0195136047.pdf

13.2 INDUCTION MACHINES 567

and the equivalent circuit may be redrawn as in Figure 13.2.5(b).R 2 ′is the per-phase standstill
rotor resistance referred to the stator, andR′ 2 [( 1 −S)/S] is a dynamic resistance that depends on
the rotor speed and corresponds to the load on the motor. [See the discussion following Equation
(13.2.6).]
When power aspects need to be emphasized, the equivalent circuit is frequently redrawn as
in Figure 13.2.5(c), in which the shunt conductancegcis omitted. The core losses can be included
in efficiency calculations along with the friction, windage, and stray-load losses.
Recall that, in static transformer theory, analysis of the equivalent circuit is often simplified
either by neglecting the exciting shunt branch entirely, or by adopting the approximation of moving
it out directly to the terminals. For the induction machine, however, such approximations might
not be permissible under normal running conditions because the air gap leads to a much higher
exciting current (30 to 50% of full-load current) and relatively higher leakage reactances.
The parameters of the equivalent circuit of an induction machine can be obtained from the
no-load(in which the motor is allowed to run on no load) andblocked-rotor(in which the rotor
of the induction motor is blocked so that the slip is equal to unity) tests. These tests correspond
to the no-load and short-circuit tests on the transformer, and are very similar in detail.

Polyphase Induction Machine Performance

Some of the important steady-state performance characteristics of a polyphase induction motor
include the variation of current, speed, and losses as the load–torque requirements change, and the
starting and maximum torque. Performance calculations can be made from the equivalent circuit.
All calculations can be made on aper-phase basis,assuming balanced operation of the machine.
Total quantities can be obtained by using an appropriate multiplying factor.
The equivalent circuit of Figure 13.2.5(c), redrawn for convenience in Figure 13.2.6, is
usually employed for the analysis. The core losses, most of which occur in the stator, as well as
friction, windage, and stray-load losses, are included in the efficiency calculations. The power-
flow diagram for an induction motor is given in Figure 13.2.7, in whichm 1 is the number of stator
phases,φ 1 is the power factor angle betweenV ̄ 1 andI ̄ 1 ,φ 2 is the power factor angle betweenE ̄ 1
andI ̄ 2 ′,Tis the internal electromagnetic torque developed,ωsis the synchronous angular velocity
in mechanical radians per second, andωmis the actual mechanical rotor speed given byωs( 1 −S).
The total powerPgtransferred across the air gap from the stator is the difference between the
electric power inputPiand the stator copper loss.Pgis thus the total rotor input power, which is
dissipated in the resistanceR′ 2 /Sof each phase so that

Pg=m 1 (I 2 ′)^2

R 2 ′

S

=Tωs (13.2.5)

Subtracting the total rotor copper loss, which ism 1 (I 2 ′)^2 R 2 ′orSPg, from Equation (13.2.5) for
Pg, we get the internal mechanical power developed,

R 1

R' 2 (1 − S)

S

+

−

V 1

jXl 1 jX'l2 R' 2

jXm E 1

I 1

I 0

I' 2

Figure 13.2.6Per-phase equivalent

circuit of a polyphase induction

motor used for performance

calculations.