Handbook of Electrical Engineering

GENERALISED THEORY OF ELECTRICAL MACHINES 485

For a squirrel cage induction motor none of the mutual and self-inductances are functions of the

rotor position.

Equation (20.2) can be applied tova,vbandvcand again toia,ibandic. The zero sequence

terms can be neglected.

The substitution exercise is very tedious, but eventually yields the following expression:-

     

vd

vq

vf

vkd

vkq

     

=











R











     

id

iq

if

ikd

ikq

     

+

     

p +ω 000

−ωp 000

00 p 00

000 p 0

0000 p

     

     

ψd

ψq

ψf

ψkd

ψkq

     

( 20. 5 )

Where, [R]=[Ra,Ra,Rf,Rkd,Rkq]Tand superscriptTmeans transpose.

Note: Since the damper circuits have no external connections and are short circuited by end rings,

the terminal voltagesvkdandvkqare zero, as shown in Figure 20.2.

c) Mutual inductances

Most authors identity the various mutual inductances in each axis of (20.4) e.g.Mab,Maf,Makd,

and then assume them to be equal as,Mdfor thed-axis andMqfor theq-axis. Some analyses have

been published in which these mutual inductances have been assumed to be unequal, particularly

when two or more damper windings have been included in each axis, see References 6, 15 and 16.

Figure 20.2 Mutually coupled circuits in the A-B-C phase andd-qaxis reference frames.