliant work of Andre Blondel to whom we all are greatly indebted in every branch of electrical engineering. PREFACE TO SECOND EDITION xiii The Heyland compensated motor has logically received its treatment in this chapter, as it is a poly-phase shunt motor as pointed out and proved by Blondel. It seemed unnecessary to treat of the windings for induction motors in view of A. M. Dudley's treatise on "Connecting Induction Motors," McGraw-Hill Book Co., 1921, and equally unnecessary to retain the two chapters on design contained in the first edition as the books of Mr. H. M. Hobart and of my former assistant, Prof. Alexander Miller Gray, * have supplied this need better than I could have done. A few chapters deal with such subjects as the improvement of power factor, as suggested by Leblanc and Kapp, the magnetic pull, and other allied subjects. Brevity in the text has been preferred to prolixity as the lesser of two evils. This book is not meant to be a work of an encyclopedic charac- ter. Nothing that I could write could, in that respect, touch the work of Arnold and LaCour. Nor is it to take the place of such admirable work as Alexander Russell's which should be read by every electrical engineer. It is essentially the work of an engi- neer, who has had the good fortune to have been actively asso- ciated with the art of electrical engineering through almost three decades and who has had a part in the development of the machines about which he writes. He thus addresses himself to his fellow-engineers, revealing the methods which he has followed in the design and construction of alternating current motors, of which literally millions of horse-power were executed under his direction, The design of electrical machinery, as of all machinery, is based upon intelligent comparison of empirical data, and the art of designing, therefore, cannot be taught without such data. The methods and principles taught in this book aim solely at creating means of effecting such comparisons. To "calculate" a machine, as the term is frequently employed, is not feasible and only principles and fundamentals can be taught in school. No apology is made for the personal references which occur in this book. The tendency to write books without references is due largely to the desire to avoid the looking-up of other writers' papers. The reader is not benefited by such treatment, as he may frequently prefer the original to the treatment of the author * "Induction Motor Design Constants." Electrical World, Dec. 30, 1911. "Electrical Machine Design," McGraw-Hill Book Co., 1913. xiv PREFACE TO SECOND EDITION whose book he is reading. Besides, a knowledge of the literature of our profession is essential to an understanding of the art and to an honest interpretation of the part played therein by our fellow-workers. My thanks are due to my secretary, Miss Gladys Naramore, A. B., Boston University, 1916, for much painstaking work, and to my friend Dr. Addams Stratton McAllister for his untiring aid, enthusiasm, and criticism. His friendship has been an in- spiration and his labors in helping me to put the book through press are beyond the rendering of thanks. To the publishers thanks are due for the successful form of the book and to Mr. John Erhardt for his efficiency and for his patience with the author. An entire chapter has had to be added to the book on account of a solution of certain problems of inversion solved in a very elegant manner by Dr. A. S. McAllister and communicated to me before publication by him. Thus has been solved a problem with which I have coped in vain these twenty-five years. To Professor Miles Walker, of the University of Manchester, England, I am indebted for numerous suggestions. This little book now goes forth as a sort of engineering testa- ment of the author's work in connection with the motors invented thirty-three years ago by his friend Mr. Nikola Tesla. Great things have been done and illumined by these theories and gigantic engineering feats have been achieved. and tho' We are not now that strength which in old days Moved earth and heaven; that which we are, we are; One equal temper of heroic hearts, Made weak by time and fate, but strong in will To strive, to seek, to find, and not to yield. (Tennyson) BOSTON, MASSACHUSETTS, B. A. BEHREND. February, 1921. PREFACE TO FIRST EDITION The literature of electrical engineering has become so vast and extensive that it is impossible for any man to keep pace with all that is written on electrical subjects. He who produces a new book that adds to the swelling tide of new publications, may justly be asked for his credentials. My justification for writing this tract will be found in the fact that, though almost all branches of applied electricity have enlisted the industry of authors, the induction motor has received comparatively little attention from competent engineers. The few whose experience and knowledge would entitle them to speak with authority on this subject are deterred from publishing by commercial reasons. I have made the induction motor the subject of early and special studies, and a comparison of my treatment of its theory with the purely analytical theories will show how far I have suc- ceeded in simplifying and elucidating so complex a subject. The graphical treatment of abstruse natural phenomena is constantly gaining ground, and I quote with satisfaction the words of so great a mathematician as Prof. George Howard Darwin, Fellow of Trinity .College, Cambridge, who says on p. 509 of the second volume of Lord Kelvin and Prof. Tait's Treatise on Natural Philosophy that "the simplicity with which complicated mechanical interactions may be thus traced out geometrically to their results appears truly remarkable." All through this little book I have endeavored to let inductive method check at every step the mathematical or graphical deduction of the results. A wide experience with mono- and poly-phase alternating-current induction motors, gained at the Oerlikon Engineering Works, Switzerland, has enabled me to do so. Thus the careful reader who is willing to profit by the experi- ence of others, will find many valuable hints and results which he can turn to account in his practice. Many induction motors have been designed on the principles laid down in this little treatise, and in no case has the theory failed to answer the questions suggested by observation. The writing of this book has been mainly a labor of love. Those who know of the troubles, cares and labor involved in xv xvi PREFACE TO FIRST EDITION writing a book and bringing it through the press, not to mention the sacrifice of personal experience by publication, will doubtless be able to appreciate this thoroughly. I wish to thank the editors of the Electrical World and Engineer for the pains they have taken with the publication of this book, and I must specially thank Mr. W. D. Weaver for the encourage- ment he has always given to me. To Mr. T. R. Taltavall, Associate Editor of Electrical World and Engineer, who has taken endless pains with the proofs of this book, I feel very much indebted. The substance of this volume was delivered in January, 1900 in the form of lectures at the University of Wisconsin, Madison, Wis., and I wish to thank Prof. John Butler Johnson, Dean of the College of Mechanics and Engineering, for the invitation as non-resident lecturer which he extended to me. To him and to Prof. D. C. Jackson I am greatly indebted for the hospitality conferred upon the stranger within their gates. SOUTH NORWOOD, OHIO, B. A. BEHREND. January, 1901. CONTENTS PAGE PREFACE TO SECOND EDITION ix PREFACE TO FIRST EDITION. . xv CHAPTER I INTRODUCTION. HISTORICAL INTRODUCTION AND BRIEF SKETCH OF THE HISTORY OF THE THEORY OF THE INDUCTION MOTOR 1 THE DEVELOPMENT OF THE THEORY OF THE SINGLE-PHASE INDUCTION MOTOR . 18 CHAPTER II THE THEORY OF FLUXES AND STRAY FIELDS FORMULA FOR INDUCED E. M. F 21 BEHREND'S AND BLONDEL'S STRAY-COEFFICIENTS 23 ELECTRIC CIRCUITS SIMULATING THE LEAKAGE PATHS OF THE MAG- NETIC CIRCUIT OF THE INDUCTION MOTOR 24 THE POLAR DIAGRAM FOR CONSTANT CURRENT 25 THE POLAR DIAGRAM FOR CONSTANT VOLTAGE . 27 CHAPTER III THE GENERAL ALTERNATING-CURRENT TRANSFORMER A. THE TRANSFORMER WITH NON-INDUCTIVE LOAD 29 The Author's Method of Accounting for Primary Resistance . ... 31 Another Method of Accounting for Primary Resistance 33 Accounting for Primary Resistance by the Method of Reciprocal Vectors 38 The Losses and Their Representation by Straight Lines 42 The Iron Losses Due to Hysteresis and Eddy Currents 44 B. THE TRANSFORMER WITH INDUCTIVE LOAD 46 C. THE TRANSFORMER WITH CAPACITY LOAD , 47 xvii xviii CONTENTS CHAPTER IV THE MCALLISTER TRANSFORMATIONS PAGE A. RESISTANCE IN SERIES WITH THE MOTOR without CORE Loss. . 50 B. REACTANCE IN SERIES WITH THE MOTOR without CORE Loss .... 51 C. IMPEDANCE IN SERIES WITH THE MOTOR without CORE Loss .... 52 D. RESISTANCE IN SERIES WITH THE MOTOR with CORE Loss 54 E. REACTANCE IN SERIES WITH THE MOTOR with CORE Loss ... 55 F. IMPEDANCE IN SERIES WITH THE MOTOR with CORE Loss .... 55 CHAPTER V THE ROTATING FIELD AND THE INDUCTION MOTOR A. THE AMPERE TURNS AND THE FIELD BELT 57 B. THE E. M. Fs. INDUCED IN THE WINDINGS 58 C. THE ELEMENTARY THEORY OF THE INDUCTION MOTOR. .... 64 D. THE SQUIRREL CAGE 69 E. THE TORQUE AND SLIP AND THE EQUIVALENCE OF MOTOR AND TRANSFORMER The Torque 74 The Slip 75 Torque Curves 76 F. HIGHER HARMONICS IN THE FIELD BELT AND THEIR EFFECT UPON THE TORQUE 77 G. EXPERIMENTAL DATA 81 H. COLLECTION OF DATA 81 CHAPTER VI THE INDUCTION GENERATOR A. THE THEORY OF TORQUE AND SLIP 82 B. STABILITY 83 C. EXPERIMENTAL DATA . 85 CHAPTER VII THE SHORT-CIRCUIT CURRENT AND THE LEAKAGE FACTOR A. THE SLOTS 87 B. THE NUMBER OF SLOTS PER POLE 89 C. CHARACTERISTICS OF ROTOR WINDINGS 89 D. TEST DATA 91 E. THE LEAKAGE FACTOR . 94 CONTENTS xix PAGE F. THE INFLUENCE OF THE AIR-GAP UPON THE LEAKAGE FACTOR . 95 G. THE INFLUENCE OF THE POLE-PITCH UPON THE LEAKAGE FACTOR . 99 H. THE DIFFERENT LEAKAGE PATHS 101 I. FURTHER EXPERIMENTAL DATA 104 K. WINDING THE SAME MOTOR FOR DIFFERENT SPEEDS 105 L. DRAWBACKS OF A HIGH FREQUENCY 107 M. HISTORICAL AND CRITICAL DISCUSSION OF THE LEAKAGE FACTOR. . 110 N. BIBLIOGRAPHY . .114 CHAPTER VIII THE DOUBLE SQUIRREL-CAGE INDUCTION MOTOR ARRANGEMENT OF SLOTS AND THE LEAKAGE PATHS 116 EQUIVALENT CIRCUITS AND FLUX DIAGRAM 117 AN EXAMPLE 119 THE POLAR DIAGRAM 120 THE TORQUE DIAGRAM 121 CHAPTER IX POLY-PHASE COMMUTATOR MOTORS Properties of Commutators A. THE ACTION OF THE COMMUTATOR 124 B. PROPERTIES OF PHASE LAG OR LEAD OF THE POLY-PHASE COMMU- TATOR 127 C. COMPARISON BETWEEN INDUCTION MOTORS WITH ROTORS SHORT- CIRCUITED THROUGH RINGS OR OF THE SQUIRREL-CAGE TYPEJ AND ROTORS SHORT-CIRCUITED THROUGH SYMMETRICAL POLY- PHASE BRUSHES 128 D. THE REFLECTION INTO THE PRIMARY CIRCUIT OF THE M. M. F. OF THE SECONDARY WITH SLIP-RING AND COMMUTATOR ROTORS. 130 E. VARIABLE AND CONSTANT SECONDARY REACTANCE OF THE COM- MUTATOR MOTOR 131 F. THE SLIP-RING COMMUTATOR TYPE AS FREQUENCY CHANGER . . 136 CHAPTER X THE SERIES POLY-PHASE COMMUTATOR MOTOR A. THE THEORY FOR CONSTANT CURRENT AND CONSTANT POTENTIAL IN THE IDEAL MOTOR 137 The Torque 141 The Slip 142 B. THE THEORY FOR CONSTANT CURRENT AND CONSTANT POTENTIAL IN THE REAL MOTOR 143 C. THE NECESSITY OF SATURATION FOR STABILITY 145 XX CONTENTS CHAPTER XI THE SHUNT POLY-PHASE A. C. COMMUTATOR MOTOR PAGE A. HISTORICAL INTRODUCTION 146 B. THE THEORY OF THE SHUNT POLY-PHASE A. C. COMMUTATOR MOTOR FOR CONSTANT POTENTIAL 147 C. DETERMINATION OF THE TOTAL PRIMARY CURRENT 149 D. SPEED REGULATION AND THE SLIP 151 E. BIBLIOGRAPHY 152 CHAPTER XII METHODS OF SPEED CONTROL A. CONCATENATION 153 B. THE POLY-PHASE MOTOR WITH SINGLE-PHASE SECONDARY. . . . 173 CHAPTER XIII METHODS OF SPEED CONTROL (Continued) C. CONCATENATION OF AN INDUCTION MOTOR WITH THE COMMUTATOR TYPES FOR THE INDUCTION OF A SLIP FREQUENCY E. M. F. . . 176 D. CHANGE OF SPEED BY CHANGING THE NUMBER OF POLES. . . . 179 CHAPTER XIV TYPES OF VARIABLE SPEED POLY-PHASE COMMUTATOR MOTORS A. THE PLAIN SHUNT MOTOR 182 B. THE MOTOR OF J. L. LA COUR 183 C. THE MOTOR OF M. OSNOS 183 D. THE MOTOR OF H. K. SCHRAGE 184 E. PLAIN SHUNT MOTOR WITH REGULATING WINDING ADDED . . . 186 CHAPTER XV METHODS OF RAISING THE POWER FACTOR OF INDUCTION MOTORS A. THE METHOD OF LEBLANC USING COMMUTATOR MACHINES FOR SECONDARY EXCITATION 187 B. THE USE OF A POLYPHASE COMMUTATOR FOR THE GENERATION OF LEADING CURRENTS 187 CONTENTS xxi PAGE C. THE METHOD OF LEBLANC INDUCING LEADING CURRENTS THROUGH RAPID OSCILLATION OF AN ARMATURE IN A MAGNETIC FIELD . . 189 D. THE SAME METHOD AS ELABORATED BY G. KAPP 195 Bibliography 196 E. THE DANIELSON-BURKE METHOD OF CHANGING THE INDUCTION MOTOR INTO A SYNCHRONOUS MOTOR . . 197 CHAPTER XVI THE MAGNETIC PULL WITH DISPLACED ROTOR A. THE FORMULA OF B. A. BEHREND 198 B. THE ACCURATE SOLUTION BY J. K. SUMEC 201 BIBLIOGRAPHY 203 CHAPTER XVII THE SINGLE-PHASE INDUCTION MOTOR A. THE TWO-MOTOR THEORY. (a) The Magnetizing and No-load Currents 205 (6) The Currents in the Armature 211 (c) The Torque and Slip 211 (d) Experimental Data 213 (e) Calculation of the Magnetizing Current of the Single-phase Motor 214 CHAPTER XVIII THE SINGLE-PHASE INDUCTION MOTOR (Continued) B. THE CROSS FLUX THEORY (a) A General Consideration of the Theory 216 (6) The Derivation of the Circle Diagram and the Locus of the Primary Currents 221 (c) Sumec's Circles for Synchronism, no Load, and Standstill . 224 (d) The Influence of the Rotor Resistance upon the Primary Current Locus 224 (e) Equivalent Circuits 226 (/) Theoretical Considerations 229 (00 The Torque 229 (h) Mechanical Output 231 (i) Rotor Copper Losses 231 xxii CONTENTS CHAPTER XIX THE SINGLE-PHASE REPULSION MOTOR PAGE A. THE NON-COMPENSATED REPULSION MOTOR (a) The General Theory 233 (6) The Speed in the Diagram 237 (c) The Torque 237 (d) The Effect of the Rotor Resistance upon the Diagram . . . 238 (e) The Effect of the Brush Shift 238 (/) Commutation 240 B. THE COMPENSATED REPULSION MOTOR OF WIGHTMAN, LATOUR, AND WlNTER-ElCHBERG (a) Connections 240 (6) The Torque 244 (c) Performance 245 (d) Leakage 245 CHAPTER XX SINGLE-PHASE COMMUTATOR MOTORS A Condensed Review A. VARIETY OP TYPES OF SERIES A. C. COMMUTATOR MOTORS . . 247 B. OPERATING CHARACTERISTICS OF DIFFERENT TYPES 250 C. METHODS OF IMPROVING COMMUTATION (a) Resistance Leads and Limits of Voltage for Commutation . 253 (6) Interpole Connections 254 D. THE SHUNT EXCITED A. C. COMMUTATOR MOTOR 255 E. THE SUPPLY OF SINGLE-PHASE POWER FROM THREE-PHASE SYSTEMS 258 APPENDIX . .261 LIST OF PORTRAITS PAGE NIKOLA TESLA Frontispiece CHARLES EUGENE LANCELOT BROWN 3 ANDR BLONDEL 23 GISBERT KAPP 65 HANS BEHN-ESCHENBURG 113 E. F. W. ALEXANDERSON 179 MAURICE LEBLANC 187 BENJAMIN GARVER LAMME . 247 XXlll THE INDUCTION CHAPTER I INTRODUCTION AND BRIEF SKETCH OF THE HISTORY OF THE THEORY OF THE INDUCTION MOTOR The Induction Motor, or Rotary Field Motor, was invented by Mr. Nikola Tesla, in 1888, a year so "memorable for the experi- mental corroboration by Hertz of Maxwell's electromagnetic waves, a piece of work so shrewdly designated by Oliver Heaviside as "a great hit." 1 The Induction Motor was also "a great hit," though many people could not see it. Engineers almost immediately seized upon its principles. Work was proceeding at Pittsburgh under Mr. George Westing- house, Mr. Tesla, Mr. Shallenberger, Mr. Scott, and Mr. Lamme. The first successful motor, however, embodying in its design and construction those characteristic features which have marked the motor during its career of 30 years, was designed most probably by Mr. C. E. L. Brown at the Oerlikon Works in Switzerland in the year 1890. I said, "most probably" as it is not impossible that that brilliant engineer, whose untimely death we deplore, Mr. Michael von Dolivo-Dobrowolsky, whose company at that time cooperated with the Oerlikon Company, was as much responsible for its creation as Mr. C. E. L. Brown. Surely, both engineers deserve the utmost credit. A 20-hp. motor, built at the Oerlikon Works and designed by Mr. C. E. L. Brown, is shown in Figs. 1 and 2. This motor was exhibited in 1891 at the Electrical Exposition in Frankfort-on-the-Main. A study of its features discloses the distributed stator winding, the small air-gap, and the squirrel-cage rotor, whose invention, I believe, is usually correctly credited to Mr. Dolivo-Dobrowol- 1 "HERTZ became quite Maxwellian after his great hit, save that, as I think, he attached rather too much importance, to the mere equations, as the representation of Maxwell's theory, to the comparative exclusion of the experimentative and philosophical basis." OLIVER HEAVISIDE, "Elec- tromagnetic Theory," Vol. iii, p. 504, "The Electrician" Printing & Publish- ing Co. Ltd., London. 1 2 INDUCTION MOTOR sky. This motor was exhibited in connection with the first alternating-current high-voltage power transmission plant in the world, the three-phase 30,000- volt experimental plant from FIG. 1. Facsimile of Figs. 3 and 4, E. T. Z., Dec. 4, 1891, of C. E. L. Brown's 20-h.p. three-phase alternating current motor. Lauffen to Frankfort, a distance of 120 km. For further his- torical references and data, I refer to my papers in the Elec- -r*ff&&**& (Facing page 2) INTRODUCTION 3 trical World and Engineer, from Nov. 16, 1901 to March 1, 1902, entitled "The Debt of Electrical Engineering to C. E. L. Brown." Figure 1 is taken from these papers. FIG. 2. Facsimile of Figs. 1 and 2, E. T. Z., Dec. 4, 1891, of C. E. L. Brown's 20-h.p. three-phase alternating current motor. It is very interesting to observe that the industrial develop- ment of machinery, whose operation is based upon the correct interpretation of scientific theory, rarely proceeds rapidly and securely until a method of interpretation of such theory has been 4 INDUCTION MOTOR devised which enables the engineer to visualize the physical processes beyond the complex texture of a stream of mathe- matical symbols. 1 However valuable the algebraization to borrow a happy term from Heaviside of physical phenomena may be, it does not supply ideas nor does it supply usually that symbolic skeleton into which the scientific imagination can weave the texture. Alternating-current phenomena are very complicated and, if quantitatively written out in equations, they appear indeed to be well-nigh incomprehensible. A clear comprehension of alter- nating-current theory was begun by a series of brilliant papers published in The Electrician, London, 1885, by Thomas H. Blakesley, entitled "Alternating Currents of Electricity." This series of 10 classical papers discussed for the first time alternating- current phenomena by means of polar diagrams, often perhaps erroneously called vector diagrams, as electromotive force and current are in these cases not vectors at all in the physical sense of the term. They are directive quantities only, because the maxi- mum value of the harmonic wave is used in their construction. I think the next landmark in the development of the theory was made by Mr. Gisbert Kapp, in two papers originally con- tributed to the British Institution of Civil Engineers and the Institution of Electrical Engineers, the latter being printed 1 See also "The Story of the Induction Motor." By B. G. LAMME, Jour- nal of the A. I.E. E., March, 1921. "The development of the Induction Motor being, in reality, an analytical problem, it did not make much headway in the 'cut and try' days of 1888 and 1889, when the Westinghouse Company was undertaking to put it into commercial form. "This brings the Induction Motor up to the present. Its history has been a most interesting one to those who are at all familiar with it. To a certain extent this type of apparatus stands apart in that its development has been due, almost entirely, to the analytical engineer. It is almost impos- sible to conceive that the Induction Motor could have been developed to its present high stage by ordinary 'cut and try' methods. Some good motors might have been obtained in that way, but they would have been accidents of design, instead of the positive results of analysis, as the art now stands. "New applications are continually leading to new developments which are worked out by the analytical designer with an assurance of success not exceeded in any other branch of the electrical art. And the result of all the elaborate theory and complicated analysis and calculation is a practical machine of almost unbelievable simplicity and reliability a standing refutation of the too common idea that complexity in theory leads to complexity in results." INTRODUCTION 5 in The Electrician, Dec. 19, 26, 1890, London. This classical paper of Mr. Kapp's explained in a simple and graphical manner the interesting phenomenon observed by Sebastian Ziani de Ferranti on his 10,000-volt concentric cables from Deptford to London. It is true that neither the work of Blakesley nor that of Kapp contained new theories or new contributions to the science, but in a sense these papers accomplished more. The cumbersome mathematical processes with which these phenom- ena had been invested by mathematicians and physicists of the time made their utilization impracticable if not impossible. Their interpretation by means of the beautiful graphical methods FIG. 3. Fig. 12, p. 447, The Electrical World, "Theory of the Transformer," by F. Bedell and A. C. Crehore. Primary resistance, no leakage, constant primary voltage. of Blakesley and Kapp gave an impulse to the entire field of electrical engineering which it could not have received without the labors of these men. In 1892, F. Bedell and A. C. Crehore published a book entitled "Alternating Currents" (Electrical World Publishers) which was followed in 1893 by a series of articles in the Electrical World. In these works, the polar diagrams were used with extreme skill and lucidity, and they were applied to all manner of problems including the theory of the alternating-current transformer. In these papers the theory of the constant-current transformer 6 INDUCTION MOTOR was developed, showing the locus of the primary e.m.f. to be the periphery of a circle, and as Fig. 3 I here reproduce their Fig. 12, p. 447, The Electrical World, June 17, 1893, showing for con- stant primary e.m.f., the polar diagram, with variation of the primary current. This diagram takes account of primary resis- tance but it does not take account of what is now usually termed the leakage. In 1894 came out the 4th edition of Gisbert Kapp's "Electric Transmission of Energy," in which a most brilliant elementary account is given of the phenomena in induction motors. The polar diagram was developed, including the primary resistance and the leakage. This diagram was given, however, only for each individual point of the load, showing no general solution of the variation of the different characteristic quantities with variation of load. It was based also on the method of represent- ing leakage through internal self -inductive e.m.fs., which is rather cumbersome. In 1895, Andre Blondel published in Eclair age Electrique, Aug. 10, 17, 24, 1895, his fundamental papers entitled "Quel- ques proprietes ge*ne"rales des champs magnetiques tournants." In these papers he developed the theory of the composition of magnetic fluxes, including the leakage fluxes, a method of conception which has since proved of tremendous value. In the same year, the present author, utilizing the conception of magnetic fluxes as developed by Blondel, proved in a simple and direct manner that with variation in load through change in the non-inductive resistance of the secondary load of a trans- former, or through change in load on the shaft of an induction motor, with constant primary e.m.f., the locus of the primary current is a circle in the polar diagram, provided the primary resultant magnetic field is constant, which is the case if the primary resistance of the transformer can be neglected. After delivering a lecture on the subject early in 1895, he published the theory later as a paper on Jan. 30, 1896, in Elektrotechnische Zeitschrift, Berlin. After the lecture, one of the learned professors present expressed his doubt as to my theory being an exact expres- sion of the facts, as he said the theory was too simple to express the complicated facts. I, therefore, wished to test the results and I soon had an opportunity to do so on a 60-hp. Oerlikon motor. Testing in those days was not a very simple matter and running a brake test and watching Siemens dyna- INTRODUCTION 7 mometers with zero reading and Cardew voltmeters was not as simple a procedure as perhaps the present-day generation may imagine, spoiled as they are by all manner of ingenious appliances for simple, direct measurements. When I felt reasonably sure that the theory was very likely correct, though corroborated by only one test, I embodied the record of the test in the paper and sent it to the E. T. Z., which was at that time the central organ for discussing such topics and there it lay until Nov. 11. 1895, when I heard from Mr. Gisbert Kapp, who was then editor of the paper, that he had accepted it. As stated before, it was printed Jan. 30, 1896. While it lay in the editorial offices, a letter came out in the E. T. Z., p. 649, 1895, by A. Heyland, discussing a motor designed by Mr. Danielson and applying to it a circle locus diagram. In his letter, Mr. Heyland referred to his paper in the E. T. Z., Oct. 11, 1894. I immediately looked up Mr. Heyland's paper, expecting to find therein the same method of reasoning and proof which I considered novel in my paper. Instead, I found a rather formid- able array of lines which I was quite unable to comprehend and which I reproduce herewith in facsimile, Fig. 4. When I received the proofs of my paper, I inserted a reference to Mr. Heyland's letter, E. T. Z., p. 649, 1895. Immediately upon the publication of my paper, it was taken up by Prof. Andre* Blondel in V Indus- trie Electrique, Feb. 25, 1896, in a paper which begins as follows: "Le diagramme fondamental des flux d'un moteur asynchrone que j'ai donne a diverses reprises, a e"te utilise^ recemment d'une maniere fort heureuse par M. Behrend, grace a la remarque qu'il a faite que si Ton suppose le F constant et fixe, Fextre'mite du vecteur < de"crit un cercle. Get auteur n'a pas cependant donne encore la solution complete. C'est celle-ci que je me propose d'exposer ici en combinant mes propres remarques avec les siennes. La theorie qui r^sulte de cette collaboration a distance permet d'embrasser d'un coup d'oeil toutes les conditions de construction et de fonctionnement, et constitue a cet e"gard le meilleur d'une e"tude de"tail!6e que j'ai publie*e re"cemment." Mr. Heyland also addressed a letter to the E. T. Z., p. 139, Feb. 27, 1896, which begins: "Mr. Behrend gives a very interest- ing derivation of my diagram . . . " and in this letter he claims his priority. In a communication to the E. T. Z. } p. 116, Feb. 13, 1896, Prof. A. Blondel writes: 8 INDUCTION MOTOR "I have read with the greatest interest the paper by Mr. Behrend (E. T. Z., 1896, No. 5, p. 63) in which he applies the diagram of magne- tic fluxes developed by me two years ago in a very happy manner to the asynchronous motor . . . " In 1895, F. Bedell and A. C. Crehore published a most inter- esting and important paper on ''Resonance in Transformer FIG. 4. Facsimile of Figs. 1, 2, and 3 of A. Heyland's article, "A Graphical Method for the Predetermination of the Transformer and Polyphase Motors," Oct. 11, 1894, E. T.Z. Circuits," in the Physical Review, May-June, 1895, Vol. ii, p. 442, in which the circle locus of the primary current of the transformer was clearly and fully treated by means of polar diagrams, including external inductance in the secondary. This paper contains the complete theory, but it does not make use of the identity of the case treated with that of a transformer with leakage. In 1896 came out "The Principles of the Trans- INTRODUCTION 9 former/' by Frederick Bedell, in which Fig. 123, p. 226, shows the circular primary current locus of a constant-potential transformer, including leakage and primary resistance, obtained by the method of reciprocal vectors from the constant current transformer diagram, which is easier to derive than the constant potential diagram. 1 Professor Blondel, in a letter dated at Paris, Sept. 19, 1903, and published in No. 40, E. T. Z., 1903, writes: "I have shown in the paper referred to 2 that Mr. Behrend and myself have a just claim to many parts of the circle diagram of the ordinary motor. In reference to Mr. Heyland's article, No. 41, E. T. Z., 1894, so repeatedly brought forward, I may say that I have re-read it again, but unfortunately I found it impossible to discover a connection between his circles and the diagram under discussion." 1 In his admirable "Direct and Alternating Current Manual," 2d Ed. New York, D. Van Nostrand Co., 1916, Dr. BEDELL says on p. 288: ''In any circuit or apparatus with constant reactance and variable power con- sumption the current will have a circle locus if the supply voltage is constant . . . This was first shown by BEDELL and CREHORE in 1892. That the induction motor nearly fulfills these conditions and that its current locus is practically the arc of a circle, was first shown by HEYLAND in 1894." A footnote states, "E. T. Z., Oct. 11, 1894; published later in book form and translated into English by ROWE and HELLMUND." The book referred to is a little volume entitled "A Graphical Treatment of the Induction Motor" by ALEXANDER HEYLAND; translated by G. H. ROWE and R. E. HELLMUND, New York, McGraw Publishing Co., 1906. In this book HEYLAND uses an entirely different method from that given by him in 1894, using only a primary leakage coefficient and thus obtaining a simple diagram in contrast to the one using coefficients of mutual and self- induction in his paper of 1894. It must also be stated that HEYLAND by no means first pointed out the identity of the theory of the alternating current transformer and the induction motor but this was first done by DR. BEHN- ESCHENBURG and by GISBERT KAPP in 1893 and 1894. 2 A. BLONDEL, L'Eclairaye Electrique, p. 137, April 25, 1903. "On me permettra de rappeler, a ce propos, que j'ai donne" il y a plu- sieurs annees la premiere epure graphique rigoureuse des flux, courants et forces electromotrices des moteurs asynchrones en fonction des coefficients de fuite de Hopkinson et des coefficients K et k. (Eclairage Electrique, 24 aout 1894, p. 364, et 19 octobre, 1895, p. 100 et 254.) Cette epure, tenant compte de la resistance du stator, conduisait une courbe repre"senta- tive elliptique. "La propriete indiquee sans demonstration par HEYLAND dans une lettre & 1'Elektrotechnische Zeitschrift, de 1895, qu'en ne"gligeant la resistance du stator, le lieu bipolaire de I'extr6mit6 du triangle de Ii et I 2 est un cercle, a 6te dSmontre'e, au moyen du diagramme des flux, par BERNARD BEHREND 10 INDUCTION MOTOR Other references are interesting as of historic importance. Henri Boy de la Tour, in his book "The Induction Motor," translation by C. 0. Mailloux, p. 123, writes: "This method, which is certainly one of the most beautiful applica- tions of graphical methods to the solution of electrical problems, is due to the work of Messrs. A. Blondel, B. A. Behrend, and A. Heyland. "Although M. Blondel may not have observed that a certain point of the diagram should move on a circumference, he has, nevertheless, in our opinion, contributed much to the discovery which was made inde- pendently and almost at the same time by Messrs. B. A. Behrend and A. Heyland, by his having given, ahead of all other authors, an exact analytical study of the operation of three-phase motors, based on a very simple diagram, which constitutes the starting-point of these two engineers." G. Kapp says, p. 459 of "Dynamos, Motors, Alternators, and Rotary Converters," 3d edition, 1902: "For practical purposes the so-called circle diagram, as elaborated by Heyland, is preferable. In the text I have chiefly followed Behrend' s work." And in the 4th edition he adds by way of consolation: "In the previous chapter the circle diagram is obtained substantially as shown in the classical papers of Heyland, Behrend, et al." Another reference is found in the following paragraph in Thomaelen-Howe's Textbook. 1 "The historical development of the circle diagram is very interesting. Heyland published the diagram in the E. T. Z. on the llth Oct. 1894, and gave further developments on pp. 649 and 823 for the year 1895. In the E. T. Z., 1896, pp. 63, Behrend developed the diagram dans TElektrotechnische Zeitschrift, du 30 Janvier 1896, p. 63, ou se trouve aussi indiqu6e pour la premiere fois la representation du travail et du couple moteur. "Quant a la representation du glissement par une echelle lineaire, elle a e"t6 indiquee pour la premiere fois dans mon article de 1' Industrie Electrique, 25 f evrier 1896, ainsi qu'un precede de correction due a la resistance negligee. "M. HEYLAND a indique plus recemment une correction graphique fort elegante, mais qui parait peu rigoureuse, comme je le montrerai prochainement." !"A Textbook of Electrical Engineering," by DR. A. THOMAELEN, Translated by GEORGE W. O. HOWE. 3d edition, 1912 (Longmans) p. 386. INTRODUCTION 11 analytically, but made a small error in the determination of the rotor current. The convenient determination of the slip and losses was given by Heyland in the E. T. Z. for 1896, p. 138. (See also Heyland's "Eine Methode zur experimentellen Untersuchung an Induktions- motoren, ' published in Voit's "Sammlung," Vol. ii, 1900). Emde corrected Behrend's error in a letter to the E. T. Z., 1900, p. 781, which opened an interesting discussion. In the " Z. fur E.," Vienna, for 1899, Ossanna gave the diagram, corrected for stator loss. (See also an article by Ossanna in the E. T. Z,, 1900, p. 712, and also by Thomaelen in the E. T. Z., 1903, p. 972.) It is interesting to note, however, that Ossanna's circle was really included in Heyland's first publication." References to the circle diagram being the work of A. Blondel, B. A. Behrend, and A. Heyland are to be found in the papers by J. Bethenod, L. Edairage Electrique, Aug., 1904, and by M. Edou- ard Roth, of Belfort, France, in L'Eclairage Electrique, Apr. to June, 1909. The interesting small volume by Dr. K. Krug 1 on the circle diagram of the induction motor contains the following instructive historical reference : "The circle diagram for the elucidation of the operation of induction motors, which was published almost simultaneously by Heyland and Behrend, took into account only approximately the iron losses and the primary copper loss. The accurate consideration of these losses was first given by Ossanna and his results were developed later in differ- ent ways by other authors. "In most of these papers the methods employed consist in a reduction or adaptation of the Heyland circle diagram, so as to take account of the primary ohmic drop as well as of the iron losses in accordance with the facts. "Original proofs of the accurate circle diagram which may be reduced to the problem of the so-called general alternating current transformer, have been given among others by Lehmann, by means of vector analysis, by La Cour by means of inversion, and by Petersen with the aid of a principle of superposition. "The following shows a solution of the general alternating current circuit by means of complex algebra." Another point of view is presented by Arnold and La Cour. 2 1 DR. KARL KRUG, "Das Kreisdiagramm der Induktionsmotoren." Ber- lin, J. SPRINGER, 1909, p. 5. 