Performance and Operation Management - Wind Energy Systems for Electric Power Generation - page 123

a) b)

Fig. 6.5 Synchronous machine model ( a ) Salient pole three-phase machine; ( b ) Transformed
windings arrangement

where
⎡
⎣

L d cos 2

γ ( L d −

⎤
⎦ =

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

+ L q sin 2

ψ α
ψ β
ψ

γ L q ) sin γ cos γ L md cos γ i i i f

γ L d sin 2 γ L md sin

+ L q cos n 2 ( L d − L q ) sin γ cos γ γ

L md cos γ L md sin γ L f f

The electromagnetic torque becomes:

2 z p i β ψ α −

i α ψ β T el = 3 (6.31)

The equation of motion of the single-inertia model is (6.15), same as for the
asynchronous machine.
The transformation into Park components follows the definition:
i i β

= cos i d

i q

i d
i q

= cos i i β

γ −

sin γ γ sin γ

↔ (6.32)

−

sin γ cos γ sin γ cos γ

Using the definition for quantities i, u and ψ , the following voltage equation is

obtained:

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎦ + d

⎡ ⎤

u d
u q
u f

R s 00
0 R s 0
00 R f

i d
i q
i f

ψ d
ψ q
ψ f

− ψ q
ψ d
0

d
dt

dt

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ + ⎣ ⎣ ⎦

(6.33)

where

ψ d
ψ f

= L d i d

i f

L md

; ψ d = L q i q

L md L f