576 ROTATING MACHINES
induction motors are usually two-pole or four-pole, rated at 2 hp or less, while slower and larger
motors can be manufactured for special purposes. Single-phase induction motors are widely used
in domestic appliances and for a very large number of low-power drives in industry. The single-
phase induction machine resembles a small, three-phase, squirrel-cage motor, except that at full
speed only a single winding on the stator is usually excited.
The single-phase stator winding is distributed in slots so as to produce an approximately
sinusoidal space distribution of mmf. As discussed in Section 13.1, such a motor inherently has
no starting torque, and as we saw in Section 12.3, it must be started by an auxiliary winding, by
being displaced in phase position from the main winding, or by some similar device. Once started
by auxiliary means, the motor will continue to run. Thus, nearly all single-phase induction motors
are actually two-phase motors, with the main winding in the direct axis adapted to carry most or
all of the current in operation, and an auxiliary winding in the quadrature axis with a different
number of turns adapted to provide the necessary starting torque.
Since the power input in a single-phase circuit pulsates at twice the line frequency, all single-
phase motors have a double-frequency torque component, which causes slight oscillations in rotor
speed and imparts vibration to the motor supports. The design must provide a means to prevent
this vibration from causing objectionable noise.
The viewpoint adopted in explaining the operation of the single-phase motor, based on
the conditions already established for polyphase motors, is known as therevolving-field theory.
(The other viewpoint,cross-field theory,is not presented here.) For computational purposes, the
revolving-field point of view, already introduced in Section 13.1, is followed hereafter to parallel
the analysis we applied to the polyphase induction motor. As stated in Section 13.1 and shown
in Section 12.3, a stationary pulsating field can be represented by two counterrotating fields of
constant magnitude. The equivalent circuit of a single-phase induction motor, then, consists of
the series connection of a forward rotating field equivalent circuit and a backward rotating one.
Each circuit is similar to that of a three-phase machine, but in the backward rotating field circuit,
the parameterSis replaced by 2−S, as shown in Figure 13.2.11(a). The forward and backward
torques are calculated from the two parts of the equivalent circuit, and the total torque is given by
the albegraic sum of the two. As shown in Figure 13.2.11(b), the torque–speed characteristic of
a single-phase induction motor is thus obtained as the sum of the two curves, one corresponding
to the forward rotating field and the other to the backward rotating field.
The slipSfof the rotor with respect to the forward rotating field is given bySf=S=ns−n
ns= 1 −n
ns(13.2.19)wherensis the synchronous speed andnis the actual rotor speed. The slipSbof the rotor with
respect to the backward rotating field is given bySb=ns−(−n)
ns= 1 +n
ns= 2 −S (13.2.20)Since the amplitude of the rotating fields is one-half of the alternating flux, as seen from Equation
(12.3.10), the total magnetizing and leakage reactances of the motor can be divided equally so
as to correspond to the forward and backward rotating fields. In the equivalent circuit shown in
Figure 13.2.11(a), then,R 1 andXl 1 are, respectively, the resistance and the leakage reactance of
the main winding,Xmis the magnetizing reactance, andR 2 ′andXl′ 2 are the standstill values of
the rotor resistance and the leakage reactance referred to the main stator winding by the use
of the appropriate turns ratio. The core loss, which is omitted here, can be accounted for later
as if it were a rotational loss. The resultant torque of a single-phase induction motor can thus be
expressed as