Environmental Engineering Reference
In-Depth Information
Accelerating (motoring) torque values are counted as positive. Comparison of
the moment of inertia of different sizes of systems is simplified when defining the
unit acceleration time, the time which would be required to bring the rotating parts
of a machine from rest to rated speed if the accelerating torque were constant and
equal to the quotient of rated active power by rated angular velocity:
ω
sN
z
p
2
Ω
N
2
P
N
τ
J
=
J
J
P
N
=
(6.16)
6.2.2.2 Modal Component Model
Now a new coordinate system is defined in which stator and rotor quantities
are referred to a single, rotating reference axis, with the advantage of eliminat-
ing the angular functions in (6.12) and (6.13). This may be done by using the
transformation:
i
1
i
2
=
e
j
ϕ
1
e
j
(ϕ
1
−
γ)
i
s
d
ϕ
1
dt
d
dt
0
where
=
ω
re f
;
=
ω
rot
=
z
p
Ω
(6.17)
0
i
r
rot
is the mechanical angular velocity of the rotor, multiplied by the
number of pole pairs. In the example of Fig. 6.1 the reference is the d-axis, also the
real axis for complex quantities.
Applying this transformation to (6.13), in which
Note that
ω
ω
ref
is an arbitrary angular speed
of the reference coordinate system and
ω
the rotational speed, we get:
u
s
0
=
R
s
0
0
R
r
i
s
¯
r
+
ψ
s
ψ
r
+
j
ω
re f
ψ
s
ψ
r
d
dt
0
(6.18)
0
(
ω
re f
−
ω
rot
)
where
ψ
s
ψ
r
=
L
s
L
m
L
m
L
r
i
s
i
r
The relevant equivalent circuit model is shown in Fig. 6.3; note the inductive
voltage drop components of transformer type (d
ψ
/
d
t
) and rotary type (j
ωψ
).
Fig. 6.3
Dynamic induction machine model
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