Consider the approximate equivalent circuit as shown in the Fig.7
In this circuit, the exciting current Io is neglected hence the exciting no load branch is not shown.
... I1 = I2r'
The total impedance is given by,
ZT = (R1e + RL')+ where RL' = R2' (1-s)/s
I1 = V1 /√((R1e + RL')2 +(X1e)2)
The power supplied to the load i.e. Pout per phase is,
Per phase Pout = I12 RL' watts per phase
... Total = 3 I12 RL'
To obtain maximum output power, differentiate the equation of total Pout with respect to variable RL' and equal to zero.
But Z1e = √(R1e2 +X1e2) = Leakage impedance referred to stator
... Z1e2 = RL'2
Thus the mechanical load on the induction motor should be such that the equivalent load resistance referred to stator is equal to the total leakage impedance of motor referred to stator.
Slip at maximum Pout : This can be obtained as,
RL' = Z1e = R2'(1-s)/s where RL' = R2/K2
... s Z1e = R2' - sR2'
... s(Z1e + R2') = R2'
This is slip at maximum output.
Expression for maximum Pout : Using the condition obtained in expression of total Pout , we can get maximum Pout.
... (Pout)max = 3 I12 Z1e as RL' = Z1e
But R1e2 + X1e2 = Z1e
1.4 Maximum Torque
In case of induction motor, the speed of the motor decreases with increase in load. Thus the maximum power output is not obtained at a slip which corresponds to maximum torque. In the previous section we have seen the condition for maximum power output. In this section we will find the condition which gives maximum torque.
The expression for torque is given by,
The condition for maximum torque can be obtained from maximum power transfer theorem. When I2r'2 R2'/s is maximum consider the approximate equivalent circuit of induction motor as shown in The Fig. 8.
 |
| Fig. 8 |
The value of Ro is assumed to be negligible. Hence the circuit will be reduced as shown below.
 |
| Fig. 9 |
The thevenin's equivalent circuit for the above network is shown in the Fig.10 across the terminals x and y.
 |
| Fig. 10 |
The mechanical torque developed by rotor is maximum if there is maximum power transfer to the resistor R2'/s. This takes place when R2'/s equals to impedance looking back into the supply source.
This is the slip corresponding to the maximum torque. The maximum torque is given by,
Substituting,
From the above expression, it can be seen that the maximum torque is independent of rotor resistance.
1.5 Synchronous Watt
The torque produced in the induction motor is given by,
Thus torque is directly proportional to the rotor input. By defining new unit of torque which is synchronous watt we can write,
T = P2 synchronous-watts
If torque is given in synchronous-watts then it can be obtained in N-m as,
Key Point : Unit synchronous watt can be defined as the torque developed by the motor such that the power input to the rotor across the air gap is 1 W while running at synchronous speed.