Environmental Engineering Reference
In-Depth Information
⎡
⎤
⎡
⎤
⎡
⎤
g
M1
g
M2
g
M3
g
1
g
2
g
3
t
11
t
12
t
13
⎣
⎦
=
⎣
⎦
⎣
⎦
t
21
t
22
t
23
(6.1)
t
31
t
32
t
33
or in shortened form
g
M
The coefficients
t
tk
of the transformation matrix can be real or complex. It is
necessary that the transformation matrix
T
is non-singular, so that the inverse rela-
tionship is valid:
g
=
T
·
g
M
=
T
−
1
·
g
If the relationship between the original quantities and the modal components is
introduced for voltages as well as for currents, then:
u
=
T u
M
;
i
=
T i
M
Transformations mainly used in the field of electrical machines are the Clarke
(
αβ
0-), the Park (dq0) and the space-phasor transformation.
6.2.1.2 Power-Invariant and Power-Variant Transformation
The power
p
expressed in terms of the original quantities is:
⎡
⎣
⎤
⎦
i
1
i
2
i
3
p
=
u
1
i
1
+
u
2
i
2
+
u
3
i
3
=(
u
1
u
2
u
3
)
=
u
T
i
∗
(6.2)
where
i
∗
denotes the conjugate complex value of
i
.
In terms of modal components the instantaneous power is expressed by:
p
=
u
M
(
T
T
T
∗
)
i
∗
M
(6.3)
The transformation is power-invariant, if
T
is chosen so that (
T
T
T
∗
)=
E
, where
E
is the unity matrix; consequently the power-independent transformation matrix
T
is unitary, so that
T
−
1
=(
T
T
)
∗
(6.4)
On the other hand, in the power-variant forms of transformation, also known
as reference-component-invariant transformations, the properties of
T
are such that
under balanced symmetrical conditions the reference component of the modal com-
ponents is equal to the reference component of the original quantities. In case of the
Clarke transformation this means (
T
T
T
∗
)=3
/
2
·
E
, see Table 6.1.
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