#### Induction motor is an asynchronous motor i.e., its speed change with a change in load. It always runs on a lagging power factor. The principle of working of an induction motor to similar to the transformer i.e., on electromagnetic induction.

The equivalent circuit of an induction motor is similar to a transformer equivalent circuit because the energy is transferred from stator to rotor is essential as a transformer operation from primary to the secondary winding. An equivalent circuit enables the performance characteristics of the induction motor. The data obtained from the equivalent circuit can be used to calculate efficiency, torque, losses, rotor output, etc. All per phase quantities are used in representing the equivalent circuit.

## Equivalent Circuit of induction Motor :

#### The various parameters used for developing the equivalent circuit of an induction motor are,- R
_{1} & X_{1} : Stator winding resistance and leakage reactance. - R
_{2} & X_{2} : Rotor winding resistance and leakage reactance at standstill ( i.e., s = 1 ). - sX
_{2} : Rotor leakage reactance at slip s ( under running condition ). - R
_{o} : No-load branch resistance and it carries working component ( I_{w} ) of no-load current I_{o} account for the losses on no-load. - X
_{o} : No-load branch reactance and it carries magnetizing component ( I_{Âµ} ) of no-load to produce the flux. - E
_{1} and sE_{2} : Stator induced emf and rotor induced emf at slip s.

From the above parameters, the equivalent circuit of an induction motor can be drawn as

_{1}& X_{1}: Stator winding resistance and leakage reactance._{2}& X_{2}: Rotor winding resistance and leakage reactance at standstill ( i.e., s = 1 )._{2}: Rotor leakage reactance at slip s ( under running condition )._{o}: No-load branch resistance and it carries working component ( I_{w}) of no-load current I_{o}account for the losses on no-load._{o}: No-load branch reactance and it carries magnetizing component ( I_{Âµ}) of no-load to produce the flux._{1}and sE_{2}: Stator induced emf and rotor induced emf at slip s.#### let us consider the actual rotor circuit of the motor.

#### From the above diagram, the rotor current I_{2} is given by,

#### Here we know that the rotor input, P_{2} is the sum of rotor copper losses P_{c} and mechanical power developed P_{m}. Thus it is possible to represent the electrical equivalent of mechanical power developed as follows,

#### To show the equivalent mechanical load ( mechanical power conversion ) in the rotor circuit. The motor equivalent circuit can be modified as,

#### Now transfer the rotor side parameters to the stator side. While shifting the rotor side parameters towards the stator side we have to divide it with value "K" ( Where K = Ratio of the effective rotor to stator turns per phase ) except the rotor current where it is multiplied with "K". When the rotor parameters are shifted they can be represented as,

- R'
_{2} = Rotor resistance referred to the stator. - X'
_{2} = Rotor reactance referred to the stator. - E'
_{2} = Rotor induced e.m.f. referred to the stator. - I'
_{2} = Rotor current referred to the stator. - R'
_{L} = Rotor equivalent mechanical load referred to the stator.

The equivalent circuit can be further modified as shown below, and it is known as Exact Equivalent Circuit as referred to the stator.

_{2}= Rotor resistance referred to the stator._{2}= Rotor reactance referred to the stator._{2}= Rotor induced e.m.f. referred to the stator._{2}= Rotor current referred to the stator._{L}= Rotor equivalent mechanical load referred to the stator.#### Where,

#### Therefore, the approximate equivalent circuit is obtained by shifting the shunt branch ( consists of R_{01} & X_{01} ) to the supply terminals as shown in the below figure. This simplification will enable easy calculations.

#### Therefore, the total resistance referred to the stator side is,

#### Similarly, the total reactance referred to the stator side is,

## Calculation of Rotor Output and Torque Using the Equivalent Circuit :

#### From the equivalent circuit, we can derive expressions for torque T, and rotor output power P_{o} of the motor.

#### From the above diagram, the power input P_{i} to the rotor is given by,

#### We can write,

*P*_{i} = SP_{i} + P_{i} - SP_{i} ( by adding and subtracting SP_{i} )

P_{i} = SP_{i} + ( 1 – S )P_{i}

This above expression shows that the rotor input power, P_{i} is the sum of voltage drop in the rotor circuit due to its resistance as SP_{i} and the equivalent resistance representing mechanical load ( 1 - S )P_{i}.

From the above, the rotor output P_{o} is given by,

*P*

P

_{i}= SP_{i}+ P_{i}- SP_{i}( by adding and subtracting SP_{i})P

_{i}= SP_{i}+ ( 1 – S )P_{i}