How ELECTRIC MOTORS

ELECTRIC MOTORS
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Dr.NaveenBansal,India,Teacher
Published Date:25-10-2017
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1 ELECTRIC MOTORS INTRODUCTION Electric motors are so much a part of everyday life that we seldom give them a second thought. When we switch on an electric drill, for example, we conWdently expect it to run rapidly up to the correct speed, and we do not question how it knows what speed to run at, or how it is that once enough energy has been drawn from the supply to bringituptospeed,thepowerdrawnfallstoaverylowlevel.Whenwe putthedrilltoworkitdrawsmorepower,andwhenweWnishthepower drawn from the mains reduces automatically, without intervention on our part. The humble motor, consisting of nothing more than an arrangement of copper coils and steel laminations, is clearly rather a clever energy converter, which warrants serious consideration. By gaining a basic understanding of how the motor works, we will be able to appreciate its potential and its limitations, and (in later chapters) see how its already remarkable performance can be further enhanced by the addi- tion of external electronic controls. This chapter deals with the basic mechanisms of motor operation, so readers who are already familiar with such matters as magnetic Xux, magnetic and electric circuits, torque, and motional e.m.f can probably aVordtoskimovermuchofit.Inthecourseofthediscussion,however, several very important general principles and guidelines emerge. These apply to all types of motors and are summarised in Section 1.8. Experi- ence shows that anyone who has a good grasp of these basic principles willbewellequippedtoweightheprosandconsofthediVerenttypesof motor, so all readers are urged to absorb them before tackling other parts of the book.S N 2 Electric Motors and Drives PRODUCING ROTATION Nearly all motors exploit the force which is exerted on a current- carrying conductor placed in a magnetic Weld. The force can be demonstrated by placing a bar magnet near a wire carrying current (Figure 1.1), but anyone trying the experiment will probably be dis- appointed to discover how feeble the force is, and will doubtless be left wondering how such an unpromising eVect can be used to make eVective motors. Wewillseethatinordertomakethemostofthemechanism,weneed to arrange a very strong magneticWeld, and make it interact with many conductors, each carrying as much current as possible. We will also see later that although the magnetic Weld (or ‘excitation’) is essential to the workingofthemotor,itactsonlyasacatalyst,andallofthemechanical output power comes from the electrical supply to the conductors on whichtheforceisdeveloped.Itwillemergelaterthatinsomemotorsthe parts of the machine responsible for the excitation and for the energy converting functions are distinct and self-evident. In the d.c. motor, for example, the excitation is provided either by permanent magnets or by Weld coils wrapped around clearly deWned projecting Weld poles on the stationarypart,whiletheconductorsonwhichforceisdevelopedareon therotorandsuppliedwithcurrentviaslidingbrushes.Inmanymotors, however, there is no such clear-cut physical distinction between the ‘excitation’ and the ‘energy-converting’ parts of the machine, and a single stationary winding serves both purposes. Nevertheless, we will Wndthatidentifyingandseparatingtheexcitationandenergy-converting functions is always helpful in understanding how motors of all types operate. Returning to the matter of force on a single conductor, we will Wrst look at what determines the magnitude and direction of the force, Force Current in conductor Figure 1.1 Mechanical force produced on a current-carrying wire in a magneticWeldN S N Electric Motors 3 before turning to ways in which the mechanism is exploited to produce rotation. The concept of the magnetic circuit will have to be explored, since this is central to understanding why motors have the shapes they do. A brief introduction to magnetic Weld, magnetic Xux, and Xux density is included before that for those