2 E. ARNOLD and J. L. LA COUR, Die Induktionsmaschinen. Berlin, J. SPRINGER, 1909, p. 65. Also French text, Les Machines Asynchrones. Premiere Partie. Les Machines d'Induction. Paris: Librairie Ch. Dela- 12 INDUCTION MOTOR "A. Heyland showed first (E. T. Z., 1895) that the locus of the current vector is a circle for constant main flux, and he gave a proof for it in E. T. Z., 1896. Behrend also derived his relation from the transformer diagram, and Blondel has referred to some relations in this diagram. The diagram is sometimes called the Heyland diagram." This brief reference to the history of the development of the theory is surely somewhat misleading, as it has been shown here that Heyland supplied no proof of the circle relation until considerably after the publication of my paper on Jan. 30, 1896, and then his proof neglected the secondary leakage. Also, it would appear that the fundamental labors of A. Blondel in this direction have been somewhat summarily put aside without the recognition due them. In the treatise of Kittler and Petersen 1 we read on p. 486 : " Heyland developed this diagram bearing his name. Through their labors in giving final form and in clarifying the diagram, Emde and Behrend have achieved preeminent merit." It is Interesting to note a letter by Mr. A. Heyland, E. T. Z., p. 61, Jan. 21, 1904, in which, after submitting a lengthy apology for the use of a single leakage field, he proceeds to say: "In respect to the query why I introduced at one time the above simplifications into the diagram (meaning especially the single leakage field), 2 allow me to say that these simplifications (sic!) were thoroughly warranted at the time. The Circle Diagram in its more complex form found little recognition 9 years ago and remained almost unknown. (The italics are my own.) It grave, 1912, p. 64. " A. HEYLAND a signale le premier (E. T. Z., 1895) que le lieu de 1'extremite du vecteur du courant etait un cercle si le flux principal est maintenu constant; il en a donne une explication en 1896 dans la E. T. Z., BEHREND a egalement deduit cette the"orie (E. T. Z., 1896); il 1'a tiree du diagramme du transformateur; BLONDEL en a deduit a son tour quelques relations. On appelle souvent ce diagramme le diagramme D'HEYLAND." ^'Allgemeine Elektrotechnik." Edited by Dr. E. KITTLER. Vol. II, "Einfiihrung in die Wechselstromtechnik." By W. PETERSEN. Stutt- gart: F. ENKE, 1909. "EMDE und BEHREND haben sich durch ihre Arbeiten um die endgtiltige Formgebung und Klaerung des Diagram mes in hervorra- gender Weise verdient gemacht." 2 "A Graphical Treatment of the Induction Motor." By ALEXANDER HEYLAND. Translated by G. H. ROWE and R. E. HELLMUND. McGraw Pub. Co., New York, 1906. The entire paper seems affected by this assumption. INTRODUCTION 13 became known only after the publication of the simplified construction of the circle, which I published in a letter to the E. T. Z., 1895, p. 649, which represented an excerpt of a paper in The Electrician in Feb. 14, 1896." I think Mr. Heyland is right that his paper of Oct. 1894 "remained almost unknown." I think I agree with Mr. Heyland that it was necessary to give a simple demonstration of the theory and a simple geometrical proof in order to introduce the Circle Diagram to the engineer. This simple demonstration and geometrical proof were first given by me, and Mr. Heyland 's prior and later publications in no wise detract from this fact. When Mr. Heyland saw that I had succeeded in giving a treat- ment which was as accurate as his own paper in 1894 but a great deal simpler, and in which nothing was neglected which he there took into account, excepting the primary resistance, he endeavored to obtain an equally simple method of treatment and he tried to prove that the secondary leakage coefficient was non- existent 1 and that that which had been viewed as secondary leakage was merely part of the primary leakage and in time phase with the primary current. Then he introdued one of the most unhappy errors which, due to his authority, has not yet completely vanished. He also introduced the circular arcs for the representation of the copper losses in the primary and secondary windings, which were superseded a few years later by the straight lines as given in Fig. 5 reproduced from Fig. 56, p. 101, of the 1st edition of the present author's book, "The Induction Motor," 1901. We have reproduced as Fig. 4, Figs. 1, 2 and 3 of Mr. Heyland's paper, p. 561, E. T. Z., Oct. 11, 1894; as Figs. 6 and 7, Figs. 2 and 3 of my own paper, pp. 63 and 64, E. T. Z., Jan. 30, 1896; and as Figs. 8 and 9, Figs. 226 and 227, pp. 224 and 228 of Silvanus P. Thompson's "Polyphase Electric Cur- rents," 2d edition, 1900. We suggest to the reader a careful study of these figures and we may then leave it safely to his judgment whether I utilized in my early paper any of Mr. Hey- land's work, or whether Mr. Heyland and the late Prof. S. P. 1 F. EMDE, E. T. Z., p. 855, 1900, where we also read: "On this occasion I wish to refer to HEYLAND'S paper, No. 41, E. T. Z., 1894, which at any rate excels in accuracy his later papers, though it lacks the 'seductive' simplic- ity and it is therefore referred to only historically." EMDE has been one of HEYLAND'S strongest admirers, and surely one of the ablest. 14 INDUCTION MOTOR Thompson used my early paper in their own work which came out after the appearance of my paper. It is true that Prof. S. P. 90* Slip C, Generator Motor FIG. 5. Facsimile of Fig. 56, p. 101, of the First Edition of B. A. Behrend's book, "The Induction Motor," New York, McGraw Publishing Co., 1901. Thompson cited my paper in the bibliography in "Polyphase Electric Currents;" it is also true that he did me the honor of INTRODUCTION 15 FIG. 6. Facsimile of Fig. 2 of B. A. Behrend's paper "On the Theory of the Polyphase Motor," Jan. 30, 1896, E. T. Z. FIG. 7. Facsimile of Fig. 3 of B. A. Behrend's paper "On the Theory of the Polyphase Motor," Jan. 30, 1896, E. T. Z. 16 INDUCTION MOTOR naming 1 my little treatise " The Induction Motor" as one of the three books on the theory of alternating-current motors which he recommended to his readers, and I would fain refrain from showing how much my work has been used by these authors if it were not for the fact that during the last 25 years I have remained silent, trusting to the fairness of authors not to deprive a fellow author of the just credit due him. FIG. 8. Facsimile of Fig. 226, p. 224, of Silvanus P. Thompson's book, "Poly- phase Electric Currents," 2d Edition, 1900. A typical case in question is the reference on p. 413 in my friend Miles Walker's " Specification and Design of Dynamo- 1 SILVANUS P. THOMPSON, "Dynamo-Electric Machinery," 7th ed., Vol. i, p. 36, London, E. & F. N. Spon, Ltd. 1904. "The theory of alternate current motors of the asynchronous and of the synchronous types has of late received much attention from various writers. The reader is referred to the author's book "Polyphase Electric Currents" (2d ed., 1900) to STEINMETZ'S "Alternating Current Phenomena" (3d ed., 1900), or to BEHREND'S "The Induction Motor" (1901). INTRODUCTION 17 Electric Machinery," Longmans, Green & Co., 1915, which shows the circle diagram as given on p. 101 of "The Induction Motor," New York, 1901, but with the following comment: "It is, therefore, convenient to reproduce here a form which is found to be very convenient in workshop use, and to give results which check sufficiently with those obtained in practice." FIG. 9. Facsimile of Fig. 227, p. 228, of Silvanus P. Thompson's book, "Poly- phase Electric Currents," 2d Edition, 1900. Reference is made to a footnote which begins: "See KarapetofFs 'Experimental Electrical Engineering/ Vol. ii, p. 166; Cramp and Smith, 'Vector Diagrams;' Graphical Treatment of the Rotating Field, R. E. Hellmund, A. I. E. E. Proceedings, p. 927, 1918, etc., etc. . ." The present form of the circle diagram, as applied to the solu- tion of induction motor and transformer problems, and the methods of its demonstration and proof, are based upon BlondeFs diagram of the composition of fluxes and upon the proof of the circular locus as developed by the present author. Mr. Hey- land's contributions to the subject, however interesting and suggestive they have been, have not survived. 18 INDUCTION MOTOR THE DEVELOPMENT OF THE THEORY OF THE SINGLE-PHASE INDUCTION MOTOR The analytical theory of the single-phase induction motor owes much to the labors of Potier, Dr. Behn-Eschenburg, Goerges, Steinmetz, and McAllister. As the analytical theory has always been somewhat abstruse, an attempt was made by the author as early as 1896 to represent the locus of the primary current through graphical analysis, and it was found that the primary current in the polar diagram could be represented by vectors drawn from a pole to the circumference of a circle. This was proved, however, only for a limited case, viz. for a motor in which the secondary resistance was partially negligible. This analysis of the operation of the single-phase induction motor by means of a proof that the primary current locus is also a circle, was given by the author in the E. T. Z., March 25, 1897. The analysis was carried through by dissolving the single oscil- lating field into two equal and oppositely rotating fields. It was assumed that the rotor resistance of the second motor with reverse torque was negligible. With these assumptions a circle represents correctly the locus of the primary current. The same analysis was repeated in the first edition of the author's book on "The Induction Motor." Utilizing the able papers of H. Goerges in the E. T. Z., 1895 and 1903, on the single-phase induction motor, in which Goerges introduced the cross field, Prof. J. K. Sumec gave a comprehen- sive and elegant graphical solution which remains perfectly simple in spite of its accuracy. His first paper was published in the "Zeitschrift fur Elektrotechnik," Vienna, No. 36, 1903. The sub- ject is most admirably treated in a little pamphlet entitled "Der einphasige Induktionsmotor," by J. K. Sumec, Nov. 20, 1904, reprinted from the "Archiv der Mathematik und Physik," Leipzig, B. G. Teubner. Treating the theory of the single-phase induction motor by means of a resolution of the oscillating field into two equal oppositely rotating fields, Dr. A. Thomaelen, in E. T. Z., 1905, p. 1,111 et seq., arrives at the same result as that given by Sumec without neglecting the rotor resistance which was the new ele- ment in Sumec's work. Dr. Thomaelen's treatment, however, is rather complex and its value consists in proving that the two methods lead to the same result. INTRODUCTION 19 This has again been proved by Arnold and LaCour in "Les Machines d'Induction," Part I, p. 149, of the French edition, Paris, Ch. Delagrave. The authors have used the method of equivalent circuits which they have employed throughout their work. It may, therefore, be safely assumed that both methods of analysis give identical results. Reference must here be made to the seventh edition, 1918, of Dr. A. Thomaelen, "Kurzes Lehrbuch der Elektrotechnik," which has just come to our attention. Throughout the treat- ment of the theory of the induction motor, both poly-phase and single-phase, Dr. Thomaelen has used the author's leakage coef- ficients, assuming apparently that they are novel and expressing his satisfaction that they give results easily and clearly. Since the present author introduced these coefficients in his first mono- graph of 1896 and as they have been used since with full credit by Messrs. Kapp and Sumec, he likes to point out again that in this work he is using the reciprocals of the coefficients which he used in his early monographs and in the first edition of this book. Thus, the coefficients are the same as those of Hopkinson, as they were adopted in 1894 by our great master, Andre Blondel. CHAPTER II THE THEORY OF FLUXES AND STRAY FIELDS The problem of problems, in the solution of which the elec- trical engineer is deeply interested, and which underlies all others, is set before us in the form of the alternating-current transformer possessing considerable leakage and a relatively large magnetizing current. A choking coil of n turns or a transformer with an open secon- dary, takes from the primary mains just so much current as is necessary to produce a magnetic field F, which balances the pri- mary voltage ei. The induced voltage e, opposite in time-phase to the impressed voltage e\ is e = -n^-10- 8 volts (1) The magnetizing current neglecting for the moment hysteresis and eddy currents lags behind the primary-impressed voltage by a quarter of a time-phase. It leads by a quarter of a time- phase over the induced counter e.m.f. We say, " It is in quadra- ture with the impressed e.m.f." The product of this current into the impressed e.m.f., integrated over the time of one com- plete period, i.e., the work done by this current, is zero. Currents in quadrature with the e.m.f. have been called by M. Dobrowolsky " watt-less" currents. Any transformer, induction motor, or other alternating-cur- rent device of any sort or description, under load or under no load, has one and only one primary magnetic field resulting from the actions, and interactions, of its current-carrying coils. The magnitude of this resultant magnetic field is such that its varia- tion produces a counter e.m.f. in phase with the impressed e.m.f., and of such magnitude as will permit the flow of the primary current through the ohmic resistance of the primary coils. The fundamental importance of this statement must be em- phasized as it is applied throughout this book. A choking coil of sectional area A, of magnetic reluctance p, of ohmic resistance r, and number of terms n, placed in a circuit 20 THE THEORY OF FLUXES AND STRAY FIELDS 21 of frequency ~, with effective (square root of mean square) current i, carries a maximum flux F, of maximum induction B : F max = A B max (2) Hds p J*Hds is the line-integral of the magnetic force H I Hds = 0.47T/ (4) which, for any closed circuit, is equal to 0.4rr times the entire magnetizing ampere-conductors. If the integral taken around a closed circuit, viz. the "magneto-motive force," is zero, no flux can result from the currents around which the integral was taken. A neglect of this fundamental conception of the theory of electro- magnetism has led to false diagrams of stray-fields. Therefore, R _ . /_>. We assume the time variation of the flux to follow a simple sine law F = F max sin co* (6) 0, = 27T~ (7) Therefore from (1) e t = -rcJ-H)- 8 (1) = co.F mox cos co/ 10~ 8 = -2?r ~ F max cos co* . 10~ 8 (8) e max = e\/2 (9) /. e = -4A4~nF max lO" 8 (10) From (8) it is apparent that e, the induced e.m.f., lags 90 time degrees behind the inducing current. The impressed e.m.f. there- fore leads the current by 90 time degrees. (See Fig. 10.) All our polar diagrams rotate in the positive direction, which is counter-clockwise by international agreement. The ohmic drop requires the addition of ir in time-phase with the flux to the impressed voltage d, requiring EI as the final resultant impressed voltage. The placing of a secondary coil on the magnetic circuit makes the device a transformer. 22 INDUCTION MOTOR If we assume that both primary and secondary coils embrace the entire flux, there being no stray or leakage fluxes, and if we assume the number of turns to be the same in both circuits, then ma* FIG. 10. FIG. 11. 61 and 62 are the e.m.fs. impressed upon both the primary and secondary circuits, respectively. If the load on the secondary circuit is non-inductive, the current i 2 is in phase with e. The primary current must be such, in phase-direc- tion and magnitude, that the resultant m.m.f. of the primary and secondary ampere-turns produces the field F in magnitude and time- phase. Knowing e\, we find F', knowing p we find the primary ampereturns for magnetization, rep- resented by i^ in Fig. 11. Adding itfi to e\ gives El, the resultant impressed voltage. There is another, and a safer way, proposed by Prof. Andre Blondel in a famous paper __ entitled, "Quelques proprietes generates des quantities in the champs magnetiques tournants" (Eclairage 12 N Electri Q Ue > 10 > 17 > 24 Au S-> 1895 )> in which the magnetic fluxes are composed as follows: If acting alone i% would produce a flux 4> 2 equal to X 2 -5- p, where Xz represents the m.m.f. of 2*2, and p the reluctance of the mag- netic circuit in common to both primary and secondary circuits; if acting alone i\ would produce a flux i equal to Xi -=- p, where Xi represents the m.m.f. of i\, and p again the reluctance fa*. (Facing page 22) THE THEORY OF FLUXES AND STRAY FIELDS 23 of the magnetic circuit in common to both primary and secondary circuits. 3> 2 vectorially subtracted from 3>i must leave F in mag- nitude and direction (Fig. 12). The great advantage of this method becomes apparent in the treatment of the theory of the transformer with leakage and in the more complex problems of double squirrel-cage motors, con- catenation, etc. Its disadvantage lies in the danger of looking upon the fictitious 3>i and $ 2 as fluxes actually in existence and having physical entity. We shall use both methods wherever they represent closely the physical phenomena. It is well known in dynamo design, as first taught us by Dr. John Hopkinson, that the flux threading the primary does not reach the secondary without leakage or stray fields. If we as- sume with Prof. A. Blondel that the ratio of the flux of the pri- mary to that which reaches the secondary is vi, where v\ is greater than 1, and the ratio of the secondary flux to that which reaches the primary is t> 2 , where v 2 is also greater than 1, then (vi l)3>i and (v 2 1)$2 represent the stray fluxes or leakage fluxes, which are in time-phase with their respective m.m.fs. or currents. In my paper E. T. Z., Jan. 30, 1896, and in the first edition of this book, I used the reciprocals of BlondeFs v's. Though un- fortunately most authors have since followed my use of these coefficients, as Silvanus P. Thompson, Gisbert Kapp, Alexander Gray, J. K. Sumec, A. Thomaelen, and others, after very careful consideration, I have become convinced that it is better, in the interest of uniformity and clearness, to give up my coefficients, which were smaller than 1, and instead to adopt BlondePs, which are larger than 1, and this practice also conforms to the disper- sion coefficients of Dr. Hopkinson's which are also greater than 1. This matter is solely a convention and in no manner affects the accuracy or correctness of our arguments or of previous papers. There is much to be said for the retention of my old coefficients as they are logical in viewing the deviation of the ratio of trans- formation at no load from the ideal ratio as the measure of the leakage. In my early notation, the primary and secondary leak- age fields were: Behrend's old Notation (11) /-->)* 24 INDUCTION MOTOR In B lenders notation: /2 = (02 ~ Blondel' s Notation (12) The utmost care is essential to avoid confusion and I believe a service is rendered by the adoption of a uniform notation. The diagram of fluxes can now be drawn directly (Fig. 13 and 14). FIG. 13. The flux diagram of the induction motor or transformer, in- cluding leakage. FIG. 14. Electric circuits sim- ulating the leakage paths of the magnetic circuit of the induction motor. F 2 induces e z + fa = X z -r- p2 = (#2 ~- 1)^2 secondary leakage flux fi = Xi -T- pi = (vi l)3>i primary leakage flux FI resultant primary flux 3> 2 = X z -T- p fictitious secondary flux $1 = Xi -4- p fictitious primary flux Ei = ei -f- iiri primary impressed voltage It is very desirable to keep in mind a picture of the corre- sponding electric currents with their e.m.fs. and distribution of resistances. In Fig. 14, Xi and X 2 represent the primary and THE THEORY OF FLUXES AND STRAY FIELDS 25 secondary m.m.fs., p, pi, P2, the reluctances of the common and leakage paths. The fluxes are entered and the diagram shows clearly how the leakage fluxes become cumulative by vectorial addition. Figures 13 and 14 should always be kept together before the mind, with the underlying assumption that the reluctance of the iron is assumed as negligible, in fact zero. FIG. 15. The polar diagram for constant current. Mr. G. Kapp originated a method before the advent of the Blondel flux diagram, which is still adhered to by Dr. Steinmetz, in which the e.m.fs. induced by the leakage fields /i and /2 are represented lagging by a quarter phase the primary and secon- dary currents. We refer to the author's Fig. 6 from his original paper of Jan. 30, 1896, in which both methods are shown in the diagram, and from which it is apparent that the flux method of Blondel is both more nearly in keeping with the physical facts and a great deal simpler in its geometrical interpretation. The results obtained by both methods are, of course, identical, though 26 INDUCTION MOTOR there seems now little warrant for retaining the older method of Mr. Kapp's as done throughout in the works of Dr. Steinmetz and in the recent textbook of Prof. R. R. Lawrence, " Principles of Alternating Current Machinery," McGraw Hill Book Co., 1916. At least the flux method should be considered beside the older conventional one. We are still concerned with the transformer. We wish to know how the magnitude and phase of the primary e.m.f. vary with constant primary current and varied secondary resistance. As Oe is proportional to the primary current, Fig. 15, we shall assume it to remain constant, neglecting for the present the primary resistance which is easy to take into account. The angle Oaw is a right angle, hence, describe a semi-circle over Oe as diameter, then by varying the secondary resistance we vary F z = Oa, which is in quadrature with and proportional to e z , the secon- dary voltage. Remember that to obtain the secondary terminal voltage we must deduct i 2 r 2 the ohmic drop in the transformer windings from e 2 . As Od = Oc -r- v\ y it follows that Od is a measure of the primary voltage. The point d divides ae in the same ratio for all configurations of the diagram, as is easily shown. ab = (v 2 bd = be ed /. ab + bd = (v 2 - -} \ v\i ad = ab + bd = (viV 2 1) - v\ .'. ad -r- ed = ViV 2 1 (13) a = ViV 2 I (14) We shall call ViV 2 1 by the Greek letter 0, which we shall see later is the most characteristic constant of a transformer or induction motor and it is usually called the Leakage Factor. [In my former notation, my old coefficients being the reciprocals of the Blondel coefficients here used, the Leakage Factor was equal to a = --- 1.1 Describing now the semi-circle edk so that Ok -T- ke = i 1) and (z; 2 1) to be large, say, vi = 1.04 and v z = 1.06, then the leakage factor a becomes (Chap. II, Eq. (14)), - 1 (14) <7 = a- = 0.1 = .834 FIG. 18. The circle diagram for constant voltage. Assume, FI = 12, and d = 120 volts D = - 1 = 120 <7 (18) With these values we construct the polar diagram, Fig. 18, whose values are shown in Cartesian coordinates in Fig. 19. The maximum power factor obtainable is cos \f/ Q = OiPi OiO _ 2(7 (19) THE GENERAL ALTERNATING-CURRENT TRANSFORMER 31 This simple relation gives at a glance the highest possible power factor for a given amount of leakage for non-inductive load. For our numerical case, we have 1 cos ^ 1.2 = 0.834 K.W.Inp,ir- 9 87 FIG. 19. Characteristic curves of transformer in Cartesian coordinates. THE AUTHOR'S METHOD OF ACCOUNTING FOR PRIMARY RESISTANCE The primary resistance adds to the impressed e.m.f. e\ the component i\r\ in phase with i\. Before we proceed further it may be advisable to call attention to the fact that, in going from the flux diagram to the current diagram, the primary current is proportional to Oe, the secondary current to be, and the resultant magnetization to Oe. In Fig. 16, therefore, if Oe is drawn to but , while Od represents the represent ii, ed represents, not open circuit current (neglecting losses) which produces the total primary flux Oc = F\. 32 INDUCTION MOTOR Instead of adding zVi to e\, and obtaining EI, and then turning the current vector Oe = i\ in the positive or counter-clockwise direction so that EI coincides again in phase with e i} apply the following simple geometrical device (Fig. 20). Make triangle Oeg similar to triangle Ohk, then eg\0e ::hk : Oh i\ . ' eg (20) FIG. 20. The author's method of taking into account the primary resistance. The new current iY must be in the time-phase Og and its mag- nitude remains equal to On = Oe if the impressed e.m.f . is raised from ei to EI. If, however, we assume the impressed e.m.f. EI to be reduced to Op, the value of e t , then the current Oe = ii is to be reduced in the ratio Og -5- Oh = e\ -4- EI, or Or -4- Oe. The magnitude of the real primary current is therefore represented by Or and its phase by Og, its magnitude and phase, therefore, by Os. The method here described is rapid and easy. It is carried THE GENERAL ALTERNATING-CURRENT TRANSFORMER 33 out in the figure. Its advantage lies in its convenient and ready application to all sorts of alternating-current problems. ANOTHER METHOD OF ACCOUNTING FOR PRIMARY RESISTANCE A simple and elegant geometrical method applicable to the circle diagram has been given by Prof. J. Sumec, 1 of the Czecho-Slovac University of Brtinn, which we shall now proceed to explain. AP . AR = OA 2 AP = As I FIG. 21. As . Ar = Ab 2 = z 2 p is a point on the original circle P is a point on the new circle corresponding to p. Instead of adding iiri to d, add i\ to , so that Oe represents 1 J. SUMEC, E. T. Z., Feb. 3, 1910. The method is due to MESSRS. STEHR AND PICHELMEYER. 3 34 INDUCTION MOTOR both ii and the resistance component to be added to- Then Fig. 21, A OP A ~ A OPc ~ ArOA AP :AO ::AO : Ar .'.AP-Ar = ffil 1 (21) As Ar = AB 2 = z 2 from a well-known property of the circle. I o A (22) Zo 2 Hence, P lies on the radius vector from A to c intersecting the circle in s, and P lies again on a circle as any arbitrary radius vector AP is always proportional to the radius vector As. Now, we have (23) Now, from Fig. 22, we have Ag:AO::AO:Ak .'. Ag-Ak = AO 2 But, from the properties of the circle, Ad-Ak= z Q 2 . Ag AO 2 *' Ad z 2 However, Agr : Ad :: AC : AC (24) AC.-AC. (25) w From inspection of Fig. 22, calling D the diameter of circle with C as center, and D the diameter of circle with C as center, Do :D ::AC : AC (26) THE GENERAL ALTERNATING-CURRENT TRANSFORMER 35 ce g a -e 36 INDUCTION MOTOR To obtain the coordinates of the center of the circle of diameter DO, which is the new locus of the primary current with full consideration of the primary resistance of the transformer, we proceed as follows : Cob :AO ::(AC - AC Q ) : AC C 6 : g :: AC :AC (27) The abscissa Ob = OC - bC is found, bC :OC ::CC Q :AC bC :OC ::(AC - AC,} : AC .: bC = OC Ob = = It is interesting to note that point g in the two diagrams of Fig. 22 remains a fixed point through which all vectors of the second- ary currents may be drawn from the corresponding points of the vector locus of the primary current with C as center. The proof of this must be left to the reader. This method has been used to consider a most instructive case, viz., that of a transformer with leakage and resistance in primary and secondary operating upon a circuit whose impressed voltage varies proportional to the frequency. At a frequency 60 cycles per second, the voltage is six times that at 10 cycles, etc. If there were no primary resistance, the locus of the primary cur- rent would be the same circle through the entire range of change of frequency and voltage, the resultant flux remaining the same. If there is primary resistance, however, its effect will be greater with reduced primary impressed voltage for the same current. THE GENERAL ALTERNATING-CURRENT TRANSFORMER 37 The characteristic circles for this case are drawn in Fig. 23 for the conditions given in the following table: FIG. 23. Variable frequency transformer. Primary current loci. Impressed voltage varies with frequency. VARIABLE FREQUENCY TRANSFORMER 61 TI er zo 2 Do r 50 2,500 3,600 69.50 15.20 41.7 40 1,600 2,700 59.30 16.30 35.6 30 900 2,000 45.00 16.50 27.0 20 400 1,500 26.65 14.70 16.0 10 100 1,200 8.35 9.17 5.0 1,100 0.00 0.00 0.0 io = 10 D = 100 a = .1 f! = 1 38 INDUCTION MOTOR These results will be found of great help in understanding Chap. XII on " Concatenation of Induction Motors." It is also of importance where induction motors are started with their generators from standstill by gradually raising the speed with constant field excitation. We shall now investigate whether it is allowable to neglect the primary resistance in practice so far as it extends to its influence upon the circle locus of the primary current. We shall calculate five values for the percentage of voltage consumed by resistance and tabulate the errors in the coordinates of the centers of the circles. We shall assume a leakage coeffi- cient a = 0.06, corresponding to a maximum obtainable power factor of 0.893. We will assume the transformer or motor to operate at this point of maximum power factor, and therefore the normal current will be 50 amp. for i = 12 amp. TABLE OF ERRORS INTRODUCED BY NEGLECTING PRIMARY RESISTANCE iiTi ei per cent TI ei ri & z * >o f $ 1 2 0.2 2,500 625 10 4 625. 255- 10 4 200.0 0.00 112.0 3 0.3 1,670 280 10 4 280. 255 -10 4 200.0 0.00 112.0 5 0.5 1,000 100 -10 4 100. 255 -10 4 199.5 3.00 111.6 7 0.7 715 51 10 4 51. 250-10* 199.0 3.58 111.3 10 1.0 500 25 10 4 25.255-10 4 198.0 4.95 109.0 ff = 0.06 io = 12 amp. D = 200 amp. ii = 50 amp. Vi =0.2 ohm e v = 500 volts From this table it appears that up to 5 per cent the effect of resistance upon the circle is negligible without a question of a doubt, while from 5 per cent to 10 per cent it seems negligible for all practical purposes, errors from other sources being vastly greater than from the neglect of primary resistance, so far as the magnitude and location of the circle are concerned. Needless to say, the losses due to primary resistance must not be neglected. How this is done is shown below. ACCOUNTING FOR PRIMARY RESISTANCE BY THE METHOD OF RECIPROCAL VECTORS In Chap. VIII of the first edition of this book the circle dia- gram of the alternating-current transformer was developed, in- THE GENERAL ALTERNATING-CURRENT TRANSFORMER 39 eluding the effect of primary resistance, using the well-known method of reciprocal vectors first applied to this problem by Prof. F. Bedell and A. C. Crehore. 1 Though principally of academic interest, we repeat here the demonstration as a useful exer- cise in the geometric interpretation of alternating-current phenomena. Oa.Ob Od.Oe = z 2 Oa.Ob = Od. Oe Oa . Od : : Oe : Ob FIG. 24. Reciprocal vectors. Figure 24 from similar triangles A Odb and A Oae we have Oa :0e ::0d :0b .'. Oa Ob = OdOe = constant (28) Also Og Oh = constant (Oc - p)(0c + p) = Oc 2 - p 2 = *o 2 .'. OaOb = OdOe = z 2 (29) where z is the length of the tangent to the circle from to c. 1 (1) F. BEDELL and A. C. CREHORE, ''Resonance in Transformer Cir- cuits," Physical Review, Vol. ii, No. 12, May- June, 1895. (2) FREDERICK BEDELL, "The Principles of the Transformer," p. 223 el. seq. New York, The Macmillan Company, 1896. 40 INDUCTION MOTOR Now, imagine a circle (Fig. 25) about C as center representing the locus of the primary impressed e.m.f. for a constant-current transformer whose current vector coincides with the ordinate OA as proved in Chap. II. If, instead of a constant current, we keep constant the impressed e.m.f., then the current of the trans- former will maintain the same phase relation to its impressed e.m.f., but its magnitude will be increased in the ratio of -j-, or -- and its phase will remain ^, so that the variable voltages for Oc \0g :: Oc': Od' Og'- Og'=0d'* Od = 500 FIG. 25. Transformation from constant current to constant voltage by means of reciprocal vectors. the constant-current transformer will be represented by vectors on the left of OA, while the variable currents for the constant potential transformer will be represented by vectors on the right of OA. Now, A Oa Ob /! = Oa' /! = Ob' ' ' Oc> ' = Og' Od' /i Od l THE GENERAL ALTERNATING-CURRENT TRANSFORMER 41 where I\ is the primary current of the constant-current trans- former. Add the last two equations, - Od) Od-Og = Od' + Og' = 20C' (30) Od-Og This equation (30) can also be written by substituting the value 1 Od' OC-Od' = OC'Og (31) Either equation can be used to calculate the center of the derived circle. C'P' OC' k* -CQ = oc = 7* = const - FIG. 26. Reciprocal vectors. We will now prove, for the sake of completeness, that the inverse of a circle is another circle. Let P, Fig. 26, be any point on the circle, P' its inverse. Let OP cut the circle again in Q. Let C be the center of the circle. Then OP OP' = k 2 , where k 2 is ei/i in the preceding argument. Now OP OQ = z 2 , where z is tangent to circle C from 0. OP' _ k 2 '' OQ ~ ^ OC' k 2 Take C' on OC such that QJT = ~ 2 > * nen 'and CP f is parallel to CQ. Therefore, C'P' OC' CQ fc 2 = -= = constant 42 INDUCTION MOTOR Therefore, P f describes a circle round C" and we have proved that the inverse of a circle is another circle. This proof, it will be noticed, is similar to that given previously. THE LOSSES AND THEIR REPRESENTATION BY STRAIGHT LINES The first and second methods showing the transformation of the resistance-less circle locus into a circle locus taking ac- count of primary resistance, yield a better insight into the transformation than does the third method. In order to render an account of the loss due to primary resistance we shall first consider the circle diagram with the FIG. 27. Accounting for secondary copper loss by means of the loss line. center of the circle on the abscissa axis, i.e., we shall assume, as we have demonstrated in Chap. Ill, that the circle locus is changed only immaterially by the presence of primary resistance of such an order as is encountered in practical apparatus. The primary copper loss is equal to ifti, the secondary copper loss is equal to e^-r^. The vector i\ = c, i Fig. 27. Obviously, a :x : :D :a b, and = o, or or or a 2 = x-D *r = x(Dr) (32) THE GENERAL ALTERNATING-CURRENT TRANSFORMER 43 In other weirds, - , which is the watt-component corresponding to the loss in the secondary copper, may be represented by the ordinates of the straight line dg. This representation is evidently applicable only so long as a = > v\ or a proportional to , which relation can be proved to hold even for the case in which primary resistance ri is taken account of. FIG. 28. Accounting for primary copper loss by means of the loss line. The primary copper loss is found as follows, Fig. 28, C 2 = a 2 + 6 2 + 2ab cosd c 2 = xD + 6 2 -f- 2bx .'