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1/42

1904.] MAGNETIC DISPE RSIO N IN INDUCTION MO TORS. 239

ON THE MAGNETIC DISPERSION IN INDUCTION

MOTORS, AND ITS INFL UE NC E ON T H E

DESIGN OF THESE MACHINES.*

By Dr. HANS BEHN-ESCHENBURG, of the Oerlikon Machine

Works, f

I.

ON T H E D ISPE RS ION -CO EFF ICE NT


2/42

240 BEHN-ESCHENBURG : ON MAGNETIC DISPERSION [Jan. 28th;

importance of the coefficient o-lies, as is known, in the limitation by it

of the maximum power-factor, and of the capacity for overload of the

motor. As is known, we have the approximate relation

COS


3/42

1904.] IN INDUCTION MOTOR S, ET C. 241

small in comparison with the magnetic resistanc e of the air-ga p. Thi s

condition may obviously always be fulfilled if we confine ourselves to

such degrees of saturation that the magnetising current is proportional

to the terminal voltage.

In formula (i), giving the definition of


4/42

242 BEHN-ESCHENBURG : ON MAGNETIC DISPERSION [Jan. 28th,

FIG. I.

or holes of the second system. In both positions let the same number

of magnetic lines be generated by the primary winding, and pass over

from the teeth of the first system into the teeth of the second system.

It is self-evident that in the former position the mutual-induction will

be exactly equal to the self-induction, since the secondary conductors

are surrounded by exactly as many magnetic lines as are the primary.

In the latter position, on the

contrary, a portion of the

magnetic lines will enclose a

smaller number of secondary

conductors than they do in

the former position.

The total amount of the

mutual-induction may be

measured in a simple way as

the sum of a set of products,

each product being the

amount of a branch of the

magneticfluxproceeding out

of a primary tooth multiplied

by the number of secondary

conductors which this branch of the flux surrounds until it again

returns into the primary system. The difference between the amounts

so reckoned of the mutual-induction in the two extreme positions

gives the loss of the mutual-induction which occurs in the second

position. This loss is equal to the difference between the self-induction

and the mutual-induction in this second position. But now, since

during the operation of the

motor, in consequence of the

slip,

the teeth of the two

systems glide past one an-

other in their relative posi-

tions,

it follows that half the

difference of the mutual-in-

ductions in the two extreme

positions will indicate the

mean value of this difference

while runn ing. If one ex-

changes the respectiveroles

of the primary and secondary

systems, the estimate

so

made

of this difference will apply

equally tothe stator system as to the rotor system of the m otor. Strictly

speaking, in this regard the windings of all the slots in their actual and

complete relations ought to be taken into consideration. All the phases

of the winding of the one system act successively and together upon all

the phases of the winding of the second system. In consequence there

occur in general at definite places in each system the known distortions

and inequalities of the magnetic field, and these are bound up with the

practical limitation of the number of current-phases to two, three, four,

J

r

H

I

i i

pi

1

|

7

\

6jU

r-

7

FIG. IA.


5/42

1904.]

IN INDUCTION MOTORS, ETC.

243

or six phases. In the case of three-phase windings this inequality may

amount to 15 per cent. But this complete investigation would entail

difficulties out of proportion to its usefulness, having regard to the

desired limits of accuracy. The principle of the phenomenon, and

also the magnitude of the determining relations, admits of being

expressed to a sufficiently close approximation in a simple investi-

gation which takes into account one phase only of the primary winding

of a three-phase motor.

Let us assume, as the first and simplest case, a motor possessing in

its primary and secondary systems three slots and three teeth per pole-

pitch. The primary winding in one phase may be represented by a

single turn, which lies in the slots 2 and 5 (see Fig. 1) of the primary

system. The secondary system has one conductor, L, in each slot.

The slots and teeth of each system are numbered progressively from

left to right. Doubtless the schematic representation of the figures

F I G .

2.

will be intelligible without further explanation. Let the arrow-heads

indicate the course of the magnetic lines, and let each arrow denote a

portion of the magnetic flux amounting to the value / .

From the figure the amount of the mutual-induction may now be

read off in the following manner, namely, that each separate partial

re-entrant magnetic flux of amount / will be multiplied by the number

of secondary conductors L which it embraces. From the middle

tooth 4 there emerge to left and right two magnetic fluxes each of

value / , each of which surrounds three conductors. Therefore the

tooth 4 contributes toward the mutual induction an amount equal to

2 x / x 3 X L. From each of the teeth 3 and 5 there emerge two

fluxes f, each of which surrounds one conductor, namely, conductors

2 and 5 respectively. These fluxes, therefore, contribute the amount

2 X 2 X / X I X L . The total mutual-induction of this system in this

position may therefore be stated as of the value :

2 / x 3 L + 2 x 2 / x 1 L = 10 ( / x L).

In Fig. IA the same system is depicted in the second position, in

which the teeth of one system stand opposite the slots of the other.

VOL. 33. .17


6/42


Let the same total magnetic flux as before pass over from each tooth

of the primary system into the secondary system ; but it must now

divide itself between two teeth of the secondary system.

