Equivalent Circuit of Induction Motor

Equivalent Circuit of Induction Motor

The operation of an Induction motor is similar to the operation of the transformer. The equivalent circuit of the induction motor is similar to the equivalent circuit of the transformer. As explain in the construction of induction motor,  there are two main parts; stator and rotor. Both parts have different circuit model. The equivalent circuit of an induction motor is drawn only for one phase.

Stator Circuit Model

Three-phase supply is given to the stator winding. The stator winding consists of winding resistance R1 and leakage reactance X1. The no-load current represents as I0 and it divides into two parts.

  1. Magnetizing current Iμ
  2. Core-less current Iw

    \[ I_0=I_\mu+I_W \]

Stator circuit model

Stator Circuit Model

In case of the induction motor, magnetizing current is large compared to the transformer. Because of the high reluctance caused by the air-gap in an induction motor. In case of a transformer, no-load current I0 is 2 to 3% of the rated current but in the case of induction motor, no-load current I0 is 25 to 40% of the rated current. The magnetizing current depends on the size of the induction motor.

Rotor Circuit Model

As explain in Working of Three-phase Induction Motor, greater the relative speed between the rotor and stator magnetic field, greater the rotor voltage. At standstill condition, the induced rotor voltage is E20 then the induced rotor voltage at any slip s is given as

    \[ E_2_s=s\:E_2_0 \]

Rotor circuit model

Rotor Circuit Model

The rotor resistance R2 is independent of the slip and constant. The reactance of induction motor depends on the inductance of rotor L2 and the frequency of voltage and current in the rotor.

(1)   \begin{equation*} \begin{align} X_2 &= 2\:\pi\:f_2\:L_2; \\ f_2 &= s\:f_1; \\ X_2 &= 2 \: \pi \: s \:f_1 \: L_2 \\ X_2 &= s\:X_2_0 \\ \end{align} \end{equation*}

Where, X20 = the standstill reactance of the rotor.

The rotor impedance

(2)   \begin{equation*} \begin{align} Z_2_s &= R_2+jX_2_s \\ Z_2_s &= R_2+jsX_2_0 \end{align} \end{equation*}

The rotor current per phase

(3)   \begin{equation*} \begin{align}  I_2_s = \frac{sE_2_0}{R_2+jsX_2_0} \end{align} \end{equation*}

(4)   \begin{equation*} \begin{align}  I_2_s = \frac{E_2_0}{\frac{R_2}{s}+jX_2_0} \end{align} \end{equation*}

These equations denoted by below figures. Equation (3) represents figure-3A and equation (4) represents figure-3B.

Rotor circuit model

Complete equivalent circuit referred to the stator

In the complete equivalent circuit of the transformer, the voltage, current, and impedance of the secondary side can be transferred to the primary side with the help of the turns ratio. Similarly, in case of induction motor also, the complete per phase equivalent circuit obtain by transferring the rotor part of the model to the stator part of the model.

Complete equivalent model referred to the stator

Complete equivalent model referred to the stator

aeff = effective turns ratio

R2’ = resistance of rotor winding per phase referred to the stator side

X20’ = standstill rotor reactance per phase referred to the stator side

(5)   \begin{equation*} \begin{align} \frac{E_2}{T_e_2} &= \frac{E'_2}{T_e_1} \\ E'_2 &= \frac{T_e_1}{T_e_2} E_2 \\ &= a_e_f_f \: E_2 \\ &= E_1 \\ I'_2&=\frac{I_2}{a_e_f_f} \\ Z'_2_0 &= a^2_e_f_f (\frac{R_2}{s}+jX_2_0) \\ R'_2 &= a^2_e_f_f \: R_2 \\ X'_2_0 &= a^2_e_f_f \: X_2_0 \end{align} \end{equation*}

Approximate equivalent circuit

An approximate equivalent circuit obtains by shifting the shunt impedance branches R0 and X0 to the input terminal as shown in below figure.

Approximate equivalent circuit

Approximate equivalent circuit

If we assume that V= E= E2’ then this approximation is true. In this circuit, the resistance representing the developed mechanical power depends on the slip. All other quantities are constant and reactance correspond to those at the fixed stator frequency f1. This circuit model is standard for all performance calculation of induction motor.

Related Article:

Torque Equation of Three-phase Induction Motor

Three Phase Induction Motor: Construction and Classification

Working of Three-phase Induction Motor

Advantages and Disadvantages of Induction Motor

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