0195136047.pdf

570 ROTATING MACHINES

an expression forI 2 ′. To that end, let us redraw the equivalent circuit in Figure 13.2.8. By applying

Thévenin’s theorem, we have the following from Figures 13.2.6 and 13.2.8:

V ̄ 1 a=V ̄ 1 −I ̄ 0 (R 1 −jXl 1 )=V ̄ 1 jXm

R 1 +j(Xl 1 +Xm)

(13.2.9)

R 1 ′′+jX′′ 1 =

(R 1 +jXl 1 )j Xm

R 1 +j(Xl 1 +Xm)

(13.2.10)

I 2 ′=

V 1 a

√

[R′′ 1 +(R 2 ′/S)]^2 +(X′′ 1 +Xl′ 2 )^2

(13.2.11)

T=

1

ωs

m 1 V 12 a(R′ 2 /S)

[R 1 ′′+(R′ 2 /S)]^2 +(X′′ 1 +X′l 2 )^2

(13.2.12)

Neglecting the stator resistance in Equation (13.2.9) results in negligible error for most induction

motors. IfXmof the equivalent circuit shown in Figure 13.2.6 is sufficiently large that the shunt

branch need not be considered, calculations become much simpler;R 1 ′′andX′′ 1 are then equal to

R 1 andXl 1 , respectively; also,V 1 ais then equal toV 1.

The general shape of the torque–speed or torque–slip characteristic is shown in Figure 13.1.6,

in which the motor region (0<S≤1), the generator region (S<0), and the breaking region (S

>1) are included for completeness. The performance of an induction motor can be characterized

by such factors as efficiency, power factor, starting torque, starting current, pull-out (maximum)

torque, and maximum internal power developed. Starting conditions are those corresponding to

S=1.

The maximum internal (or breakdown) torqueTmaxoccurs when the power delivered toR 2 ′/S

in Figure 13.2.8 is a maximum. Applying the familiar impedance-matching principle of circuit

theory, this power will be a maximum when the impedanceR 2 ′/Sequals the magnitude of the

impedance between that and the constant voltageV 1 a. That is to say, the maximum occurs at a

value of slipSmaxTfor which the following condition is satisfied:

R′ 2

SmaxT

=

√

(R 1 ′′)^2 +(X′′ 1 +Xl′ 2 )^2 (13.2.13)

The same result can also be obtained by differentiating Equation (13.2.12) with respect toS,or,

more conveniently, with respect toR′ 2 /S, and setting the result equal to zero. This calculation has

been left to the enterprising student. The slip corresponding to maximum torqueSmaxTis thus

given by

SmaxT=

R′ 2

√

(R′′ 1 )^2 +(X′′ 1 +X′l 2 )^2

(13.2.14)

and the corresponding maximum torque from Equation (13.2.12) results in

R'' 1

R' 2 (1 − S)

S

+

−

V 1 a

I' 2 jX''^1 jX'l2 R' 2 I' 2 Figure 13.2.8phase equivalent circuit for theAnother form of per-

polyphase induction motor of Fig-

ure 13.2.6.