an investigation of dual stator winding induction machines

3.7.3 Torque Equation
The electro-magnetic force developed by the machine is the only one that couples the
electrical equation with the mechanical equation. From the energy point of view, the
torque is determined by the instantaneous power transferred in the electromechanical
system.
The coenergy in a magnetic field is defined as:
J
J
c = ∑ i jλ
j W f and W f = ∫∑ i j
j=1
j=
1
W −
where, i j and λ j are the current and flux linkage of
⋅ dλ
(3.76)
j
116
th
j circuit respectively. W f is the
total field energy in the system. In the dual stator winding inductance machine, the
currents in the system include two stator currents and one rotor current. Hence for dual
stator winding induction machine, the total field energy can be 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
⋅ Ls2r
⋅ir
2
2
1 T 1 T
⋅ Lrs1
⋅iabc
+ ir
⋅ Lrs2
⋅i
xyz + ir
⋅ Lrr
⋅ir
2
2
The electromagnetic torque can be obtained from the magnetic coenergy as:
T
∂W
∂W
xyz
(3.77)
c
f
e = = −
(3.78)
∂θ
rm ∂θ
rm
where, θ rm is the mechanical angle of the rotor.
The coupling inductance between two stator winding sets is zero.
L L = 0
(3.79)
s1
s2
= s2
s1