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# Equivalent Circuit of an Induction Motor, Stator Circuit Model and Rotor Circuit Model

The operation of the induction motor is based on the principle of induction of voltages and currents in its rotor circuit from the stator circuit. The performance characteristics for steady-state conditions of the induction motor can be evaluated using the equivalent circuit of it. As the 3-phase induction motor represents a balanced load, therefore, the equivalent circuit of it is drawn only for one phase.

## Stator Circuit Model of Induction Motor

The stator circuit model or the stator equivalent circuit of the induction motor is shown in the figure.

The stator circuit model of the induction motor consists of a stator phase winding resistance R1, a stator phase winding leakage reactance X_{1}. R_{1} and X_{1} appear right at the input to the machine model. A pure inductive reactance X_{0} represents the magnetising reactance of the induction motor which has a much higher value. A non-inductive resistor R_{0} represents the core loss component of the induction motor. Therefore, the no-load current I_{0} of the induction motor is given by phasor sum of magnetising current (I_{m}) drawn by the magnetising reactance (X_{0}) and the core-loss current Iw drawn by the core-loss resistor (R0). Thus,

$$\mathrm{𝐼_0 = 𝐼_𝑚 + 𝐼_𝑤 … (1)}$$

The no-load current (I_{0}) of the induction motor is very high (about 25 to 40% of the rated current) because of the high reluctance caused by the air-gap of the induction motor.

## Rotor Circuit Model of Induction Motor

When a 3-phase supply voltage is applied to the stator windings of an induction motor, a voltage is induced in the rotor windings of the induction motor. In practice, the greater the relative motion of the rotor and the stator rotating magnetic field, the greater is the induced rotor voltage.

The greatest relative motion occurs when the rotor is stationary. This condition is known as *standstill condition* or *locked-rotor condition* or *blocked rotor condition*.

If the induced rotor voltage at standstill condition is E20, then the induced rotor voltage at any slip value is given by,

$$\mathrm{𝐸_2 = 𝑠𝐸_{20} … (2)}$$

The resistance (R_{2}) of the rotor circuit is constant and is independent of the slip in the induction motor.

The reactance of the rotor circuit of the induction motor depends upon the inductance of the rotor windings and the frequency of the rotor current. Thus, if L_{2} is the inductance of the rotor, then the rotor reactance at slip s is given by,

$$\mathrm{𝑋_2 = 2𝜋𝑓_2𝐿_2}$$

But the rotor frequency is,

$$\mathrm{𝑓_2 = 𝑠𝑓_1}$$

$$\mathrm{\therefore\: 𝑋_2 = 2𝜋𝑠𝑓_1𝐿_2 = 𝑠(2𝜋𝑓_1𝐿_2) = 𝑠𝑋_{20} … (3)}$$

Where, X_{20} is the standstill rotor reactance.

The impedance of the rotor at slip s is given by,

$$\mathrm{𝑍_2 = 𝑅_2 + 𝑗𝑋_2 = 𝑅_2 + 𝑗𝑠𝑋_{20} … (4)}$$

Therefore, the per phase rotor current of the induction motor is given by,

$$\mathrm{𝐼_2 =\frac{𝐸_2}{𝑍_2}=\frac{𝑠𝐸_{20}}{𝑅_2 + 𝑗𝑠𝑋_{20}}… (5)}$$

Using this rotor current equation, the rotor circuit model of the induction motor can be drawn as shown in the figure.

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