CHAPTER XVI. 

INDUCTION MOTOR. 

151. A specialization of the general alternating-current 
transformer is the induction motor. It differs from the 
stationary alternating-current transformer, which is also a 
specialization of the general transformer, in so far as in the 
stationary transformer only the transfer of electrical energy 
from primary to secondary is used, but not the mechanical 
force acting between the two, and therefore primary and 
secondary coils are held rigidly in position with regard to 
each other. In the induction motor, only the mechanical 
force between primary and secondary is used, but not the 
transfer of electrical energy, and thus the secondary circuits 
closed upon themselves. Transformer and induction motor 
thus are the two limiting cases of the general alternating- 
current transformer. Hence the induction motor consists 
of a magnetic circuit interlinked with two electric circuits or 
sets of circuits, the primary and the secondary circuit, which 
are movable with regard to each other. In general a num- 
ber of primary and a number of secondary circuits are used, 
angularly displaced around the periphery of the motor, and 
containing E.M.Fs. displaced in phase by the same angle. 
This multi-circuit arrangement has the object always to 
retain secondary circuits in inductive relation to primary 
circuits and vice versa, in spite of their relative motion. 

The result of the relative motion between primary and 
secondary is, that the E.M.Fs. induced in the secondary or 
the motor armature are not of the same frequency as the 
E.M.Fs. impressed upon the primary, but of a frequency 
which is the difference between the impressed frequency 


238 ALTERNATING-CURRENT PHENOMENA. 

and the frequency of rotation, or equal to the "slip," that is, 
the difference between synchronism and speed (in cycles). 
Hence, if 

N = frequency of main or primary E.M.F., 

and s = percentage slip ; 

sJV = frequency of armature or secondary E.M.F., 

and (1 — s) N= frequency of rotation of armature. 

In its reaction upon the primary circuit, however, the 
armature current is of the same frequency as the primary 
current, since it is carried around mechanically, with a fre- 
quency equal to the difference between its own frequency 
and that of the primary. Or rather, since the reaction of 
the secondary on the primary must be of primary frequency 
— whatever the speed of rotation — the secondary frequency 
is always such as to give at the existing speed of rotation a 
reaction of primary frequency. 

152. Let the primary system consist of /0 equal circuits, 
displaced angulary in space by 1 //0 of a period, that is, 
1 //„ of the width of two poles, and excited by /»0 E.M.Fs. 
displaced in phase by 1 //0 of a period ; that is, in other 
words, let the field circuits consist of a symmetrical /0-phase 
system. Analogously, let the armature or secondary circuits 
consist of a symmetrical /rphase system. 

Let 

n0 = number of primary turns per circuit or phase ; 
«a = number of secondary turns per circuit or phase ; 

a = -^ = ratio of total primary turns to total secondary turns 
n\P\ 
or ratio of transformation. 

Since the number of secondary circuits and number of 
turns of the secondary circuits, in the induction motor — as 
in the stationary transformer — is entirely unessential, it is 
preferable to reduce all secondary quantities to the primary 
system, by the ratio of transformation, a ; thus 


INDUCTION MOTOR. 239 

if E{ = secondary E.M.F. per circuit, El = aE{ 

= secondary E.M.F. per circuit reduced to primary system; 

if // = secondary current per circuit, fl= — 

= secondary current per circuit reduced to primary system ; 
if r^ = secondary resistance per circuit, rt = a2 r{ 

= secondary resistance per circuit reduced to primary system ; 
if x± = secondary reactance per circuit, xt = a2 x\ 

= secondary reactance per circuit reduced to primary system ; 
if £/ = secondary impedance per circuit, z1 = azz\ 

= secondary impedance per circuit reduced to primary system ; 

that is, the number of secondary circuits and of turns per 
secondary circuit is assumed the same as in the primary 
system. 

In the following discussion, as secondary quantities, the 
values reduced to the primary system shall be exclusively 
used, so that, to derive the true secondary values, these 
quantities have to be reduced backwards again by the factor 

a = ?*£-. 
«iA 
153. Let 

$ = total maximum flux of the magnetic field per motor pole, 
We then have 

E— V2 77-72 TV^ 10 ~8 = effective E.M.F. induced by the mag- 
netic field per primary circuit. 

Counting the time from the moment where the rising 
magnetic flux of mutual induction & (flux interlinked with 
both electric circuits, primary and secondary) passes through 
zero, in complex quantities, the magnetic flux is denoted by 


and the primary induced E.M.F., 


240 ALTERNATING-CURRENT PHENOMENA. 

where 

e= V2irrt7V<I>10-8 maybe considered as the "Active E.M.F. 
of the motor," or " Counter E.M.F." 

Since the secondary frequency is s N, the secondary in- 
duced E.M.F. (reduced to primary system) is El = — se. 
Let 

I0 = exciting current, or current passing through the motor, per 
primary circuit, when doing no work (at synchronism), 

and 

K= g -j- j 'b = orimary admittance per circuit = — . 

We thus have, 

ge = magnetic energy current, ge* = loss of power oy hysteresis 
(and eddy currents) per primary coil. 

Hence 


= total loss of energy by hysteresis and eddys, 

as calculated according to Chapter X. 
be = magnetizing current, and 
n0be = effective M.M.F. per primary circuit; 

hence ^n0be = total effective M.M.F. ; 

z 

and 

l^-n^be = total maximum M.M.F., as resultant of the M.M.Fs. 
of the /0-phases, combined by the parallelogram of 
M.M.Fs.* 

If (R = reluctance of magnetic circuit per pole, as dis- 
cussed in Chapter X., it is 

A^^ft*. 

* Complete discussion hereof, see Chapter XXV. 


INDUCTION MOTOR. 241 

Thus, from the hysteretic loss, and the reluctance, the 
constants, g and b, and thus the admittance, Fare derived. 

Let rQ = resistance per primary circuit ; 
XQ = reactance per primary circuit ; 
thus, 

•^o = ro — j XQ = impedance per primary circuit; 

rv = resistance per secondary circuit reduced to pri- 
mary system ; 

xv = reactance per secondary circuit reduced to primary 
system, at full frequency, .A7"; 

hence, 

sx! = reactance per secondary circuit at slip s; 
and 

= secondary internal impedance. 


154. We now have, 
Primary induced E.M.F., 

E = -e. 
Secondary induced E.M.F., 

Hence, 
Secondary current, 

*-$— 


Component of primary current, corresponding thereto, 
primary load current, 

7" --/, = 


Primary exciting current, 

/0 =eY=e(g+jfy; hence, 


242 ALTERNATING-CURRENT PHENOMENA. 

Total primary current, 


E.M.F. consumed by primary impedance, 


E.M.F. required to overcome the primary induced E.M.F., 

- E = e; 
hence, 

Primary terminal voltage, 
E. = e + Ez 


We get thus, in an induction motor, at slip s and active 
E.M.F. e, 

Primary terminal voltage, 


Primary current, 


or, in complex expression, 
Primary terminal voltage, 


Primary current, 


INDUCTION MOTOR. 243 

To eliminate e, we divide, and get, 

Primary current, at slip s, and impressed E.M.F., £0; 

f=^— 


or, 

/= _ j + (>i-yji _ E 

" ( 


Neglecting, in the denominator, the small quantity 
F, it is 

Z, F 


0 + r\ 


or, expanded, 

[(j^ + A'0) + r^ -f s^ (rog - 

+/ [J3 (jfo+^O + r^+JT! (xtg+r^+fx^ (xj>+ xj- 


Hence, displacement of phase between current and 
E.M.F., 
tan , = ^(^o+^ 


Neglecting the exciting current, /<„ altogether, that is, 
setting Y = 0, 
We have 

7= sEn^- 


„ S 

tan <D0 = 


244 


AL TEKNA TING-CURRENT PHENOMENA. 


155. In graphic representation, the induction motor dia- 
gram appears as follows : — 

Denoting the magnetism by the vertical vector O<b in 
Fig. 114, the M.M.F. in ampere-turns per circuit is repre- 
sented by vector OF, leading the magnetism O<& by the 
angle of hysteretic advance a. The E.M.F. induced in the 
secondary is proportional to the slip s, and represented by 
~OEl at the amplitude of 180°. Dividing ~OEl by a in the 
proportion of rt -*- sxv and connecting a with the middle b 
of the upper arc of the circle OEV this line intersects the 
lower arc of the circle at the point 7X rr Thus, OIj\ is the 
E.M.F. consumed by the secondary resistance, and OI^ 
equal and parallel to EJ^ is the E.M.F. consumed by the 
secondary reactance. The angle, E^OI^\ = ^ is the angle 
of secondary lag. 