But a mere superficial observation makes it evident.that a part of the

flux now no longer encloses any secondary conductors, and that, on

the other hand, the secondary conductor No: j is not surrounded by

any flux.

Let us count up, as for Fig. i, the amount of the induction ; then

we find for the fluxes which emerge from the middle tooth 4 the

induction values 2 / x 2 L ; for the fluxes of teeth 3 and 5 the values

2 / X 2 L - f - 2 / x o L = 2x 2 / x L. The sum of these is now 8 / x L ;

that is to say, only 80 per cent, of the m utual-induction as it was in the

first position. ' In other words, we therefore lose in this position 20 per

cent, of the total flux for the mutual-induction.

If we carry out the similar investigation for a primary and a

secondary system with six slots per pole-pitch, in which the winding

of the primary system consists of two windings lying in two (pairs of)

slots,

we then obtain, according to Figs. 2 and 2A, in the first position

a total of 3 6 / x L, in the second position 3 3 /X L. In the second

position we therefore lose about 10 per cent, of the mutual-induction,

or in the mean between the two positions about 5 per cent.

In a similar way we get for two systems with nine slots per pole-

pitch, and a primary winding of three windings distributed in three

(pairs of) slots, in the first position a total of mutual-induction of

11 9/X L ; in the second position, 114 /X L. (In this case there is

assumed for calculation a flux of 3 / in each tooth of the primary

system that is entirely surrounded by three primary windings.)

For systems with 15 slots per pole-pitch and 5 primary windings

one gets, in the first position 545 / x L ; in the second position

53 8/ X L. (In this case there is assumed a flux of 3 / i n a primary

tooth which is surrounded by all five windings.)

If now, in place of the two systems having equal numbers of slots,

we examine the case of two systems with unequal numbers of slots,

then the distribution of the magnetic fluxes through the individual

teeth takes a rather more complicated form in the different positions.

But the character of the phenomenon is quite like that of the cases

above considered. In general there can be found two positions in

which the value of the mutual-induction is respectively a maximum

and a minimum. The maximum value agrees approximately with the

value of the induction in the first position of the system with equal

numbers of slots. But in this the values are to be compared with the

primary system for equal numbers of slots, and with the secondary

system as to equal numbers of conductors. For example, if a

secondary system with 9 conductors in' 9 slots is to be compared

with a system of 15 conductors in 15 slots, then the value of the

induction in the first case must be raised in the proportion 15 :9, since

in each slot 15/9 of a conductor will be assumed.

Also in the cases of systems with different numbers of slots the

action on one another of all the phases of the primary current strictly

stated, must be taken into consideration.. Then the influence of the


7/42

1904.] IN INDUCTION MOTOKS, ETC. 245

ineq uality of the field will have a pre dom ina nt effect. Fu rth er , the

distribution of the w inding, and the winding-pitch in the two systems

must be accu rately set out for each particular case. The se influences,

however, involve very detailed expressions, and yet they exercise on the

char acter of the phen om enon and on the ma gnitud e of the relations

involved so little change, that they may be passed over in the scope of

this enquiry, the difficulty of wh ich lies rat her in its exp erim ental

part .

As an exam ple we cons ider, as in F igs. 3 and 3A, the mu tual- indu c-

tion of two systems of which the primary system has six slots per pole-

pitch, with two windings as in Fig. 2, and the secondary system 9 slots

per pole-pitch with 9 conductors.

In the first position, Fig . 3, the am oun t of th e mu tual-i ndu ction is

5 4 / X L ; in the second position, 5 2 / x L. If the secon dary nu mb er of

con ducto rs 9 is for comparison with Fig. 2 reduc ed in the prop ortion

6 : 9, then in the first position we ha ve the value S4X f X / x L =

3 6 / x L, exactly as in Fig . 2. In the seco nd position, Fig . 3A, the

F I G .

2A.

am oun t is 34*6 / X L, while in Fig . 2 t he am oun t 33 / x L was

obtained.

A similar calculation was m ade for a primary system w ith 9 slots and

3 winding s, and a second ary system with 15 slots and 15 cond uctors.

He re there was found in one position the value 19 9 / X L, in a second

position the value 196 /

X

L. For com parison with the values which

were given above for two systems with equal num bers of slots, 9 per

pole-pitch, these values must be reduced to equal numbers of conduc-

tors.

T hu s one obta ins for the system s with 9 an d 15 slots in the first

position the value 119*5 / X L ; in the second position, H7 '5 / X L ; for

the system with 9 slots in both primary and secondary we have earlier

found in the first position 119 / XL, in the second position 114 / X L.

The se c onsiderations have been set out with this com pleten ess,

beca use they afford a n insig ht into an essential el em ent of the so-called

dispersion-coefficient awhich does not arise out of ordinary mag netic

leakage, but wh ich mu st also occur in an ideally leakage-free mo tor ;

and in general the magnitude of this element will be greater than the

so-called peripheral leakage.


