# Torque Equation of Induction Motor

## 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 ϕ
_{2}) - Rotor current under running condition (I
_{2}) - 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* I_{2} cos(*ϕ*_{2})

The flux produced in the rotor is directly proportional to the stator voltage E_{2};

*ϕ* ∝ *E*_{1}

But, the transformation ratio (K) is;

Hence,

*E*_{2} ∝ *ϕ*

Rotor current I_{2};

Rotor Power factor (cos ϕ_{2});

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.

*Related Post:*

*Torque-Slip & Torque-Speed Characteristics of Induction Motor**Power, Voltage and EMF Equation of a DC Motor*

**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;

E_{1} is nearly equal to V_{1};

At starting condition, slip s = 1;

*T _{st}* ∝

*V*

_{1}^{2}Hence, the starting torque is proportional to the square of the applied stator voltage.

**Related Post:****EMF Equation of an Alternator and AC Generator**

**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 R
_{2}. 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 E
_{2}. - Inversely proportional to the rotor reactance.

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