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,
            ɸ₂  = 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 cosɸ₂        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

  

 then,

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 I_r cosɸ₂ .

where, E_r = rotor emf per phase under running condition = sE₂.  (s=slip)

            I_r = rotor current per phase under running condition

reactance per phase under running condition will be  = sX₂

therefore,

 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.