Environmental Engineering Reference
In-Depth Information
stator terminal voltage to the sum of ohmic and inductive voltage drops; capacitive
voltage drops are negligible for modelling the low frequency behaviour. The induc-
tive components are described by the time-derivative of the flux linkages.
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
u
u
0
0
R
s
000
0
R
s
00
00
R
r
0
000
R
r
i
i
i
A
i
B
ψ
α
ψ
β
ψ
A
ψ
B
d
dt
=
+
(6.12)
where
⎡
⎤
⎡
⎤
⎡
⎤
ψ
α
ψ
β
ψ
γ
−
i
i
i
A
i
B
L
s
0
L
m
cos
L
m
sin
γ
⎣
⎦
⎣
⎦
⎣
⎦
0
L
s
L
m
sin
γ
L
m
cos
γ
=
L
m
cos
γ
L
m
sin
γ
L
r
0
A
−
L
m
sin
γ
L
m
cos
γ
0
L
r
ψ
B
depend on the winding currents
i
. The coefficient matrix
contains the self-inductances
L
s
,
L
r
, and the magnetizing inductance
L
m
. Conven-
tionally, the self-inductances are the sum of magnetizing inductance and leakage
components not coupled with any other winding:
The flux linkages
ψ
L
s
=
L
m
+
L
σ
s
;
L
r
=
L
m
+
L
σ
r
.
The mutual inductances between stator and rotor consist of
L
m
multiplied by
trigonometric functions of the rotor position angle
.
When expressing stator and rotor quantities by space phasors
g
s
=(
g
α
+ j
g
β
)
and
g
r
=(
g
A
+ j
g
B
), the voltage equations assume a shortened form:
u
1
0
γ
=
R
s
0
0
R
r
i
1
i
2
+
ψ
1
ψ
2
d
dt
(6.13)
where
ψ
1
ψ
2
=
L
s
i
1
i
2
e
j
γ
L
m
·
e
−
j
γ
L
m
·
L
r
The electromagnetic torque, also called the air-gap torque is calculated from
known space phasors:
Im
i
1
i
2
·
e
−
j
γ
=
3
T
el
=
3
Im[
i
1
ψ
1
]
2
z
p
L
m
·
2
z
p
·
(6.14)
Assuming a rotating system of one inertia
J
, the equation of rotation equals the
acceleration torque with the sum of air-gap torque and load torque:
d
2
J
z
p
γ
dt
2
=
T
el
+
T
L
(6.15)
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