\ 


The secondary M.M.F. OGl is in the direction of the 
vector OIfv Completing the parallelogram of M.M.Fs. 
with OF as diagonal and OGl as one side, gives the primary 
M.M.F. OG as other side. The primary current and the 
E.M.F. consumed by the primary resistance, represented by 
OIry is in line with OG, the E.M.F. consumed by the pri- 
mary reactance 90° ahead of OG, and represented by OIxv 
and their resultant Ofz0 is the E.M.F. consumed by the 


INDUCTION MOTOR. 


245 


primary impedance. The E.M.F. induced in the primary 
circuit is OE', and the E.M.F. required to overcome this 
counter E.M.F. is OE equal and opposite to OE1. Com- 
bining OE with OIzQ gives the primary terminal voltage 
represented by vector OEy and the angle of primary lag, 
EOG 


Fig. 115. 


156. Thus far the diagram is essentially the same as 
the diagram of the stationary alternating-current trans- 
former. Regarding dependence upon the slip of the motor, 
the locus of the different quantities for different values of 
the slip s is determined thus, 


246 ALTERNATING-CURRENT PHENOMENA. 

Let £l = s£f 

Assume in opposition to O&, a point A, such that 


O A -r- 7X rx = Ev -*• /! J.*!, then 

/ir, x .£", /ir, x sE r, _, 

= - ^ = constant. 


That is, /^ lies on a half-circle with OA = — E' as 
diameter. 

That means Gl lies on a half-circle ^ in Fig. 115 with 
OC as diameter. In consequence hereof, G0 lies on half- 
circle^ with FB equal and parallel to OCas diameter. 

Thus Ir0 lies on a half -circle with DH as diameter, which 
circle is perspective to the circle FB, and Ix0 lies on a half- 
circle with IK as diameter, and IzQ on a half-circle with LN 
as diameter, which circle is derived by the combination of 
the circles Ir0 and Ixv 

The primary terminal voltage EQ lies thus on a half- 
circle e0 equal to the half-circle Iz9 and having to point 
E the same relative position as the half-circle Iz^ has to 
point 0. 

This diagram corresponds to constant intensity of the 
maximum magnetism, O®. If the primary impressed volt- 
age EQ is kept constant, the circle e0 of the primary im- 
pressed voltage changes to an arc with O as center, and all 
the corresponding points of the other circles have to be 
reduced in accordance herewith, thus giving as locus of the 
other quantities curves of higher order which most con- 
veniently are constructed point for point by reduction from 
the circle of the loci in Fig. 115. 

Torque and Power. 

157. The torque developed per pole by an electric motor 
equals the product of effective magnetism, ® / V2, times ef- 
fective armature M.M.F., F / V2, times the sine of the 
angle between both, 


INDUCTION MOTOR. 247 


If «! = number of turns, 7t = current, per circuit, with 
/rarmature circuits, the total maximum current polarization, 
or M.M.F. of the armature, is 


Hence the torque per pole, 


If q = the number of poles of the motor, the total torque 
of the motor is, 


The secondary induced E.M.F., Ev lags 90° behind the 
inducing magnetism, hence reaches a maximum displaced in 
space by 90° from the position of maximum magnetization. 
Thus, if the secondary current, Iv lags behind its E.M.F., 
Ev by angle, <av the space displacement between armature 
current and field magnetism is 


hence sin (4> fj) = cos o^ 

We have, however, 


thus, «! <$ 

substituting these values in the equation of the torque, it is 
T. 


248 ALTERNATING-CURRENT PHENOMENA. 

or, in practical (C.G.S.) units, 


is the Torque of the Induction Motor. 

At the slip s, the frequency N, and the number of poles 
q, the linear speed at unit radius is 


hence the output of the motor, 
P= TV 
or, substituted, 


is the Power of the Induction Motor. 

158. We can arrive at the same results in a different 
way : 

By the counter E.M.F. e of the primary circuit with 
current / ' = f0 + 7X the power is consumed, e I = e I0 + e 7r 
The power e I0 is that consumed by the primary hysteresis 
and eddys. The power e 1^ disappears in the primary circuit 
by being transmitted to the secondary system. 

Thus the total power impressed upon the .secondary 
system, per circuit, is 

Pi-tf, 

Of this power a part, £1fl, is consumed in the secondary 
circuit by resistance. The remainder, 

P' = fl(e-£1), 

disappears as electrical power altogether ; hence, by the law 
of conservation of energy, must reappear as some other 
form of energy, in this case as mechanical power, or as the 
output of the motor (including friction). 

Thus the mechanical output per motor circuit is 


INDUCTION MOTOR. 249 

Substituting, 


se; 
se 


it is 


hence, since the imaginary part has no meaning as power, 


and the total power of the motor, 

At the linear speed, 
at unit radius the torque is 


In the foregoing, we found 

£0 = e\ 1 + j|? + Z, Y 
or, approximately, 


or, 
expanded, 


or, eliminating imaginary quantities, 


250 ALTERNATING-CURRENT PHENOMENA. 

Substituting this value in the equations of torque and of 
power, they become, 

torque, T = 


Maximum Torque. 

159. The torque of the induction motor is a maximum 
for that value of slip s, where 


qpi r^ Eg s 
or, since T = -. — .T, . 

4 7T JV^ (>1 


for, 

ds 


expanded, this gives, 

r2 
"7 

or, st = 


Substituting this in the equation of torque, we get the 
value of maximum torque, 


That is, independent of the secondary resistance, rr 
The power corresponding hereto is, by substitution of st 
in P, 

Pt = ; 


This power is not the maximum output of the motor, 
but already below the maximum output. The maximum 
output is found at a lesser slip, or higher speed, while at 
the maximum torque point the output is already on the 
decrease, due to the decrease of speed. 


INDUCTION MOTOR. 251 

With increasing slip, or decreasing speed, the torque of 
the induction motor increases ; or inversely, with increasing 
load, the speed of the motor decreases, and thereby the 
torque increases, so as to carry the load down to the slip st, 
corresponding to the maximum torque. At this point of 
load and slip the torque begins to decrease again ; that is, 
as soon as with increasing load, and thus increasing slip, 
the motor passes the maximum torque point st, it " falls out 
of step," and comes to a standstill. 

Inversely, the torque of the motor, when starting from 
rest, will increase with increasing speed, until the maximum 
torque point is reached. From there towards synchronism 
the torque decreases again. 

In consequence hereof, the part of the torque-speed 
curve below the maximum torque point is in general un- 
stable, and can be observed only by loading the motor 
with an apparatus, whose countertorque increases with the 
speed faster than the torque of the induction motor. 

In general, the maximum torque point, st, is between 
synchronism and standstill, rather nearer to synchronism. 
Only in motors of very large armature resistance, that is 
low efficiency, st > 1, that is, the maximum torque falls 
below standstill, and the torque constantly increases from 
synchronism down to standstill. 

It is evident that the position of the maximum torque 
point, st can be varied by varying the resistance of the 
secondary circuit, or the motor armature. Since the slip 
of the maximum torque point, st, is directly proportional to 
the armature resistance, rlf it follows that very constant 
speed and high efficiency will bring the maximum torque 
point near synchronism, and give small starting torque, 
while good starting torque means a maximum torque point 
at low speed ; that is, a motor with poor speed regulation* 
and low efficiency. 

Thus, to combine high efficiency and close speed regula- 
tion with large starting torque, the armature resistance has 


252 ALTERNATING-CURRENT PHENOMENA. 

to be varied during the operation of the motor, and the 
motor started with high armature resistance, and with in- 
creasing speed this armature resistance cut out as far as 
possible. 

160. If *=:1,__ 

it is ^ = Vr02 + (xl + *0)2. 

In this case the motor starts with maximum torque, and 
when overloaded does not drop out of step, but gradually 
slows down more and more, until it comes to rest. 

If, st>l, 

then ^ > Vr02 + (^ + *0)2. 

In this case, the maximum torque point is reached only 
by driving the motor backwards, as countertorque. 

As seen above, the maximum torque Tt, is entirely in- 
dependent of the armature resistance, and likewise is the 
current corresponding thereto, independent of the armature 
resistance. Only the speed of the motor depends upon the 
armature resistance. 

Hence the insertion of resistance into the motor arma- 
ture does not change the maximum torque, and the current 
corresponding thereto, but merely lowers the speed at which 
the maximum torque is reached. 

The effect of resistance inserted into the induction motor 
is merely to consume the E.M.F., which otherwise would 
find its mechanical equivalent in an increased speed, analo- 
gous as resistance in the armature circuit of a continuous- 
current shunt motor. 

Further discussion on the effect of armature resistance 
is found under " Starting Torque." 

Maximum Power. 

161. The power of an induction motor is a maximum 
for that slip, sv, where 


INDUCTION MOTOR. 253 


expanded, this gives 

sn — - 


substituted in P, we get the maximum power, 


2 {('i + ''o) + (^ + r0)2 + (^i + *o)2} 

This result has a simple physical meaning : (i\ + r0) = r 
is the total resistance of the motor, primary plus secondary 
(the latter reduced to the primary), (x^ + x^ is the total 
reactance, and thus Vrx + r0)2 + (x^ + x0}z = z is the total 
impedance of the motor. Hence 


is the maximum output of the induction motor, at the slip, 


The same value has been derived in Chapter IX., as the 
maximum power which can be transmitted into a non- 
inductive receiver circuit over a line of resistance r, and 
impedance z, or as the maximum output of a generator, or 
of a stationary transformer. Hence : 

The maximum output of an induction motor is expressed 
by the same formula as the maximum output of a generator, 
or of a stationary transformer, or the maximum output which 
can be transmitted over an inductive line into a non-inductive- 
receiver circuit. 

The torque corresponding to the maximum output Pp is,. 


254 ALTERNATING-CURRENT PHENOMENA. 

This is not the maximum torque ; but the maximum 
torque, Tt, takes place at a lower speed, that is, greater slip, 


• since, 


-that is, st > sp. 

It is obvious from these equations, that, to reach as large 
an output as possible, r and z should be as small as possible ; 
that is, the resistances ^ + r0, and the impedances, z, 
and thus the reactances, x± + x0, should be small. Since 
r± + r0 is usually small compared with x^ -f- x0 it follows, that 
the problem of induction motor design consists in con- 
structing the motor so as to give the minimum possible 
reactances, x^ + x0. 

Starting Torque. 

162. In the moment of starting an induction motor, 
the slip is 

hence, starting current, 


Oo - 

or, expanded, with the rejection of the last term in the 
denominator, as insignificant, 


T _io11 010,io1 . 
- 8 


and, displacement of phase, or angle of lag, 

fi + r0] + *! [Jfx 4- Jf0]) - jf (r0 ^ - *0 rt) 


„ _ 
1 W° 


r0) 


INDUCTION MOTOR. 255 

Neglecting the exciting current, g = 0 = b, these equa- 
tions assume the form, 


or, eliminating imaginary quantities, 


and tan w0 = 


+ 'o 


That means, that in starting the induction motor without 
additional resistance in the armature circuit, — in which case 
^ + x0 is large compared with t\ •+• r0, and the total impe- 
dance, z, small, — the motor takes excessive and greatly 
lagging currents. 

The starting torque is 


T0= 


That is, the starting torque is proportional to the 
armature resistance, and inversely proportional to the square 
of the total impedance of the motor. 

It is obvious thus, that, to secure large starting torque, 
the impedance should be as small, and the armature resis- 
tance as large, as possible. The former condition is the 
condition of large maximum output and good efficiency 
and speed regulation ; the latter condition, however, means 
inefficiency and poor regulation, and thus cannot properly 
be fulfilled by the internal resistance of the motor, but only 
by an additional resistance which is short-circuited while 
the motor is in operation. 


256 ALTERNATING-CURRENT PHENOMENA. 

Since, necessarily, 

ri<*, 

''<•< 


and since the starting current is, approximately, 

7 =f , 
we have, Ta < 


would be the theoretical torque developed at 100 per cent 
efficiency and power factor, by E.M.F., E0, and current, /, 
at synchronous speed. 

Thus, T0<T00, 

and the ratio between the starting torque T0, and the theo- 
retical maximum torque, T^, gives a means to judge the 
perfection of a motor regarding its starting torque. 

This ratio, T0 / Tw, exceeds .9 in the best motors. 

Substituting 7 = E0 / z in the equation of starting torque, 
it assumes the form, 

7V,. 


Since 4 IT N / q = synchronous speed, it is : 

The starting torque of the induction motor is equal to the 
resistance loss in the motor armature, divided by the synchro- 
nous speed. 

The armature resistance which gives maximum starting 
torque is 


INDUCTION MOTOR. 257 


dr, 
expanded, this gives, 


the same value as derived in the paragraph on "maximum 
torque." 

Thus, adding to the internal armature resistance, r/ in 
starting the additional resistance, 


makes the motor start with maximum torque, while with in- 
creasing speed the torque constantly decreases, and reaches 
zero at synchronism. Under these conditions, the induc- 
tion motor behaves similarly to the continuous-current series 
motor, varying in the speed with the load, the difference 
being, however, that the induction motor approaches a 
definite speed at no load, while with the series motor the 
speed indefinitely increases with decreasing load. 

The additional armature resistance, t\", required to give 
a certain starting torque, if found from the equation of 
starting torque : 

Denoting the internal armature resistance by rj, the total 
armature resistance is ^ = r^ + r". 

and thus, ?A Eg rj + r" 

4 TT N (r^ + r^ + r0)2 + (Xl + *0)2 ' 
hence, 


This gives two values, one above, the other below, the 
maximum torque point. 


258 ALTERNATING-CURRENT PHENOMENA. 

Choosing the positive sign of the root, we get a larger 
armature resistance, a small current in starting, but the 
torque constantly decreases with the speed. 

Choosing the negative sign, we get a smaller resistance, 
a large starting current, and with increasing speed the 
torque first increases, reaches a maximum, and then de- 
creases again towards synchronism. 

These two points correspond to the two points of the 
speed-torque curve of the induction motor, in Fig. 116, 
giving the desired torque T0. 

The smaller value of r1" will give fairly good speed regu- 
lation, and thus in small motors, where the comparatively 
large starting current is no objection, the permanent arma- 
ture resistance may be chosen to represent this value. 

The larger value of rj' allows to start with minimum 
current, but requires cutting out of the resistance after the 
start, to secure speed regulation and efficiency. 

Synchronism. 
163. At synchronism, s = 0, we have, 


or, 


0, T=Q; 


that is, power and torque are zero. Hence, the induction 
motor can never reach complete synchronism, but must 
slip sufficiently to give the torque consumed by friction. 

Running near Synchronism. 

164. When running near synchronism, at a slip s above 
the maximum output point, where s is small, from .02 to 
.05 at full load, the equations can be simplified by neglect- 
ing terms with s, as of higher order. 


INDUCTION MOTOR. 25 £ 

We then have, current, 


or, eliminating imaginary quantities, 


angle of lag, o*i + *o , 

c2 (r_ -I- <r_\ -4- r.2 h r. 

tan w0 

T = 

or, inversely, 


A A 


that is, 

Near sychronism, the slip, s, of an induction motor, or 
its drop in speed, is proportional to the armature resistance> 
i\ and to the power, P, or torque, T. 

Example. 

165. As an instance are shown, in Fig. 116, character- 
istic curves of a 20 horse-power three-phase induction motor, 
of 900 revolutions synchronous speed, 8 poles, frequency 
of 60 cycles. 

The impressed E.M.F. is 110 volts between lines, and 
the motor star connected, hence the E.M.F. impressed per 
circuit : 

~ = 63.5 ; or EQ = 63.5. 


260 


AL TERN A TING-CURRENT PHENOMENA. 


The constants of the motor are : 

Primary admittance, Y = .1 + .4 j. 
Primary impedance, Z = .03 — .09 j. 
Secondary impedance, Zx = .02 — .085/. 

In Fig. 116 is shown, with the speed in per cent of 

•synchronism, as abscissae, the torque in kilogrammetres, 

as ordinates, in drawn lines, for the values of armature 
resistance : 


116. Speed Characteristics of Induction Motor. 


rt = .02 : short circuit of armature, full speed. 

^ = .045 : .025 ohms additional resistance. 

^ = .18 : .16 ohms additional, maximum starting torque. 

^ = .75 : .73 ohms additional, same starting torque as rt == .045. 

