5.1.1 Outline of approach

Throughout this chapter we will be concerned with how the induction motor behaves in the steady state, i.e. the supply voltage and frequency are constant, the load is steady, and any transients have died away. We will aim to develop a sound qualitative understanding of the steady-state behaviour, based on the ideas we have discussed so far (magnetic flux, m.m.f., reluctance, electromagnetic force, motional e.m.f.). But despite many similarities with the d.c. motor, most readers will probably find that the induction motor is more difficult to understand. This is because we are now dealing with alternating rather than steady quantities (so, for example, inductive reactance becomes very significant), and also because (as mentioned earlier) a single winding acts simultaneously as the producer of the flux and the supplier of the converted energy.

In the next chapter, we will extend our qualitative understanding to look at how motor performance depends on design parameters: we will again be following an approach that has served us well since the early days of the induction motor, and was developed to reflect the fact that motors were operated at a fixed voltage and frequency. It turns out that under these ‘utility supply’ conditions, the transient performance is poor, fast control of torque is not possible, and so the induction motor was considered unable to compete with the d.c. motor in controlled speed drives.

All this changed rapidly beginning in the 1970s. The full set of governing equations (describing not only the steady state but also the much more complex dynamic behaviour) had become tractable with computer simulation, which in turn led the way to understanding how the stator currents would have to be manipulated to obtain fast control of torque. The hardware for implementing rapid current control became available with the development of PWM inverters, but it was not until digital signal processing finally became cheap and fast enough to deal with the complex control algorithms that so-called ‘field-oriented’ or ‘vector’ control emerged as a practicable commercial proposition. We will defer consideration of this spectacularly successful system until later, because experience has shown that a solid grounding based on the classical approach is invaluable before we get to grips with the more demanding ideas introduced in Chapter 7.

5.2.1 Production of a rotating magnetic field

Now that we have a picture of the field, we turn to how it is produced. If we inspect the stator winding of an induction motor we find that it consists of a uniform array of identical coils, located in slots. The coils are in fact connected to form three identical groups or phase-windings, distributed around the stator, and symmetrically displaced with respect to one another. The three phase-windings are connected either in star (wye) or delta (mesh), as shown in Fig. 5.2.

The three phase-windings are connected to a balanced three-phase a.c. supply, and so the currents (which produce the m.m.f. that sets up the flux) are of equal amplitude but differ in time-phase by one third of a cycle (120 degrees), forming a balanced three-phase set.

5.2.2 Field produced by each phase-winding

The aim of the winding designer is to arrange the layout of the coils so that each phase-winding, acting alone, produces an m.m.f. wave (and hence an air-gap flux wave) of the desired pole-number, and with a sinusoidal variation of amplitude with angle. Getting the desired pole-number is not difficult: we simply have to choose the right number and pitch of coils, as shown by the diagrams of an elementary four-pole winding in Fig. 5.3.

In Fig. 5.3A we see that by positioning two coils (each of which spans one pole-pitch) 180 degrees apart we obtain the correct number of poles (i.e. 4). However, the air gap field—shown by only two flux lines per pole for the sake of clarity—is uniform between each go and return coil side, not sinusoidal.

A clearer picture of the air-gap flux wave is presented in the developed view in Fig. 5.3B, where the equally-spaced flux lines emphasise the uniformity of the flux density between the go and return sides of the coils. Finally, the plot of the air-gap flux density underlines the fact that this very basic arrangement of coils produces a rectangular flux density wave, whereas what we are seeking is a sinusoidal wave.

We can improve matters by adding more coils in the adjacent slots, as shown in Fig. 5.4. All the coils have the same number of turns, and carry the same current. The addition of the extra slightly-displaced coils gives rise to the stepped waveform of m.m.f. and air-gap flux density shown in Fig. 5.4. It is still not sinusoidal, but is much better than the original rectangular shape.

It turns out that if we were to insist on having a perfect sinusoidal flux density waveform, we would have to distribute the coils of one phase in a smoothly-varying sinusoidal pattern over the whole periphery of the stator. This is not a practicable proposition, firstly because we would also have to vary the number of turns per coil from point to point, and secondly because we want the coils to be in slots, so it is impossible to avoid some measure of discretisation in the layout. For economy of manufacture we are also obliged to settle for all the coils being identical, and we must make sure that the three identical phase-windings fit together in such a way that all the slots are utilised.

Despite these constraints we can get remarkably close to the ideal sinusoidal pattern, especially when we use a ‘two-layer’ winding (in which case the stator slots may contain turns from more than one phase winding). A typical arrangement of one phase is shown in Fig. 5.5. The upper expanded sketch shows how each coil sits with its go side in the top of a slot while the return side occupies the bottom of a slot rather less than one pole-pitch away. Coils which span less than a full pole-pitch are known as short-pitch or short-chorded: in this particular case the coil pitch is six slots, the pole-pitch is nine slots, so the coils are short-pitched by three slots.

This type of winding is widely used in induction motors, the coils in each phase being grouped together to form ‘phase-bands’ or ‘phase-belts’. Since we are concentrating on the field produced by only one of the phase-windings (or ‘phases’), only one third of the coils in Fig. 5.5 are shown carrying current. The remaining two-thirds of the coils form the other two phase-windings, as discussed in the next section.

Returning to the flux density plot in Fig. 5.5 we see that the effect of short-pitching is to increase the number of steps in the waveform, and that as a result the field produced by one phase is a fair approximation to a sinusoid.

The current in each phase pulsates at the supply frequency, so the field produced by, say, phase A, pulsates in sympathy with the current in phase A, the axis of each ‘pole’ remaining fixed in space, but its polarity changing from N to S and back once per cycle. There is no hint of any rotation in the field of one phase, but when the fields produced by each of the three phases are combined, matters change dramatically, as we will see shortly.

We should mention that although it is desirable to minimise the harmonic content of the phase m.m.f. by striving for a sinusoidal m.m.f., it is not always feasible, particularly with high pole numbers, and in smaller motor sizes. For example, an eight-pole two-layer winding in 24 slots is shown in Fig. 5.6. In this case there is only one slot per pole per phase, so the phase m.m.f. is rectangular, with a high harmonic content. Nevertheless, because the overall performance of an induction motor is dominated by the fundamental component, the simple winding is entirely satisfactory.

5.2.3 Resultant three-phase field

The layout of coils for the complete four-pole winding under discussion is shown in shown in Fig. 5.7A. The go sides of each coil are represented by the capital letters (A, B, C) and the return sides are identified by bars over the letters ($\overline{A,}$ $\overline{B,}\overline{C}$). (For the sake of comparison, a six-pole winding layout that uses the same stator slotting is shown in Fig. 5.7B: here the pole-pitch is six slots and the coils are short-pitched by one slot.)

Returning to the four-pole winding, we can see that the windings of phases B and C are identical with that of phase A apart from the fact that they are displaced in space by plus and minus two thirds of a pole-pitch respectively. Phases B and C therefore also produce pulsating fields, along their own fixed axes in space. But the currents in phases B and C also differ in time-phase from the current in phase A, lagging by one third and two thirds of a cycle respectively. To find the resultant field we must therefore superimpose the fields of the three phases, taking account not only of the spatial differences between windings, but also the time differences between the currents. This is a tedious process, so the intermediate steps have been omitted and instead we move straight to the plot of the resultant field for the complete four-pole machine, for three discrete times during one complete cycle, as shown in Fig. 5.8.

We see that the three pulsating fields combine beautifully and lead to a resultant four-pole field which rotates at a uniform rate, advancing by two pole-pitches for every cycle of the supply. The resultant field is not exactly sinusoidal in shape (though it is actually more sinusoidal than the field produced by the individual phase-windings), and its shape varies a little from instant to instant; but these are minor worries. The resultant field is amazingly close to the ideal travelling wave and yet the winding layout is simple and easy to manufacture. This is an elegant engineering achievement, however one looks at it.