8/42

246

B E H N - E S C H E N B U R G : ON

M A G N E T I C D I S P E R S I O N [Jan. 28th,

In order to obtain a view into the order of magnitude of this effect,

which we shall denote as the effect of the distribution of the winding,

or effect of the ivinclitig-coefficient, let us assemble in a Table the

Nl . 'MBER

Primary.

J

6

6

9

9

15

OF

SLOTS.

I

1

Secondary,

j

3

6

9

9

15

T

5

INDUCTION.

Maximum.

1 0

3*

3

6

119

199

545

Minimum.

8

33

34-6

114

196

538

Half-Difference

i

= Winding-coefficient.

10 per cent.

4*2

2-3

2 I

075

numerical values above obtained. As a measure of the influence of the

winding-coefficient we may regard the quotient of the difference of the

maximum and minimum values of the induction divided by the

maximum value. In order to be able to assign beforehand to these

coefficients a mean value for all possible different positions of the two

systems,

we insert in the quotient thehalf of the difference between the

maximum and minimum values.

In the same way we have to consider the combination of a limited

number of phases in the stator and rotor windings. There are slight

fluctuations, on the one hand of the self-induction of the combined

stator windings, and of the combined rotor windings, and on the

other hand of the mutual-induction between the stator and rotor wind-

ings,

fluctuations which depend on the different positions of the rotor,

and on the variations from instant to instant of the primary current.

In a motor with three-phase stator windings and three-phase rotor

windings, we must distinguish two particular positions 6f the rotor and

two particular moments in the periodical changes of the current. In

the first position the three phases of the rotor winding correspond

exactly to the three stator phases ; in the second position the rotor

phases are displaced ^ of the pole-switch. Further, the first moment in

the changes of the current is taken when the current of one phase

is at its maximum ; the second moment when it is at its zero value. If

we compare the mean value of the self-induction of the three stator

phases,

in these four cases, with the mean value of the mutual-induction

between the three stator phases and the three rotor phases, we observe

a small difference which diminishes rapidly with an increase in the

number of slots; for example, for six slots per pole this difference

may amount to 1*2 per cent., for twelve slots to 0*4 per cent. We have

here further to consider the influence of the wave-form of the primary


9/42

1904.] IN INDUCTION MOTORS, ETC.

247

currents on these effects, which

we.

put together under the designation

of " winding-coefficient."

How ever comp licated the relation betwe en the winding-coefficient

/ /

\ I

I/

\2\

u

Ul

Y

Y

\

f

v

i

;LJ/|LJ2;U;3 l_J


10/42

248 BEHN-ESCHENBURG: ON MAGNETIC DISPERSION [Jan. 28th,

with the subject, this coefficient which will denote by

a

t)

may be set

out by the expression :

' . = ' - 3 |

;

; ( 4 )

where K

x

has the meaning of a function of N to be determined

experimentally from case to case, but which generally differs but

slightly from the constant-value of unity, and in general also expresses

all those influences which may arise from the various distributions of

the winding in different parts of the phase, and from the winding

pitches . In the coefficient K

r

are also contained the effects of the

form of slots or teeth upon these phenomena, and on the influence of

the inequalities of magnetic reluctance in different positions.

In the cases hitherto considered, we have indeed discriminated

between primary and secondary systems, but it is immediately evident

that in the motor each of the two winding systems, stator or rotor, has

for the carrying out of this calculation to be regarded as at one time

acting as primary, and at one time as secondary.

The value of


11/42

1904.] IN INDUCTION MOTORS, ETC. 249

set down tentatively

for the

usual forms

of

slots

as

O T

cm. In

order

to take into acco unt the influence of the special forms of slots in

particular cases,

we

will further intro duce

a

coefficient K2, which will

requireto beexperimentally determine d. Th enwe may set :

X

H

~ o-i x bx K

2

For closed slots,inplaceof the air-slit in theperipheral surface there

isa very thin bridge of iron. Thethickness of this iron bridge will

amount to about o*imm.at the thinnest place. These iron bridges

ought, under normal running, to become completely saturated by the

stray flux ,so thatfortheir resistancewemake reckon them tentatively

to haveapermeability as low as

fi

= ioo. Ifnow the lengthof theiron

bridge

at its

thinnest place amounts to, say,

X

cm., then

the

magnetic

resistancefor theclosed slot maybe setat:

,

X

The strayfluxalong the peripheral surfaceof theiron cylinder forms

a magnetic circuit surrounding the primary coils which will be

distributed in the slots over a third of the pole-pitch. The chief

resistancein this circuit is constituted by thepaths of passageat the

openings

of all

those slots which

at the

peripheral surface include

one

primary coil. If, asbefore,Ndenotesthenumberofslotsin onepole-

pitch, then one primary coil is included or bridged over by

slot-

openings. Theresistanceof the magnetic circuit of thestrayflux is

therefore about equal

to

X

p. The

resistance

in the

path

of the

main fluxFwhich passesout of theprimary system intothesecondary

is, approximately :

R =

^T"x~V

where $is theair-gap length from iron toiron, bthe axial lengthof the

iron core,

rthe

length *

of the

pole-pitch

at the

face.

W emaydenoteby the coefficient cr

2

thequotientof the stray flux /

by themain fluxF , andobtain approximately:

^

R K

*

S

for open s lo ts ; . . . (5)

and

N

~ 2 N X r

X

X'

for closed slots (5AJ

N X r

X

X'

3. FLANK.DISPERSION (Stirnstreuung).

A second kind of magnetic dispersion which also occurs in every

*

For

motors

in

which

the

peripheral surface

is

interrupted

by

openings

of slots,the length r must be reduced byabout the total width of all the

openings of .slots within one pole-rpitch, correspond ing to. the increase of

no-load current produced by these openings.


12/42

250 BEHN-ESCHENBURG

:

ON MAGNETIC DISPERSION [Jan. 28th,

motor consists of the magnetic flux which exists outside the iron core.

Those parts of the winding which constitute the end connexions

between conductors in the slots, and which project as curved winding-

bunches or bends at the flanks of the stator and rotor cylinders, give

rise to a magneticfluxoutside the iron core-bodies. Thisfluxsurrounds

these curved connexions in such a manner generally that only a small

fraction of the flux created by the bends of the one system intersects

the bends of the other system. These bends, or end connexions at the

flanks of the motor, are in the motors of ordinary construction more or

less closely or completely surrounded by the solid iron parts which

form the housing, the casing, and the clamping-plates for the laminated

core-bodies. Yet it is possible so to choose the distance between the

winding and these iron structures that only a small part of the stray

flux created by these parts of the coils (and which we shall call flank-

dispersion) passes into iron.

In the main this stray flux is equal to the m agnetic flux which would

FIG.4.

be created by an independent group of coils of a form similar to the

two projecting bends at the two flanks, if put together as a coil. What

is necessary is therefore to determine the self-induction coefficients of

similarly constructed coils, and the coefficients of mutual induction

between such coils if placed in such positions relatively to one another

as would about correspond to the respective positions of the projecting

bends in the stator and the rotor.

There was undertaken a series of self-explanatory measurements on

variously shaped coils of this sort, away from any iron cores, in order

to obtain practically for the various forms reasonable estimates of the

influence of the lengths of the windings, the distribution of the

windings in separate coils, and the mutual-induction between the coils.

In this investigation one is chiefly concerned with two shapes of coil,

viz. :

' (/) With coils the end bends of which are straight out, or in

approximate )' the same (cylindrical) surface as that in which lie those

portions of the coils that are placed in the slots ;

(ii)

With coils the end bends of which are bent up or down out of

this surface.

Fig. 4 depicts a group of 3 straight-out coils nested against one

another; Fig. 5 a group of 3 coils having the bent ends turned up.

The details of the research of the different forms of coil may here


13/42

1904.] IN INDU CTION MOTORS, ET C. 251

be passed over. Th e results can be assembled in the following pra ctical

rules :

T h e coefficient of self-induc tion of a single coil wh ich consists of

d b

F I G.

5.

W tur ns having a mea n length of o ne turn /, is approx imately for all

forms existing in practice :

\, = 6W

S

x /.

T he coefficient of self-induction of a gro up of coils, which are laid

within one another at small distances apart as are the coils in motors,

having a total number of turns W and a mean length of turn /, amounts

approximately to

\ , = c x 6 W

2

X / ;

wh ere c varies be tw een 0 7 and 0-55 for gr ou ps of 2 to 5 coils, or on th e

average

X

s

= 3-6 W

a

X /.

If into the neighbourhood of the coils iron bodies are brought which

may represent the nearest iron parts in the neighbourhood of the bent

end s at the flanks of the moto r, then the coefficient of self-induction

will be increased about 20 per cent. Th e mutual-induction betwe en

the end ben ds of th e stator and rotor may diminis h th e value of the

self-induction by 20 to 50 per c ent, acco rding to the ar ran gem ent of

the bends.

T he coefficient of mutual- induc tion of s traigh t-out coils, wh ich are

held at the usual distance from one anothe r, am ounts to about 50 per

cent. of the coefficient of self-in ductio n ; the coefficient of m utua l-

induction betwee n a straight-ou t and a bent-u p coil, or be twee n two

coils bent-up in opposite directions, amou nts to a bout 20 per cent, of

the coefficient of self-induction.

As a mean value for the coefficient of self-induction of the end-bends,

which cause the flank-dispersion, after taking ac cou nt of the influence

of the iron masses and of the mutual induction, we may write :

\ = K

3

X 3-5 X W

2

X / ;

where the coefficient K

3

relates to the influence of the winding arrange-

ments and of the iron structures, so far as these depart in special cases

from a mean value. For /, the mean length of one wind ing of the end

bend s lying outside the slot, we de duc e from the dim ensions of the

motor an approxima tely generally valid relation, which may again in

special cases require to be reduced to a mean value by the insertion of

a coefficient.