On the same Figure is shown the current per line, in 
dotted lines, with the verticals or torque as abscissae, and 
the horizontals or amperes as ordinates. To the same 
torque always corresponds the same current, no matter 
what the speed be. 


INDUCTION MOTOR. 


261 


On Fig. 117 is shown, with the current input per line as 
abscissae, the torque in kilogrammetres and the output in 
horse-power as ordinates in drawn lines, and the speed and 
the magnetism, in per cent of their synchronous values, as 
ordinates in dotted lines, for the armature resistance ^ = .02 
or short circuit. 


20 


lase Induotio Motor. 


. 60Cyc 


110V 


Jiagram 


=.03-.09j 
z£0=J&B 


\ 


\\ 


\\ 


12 

-1 


Amperes 
150 1 200 


2,50 


300 


Fig. 117. Current Characteristics of Induction Motor. 

In Fig. 118 is shown, with the speed, in per cent of 
synchronism, as abscissae, the torque in drawn line, and 
the output in dotted line, for the value of armature resist- 
ance ?i = .045, for the whole range of speed from 120 per 


262 


ALTERNA TING-CURRENT PHENOMENA. 


cent backwards speed to 220 per cent beyond synchronism, 
showing the two maxima, the motor maximum at s = .25, 
and the generator maximum at s = — .25. 

166. As seen in the preceding, the induction motor is 
characterized by the three complex imaginary constants, 

Y0 = g0 +jbw the primary exciting admittance, 
Z0 = r0 —jx0, the primary self-inductive impedance, and 
Zi = r± — jx^ the secondary self-inductive impedance, 


Fig. 1 18. Speed Characteristics of Induction Motor. 

reduced to the primary by the ratio of secondary to pri- 
mary turns. 

From these constants and the impressed E.M.F. cot the 
motor can be calculated as follows : 

Let, 

e = counter E.M.F. of motor, that is E.M.F. induced in 
the primary by the mutual magnetic flux. 

At the slip s the E.M.F. induced in the secondary cir- 
cuit is, se 


INDUCTION MOTOR. 263 


Thus the secondary current, 


where, 


«l = -5T 


r* + Atf r? + 

The primary exciting current is, 


thus, the total primary current, 

/0 = /! + /oo = * (^i + A) 
where, 


The E.M.F. consumed by the primary impedance is, 
^ = /oZ0 = * (r0 ->0) (^ 


the primary counter E.M.F. is e, thus the primary impressed 
E.M.F., 

£, 
where, 

c\ — 
or, absolute, 

^0 = 

hence, 


This value substituted gives, 

Secondary current, 

ffi+A 
A = *b T7= 


Primary current, 

°~ 

Impressed E.M.F., 


264 ALTERNATING-CURRENT PHENOMENA. 

Thus torque, in synchronous watts (that is, the watts 
output the torque would produce at synchronous speed), 


tf + tf 

hence, the torque in absolute units, 


= = 


N (f* + r22) W 
where N= frequency. 

The power output is torque times speed, thus : 


The power input is, 


^•l2 + 

The voltampere input, 


o2 ( Vi + V,) /o2 ( Vi - V8) 


hence, 

efficiency, 

J\ _ a, (I - s) 

J? Vi + V2 

power factor, 


apparent efficiency, 


<2o 

torque efficiency, * 
a. 


./V Vi + V. 

* That 5s the ratio of actual torque to torque which would be profloced, if there were nc 
losses of energy in the motor, at the same power input. 


INDUCTION MOTOR. 265 


apparent torque efficiency,* 

rrt 

~Q0 ~ V W~+1?YT^ 


167. Most instructive in showing the behavior of an 
induction motor are the load curves and the speed curves. 

The load curves are curves giving, with the power out- 
put as abscissae, the current imput, speed, torque, power 
factor, efficiency, and apparent efficiency, as ordinates. 

The speed curves give, with the speed as abscissae, the 
torque, current input, power factor, torque efficiency, and 
apparent torque efficiency, as ordinates. 

The load curves characterize the motor especially at its 
normal running speeds near synchronism, the speed curves 
over the whole range of speed. 

In Fig. 119 are shown the load curves, and in Fig. 120 
the speed curves of a motor of the constants, 
K0 = .01 + .!/ 

z* = .i -.3> 

Z, = .1 - .3j 

INDUCTION GENERATOR. 

168. In the foregoing, the range of speed from s = 1, 
standstill, to s = 0, synchronism, has been discussed. In 
this range the motor does mechanical work. 

It consumes mechanical power, that is, acts as generator 
or as brake outside of this range. 

For, s > 1, backwards driving, P becomes negative, 
representing consumption of power, while T remains posi- 
tive ; hence, since the direction of rotation has changed, 
represents consumption of power also. All this power is 
consumed in the motor, which thus acts as brake. 

For, s < 0, or negative, P and T become negative, and 
the machine becomes an electric generator, converting me- 
chanical into electric energy. 

* That is the ratio of actual torque to torque which would be produced if there were 
neither losses of energy nor phase displacement in the motor, at the same voltampere input. 


266 


ALTERNA TING-CURRENT PHENOMENA. 


The calculation of the induction generator at constant 
frequency, that is, at a speed increasing with the load by the 
negative slip, slt is the same as that of the induction motor 
except that sl has negative values, and the load curves for 
the machine shown as motor in Fig. 119 are shown in Fig. 
121 for negative slip s{ as induction generator. 


CURV 


POWER 
4000 


"£> 


Fig. 119. 


Again, a maximum torque point and a maximum output 
point are found, and the torque and power increase from 
zero at synchronism up to a maximum point, and then de- 
crease again, while the current constantly increases. 


INDUCTION MOTOR. 


267 


Fig. 120. 


268 ALTERNATING-CURRENT PHENOMENA. 

169. The induction generator differs essentially from 
the ordinary synchronous alternator in so far as the induc- 
tion generator has a definite power factor, while the syn- 
chronous alternator has not. That is, in the synchronous 
alternator the phase relation between current and terminal 
voltage entirely depends upon the condition of the external 
circuit. The induction generator, however, can operate 
only if the phase relation of current and E.M.F., that is, the 
power factor required by the external circuit, exactly coin- 
cides with the internal power factor of the induction gen- 
erator. This requires that the power factor either of the 
external circuit or of the induction generator varies with 
the voltage, so as to permit the generator and the external 
circuit to adjust themselves to equality of power factor. 

Beyond magnetic saturation the power factor decreases ; 
that is, the lead of current increases in the induction ma- 
chine. Thus, when connected to an external circuit of con- 
stant power factor the induction generator will either not 
generate at all, if its power factor is lower than that of the 
external circuit, or, if its power factor is higher than that of 
the external circuit, the voltage will rise until by magnetic 
saturation in the induction generator its power factor has 
fallen to equality with that of the external circuit. This, 
however, requires magnetic saturation in the induction gen- 
erator, which is objectionable, due to excessive hysteresis 
losses in the alternating field. 

To operate below saturation, — that is, at constant inter- 
nal power factor, — the induction generator requires an exter- 
nal circuit with leading current, whose power factor varies 
with the voltage, as a circuit containing synchronous motors 
or synchronous converters. In such a circuit, the voltage 
of the induction generator remains just as much below the 
counter E.M.F. of the synchronous motor as necessary to 
give the required leading exciting current of the induction 
generator, and the synchronous motor can thus to a certain 
extent be called the exciter of the induction generator. 


INDUCTION MOTOR. 269 

When operating self-exciting, that is shunt-wound, con- 
verters from the induction generator, below saturation of 
both the converter and the induction generator, the condi- 
tions are unstable also, and the voltage of one of the two 
machines must rise beyond saturation of its magnetic field. 

When operating in parallel with synchronous alternat- 
ing generators, the induction generator obviously takes its 
leading exciting current from the synchronous alternator, 
which thus carries a lagging wattless current. 

170. To generate constant frequency, the speed of the 
induction generator must increase with the load. Inversely, 
when driven at constant speed, with increasing load on the 
induction generator, the frequency of the current generated 
thereby decreases. Thus, when calculating the character- 
istic curves of the constant speed induction generator, due 
regard has to be taken of the decrease of frequency with 
increase of load, or what may be called the slip of fre- 
quency, s. 

