**Principle of Operation of Induction Motor:**

The Principle of Operation of Induction Motor 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

It is seen from Eq. (9.1) that irrespective of the load conditions existing on the rotor,(P_{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 P_{r} with associated flux

density vector B* _{r}* which is responsible for production of O

_{r}rotates at synchronous speed as it is associated with balanced 3-phase currents drawn by the stator. The relative speed between B

*and the rotor causes induction of a current pattern in the shorted rotor. The torque produced by interaction of B*

_{r}*and the rotor currents would by Lenz’s law tend to move the rotor in the direction of rotation of 4 so as to reduce the relative speed. The motor is thus*

_{r}*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*because the relative speed between B

_{s}*and rotor being zero, no currents are induced in the rotor.*

_{r}Figure 9.4 shows the relative location of vectors Pr, *Br *(air-gap mmf and flux

density),l_{2} (rotor mmf) wherein Pr leads P2 by angle 8 = 90°+ *9 _{2}* (motoring action),

**02**is the angle by which rotor currents lag rotor emfs. The angle 9

_{2}however, is very small as rotor reactance is far smaller than rotor resistance (

*Reader should at this*

*point reread Sec. 5.7 subrection Induction Machine).*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 B* _{r}* moves at speed

*O*— n) with respect to rotor conductors (in the direction of fi

_{s}_{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 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 X2 = leakage reactance of rotor at standstill (rotor frequency = stator frequency, *j)*

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 B* _{r}* rotates at speed

*(n*with respect to rotor, maximum positive emf is induced in the rotor coil

_{s}— n)*aa’*(indicated by dot in conductor

*a*and cross in conductor

*a’)*when B

_{r}lies 90° ahead of the axis of the coil. Since the current in the rotor lags the emf by 0

_{2}, the current in coil

*aa’*will be maximum positive when B

_{r}has moved further ahead by angle 0

_{2}. It is at this instant of time that the rotor mmf vector will will lie along the axis of coil

*aa’.*It is, therefore, seen that

*h*

*(or*

_{r}*P*lies at an angle S = (90 + 02) ahead of 4 . Further,

_{r})*4*caused by the rotor currents of frequency f

_{2}=

*sf*rotates with respect to the rotor conductor at speed

*(n*

_{s}*—*

*n)*and at speed

*r*with respect to the stator as the rotor itself is moving in the same direction at speed

_{s}.*n*with respect to the stator. Thus F

_{r}and 4 both move at synchronous speed

*I?,*with respect to the stator and are stationary relative to each other with F

_{r}lying ahead of fr

_{2}by angle (90° + 0

_{2}). The interaction of the rotor field and the resultant field as per Eq. (5.58) creates a torque

Consider now the case of the squirrel-cage rotor with conductors spread uniformly around the rotor periphery. The rotor reaction mmf F2 is better visualized from the developed diagram of Fig. 9.6 wherein the rotor is imagined to be stationary and the B_{r} wavemoving with respect to it at slip speed *(n _{s}* —

*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 90°. If the rotor reactance is now brought into picture, the conductor current-wave and, therefore, the rotor mmf-wave would lag behind by angle 02. Thus the angle betvv sen the B

_{r}-wave and F

_{2}-wave would be (90° + 0

_{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 — 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. The quantitative relationships involved are beyond the scope of this book. Further examination of the induction motor theory in terms of wound rotor only will be undertaken.

It is seen from Eq. (9.8) that a low-reactance rotor (low 02 = tan^{-I}* jsX _{2}/R_{2}) *will generate a larger torque for given O

_{r}, F2 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 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 atatieture placed inside the stator in place of the rotor. Or, to put in another way, the tabor currents as seen from the stator have frequency *f *but have the same rms value.

The stator if vector F is located on Fig. 9.4 from the vector equation

Further, 4 s be divided into components as

where is in opposition to F and equal in magnitude and

The stator Which causes F can, corresponding to vector Eq. (9.10), be divided into component*

Here’_{m}can be recognized as file *magnetizing current *which causes the resultant *mmf F _{r}* and the resultant flux/pole, O

_{r}, while P is that component of the stator current which balances the reaction F2 of the rotor current /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 /2 has maximum positive value. For “2 to cancel F2, 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

is oppositely directed to F2, for them to cancel out while I and 12 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 Ei opposes the positive direction of while in the rotor the positive direction of /2 is in the positive direction of sE2. 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

(The reader should verify this.) This has the same sign as used in the transformer case so that the flux phasor�rand magnetizing current which creates it lags E1 by 90°. In the circuit model would therefore be drawn by the magnetizing reactance X. across El.

Remark The reader is reminded here that all the vectors (fields) are stationary with respect to one another and are rotating at ns with respect to the stator while the rotor is rotating at n with respect to the stator.