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Torque Equation of Induction Motor

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Starting Torque, Full Load Torque, Torque at Synchronous Speed & Condition for Maximum Torque Equations of an Induction Motor

An induction motor is widely used in industries as it produces good torque. The torque of an induction motor depends on the three factors;

Rotor power factor under running condition (cos ϕ₂)
Rotor current under running condition (I₂)
Part of a rotating magnetic field that induces EMF in the rotor winding (ϕ)

Therefore, the equation of torque is proportional to the above factors and equate as;

T ∝ ϕ I₂ cos(ϕ₂)

The flux produced in the rotor is directly proportional to the stator voltage E₂;

ϕ ∝ E₁

But, the transformation ratio (K) is;

Hence,

E₂ ∝ ϕ

Rotor current I₂;

Rotor Power factor (cos ϕ₂);

Now submit these values in the torque equations;

We can write the above equation by adding a proportionality constant.

This equation is known as a torque equation of induction motor.

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Full Load Torque

The slip of an induction motor depends on the loading condition. If we consider full load slip is s, the equation of full load torque is defined as the below equation.

Where k is a proportionality constant and the value of k is;

Where N_s’ is synchronous speed (RPS) = N_S / 60

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Torque at Synchronous Speed

At synchronous speed, the slip of a motor is zero (s = 0). If we put the value of slip in the torque equation, we get the torque produced by the motor is zero. Hence, the induction motor is an asynchronous motor that can be run at synchronous speed.

Starting Torque

Starting torque is defined as a torque produced at starting of the motor (consider rotor speed is zero). Hence, the speed N = 0 and slip s = 1. The starting torque is denoted as T_st. And we get the equation of starting torque by substituting s = 1 in the torque equation.

At the time of starting, the motor is in standstill condition. Therefore, starting torque is also known as standstill torque.

For an induction motor; we can write;

Put this value in torque equation;

Now, consider = constant = K

E₁ is nearly equal to V₁;

At starting condition, slip s = 1;

T_st ∝ V₁²

Hence, the starting torque is proportional to the square of the applied stator voltage.

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Condition for Maximum Torque

When a motor is running, the equation of starting torque is defined as;

It is clear from the above equation that the torque depends on the slip when a motor is in running condition. Now, we need to find the condition at which the torque is maximum. For that, we differentiate the above equation with respect to slip and compare it with zero.

Therefore,

This equation shows the value of slip that corresponds to maximum torque T_m. And it is a ratio of rotor resistance and reactance at standstill. The slip at maximum torque condition is denoted as s_m.

Now, put this value in the equation of for to find the equation of maximum torque T_m.

From the above equation, we can conclude that, the maximum torque;

Does not depend on the rotor resistance R₂. But the value of slip s_m at which maximum torque T_m occurs is directly proportional to the rotor resistance.
Directly proportional to the square of rotor voltage E₂.
Inversely proportional to the rotor reactance.

Torque Equation of Induction Motor