Performance and Operation Management - Wind Energy Systems for Electric Power Generation - page 119

Table 6.3 Properties of preferred transformations
Coordinate system

ϕ 1 ϕ 2 ω ( ω − ω rot ) ref ref

fixed to the stator 0 − γ 0 − ω rot

fixed to the rotor γ 0 ω rot 0

fixed to synchronous vector 2 π ft (2 π ft − γ ) 2 π f s · 2 π f

In practice, only a limited number of transformations according to (6.17) are of
practical interest, see Table 6.3.
Additionally, in drive technology transformations are used where the reference
axis is fixed to stator or rotor flux vector, see (6.21). Note that the special case of
steady-state operation of the induction machine is included in the equations.
When the machine is fed by a symmetrical three-phase voltage of frequency
ω s = 2 π ft , the voltage vector in αβ 0 components fixed to stator is in the power-

variant form:

⎡ ⎤

⎡ ⎤

⎦ = √ 2 U s

⎡ ⎤

u u
u 0

u a ( t )
u b ( t )
u c ( t )

cos( ω s t )

⎣ ⎦

⎣ ⎣ ⎦ →

cos( ω

s t

− 2 π / 3)

cos( ω

s t + 2

π / 3)

⎡ ⎤

cos( ω s t )

= √ 2 U s u s = √ 2 U s ·

⎣ ⎦ →

e j ω s t

sin( ω s t )

0

(6.19a)

In an arbitrary coordinate system rotating at an angular frequency ω ref the trans-

formed voltage is, also in power-variant form:
⎡ ⎤

⎦ = √ 2 U s

⎡ ⎤

u d
u q
u 0

cos[ ( ω s − ω re f ) t ]

u s = √ 2 U s · e j ( ω s − ω re f ) t

⎣ ⎣ ⎦ →

sin [ ( ω s − ω re f ) t ]

0

(6.19b)

Note that i n synchronous rotating frame, ω ref = ω s , the voltage vector is a con-

stant: u s = √ 2 U s . In steady state operation, the derivatives of fluxes in the model
(6.18) vanish, d / d t = 0. In this case the model is a proper representation of the

steady-state model described in Sect. 3.2.2.

ψ

6.2.2.3 Model in Field-Oriented Components

In induction machine drive concepts often a flux is chosen as a controlled variable,
either the stator or the rotor flux. In field-oriented control this is preferably the rotor
flux. Considering a machine controlled to run with impressed stator current, a first-
order differential equation for the rotor flux as state variable is obtained:

τ 0 r i s = 1 τ 0 r + j ω re f − ω rot

ψ r + d

ψ r
dt

L s

T 0 r = L r
R r

= T kr
σ

where (6.20)