INDUCTION MOTORS: ROTATING FIELD, SLIP and TORQUE (part 1)

INTRODUCTION

Judged in terms of fitness for purpose coupled with simplicity, the induction motor must rank alongside the screw thread as one of man kind‘s best inventions. It isn't only supremely elegant as an electro mechanical energy converter, but is also by far the most important, with something like one-third of all the electricity generated being converted back to mechanical energy in induction motors. Despite playing a key role in industrial society, it remains largely unnoticed because of its workaday role in unglamorous surroundings driving machinery, pumps, fans, compressors, conveyors, hoists and a host of other routine but vital tasks. It will doubtless continue to dominate these fix ed-speed applications, but thanks to the availability of reliable variable-frequency inverters, it's now also the leader in the controlled-speed arena.

Like the d.c. motor, the induction motor develops torque by the interaction of axial currents on the rotor and a radial magnetic field produced by the stator. But, whereas, in the d.c. motor the ‘work‘ current has to be fed into the rotor by means of brushes and a commutator,the torque-producing currents in the rotor of the induction motor are induced by electromagnetic action, hence the name ‘induction ‘ motor. The stator winding therefore not only produces the magnetic field (the ‘excitation‘), but also supplies the energy that's converted to mechanical output. The absence of any sliding mechanical contacts and the consequent saving in terms of maintenance is a major advantage of the induction motor over its d.c. rival.

Other differences between the induction motor and the d.c. motor are: firstly that the supply to the induction motor is a.c. (usually 3-phase, but induction motor rotates relative to the stator, whereas in the d.c. motor it's stationary and thirdly that both stator and rotor in the induction motor are non-salient (i.e. effectively smooth) whereas the d.c. motor stator has projecting poles or saliencies which de W ne the position of the field windings.

Given these differences we might expect to find major contrasts between the performance of the two types of motor, and it's true that their inherent characteristics exhibit distinctive features. But there are also many aspects of behavior which are similar, as we shall see. Perhaps most important from the user ‘ s point of view is that there is no dramatic difference in size or weight between an induction motor and a DC motor giving the same power at the same base speed, though the induction motor will almost always be much cheaper. The similarity in size is a reflection of the fact that both types employ similar amounts of copper and iron, while the difference in price stems from the simpler construction of the induction motor.

Outline of approach

To understand how an induction motor operates, we must first unravel the mysteries of the rotating magnetic field. We shall see later that the rotor is effectively dragged along by the rotating field, but that it can never run quite as fast as the field. When we want to control the speed of the rotor, the best way is to control the speed of the field.

Our look at the mechanism of the rotating field will focus on the stator windings because they act as the source of the flux. In this part of the discussion we will ignore the presence of the rotor conductors. This makes it much easier to understand what governs the speed of rotation and the magnitude of the field, which are the two factors that mostly influence the motor behavior.

Having established how the rotating field is set up, and what its speed and strength depend on, we move onto examine the rotor, concentrating on how it behaves when exposed to the rotating field, and discovering how the induced rotor currents and torque vary with rotor speed. In this section, we assume -again for the sake of simplicity -that the rotating flux set up by the stator isn't influenced by the rotor.

Finally, we turn attention to the interaction between the rotor and stator, verifying that our earlier assumptions are well justified. Having done this we are in a position to examine the ‘external characteristics ‘of the motor, i.e. the variation of motor torque and stator current with speed. These are the most important characteristics from the point of view of the user.

In discussing how the motor operates the approach leans heavily on first building up a picture of the main or air-gap flux. All the main characteristics which are of interest to the user can be explained and understood once a clear idea has been formed of what the flux wave is, what determines its amplitude and speed and how it interacts with the rotor to produce torque.

The use of mathematics has been kept to a minimum, and all but the simplest equivalent circuit have been avoided in favor of a physical explanation. This is because the aim throughout this book is to attempt to promote understanding not only of what happens, but also why.

The alternative approach, which is favored in most textbooks on electrical machinery, is to move quickly to a position where the machine is represented by an (fairly complicated) equivalent circuit model, which can then be used for performance prediction. The danger of this for newcomers is that they can easily be daunted by the apparent complexity of the circuit, and as a result lose sight of the key messages that ought to emerge. and although equivalent circuits can provide qualitative answers to some of the questions we will be addressing, there are other matters (such as the fact that the rotor frequency is different from the stator frequency) that are disguised in the circuit approach.

Experience has shown that to get the most benefit from an equivalent circuit, a good grasp of why the machine behaves as it does is an essential prerequisite. Armed with this knowledge the power of the equivalent circuit can be properly appreciated, so readers are urged to come to grips with the material in this section before exploring to Section 7,which can be regarded as an ‘extra‘ for those seeking a different viewpoint.

The fundamental aspects we have explored so far (magnetic flux, MMF, reluctance, electromagnetic force, motional e.m.f.) will be needed again here, just as they were in the study of the d.c. machine.

But despite their basic similarities, most reader will probably find that the induction motor is more difficult to understand than the d.c. machine. 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 working flux and the supplier of the converted energy. Readers who are unfamiliar with routine a.c. circuit theory, including reactance, impedance, phasor diagrams (but not, at this stage, ‘j’ notation) and basic ideas about 3-phase systems will have to do some preparatory work before venturing further in this section.

THE ROTATING MAGNETIC FIELD

Before we look at how the rotating field is produced, we should be clear what it actually is. Because both the rotor and stator iron surfaces are smooth (apart from the regular slotting), and are separated by a small air gap, the flux produced by the stator windings crosses the air gap radially. The behavior of the motor is dictated by this radial flux, so we will concentrate first on establishing a mental picture of what is meant by the ‘ flux wave‘ in an induction motor.

The pattern of flux in an ideal 4-pole induction motor supplied from a balanced 3-phase source is shown in Ill. 1(a).The top sketch corresponds to time t = 0; the middle one shows the flux pattern one quarter of a cycle of the mains supply later (i.e. 5 ms if the frequency is 50 Hz) and the lower one corresponds to a further quarter cycle later.

We note that the pattern of flux lines is repeated in each case, except that the middle and lower ones are rotated by 45 deg and 90 deg ,respectively, with respect to the top sketch.

The term ‘4-pole‘ reflects the fact that the flux leaves the stator from two N poles, and returns at two S poles. Note, however, that there are no physical features of the stator iron mark it out as being 4-pole, rather than say 2-pole or 6-pole. As we will see, it's the layout and interconnection of the stator coils that sets the pole number.

If we plot the variation of the radial air-gap flux density with respect to distance round the stator, at each of the three instants of time, we get the patterns shown in Ill. 1(b).The first feature to note is that the radial flux density varies sinusoidally in space. There are two N peaks and two S peaks, but the transition from N to S occurs in a smooth sinusoidal way, giving rise to the term ‘flux wave ‘.The distance from the center of one N pole to the Centrex of the adjacent S pole is called the pole-pitch, for obvious reasons.

Staying with Ill. 1(b), we note that after one quarter of a cycle of the mains frequency, the flux wave retains its original shape, but has moved round the stator by half a pole-pitch, while after half a cycle it has moved round by a full pole-pitch. If we had plotted the patterns at intermediate times, we would have discovered that the wave maintained a constant shape, and progressed smoothly, advancing at a uniform rate of two pole-pitches per cycle of the mains. The term ‘traveling flux wave‘ is thus an appropriate one to describe the air-gap field.

For the 4-pole wave here, one complete revolution takes two cycles of the supply, so the speed is 25 rev/s (1500 rev/min)with a 50 Hz supply, or 30 rev/s (1800 rev/min) at 60 Hz. The general expression for the speed of the field (which is known as the synchronous speed) N s, in rev/min is: N s = 120 f p where p is the pole number. The pole number must be an even integer, since for every N pole there must be a S pole. Synchronous speeds for commonly used pole numbers are given in the table below.

Ill. 1 (a) Flux pattern in a 4-pole induction motor at three successive instants of time, each one-quarter of a cycle apart ; (b) radial flux density distribution in the air-gap at the three instants shown in Ill. 1 (a)

We can see from the table that if we want the field to rotate at inter mediate speeds, we will have to be able to vary the supply frequency, and this is what happens in inverter-fed motors, which are dealt in Section 8.

Production of rotating magnetic field

Now that we have a picture of the field, we turn to how it's 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 Ill. 2.

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

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 MMF 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 isn't difficult: we simply have to choose the right number and pitch of coils, as shown by the diagrams of an elementary 4-pole winding in Ill. 3.

Ill. 2 Star (wye) and Delta connection of the three phase windings of a 3-phase induction motor In Ill. 3(a) we see that by positioning two coils (each of which spans one pole-pitch) 180 deg 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.

Image 1 Stator of three-phase induction motor. The semi-closed slots of the stator core obscure the active sides of the stator coils, but the ends of the coils are just visible beneath the binding tape.

Image 2 Cage rotor for induction motor. The rotor conductor bars and end rings are cast in aluminum, and the blades attached to the end rings serve as a fan for circulating.

Ill. 3 Arrangement (a) and developed diagram (b) showing elementary 4-pole, single-layer stator winding consisting of 4 conductors spaced by 90 dgr .The ‘go‘ side of each coil (shown by the plus symbol)carries current into the paper at the instant shown, while the ‘ return ‘ side (shown by the dot)carries current out of the paper.

A clearer picture of the air-gap flux wave is presented in the developed view in Ill. 3(b), where more equally spaced flux lines have been added to emphasize 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 Ill. 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 MMF and air-gap flux density shown in Ill. 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 isn't 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's impossible to avoid some measure of discretization 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 fully utilized.

Despite these constraints we can get remarkably close to the ideal sinusoidal pattern, especially when we use a ‘two-layer‘ winding.

A typical arrangement of one phase is shown in Ill. 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 almost universal in all but small 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 Ill. 5 are shown carrying current. The remaining two-thirds of the coils form the other two phase windings.