Before leaving our look at windings, it should be pointed out that the windings we have illustrated are called ‘integral-slot’ windings, where the number of slots per pole per phase is an integer: for example, the four-pole three-phase winding in 36 slots shown in Fig. 5.7A has 3 s/p/ph, while the six-pole version in Fig. 5.7B has 2 s/p/ph.

However, suppose we wanted to use a 54-slot stator for a four-pole, three-phase winding, for which the number of slots per pole per phase would be 4.5. We might guess that such a winding would not be possible, but in fact such so-called ‘fractional-slot’ windings are widely used where a standard lamination is to be used for several different pole-numbers. In the four-pole example above, the first and third phase-bands would comprise four adjacent slots, while the second and fourth would comprise five adjacent slots, an average of 4.5. These windings produce perfectly satisfactory sinusoidally-distributed travelling fields when supplied with sinusoidal currents from a utility supply.

We will come across an extreme version of fractional-slot windings later when we look at developments aimed at minimising production costs for mass-market permanent-magnet motors, particularly for use in electric vehicles. These motors are inverter-fed and their stators have relatively few slots, but with relatively high pole-numbers: consequently, the slots per pole per phase is normally fractional, and, surprisingly, is often less than unity.

5.2.4 Direction of rotation

The direction of rotation depends on the order in which the currents reach their maxima, i.e. on the phase-sequence of the supply. Reversal of direction is therefore simply a matter of interchanging any two of the lines connecting the windings to the supply.

5.2.5 Main (air-gap) flux and leakage flux

Broadly speaking the motor designer shapes the stator and rotor teeth to encourage as much as possible of the flux produced by the stator windings to pass right down the rotor teeth, so that before completing its path back to the stator it is fully linked with the rotor conductors (see later) which are located in the rotor slots. We will see later that this tight magnetic coupling between stator and rotor windings is necessary for good running performance, and the field which provides the coupling is of course the main or air-gap field, which we are in the midst of discussing.

In practice the vast majority of the flux produced by the stator is indeed main or ‘mutual’ flux. But there is some flux which bypasses the rotor conductors, linking only with the stator winding, and known as stator leakage flux. Similarly not all the flux produced by the rotor currents links the stator, but some (the rotor leakage flux) links only the rotor conductors.

The use of the perjorative-sounding term ‘leakage’ suggests that these leakage fluxes are unwelcome imperfections, which we should go out of our way to minimise. However whilst the majority of aspects of performance are certainly enhanced if the leakage is as small as possible, others (notably the large and unwelcome current drawn when the motor is started from rest directly on the utility supply) are made much worse if the coupling is too good. So we have the somewhat paradoxical situation in which the designer finds it comparatively easy to lay out the windings to produce a good main flux, but is then obliged to juggle the detailed design of the slots in order to obtain just the right amount of leakage flux to give acceptable all-round performance. (In contrast, as we will see later, an inverter-fed induction motor can avoid such issues as excessive starting current and, ideally, could be designed with much lower leakage than its utility-fed counterpart. It has to be said however that the majority of induction motors are still designed for general-purpose use, and in this respect they lose out, often unfairly, in comparison with other forms of motor that are specifically designed for operation with a drive.)

The weight which attaches to the matter of leakage flux is reflected in the prominent part played by the associated leakage reactance in equivalent circuit models of the induction motor, which we have chosen not to cover. However, such niceties are of limited importance to the user, so in this and the next chapters we will limit references to leakage reactance to well-defined contexts, and in general, where the term ‘flux’ is used, it will refer to the main air-gap field.

5.2.6 Magnitude of rotating flux wave

We have already seen that the speed of the flux wave is set by the pole number of the winding and the frequency of the supply. But what is it that determines the amplitude of the field?

