Environmental Engineering Reference
In-Depth Information
The space vector is ¯
V
m
e
−
j
θ
and it rotates clockwise along the direction of the spatial
order of
acb
; see Figure 2.7(b). These are opposite to the cases with the phase sequence of
abc
.
For the
dq
frame, the frame still rotates synchronously with the space vector. For
v
s
=
e
−
j
2
3
,
α
=
as shown in Figure 2.6(c), the space vector is
v
β
)
e
−
j
θ
g
V
m
e
j
(
θ
−
θ
g
)
v
s
=
¯
V
d
+
jV
q
=
(
v
α
+
j
=
.
Hence, the rotation matrix is
T
dq
given in (2.18) and
V
d
V
q
V
m
cos(
θ
g
−
θ
)
=
.
−
V
m
sin(
θ
g
−
θ
)
e
j
2
3
, as shown in Figure 2.7(c), the space vector is
For
α
=
v
β
)
e
j
θ
g
V
m
e
j
(
θ
g
−
θ
)
v
s
=
¯
V
d
+
jV
q
=
(
v
α
+
j
=
.
The rotation matrix is
T
dq
given in (2.17) and
V
d
V
q
V
m
cos(
θ
g
−
θ
)
=
.
V
m
sin(
θ
g
−
θ
)
In summary, when the phase sequence is consistent with the spatial order of the phases, as
shown in Figures 2.4 and 2.7, the space vector always rotates clockwise and
V
q
=
V
m
sin(
θ
g
−
θ
). These are preferred. When the phase sequence is opposite to the spatial order of the phases,
the space vector always rotates counterclockwise and
V
q
=−
V
m
sin(
θ
g
−
θ
). In all cases,
V
d
=
).
It is worth noting that, in the literature, the
d
-axis is often the reversed
q
-axis here and the
q
-axis is the
d
-axis here. This is equivalent to rotating the
dq
frame here clockwise by
2
V
m
cos(
θ
g
−
θ
rad.
Denote the
dq
frame in the literature as the
DQ
frame, then
⎡
⎤
cos
2
sin
2
V
D
V
Q
V
d
V
q
0
V
d
V
q
−
1
10
−
⎣
⎦
=
=
.
sin
2
cos
2
That is,
V
D
=−
V
q
and
V
Q
=
V
d
.
As a result, the
D
-component
V
D
is associated with the reactive power component and the
Q
-component
V
Q
is associated with the real power component.
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