Page 505 - Handbook of Electrical Engineering

GENERALISED THEORY OF ELECTRICAL MACHINES 495

the assumptions regarding symmetry and the deletion of the field winding taken into account. The
derived reactances become:-

X d = X q = X a + X m
X m X k

X = X = X = X = X a +
d q d q
X m + X k
X 2 = X (negative sequence reactance)
d
X d
T a =
ωR a

20.3.4 Derivation of an Equivalent Circuit

Equation (20.23) can be rewritten with the rotationally induced emfs correctly represented by the
rotor speed ω r instead of ω as in the case of the synchronous machine:-

     
v d R a + L a p ω r L dq Mp ω r M i d
v R a + L a p −ω r M Mp i
   −ω r L dq   
 q  =    q  (20.24)
Mp 0 R k + L k p 0
 0     i kd 
0 0 Mp 0 R k + L k p i kq
Where ω r = (1 − s)ω, ω is the frequency of the power supply, and s is the slip of the
rotor speed.
The familiar equivalent circuit for the induction motor will be developed from (20.24). The oil
industry occasionally uses variable frequency power supplies to start and run variable speed pumps
and compressors. The nominal frequency applied to the motor is ω n . The inductances in (20.24)
can be changed to their nominal reactances by using the nominal frequency ω n . The steady state
variables replace the instantaneous variables and the differential operator p is replaced by the steady
state frequency in conjunction with the j operator.

 
X d ω X md ω
R a + jω (1 − s) X dq jω (1 − s) X md
 
ω n ω n ω n ω n
 
   
 ω ω 
V d X d X md I d
 −(1 − s) X dq R a + jω −(1 − s) X md jω 

ω n ω n ω n ω n
    
 
V q 
  I q 
=  
X md X k
 0   
  I kd
0  jω 0 R k + jω 0 
 ω n ω n  I kq
 
 X md X k 
0 jω 0 R k + jω
ω n ω n
(20.25)
Where V d , V q , I d , I q , I kd and I kq are the phasor equivalents of their instantaneous variables.
For the balanced three-phase operation of the motor the following discussion applies. In the
above equation the magnitude of the q-axis variables are equal to their corresponding d-axis variables.
◦
The operator j is required in the q-axis variables to identify its 90 phase advance from the d-axis.