To answer this question we can continue to neglect the fact that under normal conditions there will be induced currents in the rotor. We might even find it easier to imagine that the rotor conductors have been removed altogether: this may seem a drastic assumption, but will prove justified later. The stator windings are assumed to be connected to a balanced three-phase a.c. supply so that a balanced set of currents flows in the windings. We denote the phase voltage by V, and the current in each phase by I_m, where the subscript m denotes ‘magnetising’ or flux-producing current.

From the discussion in Chapter 1 we know that the magnitude of the flux wave (B_m) is proportional to the winding m.m.f., and is thus proportional to I_m. But what we really want to know is how the flux density depends on the supply voltage and frequency, since these are the only two parameters over which we have control.

To guide us to the answer, we must first ask what effect the travelling flux wave will have on the stator winding. Every stator conductor will of course be cut by the rotating flux wave, and will therefore have an e.m.f. induced in it. Since the flux wave varies sinusoidally in space, and cuts each conductor at a constant velocity, a sinusoidal e.m.f. is induced in each conductor. The magnitude of the e.m.f. is proportional to the magnitude of the flux wave (B_m), and to the speed of the wave (i.e. to the supply frequency f). The frequency of the induced e.m.f. depends on the time taken for one N pole and one S pole to cut the conductor. We have already seen that the higher the pole-number, the slower the field rotates, but we found that the field always advances by two pole-pitches for every cycle of the supply. The frequency of the e.m.f. induced in the stator conductors is therefore the same as the supply frequency, regardless of the pole-number. (This conclusion is what we would have reached intuitively, since we would expect any linear system to react at the same frequency at which we excited it.)

The e.m.f. in each complete phase winding (E) is the sum of the e.m.f’s in the phase coils, and thus will also be at supply frequency. (The alert reader will realise that whilst the e.m.f. in each coil has the same magnitude, it will differ in time phase, depending on the geometrical position of the coil. Most of the coils in each phase-band are close together, however, so their e.m.f’s—though slightly out of phase—will more or less add up directly.)

If we were to compare the e.m.f’s in the three complete phase windings, we would find that they were of equal amplitude, but out of phase by one third of a cycle (120°), thereby forming a balanced three-phase set. This result could have been anticipated from the overall symmetry. This is very helpful, as it means that we need only consider one of the phases in the rest of the discussion.

So we find that when an alternating voltage V is applied, an alternating e.m.f., E, is induced. We can represent this state of affairs by the primitive a.c. equivalent circuit for one phase shown in Fig. 5.9.

The resistance shown in Fig. 5.9 is the resistance of one complete phase-winding. Note that the e.m.f. E is shown as opposing the applied voltage V. This must be so, otherwise we would have a runaway situation in which the voltage V produced the magnetising current I_m which in turn set up an e.m.f. E, which added to V, which further increased I_m and so on ad infinitum.

Applying Kirchoff’s voltage law to the a.c. circuit in Fig. 5.8 yields

$V=I_{m}R+E$

We find in practice that the term I_mR (which represents the volt drop due to winding resistance) is usually very much less than the applied voltage V. In other words most of the applied voltage is accounted for by the opposing e.m.f., E. Hence we can make the approximation

$V\approx E$

But we have already seen that the e.m.f. is proportional to B_m and to f, i.e.

$E\propto B_{m}f$

So by combining Eqs (5.3) and (5.4) we obtain

$B_m=k\frac{V}{f}$

where the constant k depends on the number of turns per coil, the number of coils per phase and the distribution of the coils.

Eq. (5.5) is of fundamental importance in induction motor operation. It shows that if the supply frequency is constant, the flux in the air-gap is directly proportional to the applied voltage, or in other words the voltage sets the flux. We can also see that if we raise or lower the frequency (in order to increase or reduce the speed of rotation of the field), we will have to raise or lower the voltage in proportion if, as is usually the case, we want the magnitude of the flux to remain constant. (We will see in Chapters 7 and 8 that the early inverter drives used this so-called ‘V/f control’ to keep the flux constant at all speeds.)

