Environmental Engineering Reference
In-Depth Information
(
)
η ===
−
P
P
P
P
1
sP
P
out
m
r
(
)
=−
1
s
(4.25)
For an induction generator (
s
negative) the power fl ow is in the reverse direction hence
approximately
in
r
r
P
P
P
P
1
out
r
η ===
−
(4.26)
(
)
1
s
in
m
Equations (4.25) and (4.26) indicate that for the conversion effi ciency to be high,
s
at full
load must be as small as possible.
Real induction generators have losses that have not been taken into account in this simpli-
fi ed analysis. The mechanical power available to produce electricity is reduced by windage
and other mechanical frictional losses within the generator. Additionally, electrical and mag-
netic losses within the rotor reduce the power that is transferred from the rotor across the air
gap to the stator. Finally, in the stator there are more losses associated with the winding
resistance and the setting up of the magnetic excitation in the shunt branch of the equivalent
circuit. As a consequence, large induction generators have effi ciencies in the region of 90%.
These extra losses will be referred to in a later section.
4.4.4 The Induction Machine Speed - Torque Characteristic
An important characteristic of any electromechanical converter is its speed-torque relation-
ship. For the induction machine the developed torque
Q
m
from Equations (4.23), (4.22) and
(4.16) is
(
)
3
P
3 1
1
−
s
P
3
P
m
r
r
Q
=
=
=
m
(
)
ω
−
s
ω
ω
r
s
s
Substituting for
2
R
s
RV
sZ
r
rs
P
=
I
2
=
r
s
2
and using the equivalent circuit of Figure 4.15 we get
2
2
R
s
I
3
R
⎡
⎢
V
RRs
⎤
⎥
⎛
⎝
⎞
⎠
r
s
r
s
Q
=
3
=
(4.27)
m
2
2
ωω
s
(
)
++
(
)
+
XX
s
s
s
r
s
r
This relationship is nonlinear and can be generalized to describe typical performances of
an induction machine by normalizing it in terms of torque and speed. Taking as normal torque
the rated torque and as normal speed the synchronous speed, the normalized relationship of
Equation (4.27) for a typical induction generator is plotted in Figure 4.16 with
R
r
as a
parameter.
Note that, as for the synchronous machine, there is a maximum or
pullout
torque beyond
which the generator will accelerate uncontrollably. However this condition is far away from
the normal operating regime. The curve for rotor resistance '
R
r
' represents the performance
of a typical induction machine with low rotor resistance and shows that the variation in speed
from zero input torque to rated torque varies by about 3-4%. For s small,
R
r
/s
is large com-
pared to
R
r
and
X
s
+
X
r
and to a good approximation Equation (4.27) can be written as
