GENERALISED THEORY OF ELECTRICAL MACHINES 495the assumptions regarding symmetry and the deletion of the field winding taken into account. The
derived reactances become:-
Xd=Xq=Xa+XmXd′=Xq′=Xd′′=X′′q=Xa+XmXk
Xm+Xk
X 2 =Xd′′ (negative sequence reactance)Ta=X′d
ωRa20.3.4 Derivation of an equivalent circuit
Equation (20.23) can be rewritten with the rotationally induced emfs correctly represented by the
rotor speedωrinstead ofωas in the case of the synchronous machine:-
vd
vq
0
0
=
Ra+LapωrLdq Mp ωrM
−ωrLdq Ra+Lap −ωrMMp
Mp 0 Rk+Lkp 0
0 Mp 0 Rk+Lkp
id
iq
ikd
ikq
(^20.^24 )
Where ωr=( 1 −s)ω,ωis 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 isωn. The inductances in (20.24)
can be changed to their nominal reactances by using the nominal frequencyωn. The steady state
variables replace the instantaneous variables and the differential operatorpis replaced by the steady
state frequency in conjunction with thejoperator.
Vd
Vq
0
0
=
Ra+Xd
ωnjω ( 1 −s)ω
ωnXdqXmd
ωnjω ( 1 −s)ω
ωnXmd−( 1 −s)ω
ωnXdq Ra+Xd
ωnjω −( 1 −s)ω
ωnXmdXmd
ωnjωXmd
ωnjω 0 Rk+Xk
ωnjω 00
Xmd
ωnjω 0 Rk+Xk
ωnjω
Id
Iq
Ikd
Ikq
( 20. 25 )
WhereVd,Vq,Id,Iq,IkdandIkqare 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 theq-axis variables are equal to their correspondingd-axis variables.
The operatorjis required in theq-axis variables to identify its 90◦phase advance from thed-axis.