Environmental Engineering Reference
In-Depth Information
Air gap
R
s
jX
s
I
s
R
r
jX
r
V
s
(1 - S)R
r
/S
jX
m
Figure 4.15
Induction machine equivalent circuit referred to the stator
With the induction machine generating, the slip is negative and the notional resistance
R
r
[(1
s
)/
s
] is also negative. This is perfectly consistent with circuit analysis. A positive
RI
2
implies irreversible conversion of electrical energy into thermal energy. A negative
RI
2
implies
the conversion of some other type of energy (in this case mechanical) into electrical.
Irrespective of whether the machine is motoring or generating the mains to which the
machine is connected supplies the reactive voltamperes absorbed by all the inductive com-
ponents of the equivalent circuit. This must be so as only positive or negative real power is
associated with the mechanical/electrical energy conversion in the resistance
R
r
(1
−
s
)/
s
. The
consequence is that induction generators always absorb reactive power from the mains.
−
4.4.3 The Induction Machine Effi ciency
If the total electrical input power per phase fed into the stator is
P
s
, the power crossing the
motor's air gap i.e. the power per phase transferred from the stator to the rotor is
PPRI
r
=−
2
.
s
s
s
{
[
(
)
]
}
=
=
(
)
All of
P
r
is dissipated in
RR
+
1
−
ss
Rsso,
P RsI
2
. Hence the rotor copper
r
r
r
r
r
r
loss is:
RI
(4.21)
2
=
sP
rr
r
Subtracting the rotor copper loss from
P
r
gives the average per-phase mechanical power
PPRI
mr
=−
2
which through substitution from Equation (4.21) gives
r
r
(
)
P
=−
1
s P
(4.22)
m
r
The developed torque
Q
m
of the motor is its total mechanical power 3
P
m
divided by the
motor shaft speed
ω
r
. Therefore
3
ω
P
m
Q
i
=
(4.23)
m
r
Substituting Equations (4.16) and (4.22) into (4.23) gives
(
)
P
31
1
−
sP
s
ag
r
Q
=
=
(4.24)
m
(
)
−
ωω
s
s
where
P
ag
=
3P
r
is the total three-phase power crossing the air gap.
Neglecting the stator copper losses and rotating mechanical losses, the effi ciency of an
induction motor is given approximately by:
