Environmental Engineering Reference
In-Depth Information
which can be decomposed into a real part
v
α
and an imaginary part
v
β
, as shown in Figure
2.4(b). That is,
v
s
=
v
α
+
¯
j
v
β
.
Note that the instantaneous values of
v
a
,
v
b
and
v
c
, not their phasors, are used to form the
v
s
. As a result, both
v
α
v
β
space vector ¯
and
are all functions of time (and frequency) as well.
e
−
j
2
3
2
For
α
=
and
k
=
3
, the voltage space vector (2.12) is
2
v
s
=
¯
k
(
v
a
+
αv
b
+
α
v
c
)
k
cos
2
k
cos
4
3
−
j
sin
2
3
3
−
j
sin
4
3
=
k
v
a
+
v
b
+
v
c
jk
√
3
2
v
b
+
√
3
2
v
c
k
2
v
c
1
2
v
b
−
1
=
v
a
−
+
−
.
Hence,
⎡
⎤
v
α
v
β
v
a
v
b
v
c
⎣
⎦
=
T
αβ
(2.13)
with the
abc
→
αβ
transformation matrix
⎡
⎤
1
2
1
2
1
−
−
⎣
⎦
.
2
3
√
3
2
√
3
2
T
αβ
=
(2.14)
0
−
The reference frame with
v
α
and
v
β
as coordinates is often referred to as the stationary
reference frame or the
αβ
frame. The transformation (2.13) is called the Clarke transform or
the
abc
→
αβ
transformation. It transforms the voltages in the natural frame into the stationary
frame or the
frame.
For balanced systems,
αβ
v
a
+
v
b
+
v
c
=
0. Hence,
⎡
⎤
v
α
v
β
v
a
√
3
3
⎣
⎦
.
=
(
−
v
b
+
v
c
)
The
α
-component in the
αβ
frame is always the same as
v
a
in the
abc
frame. In other words,
v
a
remains unchanged and stationary in the
αβ
frame. Moreover, according to (2.11),
√
3
V
m
3
cos
cos
2
3
2
3
v
β
=
θ
+
−
θ
−
=−
V
m
sin
θ.
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