Direct Power and Torque Control of AC/DC/AC Converter-Fed ...

Appendices
k - space vector described in fixed system of coordinates αβ ,
jγ
k = ke ; γ - angle
between complex space vector and α axe of stationary system of coordinates (see
Fig. A. 1. 1).
k
w
- space vector described in rotating system of coordinates with angular
frequency Ω
w
,
( γ −γ )
j w
k = .
w
ke
It is convenient to define rotating system of coordinates in special condition when,
Ω
w
= Ω m
is an angular speed of the rotor. This system of coordinates is known in
literature as dq (here is marked by
dq
m
) system:
k
w
= k
dq m
= k +
d m
jk
q m
Where
(A.1. 6)
dm
is the real axis and coincides with the direction of rotor phase a , and
q
m
the imaginary axis:
k
d m
= Re[ k ], k Im[ k ]
dq m
qm
= (A.1. 7)
dq m
It is also convenient to define rotating system of coordinates with arbitrary
angular speed Ω
K
.
Let consider that space vector k
dq m
in rotating system of coordinates dqm
is
known. It will be transformed into a rotating system of coordinates with angular
dγ
frequency Ω K
K
= , where γ
K
is an angle between real axis α of the fixed system
dt
of coordinates and real axis of the rotating one with arbitrary angular speed.
Therefore, space vector
k
dq m
can be described in rotating system of coordinates with
angular frequency Ω
K
with taking into account rotating of the both system:
( γ −γ )
− j K m
k K
= k
(A.1. 8)
dqm
e
And in respect to stationary αβ system:
k K
( γ K −γ
m ) − jγ
m − j( γ K −γ
m ) j( γ −γ
m ) − j( γ K −γ
m ) j( γ −γ
K ) − jγ
K
− j
= k e = ke
e = ke e = ke = ke
,
dq
m
Where:
k
K
(A.1. 9)
- is a space vector k described in rotating system of coordinates with arbitrary
angular frequency Ω
K
.
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