. c 2 = x(D + 26) -f 6 2 Also t'iVi = z(D + 2i )ri H Also (33) (34) (35) In words, the primary copper loss is proportional to x plus the constant *o 2 ri, and it may therefore be represented by a straight line whose ordinates, measured from the z-axis are equal to the watt-component of the primary copper loss. 44 INDUCTION MOTOR For ii = i we obtain x(D + 2*0)7-1 = .*.z =0 For ii = we obtain These simple methods for determining the copper losses were originally given by the author 20 years ago in Appendix III of the first edition of this book. If the position of the circle is such that the center does not lie on the abscissa, then the watt-components of the losses are no longer to be measured parallel to e\ but as proved above, normal to the diameter of the circle. THE IRON LOSSES DUE TO HYSTERESIS AND EDDY CURRENTS The iron losses, or the core loss, of an induction motor are due to hysteresis and eddy currents. There has been discussion regarding the most accurate manner in which to take them into account in the diagram. _ D FIG. 29. Equivalent circuits show- FIG. 30. Equivalent circuits show- ing position of core loss circuit. ing position of core loss circuit. (Al- ternate.,) In view of the fact that the hysteresis loss is likely to be pro- portional to the resultant primary field, this loss may well be assumed constant. The loss due to the eddy currents generated by this field may also be assumed constant at all loads. Losses due to stray fields are apt to be very considerable and these, therefore, would increase with increasing current load. As it is not practicable to take all these factors into account, I proposed first in 1896, and I was seconded by Prof. Blondel, to look upon these losses as though produced in a resistance shunted across the primary potential, Fig. 29. Other writers, like Steinmetz, Arnold, LaCour, McAllister, and Bragstad have, however, used a second method of an equivalent circuit, as shown in Fig. 30, in which at standstill the core loss is a minimum, gradually in- creasing with decreasing load or increasing speed. This theory THE GENERAL ALTERNATING-CURRENT TRANSFORMER 45 does not appear very reasonable, as the hysteresis loss is more likely to be dependent upon the total field rather than the com- mon field. It is true that it is much more difficult to take into account the core loss if it depended upon the potential at the terminals of the exciting shunt as it is indicated in the equivalent circuit. But the fact that it is more difficult to take it into account with this assumption, though this assumption is farther removed from the actual conditions, should be no reason why it should be considered necessary to do it. " Error which is not pleasant, is surely the worst form of wrong." FIG. 31. A third method which I have followed from time to time since 1900 assumes that the circle diagram is derived without taking into account the core losses, and that this loss is later accounted for by deducting it from the secondary output. This procedure is likely to be about as accurate as any of the previous methods and it has the merit of greater simplicity. I believe it was first suggested by my friend Heyland. Load losses should doubtless be accounted for by an increase in the primary and secondary resistances. If this is done, then it appears altogether illogical to take into account the core loss by making it depend upon the common field, which goes through 46 INDUCTION MOTOR the air-gap, and this strange conception would never have arisen but for the equivalent circuit methods which make it appear as though the voltage drop in the primary leakage reactance oc- curred outside the machine while, in reality, the flux which produces the reactance voltage , is vectorially added to the com- mon air-gap flux. If we assume the core loss which is made up of hysteresis and eddy-current losses to be constant at all loads, which is a very problematical assumption, and justifiable only on account of our profound ignorance of the causes of core loss and the magnitude of these losses, then we may assume its effect to be equivalent to a constant watt loss, whose watt-component is constant and may be represented in our diagram by a line parallel to the dia- meter of the circle. By this amount the available secondary power will be diminished. We will recur to this matter in subsequent chapters, while Fig. 31 shows these losses graphically. B. THE TRANSFORMER WITH INDUCTIVE LOAD Assume the secondary of a transformer to be closed by a circuit with resistance and inductance. Assume inductance and resistance to vary in such a manner that the power of the secondary external circuit remains constant. Then, Fig. 32, we have: F 2 the secondary resultant magnetic field which induces iz the secondary load current lagging in time by the angle ty. /2 = (vz 1)$2 = ab, secondary leakage field in phase with the secondary load current. $2 = be the fictitious secondary flux proportional to the sec- ondary m.m.f . = ae i = Oe the fictitious primary flux proportional to the primary m.m.f. /i = (vi l)$i = be the primary leakage flux. dc and Amed are similar triangles. Angle Oae is equal to a right angle plus i/% it is therefore a constant angle for a variation THE GENERAL ALTERNATING-CURRENT TRANSFORMER 47 of 1%. Hence point a moves on the arc of a circle, and point e does equally so. Od :ad : : md : de :D = Od S3 _ "' 1 1N 2 ~ - 1) -D t>l (37) (38) (39) (T-.44 Induction Motor Range with ^- ^ uctance in Sec-Circul FIG. 32. Inductance in the secondary of constant potential transformer (motor range) and capacity in secondary for induction generator range. The effect of the inductance in the secondary circuit consists in greatly reducing the maximum watt-component of the primary current, and in reducing the power factor. The capacity of a transformer is thus greatly reduced by an inductive load. C. THE TRANSFORMER WITH CAPACITY LOAD Assume the secondary of a transformer to be closed by a cir- cuit with such condenser capacity that both resistance and ca- pacity may be varied so as to keep the power factor cos ^ 2 of the external circuit constant. Then Fig. 33 shows that triangle 48 INDUCTION MOTOR Oad is similar to triangle med and eb = $ 2 , in phase with the secondary current i' 2 . Od:ad::D:de (39) (40) (41) (42) Induction Motor Range with Capacity in Secondary Circuit Induction Generator Bange Inductance in Secondary Circuit FIG. 33. Capacity in the secondary of constant potential transformer (motor range) and inductance in secondary for induction generator range. The effect of variable capacity in the secondary of a constant- potential transformer consists therefore in greatly raising its receptive capacity and in increasing the power factor. How this effect can be obtained by means of rotating apparatus is shown in Chap. XV. The effect of such apparatus upon the characteristics of the induction generator is discussed in Chap. IX. CHAPTER IV THE MCALLISTER TRANSFORMATIONS It is well known that geometrical figures can be " transformed" by means of a complex function used as an operator and that the "transformation" is frequently a solution of a problem otherwise impossible of solution. There are certain partial differential equations in physics, especially in hydrodynamics and in elec- tricity, to which these transformations have been applied success- fully. In fact, the entire fascinating subject of "conformal representation" of functions including the problem of "Merca- tor's Projection" in geography and solutions for the electrostatic capacity of conductors of different shape, is intimately connected with this subject In connection with vector diagrams as used in this book, the transformation by " reciprocal vectors" occurs most often and it has therefore been discussed in the previous chapter. It consists in simultaneously shrinking and turning a vector, and it transforms invariably one circle into another. As a general rule " point for point" reductions have been made both in the graphical treatment and in the analytical treat- ment by means of complex algebra. In a few cases, the graphi- cal treatment has had the advantage as the circle locus property permits an easy representation of the entire set of complex alge- braic equations. Where, however, a " point for point" method has to be resorted to, the advantage of the graphical method is less apparent. It has been shown in the previous chapter that the addition of resistance in series with the induction motor, the primary resistance of which has been neglected, leads again to a circle for the locus of the primary current. The center of this circle is raised above the abscissa and its diameter is smaller than that of the circle representing the performance of the motor without primary resistance. However, the process had been limited to a special case and no generalization of it had been developed. Dr. A. S. McAllister has succeeded recently in applying the same process of reasoning through which we have taken the reader 4 49 50 INDUCTION MOTOR in the development of the general circle diagram, to the general case which comprises resistance in series, reactance in series, or resistance and reactance in series with the motor. He has dis- covered a method, simple and direct, of determining the " center of inversion " from which the new circle can be located by drawing a few lines to the image of the original circle. It may be re- marked in passing that it is curious that so simple a solution has taken 25 years to be brought to light. In the accompanying figures there are treated the following cases : A and QO Mirror Plane FIG. 34. The McAllister transformations. Resistance in series with the motor, without core loss. A. Resistance in series with the motor without core loss (Fig. 34). B. Reactance in series with the motor without core loss (Fig. 35) . C. Impedance in series with the motor without core loss Figs. 36 and 37). D. Resistance in series with the motor with core loss (Fig. 38). E. Reactance in series with the motor with core loss (Fig. 39). F. Impedance in series with the motor with core loss (Fig. 40). It is interesting to observe that, if we assume the core loss to depend upon the air-gap field F only, the cases D, E, and .F show that the resultant primary locus remains a circle. If we assume the core loss to depend upon the total primary flux FI, which THE MCALLISTER TRANSFORMATIONS 51 seems to the writer the more rational assumption, then the re- sultant primary locus is also a circle. These circles, however, differ in magnitude and location. The proof of the McAllister method is very simple. It is suggested in all the diagrams. The start is made with the origi- nal circle whether this is the circle of the locus of the primary cur- rent as used for the performance of the induction motor in this book, or whether it is the locus of the primary current if the FIG. 35. The McAllister transformations. Reactance in series with the motor, without core loss. voltage on the magnetizing circuit in the " equivalent" diagram is kept constant. By reflecting this circle for the cases A and D below the ab- scissa; for B and E to the left of the ordinate; and for cases C and F below a mirror plane which forms with the abscissa an angle whose tangent is equal to X/R, similar triangles are formed by drawing rays from the " center of inversion" to the reflected point. In these rays lie the transformed points cutting the ray in the inverse proportion. 52 INDUCTION MOTOR The procedure then is the same in all six cases. Referring now specifically to Fig. 34 illustrating case "A," the circle with Cb r as radius represents the locus of the primary current if the potential on the ''magnetizing circuit" were kept constant. As we make OA equal to any radius vector meas- uring the primary current measures also the ohmic drop in the primary. Thus, as we have seen in Figs. 22 and 26, Chap. Ill, the point A or 00 becomes the center of inversion from which, 00 FIG. 36. The McAllister transformations. Impedance in series with the motor, without core loss. by means of similar triangles as used before, we obtain the new locus for the primary currents with center C in a manner indi- cated geometrically in the figure. The vectors drawn from to the circle whose center is Co represent, in the scale adopted, both the primary current and the primary ohmic drop. In the next figure (Fig. 35), which represents the case "B" in which reactance is placed in series with the motor, the process is shown in greater detail. The circle with C as center repre- sents in Op the primary current if the potential on the magnet- THE MCALLISTER TRANSFORMATIONS 53 FIG. 37. The McAllister transformations. Impedance in series with the motor, AC = Op. 54 INDUCTION MOTOR izing circuit were kept constant. As OA represents the primary impressed voltage divided by the additional reactance placed in series with the motor, Ac = Op = i\-Xi represents the react- ance drop in the circuit connected in series with the motor. Now draw the circle with C" as center which is theimage circle of C. Note that AOpOO = AAcO which triangles are simi- lar to ACO - 00. From this follows that or Op : - 00 : : Ob : 00 - b 00-b-Op = ~-i A i A and 00 FIG. 38. The McAllister transformations. Resistance in series with the motor, with core loss. This proves our proposition as it shows that the point b lies in- variably upon the ray drawn from the " center of inversion 00 to the "reflected" point p or b' which is the image of the original point p in the mirror plane OA. Point b divides 00 p in such a ratio that the product of the vectors drawn from 00 is a constant. Hence the new locus of ii is a circle with Co as center C Q b being perpendicular to 00 p. The reader would THE MCALLISTER TRANSFORMATIONS 55 FIG. 39. The McAllister transformations. Reactance in series with the motor, with core loss. FIG. 40. The McAllister transformations. Impedence in series with the motor with core loss. 56 INDUCTION MOTOR do well to draw for himself a few points and to transform them into their new positions in the vector diagram. The remaining figures referring to the six cases enumerated are sufficiently clear to require no further explanations as to their mode of derivation, provided the reader will take the pains to draw a few points in order to comprehend the principles on which this method is based. This has been done in detail in Fig. 37 illustrating with Fig. 36, the case C. Similar results to those arrived at above by the McAllister method have been obtained by successive applications of the method of reciprocal vectors, discussed in Chapter III-A, as shown by Messrs. LaCour and Bragstad, who have transformed the admittance circle diagram into an impedance circle diagram to which they have added the primary impedance, re-transform- ing the new impedance diagram back into the final admittance diagram. This method is described at length in the great work of these authors frequently referred to in this book. CHAPTER V THE ROTATING FIELD AND THE INDUCTION MOTOR A. THE AMPERE TURNS AND THE FIELD BELT In the induction motor, as invented by Mr. Nikola Tesla, two, three, or more windings lodged in slots, usually located in the stationary part or stator, are fed with alternating currents of the same frequency and voltage but of different time-phase. A FIG. 41. Distributed three-phase winding. The belt of ampere-turns and flux. belt of these windings produces a rotating field. In a short-cir- cuited rotor winding currents are induced whose interaction with the field results in the production of torque. We shall briefly examine the manner in which such a field is produced. 57 58 INDUCTION MOTOR Let I, II, and 777 be the three phases of the stator. We shall assume that the reluctance of the iron is negligible and that the induction is proportional to the line integral of the m.m.fs., and inversely proportional to the length of the gap (Fig. 41). We shall assume the current to vary according to a simple sine law. Then the intensity of the currents in 7 and 77 is each one- half that of 777. The m.m.fs. of each phase are represented by the ordinates of the curves 7, 77, and 777 respectively. Each ordinate measures the m.m.f. produced in that point of the cir- cumference where it is drawn. The adding together of the ordinates of the three curves yields the heavy line curve which is the sum of the m.m.fs. at the particular moment of time over the circumference of the air-gap. If the magnetic reluctance is the same at every point of the circumference, and the reluctance of the iron is negligible, then the magnetic induction B, produced by the m.m.f. belt shown in the heavy line in the Fig. 41, is proportional to this m.m.f. and, therefore, also represented by the ordinates of the heavy line curve. We call the total flux F, and we assume that the time variation of this flux follows a simple sine law. We shall now calculate the e.m.f. induced by this flux belt in each of the three phases. We shall assume arbitrarily that the distribution of conductors per phase is uniform, in other words, that there is an infinite number of slots. B. THE E.M.FS. INDUCED IN THE WINDINGS If all the conductors of each phase were concentrated in one slot, then the e.m.f. induced remembering that two conductors equal one complete turn would be according to Eq. (10) equal to 2.22 ^ -z-F-W~ 8 volts, where z is the total number of effective conductors per phase equal to 2n. On account of the distribu- tion of the windings over one-third of the pole-pitch, only the parts of the flux not covered with hatchings can induce an e.m.f. according to this formula, while the hatched parts of the field will have a considerably smaller effect. Let the width of the coil be 26, and n conductors in the coil spread over 26. Per unit Yl length there are, therefore, ^ conductors, hence the element dx contains dx- r conductors. The number of lines of induction THE ROTATING FIELD AND THE INDUCTION MOTOR 59 threading all the conductors in the element dx is equal to F x represented by the hatched area. Hence, de = 2.22- ^dx-F x 'lG- 8 volts P D ^ D " **x = &'n &x'-^ = 2 ! 2 2 26 = 2 - 22 jp^-^i f^ 110 - 8 e = 2.22-~-^ 26 I J 26 Jo 26 e = 2.22-~- With F = e = 2.22 B-b 2 e = 2.22 ~-^F) 10-s (43) In words, the e.m.f. induced by the field F upon a coil of the width 6 is two-thirds as great as the e.m.f. which would be induced by the same field upon a coil whose conductors are not distributed but lodged in one slot. Such a coil would not produce a triangu- lar field but a rectangular field. Therefore, the inductance of the flat coil, i.e., the number of lines or tubes of force per unit current, is one-third as large as the inductance of a coil lodged in one slot. The e.m.f. generated by the field belt in Fig. 41 can now readily be calculated. The flux of the white area is 4 The e.m.f. induced by this flux is equal to e a = 2.22-~-z- (^'^'t'b- The hatched areas represent a flux equal to 1 B The e.m.f. induced by this flux is e b = 2.22-~.z--* 60 INDUCTION MOTOR Hence e = e a + e b = 2.22 --- z(jjj.- t-b- B\lQr* wr% The total flux is /. e = .-~-z-- ZL e ?= 2. 12- 2-F-10- 8 (44) The ampere-turns in each phase which are needed to produce the induction B in the air-gap are determined by the considera- tion, which follows immediately from Fig. 41, that 2(.4-wt M \/2) = # 2A where A is the single distance, or length, between rotor and stator separated by the air-gap, n the number of conductors per pole per phase, and i M the magnetizing current. The reluctance of the iron has been neglected. Hence, B - 1 . 6A If the reluctance of the iron is not negligible, then the magnetic induction B has to be determined point for point, which can be done with the aid of a magnetizing curve. It is of importance to note that the maximum induction does not extend over a very large part of the pole-pitch, hence very high induction in the teeth may be resorted to without materially raising the magnetiz- ing current, although increasing the losses which are dependent upon B m ax in the teeth. There are numerous modes of distribution of the conductors per pole per phase. We have considered above that each phase covers one-third of the pole-pitch in a three-phase motor. The conductors may, however, spread over two-thirds of the pole- pitch. Figure 42 shows the m.m.f . belt at the time when phase III is a maximum. In order to obtain quickly an expression of the e.m.f. which such a field induces, we avail ourselves of the simple and direct method given by the author in Appendix II of the first edition of this book. Consider a winding ag, Fig. 43, spanning an arc of 180 electrical degrees, i.e., extending over the pole-pitch. In each small ele- THE ROTATING FIELD AND THE INDUCTION MOTOR 61 ment ab, or 6c, there is induced an e.m.f. de, represented graphi- cally by the small vector AB or BC, which we arbitrarily FIG. 42. Field belt of three-phase motor. Each phase spread over two-thirds of pole pitch. -A Z Effective Conductors FIG. 43. Winding covering pole pitch. represent at right angles to the element. Then ABCD G is, so to speak, the hodograph of the induced voltages, whose vector 62 INDUCTION MOTOR sum is AG. If x is the total number of conductors in the winding ag, then there are, as a result of distribution, 2 - z effective conductors only. Hence, e = \/2 ' ~ ' z F - 10~ 8 volts (46) If the winding is distributed over two-thirds the pole-pitch, or over 120 electrical degrees, Fig. 44, then the number of effective conductors is (Vz - H*) z which is equal to 0.825 z e = 1.836 ~z-P' 10~ 8 volts (47) FIG. 44. Winding covering two-thirds of pole pitch. In a quarter-phase system one-half the pole-pitch is covered by the coil, Fig. 45, therefore there are (\/2 -5- 2) * effective conductors per phase .-. e = 2~-z -F- 10- 8 volts (48) In a three-phase motor, the coils usually cover one-third of THE ROTATING FIELD AND THE INDUCTION MOTOR 63 Z Effective Conductors FIG. 45. Winding covering one-half of pole pitch. ffectfve Conductors FIG. 46. Winding covering one-third of pole pitch. 64 INDUCTION MOTOR the pole-pitch, or 60 electrical degrees (Fig. 46). Therefore, there are (1 -r- -^\ z effective conductors per pole per phase. .'. e = 2.12 ' ~-z-F 10- 8 volts (49) Ihese methods are correct whatever the shape of the field belt, as long as F equals the total flux produced. C. THE ELEMENTARY THEORY OF THE INDUCTION MOTOR The elementary phenomena of the functioning of an induction motor has been stated with admirable clearness by Prof. Gisbert Kapp, to whom we owe so much in the interpretation of the theory of electrical apparatus. As his account is also of historical interest it is here reprinted in full, as was done in Appendix I of the first edition of this book. Excerpt from Gisbert Kapp: "Electric Transmission of Energy, and its Transformation, Subdivision, and Distribution. " Fourth Edition, Thoroughly .Revised. London, Whittaker & Co., Paternoster Square, 1894, p. 301 to p. 311. "In order to be able to deal by means of simple mathematics with the working condition of a rotary field motor, we assume that the induc- tion within the interpolar space between field and armature varies according to a simple sine law. Whether this induction is due to the current in the field coils alone, or to the combined effect of field and armature currents, we need at present not stop to inquire; all we care to know is that such an induction does actually exist when the motor is at work, and that the sinusoidal field which it represents revolves with a speed corresponding to the frequency of the supply currents. Thus, if there be four field coils, and the frequency is 50, we would have a two- pole field revolving 50 times a second, or 3,000 times a minute, round the centre of the armature, and if there were no resistance to the move- ment of the latter it would be dragged round by the field at a speed of 3,000 r.p.m. It is obvious that the actual speed must be smaller. If the speed of the armature coincided exactly with that of the field, then the total induction passing through any armature coil, or between any pair of conductors on the armature would remain absolutely constant, and there would be no e.m.f., and, therefore, no current induced in the armature wires. Where there is no current there can be no mechanical force, and the armature could, therefore, not be kept in rotation. In order that there may be a mechanical force exerted, it is obviously essential that there shall be a variation in the magnetic flux passing (Facing page 64) v ? ' THE ROTATING FIELD AND THE INDUCTION MOTOR 65 through any armature coil, and that necessitates a difference in the speed of rotation between field and armature. This difference is called the 'magnetic slip' of the armature. If, for instance, the speed of the field in our two-pole motor is 50 revolutions per second, and the speed of the armature 48 revolutions per second, we would have a magnetic slip of two revolutions out of 50, or 4 per cent. In modern machines the slip at full load averages about 4 per cent, and rarely reaches as high as 10 per cent, so that good rotary field motors are in point of constancy of speed under varying load about equal to continuous shunt motors. "It was mentioned above that the motor would have a frequency of 50 revolutions at a speed only by 4 per cent short of 3,000 r.p.m. This is an inconveniently high speed for any but very small sizes. To reduce the speed is, however, quite easy. We need only increase the number, and proportionately reduce the length of the field coils. Thus, if instead of 4 coils, each spanning 90 of the circumference, we use 8 coils, each spanning 45, and connect them so as to produce two rotary fields, the speed will be reduced to one-half of its former value. By using 12 coils we obtain a six-pole motor, in which the speed will be reduced to one-third, or about 1,000 r.p.m.; with 16 coils we get down to 750 revolutions, and so on. In order to avoid unnecessary complexity we shall, however, commence the investigation on a two-pole machine, having only one re- volving field, and leaving the transition to a multi-polar ma- chine running at lower speed until the more simple case has been dealt with. "Such a machine is shown in Fig. 47. The field consists of a stationary cylinder, composed of insulated iron plates, and pro- vided close to the inner cir- cumference with holes through which the winding passes. The armature is also a cylinder made up of insulated iron plates pro- vided with holes near its outer circumference for the reception of the conductors. The use of buried conductors, although not absolutely necessary, has two important advantages first, mechanical strength and protection to the winding; and, secondly, reduction of the magnetic resistance of the air-gap, which, it will be seen later on, is an essential condition for a machine in which the difference between the true watts and apparent watts shall not be too great. The armature conductors may be connected so as to form single loops, each passing across a diam- FIG. 47. Diagram of rotor and stator of induction motor. 66 INDUCTION MOTOR eter, or they may all be connected in parallel at each end face by means of circular conductors, somewhat in the fashion of a squirrel cage. Either system of winding does equally well, but as the latter is mechan- ically more simple, we will assume it to be adopted in Fig. 47. The circular end connections are supposed to be of very large area as com- pared with the bars, so that their resistance may be neglected. The potential of either connecting ring will then remain permanently at zero, and the current passing through each bar from end to end will be that due to the e.m.f. acting in the bar divided by its resistance. It is important to note that the e.m.f. here meant is not only that due to the bar cutting through the lines of the revolving field, but that which results when armature reaction and self-induction are duly taken into account. "Let us now suppose that the motor is at work. The primary field produced by the supply currents makes ~i complete revolutions per second, whilst the armature follows with a speed of ~ 2 complete revolu- tions per second. The magnetic slip is then /***/ I /"X^/O s = - (50) If the field revolves clockwise, the armature must also revolve clock- wise, but at a slightly slower rate. Relatively to the field, then, the armature will appear to revolve in a counter clockwise direction, with a speed of ~ = ~i ~ 2 (51) revolutions per second. As far as the electro-magnetic action within the armature is concerned, we may therefore assume that the primary field is stationary in space, and that the armature is revolved by a belt in a backward direction at the rate of ~ revolutions per second. The effective tangential pull transmitted by the belt to the armature will then be exactly equal to the tangential force which in reality is trans- mitted by the armature to the belt at its proper working speed, and we may thus calculate the torque exerted by the motor as if the latter were worked as a generator backward at a much slower speed, the whole of the power supplied being used up in heating the armature bars. The object of approaching the problem from this point of view is of course to simplify as much as possible the whole investigation. If we once know what torque is required to work the machine slowly backward as a generator, it will be an easy matter to find what power it gives out when working forward as a motor at its proper speed. "Let, in Fig. 48, the horizontal a, c, b, d, a, represent the interpolar space straightened out, and the ordinates of the sinusoidal line, B, the induction in this space, through which the armature bars pass with a speed of ~ revolutions per second. We make at present no assumption THE ROTATING FIELD AND THE INDUCTION MOTOR 67 as to how this induction is produced, except that it is the resultant of all the currents circulating in the machine. We assume, however, for the present that no magnetic flux takes place within the narrow space between armature and field wires, or, in other words, that there is no magnetic leakage, and that all the lines of force of the stationary field are radial. The rotation being counter clockwise, each bar travels in the direction from a to c to b, and so on. The lines of the field are directed radially outwards in the space dac, and radially inward in the space cdb. The e.m.f. will, therefore, be directed downwards in all the bars on the left, and upwards in all the bars on the right of the vertical diameter in Fig. 47. Let E represent the curve of e.m.f. in Fig. 48, then, since there is no magnetic leakage, the current curve will coincide in phase with the e.m.f. curve, and we may represent it by the line 7. It is important to note that this curve really represents two things. In the first place, it represents the instanta- neous value of the current in any one bar during its advance from left to right; and in the second place it represents the per- manent effect of the current in all the bars, provided, however, the bars are numerous enough to permit the repre- sentation by a curve instead of a line composed of small vertical and horizontal steps. The question we have now to in- vestigate is: What is the magnetising effect of the currents which are collectively represented by the curve 7? In other words, if there were no other currents flowing but those represented by the cur.ve 7, what would be the disposition of the magnetic field produced by them? Positive ordinates of 7 represent currents flowing upwards or towards the observer in Fig. 47, negative ordinates represent downward currents. The former tend to produce a magnetic whirl in a counter clockwise direction, and the latter in a clockwise direction. Thus the current in the bar which happens at the moment to occupy the position b, will tend to produce a field, the lines of which flow radially inwards on the right of 6, and radially outwards on the left of b. Similarly the current in the bar occupying the position a tends to produce an inward field, i.e., a field the ordinates of which are positive, in Fig. 48, to the left of a, and an outward field to the right of a. It is easy to show that the collective action of all the currents represented by the curve 7 will be to produce a field as shown by the sinusoidal line A. This curve must obviously pass through the point 6, because the magnetizing effects on both sides FIG. 48. The interpolar space of the induction motor. The m.m.f. and flux belts. 