**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 I

_{2}cosɸ

_{2}OR T

**= k**ɸ I

_{2}cosɸ

_{2}.

where, ɸ = flux per stator pole,

I

_{2}= rotor current at standstill,

ɸ

_{2 }= angle between rotor emf and rotor current,

k = a constant.

Now, let E

_{2}= rotor emf at standstill

we know, rotor emf is directly proportional to flux per stator pole, i.e. E

_{2}

**∝**ɸ.

therefore, T

**∝**E

_{2}I

_{2}cosɸ

_{2 }OR T =k

_{1}E

_{2}I

_{2}cosɸ

_{2}.

### 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 =k

_{1}E_{2}I_{2}cosɸ_{2}.
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 E

_{2}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. R_{2}^{2}+ X_{2}^{2}=2R_{2}^{2 }.#### Torque under running condition

T

**∝**ɸ I_{r}cosɸ_{2 }.
where, E

_{r}= rotor emf per phase under running condition = sE_{2}. (s=slip)
I

_{r}= rotor current per phase under running condition
reactance per phase under running condition will be = sX

_{2}
therefore,

as, ɸ

**∝ E**_{2}.####
**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.