Environmental Engineering Reference
In-Depth Information
β
θ
θ
β
c
c
q
α
2
v
c
v
α
v
β
v
α
v
v
b
s
b
β
v
θ
s
g
α
2
v
θ
V
q
V
d
c
d
v
=
v
θ
θ
a
α
=
a
v
α
a
g
a
v
α
=
a
α
b
b
(a) The
abc
frame
(b) The
αβ
frame
(c) The
dq
frame
Reference frames for three-phase
acb
systems: with
α
=
e
−
j
2
3
Figure 2.6
what is discussed in the natural frame still holds. There are two different spatial diagrams
corresponding to
e
−
j
2
3
e
j
2
3
, respectively, which are the same as the case with
α
=
and
α
=
the
abc
sequence.
For the
e
−
j
2
3
αβ
frame,
v
α
is still the same as
v
a
. The transition matrices
T
αβ
for
α
=
and
e
j
2
3
T
αβ
for
α
=
are not affected, either. However, the actual
β
-component
v
β
is affected, due
e
−
j
2
3
,itis
to the change of the phase sequence. For
α
=
√
3
3
v
β
=
−
v
b
+
v
c
)
(
√
3
V
m
3
cos
cos
3
3
2
2
=
θ
−
−
θ
+
=
V
m
sin
θ.
V
m
e
j
θ
and it rotates counterclockwise along the direction of the
spatial order of
acb
; see Figure 2.6(b). For
v
s
=
Hence, the space vector is ¯
e
j
2
3
, the
α
=
β
-component
v
β
is
v
β
=−
V
m
sin
θ
.
β
β
q
b
θ
b
θ
v
v
=
v
a
α
a
α
=
a
v
α
2
α
v
θ
=
a
a
α
θ
c
θ
g
v
V
q
V
d
s
α
v
b
β
d
α
v
θ
v
v
c
b
c
2
α
v
g
β
s
c
(a) The
abc
frame
(b) The
αβ
frame
(c) The
dq
frame
Reference frames for three-phase
acb
systems: with
α
=
e
j
2
3
Figure 2.7
(preferred)
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