Torque in induction motor under running condition, condition for maximum torque at running and maximum torque equation.

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The torque in induction motor is proportional to the product of flux per stator pole and the rotor current one more term we will take in account is power factor of the rotor.

  T →  φ  *  I₂  * cos∅₂ 

Where φ  - flux per status pole
I₂ - rotor current
cos∅₂ - rotor power factor
∅₂ phase angle  rotor current and rotorEMF.

The induced rotor EMF E2 is proportional to flux per stator φ.

Therefore T = K1 * E₂ *  I₂ * cos φ2 

Where K1 is another constant.

This is torque at standstill condition.
 
Under running condition induced EMF is

T = K1 * Er *  Ir * cos ∅2 ---------------------- (1)

Where Er = rotor induced EMF per phase under running condition.

Ir = rotor current per phase under running condition.

φ = flux per stator pole.

rotor current,
Ir = Er / Zr = sE2/  √( R2² + (sX2)²)  ----------- (2)

power factor,
cos∅2 = R2 / √( R2² + (sX2)²)   ----------------(3)



So the rotor torque is, 
After putting Ir from equation (2) and cos∅2 from equation (3) in equation number (1), we we get the torque equation

T = K1 * φ * sE2 * R2/( R2² + X2²)  --------------(4)

The condition of maximum torque may be obtained if it differentiated equation number (4 )with respect to s.


X = 1/T = ( R2² + X2²) / (K1 * φ * sE2 * R2)

Since E2 is proportional to φ

X = 1/T = ( R2² + X2²) / (K1 * s E2² * R2)

After solving the above equation we get,

R2 = sX2                         ------------------ (5) 

Hence under running condition torque is maximum at that value of slip which makes rotor resistance per phase equal to rotor reactance per phase.
Hence slip at T maximum is
Smax = R2 / X2 

After putting equation (5) in equation (4)

T max= K1 E2 / (2 X2)      ---------------------       ( 6)
 
Or 

Tmax = T = K1 sE2²/( 2R2) -------------------      (7)

From equation number (6) it is clear that maximum torque is independent of rotor resistance.
But at the same time maximum torque is inversely proportional to rotor reactance per phase so it's should be a small as possible.
Although maximum torque is not dependent upon total resistance per phase but speed or slip at which maximum torque is dependent upon rotor resistance.
thanks by wearing the rotary distance we can get maximum target any value of slip.

See the diagram below.










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