GENERALISED THEORY OF ELECTRICAL MACHINES 497primary are transformed to their equivalent two-phase variables. These transformations are detailed
in References 3, 5 and 6 for example. The result is a transposition of the rows in the voltage-current
equation (20.24) and the insertion of suffices 1 and 2, 1 for the primary and 2 for the secondary (as
with static transformers). Equation (20.24) becomes:-
vd 1
vq 1
0
0
=
Mp 0 R 2 +L 2 p 0
0 Mp 0 R 2 +L 2 p
R 1 +L 1 pωrLdq Mp ωrM
−ωrLdq R 1 +L 1 p −ωrMMp
id 1
iq1
id2
iq2
(^20.^26 )
Where:R 1 =Ra,R 2 =Rk,L 1 =La,L 2 =LkandLdq=M+Lla.
Replace suffix ‘a’ with ‘1’, and suffices ‘kd’and‘kq’ with ‘2’.
The corresponding flux linkage equation, derived from (20.21), becomes:-
ψd 1
ψq1
ψd2
ψq2
=
M+Ll 1 0 M 0
0 M+Ll 1 0 M
M 0 M+Ll 2 0
0 M 0 M+Ll 2
id1
iq1
id2
iq2
(^20.^27 )
And similarly from (20.21) and differentiating:-
pψd 1
pψq1
pψd2
pψq2
=
(M+Ll 1 )p 0 Mp 0
0 (M+Ll 1 )p 0 Mp
Mp 0 (M+Ll 2 )p 0
0 Mp 0 (M+Ll 2 )p
id1
iq1
id2
iq2
(^20.^28 )
And the voltage equation (20.5) becomes:-
vd1
vq1
vd2
vq2
=
R
id1
iq1
id2
iq2
+
p 000
0 p 00
00 pω
00 −ωp
ψd1
ψq1
ψd2
ψq2
(^20.^29 )
Substituting (20.28) into (20.29) are rearranging the terms gives,
vd1
vq1
0
0
=
R 1 +(M+Ll 1 )p 0 Mp 0
0 R 1 +(M+Ll 1 )p 0 Mp
Mp ωM R 2 +(M+Ll 2 )p ω(M+Ll 2 )
−ωM Mp −ω(M+Ll 2 )R 2 +(M+Ll 2 )p
id1
iq1
id2
iq2
(20.30)
The two upper rows represent the stator and the two lower rows the rotor.