The length of the bend of one coil comprised at the two flanks is


14/42

252 BEHN-ESCHENBURG: ON MAGNETIC DISPERSION [Jan. 28th,

equal to double the length of the pole-pitch r, increased by adding

four times the distance which the end-bends project beyond the core-

body. But the length of this projection is itself approximately propor-

tional to the pole-pitch, since the coils must stand out so much the

further the more the intervening coils over which the end winding has

to span. So we put :

I= 3

r

x K

4

,

and so get approximately

X = K

s

X io X W

a

X

T

;

(6)

in which the constants K

3

and K

4

are comprised in the constant K

s

.

In order to ascertain how much this species of self-induction

contributes to the dispersion-coefficient


15/42

1904.] IN INDU CTION MOTORS, ETC. 253

and on the other hand, of the coefficient


16/42

254 BE HN -ESC HE NB UK G: ON MAGNETIC DISPER SION [Jan. 28th,

W e employ for brev ity the following symb ols :

P = num ber of poles.

N,= num ber of slots of the stator.

N

2

= num ber of slots of the roto r.

D = diame ter of the bore, in cm.


17/42

1904.] IN INDU CTION MOTORS, ETC. 255

in one motor amounts to

b

and in the other to

b',

the n the coefficients

5.

b

= 14-5.

Th e difference of th e va lues of ain the two motor types gives for

4 poles :

aV = 0*015 ; for 6 poles,

a n

d &' =

r

4 5

x l

^-

Ac cord ing to formula (8) one wo uld have

a

a

= K

3

x o*oi. B)'

comparison with the above we should obtain for these types of motors

K

3

= 1*5.

The same types of motor, but provided with a non-insulated short-

circuited winding in the rotor, gave :

Type 358 (4-pole)0 = 0*05.

Type 359 (4-pole) a' = 0*04.

a

a'

= o*oi ; K

3

= 1.

1 \

u / T A o Q

S

W it h 6 p ol es , er = 0*050.(2)

Motor lypc

838 : < 0 1 ^

>

'

Jl

(

With 8

poles,

a =

0*063.

4

D = 49 ;

0

=

0*08.

N , = 72 ; N

2

= 120.

6 = 19.


18/42

256 BEHN -ESC HE NBU RG : ON MAGNETIC DISPER SION [Jan. 28th,

Motor Type840: {

W i t h 6

P

o l e s

'

a>

= '4

2

-

Jy

* \

W ith 8 poles, V = 0-056.

D'

= 49 ; 8 = 0-08.

N '

t

= 72 ; N'

2

= 120.

b' = 28.

W i t h 6 po l e s , aa = o 'ooS.

W i t h 4 p o l e s ,


19/42

1904.] IN INDUCTION MOTORS, ET C, 257

The observations gave:

Motor Type 363 :With 8 poles.

D = 58; S = o-oo ) _ , .,, , . ,.

N, = Q6 N = 144

w

Phase-winding, a 0-054.

I '

2

^ ( Rotor with squirrel-cage,


20/42

258 BEHN -ESCHE NBU RG : ON MAGNETIC DISPER SION [Jan. 28th,

off rapidly with the increase of the short-c ircuit cu rre nt. Obviously,

we must here abando n those method s for the estimation of the charac -

teristic values of the motor which are in the diagram based upon the

assumption that


21/42

1004.] IN INDUCTION MOTOKS, ET C. 259

T he observations of th e short-circuit curr ents for the m otor under

consideration are set out in Fig. 6. Th e no-load cu rre nt am ounted in

the first case, with closed slots at 190 volts, 50 pe riods , to 80 am per es ;

in the second case to 100 am peres . Th e rema ining data run :

Motor Type

367

D = 9 0 ; 5 = c m .

N, = 144 ; N , = 180.

b = 32-5. P = 12.

Slot-breadth, n mm.

In the first case, for a short-circuit current of 700 amperes :

100 80 ,


22/42


portionately very small value of the so-called peripheral-dispersion, we

adduce the results of a motor, Type 363, the stator of which was

executed,first,with 96 closed slots; secondly, with 96 completely open

slots with an opening of 13 millimetres. The slots of the second stator

were arranged for the insertion of former-wound coils. The rotor had

in both instances 144 slots, which in the first case were closed, in the

second were slit with slits about 1 mm. wide. The iron bridges over

the closed stator and rotor slots had a thickness of o*i mm. and a

breadth of about 2 mm. The normal current of the m otor amounted

to about 200 amperes at 190

volts.

In the first case the stator winding

was carried out with two conductors per slot in star grouping ; in the

second case, with four conductors per slot joined in triangle grouping.

The curves, Fig. 7, depict the short-circuit currents in the two cases.

The no-load current amounted in the first case to 35 amperes at 200

volts ; in the second case to 53 am peres at 200 volts. The air-gap was

Amp

30 0

20 0

100

n

A

/

/

/

/

/

/

/

/

/

/

/

50

100V0IL

FIG. 7.

in the first case

0*9

mm .; in the second n mm. If reduced to equal

length of gap, and equal numbers of conductors, the no-load current in

the second case would therefore be about i*6 times greater than in the

first

case.

The short-circuit curve in the first case runs in a straight

line from about 250 amperes. For 300 amperes one obtains, in the

firs t case :

_ .35.

v

_93_

_

in the second case

The dimensions of

Chapter III. , 1),

200 300

= i

3

-

20 0

= o

-

o62.

300

the motor are (compare the last example in

Motor Type

363 :With 8 poles.

D = 58 ; b= 24.

N, = 96 ; N

2

.= 144.

Slot-breadth, 13 mm. ; slot-pitch, 22-5 mm.


23/42

1904.] IN INDUCT ION MOTO RS, ETC . 261

T he magnetic re sistance X of th e slot-opening is in the first case

considerab ly smaller than in the second case. But now althou gh the

value of the total dispersion-coefficient ais larger in the second case

than in the first case, the diminution of the peripheral-dispersion in the

second case m ust be masked by an increase of the contrib utions to a

from o ther sourc es. In pa rt, the dispersion-coefficient a

3

du e to flank-

leakag e in th e secon d case is relatively...greater in con seq uen ce of the

considera bly greate r m agne tic resistance in the second case, as

evidenced by the i*6 times grea ter norload- cur rent . Acc ording to a

calculation mad e in Chapter II I., i, for the same motor, the value for

the coefficient of flank-dispersion was foun d :

2


24/42

262 BE HN -ESC HE NB URG : ON MAGNETIC DISPERSION [Jan. 28th,

Motor Type 365 :With 8 poles.

D = 70 ; b= 30.

N ,

== 12 0; N

2

= 160.

Slots closed.

In the first case 0 = o*i ; in th e second case 5= 0*14. Us ing th e

earlier-found constants, the contribution due to flank-dispersion was

found

(1)

2 ff3

= 5 x o-i .x_ rg5

= O

-

O 2 I

.

3

(2) 2

a'

3

= 0*029.

With the short-circuit current at saturation-value there was observed

in the mean

a a = 0*009 ;

so tha t in both case s ther e may be rec kon ed a value of o*ooi for the

difference of the peripheral-disp ersions, and, therefore, for the

peripheral-dispersion itself the value

5 x 0*04 x 1*25

2

ff

, =

2

= 0*006 ;

3

4 0

2


25/42

1904.] IN INDUC TION MOTORS, ETC. 263

Fro m this we may calculate the magne tic resistance X of the closed

slot

X = 0-5,

therefore about five times gre ater than for a slot-opening of 1 mm .

All the observations set forth in this chapter show a sufficient

agreement of the observed results with the values calculated from the

theoretical considerations of Cha pter I I. 2, and lead to the inference

that in the ordinary constructions of motors with open slots the part

relatively contribu ted by the periphera l-dispersion to the total values of

the dispersion-coefficient

a

plays a very subordinate

role,

and is in any

case capable of being re presen ted by formula (5) as

_ _

ffs

- 2~N

T

X"

Fo r closed slots, in which the iron b ridg e is ma de thin e nou gh, this

dispersjpn-coefficicnt may be estima ted abo ut four times grea ter than

for slots with slits. T he theo retica l con sider ation led to a tentative

difference to be expected from the ten-fold contribution for closed

slots. But the dime nsions of the ma gne tic resistances of the slot-

apertures do not lend themselves to any precise determination.

3 . W I X D I X G - C O E F F I C I E N T S .

After having dealt in the two pre ced ing cha pter s with the two

chief sources of ma gnetic dispersion , and having established their

importance, we now finally deal with the experimental verification of

the operation describe d in Chap ter I I., 1 of the Winding-Coefficient a

t

.

Formula (4) gives the definition

N-

in wh ich the coefficient K, may be pu t as about equ al to unity.

Th is expression is distingu ished from the dispersion-coefficients r

=

and


26/42

264 BEHN-ESCHENBURG

:


For two different numbers of poles, in the case of the same motor,

one obtains two different values of the total dispersion-coefficient


27/42


of 8, r, b, P, and X are maintained alike, the number of slots alone

is changed.

For the calculations of Chapter II. , i. in the final formula (4),there

was inserted for the mean value of


28/42

266 BE HN-ES CHEN BURG : ON MAGNETIC DISPE RSION [Jan. 28th,

Motor Type

3066.

D = o,o; b=-3-

8 = o-i; X = o-i5.

N

l

= i

4 4

; N

2

= i 8 o .

Z = 162.

P = 12 ;


29/42

1904] IN INDUCTION MOTORS, ET C. 267

(b) We adduce two further examples in which for similar types of

motor, with equal num bers of poles, the num ber of slots was altered.

Motor Type

360.

D = 38.; b = 24.

Seffective = 0-08 ; P = 6.

j N , = 54 ; No= 7 2 ; X = o"2: th en * (observed) = 0054.

( N, = 108 ; N

2

= 144 ; X = o

-

i : then n'(observed) = 003 9.

