an investigation of dual stator winding induction machines

segment of end ring respectively. The calculation of the inductances is based on the
winding functions approach that has been given in Section 3.3-3.6.
The electromagnetic torque can be obtained from the magnetic co-energy as:
T
∂W
∂W
c
f
e = = −
(5.122)
∂θ
rm ∂θ
rm
where, θ rm is the mechanical angle of the rotor. The total field energy equation for the
dual stator winding induction machine is expressed as:
W
f
1
= i
2
1
+ i
2
1
+ i
2
T
abc
T
xyz
T
r
1 T
1 T
⋅ Ls1s1
⋅iabc
+ ixyz
⋅ Ls2
s2
⋅i
xyz + iabc
⋅ Ls1s2
⋅i
2
2
1 T 1 T
⋅ Ls2
s1
⋅iabc
+ iabc
⋅ Ls1r
⋅ir
+ ixyz
⋅ Ls2
r ⋅ir
2
2
1 T 1 T
⋅ Lrs1
⋅iabc
+ ir
⋅ Lrs2
⋅i
xyz + ir
⋅ Lrr
⋅ir
2
2
228
xyz
(5.123)
where, L mm ( m = s1,
s2,
r ) represents the self-inductance matrix of m winding; L mn
( n ≠ m = s1,
s2,
r ) represents the mutual inductance matrix between m winding and n
winding. Both L mm and L mn are sub-matrixes of the inductance matrix in given in
(5.121).
Only the terms in equation (5.123) which are functions of the rotor angle contribute to
the electromagnetic torque. Ignoring magnetic saturation i.e L ab = Lba
, the
electromagnetic torque becomes :
∂L
∂L
T s1r
T s2
r
Te = −iabc
⋅ ⋅ ir
− ixyz
⋅ ⋅
∂θ
rm ∂θ
rm
i
r
(5.124)