It may seem a paradox that having originally homed-in on the magnetising current I_m as being the source of the m.m.f. which in turn produces the flux, we find that the actual value of the flux is governed only by the applied voltage and frequency, and I_m does not appear at all in Eq. (5.5). We can see why this is by looking again at Fig. 5.9 and asking what would happen if, for some reason, the e.m.f. (E) were to reduce. We would find that I_m would increase, which in turn would lead to a higher m.m.f., more flux, and hence to an increase in E. There is clearly a negative feedback effect taking place, which continually tries to keep E equal to V. It is rather like the d.c. motor (Chapter 3) where the speed of the unloaded motor always adjusted itself so that the back e.m.f. almost equalled the applied voltage. Here, the magnetising current always adjusts itself so that the induced e.m.f. is almost equal to the applied voltage.

Needless to say this does not mean that the magnetising current is arbitrary, but to calculate it we would have to know the number of turns in the winding, the length of the air-gap (from which we could calculate the gap reluctance) and the reluctance of the iron paths. From a user point of view there is no need to delve further in this direction. We should however recognise that the reluctance will be dominated by the air-gap, and that the magnitude of the magnetising current will therefore depend mainly on the size of the gap: the larger the gap, the bigger the magnetising current. Since the magnetising current contributes to stator copper loss, but not to useful output power, we would like it to be as small as possible, so we find that induction motors usually have the smallest air-gap which is consistent with providing the necessary mechanical clearances. Despite the small air-gap the magnetising current can be appreciable: in a four-pole motor, it may be typically 50% of the full-load current, and even higher in six-pole and eight-pole designs.

5.2.7 Excitation power and VA

The setting up of the travelling wave by the magnetising current amounts to the provision of ‘excitation’ for the motor. Some energy is stored in the magnetic field, but since the amplitude remains constant once the field has been established, no net power input is needed to sustain the field. We therefore find that under the conditions discussed so far, i.e. in the absence of any rotor currents, the power input to the motor is very small. (We should perhaps note that the rotor currents in a real motor are very small when it is running light, so the hypothetical situation we are looking at is not so far removed from reality as we may have supposed.)

Ideally the only source of power losses would be the copper losses in the stator windings, but to this must be added the ‘iron losses’ which arise from eddy currents and hysteresis in the laminated steel cores of rotor and stator. However we have seen that the magnetising current can be quite large, its value being largely determined by the air-gap, so we can expect an unloaded induction motor to draw appreciable current from the supply, but very little real power. The VA will therefore be substantial, but the power-factor will be very low, the magnetising current lagging the supply voltage by almost 90°, as shown in the time phasor diagram (Fig. 5.10).

Viewed from the supply the stator looks more or less like a pure inductance, a fact which we would expect intuitively given that—having ignored the rotor circuit—we are left with only an arrangement of flux-producing coils surrounded by a good magnetic circuit.

5.2.8 Summary

When the stator is connected to a three-phase supply, a sinusoidally distributed, radially-directed rotating magnetic flux density wave is set up in the air-gap. The speed of rotation of the field is directly proportional to the frequency of the supply, and inversely proportional to the pole-number of the winding. The magnitude of the flux wave is proportional to the applied voltage, and inversely proportional to the frequency.

When the rotor circuits are ignored (i.e. under no-load conditions), the real power drawn is small, but the magnetising current itself can be quite large, giving rise to a significant reactive power demand from the utility supply.

5.3.1 Rotor construction

Two types of rotor are used in induction motors. In both the rotor ‘iron’ consists of a stack of silicon 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’, as shown in Fig. 5.11. The term squirrel cage was widely used at one time and the origin should be clear from Fig. 5.11. 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).

In the larger sizes the conductors will be of copper, in which case the end-rings are brazed-on. In small and medium sizes, the rotor conductors and end rings may be of copper (particularly in high efficiency motors) or most commonly die-cast in aluminium, as shown in Fig. 5.12.