Let in an induction generator, 

Y0 = gQ + j\ — primary exciting admittance, 

Z0 = r0 — jxQ = primary self-inductive impedance, 

Zi = r^ — jXj_ = secondary self-inductive impedance, 

reduced to primary, all these quantities being reduced to 
the frequency of synchronism with the speed of the ma- 
chine, N. 

Let e — induced E.M.F., reduced to full frequency. 

s = slip of frequency, thus : (1-j) N = frequency gener- 
ated by machine. 

We then have 

Secondary induced E.M.F. 
se 
thus, secondary current, 


r in 
r\ — Jsx\ 


270 ALTERNATING-CURRENT PHENOMENA. 

where, 


primary exciting current, 

In = EY0 = e 
thus, total primary current, 

/0 = /i + foo 
where, 

^1 = <*\ + £b 

primary impedance voltage, 
& = S0(r0- 

primary induced E.M.F., 


thus, primary terminal voltage, 

£0 = e(l-s) -S0(r0-j[l- s] x0) = e 
where, 

fi = ! - s ~ rA - (1 - s 
hence, absolute, 

e0 = e V^ 
and, 


Thus, 

Secondary current, 

T eO (ai 


Primary current, 

j _ eo (A + A) 

Primary terminal voltage, 

j-. ^0 \^"l 

£« = —T-, 


INDUCTION MOTOR. 
Torque and mechanical power input, 

T— P —\f nl — e°ai 
r* ~ \-e ^ ~ 7^+^ 

Electrical output, 


271 


ELECTRICAL OUTPUT P , WATTS 
1000 2000 3COO 4000 fiOOO fiOOO 7000 8000 


Fig. 122. 


Voltampere output, 
G, = < 

Efficiency, 

j 

power factor, 


272 AL TERNA TING-CURRENT PHENOMENA. 

or, 

p,j b* - V, 
= ^- = ^T^ 

In Fig. 122 is plotted the load characteristic of a con- 
stant speed induction generator, at constant terminal vol- 
tage e 0 = 110, and the constants, 

K0 = .01 + .!/ 


171. As instance may be considered a power trans- 
mission from an induction generator of constants Y0, Z0, 
Zj, over a line of impedance Z = r —jx, into a synchron- 
ous motor of synchronous impedance Zz = rz — jxz, operat- 
ing at constant field excitation. 

Let, e0 = counter E.M.F. or nominal induced E.M.F. of 
synchronous motor at full frequency ; that is, frequency of 
synchronism with the speed of the induction generator. 
By the preceding paragraph the primary current of the 
induction generator was, 


primary terminal voltage, 
E0 = e 
thus, terminal voltage at synchronous motor terminals, 


where, 

4 = fi ~ rA ~ C1 - J) *A 4 = 

Counter E.M.F. of synchronous motor, 

E2 

' 

where, 

/ = 4 - r& - (1 
or absolute, 


INDUCTION MOTOR. 


since, however, 


Z=.0|4-6j 

ULL F EQUE 
EXCIT/ 
5 VOL' 


OUTPUT OF SYNCHRONOUS, WATTS 
1000 2000 I 8000 4000 5000 


274 ALTERNATING-CURRENT PHENOMENA. 


Thus, 


Current, _ e2 (1 - j) (^ +y7;2) 

' 


Terminal voltage at induction generator, 


Terminal voltage at synchronous motor, 


and herefrom in the usual way the efficiencies, power fac- 
tor, etc. are derived. 

When operated from an induction generator, a syn- 
chronous motor gives a load characteristic very similar to 
that of an induction motor operated from a synchronous 
generator, but in the former case the current is leading, in 
the latter lagging. 

In either case, the speed gradually falls off with increas- 
ing load (in the synchronous motor, due to the falling off 
of the frequency of the induction generator), up to a maxi- 
mum output point, where the motor drops out of step and 
comes to standstill. 

Such a load characteristic of the induction generator in 
Fig. 121, feeding a synchronous motor of counter E.M.F. 
eQ = 125 volts (at full frequency) and synchronous impe- 
dance Z2 = .04 — Gj, over a line of negligible impedance 
is shown in Fig. 123. 

CONCATENATION, OR TANDEM CONTROL OF INDUCTION 
MOTORS. 

172. If of two induction motors the secondary of the 
first motor is connected to the primary of the second motor, 
the second machine operates as motor with the E.M.F. and 
frequency impressed upon it by the secondary of the first 
machine, which acts as general alternating-current trans- 
former, converting a part of the primary impressed power 


INDUCTION MOTOR. 275 

into secondary electrical power for the supply of the second 
machine, and a part into mechanical work. 

The frequency of the secondary E.M.F. of the first motor, 
and thus the frequency impressed upon the second motor, is 
the frequency of slip below complete synchronism, s. The 
frequency of the secondary induced E.M.F. of the second 
motor is the difference between its impressed frequency, 
s, and its speed ; thus, if both motors are connected together 
mechanically to turn at the same speed, 1 — s, the secondary 
frequency of the second motor is 2^—1, hence equal to 
zero at s = .5. That is, the second motor reaches its syn- 
chronism at half speed. At this speed its torque becomes 
equal to zero, the energy current flowing into it, and conse- 
quently the energy component of the secondary current of 
the first "motor, and thus the torque of the first motor be- 
comes equal to zero also, when neglecting the hysteresis 
energy current of the second motor. That is, a system of 
concatenated motors with short-circuited secondary of the 
second motor approaches half synchronism, in the same 
manner as the ordinary induction motor approaches syn- 
chronism. With increasing load, its slip below half syn- 
chronism increases. 

More generally, any pair of induction motors connected 
in concatenation divide the speed so that the sum of their 
two respective speeds approaches synchronism at no load ; 
or, still more generally, any number of concatenated motors 
run at such speeds that the sum of the speeds approaches 
synchronism at no load. 

With mechanical connection between the two motors, 
concatenation thus offers a means to operate a pair of 
induction motors at full efficiency at half speed in tandem, 
as well as at full speed in parallel, and thus gives the same 
advantage as the series-parallel control of the continuous- 
current motor. 

In starting, a concatenated system is controlled by re- 
sistance in the armature of the second motor. 


276 ALTERNATING-CURRENT PHENOMENA. 

Since, with increasing speed, the frequency impressed 
upon the second motor decreases proportionally to the de- 
crease of voltage, when neglecting internal losses in the 
first motor, the magnetic density of the second motor re- 
mains practically constant, and thus its torque the same as 
when operated at full voltage and full frequency under the 
same conditions. 

At half synchronism the torque of the concatenated 
couple becomes zero, and above half synchronism the sec- 
ond motor runs beyond its impressed frequency ; that is, 
becomes generator. In this case, due to the reversal of 
current in the secondary of the first motor, its torque 
becomes negative also, that is the concatenated couple 
becomes induction generator above half synchronism. At 
about two-thirds synchronism, with low resistance armature, 
the torque of the couple becomes zero again, and once more 
positive between about two-thirds synchronism and full syn- 
chronism, and negative once more beyond full synchronism. 
With high resistance in the secondary of the second motor, 
the second range of positive torque, below full synchronism, 
disappears, more or less. 

173. The calculation of a concatenated couple of in- 
duction motors is as follows, 

Let 

N = frequency of main circuit, 

s = slip of the first motor from synchronism. 

the frequency induced in the secondary of the first motor 
and thus impressed upon the primary of the second motor 
is, s N. 

The^peed of the first motor is (1 — s) N, thus the slip 
of the second motor, or the frequency induced in its sec- 
ondary, is 


INDUCTION MOTOR. 277 

Let 

e = counter E.M.F. induced in the secondary of the sec- 
ond motor, reduced to full frequency. 

Z0 = r0 — jxQ = primary self-inductive impedance. 

Z^ = i\ —jxv = secondary self-inductance impedance. 

Y — g +jb = primary exciting admittance of each mo- 
tor, all reduced to full frequency and to the primary by the 
ratio of turns. 

We then have, 

Second motor, 
secondary induced E.M.F., 

*(*/-!) 

secondary current, 


where, 

(2s-l)r1 


i ~ r*+ (2J-1)2^12 z ~ r*+ (2s- 

primary exciting current, 

4 = * (g +JI>} 
thus, total primary current, 

72 = 7, + 70 = e ( 
where, 


primary induced E.M.F., 

se 
primary impedance voltage, 

ft (ro — >^o) 
thus, primary impressed E.M.F., 

£3 = se + 72 (r0 -jsx0) = e (^ 
where, 


First motor, 
secondary current, 


278 ALTERNATING-CURRENT PHENOMENA. 

secondary induced E.M.F., 

£9 = 
where, 


primary induced E.M.F., 

EI = - 
where, 

s 
primary exciting current, 

total primary current, 
where, 


primary impedance voltage, 

|(>o ~> 

thus, primary impressed E.M.F., 
£0 = E, + S(r0 ->0 
where, 

^i =/i + ^o5i + *b£a 

or, absolute, 

<-„ 
and, 


V V + V 

Substituting now this value of ^ in the preceding gives 
the values of the currents and E.M.F.'s in the different 
circuits of the motor series. 