who are not familiar with the ideas involved. Magnetic field and magnetic flux Whenacurrent-carryingconductorisplacedinamagneticWeld,itexperi- encesaforce.Experimentshowsthatthemagnitudeoftheforcedepends directlyonthecurrentinthewire,andthestrengthofthemagneticWeld, andthattheforceisgreatestwhenthemagneticWeldisperpendiculartothe conductor. In the set-up shown in Figure 1.1, the source of the magnetic Weld is a bar magnet, which produces a magnetic Weld as shown in Figure 1.2. The notion of a ‘magnetic Weld’ surrounding a magnet is an abstract idea that helps us to come to grips with the mysterious phenomenon of N S Figure 1.2 MagneticXux lines produced by a permanent magnet S N S4 Electric Motors and Drives magnetism: it not only provides us with a convenient pictorial way of picturing the directional eVects, but it also allows us to quantify the ‘strength’ of the magnetism and hence permits us to predict the various eVects produced by it. ThedottedlinesinFigure1.2arereferredtoasmagneticXuxlines,or simplyXuxlines.TheyindicatethedirectionalongwhichironWlings(or small steel pins) would align themselves when placed in the Weld of the bar magnet. Steel pins have no initial magnetic Weld of their own, so thereisnoreasonwhyoneendortheotherofthepinsshouldpointtoa particular pole of the bar magnet. However, when we put a compass needle (which is itself a permanent magnet)intheWeldweWndthatitalignsitselfasshowninFigure1.2.In the upper half of the Wgure, the S end of the diamond-shaped compass settles closest to the N pole of the magnet, while in the lower half of the Wgure, the N end of the compass seeks the S of the magnet. This immediately suggests that there is a direction associated with the lines of Xux, as shown by the arrows on the Xux lines, which conventionally are taken as positively directed from the N to the S pole of the bar magnet. ThesketchinFigure1.2mightsuggestthatthereisa‘source’nearthe top of the bar magnet, from which Xux lines emanate before making their way to a corresponding ‘sink’ at the bottom. However, if we were to look at theXux lines insidethemagnet, we wouldWnd that they were continuous, with no ‘start’ or ‘Wnish’. (In Figure 1.2 the internal Xux lines have been omitted for the sake of clarity, but a very similar Weld pattern is produced by a circular coil of wire carrying a d.c. See Figure 1.6 where the continuity of the Xux lines is clear.). Magnetic Xux lines always form closed paths, as we will see when we look at the ‘magnetic circuit’,anddrawaparallelwiththeelectriccircuit,inwhichthecurrent is also a continuous quantity. (There must be a ‘cause’ of the magnetic Xux, of course, and in a permanent magnet this is usually pictured in terms of atomic-level circulating currents within the magnet material. Fortunately, discussion at this physical level is not necessary for our purpose.) Magnetic flux density Along with showing direction, the Xux plots also convey information about the intensity of the magnetic Weld. To achieve this, we introduce theideathatbetweeneverypairofXuxlines(andforagivendepthintothe paper) there is asame ‘quantity’of magneticXux. Somepeople have no diYcultywithsuchaconcept,whileothersWndthatthenotionofquanti-Electric Motors 5 fyingsomethingsoabstractrepresentsaseriousintellectualchallenge.But whether the approach seems obvious or not, there is no denying of the practicalutilityofquantifyingthemysteriousstuVwecallmagneticXux, anditleadsusnexttotheveryimportantideaofmagneticXuxdensity(B). When the Xux lines are close together, the ‘tube’ of Xux is squashed into a smaller space, whereas when the lines are further apart the same tube of Xux has more breathing space. The Xux density (B) is simply the Xux in the ‘tube’ (F) divided by the cross sectional area (A) of the tube, i.e. F B¼ (1:1) A The Xux density is a vector quantity, and is therefore often written in bold type: its magnitude is given by equation (1.1), and its direction is thatoftheprevailingXuxlinesateachpoint.Nearthetopofthemagnet inFigure1.2,forexample,theXuxdensitywillbelarge(becausetheXux is squashed into a small area), and pointing