Torque Equation of Three Phase Induction Motor

In this article, we will derive the torque equation of three phase induction motor. The torque is the turning ability of an induction motor.

When the stator of the three-phase induction motor receives a three-phase power supply, it produces a rotating magnetic field in the air gap of the induction motor. The air gap of the motor is the space or clearance between the stator and the rotor. The flux produced by the stator travels through the air gap and it links to the rotor conductor. The linking flux to the rotor produces a voltage in the rotor winding.

Do this voltage alone can rotate the motor? No

The rotating torque produces only when the current flows in the rotor. If you recall the construction of a squirrel cage induction motor, its rotor winding is star-connected and short-circuited at the end rings. In the case of slip ring induction motor the rotor circuit is connected to an external resistance for obtaining high starting torque, In both types of induction motors, thus, the rotor circuit forms a closed circuit. The voltage produced in the rotor is the driving force for the rotor current and the short-circuited rotor provides a closed path to the electric current. This way, current flows in the rotor current.

The rotor current flowing in the rotor winding interacts with the air gap flux. As a result, the torque is produced because of the interaction between the flux and the rotor current. The rotor current depends on the voltage produced in the rotor and the power factor of the rotor circuit. Thus, we can say that the torque produced in the rotor depends on flux, rotor current, and its power factor.

We can express the torque relationship with flux, rotor current, and its power factor by following mathematical expressions.

The flux produced in the air gap depends on the stator EMF (E1). The stator EMF is solely responsible for the production of magnetic flux. Thus, we can say the flux is proportional to the stator EMF.

The voltage transformation ratio(K) of the induction motor is the ratio of rotor voltage to stator voltage. We can express the relationship between the voltage transformation ratio(K), stator EMF, and rotor EMF by the following mathematical relationship.

The rotor-induced voltage(E) depends on the speed at which the motor runs. The slip of the motor shows the difference between the synchronous speed(Ns) and the actual rotor speed(N). The voltage induced in the rotor(E2) depends on the difference between the synchronous speed and the actual speed or slip(s) of the motor.

Therefore, the rotor voltage under motor running conditions is;

rotor voltage at induction motor running

The rotor current(I2) of the motor depends on the rotor voltage and impedance of the rotor. The rotor winding has resistance and reactance. The resistance of the rotor is fixed for a particular motor, however, the rotor reactance is frequency dependent. The rotor frequency reduces with speed increase, in other words, we can say the rotor reactance is slip-dependent.

Thus, the rotor reactance in motor running conditions is;

rotor reactance formula

Now, we can calculate the rotor impedance(Z2),

rotor impedance at standstill

The rotor current (I2)can be calculated by applying ohm’s law,

Induction motor rotor current formula

The power factor of the rotor circuit is the ratio of rotor resistance to rotor impedance. The power factor of the rotor circuit is;

power factor of the rotor

Putting the value rotor current I2, flux and power factor cosθ2 in the torque equation(1), we get,

torque equation of induction motor

Now, we will calculate the value of the proportionality constant(K).

Let the rotor input power, rotor copper loss, and rotor mechanical power output is P2, Pc, and Pm respectively.

The rotor copper loss is;

formula for rotor copper loss of induction motor

Rotor mechanical output

formula of mechanical output of induction motor

The ratio of Pc to Pm can be calculated by dividing equations (10) and (11),

formula showing relationship between copper loss and mechanical output of induction motor

The copper loss in the rotor circuit is;

copper loss formula of induction motor
Putting the value of I from equation(7) in equation(13), we get
derivation of copper loss of induction motor

Putting the value of copper loss(Pc) in equation(12) from equation (14), we get

formula of mechanical output in induction motor running condition

We know,

relationship between torque and angular speed of induction motor


mechanical output of induction motor with relation to its speed and torque

Putting the value of Pm from equation(16) in equation(15), we get;

formula for mechanical output of induction motor  considering slip, and speed of the induction motor

The slip of an induction motor is expressed by ;

formula for slip of induction motor

Putting value of N in equation(17), we get;

torque formula of induction motor

The synchronous speed of the motor(ns) in rps in ns can be given by the following formula.

Putting value of Ns/60 in equation(19), we get;

torque formula derivation of induction motor

Comparing equations (9) and (21), We get

value of k in torque equation of induction motor

Thus, the torque equation of the induction motor is;

torque equation of induction motor

Equation of Starting Torque

When an induction motor starts the slip of the motor is 1.

slip formula of induction motor at start

Therefore, the equation of starting torque can be obtained by simply putting the value of s = 1 in the torque equation(23) of the three-phase induction motor,

Starting torque equation of the Induction motor is;

derivation of starting torque of induction motor

Equation of Maximum Torque

The maximum torque in the induction motor occurs when the rotor resistance(R2) is equal to the product of the slip(s) and the rotor reactance(X2).

formula for maximum torque  of condition of induction motor

The slip(s) at maximum torque occurs is;

formula of slip at maximum torque of induction motor
Read – Derivation of Maximum Torque condition of Induction Motor

Putting value of slip from equation(27) in the torque equation(23), we get;

derivation of maximum torque equation of induction motor

From the above equation, it is clear that if we add external resistance to the rotor circuit it is possible to get the maximum torque at the higher slip. The resistance is cut off gradually with motor acceleration and finally, after the acceleration of the motor up to its rated speed, the rotor resistance is totally cut off. Only, rotor circuit resistance remains in the circuit, similar to the squirrel cage induction motor.

From the above equation(28), we can conclude the followings.

  1. The maximum torque depends on rotor-induced EMF at the standstill, and it is directly proportional to the square of the rotor-induced EMF.
  2. The maximum torque is inversely proportional to rotor reactance.
  3. The maximum torque does not depend on the rotor resistance.
  4. The slip at which maximum torque depends upon rotor resistance, R2. Therefore, it is possible to achieve the maximum torque at any slip by varying the rotor resistance.

Solved Problems on Torque of Induction Motor

Problem 1: A 3-phase, slip-ring, induction motor with a star-connected rotor has an induced e.m.f. of 120 volts between slip-rings at standstill with normal voltage applied to the stator. The rotor winding has a resistance per phase of 0.3 ohms and standstill leakage reactance per phase of 1.5 ohms.

Calculate the slip and rotor current per phase when the rotor is developing maximum torque.


To calculate Slip we must know the relation between resistance and reactance,

As we know, the torque of a rotor under running conditions is

torque of a rotor under running conditions

The condition for maximum torque may be obtained by differentiating the above expression with respect to slip s and after differentiating the above equation we will get,

R2 = sX2

solved problem 1 on torque of induction motor

and we also know, for a slip s, the rotor reactance will be s times the reactance at standstill.

solved problem 1- impedance of rotor

We know, for a slip s, the rotor-induced emf will be s times the induced emf at standstill.

solved problem 1- rotor induced EMF

And hence rotor current per phase will be derived as

solved problem 1- rotor icurrent

Hence the slip is 0.2 and the rotor current per phase is 33 A.

Problem 2: A three-phase, 460V, 100HP, 60Hz, 4-pole induction machine delivers rated output power at a slip of 0.05. Determine rated torque.

solved problem 2-formula of mechanical output of IM

P = Power
ω = Angular Velocity
T = Torque

formula for synchronous speed of induction motor

Ns = Synchronous speed
f = Frequency
P = Number of Pole

solved problem 1- calculation of synchronous speed

And Rotor/Rated speed will be calculated as,

solved problem 1-calculation of actual rotor speed of induction motor

Now from equation (1a)

solved problem 1- calculation of torque of induction motor

And hence rated torque of the motor is,

T = 416.6 N-m

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