Environmental Engineering Reference
In-Depth Information
at a constant symmetrical voltage, described in synchronous frame with constant
voltage. The stator circuit will be assumed in steady state, i.e. transients are ne-
glected. When neglecting also the stator resistance, the system is simplified to an
algebraic stator equation and a (complex) first-order differential equation for the
rotor:
d
ψ
r
dt
=
L
m
U
s
=
j
ω
ψ
s
;
[1 + j
τ
kr
ω
r
]
ψ
r
+
τ
kr
L
s
ψ
s
(6.24a)
s
In
dq
components, with
U
s
=
U
s
,wehave:
√
2
U
s
ω
re f
ψ
rd
ψ
rq
+
1
ψ
rd
ψ
rq
=
L
m
L
s
0
−
d
dt
−
τ
kr
ω
r
τ
kr
·
(6.24b)
τ
kr
ω
r
1
1
The electric torque is expressed equivalent to (6.14) as
√
2
U
s
ω
s
σ
3
2
z
p
L
m
T
el
=
−
L
r
·
ψ
rd
(6.25)
L
s
Together with the equation of motion (6.23) for a single inertia drive a second-
order system is obtained describing the transient machine model.
6.2.2.5 Model for Small Deviations from Steady State
In a number of problems, e.g. forced oscillations due pulsating load torque, the
quantities may be described by small deviations from a steady-state operation. A lin-
ear system in terms of deviations of flux
ΔΨ
r
and rotor pulsation
Δω
r
from a steady
state characterized by
Ψ
r0
and
ω
r0
(or
s
0
) may then be established:
ψ
r
0
=
−
j
√
2
U
s
d
Δψ
r
dt
1
[1 + jτ
kr
ω
r
0
] Δψ
r
+ τ
kr
=
−
j
τ
kr
ψ
r
0
Δω
r
where
ω
s
1 +
j
ω
r
0
τ
kr
(6.26)
The system is completed by the equation of motion which is, on the basis of
(6.22, 6.23) for small deviations:
√
2
U
s
ω
s
σ
−
τ
J
P
N
z
p
d
Δω
r
dt
3
2
z
p
L
m
=
Δ
T
el
+
Δ
T
L
where
Δ
T
el
=
−
Re(
Δψ
r
) (6.27)
L
s
L
γ
A simple solution of the linear system is obtained for steady state at no-load,
ω
r0
= 0. Using the notation of Laplace transform with
p
= differential operator, the
rotor frequency deviation as output with respect to a small load torque deviation is
described by the response:
Δω
r
Δ
ω
s
T
N
K
D
1 +
p
τ
kr
=
(6.28)
τ
J
/
K
D
+
p
2
T
L
1 +
p
τ
kr
τ
J
/
K
D
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