Ill. 4 Developed diagram showing flux density produced by one phase of a single-layer winding having three slots per pole per phase. Returning to the flux density plot in Ill. 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 fix ed 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.

Ill. 5 Developed diagram showing layout of windings in a 3-phase, 4-pole,two-layer induction motor winding, together with the flux density wave produced by one phase acting alone. The upper detail shows how the coil sides form upper and lower layers in the slots.

Resultant field

The layout of coils for the complete 4-pole winding is shown in Ill. 6(a).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 (A,B,C).(For the sake of comparison, a 6-pole winding layout that uses the same stator slotting is shown in Ill. 6(b):here the pole pitch is six slots and the coils are short-pitched by one slot.) Returning to the 4-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 4-pole machine, for three discrete times during one complete cycle, as shown in Ill. 7.

We see that the three pulsating fields combine beautifully and lead to a resultant 4-pole field, which rotates at a uniform rate, advancing by two pole-pitches for every cycle of the mains. The resultant field isn't exactly sinusoidal in shape (though it's 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 travel ling wave and yet the winding layout is simple and easy to manufacture. This is an elegant engineering achievement, however one looks at it.

Ill. 6 Developed diagram showing arrangement of 3-phase, two-layer windings in a 36-slot stator. A 4-pole winding with three slots per pole per phase is shown in (a) ,and a 6-pole winding with two slots per pole per phase is shown in (b). (a) 4-pole; (b) 6-pole

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.

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 to pass right down the rotor teeth, so that before completing its path back to the stator it's 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.

Ill. 7 Resultant air-gap flux density wave produced by a complete 3-phase, 4-pole winding at three successive instants in time. The use of the pejorative-sounding term ‘leakage‘ suggests that these leakage fluxes are unwelcome imperfections, which we should go out of our way to minimize. 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 from the mains when the motor is started from rest)are made much worse if the coupling is too good. So we have the somewhat paradoxical situation in which the designer find s it comparatively easy to layout the windings to produce a good main flux, but is then obliged to juggle the detailed design of the slots to obtain just the right amount of leakage flux to give acceptable all-round performance.

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, and is discussed in Section 7.

However, such niceties are of limited importance to the user, so in this and the next sections we will limit references to leakage reactance to well-de fined contexts, and in general, where the term ‘ flux‘ is used, it will refer to the main air-gap field.

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 3-phase a.c. supply so that a balanced set of currents flow s in the windings. We denote the phase voltage by f, and the current in each phase by I_ m, where the subscript m denotes ‘magnetizing‘ or flux producing current.

From the discussion in Section 1, we know that the magnitude of the flux wave ( B m) is proportional to the winding MMF, 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 traveling 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 mains. The frequency of the e.m.f. induced in the stator conductors is therefore the same as the supply frequency; regard less 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 realize 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 dgr) , thereby forming a balanced 3-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 ff is applied, an alternating e.m.f., E, is induced. We can represent this state of affairs by the a.c. equivalent circuit for one phase shown in Ill. 8.

Ill. 8 Simple equivalent circuit for the induction motor under no-load conditions

The resistance shown in Ill. 8 is the resistance of one complete phase winding. Note that the e.m.f. E is shown as opposing the applied voltage ff. This must be so, otherwise we would have a runaway situation in which the voltage ff produced the magnetizing current I_m which in turn set up an e.m.f. E ,which added to ff, which further increased I_ m and so on ad infinitum .

Applying Kirchoff's law to the a.c. circuit in Ill. 8 yields V = I m R + E

We find in practice that the term I m R (which represents the volt drop due to winding resistance) is usually very much less than the applied voltage ff. 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 = E [3]

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

E prop. to B_ m f [4]

So by combining equations (3) and (4) we obtain

B_ m = k (v/f) [5]

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

Equation (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 (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.

It may seem a paradox that having originally homed-in on the magnetizing current I m as being the source of the MMF 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 equation (5).We can see why this is by looking again at Ill. 8 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 MMF, 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 f. It is rather like the DC motor (see Section 3) where the speed of the unloaded motor always adjusted itself so that the back e.m.f. equaled the applied voltage. Here, the magnetizing 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 magnetizing 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 recognize that the reluctance will be dominated by the air-gap, and that the magnitude of the magnetizing current will therefore depend mainly on the size of the gap. The larger the gap, the bigger the magnetizing current. Since the magnetizing 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 magnetizing current can be appreciable: in a 4-pole motor, it may be typically 50% of the full load current, and even higher in 6-pole and 8-pole designs.

Ill. 9 Phasor diagram for the induction motor under no-load conditions, showing magnetizing current I.

Excitation power and VA

The setting up of the travelling wave by the magnetizing 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 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's running light, so the hypothetical situation we are looking at isn't 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 magnetizing 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 magnetizing current lagging the supply voltage by almost 90 dgr ,as shown in the phasor diagram (see Ill. 9).

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. This matter is explored further in Section 7.

Summary

When the stator is connected to a 3-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 from the mains is small, but the magnetizing current itself can be quite large, giving rise to a significant reactive power demand from the mains.