CHAPTER XX 
SINGLE-PHASE INDUCTION MOTORS 

177. The magnetic circuit of the induction motor at or near 
synchronism consists of two magnetic fluxes superimposed 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 power circuit, the other to the primary exciting 
circuit. In the single-phase motor the one flux is produced by 
the primary circuit, the other by the currents produced in the 
secondary or armature, which are carried into quadrature posi- 
tion by the rotation of the armature. In consequence thereof, 
while in all these motors the magnetic distribution is the same 
at or near synchronism, 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 secondary armature cur- 
rents are not carried to complete quadrature position; and thus 
only a component is 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 quadrature, 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 satisfac- 
tory at or near synchronism, their torque decreases proportionally 
with 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 electrical. 
All the electrical starting devices essentially consist in impress- 

245 


24G ALTERNATING-CURRENT PHENOMENA 

ing upon the motor at standstill a magnetic quadrature flux. 
This may be produced either by some outside e.m.f., as in the 
monocyclic starting device, or by displacing the circuits of two 
or more primary coils from each other, either by mutual induc- 
tion between the coils — that is, by using one as secondary 
to the other — or by impedances of different inductance factors 
connected with the different primary coils. 

178. The starting devices of the single-phase induction 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 in the circuits, as a condenser 
and a reactive coil, or by making the internal impedances of the 
motor circuits different, 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 induct- 
ive relation can be outside of the motor or inside, by having 
the one coil submitted to the inductive action of the other; and 
in this latter case the current in the secondary 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 com- 
bination 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 inductive 
reactance and a resistance, or an inductive and a condensive 
reactance 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. 


SINGLE-PHASE INDUCTION MOTORS 247 

179. Occasionally, 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 allowing the appli- 
cation of phase-displacing devices for producing the starting 
quadrature flux. 

Since at or near synchronism, at the same impressed e.m.f. 
— that is, the same magnetic density — the total volt-amperes 
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 exciting 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 impedance reduced to the primary is lowered, 
since in single-phase operation all secondary circuits corre- 
spond 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 reduced 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 considerably 
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 magnetic 
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. 

The exciting current of the single-phase motor consists of 
the primary exciting current or current producing the main 
magnetic flux, and represented by ^ constant admittance, 7oS 
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 excita- 


248 ALTERNATING-CURRENT PHENOMENA 

tion for the quadrature magnetic flux. This latter magnetic 
flux is equal to the main magnetic flux, $o, at synchronism, 
and falls off with decreasing speed to zero at standstill, if no 
starting device is used, or to $i = $of at standstill if by a start- 
ing 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, Fi', which changes from equality with the primary 
exciting admittance, Fo^ at synchronism to Yi^ = 0, respect- 
ively to Fi^ = tYa^ 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 admittance 
can be represented by: 

Fi' = [1 - (1 - 0 s] Toi. 

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, 

Z Z 

hence = -^ with a three-phase secondary, and = -77- with a 

two-phase secondary with impedance Zx per circuit. 

At standstill, however, the secondary circuits correspond to 
the primary circuit only with their projection in the direction 
of the primary flux, and thus as resultant only one-half of the 
secondary circuits are effective, so that the secondary impe- 

2Zi . 

dance at standstill is equal to — ^ with a three-phase, and equal 

to Zi with a two-phase, secondary. Thus the effective second- 
ary impedance of the single-phase motor changes with the speed 

and can at the slip s be represented by Zi^ = — — ^ in a 

three-phase secondary, and Zi^ = ^ in a two-phase 

secondary, with the impedance Zi 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: 

D = a,e^ (1 - s). 

{a and e see paragraph 171.) 

In a single-phase motor with a starting device which at 


SINGLE-PHASE INDUCTION MOTORS 249 

standstill produces a ratio of magnetic fluxes t, the torque at 
standstill is 

Do = tDi, 

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

Thus denoting the value -yc— = v, the single-phase motor torque 

at standstill is: 

Do =vDi= aie^v, 

and the single-phase motor torque at slip s is: 

D = aieHl - {I - v) s]. 

