Principles Of Motor Design

UNIVERSITY OF TORONTO

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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 << likely to attain if he merely regarded electrical phfnonteiiii through a cloud o/ a >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


is

determined by the intersection of the semi-

with the semi-circle

HAO,

The

as can be seen at a glance.

cosine of this angle can be expressed as follows.

HA =HD =-^>-, 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

.

,

<j>

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

=
*,

Hence

= 2.22 ~

de

.

--..

r&.^..

n -

.

ib

-L

($,.x*.(tx-\ 26

2

.

io-

[jV^ - /B^lffIJ

JL

^L.

2

I

J

2b

o

_n

.

.

I0 ,

,o

.

6 J

= 2.22 ~

e

.

.

-"

.

ifi

2

We

have *

=

.

-.

.

io-

3

,

therefore

2


(4)

This

is in

the width b

=

2.

words, is

The

e.

m.

f.

induced by the

two-thirds as large as the 13

e.

m.

* upon a coil of which would be in-

field f.

THE INDUCTION MOTOR. duced by the same field upon a lodged in only one groove. The It

case

is

not distributed, but

represented in Fig.

is

9.

not be amiss to call attention to the fact that a coil like that in

may

would produce a rectangular magnetic field twice as large as in the figure. Hence the inductance of the flat coil is

Fig. 9

that

which

coil

latter,

shown

one-third as large as the inductance of the coil lodged in one

20. The

m.

e.

f.

generated by the

field in Fig. 7

can

now

slot.

easily be

calculated.

The number is

The

m.

e.

of lines of induction represented by the white area in

equal to

Fig. 7

this flux is

produced by

f.


The hatched

= 2.22 ~ .

z

.

( -5. V 4

.

.

.

t

b

.

.

(&\ io*

/

3

areas represent a flux equal to

*""**6 2 The

e.

m.

f.

produced by

eu

= 2.22 ~ .

.

this flux is

z

(i

.

-|-

/

.

*

.

10-8

.

(B)

Hence *

The

= e + ^n = 2.22 ~

total flux



.

l

amounts

= -3-

b

.

.

.

z

(

.

p .t.b.Qt]

lO' 8

to

t

.

.

Z

(B,

hence

12

6

= 2.22 ~ .

.

.

$

10-8

.

21 (5)

21.

e

= 2.12 ~ .

.

The ampere-turns

z

.

in

$

.

io'

8

volts.

each phase which are needed to produce

the induction (B in the air-gap are determined by the consideration,

which follows immediately from 2 (o

.

4

JT

.

.

/

Fig. .

7,

1/2.

)

that

= (B

2

A

THE MAGNETIC FIELD In this equation

to

is

the magnetizing current, n the

ductors per pole and phase, and of the iron

From

is

THE POLYPHASE MOTOR.

IN

A

The

reluctance

supposed to be negligible.

this equation follows

=

/

(6)

n

22. If the reluctance of the iron ($>

A

1.6

(B 2

duction

number of con-

the air-gap in cm.

amperes.

V7=2

.

not negligible, the magnetic in-

is

has to be determined point for point, which can easily be

done with the help 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

it is

not objectionable to have an induction

of 15,000 or 16,000 in the teeth, as this high induction hardly increases the magnetizing current on account of

its

being limited to a

very small part of the surface of the pole. 23. I have not entered upon a detailed consideration of the elementary

phenomena

in the

Gisbert Kapp* and

which

it

polyphase motor, as they have been treated by

Andre Blondel* with a

would be impossible

The Formulae

B.

for

me

for the

and clearness

lucidity

to surpass.

Three-Phase Current

Motor. 24.

It

remains to prove that the theory of the general alternating cur-

rent transformer

is

directly applicable to the polyphase motor.

That

this

is true, as long as the armature of the motor is standing still, is evident. If the armature runs at synchronism, the number of its revolutions cor-

responds to the frequency of the feeding current

If the

~i.

ture runs slower than the

field,

an

armature proportional to the

e.

m.

If the

f.

is

induced

in the

the difference being

armature resistance per phase

proportional to

~'

~*.

is

rt

,

slip

arma2,

~i

then

~r

then a current will flow

The same current can be

transformer with the secondary at rest,

~!

-

if

attained in the

the resistance be thus

chosen that the relation exists 'Electric Transmission of Energy, p. 304

dix

I.

*L''Eclairagt Electriqitt, 1895.

15

and following, reprinted

in

Appen-

THE INDUCTION MOTOR.

r,

=

-^-

(7)

~'

2

r

r,

signifying the internal and external resistance of the secondary of

the transformer.

Under

same diagram represents the

this condition the

currents in size and phase in the transformer as well as in the motor.

That

must be so becomes

this

impedance of the motor a constant.

is

The impedance

clear as soon as

VV2 3

equal to

of the transformer

Vr*

+A

.

we remember

A

-f-

~

(~ x

2 ) *,

that the

A

being

equal to

is

2

~!

,

hence

=

~i ri which equation

~a) r

(~i

identical with

is

Eschenburg, Oerlikon, for having

>

Credit

(7). first

due to Dr. Behn-

is

prominently put forth

this

relation.

25. The Torque.

magnetic

field,

equal to

^

Imagine the armature to be turned against the supposed to stand still, with an angular velocity

which

is

2-

If

D

9.81 it

being the current

We

in,

.

is

D

.

(!

,)

=3

is

*

r,,

of,

each phase.

have

p />

z 2

and rt the resistance

'

if

mkg, then we have

the torque in

the

number

And

of north or south poles. 9.81

From

'

.

D

.

u2

also

= P watts.

these equations follows 9.81

9.81

.

.

D mkg .(27T. -^ 2 n D mks -^p .

16

w2 )

P

=3

z2

.

*

3

*2

THE MAGNETIC FIELD 61.6

(8)

.

THE POLYPHASE MOTOR.

IN

D mks = -^~

+ />,*,).

*r2

(3/ 2

i

we want

If

have the torque

to

8.5

The torque

Dft.

.

foot-pounds, the formula

in

ibi.

= -^-

*

(3

~i

r2

*j

+ P wat

At

26. Starting Torque.

starting

dissipated into heat

into the motor,

is

must be equal

to the ordinate of

P ;

our

is

in the

ts).

sum

therefore, proportional to the algebraic

is,

output of the motor plus the energy dissipated

is

of the

armature.

zero. All the energy that goes

therefore. 3

2

(z\

If TI

circle.

rt

and

*

it

-f-

r 2 are

rt

=)

3*

known,

the point on the circle corresponding to this case can easily be found

by trying.

A glance at our diagram running torque

;

shows us that we can

attain at starting

but the starting current will always be equal 4

never smaller than, the running current, unless the motor

under conditions, as higher or lower voltage, under load.

The energy which

27. Efficiency.

etc.,

m.

ii

i.,

the current in each phase, then

we have

obtain the output

losses,

started

from those

If e\ is the

impressed

we have

= 3'i*iw*

(9)

To

is

any and

flows into the motor neglecting

hysteresis can be taken from the diagram. e.

different

to,

and the

friction.

to deduct the

ohmic

losses, the load

Hysteresis and eddy currents may be taken

into account, since they are practically constant at all loads, as though

they were ohmic losses

no leakage loss

at

all,

in

a coil so placed upon the primary as to have

the energy wasted in this coil being equal to the

through hysteresis and eddy currents. (10)

P=3^

/!

cos *

4

We '

3

t\

rt

have, then, 3

>

i,

rt

F Q

Prof. Silvanus P. Thompson, "Polyphase Electric Currents," 213, says of a polyphase motor: "This motor starts under full load, taking less than full-load current." This is an impossible figment. In the second, vi-ry imu-h enlarged, edition of Prof. Thompson's interesting honk, this statement is changed, and we read on page 255: "This motor starts under full load, taking less than twice the full-load current." The motor should theoretically not take more than full-load current, but just as much, and in practice this is also generally obtained, as, in this case, the rotor is equipped with slide rings. first edition, p.

17

THE INDUCTION MOTOR. F

being the friction loss in watts,

Q

the loss through hysteresis and

eddy currents.

The

v

efficiency

is,

28. The most important formulas for the design of the polyphase

motor have thus been determined, and them in the form of a table.

Maximum Power (3) J/

it

be useful to give

will

Factor

.......... cos



=2a

i

-(-

Magnetic Field (5) ..............

e

=2

12

.

~

.

zv

.

.

*

8

.

io- volts.

Magnetising Current ($>

.

n

2

A V2

1.6

(6). .

Slip lip(7).

^r

=

3

Torque (8) ... .61.6

.

D m kg = -^b

(3

h *r2

+ P -watts)

Impressed Energy (9) ..............

8

= 3*1

1\

cos?

Output (10) ......... P-watts

= 3*i h costyT)

Efficiency (11) ..............

1= 3

'i *i

cos

18

t\

CHAPTER The 29.

III.

Short-Circuit Current and the Leakage Factor

An

invaluable aid in the design of polyphase motors

forded by the short-circuit current characteristic.

is

af-

In a motor hav-

ing a squirrel-cage armature, the starting current under different

voltages

is

identical with the short-circuit characteristic.

and of the

If the re-

were known, the power factor of the feeding current could be calculated; thus it would not even be necessary to use a wattmeter unless very accurate measuresistances of the armature

ments were required.

field

Theoretically, then, the short-circuit char-

acteristic is sufficient for the determination of the leakage factor; in

practice,

however,

it

will, in the

majority of cases, be unadvisable to

depend upon the short-circuit curve, on account of the corrections which become necessary. If the total resistance of the motor is

amounts

small, the lag of the current

to nearly a quarter of a period

;

the inductance of a motor at standstill (short-circuit) should always

be as small as possible; therefore, a very large current will go

through the motor at

more or

less

full voltage.

Now,

the leakage factor

is

always

dependent upon the strength of the currents which

cause the leakage, hence the leakage factor at starting with only a small resistance in the armature

smaller than structed.

In fact,

semi-circle,

therefore,

I

may be

very different from

as a rule,

the leakage factor upon which the diagram

we do

work the motor

not

in that

which corresponds to the short-circuit shall not

make much use of

the

not so

more

avail myself of

it

much matter whether

where the

characteristic.

relative value

is

for comparative purposes

the absolute value

of importance.

fluence of a closed or open slot, of the 19

con-

If,

the short-circuit curve for

the determination of the absolute value of the leakage factor, all

is

quadrant of the

is

where

I shall it

does

correct or not, but

Such questions as the

number

in-

of slots, of the air-

THE INDUCTION MOTOR. gap, and of the pole-pitch

upon the leakage factor, can all be answered by consulting the short-circuit characteristic. I shall proceed to discuss, point for point, the influence of these factors on the leakage coefficient.

THE 30.

The curves

A

and B,

SLOTS.

in Fig. 10, represent the short-circuit char-

acteristics for a closed slot of the slots of the

shape marked B.

shape marked A, and of the open

The

slots of the

armature were closed

B

120

100

200

300

400

500

600

700

900

800

FIG. IO.

in

each case.

able only

if

Curve

C

shows the

ideal short-circuit curve obtain-

the leakage paths contain no iron at

all.

Curve

D

is

the

magnetizing current reduced from the measured value of 42.2 am20

SHORT-CIRCUIT CURRENT AND LEAKAGE FACTOR. The

peres at 1900 volts to the various voltages in the diagram.

in-

crease of the magnetic reluctance of the main field through the opening of the slots proved too small to influence the magnetizing current in

any perceptible manner.

We see that for voltages above 600 the curves A, B, and C converge ir

other words, the short-circuit current at the

is

maximum

;

voltage of 1900

Hence

almost the same whether the slots are open or closed.

will be

the

full

flow of energy which can be impressed upon the motor

by no means so much dependent upon the form of the

The tendency of the closed fundamental diagram in the way shown by the

slot as

appear at first sight.

slot is to

the

full line

would change

curve in

FIG. II.

maximum power and the current for the same output yet the maximum output of the motor is hardly re-

Fig. ii.

From

factor

0.715 instead of 0.755,

is

this

58 instead of 55,

duced as the

curve follows that, though the

maximum

the broken line,

ordinate of the

is

ordinate of the semi-circle, represented by

only inconsiderably larger than the

full line curve.

excellent motors can be built with closed slots, in no

motors with open itself

slots.

slot is

cheaper?

way

inferior to

objection to closed slots, then, resolves

neglecting at present the load losses which

caused by the bridges

Which

The

maximum

If the iron bridges are kept very thin,

may

probably be

into a commercial one, viz., the question,

Labor being expensive 21

in

America,

it

is

THE INDUCTION MOTOR. cheaper to wind the coils on forms and to use open

builders of polyphase motors have preferred to to

wind by hand.

Coils

wound

The

slots.

cost

some of the leading use closed slots and

of labor being considerably smaller in Europe,

outside of the machine require

more

insulating material, and as the room is always somewhat scant in polyphase motors, the European method offers some advantages.

On

the other hand,

we have

mind

to bear in

that the greater ease

with which machine-wound coils can be exchanged, should not be underrated, and might even be bought with the loss of another ad-

A

vantage. sign

is

design

more or

is

the least

A

little

number of advantages with

it

OF SLOTS PER POLE.

we obtain the least amount we have per pole. Theoretically, have as many slots as possible. From

consideration teaches us that

of fluctuation in the field the then,

a compromise, and the best de-

number of drawbacks.

NUMBER 31.

less of

that which combines the greatest

would be advisable

more to

slots

a commercial standpoint, however, Either extreme

possible.

is

it is

advisable to have as few as

impracticable

;

we must

try to strike the

golden mean. 32.

The

number of slots upon The more conductors we have

influence of the

readily be seen.

will be the leakage field

surrounding the

with

for

slot.

the leakage can also in a slot the larger

The

active field is

our present consideration

enough accuracy whether we distribute the same number of cbnductors

or in many.

The

e.

m.

induced by this

f.

field

upon

same

the

in a

few

slots

these conductors

may, therefore, be set constant independently of the number of slots. Let us take a concrete case. If we have, for instance, 100 conductors arranged in 5 field

slots,

per slot is

to their

number.

there are in each slot 20 conductors.

The

leakage

produced by these 20 conductors, hence proportional

The

e.

m.

the 20 conductors in each

f.

induced by the leakage

slot, is

field

per slot in

proportional to 20 times 20.

22

Hence

SHORT-CIRCUIT CURRENT AND LEAKAGE FACTOR. the total is

e.

m.

f.,

induced in the 100 conductors by the 5 leakage

proportional to 20

33.

Now,

let

X

20

X5=

fields,

2000.

us arrange the 100 conductors in 10

Each

slots.

slot,

then, contains 10 conductors, the leakage field per slot being propor-

The

tional to 10.

e.

ductors in each slot

m.

e.

10 is

X

f.

induced

10

X

10

m. is

f.

in all the

= looo.

induced by the leakage

proportional to 10 times

field in

conductors in the 10 slots

In other words, the counter

the 10 con-

Hence

10. is e.

the total

proportional to

m.

f.

of leakage

twice as large in the case of 5 slots as in the case of 10 slots.

The above

calculation rests

upon the assumption that the reluctance is the same in the two cases under

of the leakage path of each slot consideration.

Though

this is true

only with some qualifications, yet

the argument clearly shows the superiority of far as leakage

34.

is

A general

be given.

To

many over few

slots, as

concerned.

rule for the

take

most favorable number of

more than

5 slots per pole

slots

and phase

can hardly

in the field

is

FIG. 12.

hardly advisable unless the pole-pitch be very large, as in motors for

low frequencies.

As

the

maximum 23

of ampere-conductors per slot 600

THE INDUCTION MOTOR. might be put down; but this should be no rigid fewer than 3 slots per pole and phase, if possible. conductors in a

slot,

the leakage field

is

rule.

Never take

you put too few not saturated, and thus the If

STARTING CURRENT

10O

300

2OO VOLTS

400

FIG. 13.

advantage of a great number of

slots

may

be balanced by this greater

leakage.

Thus we

see that in spite of

the individual

35.

To

many

rules, still

a great deal

is left

to

judgment of the designer.

enable the reader to form for himself an opinion of the ac-

curacy of the theory,

I

give the complete experimental data of a small

three-phase current motor with short-circuited armature.

SHORT-CIRCUIT CURRENT AND LEAKAGE FACTOR. The motor tween the

and

field

slots in

shall develop 20 horse-power, at a voltage of 380 be-

lines,

and a frequency of 47

p. p. s.

were closed, but the bridges were

armature and

six poles, therefore

The following

field

its

The

thin.

are represented in Fig.

synchronous speed was 940

slots in

armature

The shape of the 12. The motor had r. p.

m.

shows the starting or short-circuit current a function of the terminal volts measured between the lines table

:

Volt.

as-

THE INDUCTION MOTOR. This point Fig. 15.

lies

very near the

The diameter

maximum

of this circle

ordinate of the semi-circle in

122.5 amperes, the magnetizing

is

.current 4.5 amperes, hence

a

=o

=

.

0367

.

122.5

The maximum power COS

tf

ft

factor attainable

=

i 2<7-f- I

is,

equation (3),

= 0.93

Fig. 15 and the data given above clearly show the inaccuracy which

80

20 KILOWATTS FIG. 14.

would

arise if

we were

to use the short-circuit current as a

determining the absolute value of the leakage factor. 26

means of

SHORT-CIRCUIT CURRENT AND LEAKAGE FACTOR. 37.

1

want

to call attention to the load losses,

which are always pres-

ent in polyphase motors, the causes of which are, however,

still

vary

obscure.

The maximum

efficiency of this

motor

is

84.5 per cent.

The

losses

are: Hysteresis, eddy currents and friction

800 watts.

Ohmic Ohmic

600

loss in

primary

200

loss in secondary

Total losses

"

1600 watts. "

Output

13200 14800 watts.

Input

The energy which

the motor actually

watts, corresponding to

an additional

consumed amounted to 15600 the load loss of 6 per

loss

DIAMETER 122.5 A FIG. 15

cent of the output. load, until

it

The

load loss increases rapidly with increasing

becomes equal

to all other losses taken together.

THE INDUCTION MOTOR. Opening the

slots

has a decided tendency to diminish the load loss

considerably, therefore

it is

probable that the seat of this waste of

energy is in the bridges. 38. It has often been advocated to calculate the leakage dimensions of the

and

it

slots,

has been claimed that great accuracy

not of that opinion.

Though

at least theoretically

my

fields

from the

of the air-gap, of the pole-pitch, and so forth,

I

am

is

thus obtainable.

perfectly aware that

it is

I

am

possible

to calculate the leakage, yet I cannot close

eyes to the fact that such calculations are, of necessity, based upon

a good deal of guesswork the designer who chooses this a man groping in the dark with here and there a guiding ;

I should rather say, a will-o'-the-wisp, sometimes guiding

pa.th is like light, or, as

him

aright,

but much oftener leading him astray. I am suspicious of the a priori method it has proved utterly vicious in all departments of human ;

Upon a sound experimental foundation any mathematical superstructure may be safely reared and no desire to obtain a knowl-

knowledge.

;

edge of things, which are

in their

stood, should mislead us to build

The succeeding tors

very nature not yet clearly under-

upon the quicksand of imagined

data.

chapter will be devoted to an exposition of the fac-

which enter into the equation for the leakage

coefficient.

CHAPTER The Leakage

IV.

Factor.

are two questions of the most vital interest which in-

trude themselves THERE

How

does the

upon the designer. The first is, output a polyphase motor is capable

at every step

maximum

In other words,

of yielding, depend upon the air-gap?

if I

increase

any great extent decrease the output?

And

does a decreased air-gap increase the output of the motor?

The

the air-gap, does this to

motor

wound

second question

is this,

want

for eight poles, provided the frequency

to

wind

it

If a

is

for four poles,

and we

and the

duction in the air-gap remain the same, does the output decrease the ratio of 4

-=-

8, or,

what

inin

maximum outnow proceed to

relation exists between the

put of the motor and the number of poles?

I shall

answer these questions.

THE INFLUENCE OF THE AIR-GAP UPON THE LEAKAGE

FACTOR.

39. In order to determine the interdependence between the magnetic reluctance of the main field and the leakage factor, the following ex-

periment was made

:

The magnetic

field

the stator

of a three-phase

current motor was provided with two armatures, the diameters of

mm, and one

which were so chosen as

to create

mm.

currents were then measured as well as the

The magnetizing

short-circuit currents.

the experiment

an air-gap of

The following

0.5

1.5

tables contain the results of

:

Currents at 50

^ = 0.5 m. m.

of

THE INDUCTION MOTOR. Short-circuit Currents at 50 ~

A = 0.5 m. m.

THE LEAKAGE FACTOR. otherwise the leakage factor would have been proportional to the airgap. 40

I

I

10

40

20 I.

XL

100

(50

Volts Magnetizing Current. Short.- circuit Current.

.

0.5

120

m.m.

FIG. l6.

42.

Now,

this result is highly interesting.

shows, the energy that the motor

is

As

a glance at Fig. 18

capable of taking

in at the volt-

THE INDUCTION MOTOR. age of no,

is

the

same whether the air-gap

is

small or large. Hence..

40

30

.20

10

40

60

80

Volts Magnetizing Current. JI. Short -circuit Current. I

)

V

..

/\

^

100

120

, _ 1

5 tn.Tn

J

FIG. 17.

the overload that a

motor

This, of course, holds

is

alnc TO stand

is

independent of the air-gap.

good only of small

air-gaps.

THE LEAKAGE FACTOR. The

air-gap influences merely the strength of the magnetizing cur-

rent, but not the output.

FIG. 18.

^ V / V FIG. IQ.

The

curves of the short-circuit currents in Figs. 16 and 17 are

most straight

line

owing

al-

to the open slots in the field of our motor.

33

THE INDUCTION MOTOR. How

43. There remains to be answered the second question, leakage factor dependent upon the pitch of the poles

THE INFLUENCE OF THE POLE-PITCH UPON THE LEAKAGE 44. Before entering upon the experiments made to clear up

show deductively how the leakage

I shall attempt to

the

is

?

FACTOR. this point,

factor

may

be

expected to vary with the pole-pitch. Fig. 19 gives a view of the slots in a polyphase motor. lines

that

The broken

mark the leakage flux threading each slot. Now, let us assume we have a motor with 48 slots in the field, which we want to

wind as a two-pole,

four-pole, or eight-pole motor, the armature

being provided with a squirrel-cage winding, thus being suitable for

any number of

poles.

number of ampere-conductors per slot remains the same any number of poles, the leakage flux per slot also remains con-

If the

for

stant.

The

total

of the field

number of ampere-turns spread over

is

the circumference

then constant whether the motor has two, four, or eight

poles.

45. But the number of ampere-turns per pole to the

number of

pere-turns in the air-gap

is

inversely proportional

also inversely proportional to the

In other words, the magnetic

of poles.

is

poles; hence, the induction produced by these

field

am-

number

per pole, being propor-

tional to the product of the induction in the air-gap into the polepitch, varies inversely

46. The leakage

Hence, the

total

field,

is

as

we have

amount of leakage

pertaining to each lot

per pole

with the square of the pole-pitch.

is

also constant.

proportional to the

47. The ratio of leakage

seen, is constant for each slot.

sum

the

number of

field -=-

portional to the pole-pitch for the

main

of

all

the leakage fields

The number slots

field is

of leakage lines

per pole.

therefore inversely pro-

same number of ampere-turns per

slot.

34

THE LEAKAGE FACTOR. 48. This result

A.

is

verified

by the following

series of tests:

Three-phase current motor for 36 horse-power, 380 volts be-

tween the

lines, six poles,

42

Air-gap

~.

A = =

Pole-pitch / Volts between the lines.

0-62 30.5

m. tn. cm.