upwards, whereas on the equatorandfaroutfromthebodyofthemagnettheXuxdensitywillbe small and directed downwards. ItwillbeseenlaterthatinordertocreatehighXuxdensitiesinmotors, the Xux spends most of its life inside well-deWned ‘magnetic circuits’ madeofironorsteel,withinwhichtheXuxlinesspreadoutuniformlyto takefulladvantageoftheavailablearea.InthecaseshowninFigure1.3, for example, the cross-sectional area at bb’ is twice that at aa’, but the Xux is constant so theXux density at bb’ is half that at aa’. It remains to specify units for quantity of Xux, and Xux density. In the SI system, the unit of magneticXux is the weber (Wb). If one weber 2 of Xux is distributed uniformly across an area of 1m perpendicular to the Xux, the Xux density is clearly one weber per square metre b a a b Figure 1.3 MagneticXux lines inside part of an iron magnetic circuit6 Electric Motors and Drives 2 (Wb=m ).Thiswastheunitofmagneticfluxdensityuntilabout40years ago, when it was decided that one weber per square meter would henceforth be known as one tesla (T), in honour of Nikola Tesla who isgenerallycreditedwithinventingtheinductionmotor.Thewidespread use of B (measured in tesla) in the design stage of all types of electro- magnetic apparatus means that we are constantly reminded of the importance of tesla; but at the same time one has to acknowledge that theoutdatedunitdidhavetheadvantageofconveyingdirectlywhatXux density is, i.e. Xux divided by area. In the motor world we are unlikely to encounter more than a few milliwebers of Xux, and a small bar magnet would probably only pro- duce a few microwebers. On the other hand, values of Xux density are typicallyaround1Tinmostmotors,whichisareXectionofthefactthat althoughthequantityofXuxissmall,itisalsospreadoverasmallarea. Force on a conductor We now return to the production of force on a current-carrying wire placed in a magnetic Weld, as revealed by the setup shown in Figure 1.1. ThedirectionoftheforceisshowninFigure1.1:itisatrightanglesto both the current and the magnetic Xux density. With the Xux density horizontalandtotheright,andthecurrentXowingoutofthepaper,the force is vertically upward. If either the Weld or the current is reversed, theforceactsdownwards,andifbotharereversed,theforcewillremain upward. WeWnd by experiment that ifwe double eitherthe currentor theXux density, we double the force, while doubling both causes the force to increase by a factor of four. But how about quantifying the force? We need to express the force in terms of the product of the current and the magnetic Xux density, and this turns out to be very straightforward when we work in SI units. The force on a wire of length l, carrying a current I and exposed to a uniform magnetic Xux density B throughout its length is given by the simple expression F ¼ BIl (1:2) where F is in newtons when B is in tesla, I in amperes, and l in metres. This is a delightfully simple formula, and it may come as a surprise to some readers that there are no constants of proportionality involved inElectric Motors 7 equation1.2.Thesimplicityisnotacoincidence,butstemsfromthefact thattheunitofcurrent(theampere)isactuallydeWnedintermsofforce. Strictly,equation1.2onlyapplieswhenthecurrentisperpendicularto theWeld. If this condition is not met, the force on the conductor will be less;andintheextremecasewherethecurrentwasinthesamedirection as theWeld, the force would fall to zero. However, every sensible motor designer knows that to get the best out of the magnetic Weld it has to be perpendicular to the conductors, and so it is safe to assume in the subsequent discussion that B and I are always perpendicular. In the remainder of this book, it will be assumed that the Xux density and current are mutually perpendicular, and this is why, although B is a vector quantity (and would usually be denoted by bold type), we can drop the bold notation because the direction is implicit and we are only interested in the magnitude. The reason for the very low force detected in the experiment with the barmagnetisrevealedbyequation1.2.Toobtainahighforce,wemust haveahighXuxdensity,andalotofcurrent.TheXuxdensityattheends