68 INDUCTION MOTOR of this point are equal and opposite. For the same reason the curve must pass through a. That the curve must be sinusoidal is easily proved, as follows : Let i be the current per centimeter of circumference in fe, and let r be the radius of the armature; then the current through a conductor distant from b by the angle a, will be i cos a per centimeter of circumference. If we take an infinitesimal part of the conductor comprised within the angle da, the current will therefore be di = ir cos a da and the magnetizing effect in ampere-turns of all the currents comprised between the conductor at b, and the conductor at the point given by the angle a will be di = ir sin a (52) J and since the conductors on the other side of b act in the same sense, the field in the point under consideration will be produced by 2ir sin a ampere-turns, i being the current per centimeter of circumference at b. "Since for low inductions, which alone need here be considered, the permeability of the iron may be taken as constant, it follows that the field strength is proportional to ampere-turns and that consequently A must be a true sine curve. "When starting this investigation, we have assumed that the field represented by the curve B is the only field which has a physical exist- ence in the motor; but now we find that the armature currents induced by B would, if acting alone, produce a second field, represented by the curve A. Such a field, if it had a physical existence, would, however, be a contradiction of the premise with which we started, and we see thus that there must be another influence at work which prevents the formation of the field A. This influence is exerted by the currents pass- ing through the coils of the field magnets. The primary field must therefore be of such shape and strength, that it may be considered as composed of two components, one exactly equal and opposite to A, and the other equal to B. In other words, B must be the resultant of the primary field and the armature field A. The curve C in Fig. 48 gives the induction in this primary field, or as it is also called, the " im- pressed field," being that field which is impressed on the machine by the supply currents circulating through the field coils. It will be noticed that the resultant field lags behind the impressed field by an angle which is less than a quarter period. "The working condition of the motor, which has here been investigated by means of curves, can also be shown by a clock diagram. Let in Fig. 49, the maximum field strength within the interpolar space (i.e., number of lines per square centimeter at a and 6 of Fig. 47), be repre- sented by the line OB, and let 01 a represent the total ampere-turns due to armature currents in the bars to the left or the right of the vertical, THE ROTATING FIELD AND THE INDUCTION MOTOR 69 then OA represents to the same scale as OB the maximum induction due to these ampere-turns. We need not stop here to inquire into the exact relation between 01 a and OA, this will be explained 'later on. For the present it is only necessary to note that under our assumption that there is no magnetic leakage in the machine, OA must stand at right angles to OI a , and therefore also to OB, and that the ratio between 01 a and OA (i.e., arma- ture ampere-turns and armature field) is a constant. By drawing a vertical from the end of B and making it equal to OA, we find OC the maximum induction of the impressed field. The total ampere-turns required on the field magnet to produce this im- pressed field are found by draw- ing a line from C under the same angle to CO, as AI a forms with AO, and prolonging this line to its intersection with a line drawn through at right angles to OC. Thus we obtain OI e , the total ampere-turns to be applied to the field. The little diagram below shows a section through the machine, but instead of repre- senting the conductors by little circles as before, the armature and field currents are shown by the tapering lines, the thick- ness of the lines being supposed to indicate the density of current per centimeter of circumference at each place." D. THE SQUIRREL CAGE A rotor winding consisting of a number of bars connected in parallel, or short-circuited, by circular end rings, was invented by the late M. von Dolivo-Dobrowolsky. It is called a " squirrel- cage" 1 winding. It is a true poly-phase winding and its theory and understanding are based on the same principles as those of other poly-phase windings. French, "cage d'ecureuil;" German, "Kurzschlussanker," or "Kafig Wicklung." FIG. 49. The vector diagram of the induction motor in the elementary theory. NOTE. This is the only figure in this book in which clockwise rotation has been assumed. It was taken from Mr. Kapp's book published in 1894, before the international agreement had been reached on positive counter-clock- wise rotation. 70 INDUCTION MOTOR To fix ideas, let us look at a star-connected generator closed through a delta connected load, Fig. 50. In each phase a voltage e is induced. The delta voltages are I-II, II-III, and III-I. Calling the delta voltages E, we have E = \/3-e (53) At any point I, II, or III, the algebraic sum of the currents which meet is zero, according to Kirchhoff s First Law. This, ex- tended to our case, may be expressed in vector terms and we ii + in Non-inductive Voltage Drop in Balanced Three-phase System FIG. 50. Three-phase System FIG. 51. substitute "vector sum" for "algebraic sum." In Fig. 51 this vector sum has been drawn, from which we see that 1 I = (54) It is very important to watch directions and it is easy to commit errors. If each phase of the star-connected generator has a resistance 7*1 and no reactance, then there will be an ohmic drop ir in each phase, and in time-phase with e, as shown in Fig. 50. Then E = A/3(e - ir) (55) in accordance with Kirchhoff's Second Law that in any closed circuit the algebraic sum (vector sum) of the products of the current and resistance in each of the conductors in the circuit is equal to the e.m.f. in the circuit. In a four-phase (two-phase) system, Fig. 52, we have, if the phase voltages between neutral and outside equal e, E = \/2-e (56) for the voltages I-II, II-III, III-IV, and IV-I. THE ROTATING FIELD AND THE INDUCTION MOTOR 71 The currents are 1 (57) from Kirchhoff's First Law. i IV Four-phase System. FIG. 52. KirchofTs First Law Applied to Point I Consider, next, a six-phase system (Fig. 53). The line volt- ages are and the currents E = e I = i (58) (59) KirchofFs First Law Six-phase System Applied to Point I FIG. 53. Now, consider a true poly -phase network in which there are n phases (Fig. 54). Then the time difference between two phases is > and the voltage (with no current) between phases is In = E = 2e sin n (60) 72 INDUCTION MOTOR Apply KirchhofFs First Law and draw the current polygon, Fig. 54, whence, directly I = 2 sinl- (61) The squirrel-cage winding is such a poly-phase winding as Fig. 55 indicates, in which e is the induced e.m.f . per bar, e its current, FIG. 54. n-phase system. Kirchoff's first law applied to point I. and r its resistance, while E = 2(e ir) sin( j is obviously the ohmic drop on the two sections 2R, we have iR E = 2IR = sin ( ) W FIG. 55. The squirrel cage. e = Voltage induced in bars. E = Voltage drop in end ring. (62) THE ROTATING FIELD AND THE INDUCTION MOTOR 73 The end rings act, therefore, in such a manner as to increase the resistance r of each individual bar by the amount R A similar argument leads to a similar relation in regard to the reactance of the rings, but its application is of small importance. The relation (62) can also be obtained from the consideration that in the squirrel-cage winding the total loss is Total Loss = n(Pr + 2PR) j = i 2 sinf- . . Total Loss = m 2 / r + - \ (63) ( 2si <); from which it follows that the effect of the end rings consists in raising apparently the resistance of each bar by the amount R (64) as obtained before. Assuming again the distribution of the flux belt in the air-gap to follow a simple sine law, the e.m.fs. and the currents follow a sine law distribution at any moment of time. The e.m.fs. and currents in the end rings also are in magnitude represented by the ordinates of the curve E. The assumption of sine and cosine curves implies, of course, the tacit assumption of an in- finite number of phases. The relations obtained previously are correct for any number of phases. E. THE TORQUE AND SLIP AND THE EQUIVALENCE OF MOTOR AND TRANSFORMER The theory of the Induction Motor is the theory of the General Alternating-current Transformer. Credit for this important relation is due to Dr. H. Behn-Eschenburg, of Oerlikon, Switzer- land, who demonstrated this relation in 1893. It has become fundamental. It is evident that this relation pertains so long as the rotor is standing still. As was shown by Mr. Kapp, the theory of the field 74 INDUCTION MOTOR belt and its interactions, if represented by vectors, leads to a polar diagram in which time-phase and space-phase are interchangeable. If the armature runs in synchronism, there is no current induced in the rotor, the no-load current corresponding to the open circuit current of the transformer. If the armature lags behind the field in angular velocity, then, if ~i is the impressed frequency, and ^ 2 the frequency corresponding to co 2 , the angular velocity of the rotor, the currents induced in the rotor windings are of frequency ~i ^ 2 . If the armature resistance per phase is rz) then a current will flow t, = -' = 2.12 ( ~' ~ ~ 2) 2 F 2 10-' (65) 1 7*2 = 2.12 -^TMO-s (66) The same current will be obtained with the secondary at rest, if the external resistance is equal to Therefore, substitute s for the motor an equivalent transformer with a total internal and external resistance equal to R 2 = s The Torque. Imagine the rotor to be turned with angular velocity (coi o> 2 ) against the magnetic field, which is supposed to be at rest. If T is the torque in mkg, then we have 9.81!T-(wi - co 2 ) = (3^ 2 2 r 2 ) (67) where iz is the current, r 2 the resistance in each phase, the rotor to be assumed three-phase. If it is n-phase, substitute n for 3. CO = 27T - (68) p where p is the number of north or south poles. Also 9.81 -T-ut = P watts. From these equations follows (69) 9.81-27r.T mfca ~ - P = 3iVr 2 (70) P watta ) (71) THE ROTATING FIELD AND THE INDUCTION MOTOR 75 If we want to obtain the torque in foot-pounds, the formula is 8.5- T ft . i bs . = ^ (3i 2 2 r 8 + P waM4 ) (72) The torque is, therefore, proportional to the algebraic sum of the output of the motor plus the energy dissipated in the armature. Subtract, therefore, the primary copper loss and the core loss from the input as given by the ordinates of the circle and we obtain directly the torque T in synchronous kw. The Slip. The slip of the motor equals the loss in the rotor, divided by the output plus the loss in the rotor, Determination of the Slip from the Loss Lines FIG. 56. Torque, slip, and loss lines for the induction motor. Andre Blondel 1 suggests a beautiful mode of representing the slip by extending me, Fig. 16, to an intersection with vector e\. Then it can easily be proved that the segment OX, X being the point of intersection, is a measure of the slip. This method has to be modified if the copper loss in the primary winding is to be taken into account as diminishing the receptive capacity of the motor. Perhaps the simplest way to show graphically the slip of the motor is the following. Let ab be the output of the motor, be be the rotor loss, then slip = . Draw through g, the standstill position of the motor in which 1 ANDRE BLONDEL, "Theories Graphiques des Moteurs Polyphase^. " V Industrie Electrique, 1896, p. 77. 76 INDUCTION MOTOR the output is zero, gn parallel to df, the primary copper loss line. Then nh o ng bc:ac::ld:hl kl:hl::dl:de::nh \ng bc:ac::nh:ng q.e.d. (74) + 80 % + 00 % -HO % + 20 % NJ Q - 20 % - 40 % - 60 % -80 % -100 % Slip X) K.W. -100 Fig. 57. Torque as a function of rotor resistance, slip, and primary input. Divide ng into a percentage scale and the slip may be read off for each current. At g the slip is 100 per cent, at n it is zero. Torque Curves. The results obtained in polar coordinates shall .now be represented in Cartesian coordinates. We shall first use as abscissa the watts input of the motor, and secondly the slip, representing all characteristics as functions of these two parameters (Fig. 57). THE ROTATING FIELD AND THE INDUCTION MOTOR 77 It is interesting to note, and obvious from the diagrams, that there is a maximum torque for a given motor frame. This torque may occur at starting or at any speed. It may be varied by the resistance of the rotor, thus making it possible to start with maxi- mum torque by increasing the rotor resistance. The reader may be trusted to draw many other instructive conclusions from the diagrams. Torque may be measured conveniently in ' ' synchro- nous kw.," which is the kw. at angular velocity coi corresponding to the torque at angular velocity, o> 2 . F. HIGHER HARMONICS IN THE FIELD BELT AND THEIR EFFECT UPON THE TORQUE Examining Fig. 41 it is noticed that the field belt does not have the form of a sine wave. It consists, therefore, of a funda- mental sine wave and of higher harmonics of this fundamental. To study these effects, consider first a quarter-phase motor, 3d Harmonic Phase. I Lead! . Phase -II Phase IH-Lagi. Phase II- 1 . '. IH and II-II Rotate Backward! 5th Harmonic Phase I Leads - Phase II Phase IH Leads- Phase II- IH and II-E Rotate Forwards 7th Harmonic rbase I Leads Phase IX Phase IH Lags- Phase II-H . '. IH and II H Rotate Backwards FIG. 58. Harmonics in the field belt of a two-phase motor. 78 INDUCTION MOTOR important application of which has recently been made to the U. S. Battleship New Mexico, as discussed fully in Chap. VIII. The third harmonic combines in the two phases in such a manner that, if Phase II Fundamental is behind Phase I Fundamental, then, .Phase II 3d Harmonic is ahead of Phase I 3d Harmonic (Fig. 58). Hence, the third harmonic flux belt in a quarter- phase motor produces a backward torque. It must be emphasized that we are talking about harmonics in the flux belt and not in the e.m.f. of the supply circuit. The Third. Harmonic Torque FIG. 59. The effect of the third harmonic in the field belt on the torque of the two-phase motor. effects of the latter may be neglected as a little thought indicates. The effects of the harmonics in the flux belt, however, consist in setting up rotating fields with angular velocity corresponding to the fundamental supply frequency, whose effect is therefore the same as the superimposition of 3, 5 or 7 times the number of poles would have upon the main fundamental flux belt. We have already seen that in a two-phase motor the third harmonic acts as a brake (Fig. 59). In a two-pole motor fed from 25 cycles the synchronism of the fundamental is 1,500 THE ROTATING FIELD AND THE INDUCTION MOTOR 79 r.p.m., while the synchronism of the third harmonic takes place at 500 r.p.m., but the field rotating backwards will produce a torque curve as indicated in Fig. 59. It is quite evident from Fig. 59, that such backward torque may be extremely serious as it diminishes the starting torque. If a squirrel-cage rotor is used, the motor may be unable even to start at all. : Slip Wave from Fundamental "Wave Harmonic Belts I Torque II Torque Resulting and 3d, 5th, & 7th FIG. 60. The effects of harmonics in the field belt on the torque of a two-phase motor. Drawing the phases for the fifth harmonic, it will be seen that in the quarter-phase motor the fifth harmonic produces a for- ward torque whose synchronism occurs at +300 r.p.m. The seventh harmonic gives a backward torque whose synchronism occurs at 215 r.p.m. These three harmonics have been drawn into Fig. 60 showing the resultant torque. The dead points, which were so often observed in two-phase motors during the development stages, are particularly interesting. 80 INDUCTION MOTOR An examination of the effects of the third harmonic in a three- phase system, if the sine waves are drawn, shows that the third harmonics of the three circuits are in phase with each other, therefore, they produce no rotating field. Their effect upon the torque is therefore that of a single-phase induction motor having three times the number of poles. Its torque is shown in Fig. 61 5thH Slip 3dH FIG. 61. The effect of a third harmonic in the field belt on the torque of a three-phase motor. The torque for the 5th harmonic is shown but not added to the fundamental torque. where also the torques of the fifth and seventh harmonics are indi- cated. The fifth harmonic gives backward torque, the seventh harmonic gives a forward torque. The torque curve of the single- phase induction motor will be treated at length inChap. XVII. 1 1 This entire subject of the effects of higher harmonics, both in the field belt and in the supply circuits, was treated brilliantly by ANDR BLONDEL, as early as 1895, in his paper, -''Quelques Proprie"tes Generates des Champs Magne*tiques Tournants," L'Eclairage Electrique, Paris, to which classic and fundamental paper all readers should refer. THE ROTATING FIELD AND THE INDUCTION MOTOR 81 G. EXPERIMENTAL DATA A vast amount of experimental material has been accumulated since the first publication of this theory 25 years ago. The experimental proof on induction motors of the circle character- istic was already given in my paper of Jan. 30, 1896, on one machine only. Literally, hundreds of thousands of motors, aggregating millions of horsepower have been tested since according to this method and the theory is now solidly estab- lished. In subsequent chapters numerous experimental data are given so that we shall not here refer to the subject any more. H. COLLECTION OF DATA The most important data in the design of the three-phase induc- tion motor are here collected together: Leakage Factor: a = ViV Z 1 (14) Maximum Power Factor: Magnetic Flux: 0i = 2.12~-z r /. Air-gap A = 0.62 mm. Pole-pitch t = 30.50 cm. 100 INDUCTION MOTOR Volts between the lines Amperes, field Amperes, arma- ture Frequency 81 120 150 170 36.0 74.0 106.0 135.0 95 180 260 320 42.0 41.0 383 8.5 43.2 If we draw the short-circuit curve, we can interpolate for 380 volts the short-circuit current of 380 amp. We thus get for the leakage factor the value B. The same magnetic frame was wound for 24 hp. 190 volts between the lines, 10 poles, 50 ^. Air-gap A = 1.1 mm. Pole-pitch t = 18.3 cm. Volts between the lines Amperes, field Frequency 20.0 20.0 51 25.0 31.5 33.0 50.5 43.5 75.0 51 66.0 139.0 83.0 185.0 95.0 220.0 Magnetizing current: 31.2 amp. at 190 volts. The short- circuit current at 190 volts amounts to 470 amp. Hence, 31.2 470 0.0664 The following table shows the results of the tests: Air-gap cm. Pole-pitch cm. (7 0.062 0.110 30.5 18.3 0.0224 0.0664 SHORT-CIRCUIT CURRENT AND LEAKAGE FACTOR 101 For equal air-gaps we have ' or, in other words, ^ = ^ (80) 0"i hi The leakage factor is inversely proportional to the pole-pitch, or directly proportional to the number of poles. By the above experiments it has been demonstrated that the leakage factor is directly proportional to the air-gap, and in- versely proportional to the pole-pitch. We may, therefore, write the formula for the leakage factor, i and $2 are the fictitious primary and secondary fluxes, Xi and Xz the primary and secondary m.m.fs., p the reluctance of the common magnetic circuit, pi and p 2 the reluctances of the primary and secondary leakage circuits. From (82) and (83) i ~ ~ But = P (84) Therefore i = 1+ - (86) Pi t = 1 + p - P2 p ~ )(1 + p -) - 1 (87) Pi P2 = 5- + P - + -5- (88) Pi P2 PlP2 = p( -- 1 ) approximately (89) \Pl P2/ Now, (90) where b is the width of core of the motor. 7 = Ki-t for the end connections (91) = K r --6 for the slots (92) 104 INDUCTION MOTOR where d is depth of slot, and w width of slot. Hence, adding - -- K t + K - b p ^ w A d A r ~. 04) (93) (95) (96) where C is a factor which varies in a number of ways depending on features of design. It can be guessed at successfully by the experienced designer, but it is not amenable to sound scientific calculation. It varies between the limits of 6 and 15. I. FURTHER EXPERIMENTAL DATA From an exhaustive investigation, whose results are given in the table below and plotted in Fig. 77 the following formula has been derived: o.io 0.05 1 2o"Diam. ; 8 Poles; 12 Slott; Size .s"x 1.26* 14 'Diam.; 6 Poles ; 12 s"x 1.26" \ t-1 A =. ,: \ \ \ ^ *mi NT+T! 5 10 15 20 25 in. 30 Width of Iron FIG. 77. The leakage factor a and its dependence upon the width of the motor. , = ^ (5.1^ + 5.65) (97) in which A, t, and 6 are to be substituted in inches. SHORT-CIRCUIT CURRENT AND LEAKAGE FACTOR 105 60-Cycle Induction Motors Stators 12 slots per pole Slot 0.3 in. by 1.125 in' t = 7 . 8 in. A = . 03 on one side b Inches 2.00 3.75 3.75 5.00 6.50 7.00 7.50 19.50 0.0900 0.0640 0.0540 0.0560 0.0465 0.0430 0.0413 0.0300 It is well to keep in mind that a is approximately equal to the magnetic reluctance of the common magnetic field divided by the reluctance of the parallel leakage fields Po p = 1 4-1 Pi P2 1 Pi P2 Po (98) (99) (100) K. WINDING THE SAME MOTOR FOR DIFFERENT SPEEDS Formula (96) permits us to determine the change in the output, power factor, and so forth, of a motor wound for a different number of poles, for instance, for eight, four, or two poles. If the field has 48 or 72 slots, it can easily be wound so as to satisfy this demand. We will assume the induction in the air-gap, or, which is the same, in the teeth, to remain constant for all three cases. We will further assume that the motors are to be wound for the same voltage. Then it is clear, according to equation (49), that the total number of active conductors must be proportional to the number of poles; in other words, if the eight-pole motor has, for instance, 720 conductors, or 10 conductors in each of 72 106 INDUCTION MOTOR slots, then the four-pole motor must have 360, and the two-pole motor 180, in order to get the same induction in the air-gap. To calculate the relative value of the magnetizing current we need only know the number of active conductors n per pole, see equa- 720 tion (45). We have for n in the eight-pole motor-- - =90; in o 360 the four-pole motor -j- = 90; and in the two-pole motor 180 -- = 90. Hence, as B and n are the same in each of the three cases, it follows that the magnetizing current also remains the same. Two Poles FIG. 78. The primary current locus of the induction motor The leakage factor and the circle diagram for different numbers of poles on the same frame. As the shape and size of the slots are the same in all three cases, the factor in equation (96) for the leakage coefficient also remains the same. Hence, as the leakage factor is proportional to the quotient of the air-gap divided by the pole-pitch, we find the short-circuit current to be inversely proportional to the number of poles. This is graphically represented in Fig. 78. A glance at the diagram teaches us that the maximum energy that can be impressed upon the motor, and, therefore, also very nearly the output, vary in proportion to the pole-pitch. According to the diagram we find the leakage factor for the two-pole motor 8 8 equal to j = 0.05; for the four-pole motor equal to ~~ = 0.10; g and for the eight-pole motor equal to ^ = 0.20. The maximum power factor in each case can now be calculated with the help of formula (19). This is done in the following table: SHORT-CIRCUIT CURRENT AND LEAKAGE FACTOR 107 Number of Leakage Maximum Relative Revolutions poles factor power factor output per minute 2 0.05 0.910 80 3,000 4 0.10 0.835 40 1,500 8 0.20 0.715 20 750 The output is therefore proportional to the number of r.p.m. It may not be amiss here to remark that if the motor is ordi- narily wound for four poles, the induction in the iron above the slots may for the same frame become too high in the two-pole motor, thus increasing the magnetizing current, and possibly creating undue heating. L. DRAWBACKS OF A HIGH FREQUENCY If the circumferential speed of the armature is limited, and this is generally the case, then the pole-pitch is also limited for a given number of r.p.m. The air-gap cannot indefinitely be diminished, hence, a high frequency necessitates a large leakage factor according to formula (96). 1 We labor here under the same difficulties that we have met with in the design of alter- nators for high frequencies. It is doubtless possible to build motors for frequencies between 60 and 100, but the higher the frequency the lower will be the power factor, and the larger will be the lagging currents. It has also to be borne in mind that motors for high frequencies, if they are to be as good as those for low frequencies, must be made not inconsiderably larger. Allowing again the induction in the air-gap to be the same for different frequencies, which is a more or less challengeable proposition, it follows from formula (49) that the total number of active conductors around the circumference of the field must also be the same, for the pole-pitch is inversely proportional to the frequency, hence, the product of the frequency into the number of lines of induction per pole remains the same if the induction in the air-gap is the same. The magnetizing current, however, being proportional to the ratio of the induction B divided by the number of active 1 This is the reason why Mr. TESLA and the Westinghouse Company failed to design a successful motor between 1888 and 1890 as a frequency of 135 was then commonly used. See B. G. LAMME, "The Story of the Induction Motor," A. L E. E. Journal, March, 1921. 108 INDUCTION MOTOR conductors per pole, is thus inversely proportional to the fre- quency. The leakage factor is, according to formula (96), directly proportional to the pole-pitch, or inversely propor- tional to the frequency because the pole-pitch is, in the case under consideration, inversely proportional to the frequency hence, it follows that, as the magnetizing current has been shown to be proportional to the frequency, the diameter of the semi-circle remains constant for all frequencies. Figure 79 shows the polar diagram for the same motor, but for different frequencies. The maximum energy that the motor is capable of taking in, and, therefore, also the maximum output, is the same for 100^, 50^, or 25^. But the maximum power factor is considerably smaller for the high frequencies, as a -1GOA- FIG. 79. The leakage factor of the induction motor. The circle diagram for different frequencies. glance at the diagram shows. The following table shows the leakage factor and the power factor in relation to the frequency : Frequency Leakage factor Maximum power factor 25 50 100 0.05 0.10 0.20 0.910 0.830 0.715 I wish to call attention to the fact that the motor for the higher frequencies is here represented less unfavorably than it really is, because the induction in the air-gap has to be reduced if the motor is to be wound for a higher frequency. The im- mense lagging currents invariably bound up with the higher frequency are very clearly shown in the diagram. SHORT-CIRCUIT CURRENT AND LEAKAGE FACTOR 109 It is to be remembered that the current in the armature is dependent upon the leakage factor, since the transformation factor vi forms part of the leakage factor. (See Fig. 16.) The transformation factors v\ and v 2 are connected with a through equation (14), 1 (14) Hence, it follows, as z' 2 = A D ' ~9\ t see Fig. 16, that the current in the armature is larger for the motor running at a high fre- quency than for that running at only 25^. In our case, setting vi = v 2 , we get for v at 25~, 0.978, and at 100~, 0.912, therefore the current in the armature of the motor for 100 ~ is, for the same AD, 1.07 times larger than for 25^. This corresponds to an increased armature loss of about 14 per cent. But as the primary current is also larger for 100^ than for 25^, the arma- ture loss is still greater than here calculated. Thus to the draw- back of large lagging currents, there has to be added the further drawback of considerably larger losses. The foregoing experiments and considerations are, within my knowledge, the first attempt to deal in a rational, systematic manner with the conditions underlying the leakage in poly- phase motors. I am far from claiming for this treatment com- pleteness of conclusiveness; on the contrary, I deem it a necessity to revise it by the light of forthcoming experience. I am toler- ably confident that the main propositions will be proved true, while minor points may need some qualification. Considering the immense complexity of the phenomena in poly-phase motors, the greater or less arbitrariness which hangs about most of our assumptions which have to be made in order to be able to calculate at all, I cannot forbear from wondering that so approximate a solution can be attained at all. It may be that there are errors inherent in our fundamental assumptions which all so counteract one another as to cause the result of calculation to deviate but little from experiment and observa- tion. This view will commend itself to those who are familiar with some branch of physiology, for instance, physiological optics; here we have the testimony of Helmholtz that the eye, having "every possible defect that can be found in an optical instru- ment," yet gives us a fairly accurate image of the outer world because these various defects balance one another almost completely. 110 INDUCTION MOTOR The above remarks will be distasteful to those who have accustomed themselves to look upon only one side of a question, and who try to shut their eyes to the inevitable uncertainties that beset us in all intellectual problems. I was once taken to task by a critic for having adduced experimental evidence qualify- ing my theory, and narrowing the limits of its application, and I was told that these experiments invalidated my argument, while my intention to lay stress upon the incompleteness and the shortcomings of the theory was obviously not even thought of by my critic. Politicians and propagandists may have to hide the weak sides and spots of their arguments, but men of science are bound to point them out and to expose them. M, HISTORICAL AND CRITICAL DISCUSSION OF THE LEAKAGE FACTOR In The Electrical Engineer, London, Dec. 11, 1903, Mr. H. M. Hobart discussed the equation (96) in his usual thoughtful manner. He points out that too much weight has been placed upon the pole-pitch and that the formula might lead to motors of too large a diameter. He says : ' ( In the following article the writer wishes, in the first place, to emphasize the importance of that part of the total inductance which is due to the end con- nections, and, in the second place, to develop a simple and practi- cal method by which the best dimensions may be decided. On p. 36 of Behrend's excellent treatise on induction motors, the following formula for calculating a is given : where C is a figure which is dependent on the slot dimensions and other conditions, A is the length of air-gap, and t the pole- pitch. Behrend estimates that C varies between 10 and 15 for half -closed slots. This formula is extremely useful on account of its simplicity, especially if experimental results are available by which to decide the value of C." Mr. Hobart then proceeds to give in a table the ratio of - t and determines C accordingly. He closes with the statement: "It is, however, preferable to keep the formula as simple as possible, so that the writer thinks it better to retain Behrend's original formula, together with the values of C got from Table I and Fig. 1." SHORT-CIRCUIT CURRENT AND LEAKAGE FACTOR 111 The Institution of Electrical Engineers, London, held a meeting Jan. 14, 1904, at which Prof. Silvanus P. Thompson presented a paper prepared by Dr. H. Behn-Eschenburg, "On the Magnetic Dispersion in Induction Motors, and its Influence on the Design of these Machines." Dr. Behn-Eschenburg starts with a slightly different definition of the coefficient of leakage from that used by us. 2 , not shown in the figure would, be $2 = X% -r- p, if acting alone, and therefore if only X 2 and X 3 were acting upon the circuits, the flux < 2 + $3 = (X z + Xs) -r- p, would exist in the air-gap. These fluxes are prevented from becoming established through the existence of the m.m.f. Xi. 1 U. S. Patent No. 427,978, May 13, 1890. 2 "General Characteristics of Electric Ship Propulsion Equipments." By E. F. W. ALEXANDERSON. General Electric Review, April, 1919. 115 116 INDUCTION MOTOR To fix ideas we also represent in Fig. 88 the electric circuits to which this type of motor is equivalent, assuming a ratio of transformation of one to one. As both rotor windings have Stator Slots -t Winding I High Resistance Winding II Rotor Slots w Resistance Winding III FIG. 86. Arrangement of slots of double-squirrel cage motor. the same slip, the variation in speed corresponds to a trans- former with variable resistances, as indicated in the Fig. 88, viz., r 2 -5- s and n + s (106) ^AAAAA/WW^/W FIG. 87. The leakage paths of the magnetic circuit of the double-squirrel cage motor. We begin with Winding III. Its resultant flux is F 3 . This flux sets up the e.m.f. which sends current through the ohmic DOUBLE SQUIRREL-CAGE INDUCTION MOTOR 117 resistance of the Low Resistance Winding III. This e.m.f. is equal to e 3 = 2.12(~i - ~ 2 )z3F 3 10- 8 volts (107) FIG. 88. Equivalent transformer circuits for double-squirrel cage induction motor. FIG. 89. The diagram of fluxes of the double-squirrel cage motor. The current produced by this e.m.f. is equal to i 3 = e 3 -r- r 3 . The leakage field fs through the path of reluctance p 3 is in time- 118 INDUCTION MOTOR phase with and proportional to i 3 and it is to be estimated in the usual manner. In Fig. 89 it is represented by ab = / 3 . OB in Fig. 89 is the flux F 2 , as also indicated in Fig. 87. If X* created by i s were acting alone, it would circulate a flux 3 , Fig. 89. Likewise, the e.m.f. e% induced by F 2 , e 2 = 2.12(~i - ~ 2 )22/MO- 8 volts (108) produces a current i 2 = e 2 - r 2 , and an m.m.f. X 2 which, acting alone, would circulate a flux 2 in quadrature with F 2 , as shown in Fig. 89. Combining < 3 and $ 2 vectorially, gives f> 2 + $ 3 , represented in the diagram in line with be. As in the general flux theory of the induction motor, so here, = $3 + /3 (109) = /3 (110) = (f2 + *8) + /2 (HI) = /2 (112) / 3 = ab (113) / 2 = be (114) /i = ce (115) As before in the theory of the induction motor, it follows readily = cd (116) The diagram of Fig. 89 shows clearly and significantly the composition of the fluxes F s , / 3 , / 2 , and /i into the primary result- ant flux FI which induces the counter e.m.f., which balances the primary impressed e.m.f. As before, the primary resistance TI is neglected. It can easily be taken into account as in Chap. III. An inspection of the diagram shows the influence of the low reluctance of the Leakage Path III. To show the effect of this leakage, a complete performance of a motor has been worked out for a range of slip from zero to infinity for given motor character- istics, as follows: r 3 = 0.06 r 2 = 0.6 v 3 = 1.3 v 2 = 1.1 i = 1.1 *.-^ 7*2 33.5 DOUBLE SQUIRREL-CAGE INDUCTION MOTOR 119 These characteristics correspond closely to a large slow speed motor with the exception that the leakage is assumed somewhat larger than it would be in reality, as well as the reluctance of the main magnetic circuit. The real motor, therefore, would have a higher power factor. The following table is obtained from corresponding points carefully worked out : Slip Cosine $\ Prim, cur- rent Torque oo 0.0000 Current and Cos \l/i 50.0 0.0375 192.0 66 do not correspond 10.0 0.2250 180.0 400 to these points. 