THE INDUCTION MOTOR. 50. For equal air-gaps (TI

a\i

we have o.n

= 0.0224 = 0.0664

f-

= 0.0396

Hence,

or, in

on

0.0664

<TI

0.0396

=

1.68,

other words, _ _ " '

in'

ffi

51.

The leakage

factor

inversely proportional to the pole-pitch, or

is

number

directly proportional to the

52.

By

the above experiments

age factor

is

of poles.

has been demonstrated that the leak-

it

and inversely promay, therefore, write the formula

directly proportional to the air-gap,

We

portional to the pole-pitch. for the leakage factor,

in

which equation

c is

a factor dependent upon the shape and size of

the slots, and upon a great still

many

For

profoundly ignorant.

we

other conditions of which

can be

be

left to

practical purposes, however,

determined with satisfactory accuracy, though

will

it

still

the designer to estimate the value of c between certain limits. slots, as

shown

in Fig. 19, c varies

between 10 and

are

c

For

15.

WINDING THE SAME MOTOR FOR DIFFERENT

SPEEDS.

53. Formula (12) permits us to determine the change in the output,

power

factor,

and so

forth, of a

motor wound for a

of poles, for instance, for eight, four, or two poles.

or 72 will

slots, it

can easily be

assume the induction

teeth, to

wound

that the motors are to be

all

three cases.

wound

number

If the field has

so as to satisfy this demand.

in the air-gap, or,

remain constant for

different

for the

which

We

same

clear, according to equation (5), that the total

36

is

48

We

the same, in the

will further

voltage.

number of

assume

Then

it is

active con-

THE LEAKAGE FACTOR. ductors must be proportional to the if

motor

the eight-pole

ductors in each of 72

To

rent

see equation (6).

Hence, as

($>

;

in other

words,

order to get the same induction

180, in

in

calculate the relative value of the magnetizing cur-

we need only know

in the four- pole

poles

then the four-pole motor must have 360,

slots,

and the two-pole motor the air-gap.

number of

has, for instance, 720 conductors, or ten con-

We

the

number of

have for n

active conductors

in the eight-pole

motor =360 ^i =90; and 4

and n are the same

in the two-pole in

per pole,

720 s-

motor motor

180 2

=

90.

.

each of the three cases,

follows that the magnetizing current also remains the same.

Two

= 90;

As

it

the

Poles

FIG. 20.

shape and size of the slots are the same in in

equation (12)

Hence, as the leakage factor

is

three cases, the factor

proportional to the quotient of the air-

gap divided by the pole-pitch,

we

be inversely proportional to the

number of

represented in Fig. 20.

maximum

all

for the leakage coefficient also remains the same.

A

find the short-circuit current to

glance at the

poles.

This

is

graphically

diagram teaches us that the

energy that can be impressed upon the motor, and, there37

THE INDUCTION MOTOR. fore, also

pitch.

very nearly the output, vary in proportion to the pole-

According

to the

diagram we g

two-pole motor equal to -> O -Q-

oo

=

o.io;

= 0.05

;

find the leakage factor for the

for the four-pole

and for the eight-pole motor equal

to

-

motor equal

=

0.20.

power factor in each case can now be calculated with the help of mula (3). This is done in the following table: Number Poles.

of

to

The for-

THE LEAKAGE FACTOR. Allowing again the induction ferent frequencies, which it

is

a

in

the air-gap to be the same for dif-

more or

less challengeable proposition,

follows from formula (5) that the total

ductors around the circumference of the for the pole-pitch

is

pole remains the same

tient

number of

active con-

must also be the same,

inversely proportional to the frequency, hence

the product of the frequency into the

57.

field

if

number of

lines of induction per

the induction in the air-gap

is

the same.

The magnetizing current, however, being proportional to the quoof the induction (B divided by the number of active conductors

per pole,

is

The

thus inversely proportional to the frequency.

leak-

FIG. 21.

age factor

is,

according to formula (12), directly proportional to the

pole-pitch, or inversely proportional to the frequency

pole-pitch

is,

in the case

to the frequency

hence

because the

under consideration, inversely proportional 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.

58. Fig. 21 shows the clock diagram for the same motor, but for different frequencies.

taking

in,

The maximum energy

and, therefore, also the

~, 50 ~, or 25 ~.

But the

that the

motor

is

capable of

maximum output, is the same for 100 maximum power factor is consider39

THE INDUCTION MOTOR. ably smaller for the high frequencies, as a glance at the diagram

shows.

The following

table

shows the leakage factor and the power

factor in relation to the frequency

Frequency.

:

THE LEAKAGE FACTOR. with the conditions underlying the leakage in polyphase motors.

am

I

from claiming for this treatment completeness or conclusiveon the contrary, I deem it a necessity to revise it by the light of

far

ness

;

am

I

forthcoming experience.

main pro-

tolerably confident that the

positions will be proved true, while

minor points may need some

qualification.

63. Considering the immense complexity of the phenomena in poly

phase motors, the greater or

calculate at

all, I

in

made

order to be able to

in

cannot forbear from wondering that so approximate

a solution can be attained at

herent

which hangs about most

less arbitrariness

of our assumptions which have to be

all.

It

may

be that there are errors in-

our fundamental assumptions which

all

so counteract one little

from

itself to

those

another as to cause the result of calculation to deviate but

experiment and observation.

who

are

This view will

commend

with any branch of physiology,

familiar

physiological optics

;

here

we have

for

instance,

the testimony of Helmholtz that

the eye, having "every possible defect that can be found in an optical

instrument," yet gives us a fairly accurate image of the outer world

because these various defects balance one another almost completely. 64.

The above remarks

tomed themselves

to look

will be distasteful to those

upon only one

who have

side of a question,

shut their eyes to the inevitable uncertainties that beset us in lectual problems.

I

was once taken

to task

by a

critic for

accus-

and try

to

all intel-

having ad-

duced experimental evidence qualifying my theory, and narrowing the limits of its application, and I was told that these experiments invalidated

my

argument, while

my

intention

upon the inwas obviously not critic. Lawyers may have to hide the weak arguments, but men of science are bound to to lay stress

completeness and the shortcomings of the theory

even thought of by

my

and spots of their point them out and expose them. The formulae and rules of the preceding sides

turned to account

in

articles

the following part by applying

calculation of a motor. 41

will

now be

them

to the

CHAPTER

V.

Design of a Three-Phase Current Motor for 2OO Horse-Power.

THE

application to practice of the theory

in the pre-

expounded

ceding chapters will best be illustrated by a concrete case.

this

end

I

To

propose to calculate a three-phase current motor for

an output of 200 horse-power at 440 r. p. m. for a frequency of 60 ~. Voltage between the lines 2000. I have chosen a rather extraordinary case, the speed of 440

frequency

r.

p.

m. being comparatively low, while the

unfavorably high, in order to show that

is

sible to build

it is

yet pos-

a satisfactory motor for such conditions.

65. The following are further conditions for the design (i.)

Normal

(2.)

Maximum

:

output, 200 horse-power. torque, 400 synchronous horse-power.

The motor must be able to start with the maximum torque. 66. The torque of a motor is measured in mkg. or foot-pounds. The product of the number of mkg. into the angular velocity of the rotor, (3.)

divided by 76, yields the

of the motor.

England

is

The

number of horse-power

unit

of

slightly greater than

To

available at the shaft

horse-power used

in

America and

the unit used on the Continent

we should have to many cases, we do not care much about the absolute value of the torque, we speak of torque corresponding to a certain number of horse-power at a certain speed. The most natural speed is offered us by the speed at synchronism. If we thus

of Europe.

get "European horse-power,"

divide by 75. As, in a great

speak of a torque of 400 synchronous horse-power, this torque may be developed at any speed, but the product of it into the angular velocity of the rotor at synchronism

is

proportional to 400 horse-

This very convenient mode of expressing torque in "synchronous horse-power" has been introduced by Mr. Steinmetz; at

power.

42

THREE-PHASE CURRENT MOTOR. saw

least the author first

gentleman

in the

it

67. The motor must be

nous speed of 450 68.

The

some years ago

German Elektrotechnische

r.

p.

wound

in a short paper of this

Zeitschrift.

for 16 poles, which gives a synchro-

m.

circumferential speed of the revolving armature should not

exceed 7000

ft.

A

m.

p.

high speed

So high a speed as

greatly dependent to a cheap,

commercial design.

Had

limit.

I,

necessary in order to get a seen, the leakage factor is

this is not

Indeed,

choosing so high a speed as 7000

most favorable

is

we have

large pole-pitch upon which, as

ft.

in this

may

always favorable be urged that by

m. we have overstepped the

p.

work,

signing a cheap motor, as the market

it

may

set

myself the task of de-

here and there require,

I

should have yielded to this objection from economy and adopted a slower speed.

It is still

a

much vexed

run, the greatest commercial

lence of quality.

I

therefore prefer to

possible without giving

The motor

69.

undue regard

The

pole-pitch

The

air-gap will be

The

is

not identical with excel-

make

the

motor as good as

to cheapness.

receives a diameter of 150 cm, corresponding to a cir-

cumferential speed of 6960

70.

question whether, in the long

economy

is

equal to

made

coefficient c in

of the slots which

ft. p.

we

m.

= 29.5 cm.

^7

equal to 0.15 cm.

formula (12) may be estimated for the shape

are going to adopt at 12

leakage factor according to formula (12) a

=

12

.

0.15 - -

=

;

we

then have for the

:

0.061

29.5 71. I

20 72.

With

=

this leakage factor a

0.89

is

The motor

maximum power

factor of cos0o

=

attainable. will take

from the mains a current of about 54 am-

peres per phase at an output of 200 horse-power.

43

THE INDUCTION MOTOR. maximum

73. In order to develop a

horse-power, the in the

maximum

diagram must be equal 400 1/3

assuming the

to

= 102 amperes,

746

.

2000

.

torque equal to 400 synchronous

ordinate or the radius of the semi-circle

.

0.85

efficiency to be 85 per cent at this output.

that the diameter of the semi-circle

must be equal

74. The no-load we remember that

it

circle

current

now immediately

can

io

the quotient of

must be equal

to

<*

Hence follows

to 200 amperes.

be calculated,

it

and the diameter of the semi-

= 0.061. We thus get = 12 amperes.

t

75.

From formula if

phase,

a value for

number of conductors per pole and

(6) follows the ($>

is

We make (B, the maximum

assumed.

tion in the air-gap, equal to 5600,

induc-

and we have then

= 12 amperes. A = 0.15 cm. (B = 5600 g. *o

c.

76.

Inserting these values into the formula

_ ~

'

< 6)

we

n

The

total

number

multiplied by the

We

can

number of

now

n

A

V2

= 40

poles.

=

calculate the

with the help of formula (5). (5)

1.6

.

of active conductors per phase,

z 78.

ffi

2

get

77.

s.

16

.

40

number

=

640

It is

=2.12. ~.

e

= 2OOO volts.

z

.

*

V3

~ =60 z

= 640

$

= 1.42 44

equal to n

of lines of induction per pole

e

Hence,

z, is

Thus we have

.

io' c. g. s.

.

io- 8

THREE-PHASE CURRENT MOTOR. 79.

We

found

in the

second chapter that

*

_ _!_

b

.

.

t

.

&,

12

where b is

is

the width of the motor, and

correct only

if

FIG.

tically closed.

open

in

t

the slots in the field

As we

the pole-pitch.

22.

intend to keep the slots in the

order to be able to exchange the coils easily, 45

This formula

and the armature are prac-

field

N will

entirely

be about

THE INDUCTION MOTOR. 85 per cent smaller than according to the formula. into account,

we

b

80.

We leave in

in the

ventilation, so that the

=

17.0

taking this

cm.

middle of the iron a space of

width of the iron becomes

The

81. The" slots are represented in Fig. 22. ins.

By

find for

deep; in the

armature

i^

ins.

These

18.0

i.o

cm

for radial

cm.

slots in the field are 2

slots represent

about the

FIG. 23.

largest size that should be used in polyphase motors. is

If possible,

preferable to keep the slots smaller than those in Fig. 22.

wire used for the diameter of 0.229

The maximum

field coils is

No. 3 of B.

&

S.

wire gauge, having a

in.

induction in the iron teeth

46

is

it

The

13000.

THREE-PHASE CURRENT MOTOR. The

induction in the iron above the teeth should be no higher than

cm of iron above the slots. The main dimensions of the motor are inscribed in Fig. 23. The resistance of each phase of the field is 0.28 ohm, hot The resistance of each phase of the armature is 0.016 ohm, hot The loss through hysteresis and eddy currents is noo watts,

3000; this gives 15.0

which 530 watts

is

dissipated in the iron ring,

and 570 watts

of

in the

teeth.

82.

We are now

in possession of all the data

mine the behavior of the motor under load and This has been done

in the

following table.

given in the preceding chapters,

it

necessary to predeterat starting.

After the explanation

would be superfluous here

to enter

FIG. 24.

upon the manner in which these figures were derived partly from Fig. 24, partly from the foregoing formulas. A word, however, has to be said about the

account

With

way

sufficient

in

which the

accuracy

necessary to overcome the friction

By drawing a described, we line

loss

by

friction is taken into

we may assume

that the torque

is

constant at different speeds.

line parallel to the basis

upon which the semi-circle is between this

find in the length of the ordinates lying

and the curve which

lies

next to the semi-circle in Fig. 47

24,

a

THE INDUCTION MOTOR. measure for the torque and synchronous kilowatts available

at the

pulley of the motor.

83. For the calculation of the efficiency 7

I

have assumed that the

150

"I

ft

cos 9

/

TOO

1OO

200 KILDWATTS.CONSUM.ED

300

FIG. 25.

loss

through friction and

speeds.

That

this is

air resistance is practically constant at all

not accurately 48

so, I

need hardly remark.

THREE-PHASE CURRENT MOTOR. 84.

The

ordinates of the

full line

curve

in Fig. 24

measure the output

of the motor for an armature of small resistance, provided with external starting resistance.

85. The ordinates of the broken line curve measure the output of the

motor having a squirrel-cage armature with considerable resistance six times larger than that of the armature with external resistance in

order to start under load.

mendous reduction of

A

glance at the curves shows the tre-

the output with which

spicuous in the curves of Fig. 25.

energy

The

we have

This

vantage of dispensing with slide rings.

is

to

buy the ad-

even more con-

abscissae here represent the

consumed by the motor, while the ordinates repcurrent, the output, the efficiency, and the power factor

in kilowatts

resent the field

As

as functions of the energy consumed.

in

Fig. 24, the full line

curves represent the different quantities for the motor with external starting resistance, while the broken line curves

for the

show these

quantities

motor with squirrel-cage armature.

86. The

field current,

of course, remains unaltered, and so does the

But the output, and therefore the

power efficiency, of the motor with squirrel-cage armature are powerfully influenced by the high resistance of the armature. For a starting torque equal to the factor.

torque at normal load,

we need

a current which

as large as the current at normal load

87.

To emphasize

starting resistance

this point

;

either revolving with

rings put,

it

almost four times

once more, the abolition of an external the armature, or placed

outside the armature and being connected to

brings with

is

indeed, a poor result.

it

by means of

slide

inevitably a considerable reduction of the out-

and a lowering of the efficiency. The reduction of the output, as and 25 shows, is equivalent to a reduction of the

a glance at Figs. 24

overloading capacity of the motor, diminishing the margin of power

and thus tending to pull the motor out of step at a small temporary For these reasons European designers have abandoned the squirrel-cage armature in all larger motors.

overload.

49

THE INDUCTION MOTOR. 88. In the calculation of the efficiency the load losses have not been taken into account. efficiency, but

Data of 200

it

h. p.

is

As

I

have said before, they greatly influence the

extremely

difficult to

predetermine them for va-

Polyphase Motor for 2000 Volts, 60 Periods, 440 R. P. M.

THREE-PHASE CURRENT MOTOR. The Starting

89. In order to calculate the resistance

Resistance.

of the starting box,

it

torque, the amperes

is

advisable to plot a curve representing the

and the

volts in

of the resistance of the starting box.*

remember

the armature as a function

This can easily be done

if

we

that at starting the energy available at the shaft of the

motor, when running with the same torque, must be dissipated into heat.

We

find, therefore, the starting resistance

necessary to enable

600

300

OHMS

PB

PHASE.

FIG. 26.

the same torque to be generated in the motor, by dividing the output

by the square of the armature current. In torque, in Fig. 26, has been drawn. From starting current, the torque,

armature.

The

the curve for the

this

way

this

curve

and the voltage

we can

find the

at the slide rings of the

latter is considerable at a small starting torque,

the armature conductors must, therefore, be carefully insulated. This method was

first

recommended by Dr. Behn-Eschenburg, Oerlikon. Si

and

THE INDUCTION MOTOR. and Torque.

Slip

tion of the slip.

attainable

is

90. Lastly, Fig. 27 represents the torque as afunc-

The curves demonstrate

that the

maximum

torque

The

entirely independent of the armature resistance.

effect of a great

armature resistance consists in shifting the same

torque from a small slip to a large one.

Again, the

full line

gives the torque for the armature with a resistance of 0.016

curve

ohm

per

phase, while the broken line curve shows the torque for the squirrel 350

-

I

s:

30

50

o.

80

90

1

00*

SLIP FIG. 27.

cage armature having a resistance equivalent to six times that of the

armature with external starting resistance.

and calculation of the polyphase behooves us to glance back upon the comparatively easy motor, road by which we have reached a complete solution of a problem 91.

Having thus

finished the theory

it

which,

if

treated merely by mathematics, appears to be one of the

THREE-PHASE CURRENT MOTOR. most abstruse problems

in natural philosophy.

More and more we

begin to realize the truth of the words with which Kelvin and Tait prefaced their lucid treatise on "Natural Philosophy," that "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 read-

iness for farther progress."

92.

Time was

and that not very long ago

when

theories in which

mathematical methods were restricted to a minimum, were glibly labeled "popular," with that fine flavor of contempt that hangs about

term.

this

and

Slowly but surely graphical methods have supplanted,

will supplant in the future, highly analytical

7

methods, as

in the

graphical method the development of the idea can be followed stage

by stage, thus furnishing a means of constantly checking the process 01 reasoning

and keeping the attention of the mind concentrated

upon the development of the thought. While a long analytical argument, after starting out from certain premises, leaves us in large measure

in the

dark until the result

didactic value of graphic

methods

is

is

reached.

In

my

opinion the

vested in the difference traced

out in the above comparison. 93. These remarks must not be misconstrued. graphical treatment turns out to be far

some than IM the case

is

preferable.

many

cases the

more complicated and weari-

the analytical treatment, and

such cases the latter

In

I

need hardly say that

Whichever method

is

in

the simpler,

under consideration, deserves preference.

'One of the most complex problems in natural philosophy is the theory of the on which the greatest mathematicians from the time of Newton until today have tried their powers. This is what one of the workers in this field, Prof. tides,

and numerically its bearings un the history of the earth. "Sir William Thomson, having read the paper, told me that he thought much light might be thrown on the general physical meaning of the equation, by a comparison of the equation of conservation of moment of momentum with the energy of the system for various configurations, and he suggested the appropriateness of geometrical illustration for the purpose of this comparison. "The simplicity with which complicated mechanical interactions may be thus traced out geometrically to their results appears truly remarkable." analytically

S3

CHAPTER

VI.

The Single-Phase

and improved transformer diagram which has

simplified

THE

Motor.

stood us in good stead in understanding the phenomena in

polyphase motors, will serve our purpose equally well in the treatment of the single-phase motor. is

based upon the well-known theorem

The method here employed* first made prominent by the

and Andre Blondel, that an oscillating magnetic be replaced by two revolving fields, the

late Galileo Ferraris

field can, in all its effects,

amplitude of each of which oscillating field

is

equal to half the amplitude of the

the two fields revolve in opposite directions at a

;

frequency equal to that of the oscillating alternating 94.

A

two-pole armature revolving at a speed

ing magnetic

field

other, II, a slip of

95. Let us

field. 2

in

an

oscillat-

of the frequency ~i, has relative to the one

which we

netic field,

of ~

will call I, a slip of

+~

~i

2

~i

~

2>

mag-

relative to the

.

now consider the field II. At the immense

slip of

~

x

H

',

the secondary ampere-turns act almost exactly in an opposite direction to the primary ampere-turns

leaving only a small vectors

enough

of

;

they thus neutralize each other,

the vector of which coincides with the

primary and secondary ampere-turns just large which drives the magnetizing current

to balance the voltage

through the 96.

the

field

We

field-coils.

have dissolved the amplitude of the impressed alternating

current flowing through the primary into two components of half the

amplitude revolving in opposite directions.

having two polyphase motors, the

field-coils

This

is

equivalent to

of which are coupled in

*See the author's article on "Asynchronous Alternating- Current Motors," in the Elektrotechnische Zeitschrift, Berlin, March 25, 1897.

54

THE SINGLE-PHASE MOTOR. but in such a manner that an observer looking at the fields in

series,

the direction of the axes of the rotors, would call the one revolving clockwise, the other counter-clockwise. tors are rigidly connected.!

armatures

would

in a

The

field

The armatures I would then

clock-wise direction, while the

them

try to turn

field II, if left to itself,

The motor we have seen that

in a counter-clockwise direction.

/ would take the larger share of the voltage, since

the reactance of a polyphase motor for a great

current, that

is

great for a small

Hence, the voltage which

slip.

mo-

of both

try to turn the

comes from motor

is

through motor

7,

parison with that consumed by motor

slip,

and small

necessary to drive the II, is

small in com-

/.

97. In order, then, to draw a diagram of the single-phase motor should have to find out by trying

tween the two motors.

This

is

how

the voltage

is

we

distributed be-

We

a very wearisome procedure.

ar-

and simple solution if we remember that the resultant magnetic field in motor II will always, with very

rive at a sufficiently accurate

little

tor

7.

mo-

inaccuracy, coincide with the impressed magnetic field of

But

this

means nothing

else

than that the effect of motor //

be taken into account by considering

it

may

as an apparent increase of the

primary leakage field of motor I. 98. According to the above considerations

it

is

clear that currents

if running almost Synchronism, of course, can never be reached by the

will be induced

by the

synchronously.

field

// in the armature, even

armature unless an external force

is

applied to

its shaft.

These

ar-

mature currents react upon the primary and must, obviously, about double the magnetizing current; in other words, the single-phase

motor running

idle,

netizing current.

takes a current about twice as large as the

The

mag-

accurate relation between magnetizing cur-

rent and "idle current" depends upon the leakage factor, and can be calculated as follows.

the idle current tThe author

is

Herrmann Cahen

t.

Each of the motors is

/

and // receives one-half

the magnetizing current in motor

I,

and the

indebted for this helpful comparison to a conversation with Mr. in 1895.

55

THE INDUCTION MOTOR. voltage necessary to drive this current through the

motor 7 99.

may

is

field

coils

*i

proportional to- --

The magnetizing current of motor //, running at a slip we have for the magnetizing current

be called in, then

of 2

~

of the

single-phase motor:

im

For

in

we have

=-%- +

in

=

;

=

tn

0'

ff

*'**)

.

hence

m

i

(i _|_

<j)

Im (I3)

For


-

=

o, that is, for

+

I

a motor without leakage,

m-=-L-

For a

~f-

OA OB The

we have

:

= 0.500

=

0.05

=

0.525

we have

shows the diagram resulting from these considerations.

is

the idle current.

is

the magnetizing current.

point

C

bisects

O

A.

B'

lies

upon the semi-circle L,

B

B' divides the total primary current

A'

being the centre of the semi-circle
ff

TT^

-

Fig. 28

2

a

+2f

I

=

m

i


/

/

IT

=

4-

OB' O A' hence the locus of A'

is

''

B

I

-f-

2

2

+

2C

-=-

55

B B' G, G

being equal to

in the ratio of

0-

''

the semi-circle

BL

'

A A' L.

THE SINGLE-PHASE MOTOR. THE CURRENTS IN THE ARMATURE. 100.

The current

in the

equal to the vector .

A' B'

of the

armature of the single-phase current motor

sum

of the secondary currents

-

v\

two polyphase motors, hence equal

.

B

to

*l

is

B' and .

A' B.

Vi

A glance at Fig. 28 shows us that only pated

in the

one-half of the energy dissiarmature can be utilized for the production of the torque.

We see, namely, that at all loads the secondary currents A' B' and B'

B

represented by

remain very nearly equal, and as only motor 7

is

do-

FIG. 28.

ing useful work, the armature currents in motor II represent a loss

almost exactly equal to the loss /.

The

slip in a single-phase

in the

armature of the working motor

motor

indicates, therefore, only one-

half of the energy dissipated in the armature, hence the loss in the

armature of a single-phase motor

is

twice as large as that in a poly-

phase motor, provided the slip be equal in the

The torque can be calculated as follows Suppose the* wound in three phases, each having the resistance ry

101.

Torque.

armature

two motors.

is

The output

:

of the motor

is

P=

then *, ;,

.

:

cos

57



i,

.

r,

3

/,

rtt

THE INDUCTION MOTOR. whence follows j) m ke = -J-

(14) ..... 6i.6D

in

which equation

P-watts,

.

~i

Dmkg

is

the torque in m.kg., and

p

the

number

of north or south poles. 102.

The following reasoning

We

have for motor

and

for

motor

9.81

.

~,

:

9.81

.

A (i - s) =

3

=

3

*'

'

r> ,

a

II,

which equations

in

yields a value for

7,

DnK + is

t

u,)

* '"'

r

'

\

2

.

the angular velocity of the revolving

field,

that of the armature.

<>,

Hence, 9.81

.

(Di

- Dn

)

"2

=3^ \"2 /

or

= Plaits Let us assume ~i = 50, and ~ = *

(IS) ..................... 3

803.

To

illustrate

:

*,

r*

2

~

45

;

then

we

have

= In words, if the slip is 10 per ture amounts to 20 per cent. 104. It

is

instructive to

0.23 P-watts cent, the loss of energy in the arma-

compare with (14) the formula (7) for the

polyphase motor, which reads after some transformations, (16) ..................... 3

This

is

in

words that the

loss in per cent, setting

P

*

t\

rz

=

P-watts

.

-

~

slip in per cent is equal to the

+3

it*

r*

58

equal to 100.

armature

THE SINGLE-PHASE MOTOR. Ratio of Transformation. to cover the eral,

will be

it

We

105.

shall see that

whole circumference of the

it is

The

ratio of

not equal to the quotient of the

number

advantageous to wind but two-thirds of

transformation in this case of conductors in the

field

is

not advisable

with windings. In gen-

field

divided by the

armature. With sufficient accuracy

number

we may

it.

of conductors in the

consider the ratio of trans-

FIG. 2Q.

formation to be equal to the number of active conductors in the divided by only five-sixths of the total in the

number

field

of active conductors

armature.

EXPERIMENTAL CORROBORATION. 106.

The data given

in

the following table,

and graphically repre-

sented in the polar diagram, Fig. 29, belong to a lo-hp, single-phasi

current motor for

no

conductors in the

field

volts, 50

~, and

1500

r.

p.

m. The number of

was 120; the number of conductors 59

in the

THE INDUCTION MOTOR. armature was 312. The resistance of the

field

was

of the three phases of the armature, 0.08 ohm.

TESTS OF lo-HP MOTOR.

0.015

ohm;

of each

THE SINGLE-PHASE MOTOR. conductors per pole

tive

n,

we have

then

for the induction in the air-

gap:

n

<*=

(I7)

In this formula im

is

V/2

im 1.6.

.

A

the effective value of the magnetizing cur-

\ FIG. 30.

rent, while

motor

in

A

cm,

is /

If & is the

the air-gap.

the pole-pitch, then

*

08).

= we

Instead of a coefficient of 2.22 efficient

of 1.85; therefore,

we have e

(19)

Secondly

Fig.

31

shows

we

=

this

width of the iron of the

have, .

,


get, as will easily

~

.

1.85

.

z

of lines of induction

*

(20)

as

may

=

be seen at a glance. 61

is

.

*

.

lo-8

For the induction formula

case.

(16) holds good.

The number

be seen, a co-

the formula:

equal to b

.

t

.


THE INDUCTION MOTOR. For the

e.

m.

f.

we have

:

=

e

.

2.22

.

.

10-8

3 e

(21)

109. It will readily be seen

=

1.48

.

~

z

.

10-8

.

.

from these formulae that

it is

not advan-

tageous to use a coil spread over the whole pole-pitch.

' '

i;

'

i

I

i

I

"i

'iff

N=i

B.b.t.

1.48

\ FIG. 31.

110.

We

can

now

single-phase motor

;

proceed to study in detail the qualities of the to this purpose

we

shall devote the next chapter,

taking the iron frame of the three-phase motor designed in chapter

V, and winding

it

as a single-phase current motor.

62

CHAPTER

VII.

Calculation of a Single-Phase Current Motor.

THE

three-phase current motor, the design of which was given

in chapter

to

V,

is

reproduced

examine the behavior of

phase motor.

For

in Fig. 32.

this

simplicity's sake

motor

we

We if

shall

now

proceed

running as a single-

shall retain the winding,

and

uL_

FIG. 32.

investigate

what voltage

be remembered, III.

Field Winding.

of wire, 0.229"

;

is

most suitable for

has 16 poles,

and

is

to

work

it.

The motor,

again at 60 p. p.

128 slots, 10 conductors in each slot

resistance hot, 0.28

63

ohm.

;

as will

s.

diameter

THE INDUCTION MOTOR. Armature Winding. in three

240

2 conductors in each slot;

slots,

wound

phases in star connection; resistance of each phase, 0.016

ohm.

M.

112. E.

F.

According to (18) we have:

(18)

= = =

b /

(B

The diminution

17 cm. 29.5

cm.

5600.

of the air-gap through the open slots

is

taken into

account by reckoning only 85 per cent of the surface of the air-gap.

Thus,

we

find

:

=

4>

1

13.

io* c. g.

.

insert

100

.

=

N = 1.58 io ~ e = 1.85

8

in

.

.

.

i

z

.

42

io* c. g.

.

s.

we had

In the three-phase current motor #s

We

1.58

:

s.

formula (19) and get .

$

io"6

.

= 2250 volts.

Therefore, the terminal voltage should be equal to 2250 volts. 114. Hysteresis.

amounts

The

through hysteresis and eddy currents which 570 is dissipated in the teeth, and

loss

to 1150 watts, of

580 in the iron ring.

Magnetizing Current.

From tm

(17) follows

=

(B

.

H

We

insert the following values

A

1.6

V2

:

(B = 5600 A = 0.15 cm. n = 80. c.

and

g. s.

find

m

i

=

1

2 amperes.

64

CALCULATION OF A SINGLE-PHASE CURRENT MOTOR. The leakage

115. Leakage.

116. Idle Current.

We

coefficient is 0.061 for the iron frame,

have according to (13)

im

'

1

(13).

2

+ -f-

2

1

2 a

2

:

+ +

0.122 0.122

-

0.576 /

Loss through

117. Friction.

118.

We

in Fig. 33.

can

=

now draw our

The

20.8 amperes.

friction

and

air resistance,

2150 watts.

standard diagram. This has been done

ordinates measured between the full line curve and

FIG. 33.

the basis of the semi-circle measure the output of the

armature with small resistance. line curve

The

motor for the

ordinates between the broken

and the basis of the semi-circle measure the output of the

motor for the

squirrel cage armature.

65

THE INDUCTION MOTOR. From

this

diagram the following table has been compiled.

DATA OF

i8o-HP

SINGLE-PHASE CURRENT MOTOR.

2250 Volts, 60 P. P.

S.,

450 R. P. M.

CALCULATION OF A SINGLE-PHASE CURRENT MOTOR. 122. Fig. 34 shows the field current and the output as a function of the energy consumed.

The

full line

curves represent again the output

H

FIG. 34.

of the armature with small resistance, while the broken line curve

shows the output of the motor with 67

squirrel cage armature, the re-

THE INDUCTION MOTOR. which

sistance of

is

six times as large as that of the

armature with

external starting resistance. 123. Fig. 35 shows the cases.

The power

factor

power is,

factor

and the

of course, the

same

efficiency for the

two

In the

in either case.

calculation of the efficiency the load-losses have not been taken into

account. 124. Finally,

we have

put and the torque

in Fig. 36 a graphic representation of the out-

in watts as a function of the slip in p. p.

mo

s.,

or rath-

150

K.W. FIG. 35.

er,

of the

number of revolutions of

The broken Little

the armature expressed in

line curves refer as usual to the squirrel

need be said about these curves.

It is plain that for the

ture with a small resistance the starting torque

small at

~

2

= 40.

Such a motor

will

p. p. s.

cage armature.

is still

arma-

exceedingly

never start well without extra

resistance in the armature, whatever starting arrangement be made.

68

CALCULATION OF A SINGLE-PHASE CURRENT MOTOR. 125. ity

The effect of armature

resistance consists in reducing the capac-

of the motor, and in shifting the

maximum

torque toward the

starting point.

126. In designing the three-phase current motor,

we had made

the

condition that the motor should start with a torque equal to twice the

normal running torque. As all the starting devices that have been tried for single-phase motors are not worth very much, such a condition can, at present, not be fulfilled.

We

rate the motor, therefore,

FIG. 36.

according to kilowatts

its

most favorable

we have an

efficiency

and power

factor.

For 135

efficiency of 94 per cent and a power factor of

82 per cent, corresponding to an apparent efficiency of 77 per cent.

The motor 127.

It

will safely yield 180

is

instructive to

horse-power at 2250

compare the current

volts.

in the three

phases

of the three-phase current motor reduced to the voltage of the single-phase

phase motor.

current

The

motor

with

three-phase

the

current

in

the

single-

motor took a magnetizing cur-

69

THE INDUCTION MOTOR. rent

of

12

amperes,

corresponding

to

12

.

2000

V3

volt

-

am

peres. This divided by 2250 volts yields a magnetizing current equivalent to 18.4 amperes. The leakage factor being 0.061, we can draw

our standard diagram whence the curves in Fig. 37 are derived. 400

350

JHREE! SCO

200

150

50

150

100

200 }<..

250

300

350

400

W.

FIG. 37.

These curves clearly show that the phase current motor phase current motor.

is

only 0.65 the

maximum output of the singlemaximum output of the three-

CALCULATION OF A SINGLE-PHASE CURRENT MOTOR. STARTING ARRANGEMENTS. 128.

A

few words may

single-phase motors.

It

fitly

be added about the methods of starting

has generally been found advantageous to

use two-thirds of the pole-pitch for the main winding, and to leave the remaining third for the starting phase. as an imperfect two-phase current motor.

The motor

An

thus starts

external armature re-

S5

cb

S

a

MM in

AAA/WW FIG. 38.

sistance

is

to be

recommended

for large sizes, say, for

an output above

15 horse-power.

129.

The

starting phase receives generally either half the

convolutions as the main phase, or twice as many.

The

number of

latter

method

seems to be preferable. Figs. 38 and 39 show the switch used by the Oerlikon Engineering Works for their single-phase motors.

THE INDUCTION MOTOR. 130. In Fig. 38 the resistance or condenser

R

is,

at starting, in series

with the main phase, the auxiliary phase being directly between the terminals. 131. In Fig. 39 the resistance

phase, which

is

in series

R

is

in parallel

with the auxiliary

with the main phase.

During the throwing-over of the switch from the starting position

cS

EH ^

cT 7) AUXILIARY

vmm PHASE

FIG. 39.

to the running position, the

motor

is

temporarily disconnected from

the mains.

