EQUIVALENT CIRCUIT, ROTOR SLIP and INDUCED TORQUE OF AN INDUCTION MOTOR

THE EQUIVALENT CIRCUIT OF AN INDUCTION MOTOR

An induction motor relies for its operation on the induction of voltages and currents in its rotor circuit from the stator circuit (transformer action). This induction is essentially a transformer operation, hence the equivalent circuit of an induction motor is similar to the equivalent circuit of a transformer.

The Transformer Model of an Induction Motor

A transformer per-phase equivalent circuit, representing the operation of an induction motor is shown below:

Transformer per-phase equivalent circuit of induction motor

As in any transformer, there is certain resistance and self-inductance in the primary (stator) windings, which must be represented in the equivalent circuit of the machine. They are - R₁ - stator resistance and X₁ – stator leakage reactance. 
Also, like any transformer with an iron core, the flux in the machine is related to the integral of the applied voltage E₁. The curve of mmf vs flux (magnetization curve) for this machine is compared to a similar curve for a transformer, as shown below:

Magnetization curve

The slope of the induction motor’s mmf-flux curve is much shallower than the curve of a good transformer. This is because there must be an air gap in an induction motor, which greatly increases the reluctance of the flux path and thus reduces the coupling between primary and secondary windings. The higher reluctance caused by the air gap means that a higher magnetizing current is required to obtain a given flux level. Therefore, the magnetizing reactance X_m in the equivalent circuit will have a much smaller value than it would in a transformer.
The primary internal stator voltage is E₁ is coupled to the secondary E_R by an ideal transformer with an effective turns ratio a_eff. The turns ratio for a wound rotor is basically the ratio of the conductors per phase on the stator to the conductors per phase on the rotor. It is rather difficult to see a_eff clearly in the cage rotor because there are no distinct windings on the cage rotor.

E_R in the rotor produces current flow in the shorted rotor (or secondary) circuit of the machine.
The primary impedance and the magnetization current of the induction motor are very similar to the corresponding components in a transformer equivalent circuit.

The Rotor Circuit Model.

When the voltage is applied to the stator windings, a voltage is induced in the rotor windings. In general, the greater the relative motion between the rotor and the stator magnetic fields, the greater the resulting rotor voltage and rotor frequency. The largest relative motion occurs when the rotor is stationary, called the locked-rotor or blocked-rotor condition, so the largest voltage and rotor frequency are induced in the rotor at that condition. The smallest voltage and frequency occur when the rotor moves at the same speed as the stator magnetic field, resulting in no relative motion.
The magnitude and frequency of the voltage induced in the rotor at any speed between these extremes is directly proportional to the slip of the rotor. Therefore, if the magnitude of the induced rotor voltage at locked-rotor conditions is called,

E_R0R = sE_R0

And the frequency of the induced voltage at any slip is:

f_r = sf_e

This voltage is induced in a rotor containing both resistance and reactance. The rotor resistance R_R is a constant, independent of slip, while the rotor reactance is affected in a more complicated way by slip.
The reactance of an induction motor rotor depends on the inductance of the rotor and the frequency of the voltage and current in the rotor. With a rotor inductance of L_R, the rotor reactance is:

Rotor Reactance

where X_R0 is the blocked rotor reactance.
The rotor current flow is:

rotor current

Therefore, the overall rotor impedance talking into account rotor slip would be:

rotor impedance

In this equivalent circuit, the rotor voltage is a constant E_R0 V and the rotor impedance Z_Req contains all the effects of varying rotor slip. Based upon the equation above, at low slips, it can be seen that the rotor resistance is much much bigger in magnitude as compared to X_R0. At high slips, X_R0 will be larger as compared to the rotor resistance.

The Final Equivalent Circuit.

To produce the final per-phase equivalent circuit for an induction motor, it is necessary to refer the rotor part of the model over to the stator side. In an ordinary transformer, the voltages, currents and impedance on the secondary side can be referred to the primary by means of the turns ratio of the transformer.
Exactly the same sort of transformation can be done for the induction motor’s rotor circuit. If the effective turns ratio of an induction motor is a _eff , then the transformed rotor voltage becomes,

rotor voltage

The rotor current:

rotor current

And the rotor impedance:

rotor impedance

If we make the following definitions:

R₂ = a²_eff R_R

X₂ = a²_eff X_R0

The final per-phase equivalent circuit is as shown below:

per-phase equivalent circuit of induction motor

THE CONCEPT OF ROTOR SLIP IN AN INDUCTION MOTOR

sectional view of induction motor

The induced voltage at the rotor bar depends upon the relative speed between the stator magnetic field and the rotor. This can be easily termed as slip speed:

slip speed

Where,

 n_slip = slip speed of the machine
n_sync = speed of the magnetic field.

n_m = mechanical shaft speed of the motor.

Apart from that we can describe this relative motion by using the concept of slip:

slip

Slip may also be described in terms of angular velocity, w.

slip

Using the ratio of slip, we may also determine the rotor speed:

rotor speed

THE DEVELOPMENT OF INDUCED TORQUE IN AN INDUCTION MOTOR.

INDUCTION MOTOR

When current flows in the stator, it will produce a magnetic field in stator as such that B_s (stator magnetic field) will rotate at a speed:

synchronous speed

Where f_e is the system frequency in hertz and P is the number of poles in the machine. This rotating magnetic field B_s passes over the rotor bars and induces a voltage in them. The voltage induced in the rotor is given by:

e_ind = (v x B) l

Hence there will be rotor current flow which would be lagging due to the fact that the rotor has an inductive element. And this rotor current will produce a magnetic field at the rotor, B_r. Hence the interaction between both magnetic field would give torque:

INDUCED TORQUE

The torque induced would generate acceleration to the rotor, hence the rotor will spin.
However, there is a finite upper limit to the motor’s speed.

DEVELOPMENT OF INDUCED TORQUE IN AN INDUCTION MOTOR

Conclusion : An induction motor can thus speed up to near synchronous speed but it can never reach synchronous speed.