an investigation of dual stator winding induction machines

*
Since both ( ) C
2π
∫
0
2
*
Cs 2 ⋅ s2
and ( s2
I s2
)
( θ t)
⋅ B ( θ,
t)
dθ
= 0
, 2
I ⋅ are real number,
J (5.95)
C) The third term of T e2
Substituting the expressions of winding surface current distribution and the flux
density into the first term,
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 ω1t
−kθ
+ ( k−
P1
)
{ 2(
3C
⋅ I ) e } ⋅ Re 2 ⋅ I ⋅C
⋅e
∫ Re
s2
s2
∑
0
2
⎡
⎢
⎣
∑ ∫
⎡
⎢
⎣
∑ ∫
k
prp
2π
0
µ 0r
j
gk
2π
0
dθ
µ 0r
j
gk
2
2
⎨
⎩
k j ω2t+
ω1t
+ ( k−
P1
)
( 3C
⋅ I )( C ⋅ I ) e
2
2
s2
s2
k
R
k g
R1
217
R1
( ω t−P
θ −kθ
)
( ω t )
⎤ ⎫
dθ
⎥ ⎪
⎦ ⎪
⎬
⎤⎪
dθ
⎥⎪
⎦⎭
*
*
k j(
−ω2t
+ ω1t
+ ( k−
P1
) ωrt
+ P2θ
−kθ
)
( 3Cs
2 ⋅ I s2
)( CR
⋅ I R1)
e
R
r
2
r
⎫
⎬dθ
⎭
(5.96)
The first component of the above equation is zero unless 2 P k = − and the second
component is zero unless 2 P k = . Hence,
2π
∫
0
J
2
( θ,
t)
⋅ B ( θ,
t)
prp
⎧ µ 0r
⎪ j
⎪ − gP2
= π Re⎨
⎪ µ 0r
+ j
⎪⎩
gP2
dθ
2
2
*
( )
( ) ⎛ P2
⋅ ⋅
⎞ j ω2t+
ω1t
+ −P2
−P1
3C
I C I e
s2
D) The fourth term of T e2
2
2
s2
⎜
⎝
R
R1
( ω t )
*
*
P1
j(
−ω2t+
ω1t
+ ( P2
−P1
) ωrt
)
( 3Cs
2 ⋅ I s2
)( CR
⋅ I R1)
e
⎟
⎠
r
⎫
⎪
⎬
⎪
⎪⎭
(5.97)
Substituting the expressions of XYZ windings surface current distribution and the
flux density into the first term,