4.0 0.3300 143.0 470 3.0 0.3350 129.0 430 2.0 0.3150 116.0 370 256 1.5 0.2950 109.0 312 350 1.0 0.2500 105.0 263 500 132 0.9 0.2300 104.0 244 546 150 0.8 0.2370 104.0 244 600 165 0.7 0.2230 102.0 225 643 180 0.6 0.2350 100.0 232 710 205 0.5 0.2380 98.5 230 760 230 0.4 0.2400 96.0 235 793 280 0.3 0.2900 91.0 270 776 320 0.2 0.3700 87.0 320 635 345 0.1 0.4800 69.0 337 370 275 0.06 0.5200 54.0 276 210 190 0.0 0.0000 33.5 These results are represented in the polar diagram, Fig. 90. An analysis of this figure yields the following results: First, the locus of the primary current is no longer a circle. Secondly, if the Low Resistance Winding III did not exist, the locus of the primary current would be the circle about 0' as center, with a diameter aft = where coi then we have shown that the current leads the impressed e.m.f. Now, it makes no differ- FIG. 99. The operation of the Leblanc commutator. ence whether F is produced by the currents from the line, or whether it is set up by other means, if the brushes are short- POLY-PHASE COMMUTATOR MOTORS 133 circuited, the short-circuit corresponding to the negligible impe- dance of the line, as was so brilliantly pointed out by M. Latour. Thus the two diagrams of Fig. 99 are physically the same, and the effect of operating a commutator motor above synchronism consists in creating leading currents in it, as viewed from the stator, and thus the constant part of the secondary leakage reactance above synchronism acts like a condensance. The reader is cautioned Squirrel Cage or Slip Ring A. UVW;/J S)PS ^ Induction Motor B. M c < . "53 c M CO o O 4, 0-3 O O <^> *"* "i AAAAAAAA' Commutator Induction Motor lA^\Mf = ^ YYVVVVVV c. wvwwvwv Commutator Induction Motor after Arnold- Lacour) FIG. 100. Commutator and slip ring, or squirrel cage, types of induction motor, and their equivalent circuits. to distinguish between the total reactance and the leakage react- ance, which is a prolific source of confusion. Equivalent circuits can be drawn for the different types of induction motors as is indicated in Fig. 100, where "A" is the equivalent circuit of the slip-ring type, "B" the equivalent circuit of a commutator type assuming constant secondary react- ance, and "C" is the equivalent circuit for a commutator type 134 INDUCTION MOTOR in which both constant secondary reactance and secondary react- ance varying with the slip are indicated. Tests have been made to check the performance of these motors by E. Arnold and la Cour 1 and by L. Dreyfus and F. FIG. 101. Slip-ring type of induction motor. chronism. Space diagram below syn- Hillebrand. 2 The latter equipped a rotor with slip-rings on one side and a commutator on the other and thus recorded the standard circle diagram and the displaced circle for the same type of motor. -02 FIG. 102. Slip-ring type of induction motor. Space diagram above syn- chronism. In Figs. 101 , 102, 103, and 104 there are traced out the m.m.f . belts produced by the rotation of the resultant field F with relative angular 1 E. ARNOLD and J. L. LA COUR, Vol. V, 2, p. 221. 2 L. DREYFUS and F. HILLEBRAND, "Zur Theorie des Drehstromkollector- Nebenschlussmotors." Elektrotechnik und Maschinenbau, 1910, p. 886, POLY-PHASE COMMUTATOR MOTORS 135 velocity coi w 2 towards the rotor. For o>i > o> 2 the machine is a motor, while f or coi < oj 2 it is a generator. The m.m.f . belts indicate that, in the Slip-ring Type, viewed from the primary, the m.m.f. of the secondary, or its fictitious flux < 2 , lags behind F by + ^ 2 FIG. 103. Commutator-type of induction motor. Space diagram below synchronism. electrical degrees, because the secondary currents, being of slip frequency coi co 2 , set up a rotating magnetic field which is carried around by the rotor with angular velocity W2 in the FIG. 104. Commutator-type of induction generator. Space diagram above synchronism. direction of mechanical rotation. Therefore, the electric phase lag of the secondary current appears also as a lag in the com- bined space and time diagram. For the generator action of the slip-ring type of motor, it is 136 INDUCTION MOTOR to be noted that the secondary currents set up a magnetic field rotating in opposition to the mechanical rotation of the machine, thus the lagging current in the secondary appears in the primary and in the combined space and time diagram leading the flux F by a time or space angle of ~ + ^2- z These relations are different in the Commutator Type, in which the secondary currents are of line frequency and the coil groups between brushes are stationary. Therefore, the secondary current reflected into the primary appears as a lagging current relative to the induced e.m.f. Therefore, this consideration leads again to the curious result that, in the Commutator Type of Induction Generator running above synchronism, constant sec- ondary reactance strengthens the resultant field of the machine and it therefore acts as capacity does in the range below synchronism. F. THE SLIP-RING COMMUTATOR TYPE AS FREQUENCY CHANGER The first to suggest the use of an armature provided with a commutator on one side, on which poly-phase brushes are placed, and slip-rings on the other, appears to be Mr. B. G. Lamme, 1 who noticed that, in a rotary converter without field excitation, running below synchronism, a current of low frequency appeared at the brushes on the commutator. This low frequency dis- appeared at synchronous speed. If poly-phase currents of frequency ^i are sent through the brushes upon the slip-rings, then a magnetic field is set up rotating with angular velocity coi relative to the armature. If the armature revolves with angular velocity co 2 against the direction of rotation of the field, then in the groups between the stationary poly-phase brushes upon the commutator, there will be induced an e.m.f. of the frequency ^i ^ 2 . Hence, if the commutator of such a device were connected to the rotor of a slip-ring type of induction motor, it would receive currents of slip frequency ^i ^ 2 , and on its slip-rings it would deliver currents of the frequency ^i. An application of this interesting phenomenon is described in Chap. XIII, C. 1 B. G. LAMME, United States Patent No. 682,943. Sept. 17, 1901. Application filed July 24, 1897. CHAPTER X THE SERIES POLY-PHASE COMMUTATOR MOTOR A. THE THEORY FOR CONSTANT CURRENT AND CONSTANT POTENTIAL IN THE IDEAL MOTOR The properties of a commutator, as discussed in Chap. IX, are now to be applied to the Series Poly-phase Commutator Motor, first described in 1888 by Wilson in the British Patent No. 18,525 and by H. Goerges in the German Patent No. 61,951 of Jan. 21, 1891. Mr. H. Goerges also described the Shunt Poly-phase Commutator Motor and outlined the theory of these motors in the E. T. Z., 1891, p. 699. Figure 105 shows diagrammatically the connections of the motor. The stator windings, which are indicated here in F-connection, are in series with the delta-connected ar- mature which is rotating counter-clockwise with the angular velocity co 2 . The rotating field resulting from the action of the poly-phase currents is assumed to have a counter-clockwise rotation coi, and the brushes are shown shifted forward by an angle a in the direction of rotation. Goerges). Brush shift angle a, no trans- The neutral position of the former ' brushes, or the datum from which we count the brush-shift, is defined as that in which the current passing in series through stator and rotor produces two fictitious magnetic fields, which cancel each other, neglecting leakage. As usual we assume a two-pole magnetic structure and distributed windings in rotor and stator. The brushes slide on the commutator here assumed to be the surface of the rotor wnidings. Great attention has to be paid to the conception of time and space phases. To simplify the analysis, we assume identical 137 138 INDUCTION MOTOR windings on stator and rotor, the series connection being obtained through a series transformer whose magnetizing current and leakage we neglect for the present, Fig. 106. The rotor and stator currents are assumed equal and in time- phase, so that, with brush-shift angle a = 0, the two fictitious fields of rotor and stator would obliterate each other. If the stator windings acting alone produce a counter-clockwise or positively rotating magnetic field i, then the same current equal in magnitude and time-phase acting alone in the rotor windings, produces a magnetic field $2, whose position in space at a given moment of time is represented by the vector 2 in Fig. FIG. 106. FIG. 107. FIG. 106. Three-phase series A. C. commutator motor with series transformer- Brush shift angle a, A-Connection in stator and rotor. FIG. 107. Space diagram of fluxes in series motor with brush shift angle a. Heavy lines indicate space vectors. 107. At the same moment of time the same current produces a magnetic field $1, whose position in space is indicated by the vector 3>i. -The vector difference of 3>i and $2 is the resultant really existing rotating magnetic field F. With a = 0, there is no torque; with brush shift clockwise or negative, we obtain clockwise rotation; with brush shift a counter-clockwise or positive, we obtain counter-clockwise rota- tion of the armature, the direction of rotation reversing with the shift of the brushes. The torque is exerted in the direction of the SERIES POLY-PHASE COMMUTATOR MOTOR 139 brush shift from the defined datum a = 0, as is indicated by a simple consideration of the magnetic fluxes i and $2 and their mutual attraction. 1 Still assuming no leakage, we know that F induces in stator and rotor windings e.m.fs. which are in time quadrature with the resultant flux which embraces the windings. As the e.m.f. induced in the rotor windings must ap- pear earlier in time-phase than that induced in the stator windings, with a negative, or later with a positive, it follows that the e.m.fs. induced in the stator and rotor differ in time-phase by the same time angle a as do the space fields 3>i and < 2 by the same space angle a. Assume the resultant flux F to be pro- jected on a vertical time axis for refer- ence (Fig. 108). Assume it to be zero at a certain time. Then the voltage E a in- duced in the stator winding of the series motor is, barring leakage and resistance, in quadrature with the resultant field F \ and, as the current / is in time-phase with \ the fictitious flux $1, we now have the . \ essential elements for the determination of the complete vector diagram of the FlG i 08 .- Combined poly-phase Series motor. space and time diagram of Assume the rotor and stator resist- X^^cotr ances zero, and the rotor standing still, motor with brush shift The fictitious fluxes $1 and 2 differing angle a ' in space phase by the angle a, produce the resultant real field F. I is in time-phase with 3>i, and E a is in time quadrature with F. Thus the time lag \f/ a of the current / behind E a is determined. E a and Eb are in time-phase opposition if the brush-shift angle a = 0. They differ by the angle a in time-phase. We thus obtain E as the resultant voltage impressed upon the motor, in time quadrature with J, in the case of a resistance-less imaginary motor. This relation follows from the similar m.m.f., or flux, and e.m.f. triangles. Assuming / constant and the motor beginning to turn in the 1 See V. KARAPETOFF, "The Secomor," Trans. A. I. E. E., Feb. 16, 1918. 140 INDUCTION MOTOR direction of its rotating field, then the counter e.m.f. of rotation induced in the rotor diminishes proportionally to the slip. It disappears at synchronism. An examination of the diagram Fig. 108 shows that, below synchronism the stator takes energy from the line, while the rotor delivers energy back to it. At synchronism the stator alone takes energy from the line ; above synchronism, it is seen that both stator and rotor take energy from the line, the sum being trans- formed into mechanical energy. Such a motor is therefore described as "doubly-fed," a term widely used in the great work of the joint authorship of E. Arnold, J. L. la Cour, and A. Fraenckel. We have already seen that the time- phase of Eb depends upon the brush shift a which determines the space angle of the m.m.fs. To emphasize this important point once more, for a = 0, E a and Eb are in time-phase oppo- sition, Fig. 109. Shift the brushes clockwise, or backward in the direc- tion of the rotation of the magnetic field, and the maximum of Eb will occur earlier in time-phase by the angle a, which is now a time angle; FIG. 109. Brush shift and field rotation affect the time diagram. shift the brushes counter-clockwise, or forward in the direction of rotation of the magnetic field, and the maximum of Eb will occur later in time-phase by the angle a. We shall consider only this latter case in which the rotor and the magnetic field rotate in the same direction. Still neglecting leakage and resistance, we re-draw (Fig. 110) the time diagram of the e.m.f s. and of the current and we see at a glance that for constant current the e.m.f. triangle is OBC, in which OB is the e.m.f. induced in the primary or stator winding, BC is the e.m.f. induced in the rotor at slip s, and it is therefore under our previous assumption of equal numbers of turns equal to E a . The point B, therefore, corresponds to synchronous rotation and the range BA corresponds to speed above synchronism. The angle a at B remains constant for a full speed range, and so does the angle a \l/ a at A. If, therefore, we were to keep OC = E constant, as well as a, allowing the current to vary, it is SERIES POLY-PHASE COMMUTATOR MOTOR 141 obvious that A lies on the periphery of a circle described about OC as chord. The current 7 is always proportional to OA, therefore, I may be measured by the chord drawn from to the periphery of the circle. (Fig. 111.) The Torque. The current at starting is A S h, while the torque at starting is proportional to the product of the space quadrature component of the secondary m.m.f. into the magnetic flux F, or FIG. 110. "D," Starting; "B," Synchronism; "A," Double Synchronism. Time-phase diagram for the locus of the primary current in the series poly-phase commutator induction motor for constant potential. Currents measured from to periphery of circle. Torque = K-IF- sin (126) But in our diagram F is proportional to and in time quadrature with E a , and E a is also proportional to 7. Therefore, a Torque = Kil 2 sin ^ (127) The torque, therefore, may be represented by the square of the current and, as we have seen on previous occasions, the square of the chords in a circle may be measured graphically perpendicular to the diameter of the circle as shown in Fig. 112 by T. 142 INDUCTION MOTOR The Slip. Drop a perpendicular from A ay , the current locus at synchronism, upon the diameter of the circle OA Q . (Fig. 113.) The point of intersection a between OA and A sy S, cuts FIG. 111. Circle OAshA is locus of current I for constant E = OC. The current and the e.m.f.'s in the ideal series motor for constant potential. FIG. 112. The torque in series poly- phase commutator motor. OC = E = primary impressed voltage. For ^i = 0, OC = I and T is corresponding torque. FIG. 113. The slip in the series poly-phase commutator motor for constant potential. off A SJ/ a, which is a measure of the slip. Point S corresponds to 100 per cent slip, point A 8y to a slip per cent. SERIES POLY-PHASE COMMUTATOR MOTOR 143 Proof. Triangles ObC and OA sy a are similar. The slip is the ratio of Cb : Ob, therefore, Cb :0b ::aA sy : OA sy . aA 8V OA sy (128) At double synchronism, as is shown by inspection of the dia- gram, the power factor becomes unity. B. THE THEORY FOR CONSTANT CURRENT AND CONSTANT POTENTIAL IN THE REAL MOTOR The diagram of m.m.fs., or fluxes, remains the same if we take the leakage into consideration by using the e.m.f. induced by the FIG. 114. The time-phase diagram of the series poly-phase commutator induc- tion motor. Including resistance and leakage. leakage field which is the simplest method in this case as we must combine e.m.fs. affected by the speed. In the theory of the slip- ring or squirrel-cage induction motor, we found it simpler to employ leakage fields instead of the voltages induced by them. 144 INDUCTION MOTOR The leakage reactance of the stator windings is constant at all speeds. The leakage reactance of the rotor windings, as shown in Chap. IX, is composed of a part constant at all speeds and of another dependent upon the speed of rotation of the armature. This latter is positive at speeds below synchronism, zero at syn- chronism, and negative at speeds above synchronism. Without a serious error we may permit ourselves the license of viewing the total leakage reactance of the motor as constant, and in Power Torque 100% 80% 1 .8 .6 .4 .2 Slip FIG. 115. The torque, current, power factor and slip in the series poly-phase commutator induction motor. quadrature time-phase with the current, Fig. 114. Adding vec- torially to the e.m.f. thus obtained /(ri + r 2 ), a point G is derived at the intersection of OA and the line OH through H. The angle (a \f/a) at G is again constant. If we assume again constant primary voltage OH for the entire operating range, a circle on whose periphery lie the points 0, G, and H becomes the locus of a radius vector from 0, like OG, which is a measure of the current /. SERIES POLY-PHASE COMMUTATOR MOTOR 145 All other relations follow from this diagram as before. We have plotted in Fig. 115 in rectangular coordinates the current, the power factor, the torque, and the power as a function of the rotor slip. At speeds above synchronism the power factor approaches unity. It may be suggested to the reader to draw similar diagrams for different brush-shifts a. C. THE NECESSITY OF SATURATION FOR STABILITY The series poly-phase motor is self-exciting as a generator. If it possesses remnant magnetism, it may generate direct cur- rent as the supply circuit forms virtually a short circuit for such currents. Saturation of the motor frame or of the series trans- former may prevent these effects to a great extent. On this subject the reader may consult the following: U. S. Patent 1,164,223, Dec. 14, 1915, A. SCHERBUJS: Stabilized Commu- tator Machine. E. ARNOLD, J. L. LA COUR and A. FRAENCKEL, Die Wechselstromkom- mutatormaschinen, 1912, p. 59. V. KARAPETOFF, "The Secomor." Trans. A.I.E.E., 1918, Vol. XXXVIII, Part 1, p. 347. W. C. K. ALTES, "The Poly-phase Shunt Motor." Trans. A. I. E. E., 1918, Vol. XXXVII, Part 1, p. 385. 10 CHAPTER XI THE SHUNT POLY-PHASE A. C. COMMUTATOR MOTOR A. HISTORICAL INTRODUCTION The shunt poly-phase commutator motor as well as the series type appear to be the invention of H. Goerges who described them in the E. T. Z., 1891, p. 699. The 10 years which succeeded its invention were devoted to the practical development of the Tesla induction motor and thus the commutator type did not receive much attention. Exactly 10 years after Georges' publication, A. Heyland described again (in the E. T. Z., 1901, No. 32), the Goerges motor, showing that it can be used to compensate the watt-less component of the primary by proper brush-shift. Although Prof. Blondel contends 1 that this was a matter of course, it would have been an interesting contribution had not Mr. Heyland claimed with great emphasis that his motor was entirely different in principle from the shunt motor of Goerges. This has been disproved with great precision and clarity by Prof. Blondel in the papers cited. Mr. Heyland had suggested the use of stationary sliding contacts on the squirrel-cage rings, thus introducing an external e.m.f. at points equally spaced on the commutator, but the low resistance of these end rings acts as a powerful shunt and this arrangement proved ineffective. The author tried two independent windings with somewhat better success, and tests on a similar motor are reported by Prof. C. A. Adams. 2 It is now no longer open to doubt that Mr. Heyland's suggestion covers solely a shunt poly-phase motor with addi- tional shunts placed between the commutator bars. No practical application seems to have been made of this modification. Great activity in devising modifications and improvements of poly-phase commutator motors followed the general enthusiasm created by Mr. B. G. Lamme's single-phase railway motors. The use of these motors in order to obtain speed regulation without un- due loss in efficiency, and finally their application to the speed regu- 1 ANDRE BLONDEL, Theorie des Alternomoteurs Poly-phas6s & Collecteur. L'Eclairage Electrique, 1903, pp. 121 to 495. 2 C. A. ADAMS, Trans. A. I. E. E., 1903. 146 SHUNT POLY-PHASE A, C. COMMUTATOR MOTORS 147 lation of large induction motors with which they are concatenated, has secured for the Goerges motor a wide and interesting field. In order to obtain the appropriate voltage on the rotor of a Goerges shunt poly-phase A. C. commutator motor, it is necessary either to use a separate transformer, or to utilize the stator winding by means of taps, or to employ a regulating winding. These methods are treated in Chap. XIII where the work of Osnos, La Cour, and Schrage is given consideration. B. THE THEORY OF THE SHUNT POLY-PHASE A. C. COMMU- TATOR MOTOR FOR CONSTANT POTENTIAL Assume a stator like that of a standard induction motor in which a rotor is mounted wound like a direct-current armature equipped with a commutator. Let poly-phase current be supplied to both the rotor and the stator from the same supply circuit. We thus obtain a " doubly- fed'' type of poly-phase motor, which is called a shunt poly-phase A. C. commutator motor. The two e.m.fs. in the stator and rotor, being derived from the same supply circuit, are of the same frequency and in time-phase. If the rotor-brush position is such that the entries on both primary and secondary are opposite each other, then the m.m.f. belt of the sec- ondary in relation to that of the primary depends solely upon the time-phase of the two circuits. If the rotor brushes are shifted the brush-shift angle displaces the impressed e.m.f. of the rotor relative to the stator by the amount of the angle of shift. Thus, in Fig. 116, let E\ be the impressed e.m.f. on the stator, then, if a is the brush shift, # 2 , the impressed e.m.f. on the rotor appears displaced by the angle a relative to the impressed e.m.f. EI so far as space relations are concerned. That is to FIG. 116. Composition of e.m.f's. in the rotor of the shunt poly-phase com- mutator motor. Ei = e.m.f. impressed on stator. Ei = e.m.f. impressed on rotor. 62 = e.m.f. induced in rotor. . 148 INDUCTION MOTOR say, the m.m.f. belts due to the currents which are produced by EI and Ez, must be determined as though, in a stationary trans- former, the two e.m.fs. were displaced in time-phase by the angle a. If we assume for the moment no secondary leakage, then the resultant flux F induces an e.m.f . e 2 through the relative angular velocity o>i co 2 of the rotor in respect to F, and the vector sum of E 2 and e 2 results in E s , which produces a current which would be in phase with Es if there were no leakage in the secondary. If there is leakage, /2 would lag behind E 3 by a time angle ^ 2 - m FIG. 117. Time-phase vector diagram of the shunt poly-phase commutator induction motor. Secondary current Is is composed of currents .BCdue to e 2 and AC due to Ei. I\ = OB, I 3 = BA, i = OA. The phase relation of EI and E% depends solely upon the brush position, if they are derived from the same supply circuit. It is difficult but important to bear in mind that we are tracing rotating fields in space and that time and space-phases are utilized in the same diagram so that we can show the mutual effect of rotor and stator m.m.fs. in one diagram instead of in two. This may be confusing, but the other treatment loses in physical reality. SHUNT POLY-PHASE A. C. COMMUTATOR MOTOR 149 The m.m.f. of the rotor dueto/ 3 andthem.m.f. of thestatordue to 1 1 result in the magnetizing m.m.f. equivalent to i in the stator windings (Fig. 117). Assuming constant secondary reactance and neglecting as a good approximation that part of the secondary reac- tance which is proportional to the slip, / 3 is composed of BC due to e 2 , and of AC due to E^. Point C, therefore, is a fixed point as long as the brush-shift angle a and E 2 remain constant. Point m also is a fixed point, Od : dm being equal to vi 1, as is readily seen. Angle CBm is equal to ~ + ^2 and constant so that point B moves on the arc of a circle described over mC. The circle described about 0' as center is the primary locus of the commu- tator-induction motor with short-circuited brushes. The circle described about 0" as center is the locus of the stator current of the shunt poly-phase motor, to which has to be added, or from which has to be subtracted, vectorially, the current in the pri- mary of the transformer feeding the rotor. (Fig. 118.) C. DETERMINATION OF THE TOTAL PRIMARY CURRENT With the limiting assumption of a constant secondary lag \f/ z between the secondary total e.m.f. E% and the total secondary current Is we may now proceed to determine the total current taken from the line supplying both stator and rotor. (Fig. 118.) The stator current is OB. The rotor current is Bdv\. This current being fed into the rotor at the voltage E 2 which is smaller than EI, if the ratio of transformation is n, then the current to be added to the stator current on account of the current / 3 fed into the rotor from # 2 is , to be added to /i in such a manner that, as outside the motor Yl E z and EI are in time-phase, the phase lag between 7 3 and EI must be the same as that between 7 3 and E 2 . Thus results a simple graphical method 1 which sets off Bh = at the constant 71 brush-shift angle a, triangle dBh for all secondary currents being similar, angle dBb always being a and angle Bdh being c, also constant. Thus draw dO", make triangle dO"O f " similar to 1 A. BLONDEL, UEdairage Electrique, 1903, p. 178. 150 INDUCTION MOTOR triangle dBh, and 0'" is the center of the new circle for the total current. This total current, as was to be expected in view of the double feeding of this motor through its primary and secondary, is smaller than the stator current over the motor range of the shunt machine and it may be seen at a glance also that the rotor returns energy to the supply circuit. The whole arrangement Stator E FIG. 118. Time-phase vector diagram of the shunt poly-phase commutator in- duction motor. Circle loci of the stator current and of the total current. may be simulated by a sort of equivalent arrangement of e.m.fs., resistances, and reactances, as is shown in Fig. 119, where EI pi and - - are mechanically coupled together, spaced apart by a s space angle a. In order to obtain similarity of current and e.mJ: . relations, it is necessary to divide Ez by the dip s. SHUNT POLY-PHASE A. C. COMMUTATOR MOTOR 151 D. SPEED REGULATION AND THE SLIP It is evident that, since BC, Fig. 1 17, represents the secondary current due to e% only, and as 62 is proportional to the product of the slip into F, tgCmB is a measure of the slip. By changing E 2 and a any slip may be obtained and thus the poly-phase shunt motor acts in this respect entirely differently from the induction motor, the shunt motor having three degrees of freedom, while the induction motor has only one. 1 A counter e.m.f. may be injected into the rotor of such magnitude and phase that the L s FlG. 119. Equivalent electrical and mechanical combination simulating the action of the shunt poly-phase commutator induction motor. motor will run, say, at 20 per cent slip, of which say 4 per cent is ohmic drop, while the remaining 16 per cent is due to the in- jected e.m.f. which may be made in phase with the ohmic drop. However, incidentally, the phase of the injected e.m.f. may be shifted so as to magnetize the motor and thus to supply through the secondary the magnetizing current ordinarily supplied through the primary. As this can be done with much less K. V. A. due to the low voltage of the rotor caused by the high slip, it appears obvious that such a motor may have a very high power factor. 1 This likeness to problems in dynamics is due to PROF. V. KARAPETOFF. 152 INDUCTION MOTOR E. BIBLIOGRAPHY Before leaving the subject of these interesting motors, a short list of papers may be given chronologically. The motor was described in E. T. Z., 1891, p. 699, by H. GOERGES, by whom it was also patented Jan. 21, 1891, in the German patent No. 61,951. It was brought back to light 10 years later largely through the sensational paper by A. Heyland, E. T. Z., 1901, No. 32, in which the author described the same doubly-fed motor, calling it, however, an induction motor excited from the secondary. Through shifting the brushes any primary-phase angle may be obtained. No tests were made or described. A fundamental advance was made in the theory of these motors by PROF. A. BLONDEL, in L'Eclairage Electrique, Apr. 25, 1903 et seq., where, however, a curious assumption was made. PROF. BLONDEL assumes that to the secondary impressed voltage ' 2 the rotor offers resistance only. ET He thus composes and iz into /a and then applies to this current the flux theory of the induction motor assuming a secondary leakage field to be produced by this m.m.f. His results are thus marred by this assumption, which also implies that the secondary leakage lag diminishes with the slip of the motor, which is at best only partially true and which leads to the establishment of a specious equivalence between the squirrel-cage or slip- ring motor and the commutator-induction motor short-circuited across its brushes, a result which the tests do not seem to bear out. BLONDEL is thus led to semi-circular loci where we have arrived at arcs. The value of BLONDEL'S methods is fortunately in no way impaired by these assumptions. Three months after the appearance of the BLONDEL circle diagrams of this motor in Apr. 25, 1903, MR. HEYLAND published in the E. T. Z. No. 30, July 23, 1903, a diagram identical with BLONDEL'S diagram in spite of the curious assumption made by BLONDEL. No reference whatever appears to have been made by MR. HEYLAND to BLONDEL'S papers here referred to. An exhaustive study of the shunt motor and the derivation of a cor- rect diagram appeared in the E. T. Z., 1903, p. 368 et seq., by PROF. O. S. BRAGSTAD. M. EDOUARD ROTH, of Belfort, France, published a masterly thesis in L'Eclairage Electrique, April to June, 1909. The work of DR. E. KITTLER and DR. W. PETERSEN, Stuttgart, F. ENKE, devotes a great deal of space to these motors. Vol. V, Part 2, of E. ARNOLD, LA COUR and FRAENCKEL, Berlin, J. SPRINGER, 1912, is a mine of valuable information on alternating-current commutator machines in general. DR. F. EICHBERG'S "Gesammelte Elektrotechnische Arbeiten," 1897- 1912, Berlin; J. SPRINGER, 1914, may also be consulted. The papers by L. DREYFUS and F. HILLEBRAND in Elektrotechnik & Maschinenbau, 1910, pp. 367 et seq., may be consulted with profit. The latest contributions are the papers by N. SHUTTLE WORTH, "Poly- phase Commutator Machines and their Application," The Journal of the Institution of Electrical Engineers, Mar., 1915, arid the paper by W. C. K. ALTES, "The Polyphase Shunt Motor," Trans. A. I. E. E., 1918. CHAPTER XII METHODS OF SPEED CONTROL A. CONCATENATION The similarity in theory between an induction motor and a transformer is due to the fact that, as the secondary frequency in the rotor of the induction motor varies from full primary fre- quency at standstill to zero at synchronism, the constant resist- ance of the rotor is in effect equivalent to a variable resistance r-2, -5- s in the secondary of a transformer, where r% is the second- ary resistance of the rotor and s the slip, viz., the difference between primary and secondary frequencies divided by the primary frequency. Thus we obtain the conception of the equivalent circuits which simulate the physical phenomena of magnetizing current, leakage fields, etc. If the induced e.m.f. in the secondary of one motor at the frequency of its slip is impressed upon a second motor (we will assume it to be impressed upon the stator with a winding having a number of conductors equal to that of the rotor of the first motor), then the second motor will not operate like a standard induction motor of constant impressed voltage and constant frequency, but both its impressed voltage and frequency will vary in a peculiar manner. If the rotors of both motors are mounted rigidly on the same shaft, then they have a common mechanical angular velocity o) 2 . We thus obtain the following relations assuming the same number of poles in both motors ; and designating the angular velocity of the primary field of Motor I by coi, its slip by 81, and the slip of Motor II by s n . Sl =^^- (129) 0)1 But o) 2 = o)i(l - si) (130) The relative angular velocity at which the rotor conductors of Motor II cut through the field impressed by the e.m.f. of the secondary of Motor I is on o) 2 w 2 = i 2co2. Therefore, the slip of Motor II, 20) 2 0)1 153 /1Q1\ (131) 154 INDUCTION MOTOR And, because of equation (130), coi 2coi(l Si) - - - /iQ 3 . As the frequency of F 3 is (~i ~ 2 ) we have 6 3 = 2.12(si)~iZ 3 F 3 10- 8 (145) Therefore itfz : e 3 ::/3 r :F 3 K (146) i where K = 1 ^- 2.12~iz 3 10- 8 (147) Hence, / 3 r may be represented by a vector proportional to the ohmic drop divided by the slip Sj. It is now clear that with si = -0, fs r becomes infinite, which is the equivalent of a com- pletely open circuit of the primary of Motor II. The impressed e.m.f . of Motor II is generated by the secondary of Motor I. The relation of its phase to its current is determined by the consideration that its current is the same as the current in the primary of Motor II and its phase is the same as that of the voltage impressed on the primary of Motor II relative to its current. The current in circuits II and III being the same, the leakage fields and resistance drop fields must also be the same. Hence Og = gh = // = fc and bd = di = f 3 = fz The actual resultant magnetic field in the secondary of Motor I is now represented by hi = F 2 . The fictitious flux 3> 2 corresponding to the secondary m.m.f. of Motor I is represented by Oc = im, while hm = $1, the pri- mary fictitious flux proportional to t\. The primary leakage flux is ei = /i, and hi = F\ is the primary resultant flux generat- ing a counter e.m.f. 61 = 2.12 iZi^ilO- 8 volts (49) whose phase is in quadrature with ii and $1 or hm. An examination of the diagram shows at a glance the vectorial composition of the secondary flux ^4 with the leakage fluxes /4, /a, /2 and /i, and the resistance drop fluxes / 3 r and/ 2 r , into the resultant primary flux FI. It is most interesting and instruc- METHODS OF SPEED CONTROL 159 tive to note that this composition can take place actually in the same motor in the case of internal concatenation, which can most readily be realized by two windings with numbers of poles in the ratio of 2 : 1, as in this case the windings are mutually inde- pendent in respect of mutual induction. A similar case of great theoretical interest is the case of a poly-phase motor with a single- phase secondary which will be discussed in Chap. XII B. The diagram of the composition of fluxes, Fig. 122, neglects as usual the primary resistance of Motor I, which can be taken into consideration by the simple graphical correction given in Chap. Ill, if such correction should prove desirable. It is not necessary, in order to determine a number of points for different speeds to develop more than the flux polygon 0-a-b-d-i-e-h. The leakage flux ab = / 4 is proportional to the secondary current of Motor II. The ohmic drop ur^ is propor- tional to the flux Ft multiplied by the secondary frequency of Motor II, viz., (~i - 2~ 2 ). Therefore, equation (138) e 4 = z > 4 = 2.12(sis n )~iZ 4 /< T 4 10- 8 volts (148) t 4 = #4$4 (149) /4 = (04 - .'.t| = K 4^J (150) From (148) follows: #4/4 - 1)2.12' (152) Thus siSu being known from (152), we obtain The procedure is now as follows : Assume F* and Si ; calculate from which we obtain / 4 ; determine F A in the usual manner, eb being equal to (1 -- )$ 4 . The direction and magnitude of t>3 the primary current of Motor II thus being known from / 3 , determine the resistance drop fields Og = / 3 ' and gh = / 2 ', thus 160 INDUCTION MOTOR obtaining F 2 . The leakage field / 2 = di, and the point k are then determined as before, ik = (1 )< 2 , and hi - hk = vi, obtaining FI = hi, in quadrature with which we find e\. The angle of lag is \f/i, between $1 and e\. The construction of this diagram has been carried out for a number of values of the slip i and the results have been plotted in Figs. 123, 124, 125, 126, and 127. The torque of Motor II is proportional to the vector product of u and F 4 , it is therefore represented by the vector product of FIG. 123. Concatenation: Flux diagram motor II as generator. Leakage and resistance taken into account. F 4 and /4. The torque of Motor I is proportional to the product of the quadrature component of i 2t viz., the component md of the leakage flux / 2 into the field F 2 . This calculation has been carried out and the values of the two torques have been plotted in Figs. 125 and 126. It has been customary, in popular theories of concatenation, 1 to assume that the torque of Motor I follows, through the entire range of slip from s I = I to s I = 0, the torque curve of an ordi- 1 See F. EICHBERG, Zeitschriftfiir Elektrotechnik, Vienna, December, 1898. Also B. G. LAMME, Electric Journal, Pittsburgh, September, 1915. METHODS OF SPEED CONTROL 161 With Resistance Fio. 124. The fluxes of concatenated motors. Leakage and resistance taken into account motor II acting as generator. 2000 2000 FIG. 125. The torque curves of two concatenated motors. T\ is the torque of motor I. Tz is the torque of motor II. T\ + Tz is the resultant torque. 11 162 INDUCTION MOTOR nary induction motor at constant potential, while the torque curve of Motor II is assumed to be that of a constant potential motor whose synchronism is reached at half the synchronous 2000 G'enert'r Indicated by DIAGRAM OF FLUXES MOTOR tg P=^ FIG. 126. Concatenation: The torques and the loci of the primary current' Leakage and resistance taken into account. speed of Motor I operating by itself, in Fig. 128. It is clear from our analysis and also from the consideration of the fact that METHODS OF SPEED CONTROL 163 the torque of Motor I for s l = 0.5 must be zero, that this plausible explanation is incorrect, and misleading in regard to the physical aspects of this problem. 3 V r 3= = .21 10 FIG. 127. Concatenation: The primary current locus of two concatenated motors with leakage and resistance taken into account. FIG. 128. Convential but incorrect method of representing the torque of two concatenated motors. (3) Torque of one motor operating at double slip. (2) Torque of other motor operating at normal slip. (1) Torque of the con- catenated group. It is necessary to consider carefully the effect of the second motor on the characteristics of the group. Above half synchron- 164 INDUCTION MOTOR ous speed, Sj = 0.5, Motor II runs above its synchronous speed relative to its supply frequency ^i ^ 2 , and therefore acts throughout its range to s, = as an induction generator. Yet the effect of the resistances r 2 and r 3 consists in changing over the torque of Motor I between s r = 0.3 and s t = 0, from a gen- erator torque into a motor torque. This point is so important that we give in Fig. 123 and 124 a complete polar diagram for this condition, which should now be self-explanatory. FIG. 129. Concatenation: Resistance only and no leakage. as motor. Motor II acting From Fig. 125 it is evident that the torques of both motors are motor torques up to s t = 0.5, while from s t = 0.5 to s x = 0.3 both torques are generator torques. Between s I = 0.3 and s z = Motor I is a motor, while Motor II remains a generator. At negative slips the group acts individually and collectively as a generating unit. Figure 127 shows the polar diagram of the primary current of Motor I. The locus of this curent is a curve of the fourth power. The slip s r is marked everywhere and it is interesting to note how the current from standstill gradually diminishes to half-synchronous speed, as in a single ordinary induction motor. The magnetizing current at half -synchronous speed is almost double that of a single motor. Above this speed the group acts like an induction generator until, as a result of the effect of the secondary resistance of Motor I and the primary METHODS OF SPEED CONTROL 165 resistance of Motor II above s I = 0.21 the group becomes a motor, and the primary current, while at first increasing, is gradually choked off by these resistances until &t s I = their FIG. 130. Concatenation: Resistance only and no leakage. Motor II acting as generator. 1 2 r 2 _ 12.5 X.I Sf -.15 = 8.35 FIG. 131. Concatenation: Resistance only, FIG. 132. Concatenation: no leakage. Resistance only, no leakage. effect becomes equivalent to an open circuit of the primary of Motor II and the primary current of Motor I drops to the value of its magnetizing current. To bring out this very interesting but somewhat involved 166 INDUCTION MOTOR process more clearly, we shall consider two specific cases. First, two equal motors in concatenation, without leakage, but with resistance r 2 and r 3 ; and, secondly, two equal motors with leakage but without resistance r 2 and r 3 . Neither case corresponds to the POLAR DIAGRAM CONCATENATION RESISTANCE ONLY ,=.2 FIG. 133. Concatenation: Resistance only, no leakage. Locus of primary current. case treated fully in this chapter, but as " boundary" solutions, they afford a good insight into the operation of concatenation. We assume r 2 = r 3 and draw F*. (Figs. 129 and 130.) Deter- mine $4 = bd as before, and draw ad = 2 = $3. The resistance drop field ft = ft is shown as be = 2/J. Join ac = F z = F s and METHODS OF SPEED CONTROL 167 Qa = FI = Fa. The primary voltage e\ is in quadrature with FI, while Od = $1 represents the phase and magnitude (including a proportionality factor) of the primary current i\. Angle bad = 0, and tgO = $4 -r- F4 = s^. The resistance drop field be 2/J is also equal to VT (154) where X is a proportionality factor readily obtained as before. Figure 133 shows the polar diagram of the primary current for different slips. Its relation to the fourth degree curve of Fig. 127 is striking. Figures 131 and 132 are auxiliary diagrams used in the development of the diagram. As in the ordinary induc- tion motor without leakage, the branches of the curve are asymptotic to the ordinate axis. CONCATENATION OF Two EQUAL MOTORS Resistance Only No Leakage I tgP COS \f/i ii 2.000 7.500 0.964 143.5 1.000 2.500 0.785 65.5 0.900 2.000 0.706 59.0 0.800 1.500 0.620 54.0 0.700 1.000 0.480 50.0 0.600 0.500 0.300 48.5 0.500 0.000 0.093 49.0 0.400 0.500 -0.105 56.0 0.300 -1.000 0.200 72.5 0.250 -1.250 -0.166 87.0 0.200 -1.500 -0.895 106.0 0.175 -1.630 0.340 106.0 0.150 -1.750 0.506 94.6 0.125 -1.877 0.690 74.5 0.100 -2.000 0.700 53.5 0.050 -2.250 0.530 30.8 0.000 0.000 0.000 25.0 -0.100 -3.000 -0.535 30.6 -0.200 -3.500 -0.760 44.0 -0.300 -4.000 -0.820 53.0 -0.600 -5.500 -0.920 78.5 -1.000 -7.500 -0.960 110.5 168 INDUCTION MOTOR Figure 134 shows the current curves in polar coordinates for this limiting case, and Fig. 135 the equivalent electric circuits which are always instructive. FIG. 134. Primary current locus. Concatenation. Resistance only. Num- erals represent slip sj. Secondly, we shall now consider the action of the group of the two motors if the resistances r 2 and r 3 are neglected, the leakage, however, being fully taken into account. This is a very im- portant case as it will be shown that, for a range of slip from METHODS OF SPEED CONTROL 169 ! = 1 to s x = 0.4, which is the important range for practical purposes, this diagram becomes very simple, the locus of the primary current being again a circle. We begin again with the secondary flux, F-* of Motor II. It induces an e.m.f. e 4 = 2.12(s I s II )~iZ 4 /' 7 4lO- 8 volts (155) WV AM, FIG. 135. Concatenation: Equivalent circuits resistance only; no leakage. This e.m.f. produces a current * 4 = - (156) This current produces a leakage field / 4 = ab = (v z 4 and F 4 are combined into $3 and fa = be is the leakage field in magnitude and phase of the primary m.m.f. of Motor II. UL FIG. 136. Concatenation: Equivalent circuits leakage reactance only; no resistance. As our premise was the assumption of a negligible resistance of the windings between the two motors, see Ffg. 134 which assumption we know to be permissible only if the slip s r is not less than 60 per cent, we can directly combine the field F^ which is the resultant primary magnetic field of Motor II, with its secondary fictitious flux 3>i, and we obtain the fictitious primary field whose leakage field is /i = mu. 170 INDUCTION MOTOR The diagram is developed exactly as the flux diagram of the induction motor, and every step should be carefully thought over by the reader. (Fig. 137.) An examination- of this geometrical figure now shows the interesting and remarkable fact that, with constant impressed voltage on Motor I, neglecting its primary resistance which is permissible, the point of intersection / remains fixed for all speeds of Rotor II, and the primary flux 3>i of Motor I moves under this condition upon the periphery of the circle around D as center This can readily be proved as follows : FIG. 137. Flux diagram of two motors in concatenation. No primary and secondary resistance in motor I, and no primaary resistance in motor II. Draw as and sd and obtain an expression for sf = sd + df. (157) ed:df::F 2 :2F 2 .'. df = 2ed nd = $2(^1 1) dm = $2(^2 1) mk d = C|> 2 -f hi 111 = nd -f dm + mk . Cl = cl = (158) (159) METHODS OF SPEED CONTROL 171 X = [fli + w - -1 (160) v\j i:kl: :d:cl (161) (162) if = kl 2ed ty = * 4 (2-l) (163) a6 = * 4 (02 1) ar:ds:: : .: ds = ZQ^VM - 1) (164) df = 2ed = 2*4(1 - p) . (165) .-.*/= 2*4 i0s - (166) This equation shows that s/ is proportional to * 4 and it is there- fore a measure of $4. 0/:ce::2:l .-.0/ = 2ce ce:ck:: ei : kl F 1 .vi . ce: : : 4 - : * 4 Vi A ~ (167) A But Of =2ce /. O/ = 2^ (168) A 172 INDUCTION MOTOR From similar triangles: Qf:D:i8f:if %->* Of 2aifaX - D (169) Without Resistance FIG. 138. The fluxes of concatenated motors. Leakage only (not considering resistance). The group acting as generator. The Fig. 137 is drawn for vi = v 2 = 1.1, and FI = 49 and therefore X = vi + v z -- 01 X = 1.29 = 76 J'= 1.016 D = 74.8 METHODS OF SPEED CONTROL 173 Figure 138 shows an auxiliary diagram in which both motors are operating as induction generators. Figure 139 shows a comparison of a single motor, as one of the concatenated group, with the concatentated group, with stray coefficients vi = v 2 = 1.04, or a = .082. FIG. 139. Concatenation: The influence of leakage. Approximate primary current loci of group and of single motor. Vl =vz = 1.04 Figure 140 shows the same groups of motors with stray coeffi- cients Vi = v z = 1.1, or 2 ) may be looked upon as the impressed m.m.f. belt of a motor in which the rotor is the primary and in which the stator, short-circuited as it were through its supply circuit, forms the secondary. Thus we obtain the identical conditions of internal concatena- METHODS OF SPEED CONTROL 175 tion, and the general theory of concatenation may with propriety be applied to this subject. For a similar statement of the theory of this phenomenon, see especially E. Arnold, "Les Machines d'Induction," Paris, Ch. Delagrave, 1912, p. 184, where an experimental diagram is given of the torque of such a motor as a function of the slip, see Fig. 141, which is like Fig. 125 in the Chap. XII. A. Mr. B. G. Lamme in "Electrical Engineering Papers," p. 519, has given the same general explanation. We differ, however, from him as to the propriety of deriving the resultant torque of such a motor from two constant-potential torque curves for different slips. It is evident that, at half speed, the currents in the rotor are almost totally watt-less; therefore, there can be no torque at that speed. The method reprinted by F. Eichberg in his "Collected Papers," p. 82, is open to the same criticism. The two fictitious motors of this combination do not operate, either under constant potential, or at all similarly to the standard induction motors with short- circuited rotor as it must have become clear in discussing con- catenation in the previous chapter. The reflection of low-frequency currents into the primary of the motor through induction from the second hypothetical motor whose secondary is the main primary circuit, short-cir- cuited through the supply circuit as such, leads to disturbances in the supply circuit which make a practical application of this ingenious scheme most undesirable. Reference to this is found also in Arnold's book, loc. cit., p. 185. CHAPTER XIII METHODS OF SPEED CONTROL (Continued') C. CONCATENATION OF AN INDUCTION MOTOR WITH THE COMMUTATOR TYPES FOR THE INDUCTION OF A SLIP FREQUENCY E.M.F. All the methods devised for obtaining a change in speed by means of concatenation depend upon the generation of currents of slip frequency properly injected into the rotor of the motor which is to be controlled. (a) Poly-phase commutator motors of the series or shunt or compound type may be mounted on the same shaft with the induction motor which is to be regulated. The series commutator motor may have salient poles to obtain increased stability like the FIG. 142. "Kraemer System," Induction motor mechanically concatenated to poly-phase commutator induction motor. machines of F. Lydall, A. Scherbius, and Miles Walker. This system is called the "Kraemer System." (Fig. 142.) (6) Instead of combining the regulating machine with the main motor into one mechanical unit, they may be separated as was done by A. Scherbius. (Fig. 143.) (c) A rotary converter may be mounted on the same shaft with the induction motor, the combination operating at half speed for equal numbers of poles, and direct current power be- coming available. Method proposed by J. L. la Cour. (Fig. 144.) (d) The rotary converter may be separated from the shaft of the induction motor and a d.c. machine may be controlled on the same shaft as the induction motor by this rotary converter. (Fig. 145.) 176 METHODS OF SPEED CONTROL 111 (e) The d.c. machine on the shaft of the induction motor may be replaced by an independently running set of d.c. motor and induction generator. (Fig. 146.) FIG. 143. "Scherbius System," Induction motor electrically concatenated through poly-phase commutator motor which delivers current back into the line. D.C. Excitation l.M. D.C. FIG. 144. Combination of in- duction motor with rotary con- verter. (J. L. Lacour). FIG. 145. Speed regulation of induction motor by means of rotary converter and direct connected D. C. machine. (/) Another method was devised by Ruedenberg and it is shown in Fig. 147. 12 A. Heyland and R. 178 INDUCTION MOTOR FIG. 146. Speed regulation of induction motor by means of rotary converter and separately driven direct current induction motor generator set. FIG. 147. Combination of induction motor with commutator motor and small induction motor. (Heyland-Ruedenberg). FIG. 148. Speed regulation of induction motor by means of synchronous motor direct connected and fed from frequency changer driven by small synchronous motor. (Brown, Boveri system). (Facing page 178) METHODS OF SPEED CONTROL 179 (g) The latest and perhaps principally the simplest method is described in the German patent No. 264,673, July 24, 1910, taken out by Brown Boveri & Company. It is shown in Fig. 148, where S. M. represents a synchronous motor mounted on the shaft of the induction motor, F is a frequency changer of the type first suggested by B. G. Lamme, and S is again a small synchronous motor. D. CHANGE OF SPEED BY CHANGING THE NUMBER OF POLES A great many windings have been devised to obtain different numbers of poles with one winding. Two or more entirely separate windings wound for different numbers of poles have been used in the same slots, either with corresponding rotor windings, or with a squirrel-cage rotor. By utilizing a winding of a fractional pitch, it is possible to split it in such a manner as to arrange it for two numbers of poles, and even for four numbers. These windings are ingenious but intricate and their field of application is limited. It is important to bear in mind in designing such machines that the leakage factor of a winding of twice the number of poles is very nearly twice as great. Great attention must also be paid to the magnetizing current, otherwise it might become excessive in view of the reduction of the active conductors per pole. The subject is too broad to be treated here at length, but I shall indicate one or two methods to outline the general principle and describe one of the most effective methods used to obtain this change in the number of poles. 1. The oldest method consists in the use of two or more sets of stator coils, the pitch of which is such as to give a number of windings with a different number of poles 1 for each. Such a scheme is impracticable as it is wasteful of space and material. 2. A single winding can be used which, due to different lap caused by the coil-pitch being less than the pole-pitch, may be so connected as to produce two different numbers of poles. 3. The ingenious winding of Alexanderson, 2 by which four numbers of poles, as 6, 8, 12, and 24, may be obtained with a 1 See, for instance, B. G. LAMME, U. S. Patent No. 660,909, Oct. 30, 1900. 2 E. F. W. ALEXANDERSON. U. S. Patents Nos. 841,609 and 841,610, Jan. 15, 1907. 180 INDUCTION MOTOR winding the individual coils of which are all alike but the con- nections of which, by means of 30 leads brought out from the motor, are connected through a drum controller. Six, eight, and twelve poles are obtained by the use of only 12 leads. The . SfcEStJi i <^rr;r Vvtf- -\V/'-}-| is iU~ 3l&i o\A-t?\^fr *)ter i ^^ri 11 FIG. 149. E. F. W. Alexanderson's stator winding for three or four different numbers of poles. (From U. S. patent No. 841,609, Jan. 15, 1907.) accompanying winding diagrams, taken from Alexanderson's patent, show clearly the arrangement of the circuits. (Figs. 149 and 150.) The Oerlikon Company 1 has built motors of this kind with a J See E. T. Z., 1914, No. 31. METHODS OF SPEED CONTROL 181 double rotor, one inside the other, and each motor arranged for pole-changing. The motors are so arranged that one motor drives the other and with two sets of poles on the main motor 10 20 10 20 24 Poles 24 Poles FIG. 150. E. F. W. Alexanderson's stator winding for three or four different numbers of poles. Arrows show direction of currents in the different phases per pole. (From U. S. Patent No. 841,609, Jan 15, 1907). and six sets of poles on the auxiliary motor, it can be seen that 226 + 2 = 26 different speeds can be obtained. CHAPTER XIV TYPES OF VARIABLE SPEED POLY-PHASE COMMUTA- TOR MOTORS Having explained at great length in previous chapters the principles upon which is based the speed control of induction motors, we shall consider briefly three types of motor which embody these principles in one unit. A. The Plain Shunt Motor. This motor in its simplest form is represented by Fig. 151. The voltage on the rotor brushes as well as their phase can be varied. The brushes can be shifted. FIG. 151. Poly-phase shunt motor. Without transformer and requiring mechanical brushshift. (This motor is identical with Heyland's "Compensated. Motor," excepting for the shunts between commutator bars, later introduced and abandoned.) Both speed regulation and power factor regulation may be ob- tained. This is the Goerges motor in its simplest form. Inter- posing of transformers to give different voltages and phases suggests itself and innumerable combinations can be effected. (Fig. 152.) 182 VARIABLE SPEED POLY-PHASE COMMUTATOR MOTORS 183 B.-J. L. la Cour has added a second stator winding in order to obviate the mechanical shifting of the brushes which is detri- mental, as it incurs higher harmonics which are very serious in Stator FIG. 152. Polyphase shunt motor with transformer, requiring mechanical brush shift. connection with commutation. Separate regulation of this winding through a transformer is necessary. (Fig. 153.) FIG. 153. Motor of J. L. Lacour with additional stator winding to avoid mechanical brush shift. C.-M. Osnos, E. T. Z., Dec. 11, 1902, describes a motor in which the rotor is used as the primary and the stator as the secondary. 184 INDUCTION MOTOR The rotor is built with a commutator on one side and slip rings on the other. The groups of coils between the brushes carry currents of the slip frequency, therefore, to the stationary, form- ing the secondary, then the combination lends itself directly to speed regulation, the proper voltage necessary being obtained by a pair of brushes for each phase of the secondary. (Fig. 154.) D.-H. K. Schrage, in U. S. Patent No. 1,079,994 Dec. 2, 1913, describes a similar motor in which he has added another Rotor, or Primary Winding Acting as Regulating Winding also^ \ \ Stator or Secondary 'Winding FIG. 154. The variable speed commutator induction motor of Osnos. winding, called regulating winding, which is connected to the commutator while the slip rings are connected to an independent primary winding. Through this modification of the Osnos motor greater freedom in the choice of secondary voltages is obtained. The drawback of these arrangements seems to be that, while in the shunt-motor type with stationary primary the e.m.f. short-circuited under the brushes varies proportion- ally to the slip, and disappears at synchronism, in the Osnos and Schrage motors it remains constant over the entire range at all speeds. (Fig. 155.) VARIABLE SPEED POLY-PHASE COMMUTATOR MOTORS 185 Auxiliary Regulating Winding Stator or Secondary Winding Rotor, or Primary Winding Fio. 155. The variable speed commutator induction motor of H. K. Schrage. (U. S. Patent 1,979,994, Dec. 2, 1913.) Stator Winding FIG. 156. Shunt poly-phase commutator motor with auxiliary regulating winding on stator. (Inversion of Schrage motor.) 186 INDUCTION MOTOR The Schrage motor is in reality a kind of frequency transformer like that described in Chap. IX, F. E. If a regulating winding were added to the motor described under A, but placed on the stator instead of the rotor we obtain the inversion of the Schrage motor, Fig. 156. NOTE. In connection with the subject of commutation in these motors, briefly referred to in this chapter, see the very clear exposition by B. G. LAMME, Journal, A. I. E. E., 1920, "The Alternating Current Commutator Motor." (Facing page 187) CHAPTER XV METHODS OF RAISING THE POWER FACTOR OF INDUCTION MOTORS A. THE METHOD OF LEBLANC USING COMMUTATOR MACHINES FOR SECONDARY EXCITATION To the genius of Maurice Leblanc we owe the methods for raising the power factor of induction motors by introducing e.m.fs. of proper phase and frequency into the rotor. In U. S. Patent No. 613,204, Oct. 25, 1898, he describes a method of using two or three single-phase commutator machines excited with slip frequency in such a manner that leading currents are induced in the rotor. As a matter of historical interest, there is reproduced in facsimile the illustration from the patent specification. (Fig. 157.) The theory of these interesting machines has been treated in Chap. IX, in which it was shown how a secondary phase lag or lead affects the primary current locus, and we therefore need not repeat the subject. B. THE USE OF A POLYPHASE COMMUTATOR FOR THE GENERA- TION OF LEADING CURRENTS As indicated in Chap. VIII, B, a commutator machine without excitation, fed with polyphase currents, generates at a proper speed an e.m.f. which lags behind the exciting current. Thus the arrangement of Leblanc may be replaced by a single poly- phase commutator armature without excitation, operated only at a high enough speed above synchronism to obtain the effect required. This device has been used by Leblanc, Scherbius, Brown Boveri & Company, etc., and it is as ingenious as it is simple. Its theory has been given above. An ingenious modification of this device is used by the Brown Boveri & Company. As the stator is evidently needed solely to close the magnetic circuit, it may be made integral with the rotor, without an air-gap, and the stator forgetful of its name and connotation revolves integrally with the rotor. The saturation of the iron is high so that, after high currents are reached as the result of increasing load, the compensating 187 188 INDUCTION MOTOR No. 613,204. Patented Oct. 25, 1898. tNo Model.) M. HUTIN & M. LEBLANC. ALTERNATING CURRENT ASYNCHRONOUS MACHINE. (Application filed May 4, 1807.) 4 Sheets Sheet 3. FIG. 157. The method of Leblanc for raising the power factor of induction motors. (Facsimile of the American patent specification.) THE POWER FACTOR OF INDUCTION MOTORS 189 effect diminishes resulting in a polar diagram as indicated ap- proximately in Fig. 158. Thus the power factor is practically constant and near unity over a wide range. With Leblanc-Scherbius Rotor and Saturation Primary Current without Leblanc Rotor with without Saturation with FIG. 158. The primary current of the induction motor with or without Leblanc- Scherbius rotor. C. THE METHOD OF LEBLANC INDUCING LEADING CURRENTS THROUGH RAPID OSCILLATION OF AN ARMATURE IN A MAGNETIC FIELD Maurice Leblanc described an ingenious scheme for the gen- eration of leading currents in U. S. Patent No. 644,554, Feb. 27, 1900, with the suggestion that it be used in the secondary of a slip-ring type of induc- tion motor. Figure 159 shows the principle. A coil a a is suspended between the poles N and S where it can swing freely in the mag- netic field established in the air-gap. The low-frequency alternating current upon which an e.m.f. producing a leading current is to be impressed by the oscillation of this device, called by the inventor a ' 'recuperator," tra- verses the coil a a. The stronger the magnetic field and the lighter the frame of the coil and the smaller the frequency of the FIG. 159. Device illustrating the prin- ciple of the Leblanc "Recuperator." 190 INDUCTION MOTOR current passing through the coil a a, the greater will be the effective e.m.f . due to the oscillation causing a leading current to be induced in the circuit a a. If the moment of inertia of the coil is great, lagging current may be induced. An interesting, but not very practical, method of giving concrete shape to this idea is shown in Fig. 160, taken from Le- blanc's patent. A disc swings in a strong magnetic field into which current is conducted by means of the ring R and collected at the circumference through a mercury-trough M. The theory is interesting and it is clearly stated by Leblanc and Kapp (see later). The fundamental idea consists in making a device, like a coil spring, which through its oscillations induces an e.m.f. in the circuit from which originates the forced frequency. The effect of such a coil spring is like the effect of the "recuperator." FIG. 160. A possible practical form suggested by Leblanc for his " Recuperator." In a general way it is apparent that we can devise two types of dynamic systems. One, in which a heavy mass is set into oscillation; another, in which a very small mass attached to a powerful spring, oscillates. It is shown in the theory of dynamics that the velocity of the oscillating system is a maximum in the first case when it is a minimum in the second. If, therefore, e.m.fs. can be induced by the swinging coil, in the former case it may be expected that a lagging current will result, while in the latter case a leading current will be induced. In one-half of a period, the low-frequency exciting current rises from zero to a maximum and declines to zero; during this same interval of time the magnet NS swings from its position of equilibrium in the plane YY to its extreme right position and back again to its position of equilibrium. (Fig. 161.) The maximum velocity with which the magnet sweeps by the low-frequency exciting winding is reached when the magnet passes its plane of equilibrium. A maximum of kinetic energy THE POWER FACTOR OF INDUCTION MOTORS 191 is stored in the magnet at this time which is given up to the low- frequency exciting winding during the swing of the magnet to its extreme right position. Thus, energy is transferred from the moving coil to the exciting circuit, the induced e.m.f. de- creasing from a maximum to zero while the exciting current increases from zero to a maximum. On the return swing the low-frequency exciting circuit transfers energy to the magnet which had yielded up all its energy when it reached its extreme right position. Thus, the magnet induces Excitatio FIG. 161. Principle of Leblanc "Recuperator" and Kapp "Vibrator." an e.m.f. in the exciting winding which is in leading quadrature with the exciting current. If the magnet were allowed to swing beyond the x-axis, it would operate eventually as a synchronous motor. In fact, the " recuperator" is identical with an over-excited synchronous motor. If the mass of the magnet was great and the field weak, it would act like an under-excited synchronous motor. Rotation is merely a special case of the phenomenon of oscillation. The elementary theory, omitting the effect of mechanical or electrical damping, in the oscillations of the magnet, may be given as follows : 192 INDUCTION MOTOR Let the moment of inertia of the magnet in cm units be 7. Let 6 be the angle which at any moment of time the magnetic axis of the oscillating body forms with the magnetic axis of the stator. The stator is excited with low-frequency currents obtained from the slip rings of the induction motor in the circuit of which the "recuperator" is connected. Let -JT be the angular velocity of the swinging magnet. This magnet swings in such a manner that it moves through a total angle of 40 mo * in the time T, in which a complete cycle is passed through by the low-frequency exciting currents. The mean angular velocity of the rotor is therefore immediately obvious since /rlf)\ T 4-0 f.firt. *- ~ ~ ~t/ rnn.r. Hence, 46 max -co dt/ m 27T Also, as the oscillations are assumed to take place according to a simple sine law, A (181) (182) Z = B*A dynes irRb d b -d a Numerical Example: B = 5,900 c.g.s. A = 9,350 cm. 2 A = 0.1 cm. d = 0.36 cm. 5,900 2 X 9,350 2X0.1 Z = 8 0.36 dynes 9.81 X 10 5 dynes = 1 kg. = 7,400 kg. THE MAGNETIC PULL WITH DISPLACED ROTOR 201 B. THE ACCURATE SOLUTION BY J. K. SUMEC J. K. Sumec 1 has given the integral of equation (174) and obtained an elegant solution. He starts with the author's equation (174). The vertical component is 2 - ' ' Taking points diametrically opposite and simplifying 4- cos 2 a cos a cos a 5 2 (A \ 2 / A \' 2 r /A\ 2 n !--*) (1+-C080,} [l-(-)c OS *J A cos 2 i -$-] A A / cos' ^ Substituting u = tga du = (1 -+ 1 1 cos 2 a I 1 + tg z a 1 + u 2 COS 2 adct (185) o This integral is a rational algebraic function of the form / / o a = J ~ (t) 2 (186) o . 1 J. K. SUMEC, Zeitschrift fur Elektrolechnik, Vienna, Dec. 18, 1904. 202 INDUCTION MOTOR which can be solved readily by trigonometric substitution x = Va tan (187) dx = V~a ^L (188) Va- d0 I ( a + a ^Y^e = I V COS 2 07 JQ COS 2 \/a- cos 2 0-d0 (189) = cos 2 0-d0 a |- sin 20] I -\-^= tan- 1 -4= + r-il I (190) 2alV Va a + x 2 ir 1 TT Substituting this evaluated integral in (184) 5 2 . 2A 1 (191) (192) (193) This is Prof. J. K. Sumec's very elegant result which but for the last term is identical with the author's formula (181). The factor [' - e) : is evaluated for different ratios of -r and the following table is obtained which is also taken from Prof. Sumec's paper. A 8 0.1 0.2 0.3 0.4 0.5 1 1.015 1.063 1.152 1.30 1.54 [-']" THE MAGNETIC PULL WITH DISPLACED ROTOR 203 For eccentricities less than 25 per cent the errors in using the author's formula are negligible. At 30 per cent eccentricity the error is 15 per cent, and at 50 per cent eccentricity, which is not at all unusual in induction motors, the error is 54 per cent. On this account this subject has been included in this book. BIBLIOGRAPHY The paper by E. ROSENBERG, Trans. A. I. E. E., 1918, Vol. XXXVII, p. 1417, with a full bibliography by ALEXANDER MILLER GRAY and J. G. PERTSCH, JR., entitled "Critical Review of the Bibliography on Unbalanced Magnetic Pull in Dynamo-electric Machines," should be consulted by the reader. Also the papers by B. A. BEHREND, "On the Mechanical Forces in Dynamos Caused by Magnetic Attraction," Trans. A. I. E. E. } Nov. 23, 1900, p. 617. And J. K. SUMEC, loc. tit. CHAPTER XVII THE SINGLE-PHASE INDUCTION MOTOR "The single-phase motor has been the subject of perhaps more theo- retical speculation than any other dynamo-electric machine, and the reason for this is, undoubtedly that it is in its functioning, the most complicated of all dynamo-electric machines, although in its structure it is the simplest of them all." 1 The simplified and improved transformer diagram which has served us well in the understanding of the phenomena in poly- phase motors, serves our purpose equally well in the treatment of the single-phase motor. We shall employ two methods of dealing with this problem. A. Galileo Ferraris and Andre Blondel have made use of FresneFs theorem that an alternating or oscillating field of force may be replaced by two equal and oppositely rotating fields the amplitude of each of which being equal to one-half the maximum amplitude of the alternating field. These two fields rotate in opposite directions at an angular velocity equal to 2?r times the frequency of the alternating field, assuming a two-pole field as has been done throughout this volume. Employing Fresnel's theorem, the author 2 developed the following simple vector diagram which has proved amply accurate for all the purposes of engineering applications. A two-pole rotor, revolving at an angular velocity correspond- ing to a frequency ^2 in an oscillating magnetic field whose frequency of oscillation is ^i, has relative to field I of Fresnel's two fields a slip - = s and relative to the other field II a ~i / * > " / i | ^*^9 slip -- = 2- s. ~i Consider the field II. At the great slip - the primary ^i and secondary ampere-turns act almost in space opposition. They would act exactly in space opposition if the ohmic rotor 1 E. F. W. ALEXANDERSON, Trans. A. I. E. E., Part I, p. 691, 1918. 