Thus the output torque of an induction motor is given by
Pmech
1 2 Rr
1 Rr
5
Ir ð1 2 sÞ 5 Ir2
ωmech s
ωs s
ωmech
Tmech 5
(3.32)
where ωmech 5 ð1 2 sÞωs and ωs represents the synchronous angular frequency.
We need to express this torque equation in terms of the input voltage instead
of a nonmeasurable rotor current. The rotor current can be easily obtained from
the approximate equivalent circuit given in Fig. 3.26, in which the core loss
resistance is ignored and the magnetizing inductance is shifted to the input side
assuming Rs 1 jXls {ωs Lm .
From this circuit, the rotor current is given by
Ir 5 V
s
Rr
Rs 1
1 jðXls 1 Xlr Þ
s
(3.33)
By substituting this rotor current into Eq. (3.32), the output torque (per phase)
is expressed as
Torque: Tmech 5
1
Vs2
Rr
2
ωs
s
Rr
Rs 1
1 jðXls 1Xlr Þ2
s
(3.34)
This is the steady-state average torque per phase for a given input voltage Vs ,
which is a function of the slip. The torque for a given slip is proportional to the
square of the stator input voltage. The torque calculated by using this approximate
circuit varies within 5% from that of the full circuit [2]. The shaft torque, which
is available to the load at the shaft of the motor, can be obtained by subtracting
the friction and windage torques from this output torque.
Let us examine the speedtorque characteristics of an induction motor based
on Eq. (3.34). In the low-slip region, which is the normal operating range of
an induction motor, the impedance of equivalent circuit parameters shows the
following relations as
Rs 1
Rr
cXls 1 Xlr
s
and
Rr
cRs
s
(3.35)
Thus the torque of Eq. (3.34) can be simplified as
Tmech 1 Vs2
s - Tmech ~ s
ωs Rr
(3.36)
In this case, it is noteworthy that the output torque increases linearly
with the slip s. On the other hand, in the low-speed range with larger values of
the slip,
Rs 1
Rr
{Xls 1 Xlr
s
(3.37)
Thus the torque varies almost inversely with the slip s as
Tmech 5
1
Vs2
Rr
1
- Tmech ~
ωs ðXls 1Xlr Þ2 s
s
(3.38)
There is a speed at which the maximum torque, often referred to as pull-out
torque or breakdown torque, is developed. The slip at the maximum torque can
be obtained by solving dTmech =ds 5 0 as
Rr
smax 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
Rr 1 ðXls 1Xlr Þ2
(3.39)
It is important to note that the rotor resistance Rr is an important parameter
in determining the slip at which the maximum torque occurs. From smax ,
the maximum torque can be given as
Tmax 5
1
Vs2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ωs R 1 R2 1 ðX 1X Þ2
s
ls
lr
s
(3.40)
The value of the maximum torque is independent of the rotor resistance,
which determines the maximum slip value. The speedtorque curve of a typical
three-phase induction motor is shown in Fig. 3.27. This curve shows the speed
versus the output torque when an induction motor is started with full voltage.
The normal operating range of an induction motor is near the synchronous speed,
confined to less than 5% slip. In this low-slip region, the output torque increases
linearly with the increasing slip. The slip increases approximately linearly with the
increased load, and thus the rotor speed decreases approximately linearly with
the load.
3.1.4.4 Stable operating point
When a motor is driving a mechanical load, the motor will operate in a steady
state at a speed at which the torque developed by the motor is equal to the torque
required by the load. As it can be seen from the motion equation in Eq. (3.41),
when the motor torque Tmotor exceeds the load torque Tload , the motor speed will
increase. When the motor torque is less than the load torque, the motor speed
will decrease. Thus the equilibrium point will be the speed at which the motor
torque equals to the load torque.
Tmotor 2 Tload 5 ðJmotor 1 Jload Þ
dω
dt
(3.41)