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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