an investigation of dual stator winding induction machines

2π
∫
0
=
J
2π
( θ,
t)
⋅ B ( θ,
t)
⎧
⎪
1 ⎪ k
= Re⎨
2 ⎪
⎪
+
⎩
j(
ω t−
P θ ) ⎧ jµ
r
2 2
0
k j ω2t−kθ
+ ( k−
P2
)
{ 2(
3C
⋅ I ) e } ⋅ Re 2 ⋅ I ⋅C
⋅e
∫ Re
s2
s2
∑
0
2
⎡
⎢
⎣
∑ ∫
⎡
⎢
⎣
∑ ∫
k
qrq
2π
0
µ 0r
j
gk
2π
0
dθ
µ 0r
j
gk
2
2
⎨
⎩
k j 2ω2t+
( k−
P2
)
( 3C
⋅ I )( C ⋅ I ) e
2
2
s2
s2
k
R
k g
R2
218
R2
( ω t−P
θ −kθ
)
*
*
k j(
( k−
P2
) ωrt
+ P2θ
−kθ
)
( 3Cs
2 ⋅ I s2
)( CR
⋅ I R2
) e
R
r
2
( ω t )
⎤⎫
dθ
⎥⎪
⎦⎪
⎬
⎤⎪
dθ
⎥⎪
⎦⎭
r
⎫
⎬dθ
⎭
(5.98)
The first component of the above equation is unless 2 P k = − and the second
component is zero unless 2 P k = . Hence,
2π
∫
0
J
2
( θ,
t)
⋅ B ( θ,
t)
qrq
⎧ µ 0r
⎪ j
⎪ − gP2
= π Re⎨
⎪ µ 0r
+ j
⎪⎩
gP2
2
2
dθ
2
2
*
( )
( ) ⎛ P2
⋅ ⋅
⎞ j 2ω2t−2
P2ω
rt
3C
I C I e
( )( ) ⎪ ⎪
⎫
s2
s2
⎜ R R2
⎟
⎝ ⎠
⎪
⎬
*
*
P2
3Cs
2 ⋅ I s2
CR
⋅ I R2
⎭
5.3 Equation Summary
(5.99)
The voltage and torque equations of the dual stator winding induction machine due to
fundamental current components are now summarized.
5.3.1 Voltages in the ABC Winding Set
If only the fundamental component is considered, the voltages that can be found at
the terminals of the ABC winding set are: