an investigation of dual stator winding induction machines

all the possible harmonic currents are to be included in the analysis. When the space
harmonics in the rotor circuits are ignored, a simplified reference frame transformation
given in [1.10] is proposed which drastically reduces the computation time.
A general arbitrary reference frame transformation matrix and its inverse for the n
phase system expressed in complex variable form are [3.12]:
T
T
where,
( θ )
( θ )
⎡ 1 2
⎢ jθ1
⎢
e
2 jθ2
= ⎢ e
n ⎢
⎢ M
⎢ jθn−
⎣e
−1
1
⎡ 1 2
⎢ − jθ1
⎢
e
− jθ2
= ⎢ e
⎢
⎢ M
⎢ − jθn
−
⎣e
1
a
1 2
1
a e
2
e
jθ1
jθ2
a
a
1 2
2
4
e
e
jθ1
jθ2
L
L
L
118
( n−1)
( n−1)
jθ1
jθ2
( ) ( ) ( ) ⎥ ⎥⎥⎥⎥⎥
M M O M
2
n−1
jθn−1
2 n−1
jθn−1
n−1
jθn
−1
a e a e L a e ⎦
a
a
1 2
−1
− jθ
−2
jα
a = e , α 2π
n
is an initial condition.
e
e
1
− jθ
2
a
a
1 2
−2
−4
e
e
− jθ
1
− jθ
2
a
a
L
L
L
2
1 2
e
e
−
−2
⎤
1 2
( n−1)
( n−1)
− jθ
1
− jθ
( ) ( ) ( ) ⎥ ⎥⎥⎥⎥⎥
M
M O M
2
− n−1
− jθn−1
−2
n−1
− jθn−1
− n−1
− jθn
−1
a e a e L a e ⎦
a
a
e
e
2
⎤
(3.83)
(3.84)
= , θ i = ωi
t + θ0i
, ω i is the reference frame angular speed and θ 0i
The transformation matrixes and their inverses for 3 phases, 5 phases and 6 phases
systems are given in equation (3.85), (3.86) and (3.87) respectively.
⎡1
2 1 2 1 2 ⎤
⎢
⎛ 2π
⎞ ⎛ 4π
⎞
2
j⎜
θ
⎥
1+
⎟ j⎜
θ1+
⎟
jθ
( ) = ⎢ 1 ⎝ 3 ⎠ ⎝ 3 ⎠
T θ e e e ⎥
(3.85a)
3 ⎢
⎛ 4π
⎞ ⎛ 2π
⎞
j⎜
θ + ⎟ ⎜ + ⎟ ⎥
2
j θ2
jθ
⎢ 2 ⎝ 3 ⎠ ⎝ 3 ⎠ ⎥
⎣
e e e
⎦
⎡ 1 2 1 2 1 2 ⎤
⎢
⎛ 2π
⎞ ⎛ 4π
⎞
− j⎜
θ
⎥
1+
⎟ − j⎜
θ1+
⎟
−1
− jθ
( ) = ⎢ 1 ⎝ 3 ⎠ ⎝ 3 ⎠
T θ e e e ⎥
(3.85b)
⎢
⎛ 4π
⎞ ⎛ 2π
⎞
− j⎜
θ + ⎟ − ⎜ + ⎟ ⎥
2
j θ2
− jθ
⎢ 2 ⎝ 3 ⎠ ⎝ 3 ⎠ ⎥
⎣
e e e
⎦