180. In the single-phase motor considerably more advan- 
tage is gained by compensating for the wattless magnetizing 
component of current by capacity than in the polyphase motor, 
where this wattless component of the 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 condenser of higher 
frequency may lower the power-factor more than the compen- 
sation for the wattless component of the fundamental wave 
raises it, as will be seen in the chapter 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 circuit, 
but in a tertiary circuit; that is, in a circuit stationary with 
regard to the primary impressed circuit but submitted to in- 
ductive action by the revolving secondary circuit. 

In this case the condenser is supplied with an e.m.f. trans- 
formed twice, from primary to secondary and from secondary 
to tertiary, through multitooth structures in a uniformly re- 
volving field, and thus a very close approximation 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 power current only, and thus 
the efficiency of the motor is essentially increased. 


250 ALTERNATING-CURRENT PHENOMENA 

The tertiary circuit may be at right angles to the primary, 
or under any other angle. Usually it is applied on an angle 
of 45° to 60°, so as to secure a mutual induction between tertiary 
and primary for starting, which produces in starting in the con- 
denser a leading current, and gives the quadrature magnetic 
flux required. 

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

Let Yq = go — jbo = primary exciting admittance of the motor 
per delta circuit. 

Zo = ro -{■ jxo = primary self-inductive impedance per delta 
circuit. 

Zi = I'l -|- jxi = secondary self-inductive impedance per delta 
circuit reduced to primary. 

Let 

i^3 = 6^3 + jbs = admittance of the condenser connected be- 
tween terminals 1 and 3. 

If then, as single-phase motor, 

t = ratio of auxiliary quadrature flux to main flux in 

starting, 
h = ratio of e.m.f. generated in condenser circuit to 
e.m.f. generated in main circuit in starting, 
_ starting torque 
aie^ in starting 

Operating single-phase 

Fo^ = 1.5 Yo = 1.5(^0 — jbo) = primary exciting admit- 
tance; 
Fii = 1.5 Fo[l - (l-t)s] 

= 1.5 (go — jbo) [1 — (1 — t) s] = secondary exciting 
admittance at slip s; 

2Zo 2 (ro 4- jxo) . u- A ,■ • 

Zo^ = — o~ = ^ = primary seli-mductive impe- 
dance; 

7 1 (1 + ^) 7 (1 + S) . , • N , .f 

Zi' = ^ Zi = ^ (ri -|- jsxi) — secondary seli- 

inductive impedance; 

Zi^ — —^ = ^ = tertiary self-inductive impe- 
dance of motor. 


SINGLE-PHASE INDUCTION MOTORS 251 

Thus, 

Yi = r- = total admittance of tertiary circuit. 

Since the e.m.f. generated in the tertiary circuit decreases 
from e at synchronism to he at standstill, the effective tertiary 
admittance or admittance reduced to a generated e.m.f., e, is 
at slip s, 

F4I = [1 - (1 - h)s]}\. 
Let then, 

e = counter e.m.f. of primary circuit, 
s = slip. 
We have, 
the secondary load current, 

J se S se / • N 

Zi^ (1 + s) (ri + jsxi) 

the secondary exciting current, 

/ii = eYi' = 1.5eFo[l - (1 - t) [s; 

the secondary condenser current; 

/, = er,i = eYi[l - (1 - /i) s]; 

thus, the total secondary current, 

I' = h +/1I + /4; 
the primary exciting current, 
W = eYo' = 1.5 e Fo, 
thus, the total primary current, 

/o = /I + h' = /i + /4 + Ii' + lo' = e(6i - M; 
the primary impressed e.m.f., 

Eo = e + Zo^/o = e(ci - JC2) ; 

thus, the main counter e.m.f., 

Eo 
e 


Cl - JC2 

or, 

eo 


E = 


Cl - JC2 


252 ALTERNATING-CURRENT PHENOMENA 
and the absolute value, 


hence, the primary current, 

J eo(fei - jbj) 

-«o — '• , 

Ci - JC2 

or, 


^0 = eo 


lb? 


-\-b, 


The volt-ampere input, 
the power input, 

the torque at slip s, 

D = Z)' [1 - (1 - .) s] = -1^ [1 - (1 - f) s], 

Ci -f- C2 

and the power output, 

P = D{1 - s) 


,21:2(1 - s) [1 - (1 - v) s], 

Ci -)- C2 


and herefrom in the usual manner may be derived the efficiency, 
apparent efficiency, torque efficiency, apparent torque efficiency, 
and power-factor. 

