# Equivalent circuit of induction motor (for 3 phase and single phase)

May 18, 2018

## Equivalent circuit of induction motor

The Equivalent circuit of induction motor … We have repeated over and over that we use the induction motor everywhere in the industrial and domestic applications and that makes it very important to predict the behavior of the induction machine under various operating conditions. With the help of the equivalent circuit of induction motor that became easy.

Before deepen in the equivalent circuit of induction motor we need to know some important things.

The induction motor works on the same principle of the transformer; When we supply the stator with an EMF a voltage induced in the rotor as a result of electromagnetic induction.

So, we can say that the induction motor is a transformer with rotating secondary.

The stator winding of the induction motor resembles the primary of the transformer; The rotor of the induction motor resembles the secondary of the transformer.

And as the induction motor runs below the synchronous or full load speed there will be a relative difference between the speed of rotation of the motor and the synchronous speed. This relative difference is known as the slip (s).

While:

Ns: the rotation synchronous speed. so,

while f: the supply voltage frequency.

And, P: number of poles of the machine.

You may ask why I say that now, In electric machines, all information is connected with the all types even it’s a transformer, Ac, Dc, motor, or even generator. And to understand the equivalent circuit we should tie everything with the other.

### Equivalent circuit of three phase induction motor:

The equivalent circuit illustrates the various parameters of the machine and the losses may occur with the machine.

The induction motor has an induction and a resistor which cause losses.

The winding resistance causes copper losses in the windings, the windings inductive resistance cause a voltage drop (power factor).

And  the induction motor, especially the three-phase induction motor, has two  types of equivalent circuits which are:

#### Exact equivalent circuit:

From the above figure we have:

R1: the stator winding resistance.

X1: the stator winding inductance.

Rc: the copper loss component.

Xm: the winding magnetizing reactance.

R2/s: the rotor power and it includes the output mechanical power and the rotor copper losses.

So, When we draw this circuit as referred to the stator it will be:

while :

R’2: the resistance of the rotor referred to the stator winding.

X’2: the inductance of the rotor referred to the stator winding.

R2(1-s)/s: the resistance used to show the power that is converted to useful power or mechanical output power.

#### Approximate equivalent circuit:

In the approximate circuit, we delete one node to simplify the calculation.

We shift the shunt branch towards the primary side.

And what helps us to do that is the less voltage drop between the stator resistance and inductance, and also the difference between the supply voltage and the induced voltage is not much.

But there are some reasons make it not appropriate to do that, these reasons are:

• In the induction motor, we have a larger inductance of stator and rotor.
• Also, We use distributed windings in the induction motor.
• There is an air gap in the magnetic circuit of the induction motor which made the current larger compared to the transformer.

These reasons make it better to use the exact equivalent circuit, not the approximated.

So, We can only use the approximate analysis for large motors.

#### Power relation of the equivalent circuit:

Also, from the figure we illustrate the power input and output from the induction motor:

• The input power: which is directed by the stator and it equals (3V1I1Cos Ө).

while: V1: the voltage applied to the stator.

and, I1: the current drawn by the stator, Cos Ө: the stator power factor.

• Rotor input, power input, stator copper losses, and stator iron losses.
• The rotor copper losses: and they equal (slip* rotor input power).
• Developed power: which equals {(1-s) * rotor input power}.

### Equivalent circuit of a single phase induction motor:

The equivalent circuit of the single-phase differs than of the three-phase because the single phase motor circuit is given by double revolving field theory.

We should resolve the stationary pulsating magnetic field into two rotating fields; the two fields equal in magnitude but opposite in direction which made the induced net torque at standstill equals zero.

And we have a forward rotation known as the rotation with slip(s), and a backward rotation (2-s).

So, the equivalent circuit presented as:

while:

Zf: shows the forward impedance.

Zb: shows the backward impedance.

Note the sum of forward and backward slip = 2 …  So, we say that the backward slip is (2-s).

R1: the stator winding resistance.

X1: The stator winding inductive resistance.

Xm: the magnetizing reactance.

R’2: the rotor reactance referred to the stator.

X’2: the rotor inductive reactance referred to the stator.

#### Calculation of power of equivalent circuit:

• We first find the forward slip (Zf) and backward slip (Zb).
• We find the stator current which calculated from (stator voltage/total circuit impedance).
• And the input power: which is (stator voltage*stator current*Cos (Ө)); where:

Ө: the angle between the stator current and voltage.

• The power developed(Pg): which illustrates the difference between the forward field power and the backward power.
• And the rotor copper losses: Pg*slip.
• finally, the output power: (Pg-s*Pg-Rotational losses). The rotational losses include the friction loss, windage loss, and core loss.
• Also, we can calculate the efficiency by dividing the output power by the input power.

A short example will make it easy to understand what we say;

Also, the Equivalent circuit of induction motor may also help us to:

• Choose the power supply for the circuits we designed.
• Also, Know the temperature coefficient.
• Illustrate noise.
•  and Find the necessary parameters needed.