Torque equation of three phase induction motor

Torque of a three phase induction motor is proportional to flux per stator pole, rotor current and the power factor of the rotor.

T   ɸ I2 cosɸ2      OR      T = k ɸ I2 cosɸ2 .
where, ɸ = flux per stator pole,
            I2 = rotor current at standstill,
            ɸ = angle between rotor emf and rotor current,
             k = a constant.

Now, let E2 = rotor emf at standstill
we know, rotor emf is directly proportional to flux per stator pole, i.e. E2 ɸ.
therefore,  T E2 I2 cosɸ       OR      T =k1 E2 I2 cosɸ2.

Starting torque

The torque developed at the instant of starting of a motor is called as starting torque. Starting torque may be greater than running torque in some cases, or it may be lesser.
We know, T =k1 E2 I2 cosɸ2.

let, R2 = rotor resistance per phase
      X2 = standstill rotor reactance

Therefore, starting torque can be given as,
The constant k1 = 3 / 2πNs

 Condition for maximum starting torque

If supply voltage V is kept constant, then flux ɸ and E2 both remains constant. Hence,
Hence, it can be proved that maximum starting torque is obtained when rotor resistance is equal to standstill rotor reactance. i.e. R22 + X22 =2R22 .

 Torque under running condition

T  ɸ Ir cosɸ2 .
where, Er = rotor emf per phase under running condition = sE2.  (s=slip)
            Ir = rotor current per phase under running condition
reactance per phase under running condition will be  = sX2
 as, ɸ ∝ E2.

Maximum torque under running condition

Torque under running condition is maximum at the value of slip (s) which makes rotor reactance per phase equal to rotor resistance per phase.