The derivation of the constants, t, h, v, which have to be 
determined before calculating the motor, is as follows: 

Let eo = single-phase impressed e.m.f., 

Y = total stationary admittance of motor per delta circuit, 
E3 = e.m.f. at condenser terminals in starting. 

In the circuit between the single-phase mains from terminal 
1 over terminal 3 to 2, the admittances, Y + Y3, and Y, are con- 
nected in series, and have the respective e.m.fs., E3 and €q — E3. 
It is thus, 

It -{- Y3 -i- Y = Co — E3 -i- E3, 

since with the same current in both circuits, the impressed 
e.m.fs. are inversely proportional to the respective admittances. 
Thus, 

E3 ^2 Y + Y ^ ^°^^*' ~ •^^^'^^' 


SINGLE-PHASE INDUCTION MOTORS 


253 


and the quadrature e.m.f. is 


eoh2, 


hence, 
and 


^3 = eoVhJ^Fh?, 


h = Vhi^Th?- 
Since in the three-phase e.m.f. triangle, the altitude corre- 
sponding to the quadrature magnetic flux = 'z~~7--'' and the 

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

t = ^' = 1.155/12. 
V3 

From t, V is derived as shown in the preceding. 
182. The most frequently used starting device of single-phase 
induction motors (with the exception of fan motors, in which the 


e,l 


El, li.Yi 


Eo.U.Yj 


Fig. 128. 

shading coil is commonly used) is the monocyclic starting device. 
It consists in producing externally to the motor a system of 
polyphase e.m.fs. with single-phase flow of energy, and im- 
pressing it upon the motor, which is wound as polyphase, usually 
three-phase motor. 

Such a polyphase system of e.m.fs. with single-phase flow of 
energy has been called a monocyclic sj^stem. It essentially 
consists, or can be resolved into, a main or energy e.m.f., in 
phase with the flow of energy, and an auxiliary or wattless e.m.f. 
in quadrature thereto. 

If across the single-phase mains of voltage, e, two impedances 
of different inductance factors, of the respective admittances, 
Fi and F2, are connected, the voltages, Ei and E2 of these im- 


254 ALTERNATING-CURRENT PHENOMENA 

pedances are displaced from each other, thus forming with the 
main voltage, e, a voltage triangle, or a more or less distorted 
three-phase system, as shown in Fig. 128. 

Connecting now a three-phase induction motor with two of 
its terminals, 1 and 2, to the single-phase mains a, and h, and 
with its third terminal 3 to the common connection, c, of the two 
impedances, a quadrature flux is produced in this motor, by the 
traverse voltage, E3, of the monocyclic triangle, Fig. 128. 

It is then: 

Ei + E2 = e (1) 

E2 — El = Ez (2) 

hence: 


El = ^ — E3 

E2 = 2 ~^ ^^ 


(3) 


Let now, in Fig. 128. 
Y = effective admittance of motor between terminals 1 and 

2 at standstill. 
F3 = effective admittance of motor for the quadrature flux, 

from terminal 3 to middle between 1 and 2. 

As the voltage of this latter admittance is -^\/3^ the altitude 

of the three-phase motor triangle, and as the magnetic flux is the 
same in all directions, in the polyphase motor, and the effective 
admittances are proportional to the square of the voltage, it is: 


F3 -^ F = e^ ^ (I V3) 


hence: 


-4 


F3 = |f 


Denoting the currents and voltages in the direction as shown 
by the arrows in Fig. 128, it is: 

h = h - h (4) 

and: 

h = F3^3 = I F^3 (5) 


SINGLE-PHASE INDUCTION MOTORS 255 


(6) 


(By equation (3)) substituting (5) and (6) into (4), gives, after 
transposing: 

• ^1-1+ Y, + I F (^> 

Substituting (7) into (3), (5), (6) then gives the voltages and 
currents: 

El, E2, I:i, Ii, I2 

The current traversing the motor from terminal 1 to terminal 2 
is 

/' = eY (8) 

and upon this superimpose the return of the current I3, so that 
current 

I\ ^eY + \h (9) 

leaves terminal 2, and current 

I\ = eY -\ h (10) 

enters terminal 1. 

The total current taken by the motor and starting device from 
the single-phase mains then is: 

/ = /i + /'i 1 

(11) 


= /2 + /'2 

and herefrom follows the volt-ampere input: 

Q = el (12) 

while on polyphase supply, the volt-ampere input is: 

Qo ^2 el' = 2e^Y (13) 

thus the ratio of volt-ampere inputs is: 

The ratio of the starting torque of the motor with the monocyc- 
lic starting device, to that of the same motor on three-phase 


256 ALTERNATING-CURRENT PHENOMENA 

supply, is the ratio of the quadrature fluxes, which is proportional 
to the quadrature voltages: 

~tV3~ I Vs{r.+ Y. + lY}l ('■" 

where the index, j, denotes, that only the quadrature term of the 
expression is efi'ective in producing torque. 

The ratio of the apparent starting torque efficiencies thus is : 

s = - (16) 

q 

183. Usually a resistance and a reactance are used as the two 
impedances of the monocyclic starting device, as the cheapest, 
though the triangle produced thereby has a low altitude, £'2, and 
starting torque and torque efficiency thus are comparatively low. 

Let as illustration, in the three-phase motor, Figs. 122 and 123, 
a resistance-reactance starting device be used of the values: r = 
1 ohm, and a; = 1 ohm hence: 

F. = i = 1 

r 

X •' 

In this motor, at standstill, it is, per delta circuit: 

(a) Without start- (h) With secondary 
ing resistance : resistance in- 

creased ten fold: 
Voltage: e = 110 volts 

Current: i = 176 amp. 8.97 amp. 

Torque: D = 2.93 syn. kw. 7.38 syn. kw. 

Power-factor: p = 0.313 0.835 

Hence the current, 

vectorially: / = 55 - 167 j 75 - 49 j 

and the admittance, per motor 

circuit: Y' = 0.5 - 1.52 j 0.68 - 0.45 j 

Hence, the effective admittance, between two motor terminals 1 
and 2: 

Y = 1.5 F' = 0.75 - 2.28 j 1.02 - 0.67 i 

Herefrom follows: 

Quadrature voltage: ^3 = - 5.5 + 16.3 j 2.7 + 25.5 j 


SINGLE-PHASE INDUCTION MOTORS 257 

Relative starting 

torque: t = 0.172 0.268 

Starting torque: 3 tD — 1.52 syn. kw. 6.73 syn. kw. 

As seen, with starting resistance in the secondary circuit, a 
fairly good starting torque is given by this device; but with 
short-circuited armature, the starting torque is low. 

184. The greater the difference in the inductance factors of 
the two impedances in the starting device, the higher values of 
quadrature voltage, E3, and thus of starting torque are available. 

The combination of inductance and capacity thus gives the 
highest torque, and by such combination, true three-phase rela- 
tion can be secured, that is, the conditions brought about: 

El = E2 = 6 

The starting by condenser in the tertiary circuit, of a three- 
phase motor, can be considered as a special case of the mono- 
cyclic starting device, for Yi = 0 and F2 = capacity susceptance. 

A further extension of the monocyclic starting device is, to use 
another induction motor, which is running at speed, to supply 
the quadrature voltage, E3. 

Thus, if a number of single-phase induction motors are oper- 
ated near each other, as in the same factory, etc., they can all be 
made self-starting — except the first one — by connecting their 
third terminals together. That is, connecting a number of three- 
phase induction motors, with two of their terminals, 1, 2 to 
single-phase mains a, h, and connecting all their third terminals, 3, 
with each other by an interconnecting main, c, then, as soon as 
one of the motors is running, all the others can be started by 
drawing quadrature voltage and current from the one which is 
running. 

This is a convenient means of operating single-phase induction 
motors self-starting without separate starting devices. It has 
the further advantage, that an overloaded motor begins to draw 
current over the interconnecting circuit, c, from the other motors, 
as phase converters, and the maximum output of the individual 
motors thereby is increased far beyond that of the motor as 
single-phase motor, near to that as three-phase motor. 

As single-phase motors, especially with armature resistance, 
when once started and when not loaded, speed up from low speed 

17 


258 ALTERNATING-CURRENT PHENOMENA 

to full speed, the first motor in such monocyclic interconnecting 
system can be started by hand, after taking its load off. 

For further discussion on the theory and calculation of the 
single-phase induction motor, see American Institute Electrical 
Engineers Transactions, January, 1898 and 1900. 


SECTION V 

SYNCHRONOUS MACHINES