Environmental Engineering Reference
In-Depth Information
Table 6.1
Transformation matrices in the power-variant form I
×
i
α
i
0
T
[
i
d
i
q
i
0
]
T
[
i
s
i
z
i
0
]
nr
[
i
s
i
z
i
0
]
r
i
β
×
×
×
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
1
0
1
e
j
ϑ
e
−
j
ϑ
i
a
i
b
i
c
c
1
−
s
1
1
112
a
2
2
1
/
2
√
3
/
21
1
2
1
2
a
2
e
j
ϑ
a
e
−
j
ϑ
−
c
2
−
s
2
1
a
2
2
√
3
/
21
a
2
c
3
−
s
3
1
a
2
a
e
j
ϑ
a
2
e
−
j
ϑ
−
1
/
2
−
2
c, s
are as in (6.6)
ϑ
is e.g. as in (6.9)
Table 6.2
Transformation matrices in the
power variant form II
[
i
a
i
b
i
c
]
T
×
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
i
i
i
0
1
−
1
/
2
−
1
/
2
0
√
3
/
2
−
√
3
/
2
1
/
21
/
2
/
2
2
3
=
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
i
d
i
q
i
0
c
1
c
2
c
3
−
s
1
−
s
2
−
s
3
1
/
2
/
2
/
2
2
3
=
⎡
⎣
⎤
⎦
nr
⎡
⎣
⎤
⎦
1
a a
2
1
a
2
a
1
/
2
/
2
/
2
i
s
i
z
i
0
2
3
=
⎡
⎣
⎤
⎦
r
⎡
⎣
⎤
⎦
e
−
j
ϑ
a
e
−
j
ϑ
a
2
e
−
j
ϑ
i
s
i
z
i
0
2
3
e
j
ϑ
a
2
e
j
ϑ
ae
j
ϑ
=
1
/
2
/
2
1
/
2
and 6.2 in the power-variant version. The tables indicate how original components
are deduced from modal components, and vice versa, for the currents as example.
Transformation matrices to connect different modal systems may be deduced from
the Tables; they are listed in [IEC62428].
Note that the matrices regarding
0- and dq0-transformation consist of real
elements. The two-phase components may, however, be combined in complex form,
e.g.
g
=
g
α
+ j
g
β
.
In the power-variant form (6.4) is no longer valid, and specific constant factors
have to be applied for power calculation from modal components.
Figure 6.1 illustrates the application to AC machines in general. Figure 6.1a
models a cross-section of a two-pole, three-phase machine with a smooth air-gap
between stator and rotor. The stator carries the windings a, b, c; with 120
◦
displace-
ment between any two of them. On the rotor also a three-phase system with the
windings k, l, m is assumed. In the figure the rotor position angle is
αβ
γ
, the electrical
angle between the reference axes a of the stator and k of the rotor.
Figure 6.1b shows the coordinates as assigned to the modal components defined
above. The Clarke transformation (6.4) assigns stator components
to the orig-
inal components a, b, c. The reference axis of the coordinate system remains the
α
,
β
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