an investigation of dual stator winding induction machines

Only terms in equation (3.77) which are the functions of the rotor angle can
contribute the electromagnetic torque. So applying (3.78) to (3.77), the electromagnetic
torque can be expressed as:
T
e
1
= − i
2
1
− i
2
T
abc
T
r
∂L
⋅
∂θ
∂L
⋅
∂θ
rs1
rm
s1r
rm
⋅i
⋅i
abc
r
1
− i
2
1
− i
2
T
r
T
xyz
For a linear magnetic circuit,
ij
ji
∂L
⋅
∂θ
∂L
⋅
∂θ
rs2
rm
s2r
rm
⋅i
⋅i
xyz
r
117
(3.80)
L = L
(3.81)
Hence the torque equation is simplified as:
T ∂L
∂L
Te = −iabc
⋅
⋅
∂θ
s1
r
T s2
r
⋅ ir
− ixyz
⋅ ir
(3.82)
rm ∂θ
rm
3.8 Complex Variable Reference Frame Transformation
The circuit model derived in the previous sections can be used to simulate the
dynamic and steady state characteristics of the machine. Unfortunately, the model of the
machine is complicated due to the time-varying mutual inductances and it is desirable to
simplify it to simplify computation. The q − d reference frame transformation for three-
phase electric machines is widely and traditionally used to simplify phase models of
electric machines to facilitate their analysis and control, since they eliminate the time
variance of the mutual inductances. However, for multi-phase systems including space
harmonics, the proper reference frame transformation is complicated [3.11]. Since the
rotor model is actually a n phase system, where n is the number of rotor bars, an
appropriate n x n arbitrary reference transformation for multi-phase system is required if