The

last chapter

we

shall devote to a

more general and broader

treatment of the polar diagrams of the general alternating-current transformer.

72

CHAPTER

VIII.

The Polar Diagrams

of the General AlternatingCurrent Transformer.

this last chapter

it is

my aim

and lucid

to present, in as simple

IN a manner as possible, the general theory of the alternating-current transformer, taking also the resistance of the primary into

The

account.

results of this consideration are partly

the road by which they can be reached,

serve

some consideration.

shall

I

is

known, but as

a very easy one,

make use

in

my

may

it

de-

treatment of the

problem of the theorem of reciprocal vectors, known

in

kinematics

and geometry as the theorem of inverse points, and I will here make good a sin of omission, of which I became aware only when by chance glancing over the pages of Dr. Bedell's book, "The Principles of the Transformer,"

in

November,

1900.

As

early as

1893

Messrs. Bedell and Crehore gave the theoretical proof of the fact that the locus of the primary voltage in a transformer at constant

current

is

a semi-circle,* and in his paper on "Transformer Dia-

grams Experimentally Determined," read in

at the Electrical

the constant-current transformer that the potential at

can be represented by chords

gram to

Congress

Chicago, 1893, Dr. Bedell gave also the experimental proof for

in a semi-circle.

To

its

terminals

develop the dia-

for the constant potential transformer was, however, reserved

European

physicists.

DIAGRAM OF FLUXES AND MAGNETOMOTIVE FORCES. 132.

The

principle of the conservation of energy requires thaf the

magnetomotive forces

,Yi

and Xi of the primary and secondary of the

transformer tend to magnetize the core in opposite directions. 'See a series of ten World, May, 1893, and

article*

X

on the "Theory of the Transformer," Electrical

later.

73

THE INDUCTION MOTOR.

X may be considered to be magnetic cells producing, as it were, magnetomotive forces Xi and X*. The reluctance of the circuit com-

and

x,

Q

Fig. 40.

mon

to both be R, the reluctance of the stray-fields that naturally

form about Xi and

Xt, be PI

and p2 74

.

Let us

call

the flux flowing

ALTERNATING-CURRENT TRANSFORMER.

X

through Xi and

We

t,

F, and the leakage-fluxes fa and ^, respectively.*

have then, (i)

Xi

(2)

X,

X

Treating

t

= =

t lPl
P,

and Xi not as scalar quantities but as

vectors,

we may

write,

X

(3)

l

The phase

133.

X*

=

F

of the leakage-fields fa and fa

.

is

R.

same as

the

that of

the magnetomotive forces which produce them, hence they are in

phase with the currents. 134. Let Ei, Fig. 40, represent the voltage

at the terminals of a

non-

inductive resistance in the secondary of the transformer, then the current will be in phase with

by

produce

E The 3.

magnetic

field

which

variation, is Ft, in quadrature with

its

is

of induction in phase with the secondary current, whose

quadrature with the current, are represented by 0. treat the case in

and next we

which these

required to

E* The e.

m.

We

f.

lines is in

shall first

lines are all within the transformers,

shall deal with the case of an external inductance or ca-

pacity.

135.

The

flux F, as

mentioned above, links together the primary and

The vector-sum of 0, and F is equal to F. we neglect for the moment the lag of phase between coils.

secondary 136. If

magnetomotive force and the

flux,

the magnetizing current

phase with the flux F.

is in

the

then the magnetomotive force of It

may be

rep-

resented by the vector X. 137.

X

must be the vector-difference between

138. Xi,

The

139.

X

2,

and

X

have

A'i

and X*.

real existence.

leakage-flux fa of the primary

is in

phase with the pri-

mary magnetomotive force X and may be represented in the diagram by fa. The vector-sum of fa and F is equal to the total primary t,

flux

F

t.

The notation of

fluxes in this chanter is different from that of the preceding main flux being called /' instead of . I hope that this reference prevent readers from making a mistake.

chapters, the will

75

THE INDUCTION MOTOR. Let us now choose our scale for X, Xi and Xt so that

X

is

equal to

F; then, Fig. 41, O~A = Xi, and O~B = X,. THE DIAGRAM FOR THE CONSTANT-CURRENT TRANSFORMER. 140. Let us keep

OA

= Xi

still

and constant, then we see

at

a

50 AMP.

80

100

VOLTS

Fig. 41.

glance that the point

G will move in the semi-circle Oi. G C = fa, is alA C = O B = X because of equation (2).

ways a constant portion of

2,

76

ALTERNATING-CURRENT TRANSFORMER. y and A very useful way of defining is that

=

0,

.

<j>

i

t

of A.

P\

who

Heyland,

writes, *!

Here

=

''

-*V

obviously the magnetic conductivity or reluctivity of

is

TJ

the leakage-path.

Similarly he writes, 6l

According

we

chapters,

^

define

and

=

r,

X^

.

which

to the notation

_=

have used

I

=

~0~A'

the preceding

AG

by saying that

0,

in

*-,

and

_

v\

G mo\*es in the circle O\, C, which divides A G in the ratio A C A G :: Vt I, also moves in a circle, which is determined by C' OA v I. dividing O A in the ratio AC' 141.

If

:

:

: :

:

142. It

:

just as simple to find the locus of the primary flux Fi,

is

by our assumption, the primary current

since,

field 0i is also constant.

circle

Ot

U A'

::vi

143.

to Ot,

is

constant, the leakage-

have, therefore, only to transfer the semi-

Oi Ot being equal to

lF

or, to

it

put

differently,

OA

:

i.

:

The primary e.

resistance can easily be taken into account a

that the drop caused by the

may imagine lent to the

We

m.

f.

produced by a magnetic

ohmic resistance field

This

lagging behind the current by 90 degs.

is

we

equiva-

of constant magnitude, field is

represented by

D~M.

The

O

locus of the primary field

Ot being equal to 144.

The

therefore, the semi-circle Ot,

is,

D M.

potential at the terminals, necessary to drive a current

proportional to X, through the transformer,

From O M,

the potential at the terminals

is is

proportional to calculated

O M.

from the

formula 100 e in

which k

tors,

is

=

k

~

z

F

generally equal to 2.22,

and equal

to 4.44,

if

z

is

the

if

io"

s

volts, is

the

number of conduc-

number of convolutions.

77

THE INDUCTION MOTOR. The

145.

now

position of the semi-circle can

easily be determined,

by the use of complex imaginary numbers, or graphically, as

either

I prefer.

The

ratio

O D'

:

A' D' can be found

O& = -^ 7

A' D'

=

X^

,

~^'~

A'D'

-v^

v*

v* v*

146. This constant ratio

by

have:

v2 hence

.

OD'

it

We

ATiy

i

former, and denote

at once.

we

leakage factor of the trans-

call the


I

(4).

In Heyland's notation

= 147.

To

(i

+

^i)

recapitulate

ondary resistance

is

:

we should have (i

+-

i'=

2)

:

Ti

+

T2

+

ri TI

In a constant-current transformer whose sec-

varied from naught to open-circuit, the terminal

voltage varies in such a manner that the vector of the field to which it is

proportional and with which

it is

O

the semi-circle, determined by circle is perfectly defined if the

t

in quadrature,

as centre.

The

has for

its

locus

position of this

primary resistance and the leakage

known.

factor are

THE CONSTANT-POTENTIAL TRANSFORMER. 148.

If, in

Fig. 41,

we wanted

to

know what

current would flow

through the transformer at a certain difference in phase between

and

O A,

if

OM

were n-times as large as

simply reason that, the counter times as if

we

much

kept

OM

e.

m.

f.

in the diagram,

being n-times as large, only

current could flow through the transformer. constant, varying only

78

OM

we should

its

phase relative to

Hence,

O A,

the

ALTERNATING-CURRENT TRANSFORMER. current would vary in inverse proportion to the magnitude of the

we kept O

If

field-vectors.

M not only constant, but also OA

position, the locus of the vector

This

is

a very

method of

and

fertile principle,

it

would

was

in the

same

also be a semi-circle.

called

by Dr. Bedell the

reciprocal vectors.

Let, in Fig. 42, the semi-circle having Oi as centre, represent the

locus of the primary field of the constant-current transformer,

being the primary current

A

O

and

to 50,

current

current I,

2, 3,

is

04

Oi

O 1*

be numerically equal to 20,

i to 60.

O

I*

=

50 produces a

the flow of the current. -

current of

Let

X

50

=

represented by

Hence,

if

field

equal to

the field

is

OI

only

=60, hindering

O4

= 20,

then a

150 can flow through the transformer.

O 4'.

It

This

can easily be shown that the points

4 correspond point for point to the points

i', 2', 3', 4',

the

angle 301 being equal to 2'oi. Fig. 43 represents the circle O/, reciprocal to 0*.

equal to angle

A

t

'

A

Angle

OA

is

A.

concrete case will bring the matter into a

149.

rent

O

more palpable form.

transformer or induction motor requires a magnetizing cur-

=5

amp.

We

assume

Vi

= 0.91 79

and

Vt

= 0.80.

The constant

THE INDUCTION MOTOR. no

potential of the transformer be

mary be

2 ohms.

The

All that remains to be done stant-potential transformer.

O2

is

The

volts.

is

to construct the semi-cii*cle.for the con-

Oi, Fig. 44, is a

no

X

volts,

5

=

equivalent to

field

a field equivalent to 31.5 volts; hence,

constant at

resistance of the pri-

semi-circle Ot can then easily be constructed.

if

will flow

174 amp.

no volts.

the potential

is

kept

through the

DIAGRAM ILLUSTRATING METHOD OF RECIPROCAL VECTORS. Angle

0^0 A,- Angle

4

Angle M'O'A,- Angle

M

A.- Angle

L

Angle

0~M

.

L'

OM'- OD

-

6~D'-

O

OL

.

A,

;

A,

;

A,

.

O~L

Fig. 43-

transformer at the same phase-angle.

We

thus get point

centre of the semi-circle can at once be. found, points

determined, as angle Ot 150.

The

OA

is

equal to angle

I

2.

The

and 2 being

O A.

ordinates of the semi-circle Ot represent the watt-current

of the transformer, the ordinate of point

80

I

being equal to

2

5

X2

-f-

no

ALTERNATING-CURRENT TRANSFORMER.

= 0.455,

and the ordinate of 2 being equal

to 174*

X

2

-r-

no

=

5.5

amp.

Oi; O2, represent

To

directly the

find the secondary currents

in Fig. 45.

OA

is

the primary m. m.

OM

is

primary currents. I have redrawn the same diagram

the primary current, or, to be

proportional to the terminal voltage of SCALE FOR

more

strictly correct,

f.

10 volts in our case.

SCALE FOR FIELDS

M. M. T.

01234

1

5

AMP.

20

40

60

80

100

VOLTS

Fig. 44-

MD

is

proportional to

ti

n, the ohmic drop in the primary.

It is

would be equivalent to the throttling action of the ohmic resistance must be in quadrature with the voltage necessary to overcome it

drawn perpendicular

O D,

then,

is

to Xi, as the field that

the primary flux F\. 81

THE INDUCTION MOTOR. CD

is

the primary leakage-field

C~D

CG

is

^

--

=

i

6~A

i

A~C

the secondary leakage-field ^ 2 .

C~G

--

=

O G is the secondary field F

By means of the scale

z.

secondary voltage can at once be determined. SCALE FOR 1

2

45

for the fields the

must be borne

in

SCALE FOR FIELDS

M. M. F.

3

It

AMP.

20

OA-X,; "oLT-F,;

40

80

60

AC-X 2 OC-F;

;

100

VOLTS

OOX "OG-F2

J 7

OME, Fig. 45-

mind

that

OG

is

terminals, but that 151.

in

Though

not proportional to the voltage at the secondary it

includes the ohmic drop in the secondary.

the preceding considerations offer no great difficulty

understanding them clearly, yet the transformer diagram becomes

surprisingly simple

if

we

neglect the resistance of the primary.

not only the diagram, but also

its

82

evolution,

And

become so perspicuous

ALTERNATING-CURRENT TRANSFORMER. that

it

seems peculiar that

has taken such a long time to arrive at

it

this solution.

