an investigation of dual stator winding induction machines

Re
j(
ω ) ⎧
1t−
P1θ
µ 0r
j(
ω2t−
P2θ
)
{ 2(
3C
⋅ I ) e } ⋅Re
j [ 2(
3C
⋅ I ) e ]
1 µ 0r
= −
2 gP
2
s1
2
s1
2
3C
s1
⋅ I
s1
3C
s2
⎨
⎩
⋅ I
s2
gP
Equation (5.76) can be written as:
2π
⎧ µ 2
0r
⎪∫
j 2
1 ⎪ gP 0 2
Re⎨
2π
2 ⎪ µ 0r
⎪
+ ∫ j 2
⎩ gP 0 2
2
⎡sin
⎢
⎣+
sin
s2
( )
( ) ⎥ ω1t
+ ω2t
− P1θ
− P2θ
+ θ1
+ θ 2 ⎤
ω2t
−ω1t
− P2θ
+ P1θ
+ θ 2 −θ1
⎦
213
s2
j(
ω1t
+ ω2t−
P1θ
−P2θ
)
( 3C
⋅ I )( 3C
⋅ I ) e
2
s1
s1
s2
s2
*
*
j(
ω2t−ω1t
+ P1θ
−P2θ
)
( 3Cs1
⋅ I s1
)( 3Cs
2 ⋅ I s2
) e
⎫
⎬
⎭
⎫
dθ
⎪
⎪
⎬
dθ
⎪
⎪
⎭
(5.80)
(5.81)
The above equation will be zero unless the pole numbers of two stator winding meet
any of the following conditions.
1 2 P P = or P1 = −P2
Since dissimilar pole numbers are chosen in this special machine,
2π
∫
0
1
( θ t)
⋅ B ( θ,
t)
dθ
= 0
J (5.82)
, 2
C) The third term of T e1
For the th
i rotor loop, substituting the expressions of the surface current distribution
and the flux density due to the ABC winding set into the third term,
2π
∫
0
J
1
( θ,
t)
⋅ B ( θ,
t)
prpi
⎧
⎪
1 ⎪ k
= − rl Re⎨
2 ⎪
⎪
+
⎩
⎡
⎢
⎣
∑ ∫
⎡
⎢
⎣
∑ ∫
k
2π
0
dθ
µ 0r
j
gk
2π
0
µ 0r
j
gk
2
2
k
j 2ω1t
+ ( k −P1
) ωrt
+ k ( i−1)
( 3C
⋅ I )( C ⋅ I ) e
2
2
s1
s1
R
iR1
( α −Pθ
−kθ
)
*
*
k j(
( k −P1
) ωrt
+ k ( i−1)
α r + P1θ
−kθ
)
( 3C
s1
⋅ I s1
)( CR
⋅ I R1
) e
r
1
⎤⎫
dθ
⎥⎪
(5.83)
⎦⎪
⎬
⎤⎪
dθ
⎥⎪
⎦⎭