Th e no-load cur ren ts in the two cases w ere approx imately alike.

In the first case eacli slot he ld four con duc tors ; in the second case,

two conductors.

The difference of the peripheral leakage was reduced to zero by

the slit in the slots being in the first case dou ble as wide a s in the

second case.

Therefore we have :

a a = 2(


30/42

268 BEHN-ESCHENBURG

:


If, following our earlier calculation, we estimate the peripheral-disper-

sion of case

(i)

as four times greater, we get

2 (


31/42


winding elements in the stator and rotor, which alter the uniformity of.

the magnetic field,

(v)

in consequence of particular winding-pitches of

the coils in stator and rotor which affect the coefficients of

self-

induction and mutual induction of these elements, (vi)in consequence

of diverse actions which the particular dimensions of slots and air-gap

exercise upon the reluctance of the magnetic circuit of which the

magnetic system of the stator and rotor consists.*

Let it be assumed that the dispersion-coefficient amay be deduced

with extreme accuracy from the constructive data, on the basis of the

concluding formula, then there remains as the final task for the con-

structor, using this value of


32/42

270 BEHN -ESC HE NB UR G: ON MAGNETIC DISPERSIO N [Jan. 28th,

the ohmic resistance by

n,

the voltage-drop

]r

by

c

; so then we obtain

the connexion between

n

the voltage of the supp ly

mains E

o

, that of the

reduced voltage E, and

the current J directly

from the figure.

W e designa te by S

e the slip of the mo tor, by

FIG. y. r

s

the ohmic resistance

of one phase of the

secondary system, by

m

the transformation-ratio of the windings of the

prim ary and second ary systems. J

o

denotes the magnetising current,

wh ich, for simplicity, we will reg ard as coin cide nt with the no-load

cur ren t. An expression wh ich often recu rs in the theory we will

write, for brevity

S E . _

n (I.

0

r~ in

Then we have for the primary current

T = T

Ji

Z-

( )

for the secondary current

J

3

= J, X m Xa -V-L^-Z : (2)

and


33/42

1904,]

IN INDUCTION MOTORS, ETC .

27 1

The input of power is :

3 EJ

0

(i r

A = 3E |, cos

$

=

The torque developed, including that which is used in producing

the no-load work (friction, etc.) is in kilogrammetres :

D =

A

- ;

T

o

x i

03

'

where T

o

signifies the no-load speed.

Third Point

J i -

- T " '

CO S (j) = I 2 cr ;

A = -2-V " (1 -

2

IT) ;

This point gives the load with maximum power-factor. Now a

rationally-built motor will obviously be so dimensioned that its

normal load approximately corresponds to this point, always provided

that the conditions of capacity for overload do not conflict with it.

Fourth Point

_

r

5

J = -

co s $ = 12'3

cr;

J J

2m

~

( 2 < r '

(7)

c

A = - -- .- i XT-5(14 '6cr) I

Fifth Point

_

J_

.

a

T Jf

J

. -- ;

s/2

(8)

A : = J" ( j _

ff

)

2 ff '

This point corresponds to the maximum torque which the motor

can exert.

If we denote by D the torque which corresponds to the third

point, at ideal normal load, then the maximum torque D

OT

is related

to the normal torque according to the expression :

(9)


34/42


This

is

the maximum capacity

for

overload

of a

motor whose

normal load corresponds to the third point.

The torque at the second point isvery nearly equal to the half of

the torque of third point, and that at the fourth point nearly 1-5 times

that of the third point.

Now these five points determine the characteristic performance oi:

the motor with adequate precision so far aspractical design is

concerned. For this purpose the influence of the primary resistance,

cau easily be subsequently taken into account as a correcting term by

reference toFig. 8, since the voltage Eused in the formulas may be

reckoned from the supply voltage E, and the voltage-drop

e.

So long

as

e

is small

in

comparison with E, then from the figure we have

approximately :

E = E

o

J

x

r

x

c o s


35/42

1904.

]

IN INDUCTION MOTORS, ETC .

273

The magnetising current J

o

can only be exactly calculated if the

numbers and dimensions of the slots are known in addition to the

principal dimensions and winding data. The depen dence of the

exact distribution of the magnetic field upon the number of slots has

been repeatedly discussed by others. The influence on the magnetising

curren t of a voltage curve which departs from the simple sine form

will not be here regarded.

EXAMPLE I.TO find the characteristic curves of a q-pole 5 -HP.

motor, of which we have observe d t he following data :

At no-load,\vith E = 2oovolts;

J


36/42

274 BEHN-ESCHENBURG: ON MAGN ETIC DIS PERS ION [Jan. 2Slh,

These values

of

torque, current, power-factor,

and

efficiency,

are

calculated without taking into acco unt the drop of voltage due to

the primary resistance /',. Now we have to correct these values in

accordance with the diagram Fig. 8 and formula

(TO).

The corrected

Tableis asfollows:

POINT.

I

2

3

4

5

TO R Q U H .

0-13

i '5

3'

3 "9

6-8

CURRENT.

2

5

8-8

11-

9

27

POWKR-FACTOR.

O-I8

0-852

0-905

o-886,

74

SLIP.

0

0-92

1-87

3

> T

5

8-6

EKFICIKXCI

0

9 0

92

86

For larger motors these corrections are obviously much smaller,

since the loss in the primary copper is relatively smaller.

A few further formulae are needed to complete the set for the

calculation

of

motors.

The magnetising current of three-phase motors can be estimated,

with a precision practically sufficient for the purpose of design, from

the expression

where o is the air-gap, W the number of conductors of one phase

within one pole-pitch,and B thevalueof theamplitudeof the maximum

flux-densityin thegap. Inthis expression the re isassumed a customary

width of aperture of slots,and inaddition an increase of the magnetic

resistance

of the

air-gap

due to the

iron teeth, amounting

to

about

20percent. If wedenoteby F theuseful flux through onepole-pitch,

and by D the diameter of the bore, then B is defined * by the

equation

B - -

3 X F P

(i )

_ E x 10

8

.

r

,

1

~~2-2 X / PW

K

*{

where

/ is the

frequency

-HW V


37/42


38/42

to

to

o

to

NO

O N

o

VI

O l

o

O N

o

H

O i

O

O i

O i

V J

O

O

O

-vj

V J

O

o

O l

O J

to

4^

o

NO

NO

O

O

o

o

00

1i

o

ON

CO

o

o

ON

o

o

O J

NO

oo

to

4^

O

M

NO

NO

M

o

o

0 0

o

o

00

o

O i

o

0 0

OJ

O i

o

a

=-

o

o

v O

O J

to

o

O J

O i

o

NO

o

O l

to

v O

o

1-1

o

o

O J

O

O J

v O

o

5

i -

o

o

O J

M

O J

to

o

to

O J

O N

HI

O J

ON

o

O l

CO

o

M

o

O J

O l

O J

O l

o


39/42


second; and JL

o

is the specific load, or number of ampere-con-

ductors per centimetre of periphery. This formula (14) is an important

and very practical formula for electric machines of all types; but

for continuous-current machines the coefficient n may be replaced

1

If now the problem is put of designing a motor for an output of

A watts, with P poles, / cycles per second, then the product of volt-

amperes A', which the motor at normal load will take, is given with

close practical approximation by

A = A' x i| x cos p.

The normal current corresponding to A' is

F

so the problem is so to build the motor that the normal load is

coincident with the load at the maximum power-factor. Then we

must have

= ] X Ja (15)

The weight and size of the motor is fairly determined by the total

flux P F ; and, by formula (14), this is so much the smaller the greater

the peripheral velocity, and the greater the number of ampere-con-

ductors per centimetre of periphery.

By transposition of formula (12) we have

P F = #7 rD 6B (16)

For reasons of construction it is in general not possible to arrange

more than 300 ampere-conductors in 1 centimetre of periphery, and,

moreover, mechanical difficulties do not admit of a peripheral speed

exceeding 4,000 centimetres per second. The gap-density B is limited

by the saturation of the teeth, which ought not to exceed the limit

beyond which the magnetising current increases faster than the flux-

density. In order to afford a large winding space in the slot the teeth

must be kept narrow. The air-gap

d

must, for mechanical reasons, not

be made less than about ^ of D. We will design the motor with

the moderate values: U =

1,500,

J1^=150, for motors of less than

10 H.P.; and U = 2,500, J

L

o

=

250 for motors exceeding 100 H.P.

Then we at once can arrive at P F, and from it at the product

b

B.

Having JL

o

and U, W is determined. But, according to formula (n) ,

W and B are connected with one another by the prescribed no-load

current J,,, and so all the dimensions are thus determinate.

From the earlier discussion respecting ait is known that adistinctly

decreases as the number of slots is increased ; but a large number of

slots can in general be accommodated only in a large pole-pitch ; and

further, adiminishes asbthe core-length is increased. One part of


40/42

276 BEHN-ESCHENBURG : ON MAG NETIC DISP ERS ION [Jan. 28th,

depends on 6 has been shown to be equal to 6 d-r-b; therefore for

S

= o

f

i, it follows that.we . must have

b.=

30 if this parfcof the dispersion

is to have a value equa l to tha t of t he first par t: Now by chan gin g the

dimensions here and there, and balancing the difficulties and profits

of one alteration in the dimensions against those of another, we find by

successive ap pro xim atio ns th e m ost econ om ic value of


41/42

1904.] ' IN INDU CTION MO TORS , ET C. 277

Now the magnetising current ]

0

is determined by the condition :

Jo= J X \/a = o

-

86 ampere.

For motors of a size so small as this, we apply in desig ning the

me an values : - .

U = 1500 cm. per sec .; J L

o

= 150 am ps, per cm .; 8= 0*05c m.

W e will take D

20 cm . For mu la (17) gives B = 4200.

T o fulfil formula (11), we calc ulate as num ber of stator co nd uc tor s

per phase per pole

w

_

_ O-Q5

o"86xi

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