an investigation of dual stator winding induction machines

The corresponding flux densities in the air gap induced by this current distribution are
expressed as:
B
prpi
⎧ jµ
r
( ω ) ⎫ rt
, = ⎨∑
iR1
R
⎬
(5.48)
⎩ k k g
⎭
0
k j ω1t
−kθ
+ k ( i−1)
α r + ( k −P1
)
( θ t)
Re 2 ⋅ I ⋅ C ⋅ e
The electric fields induced at the stator surface become:
E
prpi
2
⎧ jµ
ω r
( ω ) ⎫
rt
, = ⎨∑
iR1
R
⎬ (5.49)
⎩ k k g
⎭
0 1
k j ω1t
−kθ
+ k ( i−1)
α r + ( k −P1
)
( θ t)
Re − 2 ⋅ I ⋅ C ⋅ e
The induced EMFs in the stator windings due to rotor currents in the th
i loop are
given as the product of the electric field with the winding distributions of the phase
windings. For phase A, the induced EMF is given as:
2π
() t = rl C ( θ ) ⋅ E ( θ t)
∫
prpi A , dθ
(5.50)
u prpi
0
Substituting (5.1) and (5.49) into (5.50) and integrating,
u
prpi
() t
⎧
⎪
⎪ k
= rl Re ⎨
⎪
⎪
+
⎩
2π
∑ ∫
∑ ∫
k
C
e
− jP1θ
s1
2π
*
jP1θ
Cs1
e
0
µ 0ω1r
⋅
jk g
µ 0ω1r
⋅
jk g
The first term in (5.51) is zero unless,
k = −P1
0
2 ⋅ I
The second term in (5.51) is zero unless,
k = P1
2
2
iR1
2 ⋅ I
205
⋅C
iR1
k
R
⋅C
Then the induced EMF equation is simplified as:
⋅ e
k
R
( ω t −kθ
+ k ( i−1)
α + ( k −P
) ω t )
j
⋅ e
1
( ω t −kθ
+ k ( i−1)
α + ( k −P
) ω t )
j
1
r
r
1
1
r
⎫
dθ
⎪
⎪
⎬
dθ
⎪
⎪
⎭
r
(5.51)