2 B. A. BEHREND, "Asynchronous Alternating Current Motors," E. T. Z., March 25, 1897. 204 .THE SINGLE-PHASE INDUCTION MOTOR 205 resistance could be neglected. In order to simplify the under- standing of the theory of this motor we shall assume,. which is admissible without great error, that so far as the field II is concerned the rotor resistance is negligible and rotor and stator ampere-turns are in phase in space, the stator ampere-turns being just enough larger than the rotor ampere-turns to magnetize the core to the extent of producing a field which balances the voltage required to pass through the stator of field II the current the ampere-turns of which we have considered here. The primary ampere-turns of the single-phase motor have been resolved into two oppositely rotating components of one- half the amplitude. This resolution is equivalent to two poly-phase motors whose stator windings are connected in series while the rotor windings are common to both stators. The rotor being in- tegral, the rotor torques act in opposite directions. The field I tries to turn the rotor in a clockwise direction, while the field II tries to turn the rotor in a counter-clockwise direction. The motor I takes the larger share of the voltage since the apparent reactance of an induction motor is high at a small slip, and low for a large slip. Hence, as the stators of I and II carry the same current, viz., one-half the total current, the voltage impressed on the stator of motor II is very small. If we neglect in the operation of motor II the rotor resistance, it has been pointed out that the effect of motor II consists mainly in the reactance effect of this motor. The voltage impressed upon motor I is diminished by the voltage required by motor II and this voltage is always proportional to the stator current of motor I and in time quadrature with it. Therefore, the effect of motor II may be taken into account by correspondingly increasing the primary leakage of motor I. The resultant vector diagram of a single-phase induction motor may now be developed. a. THE MAGNETIZING AND NO-LOAD CURRENTS /"X^/1 -H /^M^O At synchronism, the slip of motor II is = 2. The same current passes through the stators of motor I and II. With no leakage and no resistance in the rotor and stator of motor II, the voltage impressed upon motor II would be zero and therefore, the full voltage would be impressed upon motor I. Hence, the magnetizing current would be exactly doubled. 206 INDUCTION MOTOR As the magnetizing current is proportional to the impressed voltage it is admissible to write for the case above, designating the magnetizing current of the single-phase induction motor by z' M and its no-load current by i i = 2i M + O = 2t M (195) where z' M is the magnetizing current of either motor I or II. However, taking leakage into account, we have the current 2 passing through each stator I and II. The impressed voltage fx (?,-!) tsCV (,-*>*/< Fio. 167. Vector diagram of the two-motor theory of the single-phase induction motor. (The two motors in series.) on motor I at no load is proportional to ", while the impressed voltage on motor II is proportional to 2", where z'J, 1 is the magnet- izing current of motor II. Both magnetizing currents must equal the magnetizing current of the single-phase induction motor. We shall draw a flux diagram and a current diagram to make this clear. These diagrams may be compared advantageously with the corresponding diagrams for the equivalent circuits. An examination of the flux and current diagrams reveals the following simple relations: (Figs 167, 168, 169, 170, 171, and 172). THE SINGLE-PHASE INDUCTION MOTOR 207 OK = BG = FIG. 168. The time-phase vector diagram of the fluxes of the single phase induction motor. The two motor theory. FIG. 169. The circle diagram of the single-phase induction motor. The two-motor theory (after Behrend). 208 INDUCTION MOTOR l /2-X 2 FIG. 170. The complete and theoretically exact equivalent circuits of the single-phase induction motor in the two motor theory. FIG. 171. First approximation of equivalent circuits of single-phase induction motor in the two motor theory. THE SINGLE-PHASE INDUCTION MOTOR 209 The ratio of the magnetizing current of the second motor i" to the total current of this motor at no load is i" -f- 5-. Now, tf : 2 is equal to the ratio of the sum of the leakage fields of the second motor to the total flux %$\v\. Hence, f\ l = ( Vl - (196) = 0(^2 1) approximate Vi (197) FIG. 172. Second approximation of equivalent circuits of single-phase induction motor in the two motor theory. Admittance y negligible in comparison with 1/0:2. Therefore, - 1 (198) assuming vi = v z which is permissible for the second motor. A simple transformation now yields . (199) The total magnetizing current of the single-phase motor is the 14 210 INDUCTION MOTOR sum of the magnetizing currents of motor I and motor II. *. - - >" ' (2) In speaking loosely of "adding magnetizing currents" of two motors in series, we mean of course that the voltages impressed upon each motor may be added and to the sum of these voltages corresponds a magnetizing current equal to the sum of the magnetizing currents of each motor. The same result is readily read off the Fig. 169 if we divide OD by OC. We obtain OD + OC = (zvi - -}- = (2i!; a - 1) For (7 = 0, that is, for a motor without leakage, we have: A common value for a is 0.05 when 1 /i -*- IQ = 2 The point G, Figs. 168 and 169, is determined in the same manner as in the theory of the general alternating current transformer, from the consideration that the ratio between the magnetizing current, proportional to OB, and BG, is equal to a- = ViVz 1. It is clear that the point C moves also on a circle as the point D, which moves on the circumference of the circle BHD, divides OC in a constant ratio. (Fig. 168.) Thus the locus of C is the circle KGC, where (203) and OB = i (204) and BG = ip + 2 ) = ~^ (208) and for motor II 9.81-Z> n (ui + co 2 ) = ^~ (209) where coi = 2 - and P co 2 = 27T-n the angular velocity of the rotor at n revolutions per second. Hence, 9.81 (Di - Z>n)2 = 3^ 2 V 2 - \ (210) (211) L x _ 2/ (212) Writing S for we obtain r 1 n (213) To illustrate, Let us assume ~i = 50, and ~ 2 = 45, then we have = (j.Zo r watts In words, if the slip is 10 per cent, the loss of energy in the armature amounts to 23 per cent of P or approximately twice the percentage of the slip, if figured upon P + 3i' 2 V 2 . It is instructive to compare (212) with the similar one in the poly-phase induction motor, which develops after some trans- formations : " " (214) THE SINGLE-PHASE INDUCTION MOTOR 213 In words the slip in per cent is equal to the ratio of the armature / "*~ / 1 loss to the entire secondary power P. ~2 In Fig. 173 the output and torque in watts and synchronous watts as a function of the rotor speed expressed in cycles per second have been represented as calculated from the results given in this chapter. The heavy lines represent a rotor with small resistance, while the broken lines represent a rotor with fairly large resistance. These curves represent actual conditions. 250 200 150 FIG. 173. Output and torque curves of single-phase induction motor as a function of the speed. d. EXPERIMENTAL DATA There are reproduced here, from the first edition of this book, the characteristics of a 10-hp. single-phase motor for 110 volts, 50~, and 1,500 r.p.m. The total number of conductors in the field was 120; the total number of conductors in the armature was 312. The resistance of the field was 0.015 ohm; of each of the three phases of the armature, it was 0.08 ohm. It is instructive to calculate the armature loss in watts for the greatest load. We find this to be 55 2 .024 = 730 watts, corresponding, according to equation (213), to a slip of 2.1 per cent. The discrepancy between this and the measured value of 2.66 per cent, is due probably mainly to the difficulty of measur- ing a small slip. 214 INDUCTION MOTOR TESTS OF 10-HP. MOTOR o b C u d} CQ ,-* ,^ 2 cr B> a2 11 02 Is ^ D- -* 1 o si ft 2 .2* o> S frt g ^ ff 03 & o 3 tf GQ < system," i.e., the speed field, and p x the total reluctance of the speed field including its leakage field. The total flux follows directly from the area and distri- bution of the induction. If there is no resistance in the rotor, or if the reluctance of the speed field circuit were zero, then the time-phase of the speed flux F a would be in quadrature with the impressed e.m.f. of rotation, FIG. 177. The e.m.f.'s in the speed field and their time-phases. of the speed field. Production e F s, lagging 90 degrees behind this e.m.f. (We use the subscripts FT, FS, FSS and FST to indicate that an e.m.f. is due to the flux F or the flux F 8 , and produced by transformation T or by speed rotation s .) Exactly as in an open-circuited transformer and the speed field is always in the open-circuited condition by impressing upon the speed field circuit the e.m.f. e FS a current flows which, through its rate of change, induces a counter e.m.f. in time quadrature and lagging 90 degrees behind the flux produced by the current which is in time-phase with the flux. (See Fig. 177.) F 8 is THE SINGLE-PHASE INDUCTION MOTOR 219 the speed field in time-phase with i x which is the current. We designate the current with the subscript x as its m.m.f. acts in space in the X-axis. Similarly, as in the transformer on open circuit, OA = e FS is the impressed e.m.f. generated by rotation in the main field Fs AB = i x r z is the ohmic drop in the rotor. AC = e Ffi s is the counter e.m.f. produced by the rate of change of i x . This counter e.m.f. may be considered as composed of two e.m.f s. if we wish to resolve the total field in the speed field or x-axis produced by i x into a leakage field and an air-gap field. However, this is entirely unnecessary as the speed-field circuit is in the position of a choke coil. Let XQ be the total reactance of the speed-field circuit including its leakage effects and let it be inversely proportional to its reluctance p x , then AB = p 9 T = ixXo Hence, OB + AB = i x rz -r- i x x .-. tgi = ?? = JL. = Constant (220) 7*2 l2px It appears, therefore, from this very general inspection, that the time-phase angle between the speed e.m.f. OA = e FS and the speed flux F s is constant. Since OA must be in time-phase with F 2) it appears that the time-phase angle between the main field and the speed field is constant at all loads and speeds. Considering now the main field, or the Y-system of the rotor in space, we note at once that there will be induced in it first, an e.m.f. of transformation e F T and, secondly, an e.m.f. of rotation produced by the cutting of the rotor winding through the speed field F 8 . This e.m.f. we denote by e FsS . The former is in time quadrature with F 2 , while the latter is in time-phase with F 8 . (Fig. 178.) As e F T is due to the rate of change of F 2 , it lags in time 90 degrees behind F 2 , while e Fs s produced by rotation in the speed flux must be opposed to e FT , their vector difference being equal to the drop due to the resistance and local leakage reactance of the rotor. In the theory of the poly-phase induction motor we have 220 INDUCTION MOTOR replaced with advantage the local leakage reactances by their respective leakage fluxes. The same will be done here. We shall consider from now on, unless specifically stated otherwise, that F 2 represents as in the theory of the induction motor the net flux actually passing through the rotor winding so that, vectorially, F 2 = F /2, where / 2 is the local secondary leakage flux in the transformer field or F-axis. G FS sin e.m.f.generated by speed rotation Fio. 178. Single-phase induction motor. Time-phase diagram of the main field and the speed field and their e.m.f's. If this is done, then, as in the theory of the induction motor, we may draw the following diagram in which the resultant of e FS and e FsT is used to overcome only the resistance of the rotor winding the same applying to the resultant of e FT and e PaS . Very simple relations can now be derived for the speed flux in terms of the main flux. These follow directly from the geometrical figure. At standstill A^L is zero arid N lies at L. At THE SINGLE-PHASE INDUCTION MOTOR 221 synchronism N lies at P and above synchronism the motor turns into a generator. (Fig. 178.) S is the ratio of co 2 -r- coi 0)1 = 2-7T' 'i The reader is cautioned that we have designated the slip s = /****' 1 /^XN/O /^XM'O with a small s, and the ''speed" with a large $ = 1 - s. If the direction of rotation be reversed, i.e., if S is made nega- tive, S 2 remains positive and, therefore, the range from L to N and to P will be resumed. This is strikingly illustrated by the fact that a single-phase induction motor runs in either direction dependent on the direction of the impulse which is necessary to make it start. (b) THE DERIVATION OF THE CIRCLE DIAGRAM AND THE Locus OF THE PRIMARY CURRENT It is interesting and important to investigate the locus of the primary current under different loads and speeds. (Fig. 179) Let ab be the secondary leakage flux in time phase with MN = i y rz of the previous figure. Let OA = Fz be the NET transformer flux, whose space- phase is the -axis.' Let be = NM = 3> 2 be the " fictitious" secondary flux proportional to i y (i y corresponds to iz in the theory of the transformer and induction motor). ab -r- be = (vz 1) ac = vz$z (221) Then Oc = $1, the "fictitious" primary flux, proportional to ii. cM -r- Oc = (vi - 1) OM = v&i (222) vi and v z are here as always Dr. John Hopkinson's stray coeffici- ents. Draw MG parallel to Oa = Fz, to the intersection G with the extension of ON = F\. Then the reader can readily prove that ON = = Vl v 2 - 1 (223) 222 INDUCTION MOTOR This is our old well-known leakage coefficient of the transformer. It is instructive now to draw the triangle LMN which is taken from our previous figure. We have chosen our scale for Tjl this triangle arbitrarily so that NM = i y = -. As all quanti- ties are entered into the figure, we shall let the reader think for himself. FIG. 179. The primary current locus of the single-phase induction motor. (ML perpendicular to Oa See Fig. 178). Draw LN to the intersection H with the extension of GM establishing at H. the angle . This angle we have seen to be constant and equal to to-^r Thus H lies on the arc of a circle with NG as a chord. Through M draw MK parallel to HN ', establishing the angle at M, so that the locus of M is the circle over KG as a chord. THE SINGLE-PHASE INDUCTION MOTOR 223 The center C of the circle KGM is readily seen to be determined by the angle where R w is the resistance in the equivalent circuit to produce the current i w under an impressed voltage coF 2 . Hence .- . ft. = . f } (232) Apply the same reasoning to the watt-less circuit. Mn = iwi'fz = co \Fz S 2 sin cos A 1 CO]/' 1 2 Also i w i = -=r- where X w i is the reactance to produce the same watt-less current TT tr COlT 2 Hence X w i = = .8' sin 7 cos{ (233) And <0{= ^J- (234) (235) 228 INDUCTION MOTOR We thus obtain the following circuits which simulate the charac- teristics of the single-phase induction motor, Fig. 183. If the reluctance of the speed field circuit is infinite, the machine does not operate as a motor as it is permanently short- circuited. If the reluctance of the speed-field circuit is zero, then the machine resembles somewhat a poly-phase induction motor as the extra watt-less circuit is open and sin 2 = 1. Such a motor would show only a different speed characteristic from a two-phase induction motor. , Note. The assumption that the effect of the leakage field in the rotor may be taken into account by assuming that the leakage of the m.m.f. of the currents of the F-system only need be con- 32 5 Fio. 183. The equivalent circuits of the single-phase induction motor in the cross flux theory. sidered, is an entirely arbitrary one. It is a very reasonable one and it appears to answer all practical considerations and to lead to a comparatively lucid picture of the phenomena within the machine. However, as we have seen in Chap. IX, E, in discussing the secondary leakage reactance of the induction motor with slip rings and with commutator, there are here also local leakage fields which must induce e.m.fs. in the conductors of the rotor in such a manner that the leakage fields produced by the cur- rents in the speed field affect the conductors in the transformer field. This can most readily be seen if we study for a moment the conditions near synchronism. At synchronism there exists a nearly perfect rotating field created by the interaction of the X-currents which are in perfect space quadrature and nearly in time quadrature with the F-currents. THE SINGLE-PHASE INDUCTION MOTOR 229 That portion of this field which does not reach the primary, cuts the rotor conductors at slip frequency, thus the effect of the speed-field currents consists in diminishing the leakage reactance of the F-system in the rotor. (f.) THEORETICAL CONSIDERATIONS Consider a single turn in the rotor. Denote the flux at time t in the F-system with f y and the flux at the same time in the ^-system with / ft = Fy-sin co* (236) fi = F x -sm (co* - & (237) Let the single turn form the angle a. with the F-axis, then the flux at time t passing through the coil is ft = #cosa+,#sina (238) = Fy-cos a-sinco* + F x -sm a-sin(co* ) The e.m.f. induced in the coil at a rotor speed of $co is g dft da dt dadt = [ wF y cos co* SuF x sin (co* )]cos a + [SwFy sin co* coFa- cos (co* )]sin a which may be written e a = e y cos a + e x sin a as we may write e y = uFy cos co* SuF x sin (co* ) e x = SuF x sin co* - a>F x cos (co* - Q ' The equations show that the mode of consideration of two space fields in each of which two e.m.fs. are active, a conclusion which we reached at the outset of this chapter on general physi- cal principles, is consistent with a more careful mathematical analysis. (g.) THE TORQUE The e.m.f. induced in a single rotor turn at the speed co$ is, e a8 = ~^~ = (/i-sin a -/j-cos )o OCX Ol Through this conductor there flows a current ia = iy-cos a + if'Sm a. 1 Throughout this paragraph we denote the "speed field" F s by F x , and F 2 by F tf . 230 INDUCTION MOTOR Hence, the work done upon this conductor e a8 i-dt = (f x cos a f y sin a)(^-cos a + z'^sin oi)Swdt Carrying out the trigonometrical multiplication and summing up the work over the total number of rotor conductors z 2 , we obtain dW = S(/j i y cos 2 - ft i x -sm 2 a)S'wdt = ^(fx %l -f v i)So>dt, and the instantaneous torque, "^w*-* w (240) At any moment the instantaneous torque T t consists of two quantities: A positive torque produced by the interaction of the "speed field" with the " transformer currents," and a negative torque produced by the interaction of the "transformer field" with the " speed field currents." Let be the angle of time lag of the ^/-current relative to the main transformer-field F 2 , then we have T t = ^z 2 [F x i y sin (coZ - )-sin (at - 6) - F v i x sin at-sin (at - ()] (241) = ^z 2 [^'j,{cos '(0 - Q - cos (2ut - d - )} - F v i f {cos { - cos (2ut - {)}] The resultant momentary torque, composed of a positive and a negative part, pulsates with double frequency as each part pulsates in this manner. The mean torque is therefore: T = iz 2 [F x i y cos (0 - & - F v i x cos ] (242) From Fig. 178 F x = SFy-siu & i x = Fy-cos i v cos (0 - Q = -F y sin ^(1 - S 2 ) 7*2 By substitution in (242) above T = i ^aF v 2 S[(l - S 2 ) sin 2 ^ - cos 2 ^] 4 TZ = i -coF, 2 S[(2 - ^ 2 ) sin 2 ? - 1] (243) THE SINGLE-PHASE INDUCTION MOTOR 231 These are the equations as given by Goerges and Sumec. From this equation follows: 1. For S = and for = V 2 ~ sln the torque becomes zero, viz., at starting and slightly below synchronism. These conditions we have already noted from a general consideration of the physical phenomena. 2. Increasing the rotor resistance diminishes and therefore sin thus the torque decreases rapidly as we have seen clearly in Sumec's circle diagram. (h) MECHANICAL OUTPUT From the equation for the torque T we have the output P = SuT 2 [S 2 (l - S 2 ) sin 2 - S 2 cos 2 ] T: 7*2 The output due to the ^/-currents is proportional to, see Fig. 178, MN-LN-cos (NML) = NP-NL (244) The energy loss per second due to the ^-currents is equal to i* 2 T 2 . E x = S-MP _ ~ LP MP* = PH-PL so that E 2 X = PH-LN (245) Hence the NET output is proportional to the difference of (244) and (245), P = NPLN - PH-LN = LN(NP - PH) = LN NQ (246) (i) ROTOR COPPER LOSSES In each conductor there is dissipated into heat in time dt r z ildt = ri(iy cos a + i l x sin a)*dt In summing up over the circumference of the rotor r 2 S 2 cos 2 * + i sm*a)dt = 232 INDUCTION MOTOR Or, by integration, per unit time Again from our figure rst v 2 = PM* + PN 2 2 [(1 - >S 2 ) 2 sin 2 ^ + cos 2 fl (247) Hence the total rotor loss 2 [(l - S 2 ) 2 sin 2 ^ + (1 + >S 2 ) cos 2 ^] (248) CHAPTER XIX THE SINGLE-PHASE REPULSION MOTOR A. THE NON-COMPENSATED REPULSION MOTOR The theory of the single-phase repulsion motor can be ap- proached successfully in the same way as we have treated the poly-phase and single-phase induction motors. In fact, the use of the leakage fluxes shows their great effectiveness in their ap- plication to these interesting motors. Throughout the treatment of the theory of alternating-current commutator machines it must be borne in mind that the phe- nomena occurring in the short-circuited coils under the brushes vitiate to a great extent all theoretical considerations. Any attempt to take the effects of short-circuit into account proves disastrous as the complications that have to be introduced into the theory befog utterly the mind that desires to obtain a general understanding of the characteristics of these motors. We intend to limit our discussion to an ideal hypothetical motor without core losses and without losses in the coils under the brushes or other unpleasant confusing elements introduced by the process of commutation. However, the performance of this ideal motor is very instructive and it offers a clue to the understanding, and a good sign post to the designer, of the real motor. (a) The General Theory. A treatment of the theory of the repulsion motor consistent with the general theory which we have followed in this book, was given almost simultaneously by M. Osnos 1 and Andre Blondel. 2 Both papers are beautiful 1 M. OSNOS, E. T. Z., Oct. 29, 1903. 2 A. BLONDEL, L'Eclairage Electrique, Dec. 12 and 26, 1903. 233 234 INDUCTION MOTOR contributions to the simple elucidation of the complex phenomena in the ideal repulsion motor. Figure 184 shows diagrammatically the circuits of the repul- FIG. 184. Single-phase repulsion motor and diagram of space phases. sion commutator induction motor. The axis Y-Y is the axis of the stator field in space, while B-B is the axis of the short- circuited brushes on the commutator of the rotor. At the outset, we warn the reader that he should distinguish carefully in his mind between space-phases and time-phases. The only way in which the short- circuited rotor can react upon the stator is by means of a field whose space axis lies in the Y-Y axis. Hence any rotor field reacting upon the stator is going to appear multi- plied by cos a, the cosine of the angle of brush shift. Again, the total field in the stator, appearing in the axis Y-Y, is not able to react upon the rotor. Only such part as falls into the brush axis B-B can affect the rotor. Let, as before, v&i be the total "fictitious" primary field, including the primary leakage fields, due to the primary m.m.f. Then, the interaction between v&i and $2 cos a from the rotor must FIG. 185. The flux diagram of the single-phase repulsion motor. (All time-phases.) THE SINGLE-PHASE REPULSION MOTOR 235 produce, or leave over, so to speak, the real field FI which balances the impressed primary voltage. Now, all time-phases, Fig. 185. OB = v&i AB = $2 cos a OA = F l In order to find the rotor fluxes, we have to consider that the interaction of v&i cos a and v^z leaves over the resultant rotor flux F 2 = OD. These relations are those discussed again and again in this volume slightly altered quantitatively only by the mechanical configuration of the interacting systems. As in the previous chapter we considered the resultant voltage produced by the action of the "transformer" field and of the " speed" field, so we consider here the similar e.m.fs. The resultant "transformer" field in the axis B-B is F 2 which, through the periodic rate of change, induces a transformer voltage in time quadrature with F 2 . This voltage we shall designate with e F2T to indicate, as in the cross-flux theory of the single-phase induction motor, that it is induced by F% and by the same process as that of transformation. At right angles in space to the brush axis B-B there exists a "speed" field F s = v&i sin a whose time-phase coincides with the time-phase of 3>i. Through rotation at angular velocity &oi, where S is again the speed, S = , there is induced a "speed" 0)2 e.m.f . e Ff s = SuiF a = Suiv&i sin a, which if composed vectorially with ep 2T gives a resultant which must be equal to the ohmic drop in the rotor all leakages having been taken into account through their respective leakage fields. (Fig. 186.) It is now evident, and we remind the reader of the procedure in the previous chapter, that the angle at K which is equal to 90 , is constant as DK : OD is constant. Hence the angle at L which is equal to 180 is constant. Draw O'D parallel to OA, then O'L is proportional to OB and therefore to ii, the primary current. As L lies on the arc O'LG vith C as center, the primary current locus is the arc O'LG. 0'L:LD::OB:AB v&i:$2 cos a 236 INDUCTION MOTOR r\u . . U L = v\v<2, COS a cos a COS a 2 cos a 2 cos a FIG. 186. The diagram of the single-phase repulsion motor. NOTE. In order not to lose in simplicty of the diagram, we have assumed rz = 1, so that the scale for e FsS and e P2T and LD could be made the same. O'L = t, LD = cos a (249) (250) THE SINGLE-PHASE REPULSION MOTOR 237 From simple proportionalities follows ( J - T?) (251) U (JT \ ViVz I Also = - 1 (252) DG cos 2 Hence, the leakage coefficient of the repulsion motor is similar to that of the standard induction motor excepting that it depends also upon the brush-shift angle a. If a = 0, then O'D as is to be expected. (b) The Speed in the Diagram. We have found that i z rz is the vector resultant of e FaS and e FzT - eps LK = Sui&i sin a Now, $> i cos a-sin = MD Also LK-cos = LM T-ToTl /"O S co LjK - -LlcilCc 3>i sin a LM nna fc. MD sin /v cos a sin frcot (253) The speed of the motor is therefore equal to the tangent of the angle Z LDM . This is zero at the point M, hence M s is the standstill point. It approaches a maximum, but not infinity, as the primary current decreases. (c) The Torque. The torque is proportional to the product of the "speed" field F s into the rotor current component in time- phase with the time-phase of the "speed" field, this latter being equal to the time-phase of the primary current, neglecting hys- teretic lag. Hence, we may write T = Const. sin a LD ^ cos DLK Vi cos a = O'LLDtgacos DLK 238 INDUCTION MOTOR Construct a semi-circle over O'D as diameter, then LN = LD cos DLK Hence, T = O'LLN-tga (254) M. Osnos, to whom this relation was first due, Blondel and myself having borrowed it from him, points out that, for a given primary current O'L, the rapid decrease in torque is due rather to the increased phase lag between the rotor current and the primary current, than to the decrease in the rotor current itself. The torque, as determined here, vanishes at the point P, hence the maximum speed the motor is capable of obtaining, barring losses, is reached at this point. The repulsion motor, therefore, does not run away like the series motor. (d) The Effect of the Rotor Resistance upon the Diagram. From the physical relations and the diagram we have = = P -r 2 . Const, where t'J is the magnetizing current of the rotor producing the field F^t and p the reluctance of the air-gap path. Hence, the distance of the center C from the abscissa O'G is a measure of, and proportional to, the rotor resistance. This result is similar to that obtained in the previous chapter in the theory of the single-phase induction motor. Thus a number of circles may be drawn, as in Fig. 187, show- ing that the primary current locus of the repulsion motor is a semi-circle only if the rotor resistance is zero. For an infinite rotor resistance, the locus is a straight line and no power can be developed by the machine. For resistances between zero and infinity, the loci curves are arcs of circles whose centers lie on the same vertical line CC r . Rotor resistance control can be used for speed regulation as the diagram indicates. The maxi- mum speed and the starting torque and starting current can be regulated in the same manner. (e) The Effect of the Brush Shift. Rocking the brushes and changing the angle a changes the ratio of the magnetizing current to the length of the chord of the arc of the locus of the primary current. It does not affect the angle . We have seen that O'D vivt DG : ~cos 2 a THE SINGLE-PHASE REPULSION MOTOR 230 O'--r 2 =8 FIG. 187. The influence of rotor resistance upon the primary current locus of the repulsion motor. Fixed brush position. (After Osnos). FIG. 188. The influence of the brush shift on the primary current locus of the repulsion motor fixed rotor resistance. 240 INDUCTION MOTOR This expression is a minimum, and therefore DG a maximum the smaller the angle a and the nearer cos a = 1. For a = 0, we obtain for this ratio ViV 2 1. For a = -, we obtain for this ratio 0. 2i Figure 188 shows these circles very much as given in the brilliant paper by M. Osnos already repeatedly referred to. The points M s marking the starting point lie on a circle as indicated. The diagram suggests the mode of regulating the speed and torque by means of mechanical brush-shift used at one time extensively by the Brown Boveri Company for railway motors. (f) Commutation. The repulsion motor develops an elliptical rotating field as it gains in speed. At or near synchronous speed this field rotates with little or no slip relative to the rotor, hence the injurious effect of commutation through short-circuit currents induced in the coils under the brush is less in these motors than in motors of the series type. However, in starting, there is no advantage in the repulsion motor over the series motor and it is in starting that the trans- former effects in the coils under the brush are most injurious. It is doubtless on this account that the career of the repulsion motor as a railway motor has been rather brief. A performance curve of an actual motor will be given in the Chapter on Commutator Motors. B. THE COMPENSATED REPULSION MOTOR OF WIGHTMAN, LATOUR, AND WINTER-EICHBERG (a) While the compensated type of repulsion motor in which the "speed" field is produced by passing the primary current through the rotor by means of an additional set of brushes, and leaving off the field coil, has ceased to be of practical importance, it still offers an object of great interest from the point of view of its theory. It should be entirely clear what this motor really is, but to avoid misunderstanding let the two types of repulsion motor be placed side by side. Supposing we wish to eliminate the field coil F in the connec- tion diagram, Fig. 189, of the ordinary repulsion motor in order THE SINGLE-PHASE REPULSION MOTOR 241 to get rid of the self -inductive effect which causes a voltage drop and a lowering of the power factor of the motor. The idea immediately suggests itself to utilize the armature itself for the purpose of exciting the " speed" field and to conduct the primary FIG. 189. Repulsion motor and compensated repulsion motor. current into the armature through a set of brushes in mechanical quadrature with the power brushes. It remains for us to take stock of the gain or loss resulting from the abolition of a field coil on the stator and the addition of an extra set of brushes on the rotor. X FIG. 190. Connections and space phases of compensated repulsion motor. Let us neglect leakage provisionally in order to get under way and to obtain a general idea of the phenomena within the motor. We make our usual assumption that the active number of conductors on the stator and rotor is equal, z\ = z^. Then we have again in space two fields, at right angles to each other, HI 242 INTRODUCTION MOTOR one being the " transformer " field F in the F-axis, the other being the " speed" field F 8 in the X-axis. Neither can react upon the other. (Fig. 190.) It follows that the resultant magnetization in the F-axis which is due to the m.m.f. of the stator iiZi and the opposing m.m.f. i&z of the rotor, must produce a field which balances the part of the impressed voltage be- tween the points A and B. There must be added to AB the voltage drop BC, in order to obtain AC, which is the total impressed voltage. Thus AH is the primary m.m.f. and HM the secondary m.m.f. result- ing in AM the magnetizing m.m.f. However, the current cre- ating the "speed" field in the X-axis induces through transformation an e.m.f. coiF s which lags in the time- phase diagram Fig. 191 a quarter-phase behind the field. Let DG be this e.m.f. When the motor stands still we have to add the e.m.f. DG with sign reversed to AB, obtaining BC' the total impressed e.m.f. Now, assume the rotor to turn as a result of the torque produced between the current iz and the "speed" field F s There will be induced in the rotor four e.m.fs. acting in two pairs, in mechanical space quadrature. First, as in the case of the repulsion motor, there are two e.m.fs. set up in the rotor in the F-axis. The e.m.f. of trans- formation proportional to u\F and independent of the speed, and the speed e.m.f. Su\F s due to the rotor cutting through the "speed" fluxF s produced by the primary current flowing through the exciting brushes in the X-axis. The resultant e.m.f. FIG. 191. Time-phase diagram of compen- sated repulsion motor. (No leakage.) THE SINGLE-PHASE REPULSION MOTOR 243 must fall in the direction HM and, if we assume r 2 = 1, then HM is a measure of this e.m.f. Secondly, between the exciting brushes in the rotor in the X- axis there is set up by rotation in the "transformer" field F a "speed" e.m.f. proportional to and in time-phase with F. The direction of this e.m.f. must follow Lenz's law and be opposed to the transformer e.m.f. DG. Let GL = Sc^F be this "speed" e.m.f., then DL is the result- ant e.m.f. which must be made up by an equal and opposite e.m.f. BC so that AC is the total impressed e.m.f. upon the motor. Now, following a sug- gestion made by M. Osnos, 1 we may intro- duce a hypothetical or imaginary " transfor- mer" field, to the rate of change of which there is due the "real speed" e.m.f. GL which is produced through rotation in the field F. (Fig. 192.) S UI F As We wish to Sim- FIG. 192. Time-phase diagram of compensated motor. (No leakage.) (After Osnos.) or $0=5* tffMANtgy-S N Ulate a field which would induce, if it existed, by transformation the e.m.f. we have = coi $0 = SF Hence, make MN = $o = SF, so that the tangent of angle MAN = tgy = S, then in the force polygon DNMA we find DN as the resultant field which we have usually designated 1 E. T. Z., November 12, 1903. 