Research on Vector Control and PWM Technique of Five-Phase PMSM - Intelligent Computing for Sustainable Energy and Environment - page 268

Where θ i = θ

iα , α =2 / 5 π . Ψ m 1 is the amplitude of fundamental flux, Ψ m 3 the
amplitude of third harmonic flux.

−

2.2 Transformation Matrix

The static decoupling transformation matrix on vector space of the five-phase
motor is

⎡
⎣

⎤
⎦

1 cosα cos 2 αcos 3 αcos 4 α
0 sinα sin 2 αsin 3 αsin 4 α
1 cos 3 αcos 6 αcos 9 αcos 12 α
0 sin 3 αsin 6 αsin 9 αsin 12 α

(0) = 2
5

T

(4)

1 / 2 1 / 2 1 / 2 1 / 2 1 / 2

The variables are mapped to the static coordinate α 1- β 1 subspace and α 3- β 3
subspace. The amplitude doesn't change after the transformation. The fifth vari-
able is zero because the neutral point is isolated.
The variables in the static coordinate are needed to be transformed into ones
in rotating coordinate. The rotating transformation matrix is

⎡ ⎤

cosθ sinθ 0 0 0

⎣ ⎦

− sinθ cosθ 0 0 0

R ( θ )= 0 0 cos 3 θsin 3 θ 0 (5)

0 0 − sin 3 θcos 3 θ 0

0 0 0 0 1

( θ ), the rotating transformation matrix between natural
coordinate and rotating coordinate is

T R

From the (0) and

⎡
⎣

⎤
⎦

cosθ 0 cosθ 1 cosθ 2 cosθ 3 cosθ 4

sinθ 4
cos 3 θ 0 cos 3 θ 1 cos 3 θ 2 cos 3 θ 3 cos 3 θ 4
−

− sinθ 0 − sinθ 1 − sinθ 2 − sinθ 3 −

(0) = 2
5

T ( θ )= R ( θ ) T (6)

sin 3 θ 0 − sin 3 θ 1 − sin 3 θ 2 − sin 3 θ 3 − sin 3 θ 4

1 / 2 1 / 2 1 / 2 1 / 2 1 / 2

2.3 Model in Rotating Coordinate

The voltage equation in rotating coordinate of five-phase PMSM is

R s i dqs + d u dqs = T ( θ ) u s =

dt Ψ dqs − Ω dqs Ψ dqs

(7)

where u dqs is the rotating voltage matrix,

u dqs = T ( θ ) u s =[ u d 1 u q 1 u d 3 u q 3 u 0 ]

i dqs is the rotating current matrix,

i dqs = T ( θ ) i s =[ i d 1 i q 1 i d 3 i q 3 i 0 ]