Principles of Motor Design

UNIVERSITY OF TORONTO

LIBRARY

Acme Under

Library Card Pocket " Ref. Index

Pat.

File."

Made by LIBRAET BUfiEAU

PROPERTY OF ELECTRICAL LABORATORY, FACULTY OF APPLIED SCIENCE. Date

LABORATORY, PROPERTY OF ELECTRICAL SCIENCE. FACULTY OF APPLIED

Date..

PROPERTY OF ELECTRICAL LABORATOfiT" FACULTY OF APPLIED SCIENCE.

I).-.

THE INDUCTION MOTOR A Short

Theory and Design, with Numerous Experimental Data

Treatise on

its

and Diagrams

B.

BEHREND

A.

i

M

i-ii!

her lust.

C.

E.,

Member

Inst.

E.

.,

Germany; Member

E. E., Switzerland', Associate Member American Inst. /.'. ; .Formerly Assistant Chiff Electrician of the

Oerlikon

Engineering Switzerland

lust.

K.

Works,

" The absence of analytical difficulties allows attention to be mart easily con,entrated an the physical aspects / the question, and thus girts the student a mure rii'iii idea and a mure inanaxeable grasp oj the subject than hf would ta lyt ica I symbols." I.

NEW YORK Mi-GKAW PUBLISHING COMPANY 114

I.

I

B H K

T Y

STREET

-I.

TlloM-'is.

COPYRIGHTED,

1901,

BY

ELECTRICAL WORLD AND ENGINEER (INCORPORATED)

TO MY FRIEND AND TEACHER

MR. I

GISBERT KAPP

INSCRIBE THIS WORK.

PREFACE. The

literature of electrical engineering has

become so vast and ex-

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 tensive that

impossible for any

is

it

My

his credentials.

justification for writing this tract will be

though almost

in the fact that,

all

motor has received

enlisted the industry of authors, the induction

comparatively

attention from competent engineers.

little

found

branches of applied electricity have

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

purely analytical theories will show plifying and elucidating so

how

complex a

far

I

its

theory with the

have succeeded

subject.

The

in

sim-

graphical treat-

ment of abstruse natural phenomena is constantly gaining ground, I quote with satisfaction the words of so great a mathematician

and

as Prof. George bridge,

who

Howard Darwin, Fellow

says on

p.

of Trinity College,

Cam-

509 of the second volume of Lord Kelvin

and Prof. Tail'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

method check of the results.

little

book

at every step the

A

I

have endeavored to

let

inductive

mathematical or graphical deduction

wide experience with mono- and polyphase

alter-

nating current induction motors, gained at the Oerlikon Engineering

Works, Switzerland, has enabled me reader

many

who

is

willing to profit

to

do

so.

Thus

the careful

by the experience of others, will find

valuable hints and results which he can turn to account in his

Many

practice.

down

ciples laid

induction motors have been designed on the prinin this little treatise,

and

in

no case has the theory

answer the questions suggested by observation. The writing of this book has been mainly a labor of love.

failed to

who know

Those

of the troubles, cares and labor involved in 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.

has always given to me. of ELECTRICAL

W. D. Weaver for the encouragement he To Mr. T. R. Taltavall, Associate Editor

WORLD AND ENGINEER, who has taken I feel very much indebted.

endless pains

with the proofs of this book,

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 Col-

lege of Mechanics and Engineering, for the invitation as non-resi-

dent lecturer which he extended to me.

Jackson

I

am

To him and

to Prof. D. C.

greatly indebted for the hospitality conferred

stranger within their gates.

upon the

CONTENTS. PARA-

CHAPTER. I.

II.

PACE.

The General Alternating-Current Transformer A. The Character of the Magnetic Field in the

III.

IV.

V.

The Formula

for

the Three-Phase-Curent

The Short-Circuit Current and The Leakage Factor Design

of

a

Motor..

the Leakage Factor...

Three-Phase-Current

Motor

for

VII. VIII.

of a

Appendix

of

Transformer

rent 1

Appendix II Appendix III

24-28

19

29-38

29

39-64

54

Single-Phase-Current

The Polar Diagrams

18-23

15

42

The Single-Phase Motor Calculation

ii

200

Horse-Power VI.

the

General

1-17

Poly-

Motor

phase B.

i

GRAPH.

Motor

65-93 94-

1 1

o

63

111-131

73

132-162

Alternating-Cur-

89 96 100

THE INDUCTION MOTOR CHAPTER The General 1.

The problem

engineer

is

I.

Alternating Current Transformer.

of problems, in the solution of which the electrical

deeply interested, and which underlies

all others, is set

be-

fore us in the form of the alternating current transformer possessing

considerable leakage and a relatively large magnetizing current. 2. A transformer with an open secondary takes from the primary mains just so much current as is necessary to produce a magnetic field which can balance the primary voltage. This current neglect-

moment hysteresis and eddy currents lags behind the primary voltage by a quarter of a phase hence the work done by this current is zero, and the magnetizing current is therefore a "wattless" ing for the

;

current.

This consideration

leakage.

The magnetizing

the sense in which this term

about this

say,

true only for a transformer without

is

generally used.

We

shall learn

to

and reaction of the

primary and the secondary system of the transformer, permitting a larger current to flow.

mary

is

the secondary of the transformer be closed through a resist-

ance, then the impedance represented by the action

make

more

Chapter VIII.

you throw a non-inductive load upon the secondary, that

3. If if

in

is

current need not be a wattless current in

the is

assumption

that

the

transmitted without loss

If,

is

diminished,

for didactic purposes,

we

whole magnetic flux of the priinto the secondary, and vice versa,

then the vector of the primary current must be composed of two vectors, the one representing the magnetizing or wattless current,

lagging behind the terminal volts by a quarter of a phase, and the other representing the watt

irrent i

and being

in

phase with the

ter-

THE INDUCTION MOTOR. minal

volts.

resistance

is

Thus the vector of the primary current for any external determined by the locus of the point A, Fig. i, which is

the straight line

AD

parallel with the vector of the impressed

The energy consumed by

the transformer

&

(i) 4.

The

gram

is

=e

e.

m.

.

i

cos

fi

introduction of leakage into the transformer changes the dia-

as follows

The

:

through the primary

coil

total

number of

lines of induction passing

must remain constant

voltage remains constant, neglecting for the sistance of the coil.

The magnetomotive

as long as the terminal

moment

force of the

the ohmic re-

main current

produces a stray-field proportional to the driving current; this

added vectorially magnetic

line,

*A

field

to the

main magnetic

is

that

A

field,

field

generates the constant

The result of these acdoes no longer move in a straight

included by the primary

and reactions

tions

f.

given by the equation

coil.

but in a semi-circle described upon the prolongation of

OD

C).

writer worked out the theory here given in the summer of 1895, and sent the paper to the Elektrotechnische Zeitschrift, Berlin, where it was published in February, 1896. Meanwhile Mr. A. Heyland, in some letters to the above-named paper, used the same diagram without, however, giving any proof. When Mr. Heyland's letters were published I inserted a note in my MS. referring to them. I have since, whenever I had an opportunity, given Mr. Heyland ample credit for his priority, and I have done it with satisfaction, as I really admired some of his later papers very much. Mr. Steinmetz informed me some time ago that he had found this relation as early as 1893, but that commercial reasons prevented him from pubhistorical

lishing.

remark may not be out of place here.

The present

GENERAL ALTERNATING CURRENT TRANSFORMER. It is

'

of extreme importance for us to clearly understand these rela-

form the basis for

tions as they

5. In Fig. 3

mary,

or, in

OA

is

all

further reasoning.

the vector of the magnetomotive force.of the pri-

other words, the total number of lines of force (not in-

duction) produced by the primary current, and corresponding to the

number of ampere-turns. Not

all

the lines of induction which the pri-

mary current generates can reach

the secondary of the transformer.

FIG. 3.

Let us assume that the amount Vi

.

O A,

lines

Vi

from the primary

of the total

vt

.

OB

1

AA

number of

the

is

sum

of

and

i

lines that

OB

equal to

and measuring the

OB

1

loss of

represent the vector

lines of induction of the secondary,

number of

A

.i 1

lost,

to the secondary. Let

and

OB=

extend into the primary, vt being

again a coefficient smaller than one, then tor

h OA* being

1

being a factor smaller than one

must be equal

we

see at once that the vec-

to the vector of the

magneto

THE INDUCTION MOTOR. up the magnetizing current. The vector of magnetomotive force is represented by the line O C.

motive force which this

The

6.

sets

lines of force

which are common

to both

primary and secO A and B

ondary, are the effect of the two magnetizing forces

;

while the lines of induction which pass through the secondary only,

can be found as the resultant of

must be perpendicular tions of

O

understood

= Xi =X

OA OB

OA

2

1

OD

is

O B,

as

OB

is

OB

l .

This resultant,

produced through the

O

G,

oscilla-

list

will help to

make

the diagram

more

clearly

:

is

the magnetizing force of the primary.

is

the magnetizing force of the secondary.

OA

=

0~B

=

the field balancing the terminal voltage,

8. It will readily be seen that,

sistance of the primary racy,

and

G.

The following

7.

to

OA

OD

is

constant

Z

if

may

if

the drop caused by the

be neglected without too

the terminal voltage

HA D= Z HOG ~C~K

C~G

Z

6

OK

OK X 2

=

vj)

(i

-*(-=--') O~D

.

4

Vl

is so.

ohmic

much

We have,

re-

inaccuFig. 3,

GENERAL ALTERNATING CURRENT TRANSFORMER. Hence,

.

X, = -^ OD = ** (v

f

sin &

rTn \

In words, this means that Xi semi-circle described

LD

upon

ITD

=

-jr *

\

v*

may

be represented as a chord in a

.

( \Z/!

If

we want

to take

and having a diameter

as basis,

O~D

-)

A/

Vt

from the diagram Xi

directly without having to

FIG. 4.

multiply by

v\,

we have

to join

A

and

K

by a

circle.

For a more de-

tailed treatment of all these points I refer the reader to 9.

We call the quotient

-

LD

Chapter VIII.

the leakage factor a of the transformer,

and have therefore I

(2),

The leakage

coefficient a is the

most important factor 5

in the

theory

THE INDUCTION MOTOR. of the alternating current transformer, and a successful design must endeavor to keep a as small as possible. The determination of a will

be treated of 10.

in a

chapter devoted entirely to the leakage factor.

The following

table contains the results of a series of measure-

ments as a corroboration of the theory. The data were taken from a three-phase current motor, the armature of which was standing

still

;

the-

whole apparatus was thus acting as a transformer with considerable leakage.

The

field

contained 36 closed

resistance of each phase 0.045 ohms.

round

slots, 7

holes, 3 conductors in each hole;

0.172 ohms.

Number

of poles,

PRIMARY CIRCUIT.

6.

conductors in each slot

The armature

;

contained 90

resistance at each phase

Frequency, 48

.

GENERAL ALTERNATING CURRENT TRANSFORMER.

We

13.

should make a great mistake were

we

to

assume that

this

value would give us a leakage factor a true to reality, since in our case,

where the

slots in armature and field are closed, v depends greatly upon the saturation of the thin iron bridges closing the slots. The

saturation of these bridges

is

dependent upon the intensity of the

FIG. 5.

current

;

beyond a certain intensity

constant.

Assuming a

z/j

v,

to be equal to

=_

i

and therefore a

vt, we should get

= 0.235, instead of

0.90-0.00 as follows from the diagram. 14.

It

may

'-$

1

68

is

practically

for

= 0.098,

be advisable to emphasize that in the derivation of the

ohmic resistance of the primary has not been taken into account. As this point is of extreme theoretical and practical impordiagram the

tance,

we have

to dwell

on

it

at

some

length.

THE INFLUENCE OF THE RESISTANCE OF THE PRIMARY UPON THE DIAGRAM. 15.

The

semi-circle

L

i' 2' 3' 4'

D, Fig.

current for a constant terminal voltage

5,

represents the locus of the

OE, upon

the primary resistance be negligible. 'The arc 7

I

the assumption that 2 3 4

is

the locus of

THE INDUCTION MOTOR. the current

if

we assume

that the drop through

ohmic resistance

in

the primary amounts for point 4' to 10% of O E. Finally the ordinates of the curve i* 2" 3" 4" represent the amount of watt-com-

ponent of the current that

would be superfluous here

available in the secondary circuit.

is

It

anything about the manner in which

to say

these curves have been plotted, as everyone familiar with polar dia-

grams

will readily understand

It is of

assumed

importance to note

to have a value

deviate at

from each

smaller than

in

it is

son with LD, and to

draw a diagram

our

if

L

i' 2' 3' 4'

If

other.

In reality,

all.

lines in the figure.

though the primary resistance was

which exceeds about

in practice, yet the curves

tively little

from the

it

that,

OD

figure.

times the real value

2 '3 4 deviate compara-

I

OD were zero, then they would not LD is almost always considerably If, however, OD is large in compari-

+-

the primary resistance

like Fig. 5.

five

D and

is

considerable,

we have

This will be the exception and not the

rule. 16.

