Environmental Engineering Reference
In-Depth Information
In case of permanent magnet (PM) excited machines the notion of an impressed
exciter current is used, and
ψ
p
=
L
md
i
f
is defined as the inductor pole flux, which is obviously a constant for a given
machine.
Now the expression for the torque becomes:
T
el
=
3
ψ
d
−
2
z
p
(
i
q
i
d
ψ
q
)
(6.34)
In grid operation, when the machine is fed from a symmetrical system of fre-
quency
and d, q compo-
nents is described by (6.19), where the definition contains the load angle
ω
s
= 2
π
f
, the voltage component representation in
α
,
β
, which
in steady-state operation is a constant, due to the then synchronous rotor speed.
u
u
ϑ
cos(
u
d
u
q
=
√
2
U
s
sin
;
=
√
2
U
s
ω
s
t
)
ϑ
s
t
+
2
(6.35)
·
→
·
γ
−
ω
where
ϑ
=
sin(
ω
s
t
)
cos
ϑ
β
6.2.3.2 Transient Model
A suitable means to calculate several non-steady regimes of a synchronous machine
is the transient model. The approach is to consider steady state stator equations,
which means neglecting transients decaying with short-circuit time constants. A fur-
ther simplification is made by neglecting the stator resistance voltage drop, giving:
U
d
U
q
=
−
ψ
q
ψ
d
ω
s
In the remaining non-linear first order rotor differential equation the transient
inductor e.m.f., usually called
e
p
, is chosen for state variable:
X
d
u
p
+
1
de
p
dt
+
e
p
X
d
X
d
X
d
τ
d
−
·
U
s
cos
ϑ
;
(6.36)
where
u
p
=
u
f
X
md
R
f
ω
s
X
md
e
p
=
;
X
f
ψ
f
and
X
d
,
τ
d
are as defined with (6.47).
The parameters are now, besides the direct axis synchronous reactance
X
d
=
s
L
d
, the direct axis transient reactance
X
d
=
s
L
d
and direct axis transient time-
ω
ω
τ
d
. The electromagnetic torque becomes:
constant
U
s
·
1
X
d
−
sin(2
)
e
p
X
d
U
s
2
3
z
p
2
1
X
q
T
el
=
−
·
sin
ϑ
−
ϑ
(6.37)
ω
s
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