**Principle of Operation of Induction Motor:**

The Principle of Operation of Induction Motor in Figure 9.4 shows the cross-sectional view of an induction motor. The stator is fed from a 3-phase supply of voltage V/phase and frequency f Hz. The rotor is wound 3-phase for as many poles as the stator and is short-circuited. It is assumed that the stator resistance and leakage reactance are both negligible so that

where

- E
_{1 }= stator induced emf/phase - k
_{w1 }= stator winding factor - N
_{ph1}(series) = stator series turns/phase - Î¦
_{r }= resultant air-gap flux/pole

It is seen from Eq. (9.1) that irrespective of the load conditions existing on the rotor,Î¦_{r}, the flux/pole established in the air-gap is constant, related to the applied voltage in view of the assumption made.

The mmf vector FÌ…_{r} with associated flux density vector BÌ…_{r} which is responsible for production of Î¦_{r}_{Â }rotates at synchronous speed as it is associated with balanced 3-phase currents drawn by the stator. The relative speed between BÌ…_{r} and the rotor causes induction of a current pattern in the shorted rotor. The torque produced by interaction of BÌ…_{r} and the rotor currents would by Lenz’s law tend to move the rotor in the direction of rotation of BÌ…_{r} so as to reduce the relative speed. The motor is thus **self-starting** and the rotor acquires a steady speed n < n_{s} depending upon the shaft load. It may be noted that no torque is produced at n = n_{s} because the relative speed between BÌ…_{r}Â and rotor being zero, no currents are induced in the rotor.

Figure 9.4 shows the relative location of vectors FÌ…_{r}, BÌ…_{r}(air-gap mmf and flux density),FÌ…_{2} (rotor mmf) wherein FÌ…_{r} leads FÌ…_{2} by angle Î´ = 90Â°+ Î¸_{2} (motoring action), Î¸_{2}**Â **is the angle by which rotor currents lag rotor emfs. The angle Î¸_{2} however, is very small as rotor reactance is far smaller than rotor resistance. The stator mmf vector is then given by

is located on the vector diagram. At the instant at which the diagram is drawn, the stator and rotor phase a (shown as single coil) currents are maximum possible. The reader may verify the location of vectors from the phase a currents applying the right hand rule.

**Slip and Frequency of Rotor Currents:**

With reference to Fig. 9.4, it is easily observed that BÌ…_{r} moves at speed (n_{s}Â – n) with respect to rotor conductors (in the direction of BÌ…_{r}). This is known as **slip speed**. The slip is defined as

Obviously s = 1 for n= 0, i.e. for the stationary rotor and s = 0 for n = n_{s}, i.e. for the rotor running at synchronous speed.

The frequency of currents induced in the rotor is

The normal full-load slip of the Principle of Operation of Induction Motor is of the order of 2%-8%, so that the frequency of the rotor currents is as low as 1-4 Hz.

The per phase rotor emf at s = 1 (standstill rotor) is given by

At any slip s, the rotor frequency being sf, the rotor induced emf changes to sE_{2}. Consider now the impedance of the rotor circuit

where

- X
_{2}Â = leakage reactance of rotor at standstill (rotor frequency = stator frequency, f)

When the rotor runs at slip s, its frequency being sf, its impedance changes to

It is, therefore, seen that the frequency of rotor currents, its induced emf and reactance all vary in direct proportion to the slip. Figure 9.5 shows the rotor circuit at slip s. The phase angle of the circuit is

**Rotor MMF and Torque Production:**

In Fig. 9.4 as the resultant flux density vector BÌ…_{r}Â rotates at speed (n_{s} â€” n) with respect to rotor, maximum positive emf is induced in the rotor coil aa’ (indicated by dot in conductor a and cross in conductor a’) when BÌ…_{r} lies 90Â° ahead of the axis ofÂ the coil. Since the current in the rotor lags the emf by Î¸_{2},Â the current in coil aa’ will be maximum positive when BÌ…_{r} has moved further ahead by angle Î¸_{2}. It is at this instant of time that the rotor mmf vector FÌ…_{2 }will will lie along the axis of coil aa’. It is, therefore, seen that BÌ…_{r} (or FÌ…_{r}) lies at an angle Î´ = (90 + Î¸_{2}) ahead of FÌ…_{2}. Further, FÌ…_{2 }caused by the rotor currents of frequency f_{2} = sf rotates with respect to the rotor conductor at speed (n_{s} â€” n) and at speed n_{s}. with respect to the stator as the rotor itself is moving in the same direction at speed n with respect to the stator. Thus FÌ…_{r} and FÌ…_{2} both move at synchronous speed n_{s }with respect to the stator and are stationary relative to each other with FÌ…_{r} lying ahead of FÌ…_{2} by angle (90Â° + Î¸_{2}). The interaction of the rotor field and the resultant field creates a torque in the direction of rotation of FÌ…_{r}.