Thus we have learned

that the influence of the ohrnic resistance

upon the locus of the current

in

is,

most

but the energy dissipated in the resistance, into account

;

this

practical cases, negligible;

cases to be taken

is in all

can be done by deducting the watt-component cor-

responding to the ohmic loss from the ordinates of the semi-circle

LD.

We thus arrive at a curve similar to

i" 2" 3" 4".

GENERAL CONCLUSIONS AND SUMMARY. 17.

We

are

now

enabled, with the help of the diagram, to solve any

We

shall,

many

prob-

question pertaining to the alternating current transformer. in a later chapter, discuss in detail for a concrete case the

lems of interest which this diagram permits us to solve; here we shall merely summarize the main conclusions at which we have arrived.

In Fig. 6

ing current

OA to,

represents the primary current

and

AD

is

equal to

vl

2 z'

2

in

ti,

OD

which

HI

the magnetiz-

and

n* are the

7?j

number of turns

in the

primary and the secondary, respectively. 8

GENERAL ALTERNATING CURRENT TRANSFORMER. The circle

smallest lag

LD

-, 20 HA = cos HO

It is

=

(T

OO



20

I

I

+ .

(3)-

20+1

This equation enables us to predict the maximum power factor if the leakage coefficient a is known. I will here premise that the starting current furnishes a value for the determination of the diameter of

8

8 FIG. 6.

the semi-circle, while the magnetizing current can always easily be

measured.

This method of determining the

maximum power

factor

-

THE INDUCTION MOTOR. attainable, thus

account of

The

full

recommends

itself

not only to the designer, but, on

and accuracy, also to the customer. curve / of higher order in Fig. 6, which can easily be

its

simplicity

constructed, represents the power factor as a function of the

input

;

the dotted line // shows the

without any leakage.

10

power power factor for a transformer

CHAPTER A.

II.

The Character

of the Magnetic Field Polyphase Motor.

18. The magnetic field duced by three windings,

a three-phase current motor

in I,

in

and

II,

III

III, Fig. 7.

the is

pro-

If the current in

I

FIG. 7.

Ill

is

a

maximum, and

if

the currents vary acording to a simple

sine curve, then the currents in

I

and II

II are

each equal to half the cur-

THE INDUCTION MOTOR. rent in III.

The magnetomotive

by the ordinates of the curves

forces of each phase are represented

I, II,

and

III respectively.

Each ordinate

measures the magnetomotive force produced in that place of the circumference in which it is drawn. The adding up of the three curves

drawn below.

yields the thick line curve If the

magnetic reluctance

ference, in other words,

if

is

the

same

at every point of the circum-

the reluctance of the iron

is

negligible,

then the flux, produced by the magnetomotive force represented by the thick line curve,

Hence

We

flux.

is

proportional to that magnetomotive force.

the thick line curve call the total

may

be taken as

number of

"a

representation of the

and we

lines of induction $,

as-

II

FIG. Q.

FIG. 8.

sume

now

that this flux varies according to a simple sine law.

proceed to calculate the

e.

m.

f.

induced by this

field

We

shall

upon each

phase.

We have

tacitly

assumed

in a practically infinite case, yet this

that the coils

number of

assumption

may

slots.

I, II,

and

Though

III are distributed this

cannot be the

safely be made for our present pur-

pose. 19.

It is

obvious

centrated in one

that, if the

convolutions of each phase were

alj

con-

slot, the effect of the oscillation or the traveling of

12

THE MAGNETIC FIELD would be

the field

to set

up an

THE POLYPHASE MOTOR.

IN

e.

m.

equal to 2.22

f.

~z

.

,



10"* volts;

however, through the distribution of the winding, only the parts of the flux not covered with hrtchings can produce an

by this formula, while the hatched parts of the

The induced

siderably smaller effect.

follows

:

The width

of the coil

2b

is

Per unit length there

spread over 2b.

m.

f.

expressed

have a con-

can be calculated as

f.

conductors -

are, therefore,

- conductors.

in the coil

conductors,

hence the element d

x

lines of induction

threading the conductors in the element

equal to

contains d

.

2b

We

represented by the hatched area.

**

= 2.22 ~

de

x

m.

e.

there are

;

e.

field will

.

dx

.

.

**

.

The number

of

dx

\t

have, therefore,

10-8 volts.

.

2 b

=