244 INDUCTION MOTOR with FI and with which the impressed e.m.f. A C is in time quadrature. It is important to note that, if z\ is not equal to 22, as would be the rule in an actual motor, then, as AD represented the "speed" field in time-phase the " speed" voltage being measured by its magnitude, with, for instance, 2 2 = \z\, the speed voltage would be S<*>iF s -%Zi} and F 8 being itself proportional to 2 2 , it follows R----L FIG. 193. Polar, or time-phase, diagram of compensated repulsion motor. (Including leakage.) that the "speed" voltage is proportional to (-) . Hence, AD must be multiplied with this ratio in order that it can be com- posed with AM and with the hypothetical field MN into one resultant DN in time quadrature with e\. (b) The Torque. The motor has two torques, a positive and a negative one. THE SINGLE-PHASE INDUCTION MOTOR 245 The positive torque is T l = DA HM cos d (256) The negative torque is T 2 = -DA- AM cost (257) as is readily seen from Fig. 192. As the speed increases, these torques become more and more equal to one another. The motor cannot run away and it is in this respect like the repulsion motor. (c) Performance. As will be shown from an actual perform- ance curve in Chap. XX, the motor acts like a repulsion motor with the same advantages as to commutation, approaching a rotating field at speeds near synchronism. Its power factor increases with the speed. (d) Leakage. The leakage is taken into account in Fig. 193. As usual we lay down the total primary " fictitious" flux which would be produced if the primary m.m.f. acted alone on the magnetic circuit. This is AH' = ViAH. EM' = v 2 HM AM = F\ AM' = F 2 M'N' = o = SF* DN = FI which is not a real field, the real primary field in the F-axis being AM = F\, DN = FI being a hopothetical field as the "speed" voltage induced by rotation in the field F% is not represented by a field in the 7-axis. To obtain the real physical conception of the phenomena in the motor, it is well to adhere to Figs. 191 and 192. CHAPTER XX SINGLE-PHASE COMMUTATOR MOTORS A CONDENSED REVIEW With the appearance in 1902 of Mr. B. G. Lamme's paper on the application of single-phase commutator motors to railway work, 1 a new impetus was given to the inventors and engineers the like of which had not been witnessed since Tesla's great invention. Though the particular installation referred to in the paper was never actually executed, though the motor was, in Steinmetz's 2 expression, "our old friend" the single-phase com- mutator motor, yet there was infused new hope and energy into the railway field. The feverish activity which absorbed the engineering community for 10 years after the reading of Mr. Lamme's paper resulted in the creation of innumerable types which, while they varied only slightly in the manner of their operation, attracted attention altogether beyond their intrinsic value and interest. Nineteen years have passed and a more sober frame of mind has superseded the fond dreams of that early period. A few types have survived among them particularly the conductively compensated series motor with interpoles in shunt connection and the repulsion induction motor already treated in Chap. XIX. 1 B. G. LAMME, "Washington, Baltimore & Annapolis Single-phase Railway," Trans. A. I. E. E., September, 1902. 2 "I believe we can congratulate ourselves then that here is published the record of some work done in the direction of developing apparatus, giving the proper characteristics for alternating current railway work. I must confess, however, that I have been somewhat disappointed in reading this paper, by seeing that after all this new motor is nothing but our old friend the continuous current series motor adapted to alternating currents by laminating the field. Now, I remember this type of motor very well because I was associated with Mr. EICKEMEYER in 1891 and 1892, and we spent a very great deal of time in building alternating current series motors, investigating their behavior, and trying to cure them of their inherent vicious defects." MR. C. P. STEINMETZ'S discussion of MR, LAMME'S paper, Trans., A. L E. E., Sept. 6, 1902. 246 (Facing page 246) SINGLE-PHASE COMMUTATOR MOTOR 247 A. VARIETY OF TYPES OF SERIES A. C. COMMUTATOR MOTORS In the accompanying diagrams a number of connections are shown representing a few of the very large number of schemes which have been brought to light as a result of the feverish in- ventive activities in this direction. Many of these connections are quite old and interest in them was revived when single-phase motors again commanded atten- tion as a result of the work of Mr. Lamme and Mr. Westinghouse. For instance, the interesting compensated motor later re- invented by Marius Latour and Winter and Eichberg, is described fully in the U. S. patent by M. J. Wightman, No. 476, 346, dated June 7, 1892, and assigned to the Thomson-Houston Elec- tric Company. Mr. Lamme's first motors were not compensated. They had a field winding of few turns and a magnetic frame of low reluct- ance. To handle the transformer current so troublesome in starting and at low speeds, induced in the coils short-circuited under the brushes, Mr. Lamme used resistance leads ingeniously embedded in the slots side by side with the conductors. We refer the reader to U. S. patents No. 758, 667, May 3, 1904, and to No. 758,668 of the same date. It is very easy and very human and natural to remark, as was done at the time when Mr. Lamme read his epoch-marking paper that there was little that was new in the system he described. That statement may be admitted. Yet no one had used these old ideas and no one had designed a workable single-phase com- mutator motor. I think the present author may claim fairly for himself that he never became a single-phase enthusiast. The limitations of the system were too deeply forced upon his attention in his work done in the middle nineties. At no time did he share the enthusiasm expressed, for instance, by Mr. Steinmetz in the following telling passage: 1 "If I may be permitted to take a look into the future, although we do not know what to-morrow will bring, I think the system of the future will be the single-phase system. Where the power is transmitted over a long distance by an overhead wire, the ground can be used as the return conductor. . . . But if we use a single-phase current in the power transmission of the future, then we will have to learn many things." 1 "Proceedings of the International Electrical Congress held in the City of Chicago. Published by the A. I. E. E., New York, 1894, p. 437. 248 INDUCTION MOTOR The present writer sounded a note of scepticism and caution in a paper in Gassier' s Magazine, May, 1907, where he said : " There is, in the realm of ideas, a distinct difference between 'nat- ural' ideas and 'forced' ideas. The natural idea may be likened to a plant growing under favorable conditions and adapting itself to its environment; the forced idea may be likened to a plant raised in a hot- house, with the exclusion of such conditions as might have a tendency to prevent its development. The natural idea will survive; the forced idea will go to the wall ; but it is often only after extended experiments conducted on a large scale have been laboriously completed that we realize that an idea has been followed out which could have lived only under particularly favorable conditions, such as are not usually found in practical operation. In contemplating the history of the develop- ment of the utilization of alternating currents, the single-phase system has appeared to be an almost ideally simple system. It is only too obvious that, if power could be safely transmitted and utilized in an economical manner, and by means of simple mechanical apparatus, the generation and utilization of single-phase currents would soon replace the poly-phase system. Such attempts were made 15 years ago, and, after considerable effort, most engineers abandoned hope in developing a practical system of transmission of energy by means of single-phase currents. The experience gained during the past 15 years with -poly- phase currents and the many opportunities afforded the engineer for comparing the single-phase machinery, generators, and motors, with their poly-phase cousins, have led to an attitude of skepticism towards single-phase current. The very much reduced output of both generators and motors if operated single-phase; the reduced efficiency; the impaired regulation; the increased heating, and the lesser stability of single-phase motors and generators, connected with the increased cost as produced by the greater amount of material required ; these form the main reasons for inducing me to call the recent attempts which have been made in the utilization of single-phase current, a forced idea." And later in the same paper the present author remarked : " Those gigantic experiments to be conducted on the New York, New Haven & Hartford Railroad are being watched with the respect due an enterprise of such magnitude, and with the hope that, even if the sys- tem should not be all that its ingenious designers had expected, it may yet lead on to ideas which will finally solve the problem of electrically operating the present steam railways of the world." Starting with the non-compensated series motor (1) we pro- ceed to the type with conductively compensated armature (2) SINGLE-PHASE COMMUTATOR MOTOR 249 and then the inductively compensated connection suggests itself (3). (Fig. 194.) TYPES OF SINGLE- PHASE COMMUTATOR.MOTORS (1) Plain Series Motor (2) Conductively Compensated Series Motor lEickemeyer, Lamme) ( 3) Inductively Compensated Series Motor (Eickemeyer) (4) Inverted Repulsion Motor, Series Motor with Secondary Excitation (5) Repulsion Motor (6) Repulsion Motor (7) Series Repulsion (8) Series Repulsion with Stator E-xcitation Motor with Secondary Motor, with Primary (Atkinson) Excitation , A , .. Ex,citatio? ('Doubly-Fed ) (9) Compensated Repulsion Motor (Wightman, Latour,) (Winter,Eich^erg) (10) Rotor-Excited Series Motor with Conductive Compensation (11) Rotor-Excited Series Motor with Inductive Compensation (12) Series Motor with Rotor Excitation and Compensation (McAllister) (13) Induction Repulsion Motor (Atkinson) (14) Commutator Induction Motor (.Atkinson ) (15) Repulsion Motor (16) Single-Phase Unity (Deri,Latour) Power Factor Motor (Wagner-Fynn) (17) Shunt Motor (Creedy) (IS) Shunt Motor (Fynn) (19) Series Motor Conductively Compensated with Interpole in Shunt FIG. 194. Single-phase commutator motors. A variety of connections. As the next step we may excite both the field and the com- pensating coil inductively (4). By exciting the field and compensating windings from the bus 250 INDUCTION MOTOR bars and short-circuiting the rotor we obtain one of the many types 1 of repulsion induction motor (5). By splitting field and compensating winding, we obtain Figs. (6) and (7). By feeding both the field and compensating windings in series from one potential and the armature from another we obtain the interesting "doubly-fed" motor of Latour 2 and Alexanderson. 3 By exciting the field, which is located on the stator in the repulsion motor (5), in the rotor proper, by means of an additional set of brushes at right angles to the short-circuited brushes, we arrive at the Wightman-Latour-Winter-Eichberg compen- sated motor (9) the chief advantage of which lies in better com- mutation and higher power factor than in the repulsion induction motor. The extra brushes are a grave mechanical drawback and they would seem to be too high a price to pay for the advan- tage obtained. (10), (11), (12) the McAllister connection, and (13) are modifications of (9). (14) is a single-phase commutator induction motor, also due to Atkinson, I believe. Deri and Latour suggested (15), Fynn suggested (16), Greedy suggested (17), and (18) is again a Fynn motor. (19) appears to represent the chief survivor of this great host of types. It is a plain series conductively compensated single- phase commutator motor with interpoles excited from the impressed voltage so as to produce a commutating field of the right time-phase. If motors of this type are started with direct current and operated on single-phase alternating current, then commutation will be very satisfactory. B. OPERATING CHARACTERISTICS OF DIFFERENT TYPES In order to give a general idea of the performance of some of the types of single-phase commutator motor which have been enumerated here we reproduce from tests the following: 1 See LLEWELYN BIRCHALL ATKINSON, Proceedings, Institution of Civil Engineers of Great Britain, Feb. 22, 1898. "The Theory, Design and Work- ing of Alternate-Current Motors." 2 M. C. A. LATOUR, U. S. Patent No. 841,257, Jan. 15, 1907. 3 E. F. W. ALEXANDERSON, U. S. Patent, No. 923,754, June 1, 1909. SINGLE-PHASE COMMUTATOR MOTOR 251 Figure 195 shows the characteristics of a repulsion motor like (5). 500. SC 300- 200_ 20_]40.| 4 100-1 10-20- 20 "Vj<. ot Single Phase R'w'y Mojtor I" 'O Repulsion-Motor hConnection- I- Operating on 200 Volts', 25 1 Cycles Brushea Shi Rotation, o 19 4(Elec.) Ampcies Line I 100 j 120 140 FIG. 195. Operating characteristics of a single-phase repulsion motor. FIG. 196. Operating characteristics of a single-phase repulsion motor. Latour connections. Figure 196 shows the performance characteristics of a com- pensated repulsion motor with Latour connections like (9). 252 INDUCTION MOTOR Figure 197 shows the performance characteristics of the same motor connected as suggested by A. S. McAllister in (12). L4U 1400 > 100- "I 03 I s 40> 100- "I 80 70-| 60^ 59-2" 40 1 20- 1000 "fc pj a 600- 400- 20 X -Po ^y \ / 18 X X 5 ^1 *Jo >x x * rse- 'owe A. 0) 16-* \ ^ s. "1 14-i, and the other part a single-phase stationary field lying along the major axis of the ellipse, given by the expression x z = c-cos = Now, the single-phase field can be broken up into two oppositely-rotating fields : 2/2 = 2/4 + 2/5 c- sin (at + 260 INDUCTION MOTOR Adding Xi to x* and y\ to 2/4, we get: cos w * sin w * These form a uniform field rotating forwards, while x& = %c-cos ( cot) and 2/5 = H^'sin ( <*t) give us a uniform field rotating backwards. Thus we see that the magnetic field produced in a poly-phase machine having symmetrically disposed coils carrying an unbalanced load can be re- garded as the resultant of two uniform fields of different amplitudes rotating in opposite direction." The general principle enunciated in Oilman & Fortescue's paper, viz., that "an unbalanced poly-phase system can be re- solved into two balanced systems of positive and negative phase- rotation," does not appear to have been proved rigorously in that paper. The subject is of increasing importance and would have merited more space than this passing reference. APPENDIX It became incumbent upon the writer more than 20 years ago, to appear as though he gave countenance to the infringement of the fundamental Tesla patents. A large number of induction motors designed by him during the life of these patents, which constituted a plain infringement of Tesla's inventions, have no doubt been pointed to as an indication that he either did not believe in the validity of these patents or that he deliberately became a party to their infringement. The Company of which, at the period referred to, he was Chief Engineer owed its growth and development largely to his personal efforts in the design and development of electrical machinery and to his success in organizing an effective engineering staff, con- sisting of a number of eminent men among whom were David Hall, A. B. Field, W. L. Waters, Bradley T. McCormick, H. A. Burson, Alexander Miller Gray, R. B. Williamson, Carl Fech- heimer and others. In due course the owners of the Tesla patents proceeded against our company and in the long litigation which followed the writer's position was at times embarrassing and disagreeable. By way of epilogue, he begs leave to publish now, with the bitterness of the controversy abated, a letter addressed to the patent counsel of his Company: Cincinnati, Ohio, May 23d, 1901. MR. ARTHUR STEM, PATENT ATTORNEY, CITY. My dear Sir: Enclosed please find my notes on the Record of Final Hearing in the suit of Westinghouse Electric & Mfg. Co. vs. the New England Granite Co. You will see that I am now, even more than I have been before, of the opinion that it is not possible for us to bring forth arguments that could go to show the invalidity of the Tesla Patents in suit. While I am, as engineer in charge, perfectly willing to give you all the technical assistance in my power that you might need or ask for, I cannot undertake to give expert evidence in this case in favor of my concern, as such evidence would be against my better convictions in this case. As, during my last call at your office, you intimated my being one of the experts, I think it best to let you know as early as possible that I cannot undertake this duty. 261 262 INDUCTION MOTOR Model maker, Mr. W. J. Schultz, called at our office yesterday and I gave him all the necessary instructions for making the devices that we had deemed advisable to make for this suit. Mr. Schultz is thus prepared to let us have his bid on them and this will be submitted to our management. I remain, Yours very truly, (Signed) B. A. BEHREND, CHIEF ENGINEER, ETC. It was a matter of gratification to the writer that almost 20 years later, when he was a member of the Edison Medal Com- mittee, he was able to propose the name of Tesla for the award of the Edison Medal and upon the occasion of the presentation of the Medal to express his great admiration for the medallist's creative work. There is published herewith, as the closing chapter of a long story, the writer's address as it was delivered on that occasion. To PROFESSOR ANDRE BLONDEL OF L'ECOLE NATIONALS DES FONTS ET CHAUSSEES OF PARIS An honorary member of our Institute, whose brilliant and inspiring work a generation ago laid the theoretical foundation of the development of alternating- current engineering practice; to whose generous support of my own modest labors, over a score of years ago, I owed recognition; in the hour of his country's trial, I beg leave to inscribe these remarks. Hands across the sea, may a happier future dawn before us! B. A. BEHREND. Address Mr. Chairman: Mr. President of the American Institute of Electrical Engi- neers: Fellow Members: Ladies and Gentlemen: DY an extraordinary coincidence, it is exactly twenty-nine years ago, to the very day and hour, that there stood before this Institute Mr. Nikola Tesla, and he read the following sentences: "To obtain a rotary effort in these motors was the subject of long thought. In order to secure this result it was necessary to make such a disposition that while the poles of one element of the motor are shifted by the alternate currents of the source, the poles produced upon the other elements should always be maintained in the proper relation to the former, irrespective of the speed of the motor. Such a condition exists in a continuous current motor: but in a synchronous motor, such as described, this condition is fulfilled only when the speed is normal. "The object has been attained by placing within the ring a properly subdivided cylindrical iron core wound with several independent coils APPENDIX 263 closed upon themselves. Two coils at right angles are sufficient, but a greater number may be advantageously employed. It results from this disposition that when the poles of the ring are shifted, currents are generated in the closed armature coils. These currents are the most intense at or near the points of the greatest density of the lines of force, and their effect is to produce poles upon the armature at right angles to those of the ring, at least theoretically so; and since this action is entirely independent of the speed that is, as far as the location of the poles is concerned a continuous pull is exerted upon the periphery of the armature. In many respects these motors are similar to the continuous current motors. If load is put on, the speed, and also the resistance of the motor, is diminished and more current is made to pass through the energizing coils, thus increasing the effort. Upon the load being taken off, the counter-electromotive force increases and less current passes through the primary or energizing coils. Without any load the speed is very nearly equal to that of the shifting poles of the field magnet. "It will be found that the rotary effort in these motors fully equals that of the continuous current motors. The effort seems to be greatest when both armature and field magnet are without any projections." Not since the appearance of Faraday's "Experimental Researches in Electricity" has a great experimental truth been voiced so simply and so clearly as this description of Mr. Tesla's great discovery of the generation and utilization of poly-phase alternating currents. He left nothing to be done by those who followed him. His paper contained the skeleton even of the mathematical theory. Three years later, in 1891, there was given the first great demonstration, by Swiss engineers, of the transmission of power at 30,000 volts from Lauffen to Frankfort by means of Mr. Tesla's system. A few years later this was followed by the development of the Cataract Construction Company, under the presidency of our member, Mr. Edward D. Adams, and with the aid of the engineers of the Westinghouse Company. It is interesting to recall here to-night that in Lord Kelvin's report to Mr. Adams, Lord Kelvin recom- mended the use of direct current for the development of power at Niagara Falls and for its transmission to Buffalo. The due appreciation or even enumeration of the results of Mr. Tesla's inventions is neither practicable nor desirable at this moment. There is a time for all things. Suffice it to say that, were we to seize and to eliminate from our industrial world the results of Mr. Tesla's work, the wheels of industry would cease to turn, our electric cars and trains would stop, our towns would be dark, our mills would be dead and idle. Yea, so far reaching is this work, that it has become the warp and woof of industry. The basis for the theory of the operating characteristics of Mr. Tesla's rotating field induction motor, so necessary to its practical development, was laid by the brilliant French savant, Professor Andre" Blondel, and by Profes- sor Kapp of Birmingham. It fell to my lot to complete their work and to coordinate by means of the simple "circle diagram" the somewhat mysterious and complex experimental phenomena. As this was done twenty-one years ago, it is particularly pleasing to me, upon the coming of age of this now universally accepted theory tried out by application to 264 INDUCTION MOTOR several million horsepower of machines operating in our great industries to pay my tribute to the inventor of the motor and the system which have made possible the electric transmission of energy. His name marks an epoch in the advance of electrical science. From that work has sprung a revolution in the electrical art. We asked Mr. Tesla to accept this medal. We did not do this for the mere sake of conferring a distinction, or of perpetuating a name; for so long as men occupy themselves with our industry, his work will be incorporated in the common thought of our art, and the name of Tesla runs no more risk of oblivion than does that of Faraday, or that of Edison. Nor indeed does this Institute give this medal as evidence that Mr. Tesla's work has received its official sanction. His work stands in no need of such sanction. No, Mr. Tesla, we beg you to cherish this medal as a symbol of our gratitude for a new creative thought, the powerful impetus, akin to revolu- tion, which you have given to our art and to our science. You have lived to see the work of your genius established. What shall a man desire more than this? There rings out to us a paraphrase of Pope's lines on Newton: Nature and Nature's laws lay hid in night: God said, "Let TESLA be," and all was light. New York City May 18, 1917. INDEX Reference to Pages Adams, Comfort, A., 114, 146 Adams, Edward D., 263 Alexanderson, E. F. W., 179, 181, 204, 250 and Hill 258 on U. S. Battleship New Mexico 115 Altes, W. C. K., 145, 152 Arnold, E., and J. L. la Cour 11, 19, 114, 134, 145, 152, 175 Ampere-turns 57 Atkinson, L. B . , 250 B Bedell, F., early papers 5, 8, 9 Behn-Eschenburg, H., 18, 111, 112, 113, 254 his connection of shunt motor fields 255 Behrend, B. A 203, 204 criticism of his leakage factor by Hobart 110 earliest paper on discovery and proof of circle diagram 6 excentric magnetic pull , 198 on single-phase system in Gassier 's Magazine 248 stray-coefficients 24 "the Debt of Electrical Engineering to C. E. L. Brown," 3 BSthenod, J 11 Blakesley, Thomas H., Reference to early papers 4, 5 Blondel, Andre 146, 149, 152, 233 Blondel's stray-coefficients 24 early papers 6 references , takes up Behrend's circle diagram 7 Bouasse, Henri, 197 Boy de la Tour, Henri 10 Bragstad, O. S 152 Bridges, effect of slot bridges on short circuit current 88 Brown, Boveri Company 178, 187 Brown, C. E. L 1 "Debt of Electrical Engineering to," 3 squirrel cage induction motor of 1891 3 Burson, H. A., 261 265 266 INDEX Cassier's Magazine, quotation from 248 V Circle diagram, first experimental corroboration of 94 for air-gaps of different lengths 93 for different frequencies, ; . . IQg for series polyphase commutator motor . 143 Commutation, methods of improving 253 Commutator motors, polyphase, properties of 124 comparison with squirrel cage 128 local leakage reactance of 126 shunt polyphase 146 types of variable speed . 182 variable and constant secondary reactance of 131 Compensated repulsion motor 240 diagram of connections 241 time phase diagram 241 Concatenation 153 comparison of group with single motor 173 equivalent circuits 155, 169 polar loci 162, 163, 166, 168 torque curves 161 vector diagrams 157 Constant current transformer, historical reference 5 circle diagram of . '. 25 Constant voltage transformer 27 Copper loss, accounting for secondary 42 accounting for primary 43 Correction of power factor 188 Greedy, F., 255 his motor 250 Crehore, A. C., see also Bedell, F 5 Cross-flux theory, of single-phase induction motor 216 equivalent circuits of 228 D Danielson, E., early motor design 7 -Burke 197 Darwin, Sir George Howard Preface Dead points in torque curve 79 Deri-Latour motor 249 Dobrowolsky, M. von Dolivo-, 1, 115 term "wattless current," 20 term "wattless component," 29 Double squirrel cage induction motor 115 flux diagram of 1 117 polar diagram of 120 torque curves of 121, 122 INDEX 267 Doubly-fed motors 147, 249 Dreyfus, L., and F. Hillebrand 134, 152 Dudley, A. M., Preface E Edison medal, address of B. A. Behrend 262 Eichberg, F 152, 175 compensated repulsion motor of 240 Eickemeyer, R., 246 Elementary theory of induction motor 64 et seq. Equivalent circuits, comparison of squirrel cage with commutator rotor 133 for double squirrel cage induction motor 117 in concatenation 155, 169 of single-phase induction motor 208, 209 of single-phase induction motor in the cross-flux theory 228 F Fechheimer, Carl J. , 261 Ferranti, Sebastian Ziani de, 10,000 volt line Deptford to London 5 Ferraris, Galileo 204 Field, A. B., Preface 258, 261 Field belt 57 Fortescue, C. Le G. , 258, 260 Frankfort-on-the-Main, Electrical Exposition, 1891 1 Frequency, drawbacks of high 107 Fresnel's theorem 204 Fynn, Val. , 249 G Generator, see Induction Generator 82 Gilman, R. E 260 Goerges, H 137, 152 his shunt poly-phase commutator motor 182 Goldschmidt, R 114 Gray, Alexander Miller, Preface 114, 203, 261 Use of Behrend's stray-coefficients 23 Guillet, A 197 H Hall, David 261 Harmonics, in flux belt 78 Heaviside, Oliver 1 "Algebraization," 4 Hellmund, R. E., Preface 114 Translation of Heyland's paper 9, 12 Helmholtz, quoted 110 Hertz, Heinrich 1 268 INDEX Heyland, Alexander 7, 13, 152, 177 his compensated induction motor 182 Hobart, H. M., on leakage factor 110, 111, 112 Hodograph of induced voltages in distributed windings 61 Hopkinson, Dr. John 23 Huxley, T. H Preface I i Ideas, forced and natural 248 \ I Induction motor, elementary theory of 63 \J I equivalent to transformer 73 \J torque of 74 v/ with commutator rotor, circle diagram of 132 Induction generator 82 et seq. tests on 85 Interpole shunts 254 Interpoles in single-phase motors 254 J Jackson, D. C Preface Johnson, John Butler Preface K Kapp, Gisbert 196 definition of v open circuit current 29 "Electrician" paper, 1890 5 "Electric Transmission of Energy," 4th edition 6 his vibrator 191 Institution of Civil Engineers paper, 1885 4 on elementary theory of induction motor 63 Karapetoff, V 151 the secomor 145 Kelvin's report on Cataract Construction Co.'s Development 263 Kirchhoffs laws applied to squirrel cage 70, 71 Kittler, E., and W. Petersen 12, 152 Kraemer system 176 Krug, Dr. Karl 11 L La Cour, J. L 134, 176, 183 Lamb, Horace . 196 Lamme, B. G.. 1, 136, 179, 186, 246, 247, 253 concatenation 175 quoted 107 "The Story of the Induction Motor," quoted 4 Latour, M. C. A 250 compensated repulsion motor of 240 his motor, tests of 251 INDEX 269 Lawrence, R. R 26 Leakage coefficient 26 Leakage factor 86, 104 as affected by frequency 107 for different numbers of poles 106 historical and critical discussion of 110 influence of pole-pitch on 99 of single-phase induction motor 210 the effect of the air-gap on 95 Leakage flux, conventional but erroneous representation 102 Leakage paths of double squirrel cage motor 116 Leblanc, Maurice 187, 189 Scherbius rotor- 189 Lehmann 11 Load loss 94 Locked current, see short circuit current 86 Loss, iron, as shown in equivalent circuits 44 Loss lines 42, 43 V Losses, as represented in the circle diagram 45 Lydall, F 176 M Magnetic pull, unbalanced 198 Magnetizing current as affected by high tooth induction 60 Mailloux, C. 10 Maxwell, James Clerk. 1 Me McAllister, A. S 114, 217, 216, 252 his transformations 49 et seq. McCormick, Bradley T 261 N New Mexico, United States Battleship 78, 115 Newton, cited 264 New York, New Haven, & Hartford railroad 248 O Oerlikon Company '. 1, 180 Osnos, M 183, 233, 239 Time phase diagram of compensated repulsion motor 243 Ossanna 11 P Pennsylvania Railroad 258 Perry, John 196 , 270 INDEX Pertsch, J. G . 203 Pole-pitch, its influence on the leakage factor 99 Poly-phase commutators 129, 130 Poly-phase commutator for generation of leading currents 187 Potier, Alfred 18, 216 Power, single-phase from three-phase systems 258 Power factor, maximum power factor depending on leakage 30 Prescott, J 196 Pupin, M. 1 82, 211 Q Quarter-phase system, e.m.fs. induced in 62 R Recuperator 189, 190 Repulsion motor, single-phase 233 commutation 240 effect of brush-shift 238 effect of resistance 238 space-phase diagram . . , 234 speed in the diagram 237 time-phase diagram 236 torque 237 Resistance, primary, its effect on the operating characteristics of the motor 31 correction for it by Behrend's method 32 correction for it by method of reciprocal vectors 38 correction for it by Sumec et al , 33 Resistance leads 253 Rosenberg, E 203 Roth, Edouard 11, 152 Routh, Edward John 196 Ruedenberg, R 177 Russell, Alexander Preface S Saturation, affects short-circuit current 87 necessity of, in series poly-phase commutator motor 145 Scherbius, A 145, 176 Schrage, D. H. K 184 Scott, Charles F 1 Series poly-phase commutator motor 137 brush-shift in 140 slip of 142 space-phases and time-phases 139 torque of 141 Shallenberger 1 Aai INDEX 271 i/ Short-circuit current 86 affected by saturation 87 circle diagram of 148 error introduced by its use 94 Shunt poly-phase commutator motor total current of 150 equivalent model of 151 Shuttleworth, N 152 Single-phase commutator motors 246 different types of 249 Single-phase induction motor 204 circle diagram of 207 cross flux theory of 216 equivalent circuits of 208 experimental data of 213 leakage factor of 210 magnetizing current 214 magnetizing current of 205 no load current of 210 two-motor theory 205 Single-phase secondary, poly-phase motor with 173 Slepian, J 216 Slip, determination of 75 in the elementary theory 66 in single-phase induction motor 211 Slip-ring type, comparison with commutator type 135 Slots, closed 91 closed, short-circuit current with 92 in double squirrel cage motor 116 number of slots 89 open and closed, affect short-circuit current 87 .Space-phase, interchangeable with time-phase 74 \s Speed control, methods of 153 Squirrel cage, effect of end rings 73 first squirrel cage motor of C. E. L. Brown's 3 in the elementary theory 66 theory of 69 Stability of induction machine as motor or generator 83 Steinmetz, C. P 246, 247 Sumec, J. K 18, 201, 203 Formula of excentric magnetic pull by 201 his circles of single-phase induction motor 224, 225, 226 Uses Behrend's stray-coefficient 24 T Tesla, Nikola 1, 107 his patents 261 Tests of 20 H.P. motor made in 1896 93 Thomaelen 10, 18, 19 / 272 INDEX Thompson, Silvanus P 14, 16 on leakage factor m Time-phase interchangeable with space-phase 74 Torque curves of series poly-phase commutator motor 144 Torque, dead points in 79 in single-phase induction motor 211 of induction motor 74 Transformer, equivalent to induction motor 73 with capacity load 47 with inductive load 46 V Vectors, reciproca. 38 et seq. term misued for e.m.f. and current 4 Vibrator 191 theory of 192 W Wagner-Fynn motor ' 249 Walker, Miles 16, 176, 258 Washington, Baltimore & Annapolis single-phase railway 246 Waters, W. L 261 Weaver, William D Preface Westinghouse Company, Kapp vibrator built by 194 their single-phase motor 253 Westinghouse, George 247 Wightman, M. J 247 compensated repulsion motor of 240 Williamson, R. B 261 Wilson, British patent of 137 "TS BOOK IS 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. REC'D :.. APR 1 1 1961 REC'D LD JUN 5 '65 -H AM YC 19463 UNIVERSITY OF CALIFORNIA LIBRARY