of a bar magnet is low, perhaps 0.1 tesla, so a wire carrying 1 amp will experienceaforceofonly0.1 N/m(approximately100 gmwt).Sincethe Xux density will be conWned to perhaps 1 cm across the end face of themagnet,thetotalforceonthewirewill beonly1 gm.Thiswouldbe barely detectable, and is too low to be of any use in a decent motor. So how is more force obtained? The Wrst step is to obtain the highest possible Xux density. This is achieved by designing a ‘good’ magnetic circuit, and is discussed next. Secondly, as many conductors as possible must be packed in the space wherethemagneticWeld exists,andeach conductormustcarryas much currentasitcanwithoutheatinguptoadangeroustemperature.Inthis way, impressive forces can be obtained from modestly sized devices, as anyone who has tried to stop an electric drill by grasping the chuck will testify. MAGNETIC CIRCUITS So far we have assumed that the source of the magnetic Weld is a permanent magnet. This is a convenient starting point as all of us are familiarwithmagnets,evenifonlyofthefridge-doorvariety.Butinthe majority of motors, the working magnetic Weld is produced by coils of wire carrying current, so it is appropriate that we spend some time looking at how we arrange the coils and their associated iron ‘magnetic circuit’ so as to produce high magnetic Welds which then interact with othercurrent-carryingconductorstoproduceforce,andhencerotation.8 Electric Motors and Drives Figure 1.4 MagneticXux lines produced by a straight, current-carrying wire First, we look at the simplest possible case of the magnetic Weld surrounding an isolated long straight wire carrying a steady current (Figure 1.4). (In the Wgure, the þ sign indicates that current is Xowing into the paper, while a dot is used to signify current out of the paper: these symbols can perhaps be remembered by picturing an arrow or dart, with the cross being the rear view of the Xetch, and the dot being the approaching point.) The Xux lines form circles concentric with the wire, the Weld strength being greatest close to the wire. As might be expected, the Weld strength at any point is directly proportional to the current.TheconventionfordeterminingthedirectionoftheWeldisthat the positive direction is taken to be the direction that a right-handed corkscrew must be rotated to move in the direction of the current. Figure1.4issomewhatartiWcialascurrentcanonlyXowinacomplete circuit, so there must always be a return path. If we imagine a parallel ‘go’ and ‘return’ circuit, for example, the Weld can be obtained by superimposing the Weld produced by the positive current in the go side with the Weld produced by the negative current in the return side, as shown in Figure 1.5. We note how the Weld is increased in the region between the conduc- tors,andreducedintheregionsoutside.AlthoughFigure1.5strictlyonly applies to an inWnitely long pair of straight conductors, it will probably notcomeasasurprisetolearnthattheWeldproducedbyasingleturnof wireofrectangular,squareorroundformisverymuchthesameasthat shown in Figure 1.5. This enables us to build up a picture of the Weld Figure 1.5 MagneticXux lines produced by current in a parallel go and return circuitElectric Motors 9 a b Figure1.6 Multi-turncylindricalcoilandpatternofmagneticXuxproducedbycurrent inthecoil.(Forthesakeofclarity,onlytheoutlineofthecoilisshownontheright.) thatwouldbeproducedinair,bythesortofcoilsusedinmotors,which typicallyhavemanyturns,asshownforexampleinFigure1.6. The coil itself is shown on the left in Figure 1.6 while theXux pattern produced is shown on the right. Each turn in the coil produces a Weld pattern,andwhenall theindividualWeld componentsaresuperimposed weseethattheWeldinsidethecoilissubstantiallyincreasedandthatthe closedXux paths closely resemble those of the bar magnet we looked at earlier. The air surrounding the sources of the Weld oVers a homoge- neous path for the Xux, so once the tubes of Xux escape from the concentrating inXuence of the source, they are free to spread out into thewholeofthesurroundingspace.Recallingthatbetweeneachpairof Xux lines