* At s = 0 these terms/i and/s become indefinite, and thus at and very near synchronism 
have to be derived by substituting the complete expressions fory^ andy"2. 


INDUCTION MOTOR. 279 

In the second motor, the torque is, 

T2 = [,/J = ^ 
hence, its power output, 

/»,= (!- s) r2 = (1 - s) <?ai 
The power input is, 


hence, the efficiency, 

PS (1 - s) fa, 


the power factor, 


etc. 

In the first motor, 
the torque is, 


the power output, 

PI = 71 (1 - j) 

= ^ (1 - ,) (/^ -h/A) 

the power input, 

P1 = 


Thus, the efficiency, 

^ (1 - Q (/A +/A) 


+ ^2) - (^ + 
the power factor of the whole system, 


280 ALTERNATING-CURRENT PHENOMENA. 

the power factor of the first motor, 


the total efficiency of the system, 


etc. 


f /ff. 724. Concatenation of Induction Motors. Speed Curves. 
Z=.1— .3/ K=.01 + .l> 

174. As instance are given in Fig. 124, the curves of 
total torque, of torque of the second motor, and of current, 
for the range of slip from s = + 1.5 to s = — .7 for a pair 
of induction motors in concatenation, of the constants : 

Z0 = Z, = .1 - .Bj 


As seen, there are two ranges of positive torque for the 
whole system, one below half synchronism, and one from 
about two-thirds to full synchronism, and two ranges of 


INDUCTION MOTOR. 


281 


negative torque, or generator action of the motor, from half 
to two-third synchronism, and above full synchronism. 

With higher resistance in the secondary of the second 
motor, the second range of positive torque of the system 
disappears more or less, and the torque curves become as 
shown in Fig. 125. 


001 

| | 
CATENATION jOF IN 

SUCTION MOTORS. 

L 

j SPEED CURVES 
|z=.|— .3,j Y4=.OI 

H-.l 

it 

rag 

RE! 

. IN S 

;COND 

kRY 0 

' SECO 

NO MC 

TOR. 

| 

H 
8000 

6000 

- 


4000 

\ 

2000 

1 

— 

— 

— 

""-s. 

\ 

I 

0 

M 

\\ 

\ 

-2000 

\\ 

X 

^ 

-4000 

£ 

/ 

f 

-60C( 

./ 

-8000 

1 

0 

9 

s 

. 

6 

j 

4 

3 

2 

j 

„ 

Fig. 125. Concatenation of Induction Motors. Speed Curves. 


SINGLE-PHASE INDUCTION MOTOR. 

175. The magnetic circuit of the induction motor at or 
near synchronism consists of two magnetic fluxes super- 
imposed upon each other in quadrature, in time, and in 
position. In the polyphase motor these fluxes are produced 
by E.M.Fs. displaced in phase. In the monocyclic motor 
one of the fluxes is due to the primary energy circuit, the 
other to the primary exciting circuit. In the single-phase 


282 AL TERN A TING-CURRENT PHENOMENA. 

motor the one flux is produced by the primary circuit, the 
other by the currents induced in the secondary or armature, 
which are carried into quadrature position by the rotation 
of the armature. In consequence thereof, while in all these 
motors the magnetic distribution is the same at or near syn- 
chronism, and can be represented by a rotating field of 
uniform intensity and uniform velocity, it remains such in 
polyphase and monocyclic motors ; but in the single-phase 
motor, with increasing slip, — that is, decreasing speed, — 
the quadrature field decreases, since the induced armature 
currents are not carried to complete quadrature position ; 
and thus only a component available for producing the 
quadrature flux. Hence, approximately, the quadrature flux 
of a single-phase motor can be considered as proportional to 
its speed ; that is, it is zero at standstill. 

Since the torque of the motor is proportional to the 
product of secondary current times magnetic flux in quad- 
rature, it follows that the torque of the single-phase motor 
is equal to that of the same motor under the same condition 
of operation on a polyphase circuit, multiplied with the 
speed ; hence equal to zero at standstill. 

Thus, while single-phase induction motors are quite sat- 
isfactory at or near synchronism, their torque decreases 
proportionally to the speed, and becomes zero at standstill. 
That is, they are not self-starting, but some starting device 
has to be used. 

Such a starting device may either be mechanical or elec- 
trical. All the electrical starting devices essentially consist 
in impressing upon the motor at standstill a magnetic quad- 
rature flux. This may be produced either by some outside 
E.M.F., as in the monocyclic starting device, or by displa- 
cing the circuits of two or more primary coils from each 
other, either by mutual induction between the coils, — that 
is, by using one as secondary to the other, — or by impe- 
dances of different inductance factors connected with the 
different primary coils. 


INDUCTION MOTOR. 283 

176. The starting-devices of .the single-phase induc- 
tion motor by producing a quadrature magnetic flux can be 
subdivided into three classes : 

1. Phase-Splitting Devices. Two or more primary 
circuits are used, displaced in position from each other, and 
either in series or in shunt with each other, or in any other 
way related, as by transformation. The impedances of 
these circuits are made different from each other as much 
as possible, to produce a phase displacement between them. 
This can be done either by inserting external impedances 
into the circuits, as a condenser and a reactive coil, or by 
making the internal impedances of the motor circuits differ- 
ent, as by making one coil of high and the other of low 
resistance. 

2. Inductive Devices. The different primary circuits 
of the motor are inductively related to each other in such a 
way as to produce a phase displacement between them. 
The inductive relation can be outside of the motor or inside, 
by having the one coil induced by the other ; and in this 
latter case the current in the induced coil may be made 
leading, accelerating coil, or lagging, shading coil. 

3. Monocyclic Devices. External to the motor an 
essentially wattless E.M.F. is produced in quadrature with 
the main E.M.F. and impressed upon the motor, either 
directly or after combination with the single-phase main 
E.M.F. Such wattless quadrature E.M.F. can be produced 
by the common connection of two impedances of different 
power factor, as an inductance and a resistance, or an in- 
ductance and a condensance connected in series across the 
mains. 

The investigation of these starting-devices offers a very 
instructive application of the symbolic method of investiga- 
tion of alternating-current phenomena, and a study thereof 
is thus recommended to the reader.* 

» See paper on the Single-phase Induction Motor, A.I.E.E. Transactions, 1898. 


284 ALTERNATING-CURRENT PHENOMENA. 

177. As a rule, no special motors are built for single- 
phase operation, but polyphase motors used in single-phase 
circuits, since for starting the polyphase primary winding is 
required, the single primary coil motor obviously not allow- 
ing the application of phase-displacing devices for produ- 
cing the starting quadrature flux. 

Since at or near synchronism, at the same impressed 
E.M.F. — that is, the same magnetic density — the total 
voltamperes excitation of the single-phase induction motor 
must be the same as of the same motor on polyphase circuit, 
it follows that by operating a quarter-phase motor from 
single-phase circuit on one primary coil, its primary excit- 
ing admittance is doubled. Operating a three-phase motor 
single-phase on one circuit its primary exciting admittance 
is trebled. The self-inductive primary impedance is the 
same single-phase as polyphase, but the secondary impe- 
dance reduced to the primary is lowered, since in single- 
phase operation all secondary circuits correspond to the 
one primary circuit used. Thus the secondary impedance 
in a quarter-phase motor running single-phase is reduced to 
one-half, in a three-phase motor running single-phase re- 
duced to one-third. In consequence thereof the slip of 
speed in a single-phase induction motor is usually less than 
in a polyphase motor ; but the exciting current is consider- 
ably greater, and thus the power factor and the efficiency 
are lower. 

The preceding considerations obviously apply only when 
running so near synchronism that the magnetic field of the 
single-phase motor can be assumed as uniform, that is the 
cross magnetizing flux produced by the armature as equal 
to the main magnetic flux. 

When investigating the action of the single-phase motor 
at lower speeds and at standstill, the falling off of the mag- 
netic quadrature flux produced by the armature current, the 
change of secondary impedance, and where a starting device 
is used the effect of the magnetic field produced by the 
starting device, have to be considered. 


INDUCTION MOTOR. 285 

The exciting current of the single-phase motor consists 
of the primary exciting current or current producing the 
main magnetic flux, and represented by a constant admit- 
tance F,,1, the primary exciting admittance of the motor, and' 
the secondary exciting current, that is that component of 
primary current corresponding to the secondary current 
which gives the excitation for the quadrature magnetic flux. 
This latter magnetic flux is equal to the main magnetic flux 
3>0 at synchronism, and falls off with decreasing speed to 
zero at standstill, if no starting device is used or to 4^ = /<£0 
at standstill if by a starting device a quadrature magnetic 
flux is impressed upon the motor, and at standstill t = ratio- 
of quadrature or starting magnetic flux to main magnetic 
flux. 

Thus the secondary exciting current can be represented 
by an admittance Y* which changes from equality with the 
primary exciting admittance Y^ at synchronism, to Y* = 0, 
respectively to Y^ — t Y^ at standstill. Assuming thus that 
the starting device is such that its action is not impaired by 
the change of speed, at slip s the secondary exciting admit- 
tance can be represented by : 

Y* = [!-(!-/) j] Fo1 

The secondary impedance of the motor at synchronism 
is the joint impedance of all the secondary circuits, since all 
secondary circuits correspond to the same primary circuit, 

hence = -^ with a three-phase secondary, and = -^ with a 

two-phase secondary with impedance Z1 per circuit. 

At standstill, however, the secondary circuits correspond 
to the primary circuit only with their projection in the direc- 
tion of the primary flux, and thus as resultant only one-half 
of the secondary circuits are effective, so that the secondary 
impedance at standstill is equal to 2 Zl / 3 with a three-phase, 
and equal to Z^ with a two-phase secondary. Thus the 
effective secondary impedance of the single-phase motor 


286 ALTERNATING-CURRENT PHENOMENA. 

changes with the speed and can at the slip s be represented 

by Zf = - -- -^ — - in a three-phase motor, and Z{ = - - <p — -1 

in a two-phase motor, with the impedance Z^ per secondary 
circuit. 

In the single-phase motor without starting device, due to 
the falling off of the quadrature flux, the torque at slip s is : 

T = a^ (I - s) 

In a single-phase motor with a starting device which at 
standstill produces a ratio of magnetic fluxes t, the torque at 
standstill is ; 

TQ = /7I 

where 7^ = total torque of the same motor on polyphase 
circuit. 

. Thus denoting the value —~ = v 
&f 

the single-phase motor torque at standstill is : 


and the single-phase motor torque at slip s is : 
T = of [1 - (1 - v) s] 

178. In the single-phase motor considerably more 
advantage is gained by compensating for the wattless mag- 
netizing component of current by capacity than in the 
polyphase motor, where this wattless current is relatively 
small. The use of shunted capacity, however, has the dis- 
advantage of requiring a wave of impressed E.M.F. very 
close to sine shape ; since even with a moderate variation 
from sine shape the wattless charging current of the con- 
denser of higher frequency may lower the power factor 
more than the compensation for the wattless component of 
the fundamental wave raises it, as will be seen in the chap- 
ter on General Alternating Current Waves. 

Thus the most satisfactory application of the condenser 
in the single-phase motor is not in shunt to the primary 


INDUCTION MOTOR. 287 

circuit, but in a tertiary circuit ; that is, in a circuit stationary 
with regard to the primary impressed circuit, but induced 
by the revolving secondary circuit. 

In this case the condenser is supplied with an E.M.F. 
transformed twice, from primary to secondary, and from 
secondary to tertiary, through multitooth structures in a 
uniformly revolving field, and thus a very close approxi- 
mation to sine wave produced at the condenser, irrespective 
of the wave shape of primary impressed E.M.F. 

With the condenser connected into a tertiary circuit of 
a single-phase induction motor, the wattless magnetizing 
current of the motor is supplied by the condenser in a 
separate circuit, and the primary coil carries the energy cur- 
rent only, and thus the efficiency of the motor is essentially 
increased. 

The tertiary circuit may be at right angles to the pri- 
mary, or under any other angle. Usually it is applied on an 
angle of 60°, so as to secure a mutual induction between 
tertiary and primary for starting, which produces in start- 
ing in the condenser a leading current, and gives the quad- 
rature magnetic flux required. 

179. The most convenient way to secure this arrange- 
ment is the use of a three-phase motor which with two of 
its terminals 1-2, is connected to the single-phase mains, 
and with terminals 1 and 3 to a condenser. 

Let YQ = g0 -\-jb0 = primary exciting admittance of the 
motor per delta circuit. 

Z0 = r0 — jxQ = primary self-inductive impedance per 
delta circuit. 

Z^ = i\ —jx^ = secondary self-inductive impedance per 
delta circuit reduced to primary. 

Let 

Ys = gs — jb9 = admittance of the condenser connected 
between terminals 1 and 3. 


288 ALTERNATING-CURRENT PHENOMENA. 

If then, as single-phase motor, 

/ = ratio of auxiliary quadrature flux to main flux in 
starting, 

h = ratio of E.M.F. induced in condenser circuit to 

E.M.F. induced in main circuit in starting, 
starting torque 


It is single-phase 

Fo1 = 1.5 Y0 = 1.5 (£•„ +/£0) = primary exciting admit- 

tance, 
Y? = 1.5 Y0 [1 - (1 - 0 s] 

= 1.5 (g0 +/£<)) [1 — (1 — 0 J] = secondary exciting 

admittance at slip s. 

Z0l = ?^° = 2fo~^*o) = primary self-inductive impe- 
o o 

dance. 

Zxi = £L±^ Zi = ^L + ^ (ri -jsxj = secondary self- 
o o 

inductive impedance. 

Z,1 = ^ = 2 (r° ~ ***> = tertiary self-inductive impe- 
o o 

dance of motor. 
Thus, 

Y4 = -^r - T- = total admittance of tertiary circuit. 


Since the E.M.F. induced in the tertiary circuit decreases 
from e at synchronism to he at standstill, the effective ter- 
tiary admittance or admittance reduced to an induced E.M.F. 
e is at slip s 

Y? = [!-(!-*) s] Y4 
Let then, 

e = counter E.M.F. of primary circuit, 
s = slip. 


INDUCTION MOTOR. 289 


We have, 
secondary load current 

3se 


(1 + s) (r, -jsx,) 
secondary exciting current 

secondary condenser current 
thus, total secondary current 
primary exciting current 


thus, total primary current 

/o = 71 + /o1 
= /, + /, + 

= ' (*i + A) 
primary impressed E.M.F. 


thus, main counter E.M.F. 


or, 


and, absolute 


V^2 + c* 
hence, primary current 


T_slW + % 

J* - e° v f* + ^ 


290 ALTERNATING-CURRENT PHENOMENA. 

voltampere input, 

Qo = **!» 
power input 


*t — Oo — O 2 , 2 

6j T '2 

torque at slip .$• 

2^= r1 [i - (i - v) s] 


and, power output 


and herefrom in the usual manner the efficiency, apparent 
efficiency, torque efficiency, apparent torque efficiency, and 
power factor. 

The derivation o.* the constants /, //, v, which have to be 
determined before calculating the motor, is as follows : 

Let <?0 = single-phase impressed E.M.F., 

Y — total stationary admittance of motor per delta cir- 

cuit, 
Ez = E.M.F. at condenser terminals in starting. 

In the circuit between the single-phase mains from ter- 
minal 1 over terminal 3 to 2, the admittances Y + Y8, and Y, 
are connected in series, and have the respective E.M.Fs. E^ 
and e0 - Ey It is thus, 

Y+ Ys+ Y=e0-£t+£s, 

since with the same current passing through both circuits, 
the impressed E.M.Fs. are inverse proportional to the re- 
spective admittances. 

Thus, 


INDUCTION MOTOR. 291 

and quadrature E.M.F. 

hence 
thus 


Since in the three-phase E.M.F. triangle, the altitude 
corresponding to the quadrature magnetic flux = — y= , and 

the quadrature and main fluxes are equal, in the single-phase 
motor the ratio of quadrature to main flux is 

/ = — 2 = 1.155 Aa 

V3 

From /, v is derived as shown in the preceding. 

