Environmental Engineering Reference
In-Depth Information
where
L
s
=
L
+
M
, and the field flux linkage can be rewritten as
f
=
L
f
i
f
+
M
f
i
,
cos
θ
,
(18.2)
where
·
,
·
denotes the conventional inner product. The second term
M
f
i
,
cos
θ
is constant
if the three phase currents are sinusoidal (as functions of
) and balanced. Assume that the
resistance of the stator windings is
R
s
, then the phase terminal voltages
θ
v
c
]
T
v
=
[
v
a
v
b
can
be obtained from (18.1) as
d
dt
d
i
dt
+
v
=−
R
s
i
−
=−
R
s
i
−
L
s
e
,
(18.3)
e
c
]
T
where
e
=
[
e
a
e
b
is the back EMF due to the rotor movement given by
M
f
d
i
f
θ
sin
˙
e
=
M
f
i
f
θ
−
dt
cos
θ.
(18.4)
Similarly, according to (18.2), the field terminal voltage is
d
f
dt
,
v
f
=−
R
f
i
f
−
(18.5)
where
R
f
is the resistance of the rotor windings. However, this is not used here because the
field current
i
f
, instead of
v
f
, is used as an adjustable constant input. This completes modelling
the electrical part of the machine.
18.1.2 Mechanical Part
The mechanical part of the machine is governed by
J
¨
D
p
˙
θ
=
T
m
−
T
e
−
θ,
(18.6)
where
J
is the moment of inertia of all parts rotating with the rotor,
T
m
is the mechanical
torque,
T
e
is the electromagnetic toque and
D
p
is a damping factor.
T
e
can be found from the
energy
E
stored in the machine magnetic field,
i.e.
,
1
2
i
1
2
i
f
f
=
,
+
E
1
2
1
2
i
f
(
L
f
i
f
+
=
i
,
L
s
i
+
M
f
i
f
cos
θ
+
M
f
i
,
cos
θ
)
1
2
1
2
L
f
i
f
.
=
i
,
L
s
i
+
M
f
i
f
i
,
cos
θ
+
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