# Torque Production in Induction Motor

In this article we begin with a brief description of rotor types, and introduce the notion of ‘slip’, before moving onto explore how the torque is produced, and investigate the variation of torque with speed. We will find that the behaviour of the rotor varies widely according to the slip, and we therefore look separately at low and high values of slip.

Throughout this article we will assume that the rotating magnetic field is unaffected by anything which happens on the rotor side of the air-gap. Later, we will see that this assumption is pretty well justified.

## Rotor Construction

Two types of rotor are used in induction motors. In both the rotor ‘iron’ consists of a stack of steel laminations with evenly spaced slots punched around the circumference. As with the stator laminations, the surface is coated with an oxide layer, which acts as an insulator, preventing unwanted axial eddy currents from flowing in the iron.

The cage rotor is by far the most common: each rotor slot contains a solid conductor bar and all the conductors are physically and electrically joined together at each end of the rotor by conducting ‘end-rings’ (see Figure 1).

The conductors may be of copper, in which case the end-rings are brazed-on. Or, in small and medium sizes, the rotor conductors and end-rings can be die cast in aluminium. The term squirrel cage was widely used at one time and the origin should be clear from Figure 1.

The rotor bars and end-rings are reminiscent of the rotating cages used in bygone days to exercise small rodents (or rather to amuse their human captors). The absence of any means for making direct electrical connection to the rotor underlines the fact that in the induction motor the rotor currents are induced by the air-gap field.

It is equally clear that because the rotor cage comprises permanently short-circuited conductor bars, no external control can be exercised over the resistance of the rotor circuit once the rotor has been made. This is a significant drawback that can be avoided in the second type of rotor, which is known as the ‘woundrotor’ or ‘slipring’ type.

In the wound rotor, the slots accommodate a set of three phase-windings very much like those on the stator. The windings are connected in star, with the three ends brought out to three sliprings (see Figure 2). The rotor circuit is thus open, and connection can be made via brushes bearing on the sliprings.

In particular, the resistance of each phase of the rotor circuit can be increased by adding external resistances as indicated in Figure 2. Adding resistance in appropriate circumstances can be beneficial, as we will see.

Cage rotors are usually cheaper to manufacture, and are very robust and reliable. Until the advent of variable-frequency inverter supplies, however, the superior control which was possible from the slipring type meant that the extra expense of the wound rotor and its associated control gear were frequently justified, especially for high-power machines. Nowadays comparatively few are made, and then only in large sizes.

## Slip of Induction Motor

A little thought will show that the behaviour of the rotor depends very much on its relative velocity with respect to the rotating field. If the rotor is stationary, for example, the rotating field will cut the rotor conductors at synchronous speed, thereby inducing a high e.m.f. in them.

On the other hand, if the rotor was running at the synchronous speed, its relative velocity with respect to the field would be zero, and no e.m.f.’s would be induced in the rotor conductors.

The relative velocity between the rotor and the field is known as the slip. If the speed of the rotor is N, the slip speed is N_{s} – N, where N_{s} is the synchronous speed of the field, usually expressed in rev/min. The slip (as distinct from slip speed) is the normalised quantity defined by

s = (N_{s} – N)/N_{s} ………(equation 1)

Slip is usually expressed either as a ratio as in equation (1), or as a percentage.

A slip of 0 therefore indicates that the rotor speed is equal to the synchronous speed, while a slip of 1 corresponds to zero speed.

## Rotor Induced E.M.F., Current and Torque

The rate at which the rotor conductors are cut by the flux – and hence their induced e.m.f. – is directly proportional to the slip, with no induced e.m.f. at synchronous speed (s = 0) and maximum induced e.m.f. when the rotor is stationary (s = 1).

The frequency of rotor e.m.f. is also directly proportional to slip, since the rotor effectively slides with respect to the flux wave, and the higher the relative speed, the more times in a second each rotor conductor is cut by a N and a S pole.

At synchronous speed (slip = 0) the frequency is zero, while at standstill (slip = 1), the rotor frequency is equal to the supply frequency. These relationships are shown in Figure 3.

Although the e.m.f. induced in every rotor bar will have the same magnitude and frequency, they will not be in phase. At any particular instant, bars under the peak of the N poles of the field will have maximum positive voltage in them, those under the peak of the S poles will have maximum negative voltage (i.e. 180^{o} phase shift), and those in between will have varying degrees of phase shift.

The pattern of instantaneous voltages in the rotor is thus a replica of the flux density wave, and the rotor induced ‘voltage wave’ therefore moves relative to the rotor at slip speed, as shown in Figure 4.

Since all the rotor bars are short-circuited by the end-rings, the induced voltages will drive currents along the rotor bars, the currents forming closed paths through the end-rings, as shown in the developed diagram (see Figure 5).

In Figure 5 the variation of instantaneous e.m.f. in the rotor bars is shown in the upper sketch, while the corresponding instantaneous currents flowing in the rotor bars and end-rings are shown in the lower sketch.

The lines representing the currents in the rotor bars have been drawn so that their width is proportional to the instantaneous currents in the bars. The axial currents in the rotor bars will interact with the radial flux wave to produce the driving torque of the motor, which will act in the same direction as the rotating field, the rotor being dragged along by the field.