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 which can be avoided in the second type of rotor, which is known as the ‘wound-rotor’ 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 slip-rings (Fig. 5.13). 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 Fig. 5.13. Adding resistance can be beneficial in some circumstances, 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 slip-ring 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. But many old motors remain in service, so they are included in Chapter 6.

5.3.2 Slip

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 speed. 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

$s=\frac{N_S-N}{N_S}$

and is usually expressed either as a ratio as in Eq. (5.6), 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. (When tests are performed on induction motors with their rotor deliberately held stationary so that the slip is 1, the test is said to be under ‘locked-rotor’ conditions. The same expression is often used loosely to mean zero speed, even when the rotor is free to move, e.g. when it is started from rest.)

5.3.3 Rotor induced e.m.f. and current

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 the rotor e.m.f. (the slip frequency) 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 slip frequency is zero, while at standstill (slip = 1), the slip frequency is equal to the supply frequency. These relationships are shown in Fig. 5.14.

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° 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 Fig. 5.15.

All the rotor bars are short-circuited by the end-rings, so 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 (Fig. 5.16).

In Fig. 5.16 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.

5.3.4 Torque

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. The fact that motor action is only possible if the speed is less than the synchronous speed explains why the induction machine is described as ‘asynchronous’. 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.

5.3.5 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 Fig. 5.17.

To calculate the torque we first need to evaluate the ‘BI_rl_r’ product (see Eq. 1.2) in order 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 Fig. 5.17 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

$T=\mathit{KB}{I}_r$

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 Fig. 5.17 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 (See Eq. 5.5). The rotor e.m.f. (and hence I_r) is then proportional to slip, so we can see from Eq. (5.7) 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 Fig. 5.18.

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 Fig. 5.18.

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 Fig. 5.18.

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 Fig. 5.18.

The torque-slip (or torque-speed) characteristic shown in Fig. 5.18 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. We note that, in this normal operating region, the torque-speed curve is very similar to that of a d.c. motor (see Fig. 3.9).

5.3.6 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 Fig. 5.19.

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 ϕ_r in Fig. 5.20.

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 Eq. (5.7), to obtain a more general expression for torque as

$T=\mathit{KB}{I}_r\mathit{cos}{\varphi }_r$

Eq. (5.7) is merely a special case of Eq. (5.8), which only applies under low-slip conditions where cos ϕ_r ≈ 1.

For most cage rotors, it turns out that as the slip increases the term cos ϕ_r reduces more quickly than the current (Ir) 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 Fig. 5.21. 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.

5.3.7 Generating—Negative slip

When we explored the steady-state characteristics of the d.c. machine (see Section 3.4) we saw that at speeds less than that at which it runs when unloaded the machine acts as a motor, converting electrical energy into mechanical energy. But if the speed is above the no-load speed (e.g. when driven by a prime-mover), the machine generates and converts mechanical energy into electrical form.

The inherently bi-directional energy converting property of the d.c. machine seems to be widely recognised. But in the experience of the authors the fact that the induction machine behaves in the same way is far less well accepted, and indeed it is not uncommon to find users expressing profound scepticism at the thought that their ‘motor’ could possibly generate.

In fact, the induction machine behaves in essentially the same way as the d.c. machine, and if the rotor is driven by an external torque such that its speed is above the synchronous speed (i.e. the slip becomes negative), the electromagnetic torque reverses direction, and the power becomes negative, with energy fed back to the utility supply. It is important to note that, just as with the d.c. machine, this transition from motoring to generating takes place naturally, without intervention on our part.

When the speed is greater than synchronous, we can see from Eq. (5.6) that the slip is negative, and in this negative slip region the torque is also negative, the torque-speed curve broadly mirroring that in the motoring region, as shown in Fig. 5.22. We will discuss this further in Chapter 6, but it is worth noting that for both motoring and generating continuous operation will be confined to low values of slip, as indicated by the heavy line in Fig. 5.22.

