Environmental Engineering Reference
In-Depth Information
The following paragraphs describe the main transformations used for investigat-
ing electrical machine performance, in power-invariant form.
6.2.1.3 Transformation into
αβ
0
-Components (Clarke Transformation)
Original components g
a
,
g
b
,
g
c
are transformed to orthogonal components g
α
,
g
β
,
g
0
by using the real transformation matrix
T
α
:
⎡
⎤
1
/
√
2
1
0
=
2
3
√
3
/
2
/
√
2
⎣
⎦
−
1
/
2
T
α
(6.5)
−
√
3
/
2
/
√
2
−
1
/
2
The original components of the three-phase system are mapped by two compo-
nents of a two-phase system, plus a zero-sequence component.
The
αβ
0-transformation is usually applied for calculations in stator reference
frame.
6.2.1.4 Transformation into dq0-Components (Park Transformation)
Original components g
a
,
g
b
,
g
c
are transformed to orthogonal components g
d
,
g
q
,
g
0
by using the real transformation matrix
T
d
:
⎡
⎣
1
/
√
2
⎤
⎦
c
1
−
s
1
T
d
=
2
3
1
/
√
2
−
c
2
s
2
(6.6)
1
/
√
2
c
3
−
s
3
where
c
1
= cos
ϕ
;
c
2
= cos(
ϕ
−
2
π
/
3);
c
1
= cos(
ϕ
+ 2
π
/
3)
s
1
= sin
ϕ
;
s
2
= sin(
ϕ
−
2
π
/
3);
s
1
= sin(
ϕ
+ 2
π
/
3)
Similar as above, 2 components and the zero-sequence component are created;
however the coordinate system is turned by angle
ϕ
which for a rotating machine is
a time-dependent variable.
The dq0-transformation is usually applied for calculations in rotating reference
frame, especially when the reference frame is fixed to the rotor.
6.2.1.5 Transformation into Space Phasors
Space phasors or space vectors are complex quantities representing a two-phase
system;
plus
the
zero-sequence
component
required
to
ensure
a
reversible
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