Consider now the case of the squirrel-cage rotor with conductors spread uniformly around the rotor periphery. The rotor reaction mmf F_{2} is better visualized from the developed diagram of Fig. 9.6 wherein the rotor is imagined to be stationary and the B_{r} – wave moving with respect to it at slip speed (n_{s} â€” n). Let the rotor reactance be considered negligible so that the conductor (shorted) currents are in-phase with the conductor emf s. The conductor current pattern is, therefore, sinusoidally distributed and is in space phase with B_{r}-wave and moves synchronously with it. The rotor mmf-wave is a stepped-sinusoidal with the same number of poles as the B_{r}-wave moving synchronously with it. Its fundamental (F_{2}) shown in Fig. 9.6 lags B_{r}-wave by 90Â°. If the rotor reactance is now brought into picture, the conductor current-wave and, therefore, the rotor mmf-wave would lag behind by angle Î¸_{2}. Thus the angle between the B_{r}-wave and F_{2}-wave would be (90Â° + Î¸_{2}), the same as in the wound rotor. A squirrel-cage rotor, therefore, inductively reacts in the same way as a wound rotor except that the number of phases is not obvious â€” one can consider it to have as many phases as bars/pole. A squirrel-cage rotor can always be replaced by an equivalent wound rotor with three phases.

It is seen from Eq. (9.8) that a low-reactance rotor (low Î¸_{2}Â = tan^{-1}Â jsX_{2}/R_{2})Â will generate a larger torque for given Î¦_{r},F_{2} and s. A squirrel-cage motor is superior in this respect as compared to a wound-rotor motor as the cage rotor has lower reactance since it does not have winding overhang.

One very important observation that can be made in Principle of Operation of Induction Motor here is that while the rotor currents have a frequency sf, the mmf (F_{2}) caused by them runs at synchronous speed with respect to the stator. In other words, the reaction of rotor currents corresponds to the stator frequency (f) currents flowing on an equivalent stationary cylindrical structure placed inside the stator in place of the rotor. Or, to put in another way, the rotor currents as seen from the stator have frequency f but have the same rms value.

The stator mmf vector FÌ…_{1} is located on Fig. 9.4 from the vector equation

Further, FÌ…_{1} can be divided into components as

where FÌ…_{2 }is in opposition to FÌ…_{2} and equal in magnitude and

The stator current which causes FÌ…_{1} can, corresponding to vector Eq. (9.10), be divided into components

Here IÌ…_{m} can be recognized as file magnetizing current which causes the resultant mmf F_{r} and the resultant flux/pole, Î¦_{r}, while Iâ€²_{2} is that component of the stator current which balances the reaction F_{2} of the rotor current I_{2}.

Figure 9.4 also shows the relative location of stator coil AA’ and the positive direction of current in it. This instantaneous vector picture holds when I_{2} has maximum positive value. For Fâ€²_{2} to cancel F_{2}, the stator current component which balances the rotor mmf must be in phase with the rotor current as seen from the stator.

In terms of magnitudes Fâ€²_{2 }is oppositely directed to F_{2}, for them to cancel out while Iâ€²_{2} and I_{2} must obey the proportionality of Eq. (9.13) and must be in phase.

Further, by reference to Fig. 9.4, it can easily be seen that in the stator the positive direction of emf E_{1} opposes the positive direction of I_{1}, while in the rotor the positive direction of I_{2} is in the positive direction of sE_{2}. This is analogous to the transformer case.

With the direction of positive current in the stator coil AA’ marked as in Fig. 9.4 and the direction of the coil axis indicated, the law of induction which will give positive emf in opposition to current is

This Principle of Operation of Induction Motor has the same sign as used in the transformer case so that the flux phasor Î¦_{r} and magnetizing current which creates it lags E_{1}Â by 90Â°. In the circuit model I_{m }would therefore be drawn by the magnetizing reactance X_{m}Â across E_{1}.