an investigation of dual stator winding induction machines

Substituting (5.31) and (5.43) into the above equation yields,
J
r 2i
⎧
'
'
k j s2ω2t
−k
θ −(
i−1)
( θ t)
Re 2 ⋅ I ⋅C
⋅ e
( [ α ] ) ⎫ r
, = ⎨∑
iR2
R
⎬
(5.59)
⎩ k
⎭
Transform the above equation into the stationary reference frame,
J
r 2i
⎧
( ω ) ⎫ rt
, = ⎨∑
iR2
R
⎬
(5.60)
⎩ k
⎭
k j ω2t
−kθ
+ k ( i−1)
α r + ( k −P2
)
( θ t)
Re 2 ⋅ I ⋅C
⋅ e
The corresponding flux densities in the air gap induced by these currents are
expressed as:
B
qrqi
⎧ jµ
r
( ω ) ⎫ rt
, = ⎨∑
iR2
R
⎬
(5.61)
⎩ k k g
⎭
0
k j ω21t−kθ
+ k ( i−1)
α r + ( k −P2
)
( θ t)
Re 2 ⋅ I ⋅C
⋅ e
The electric field induced at the stator surface is:
E
qrqi
2
⎧ jµ
ω r
( ω ) ⎫
rt
, = ⎨∑
iR2
R
⎬ (5.62)
⎩ k k g
⎭
0 2
k j ω2t
−kθ
+ k ( i−1)
α r + ( k −P2
)
( θ t)
Re − 2 ⋅ I ⋅ C ⋅ e
Then the EMF induced in the phase X of the XYZ winding set is obtained by
multiplying the electric field with the winding distribution of the phase X as:
2π
() t = rl C ( θ ) ⋅ E ( θ t)
∫
u qrqi
qrqi X , dθ
(5.63)
0
Substituting (5.18) and (5.62) into (5.63) and integrating,
u
qrqi
() t
⎧
⎪
⎪ k
= rl Re ⎨
⎪
⎪
+
⎩
2π
∑ ∫
∑ ∫
k
0
C
e
− jP2θ
s2
2π
*
jP2θ
Cs2
e
0
µ 0ω2r
⋅
jk g
µ 0ω2r
⋅
jk g
The first term in (5.64) is zero unless,
k =
−P2
2
2
2 ⋅ I
R2
2 ⋅ I
208
⋅C
R2
k
R
⋅C
⋅ e
k
R
( ω t −kθ
+ k ( i−1)
α + ( k −P
) ω t )
j
⋅ e
( ω t −kθ
+ k ( i−1)
α + ( k −P
) ω t )
j
2
2
r
r
2
2
r
⎫
dθ
⎪
⎪
⎬
dθ
⎪
⎪
⎭
r
(5.64)