That the locus of the vector of the primary m. m. f. must be a semi-circle, follows directly from the diagram of Fig. 41. This circle is

reproduced in Fig.

magnetomotive 152.

OC

is

open-circuit,

The

46.

thick lines

show the

triangle of the

forces.

the magnetizing current.

OC

becomes equal to

If the transformer runs

O K. We

on

see, therefore, that the

20 40 60

80 1OO VOLTS

Fig. 46.

magnetizing current

is

not constant for

all

the diminution of the secondary resistance. the primary magnetic field Fi,

F

and the

leakage-field

ft,

which

F

is

is

loads, but decreases with

What remains constant

composed

of the

common

is

field

constantly diminishing, while ft

is

increasing, their vector-sum being constant.

153.

K

drawing a

divides line

O~D

in a constant ratio,

through K,

for instance,

83

A

O~K

=v

, .

O~D.

K, and producing

Hence, it

until

it

THE INDUCTION MOTOR. intersects the semi-circle in G, yield us all the data of the transformer

that

we

are interested

in.

INDUCTANCE IN SECONDARY. is no difficulty in drawing the diagram for a transformer on an inductive load of constant power-factor. The develworking

154.

There

INDUCTANCE

IN

SECONDARY.

/ /

A/!/

a -77^7-1-0.66

Fig. 47-

opment of the diagram is of the simplest if we neglect the primary Afterwards it resistance, which is generally perfectly permissible.

ALTERNATING-CURRENT TRANSFORMER. is

easy to correct the diagram with regard to the ohmic resistance of

the primary, 155. Let

should be desired.

if this

O

Q, Fig. 47, be the e. m. f. necessary to overcome the ohmic resistance of the secondary circuit, including the resistance of

ON

the coils of the transformer, and

come

equal to the

e.

m.

f.

m.

e.

f.

required to drive the current

The

secondary of the transformer.

E

in

quadrature with

is

the secondary m. m.

CG

t.

primary leakage scale of Xi and X 2

OK

is

is

required to over-

OP

necessary to do this

field

is

the primary m. m.

f.

common magnetic

the

is

O

is

X\.

CD

field,

constant,

OK

is

G,

CA the

and the

so chosen as to give a resultant equal to

OD

is

through the whole

the leakage field of the secondary,

t,

OC

constant as

It follows at

is

X OA

f.

field.

F.

the

the external inductance in the secondary circuit, then

being equal to Vi

O C= O D.*

.

once from the diagram,

AK X, AK = :

:

:

X,

:

-~ Vl

v,

.

X,.

P O N, G O G K remains constant and equal to 180 O G K. If the diagram is constructed for one point, the locus of G is determined. 156. To determine the locus of A we have to consider the ratio between G K and A K, which we have called As

angle

moves

in the arc

KC = For

A K we

have a

Vi

.

X,

(i

vj

Xi, hence

G~K = = --

A ~K

-

--

vt

i

=

*i

i

--

^

I

This diagram is identical with that given by Herr Emde in the EltktrottchI refer the reader to Herr Erode'* important ische Zeitschrift, Oct. n, 1900. contributions on this subject, as well as to his valuable criticism of my paper of also Herrn Heubach's, Kuhlmann's, and 1896 on the general transformer. See "al* ITC. Sumec's letters on the same subject there.

85

THE INDUCTION MOTOR. Hence the

locus of

A

is

an arc the chord of which

is

determined by

the ratio.

O~K (5).

157. is

The

centre of this arc

Q O P, Q O P.

equal to

equal to

is

KL

^

Vi

determined by the angle Os

the angle of lag in the secondary.

K L, which O K is

Angle Oi

CAPACITY IN THE SECONDARY. 158. Fig. 48

is

constructed for a capacity in the secondary in series

with the resistance. secondary

field,

P O B is O A K is the

triangle

find exactly as before

OL Bearing 159.

P

in

mind

triangle of the m. m.

that angle

=

a

the

We

O GK

l

i

is

constant and equal to angle

follows at once that the locus of

it

is

ff.

:

OK =

N O P,

the secondary lead, Ft

Angle

Again angle

3

KL

is

G and

that of

equal to angle Oi

O K,

A

are circles.

equal to angle

Q, the angle of the secondary lead.

HYSTERESIS AND EDDY CURRENTS. 160.

A

word about

the

way

should be taken into account. loads

which they are

in

which hysteresis and eddy currents

Assuming them

to be constant for all

not, as, if the leakage-path

is

slightly saturated,

the leakage-flux becomes greater with larger currents, and the greater loss in the leakage-path

may outdo

the decrease of the loss in the

main

seems most logical to take these losses into account by assuming a lag between the common field O C and the magnetizing current. This lag diminishes the secondary current. Draw a line parfield

it

allel

with

in watts

L D, the distance of which from L D being equal to the loss through hysteresis and eddies divided by the primary volt-

age, then the secondary currents

and the semi-circle

mus* be measured between

O*.

86

this line

ALTERNATING-CURRENT TRANSFORMER. 161.

I

wish to impress upon the reader that there

contribution that

can claim specially

I

my

own.

I

is

little in

this

have merely com-

bined the diagrams of Kapp, Steinmetz, and Blondel, simplifying

wherever

it

was

possible.

The

application of the principle of recipro-

manner

cal vectors* enables us in a surprisingly simple

the intricate

phenomena

in the

to trace out

transformer for constant potential,

CAPACITY

IN

if

SECONDARY.

t/,-0.80

V.-0.75

-

P

Q Fig. 48.

we want at the

to include the

primary resistance.

diagram of the general transfomer

dell's in so far as

duction, a

is

My

he uses the coefficients of

method which

I

method of arriving from Dr. Be

different self-

and mutual

in-

cannot advocate.

'The method of reciprocal vectors was admirably treated in 1878 by Prof. W. K. Clifford in the chapter on "Pedal and Reciprocal Curves" in his work on "EleBut Dr. Bedell first applied the principle to the transformer.

ments of Dynamic."

87

THE INDUCTION MOTOR. 162.

A

very beautiful and simple diagram can be drawn, showing

in polar co-ordinates the locus of the primary current for a

lag,

non-

inductive load, and a lead in the secondary, as for the same trans-

former or motor

KL

is

a constant

if

O K

is

constant.

Neglecting

the primary resistance, a diagram can be constructed for constant terminal voltage by erecting arcs over

K L

Arcs

as chord.

flatter

than the semi-circle correspond to an inductance in the secondary, the semi-circle corresponds to a non-inductive load, and an arc whose inscribed angle

is

smaller than a right angle, standing on

K L

as

chord, determines the locus of the primary current for a condenser in the secondary.

As

the secondary terminal voltage

is

determined

most simple and beautiful manner the pheby nomena of resonance and kindred phenomena may at a glance be the circle Oi, in the

qualitatively

and quantitatively understood. who will find no

construction to the reader,

But

I

must leave the

difficulty in building

the diagram synthetically with the help of Figs.

46, 47,

worthy of notice how injurious an inductance

in the

and

48.

up

It is

secondary

is

with regard to the

the transformer or motor

is

capable of taking

a condenser in the armature

maximum energy in, and how much

of the motor would increase the power of the motor to do work.

APPENDIX The following phenomena tric

presentation by Mr. Gisbert

in the

induction motor

is

and most concise

of the elementary

Kapp

reprinted from his book, "Elec-

Transmission of Energy," because,

the clearest

I.

in the author's opinion,

it

is

logical evolution of the principles un-

derlying the theory set forth in the preceding pages.

It will

repay

the student to go over Mr. Kapp's presentation of the subject, and

he

will

understand the vector diagram much more clearly after hav-

ing become thoroughly familiar with this extract from the

work of

a master of the art of exposition.

Extract from Gisbert Kapp's Electric Transmission of Power On the Induction Motor. armature conductors may be connected so as to form single loops, each passing across a diameter, or they may all be con-

THE

nected

in parallel at

somewhat

each end face by means of circular conduc-

in the fashion of a squirrel cage.

Either system of winding does equally well, but as the latter is mechanically more simple, we will assume it to be adopted in Fig. 49. The circular end contors,

nections are supposed to be of very large area as compared witli 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

to note that the

e.

m.

f.

here meant

its is

resistance.

cutting through the lines of the revolving sults

when armature

reaction

It is

important

not only that due to the bar field,

but that which re-

and self-induction are duly taken into

account.

89

THE INDUCTION MOTOR. Let us now suppose that the motor is at work. The primary field produced by the supply currents makes ~j complete revolutions per second, whilst the armature follows with a speed of ~ a complete revolutions per second.

The magnetic

slip 5 is

If the field revolves clockwise, the armature

wise, but at a slightly slower rate.

then

must

also revolve clock-

Relatively to the

field,

then, the

FIG. 49.

armature will appear to revolve

in a

counter clockwise direction, with

a speed of

revolutions per second.

the armature

mary by

field is

is

The

far as the electro-magnetic action within

we may

stationary in space,

a belt in a

second.

As

concerned,

backward

therefore assume that the pri-

and that the armature

direction at the rate of

effective tangential pull transmitted

90

~

is

revolved

revolutions per

by the

belt to the

APPENDIX armature

will then be exactly equal to the tangential force

by the armature

reality is transmitted

in

I.

to the belt at its

which 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 in

speed, the

whole of the power supplied being used up

heating the armature bars.

lem from

view

this point of

The is,

possible the whole investigation.

required to

work

object of approaching the prob-

much we once know what torque

of course, to simplify as If

the machine slowly

backward

be an easy matter to find what power

forward as a motor Let

as a generator,

gives out

it

as is

will

it

when working

at its proper speed.

in Fig. 105 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

~

We

revolutions per second.

how

assumption as to

this induction

is

make

no

at present

produced, except that

the resultant of all the currents circulating in the machine.

it

We

is

as-

sume, however, for the present that no magnetic flux takes place within the narrow space between armature and

other words, that there

is

of force of the stationary

wires, or, in

field

no magnetic leakage, and that field

The

are radial.

all

the lines

rotation being counter

clockwise, each bar travels in the direction from a to c to

b,

and so on.

The

in

the space

d a

lines of the field are directed radially c,

and radially inward

fore, be directed

in the

downwards

space c b d.

in all

the bars

outwards

The

e.

on the

m.

left,

f.

will, there-

and upwards Fig. 49. Let

on the right of the vertical diameter in E represent the curve of e. m. f. in Fig. 50, then, since there in all the bars

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

to note that this curve really represents

place

it

two

I.

It is

things.

important

In the

fir-t

represents the instantaneous value of the current in any one

bar during

its

advance from

left to right

;

and

represents the permanent effect of the current in 01

in the all

second place,

it

the bars, provided,

THE INDUCTION MOTOR. however, the bars are numerous enough

to

permit the representation

by a curve instead of a line composed of small vertical and horizontal steps.

The question we have now

netizing effect of the currents

the curve

I ?

In other words,

to investigate

which are if

what

is

the

magby

collectively represented

i,

what would be the disposition

produced by them?

field

:

there were no other currents flowing

but those represented by the curve of the magnetic

is

Positive ordinates of

I

represent currents flowing upwards or towards the observer in Fig.

FIG. 50.

49, 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.

moment

Thus

the current in the bar

which happens

at the

produce a

the lines of which flow radially inwards on the right

field,

to

occupy the position

and radially outwards on the

b, will

tend to

of

b,

in

the bar occupying the position a, tends to produce an inward

92

left

of

b.

Similarly the current

APPENDIX field,

the

i.

show

a

e.,

left

I.

the ordinates of which are positive, in Fig. 105, to

field

of a, and an outward

the right of

field to

It is

a.

easy to

that the collective action of all the currents represented by the

field as shown by the sinusoidal line A. This curve must obviously pass through the point b, because the magnetizing effects on both sides of this point are equal and opposite.

curve / will be to produce a

For the same reason the curve must pass through

must be sinusoidal

per centimetre of circumference in

armature angle

b,

and

let r

That the curve

a.

easily proved, as follows: Let

is

i

be the current

be the radius of the

then the current through a conductor distant from b by the

;

be

a, will

i

cos a per centimetre of circumference.

If

we take

an infinitesimal part of the conductor comprised within the angle d o, the current will therefore be di

=

i

r cos a

d

a,

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

=

i

r sin a,

i

and since the conductors on the other side of b the

field in the

point under consideration

sin a ampere-turns,

ference at

act in the

will be

same sense,

produced by 2

r

being the current per centimetre of circum-

i

b.

Since for low inductions, which alone need here be considered, the permeability of the iron field

A

strength

must be

When

is

may

proportional to ampere-turns,

starting this investigation,

in

induced by

it

follows that the

and that consequently

a true sine curve.

represented by the curve existence

be taken as constant,

the

B

motor

would,

;

if

sented by the curve A.

B

but

is

we had assumed

that the field

the only field which had a physical

now we

find that the

armature currents

acting alone, produce a second field, repre-

Such a

field, if it

had a physical existence,

would, however, be a contradiction of the premise with which 93

we

THE INDUCTION MOTOR. started,

and we see thus that there must be another influence

which prevents the formation of the

field

A.

at

This influence

work is

ex-

erted by the currents passing through the coils of the field magnets.

FIG.

51.

The primary field must therefore be of such shape and strength, that it may be considered as composed of two components, one exactly 94

APPENDIX

I.

equal and opposite to A, and the other equal to B. In other words, must be the resultant of the primary field and the armature fielc A.

B

1

The curve C as

is

it

in Fig. 50 gives the induction in this

primary

or

field

also called, the "impressed 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

The working

is

less

than a quarter period.

condition of the motor, which has here been investi-

gated by means of curves, can also be shown by a clock diagram. in Fig. 106, the (i.

maximum

number of

e.,

and b of

lines per square centimetre at a

be represented by the line

O

B, and

let

O

right of the vertical, then

maximum

O A

Fig. 104,

Ia represent the total

pere-turns due to armature currents in the bars to the

the

Let

strength within the interpolar space

field

left

am-

or the

B

represents to the same scale as

We

induction due to these ampere-turns.

O

stop here to inquire into the exact relation between

la

need not

and

O

A,

For the present it is only necessary note that under our assumption of no magnetic leakage in

this will be explained later on.

to

the machine, fore also to

O A must stand O B, and that the

at right angles to

ratio

between

O

O

la

and there-

Ia ,

and

O A

(i.

e.,

armature ampere-turns and armature field) is a constant. By drawing a vertical from the end of B and making it equal to O A, we find

O C

the

maximum

induction of the impressed

pere-turns required on the

field.

The

total

am-

magnet to produce this impressed field are found by drawing a line from C under the same angle to C O, as A Ia forms with A O, and prolonging this line to its intersection with a line drawn through O at right angles to O C. Thus we

O Ic the total ampere-turns to be applied to the field. The diagram below shows a section through the machine, but in-

obtain little

field

,

stead of representing the conductors by

armature and field currents are

little circles

shown by

as before, the

the tapering lines, the

thickness of the lines being supposed to indicate the density of current per centimetre of circumference at each place.

95

APPENDIX The following

is

II.

and elegant method of arriving

a very simple

graphically at the results of the integrations in Sections 19, 20,

and

Consider a winding ag, Fig. that

108

109.

is all

52,

over the pole-pitch.

infinitesimally small arc, there

extending over an arc of 180 degs.,

In each element, extending over an is

an

FIG.

call dc,

e.

m.

f.

induced which we shall

52.

represented by the infinitesimally small vector

dicular to the element of the winding ab. that the vector-sum, or the expression

the algebraic to the arc

sum

of the

ABC D

e.

m.

f.'s

(

We

de, is equal to

induced

AB

perpen-

can see at a glance

in the

AG, and

elements

is

that

equal

G, since in the latter case the elements have to

96

APPENDIX be added independently of the algebraic

sum

of the

II.

their phase relation.

m.

e.

f.'s

~

of

all

z *

But we know that

the elements

is

equal to

io' 8 volts,

V/2

hence we

induced

may

is

conclude that

equal to

AG

-r-

in

a distributed winding the

A B CD G

m.

f.

This

is,

e.

multiplied by

e.

= V/2 ~

io~* volts.

FIG. 53-

Z *

however, equal to This formula sentatfon

10"'

z *

2

I

differs slightly

we have assumed

from (21) as

in

our graphic repre-

a sinusoidal field the lines of which are

cutting the winding, whereas fields with straight contours have been

considered in Sections If the coil covers

19,

20 and 108.

an angle of 120 degs., (Fig. 53), the 97

e.

m.

f.

in-

THE INDUCTION MOTOR. it

duced in that

is

it is

equal 10-7= V2

~

z *

IO"8

=

icr8 multiplied

A G-^ABCDG,

by

to say, equal to

z *

For a

coil

1.836

~

z

volts.

extending over an angle of go degs. (Fig. 54),

we

find

FIG. 54.

at e.

once the co-efficient entering into the general equation for the

m.

f.

to be

2

1/2

Let the

coil

cover an angle of 30 degs. (Fig. 55), and

once for the co-efficient 7T

V/2

2

7T

6

1/2

as found in Section 20.

98

we have

at

APPENDIX The foregoing

is

II.

again an illustration of the ease with which

graphic methods lead to results otherwise attainable only by lengthy integrations.

FIG. 55-

In Appendix III.

we

shall describe a

account the ohmic losses that adopted in Figs.

5,

24,

in field

and

way

in

which

to take into

and armature much simpler than

33.

99

APPENDIX Figs.

5, 24,

and

can be greatly simplified

.33

corresponding to the C~R losses

For

it

if

the watt

DL

nates between

RC DL

2

from the semi-

instead of

can easily be proved (Fig. 56) that

tween RCi and

component

primary and secondary of the

in the

rotatory transformer are set off from circle.

III.

are exactly proportional to

ordinates be-

.the

CV? 2

while the ordi-

,

and RCi represent, with extremely

little

inac-

curacy, the watt component corresponding to the ohmic loss in the

The proof

primary, C*Ri. dinate between

of this

RC

RCi and

2,

Co

where

m

is

is

2

b be the or-

let

we have

= mbo,

a constant.

Calling the projection of Co on

then

very simple, for

or a current Co, then

DL,

and that of

do,

C

on DL,

a,

we have

DN = C = a 2

We

2

also have

=

a

(DL

a)

Hence follows

C*

-a

=

In a similar

-

.

manner

b

it

.

DL

(a)

can be proved that the ohmic loss in the

primary may be represented by the ordinates between

RD

DL

and RCi,

being equivalent to the ohmic loss produced by the current OR.

It is

now

evident that the output

P

the ordinates between the circle and

tween the

circle

and

Rd

of the motor

RC

2,

represent the torque

100

is

represented by

while the ordinates be-

D

of the motor.

APPENDIX

III.

i90;. Slip

Generator

roi

THE INDUCTION MOTOR. The by the

equal to the ratio of the loss in the armature divided

slip is

total

energy developed

ratio of the ordinates

between the

circle

in the secondary,

hence equal to the

between RCi and RCi, divided by the ordinates

and RC\.

Mr. Heyland has shown that the diagram may be used to represent the action of the motor when running at a speed above synchronism, i.

e.,

line, I

at a negative slip.

The -motor then

gives energy back upon the

requiring mechanical energy to drive

it.

The

simplification that

have just introduced, by which the ohmic losses can be taken into

account by merely drawing two straight

lines,

once the current, d, the torque D, the output cos

<j>,

and the

slip

5",

enables us to plot at

P

,

the

power

factor

in rectangular co-ordinates as functions of the

electrical

energy consumed by the motor, and as functions of the

electrical

energy given back by the generator.

Fig. 56 illustrates these curves, the plotting of

few minutes.

102

which takes but a

INDEX. (The numbers

refer to the pages.)

Ampere-Conductors P er Slot, Their Influence on Leakage Ampere-Conductors per Slot Armature Currents Li Different Frequencies

24 23 40

Bedell, Dr. Fred Bedell & Crehore

87 73

Behn-Eschenburg, Dr. Hans Blondel, Prof.

C ahen,

51

Andre

15

Hermann

35

Calculation of Single-Phase Motor, Chapter VII Capacity in Secondary of Transformer

Character of Magnetic Field Characteristic Curves of Single-Phase Motor Clifford,

W.

K

Constant-Current Transformer

Comparison of Single-Phase and Three-Phase Motor Conductors, Number of, per Slot Constant Potential Transformer Corroboration of Theory Counter-E. M. F. in Three-Phase Motor Counter-E. M. F. in Single-Phase Motor

D arwin,

Prof. George Howard Design of 2OO-hp Three-Phase Motor Determination of Characteristic Curves of 2OO-hp Motor

85.

63 86 1 1

60. 67, 68,

69 87 76 70 24 78 5

13.14 61,62 53

42 47,48

E fficiency

17

Emde,

85

Fritz

Exciting Current

;

Its

Determination 103

15

THE INDUCTION MOTOR.

f ormulae

of Induction Motor .............................. Frequency, Drawbacks of High ..............................

38

General Alternating-Current Transformer, Chapter I ........ Graphic Integration, Appendix II ............................

96

H elmholtz, Quoted

15

i

.........................................

41

Heubach ...................................................

85

A

................................................ 2, 102 Heyland, Hysteresis and Eddy Currents .............................. 87 in Secondary of Transformer .................... Induction Generator, Appendix III ...........................

I nductance

K

Gisbert

a PP>

84 100

............... ............................. Induction Motor, Appendix 1 ....................... Kuhlmann .................................................

15

:

On

L eakage

89 85

...................................................

2

Leakage Coefficient ........................................ Leakage Coefficient Its Theoretical Predetermination ........ Leakage Factor, Chapter IV ................................ Leakage Factor, as Dependent on Air-Gap and Pole-Pitch. .. .32, Leakage Factor, Influence of Air-Gap on .................... 29, Leakage, How Influenced by Pole-Pitch ...................... ;

5

28

29 36

30 34

Load Losses ...............................................

27

Current .................................. i, 4, Factor ...................................

14, 15

Maximum Power

18

Current, Relation Between No-Load and Magnetizing Current in Single-Phase Motors .....................

Qhmic Output

Resistance of Primary ............................. ....................................................

Output, How Dependent on Speed .......................... Overload, Dependence on Air-Gap .......................... 104

56

4,

7 18

38

32

INDEX.

P olar

Diagrams of General Alternating-Current Transformer,

Chapter VIII

23 34

Pole- Pitch, Its Influence on Leakage Factor Power Factor Its Relation to Leakage Coefficient

9

;

Power Factor;

How

Primary Resistance Primary Resistance

R elation S

;

;

Affected by Open or Closed Slots Its Influence on Diagram

21

How

81

8

Taken Into Account

Between Leakage Factor and Power Factor

26

emi-Circle Diagram

5

Semi-Circle Diagram for Single- Phase Motors Short-Circuit Current

60 19

.'

Single- Phase Motor, Chapters VI. and Slip and Resistance

VII

54 16,

and Torque Curves Slip and Single-Phase Motor Slots,

Slots

;

58 22

Number per Pole Their Number as Influencing Leakage

Comparison Between Open and Closed Speeds, Winding a Motor for Different Squirrel Cage Armature Starting Arrangement for Single-Phase Motors

23 20

Slots,

Steinmetz, C.

P

Sumec

Thompson,

Silvanus

18

52

Slip

36 49 71,72 2 42 85

P

17

Three- Phase Motor, Experimental Data of

25

Torque Torque Curves of Single- Phase Motor

16,

Vectors, Reciprocal

79,

105

17

57

80

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