We note that slip is essential to this mechanism, so that it is never possible for the rotor to catch up with the field, as there would then be no rotor e.m.f., no current and no torque.

Finally, we can see that the cage rotor will automatically adapt to whatever pole number is impressed by the stator winding, so that the same rotor can be used for a range of different stator pole numbers.

## Rotor Currents and Torque – Small Slip

When the slip is small (say between 0 and 10%), the frequency of induced e.m.f. is also very low (between 0 and 5 Hz if the supply frequency is 50 Hz). At these low frequencies the impedance of the rotor circuits is predominantly resistive, the inductive reactance being small because the rotor frequency is low.

The current in each rotor conductor is therefore in time phase with the e.m.f. in that conductor, and the rotor current wave is therefore in space phase with the rotor e.m.f. wave, which in turn is in space phase with the flux wave. This situation was assumed in the previous discussion, and is represented by the space waveforms shown in Figure 6.

To calculate the torque we first need to evaluate the ‘BI*l*_{r}’ product to obtain the tangential force on each rotor conductor. The torque is then given by the total force multiplied by the rotor radius.

We can see from Figure 6 that where the flux density has a positive peak, so does the rotor current, so that particular bar will contribute a high tangential force to the total torque. Similarly, where the flux has its maximum negative peak, the induced current is maximum and negative, so the tangential force is again positive.

We don’t need to work out the torque in detail, but it should be clear that the resultant will be given by an equation of the form

T = kB*I*_{r} ………(equation 2)

Where, B and *I*_{r }denote the amplitudes of the flux density wave and the rotor current wave, respectively.

Provided that there are a large number of rotor bars (which is a safe bet in practice), the waves shown in Figure 6 will remain the same at all instants of time, so the torque remains constant as the rotor rotates.

If the supply voltage and frequency are constant, the flux will be constant. The rotor e.m.f. (and hence *I*_{r}) is then proportional to slip, so we can see from equation (2) that the torque is directly proportional to slip.

We must remember that this discussion relates to low values of slip only, but since this is the normal running condition, it is extremely important.

The torque–speed (and torque/slip) relationship for small slips is thus approximately a straight-line, as shown by the section of line AB in Figure 7.

If the motor is unloaded, it will need very little torque to keep running – only enough to overcome friction in fact – so an unloaded motor will run with a very small slip at just below the synchronous speed, as shown at A in Figure 7.

When the load is increased, the rotor slows down, and the slip increases, thereby inducing more rotor e.m.f. and current, and thus more torque. The speed will settle when the slip has increased to the point where the developed torque equals the load torque – e.g. point B in Figure 7.

Induction motors are usually designed so that their full-load torque is developed for small values of slip. Small ones typically have a full-load slip of 8%, large ones around 1%.

At the full-load slip, the rotor conductors will be carrying their safe maximum continuous current, and if the slip is any higher, the rotor will begin to overheat. This overload region is shown by the dotted line in Figure 7.

The torque–slip (or torque–speed) characteristic shown in Figure 7 is a good one for most applications, because the speed only falls a little when the load is raised from zero to its full value.

## Rotor Currents and Torque – Large Slip

As the slip increases, the rotor e.m.f. and rotor frequency both increase in direct proportion to the slip. At the same time the rotor inductive reactance, which was negligible at low slip (low rotor frequency) begins to be appreciable in comparison with the rotor resistance. Hence, although the induced current continues to increase with slip, it does so more slowly than at low values of slip, as shown in Figure 8.

At high values of slip, the rotor current also lags behind the rotor e.m.f. because of the inductive reactance. The alternating current in each bar reaches its peak well after the induced voltage, and this in turn means that the rotor current wave has a space-lag with respect to the rotor e.m.f. wave (which is in space phase with the flux wave). This space-lag is shown by the angle ɸ_{r} in Figure 9.

The space-lag means that the peak radial flux density and peak rotor currents no longer coincide, which is bad news from the point of view of torque production, because although we have high values of both flux density and current, they do not occur simultaneously at any point around the periphery.

What is worse is that at some points we even have flux density and currents of opposite sign, so over those regions of the rotor surface the torque contributed will actually be negative.

The overall torque will still be positive, but is much less than it would be if the flux and current waves were in phase. We can allow for the unwelcome space-lag by modifying equation (2), to obtain a more general expression for torque as

T = kB*I*_{r} cos ɸ_{r} ……(equation 3)

Equation (2) is merely a special case of equation (3), which only applies under low-slip conditions where cos ɸ_{r} ∼1.

For most cage rotors, it turns out that as the slip increases the term cosɸ_{r }reduces more quickly than the current (I_{r}) increases, so that at some slip between 0 and 1 the developed torque reaches a maximum value. This is illustrated in the typical torque–speed characteristic shown in Figure 10.

The peak torque actually occurs at a slip at which the rotor inductive reactance is equal to the rotor resistance, so the motor designer can position the peak torque at any slip by varying the reactance to resistance ratio.

We will return to the torque–speed curve after we have checked that when we allow for the interaction of the rotor with the stator, our interim conclusions regarding torque production remain valid.