0195136047.pdf

13.1 ELEMENTARY CONCEPTS OF ROTATING MACHINES 561

Torque

2 1 0 − 1

−ωs RS 0 P ωs Speed 2 ωs

Slip

Braking

region

Motor

region

Generator

region

Q

Figure 13.1.6General form of

torque–speed curve (or torque–

slip characteristic) for a poly-

phase induction machine.

in the home, office, and factory. For the sake of simplicity, let us consider a single-phase induction
motor with a squirrel-cage rotor and a stator carrying a single-phase winding, connected to a single-
phase ac supply. The primary mmf cannot be rotating, but, in fact, it is pulsating in phase with
the variations in the single-phase primary current. It can be shown, however, that any pulsating
mmf can be resolved in terms of two rotating mmfs of equal magnitude, rotating in synchronism
with the supply frequency but in opposite directions (see Section 12.3). The wave rotating in the
same direction as the rotor is known as theforward-rotating wave,whereas the one rotating in
the opposite direction is thebackward-rotating wave.Then the slipSof the machine with respect
to the forward-rotating wave is given by

Sf=

ωs−ωm

ωs

(13.1.9)

which is the same as Equation (13.1.6). The slipSbof the machine with respect to the backward-
rotating wave, however, is given by

Sb=

−ωs−ωm

ωs

= 2 −Sf (13.1.10)

SfandSbare known as forward (or positive-sequence) slip and backward (or negative-sequence)
slip, respectively. Assuming that the two component mmfs exist separately, the frequency and
magnitude of the component emfs induced in the rotor by their presence will, in general, be
different becauseSfis not equal toSb. Then the machine can be thought of as producing a steady
total torque as the algebraic sum of the component torques. The equivalent circuit based on
therevolving-field theoryis pursued in Section 13.2. At standstill, however,ωm =0 and the
component torques are equal and opposite; no starting torque is produced. Thus, it is clear that
a single-phase induction motor is not capable of self-starting, but it will continue to rotate once
started in any direction. In practice, additional means are provided to get the machine started
(usually as an asymmetrical two-phase motor) from a single-phase source, and the machine is
then run as a single-phase motor. An approximate shape of the torque–speed curve for the single-
phase motor can readily be obtained from that of the three-phase machine shown in Figure 13.1.6.
Corresponding to a positive slip ofSf=OPin Figure 13.1.6, the positive-sequence torque isPQ.
Then, corresponding toSb= 2 −Sf=OR, the negative-sequence torque isRS. The resultant
torque is given byPQ−RS. This procedure can be repeated for a range of slips 1≤S≤ 0
to give the general form of the torque–speed characteristic of a single-phase induction motor, as
shown in Figure 13.1.7.