For further discussion on the Theory and Calculation of 
the Single-phase Induction Motor, see American Institute 
Electrical Engineers Transactions, January, 1900. 


SYNCHRONOUS INDUCTION MOTOR. 

180. The induction motor discussed in the foregoing 
consists of one or a number of primary circuits acting upon 
a movable armature which comprises a number of closed 
secondary circuits displaced from each other in space so as 
to offer a resultant circuit in any direction. In consequence 
thereof the motor can be considered as a transformer, having 
to each primary circuit a corresponding secondary circuit, 
— a secondary coil, moving out of the field of the primary 
coil, being replaced by another secondary coil moving into 
the field. 

In such a motor the torque is zero at synchronism, posi- 
tive below, and negative above, synchronism. 

If, however, the movable armature contains one closed 
circuit only, it offers a closed secondary circuit only in the 
direction of the axis of the armature coil, but no secondary 
circuit at right angles therewith. That is, with the rotati .n 


292 ALTERNATING-CURRENT PHENOMENA. 

of the armature the secondary circuit, corresponding to a 
primary circuit, varies from short circuit at coincidence of 
the axis of the armature coil with the axis of the primary 
coil, to open circuit in quadrature therewith, with the 
periodicity of the armature speed. That is, the apparent 
admittance of the primary circuit varies periodically from 
open-circuit admittance to the short-circuited transformer 
admittance. 

At synchronism such a motor represents an electric cir- 
cuit of an admittance varying with twice the periodicity of 
the primary frequency, since twice per period the axis of the 
armature coil and that of the primary coil coincide. A vary- 
ing admittance is obviously identical in effect with a varying 
reluctance, which will be discussed in the chapter on reac- 
tion machines. That is, the induction motor with one 
•closed armature circuit is, at synchronism, nothing but a 
reaction machine, and consequently gives zero torque at 
synchronism if the maxima and minima of the periodically 
varying admittance coincide with the maximum and zero 
values of the primary circuit, but gives a definite torque if 
they are displaced therefrom. This torque may be positive 
or negative according to the phase displacement between 
admittance and primary circuit ; that is, the lag or lead 
of the maximum admittance with regard to the primary 
maximum. Hence an induction motor with single-armature 
circuit at synchronism acts either as motor or as alternat- 
ing-current generator according to the relative position of 
the armature circuit to the primary circuit. Thus it can be 
called a synchronous induction motor or synchronous in- 
duction generator, since it is an induction machine giving 
torque at synchronism. 

Power factor and apparent efficiency of the synchron- 
ous induction motor as reaction' machine are very low. 
Hence it is of practical application only in cases where a 
small amount of power is required at synchronous rotation, 
and continuous current for field excitation is not available. 


INDUCTION MOTOR. 293 

The current induced in the armature of the synchronous 
induction motor is of double the frequency impressed upon 
the primary. 

Below and above synchronism the ordinary induction 
motor, or induction generator, torque is superimposed upon 
the synchronous induction machine torque. Since with the 
frequency of slip the relative position of primary and of 
secondary coil changes, the synchronous induction machine 
torque alternates periodically with the frequency of slip. 
That is, upon the constant positive or negative torque be- 
low or above synchronism an alternating torque of the fre- 
quency of slip is superimposed, and thus the resultant 
torque pulsating with a positive mean value below, a nega- 
tive mean value above, synchronism. 

When started from rest, a synchronous induction motor 
will accelerate like an ordinary single-phase induction mo- 
tor, but not only approach synchronism, as the latter does, 
but run up to complete synchronism under load. When 
approaching synchronism it makes definite beats with the 
frequency of slip, which disappear when synchronism is 
reached. 

THE HYSTERESIS MOTOR. 

181. In a revolving magnetic field, a circular iron disk, 
or iron cylinder of uniform magnetic reluctance in the 
direction of the revolving field, is set in rotation, even if 
subdivided so as to preclude the induction of eddy currents. 
This rotation is due to the effect of hysteresis of the revolv- 
ing disks or cyclinder, and such a motor may thus be called 
a hysteresis motor. 

Let / be the iron disk exposed to a rotating magnetic 
field or resultant M.M.F. The axis of resultant magneti- 
zation in the disk / does not coincide with -the axis of the 
rotating field, but lags behind the- latter, thus producing a 
couple. That is, the component of magnetism in a direction 
of the rotating disk, /, ahead of the axis of rotating M.M.F., 
is rising, thus below, and in a direction behind the axis 


294 AL TERN A TING-CURRENT PHENOMENA. 

of rotating M.M.F. decreasing; that is, above proportion- 
ality with the M.M.F., in consequence of the lag of magnet- 
ism in the hysteresis loop, and thus the axis of resultant 
magnetism in the iron disk, /, does not coincide with the 
axis of rotating M.M.F., but is shifted backwards by an 
angle, a, which is the angle of hysteretic lead in Chapter 
X., § 79. 

The induced magnetism gives with the resultant M.M.F. 
a mechanical couple, — 


T= mF& sin a, 

where 

F= resultant M.M.F., 

<£ = resultant magnetism, 

a = angle of hysteretic advance of phase, 

m = a constant. 

The apparent or voltampere input of the motor is, — 
Q = mF®. 

Thus the apparent torque efficiency, — 

T 

2 = sma, 

and the power of the motor is, — 

P = (1 — s) T= (1 — s) m F<$> sin a, 
where 

s = slip as fraction of synchronism. 

The apparent efficiency is, — 

P 

- = (!_*) sin a. 

Since in a magnetic circuit containing an air gap the 
angle a is extremely small, a- few degrees only, it follows 
that the apparent efficiency of the hysteresis motor is ex- 
tremely low, the motor consequently unsuitable for produ- 
cing larger amounts of mechanical work. 


INDUCTION MOTOR. 295 

From the equation of torque it follows, however, that at 
constant impressed E.M.F., or current, — that inconstant 
F, — the torque is constant and independent of the speed ; 
and therefore such a motor arrangement is suitable, and 
occasionally used as alternating-current meter. 

The same result can be reached from a different point 
of view. In such a magnetic system, comprising a mov- 
able iron disk, /, of uniform magnetic reluctance in a 
revolving field, the magnetic reluctance — and thus the dis- 
tribution of magnetism — is obviously independent of the 
speed, and consequently the current and energy expenditure 
of the impressed M.M.F. independent of the speed also. If, 
now, — 

V '= volume of iron of the movable part, 
B = magnetic density, 
and 77 = coefficient of hysteresis, 

the energy expended by hysteresis in the movable disk, /, is 
per cycle, — 

IV, = V^B™, 

hence, if N= frequency, the energy supplied by the M.M.F. 
to the rotating iron disk in the hysteretic loop of the 

M.M.F. is, — 

P = 


At the slip, s N, that is, the speed (1 — s) N, the energy 
xpended by hysteresis in the rotating disk is, however, — 


Hence, in the transfer from the stationary to the revolv- 
ing member the magnetic energy, — 


has disappeared, and thus reappears as mechanical work, 
and the torque is, — 

'-p^iprW' 

that is, independent of the speed. 


296 AL TERNA TING-CURRENT PHENOMENA. 

Since, as seen in Chapter X., sin a is the ratio of the 
energy of the hysteretic loop to the total apparent energy, 
in voltampere, of the magnetic cycle, it follows that the 
apparent efficiency of such a motor can never exceed the 
value (1 — s) sin a, or a fraction of the primary hysteretic 
energy. 

The primary hysteretic energy of an induction motor, as 
represented by its conductance, g, being a part of the loss 
in the motor, and thus a very small part of its output only, 
it follows that the output of a hysteresis motor is a very 
small fraction only of the output which the same magnetic 
structure could give with secondary short-circuited winding, 
as regular induction motor. 

As secondary effect, however, the rotary effort of the 
magnetic structure as hysteresis motor appears more or less 
in all induction motors, although usually it is so small as to 
be neglected. 

If in the hysteresis motor the rotary iron structure has 
not uniform reluctance in all directions — but is, for in- 
stance, bar-shaped or shuttle-shaped — on the hysteresis 
motor effect is superimposed the effect of varying magnetic 
reluctance, which tends to accelerate the motor to syn- 
chronism, and maintain it therein, as shall be more fully 
investigated under " Reaction Machine " in Chapter XX. 


ALTERNATING-CURRENT GENERATOR. 297