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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
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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
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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
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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.
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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
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244 BEHN-ESCHENBURG : ON MAGNETIC DISPERSION [Jan. 28th,
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
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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.
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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
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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
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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
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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.
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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
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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
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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
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and on the other hand, of the coefficient
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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.
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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.
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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 ,
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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,
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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
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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 ,
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260 BEHN-ESCHENBURG : ON MAGNETIC DISPERSION [Jan. 28th,
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.
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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
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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
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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
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264 BEHN-ESCHENBURG
:
ON MAGNETIC DISPERSION [Jan. 28th,
For two different numbers of poles, in the case of the same motor,
one obtains two different values of the total dispersion-coefficient
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1904.] IN INDUCTION MOTORS, ETC. 265
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
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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 ;
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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(
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268 BEHN-ESCHENBURG
:
ON MAGNETIC DISPERSION [Jan. 28th,
If, following our earlier calculation, we estimate the peripheral-disper-
sion of case
(i)
as four times greater, we get
2 (
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1904.] IN INDUCTION MOTORS, ETC. 269
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
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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
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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)
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272 BEHN-ESCHENBURG : ON MAGNETIC DISPERSION [Jan. 28th,
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
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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
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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
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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
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1904.] IN INDUCTION MOTORS, ETC. 275
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
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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
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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