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 = R₁ + j X₁

Where                  R₁ = Stator resistance

                             X₁ = Stator reactance

And                      X₂  = rotor reactance referred to stator

                             R₂ = rotor resistance referred to stator

       Hence the impedance of each rotor is r₂ + j x₂   where

                            x₂ = X₂/2

       The resistance of forward field rotor is r₂/s while the resistance of backward field rotor is r₂ /(2 - s). The r₂ 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.

Fig. 1  Equivalent circuit without core loss

       Now the impedance of the forward field rotor is Z_f  which is parallel combination of  (0 + j x_o ) and (r₂ /s) + j x₂

 

        While the impedance of the backward field rotor is Z_b which is parallel combination of  (0 + j x_o) and (r₂ / 2-s) + j x₂.

       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  = Z₁  + 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  = /((r₂/s) + j x₂) where V_f  = I₁  x Z_f 
and                         I_2b  = /((r₂/2-s)  + j x₂)
                               P_f  = Power input to forward field rotor
                                   = (I_2f)² (r₂/s)  watts
                               P_b = Power input to backward field rotor
                                    = (I_2b)² (r₂/2-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 /(2πN/60) N-m
and                        Tb = backward torque = P_b /(2πN/60) N-m
                               T = net torque = T_f  - T_b 
while                       T_sh  = shaft torque = P_out /(2πN/60) N-m
                              %η = (net output / net input) x 100
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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.

Fig. 2 Equivalent circuit with core loss

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  ║( r₂/s + j x₂ )
       All other expressions remains same as stated earlier in case of equivalent circuit without core loss.