Environmental Engineering Reference
In-Depth Information
transformation algorithm. For real original components the properties are such that
the second component is the conjugate complex of the first.
Original components
g
a
,
g
b
,
g
c
are transformed to components
g
s
,
g
z
=
g
s
,
g
0
by using the complex transformation matrix
T
s
:
For a non-rotating frame and with unchanged reference axis (direction a = real
axis), the transformation is defined by:
⎡
⎤
111
a
2
1
√
3
⎣
⎦
a
1
T
s
=
(6.7)
a
2
a
1
where
a
=
e
j
2π
/
3
;
a
2
=
a
∗
;1+
a
+
a
2
= 0
For a rotating frame, where the coordinate system is turned by an angle
ϑ
,the
transformation is defined by:
⎡
⎣
⎤
⎦
e
j
ϑ
e
−
j
ϑ
1
1
√
3
a
2
e
j
ϑ
a
e
−
j
ϑ
1
T
s
=
(6.8)
a
e
j
ϑ
a
2
e
−
j
ϑ
1
is often chosen to define a field vector as reference direc-
tion; the value of the reference vector is then a real. On the other hand, the reference
frame is fixed to the rotor of a machine rotating at instantaneous velocity
Note that in practice
ϑ
Ω
by
choosing:
=
ϑ
Ω
(
t
)
dt
(6.9)
6.2.1.6 Transformation into Symmetrical Components
(Fortescue Transformation)
The transformation transforms complex original quantities into complex symmet-
rical components
g
(1)
,
g
(2)
,
g
(0)
. The transformation matrix is the same as (6.6),
applicable to map original phasors of a three-phase system by positive-sequence,
negative-sequence and zero-sequence components.
This transformation is mentioned for completeness; its application is normally
for unsymmetrical states of distribution grids.
6.2.1.7 Transformations and Reference Frames
In practice the transformation matrices are often used in power-variant form. The
matrices for
0-transformation, dq0-transformation, sz0-transformation with non-
rotating and rotating frame and their respective inverse are given in Tables 6.1
αβ
Search WWH ::
Custom Search