5.4.1 Reduction of flux by rotor current

We should begin by recalling that we have already noted that when the rotor currents are negligible (s = 0), the e.m.f. which the rotating field induces in the stator winding is very nearly equal to the applied voltage. Under these conditions a reactive current (which we termed the magnetising current) flows into the windings, to set up the rotating flux. Any slight tendency for the flux to fall is immediately detected by a corresponding slight reduction in e.m.f. which is reflected in a disproportionately large increase in magnetising current, which thus opposes the tendency for the flux to fall.

Exactly the same feedback mechanism comes into play when the slip increases from zero, and rotor currents are induced. The rotor currents are at slip frequency, and they give rise to a rotor m.m.f. wave, which therefore rotates at slip speed (sN_s) relative to the rotor. But the rotor is rotating at a speed of (1–s)N_s, so that when viewed from the stator, the rotor m.m.f. wave always rotates at synchronous speed, regardless of the speed of the rotor.

The rotor m.m.f. wave would, if unchecked, cause its own ‘rotor flux wave’, rotating at synchronous speed in the air-gap, in much the same way that the stator magnetising current originally set up the flux wave. The rotor flux wave would oppose the original flux wave, causing the resultant flux wave to reduce.

However, as soon as the resultant flux begins to fall, the stator e.m.f. reduces, thereby admitting more current to the stator winding, and increasing its m.m.f. A very small drop in the e.m.f. induced in the stator is sufficient to cause a large increase in the current drawn from the supply because the e.m.f. E (see Fig. 5.9) and the supply voltage V are both very large in comparison with the stator resistance volt-drop, IR. The ‘extra’ stator m.m.f. produced by the large increase in stator current effectively ‘cancels’ the m.m.f. produced by the rotor currents, leaving the resultant m.m.f. (and hence the rotating flux wave) virtually unchanged.

There must be a small drop in the resultant m.m.f. (and flux) of course, to alert the stator to the presence of rotor currents. But because of the delicate balance between the applied voltage and the induced e.m.f. in the stator the change in flux with load is very small, at least over the normal operating speed-range, where the slip is small. In large motors, the drop in flux over the normal operating region is typically less than 1%, rising to perhaps 10% in a small motor.

The discussion above should have answered the question as to how the stator knows when to supply mechanical power across the air-gap. When a mechanical load is applied to the shaft, the rotor slows down, the slip increases, rotor currents are induced and their m.m.f. results in a modest (but vitally important) reduction in the air-gap flux wave. This in turn causes a reduction in the e.m.f. induced in the stator windings and therefore an increase in the stator current drawn from the supply. We can anticipate that this is a stable process (at least over the normal operating range) and that the speed will settle when the slip has increased sufficiently that the motor torque equals the load torque.

As far as our conclusions regarding torque are concerned, we see that our original assumption that the flux was constant is near enough correct when the slip is small. We will find it helpful and convenient to continue to treat the flux as constant (for given stator voltage and frequency) when we turn later to methods of controlling the normal running speed.

It has to be admitted, however, that at high values of slip (i.e. low rotor speeds), we cannot expect the main flux to remain constant, and in fact we would find in practice that when the motor was first switched on to the utility supply (50 or 60 Hz), with the rotor stationary, the main flux might typically be only half what it was when the motor was at full speed. This is because at high slips, the leakage fluxes assume a much greater importance than under normal low-slip conditions. The simple arguments we have advanced to predict torque would therefore need to be modified to take account of the reduction of main flux if we wanted to use them quantitatively at high slips. There is no need for us to do this explicitly, but it will be reflected in any subsequent curves portraying typical torque-speed curves for real motors. Such curves are of course used when selecting a motor to run directly from the utility supply, since they provide the easiest means of checking whether the starting and run-up torque is adequate for the job in hand. Fortunately, we will see in Chapter 7 that when the motor is fed from an inverter, we can avoid the undesirable effects of high-slip operation, and guarantee that the flux is at its optimum value at all times.