Environmental Engineering Reference
In-Depth Information
β
θ
β
θ
q
c
c
α
2
v
2
α
v
V
q
c
c
v
v
α
v
β
v
β
α
v
v
α
v
b
s
b
s
b
α
2
v
α
=
a
c
θ
v
v
=
v
θ
α
=
a
a
α
v
V
d
α
a
θ
g
a
d
θ
g
b
b
(a) The
abc
frame
(b) The
αβ
frame
(c) The
dq
frame
Reference frames for three-phase
abc
systems: with
α
=
e
−
j
2
3
Figure 2.4
(preferred)
for a balanced system, which means that there are only two independent variables and that the
three-phase balanced voltages can be expressed on a two-dimensional space.
Another possible way to spatially distribute the three phases
a
,
b
and
c
is shown in Figure
2.5(a), with
e
j
2
3
. It is worth noting that the spatial order of the phases should not be
confused with the phase sequence. They may or may not be the same.
α
=
2.3.2
) Frame
Taking into account the spatial feature of the voltages on a spatial diagram, the vectors having
the instantaneous voltage values in the natural frame (2.11) can be adopted to form a space
vector
Stationary Reference (
αβ
2
v
s
=
¯
k
(
v
a
+
αv
b
+
α
v
c
)
,
(2.12)
β
β
q
θ
b
b
θ
θ
d
g
V
d
v
v
=
v
θ
v
a
α
a
g
α
θ
θ
α
2
v
=
a
α
=
a
a
α
c
v
s
v
α
v
v
s
α
b
β
β
V
q
α
v
b
c
b
c
2
α
v
2
α
v
c
c
(a) The
abc
frame
(b) The
αβ
frame
(c) The
dq
frame
Reference frames for three-phase
abc
systems: with
α
=
e
j
2
3
Figure 2.5
Search WWH ::
Custom Search