# Equivalent Circuit Of Single Phase Induction Motor

The double revolving field theory can be effectively used to obtain the equivalent circuit of a single phase induction motor. The method consists of determining the values of both the fields clockwise and anticlockwise at any given slip. When the two fields are known, the torque produced by each can be obtained. The difference between these two torques is the net torque acting on the rotor.

Imagine the single phase induction motor is made up of one stator winding and two imaginary rotor windings. One rotor is rotating in forward direction i.e. in the direction of rotating magnetic field with slip s while other is rotating in backward direction i.e. in direction of oppositely directed rotating magnetic field with slip 2 – s.

To develop the equivalent circuit, let us assume initially that the core loss is absent.

**1. Without core loss**

Let the stator impedance be Z Ω

Z = R_{1} + j X_{1}

Where R_{1} = Stator resistance

X_{1 }= Stator reactance

And X_{2 } = rotor reactance referred to stator

R_{2} = rotor resistance referred to stator

Hence the impedance of each rotor is r_{2} + j x_{2} where

x_{2} = X_{2}/2

The resistance of forward field rotor is r_{2}/s while the resistance of backward field rotor is r_{2} /(2 – s). The r_{2} value is half of the actual rotor resistance referred to stator.

As the core loss is neglected, R_{o} is not existing in the equivalent circuit. The x_{o} is half of the actual magnetising reactance of the motor. So the equivalent circuit referred to stator is shown in the Fig.1.

Now the impedance of the forward field rotor is Z_{f }which is parallel combination of (0 + j x_{o} ) and (r_{2}/s) + j x_{2}

While the impedance of the backward field rotor is Z_{b }which is parallel combination of (0 + j x_{o}) and (r_{2} / 2-s) + j x_{2}.

Under standstill condition, s = 1 and 2 – s = 1 hence Z_{f }= Z_{b }and hence V_{f }= V_{b }. But in the running condition, V_{f }becomes almost 90 to 95% of the applied voltage.

**. ^{.}. **Z

_{eq }= Z

_{1 }+ Z

_{f }+ Z

_{b }= Equivalent impedance

Let I

_{2f }= Current through forward rotor referred to stator

and I

_{2b }= Current through backward rotor referred to stator

**.**I

^{.}._{2f }= /((r

_{2}/s) + j x

_{2}) where V

_{f }= I

_{1 }x Z

_{f }

and I_{2b }= /((r_{2}/2-s) + j x_{2})

P_{f }= Power input to forward field rotor

= (I_{2f})^{2} (r_{2}/s) watts

P_{b }= Power input to backward field rotor

= (I_{2b})^{2} (r_{2}/2-s) watts

P_{m }= (1 – s){ Net power input}

= (1 – s) (P_{f }– P_{b }) watts

P_{out }= P_{m }– mechanical loss – core loss

**. ^{.}. **T

_{f }= forward torque = P

_{f }/(2?N/60) N-m

and Tb = backward torque = P

_{b }/(2?N/60) N-m

T = net torque = T

_{f }– T

_{b }

while T

_{sh }= shaft torque = P

_{out }/(2?N/60) N-m

%? = (net output / net input) x 100

Must Read

- Double revolving field theory

**2. With core loss**

If the core loss is to be considered then it is necessary to connect a resistance in parallel with, in an exciting branch of each rotor is half the value of actual core loss resistance. Thus the equivalent circuit with core loss can be shown as in the Fig. 2.

Let Z_{of }= Equivalent impedance of exciting branch in forward rotor

= r_{o}?(j x_{o} )

and Z_{ob }= Equivalent impedance of exciting branch in backward rotor

= r_{o}?(j x_{o} )

**. ^{.}. **Z

_{f }= Z

_{of }?( r

_{2}/s + j x

_{2})

All other expressions remains same as stated earlier in case of equivalent circuit without core loss.