an investigation of dual stator winding induction machines

where ( θ )
n A is the average of the turn function. Equation (3.26) is similar to the
definition of the winding function traditionally used for constant air gap length.
To simplify the integration in the winding function calculation, the inverse of the air
gap is approximated as:
g
where,
where,
1
( θ,
θ )
rm
= A
0
0
+
∑ ∞
i=
1
2
3
A cos
1
A0
= , Ai
=
g 1−
a
i
' ( iθ
− iθ
+ iγ
)
g
0
rm
2
1−
a
2
3
⎛
⎜1
− 1−
a
⋅
⎜ a3
⎝
95
2
3
i
⎞
⎟ ,
⎟
⎠
2
2
a3 a1
2a1a2 cos rm a2
+
+ = θ , ⎟ ' ⎛ a ⎞
2 sinθ
rm
θ = ⎜
rm arctan
.
⎝ a1
+ a2
cosθ
rm ⎠
The other approximations of the inverse of the air gap function are given as:
1
( θ,
θ )
g rm
=
2 ⎡⎛
ρ ⎞
⎢ ⎜
⎜1+
⎟ + ρ cos
1 ⎢⎝
2 ⎠
g ⎢ 3
0 ρ 3 ⎢+
cos
⎣ 4
0
2
ρ
2
2
( θ − σ ) + cos ( θ − σ )
( ) ⎥ ⎥⎥⎥
θ − σ + L
⎤
⎦
(3.27a)
(3.27b)
2 2
α + β
⎛ β ⎞
ρ = , σ = arctan ⎜ ⎟ , α and β are the displacement of rotor center
g
⎝ α ⎠
respected to the stator central point.
If the first two terms in (3.27a) are used in the approximation, the inverse of the air
gap length function is expressed as:
g
1
,
( θ θ )
rm
= A + A cos
0
1
' ( θ −θ
+ γ )
rm
(3.28)
When the expression for the inverse of the air gap is used in the general winding
function equation, a new simplified expression of winding function will results: