# Stable Operating Regions of Induction Motor

In addition to having sufficient torque to start the load it is obviously necessary for the motor to bring the load up to full speed. To predict how the speed will rise after switching on we need the torque–speed curves of the motor and the load, and the total inertia.

By way of example, we can look at the case of a motor with two different loads (see Figure 1) . The solid line is the torque–speed curve of the motor, while the dotted lines represent two different load characteristics.

Load (A) is typical of a simple hoist, which applies constant torque to the motor at all speeds, while load (B) might represent a fan. For the sake of simplicity, we will assume that the load inertias (as seen at the motor shaft) are the same.

The speed–time curves for run-up are shown in Figure 2. Note that the gradient of the speed–time curve (i.e. the acceleration) is obtained by dividing the accelerating torque Tacc (which is the difference between the torque developed by the motor and the torque required to run the load at that speed) by the total inertia.

In this example, both loads ultimately reach the same steady speed, N (i.e. the speed at which motor torque equals load torque), but B reaches full speed much more quickly because the accelerating torque is higher during most of the run-up.

Load A picks up speed slowly at first, but then accelerates hard (often with a characteristic ‘whoosh’ produced by the ventilating fan) as it passes through the peak torque–speed and approaches equilibrium conditions. It should be clear that the higher the total inertia, the slower the acceleration and vice versa.

The total inertia means the inertia as seen at the motor shaft, so if gearboxes or belts are employed the inertia must be ‘referred’.

An important qualification ought to be mentioned in the context of the motor torque–speed curves shown by the solid line in Figure 1. This is that curves like this represent the torque developed by the motor when it has settled down at the speed in question, i.e. they are the true steady-state curves.

In reality, a motor will generally only be in a steady-state condition when it settles at its normal running speed, so for most of the speed range the motor will be accelerating.

In particular, when the motor is first switched on, there will be a transient period of a few cycles as the three currents gradually move towards a balanced 3-phase pattern. During this period the torque can fluctuate wildly and the motor can pick up significant speed, so the actual torque may be very different from that shown by the steady-state curve, and as a result the instantaneous speeds can fluctuate about the mean value.

Fortunately, the average torque during run-up can be fairly reliably obtained from the steady-state curves, particularly if the inertia is high and the motor takes many cycles to reach full speed, in which case we would consider the torque–speed curve as being ‘quasi-steady state’.

## Harmonic Effects – Skewing

A further cautionary note in connection with the torque–speed curves shown in this and most other books relate to the effects of harmonic airgap fields. Despite the limitations imposed by slotting, the stator winding magnetic flux (MMF) is remarkably close to the ideal of a pure sinusoid.

Unfortunately, because it is not a perfect sinusoid, Fourier analysis reveals that in addition to the predominant fundamental component, there are always additional unwanted ‘space harmonic’ fields. These harmonic fields have synchronous speeds that are inversely proportional to their order.

For example a 4-pole, 50 Hz motor will have a main field rotating at 1500 rev/min, but in addition there may be a fifth harmonic (20-pole) field rotating in the reverse direction at 300 rev/min, a seventh harmonic (28-pole) field rotating forwards at 214 rev/min, etc.

These space harmonics are minimised by stator winding design, but can seldom be eliminated. If the rotor has a very large number of bars it will react to the harmonic field in much the same way as to the fundamental, producing additional induction motor torques centred on the synchronous speed of the harmonic, and leading to unwanted dips in the torque speed, typically as shown in Figure 3.

Users should not be too alarmed as in most cases the motor will ride through the harmonic during acceleration, but in extreme cases a motor might, for example, stabilise on the seventh harmonic, and ‘crawl’ at about 214 rev/min, rather than running up to 4-pole speed (1500 rev/ min at 50 Hz), as shown by the dot in Figure 3.

To minimise the undesirable effects of space harmonics the rotor bars in the majority of induction motors are not parallel to the axis of rotation, but instead they are skewed (typically by around one or two slot pitches) along the rotor length. This has very little effect as far as the fundamental field is concerned, but can greatly reduce the response of the rotor to harmonic fields.

Because the overall influence of the harmonics on the steady-state curve is barely noticeable, and their presence might worry users, they are rarely shown, the accepted custom being that ‘the’ torque–speed curve represents the behaviour due to the fundamental component only.

## High Inertia Loads – Overheating

Apart from accelerating slowly, high inertia loads pose a particular problem of rotor heating, which can easily be overlooked by the unwary user. Every time an induction motor is started from rest and brought up to speed, the total energy dissipated as heat in the motor windings is equal to the stored kinetic energy of the motor plus load.

Hence with high inertia loads, very large amounts of energy are released as heat in the windings during run-up, even if the load torque is negligible when the motor is up to speed. With totally enclosed motors the heat ultimately has to find its way to the finned outer casing of the motor, which is cooled by air from the shaft-mounted external fan. Cooling of the rotor is therefore usually much worse than the stator, and the rotor is thus most likely to overheat during high inertia run-ups.

No hard and fast rules can be laid down, but manufacturers usually work to standards which specify how many starts per hour can be tolerated. Actually, this information is useless unless coupled with reference to the total inertia, since doubling the inertia makes the problem twice as bad.

However, it is usually assumed that the total inertia is not likely to be more than twice the motor inertia, and this is certainly the case for most loads. If in doubt, the user should consult the manufacturer who may recommend a larger motor than might seem necessary simply to supply the full-load power requirements.

## Steady State Rotor Losses and Efficiency

The discussion above is a special case, which highlights one of the less attractive features of induction machines. This is that it is never possible for all the power crossing the air-gap from the stator to be converted to mechanical output, because some is always lost as heat in the rotor circuit resistance.

In fact, it turns out that at slip s the total power (Pr) crossing the air-gap always divides so that a fraction sPr is lost as heat, while the remainder (1 – s)Pr is converted to useful mechanical output. Hence, when the motor is operating in the steady state the energy conversion efficiency of the rotor is given by

ɳr = Mechanical output power/Rated power input to rotor = (1 – s) (6:2)

This result is very important, and shows us immediately why operating at small values of slip is desirable. With a slip of 5% (or 0.05), for example, 95% of the air-gap power is put to good use. But if the motor was run at half the synchronous speed (s = 0.5), 50% of the airgap power would be wasted as heat in the rotor.

We can also see that the overall efficiency of the motor must always be significantly less than (1 – s), because in addition to the rotor copper losses there are stator copper losses, iron losses and windage and friction losses. This fact is sometimes forgotten, leading to conflicting claims such as ‘full-load slip = 5%, overall efficiency = 96%’, which is clearly impossible.

## Steady-State Stability – Pullout Torque and Stalling

We can check stability by asking what happens if the load torque suddenly changes for some reason. The load torque shown by the dotted line in Figure 4 is stable at speed X, for example: if the load torque increased from Ta to Tb, the load torque would be greater than the motor torque, so the motor torque would decelerate.