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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-

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-

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