there is an equal amount of Xux, we see that because the Xux lines spread out as they leave the conWnes of the coil, theXux density is much lower outside than inside: for example, if the distance ‘b’ is say four times ‘a’ theXux density B is a quarter of B . b a Although the Xux density inside the coil is higher than outside, we wouldWndthattheXuxdensitieswhichwecouldachievearestilltoolow tobeofuseinamotor. WhatisneededWrstlyisawayofincreasingthe Xux density, and secondly a means for concentrating the Xux and pre- venting it from spreading out into the surrounding space. Magnetomotive force (MMF) OneobviouswaytoincreasetheXuxdensityistoincreasethecurrentin the coil, or to add more turns. WeWnd that if we double the current, or10 Electric Motors and Drives the number of turns, we double the totalXux, thereby doubling theXux density everywhere. We quantify the ability of the coil to produce Xux in terms of its magnetomotive force (MMF). The MMF of the coil is simply the product of the number of turns (N) and the current (I), and is thus expressed in ampere-turns. A given MMF can be obtained with a large number of turns of thin wire carrying a low current, or a few turns of thickwirecarryingahighcurrent:aslongastheproductNIisconstant, the MMF is the same. Electric circuit analogy We have seen that the magnetic Xux which is set up is proportional to the MMF driving it. This points to a parallel with the electric circuit, where the current (amps) that Xows is proportional to the EMF (volts) driving it. In the electric circuit, current and EMF are related by Ohm’s Law, which is EMF V Current¼ i:e: I ¼ (1:3) Resistance R ForagivensourceEMF(volts),thecurrentdependsontheresistanceof thecircuit,sotoobtainmorecurrent,theresistanceofthecircuithasto be reduced. Wecanmakeuseofanequivalent‘magneticOhm’slaw’byintroduc- ing the idea of reluctance (R). The reluctance gives a measure of how diYcultitisforthemagneticXuxtocompleteitscircuit,inthesameway thatresistanceindicateshowmuchoppositionthecurrentencountersin the electric circuit. The magnetic Ohm’s law is then MMF NI Flux¼ i:e: F¼ (1:4) Reluctance R Weseefromequation1.4thattoincreasetheXuxforagivenMMF,we need to reduce the reluctance of the magnetic circuit. In the case of the example(Figure1.6),thismeanswemustreplaceasmuchaspossibleof the air path (which is a ‘poor’ magnetic material, and therefore consti- tutes a high reluctance) with a ‘good’ magnetic material, thereby reduc- ing the reluctance and resulting in a higherXux for a given MMF. The material which we choose is good quality magnetic steel, which for historical reasons is usually referred to as ‘iron’. This brings several very dramatic and desirable beneWts, as shown in Figure 1.7.Electric Motors 11 Iron Coil Leakage flux Air-gap Figure 1.7 Flux lines inside low-reluctance magnetic circuit with air-gap Firstly, the reluctance of the iron paths is very much less than the air paths which they have replaced, so the total Xux produced for a given MMF is very much greater. (Strictly speaking therefore, if the MMFs andcross-sectionsofthecoilsinFigures1.6and1.7arethesame,many moreXuxlinesshouldbeshowninFigure1.7thaninFigure1.6,butfor the sake of clarity similar numbers are indicated.) Secondly, almost all the Xux is conWned within the iron, rather than spreading out into the surrounding air. We can therefore shape the iron parts of the magnetic circuitasshowninFigure1.7inordertoguidetheXuxtowhereveritis needed.AndWnally,wesee thatinsidetheiron,theXuxdensityremains uniformoverthewholecross-section,therebeingsolittlereluctancethat there is no noticeable tendency for the Xux to crowd to one side or another. Before moving on to the matter of the air-gap, we should note that a question which is often asked is whether it is important for the coils to be wound tightly onto the magnetic circuit, and whether, if there is a multi-layer winding, the outer turns are as eVective as the inner ones. The answer, happily, is that the total MMF is determined solely by the number of turns and the current, and therefore every complete turn makes the same contribution to the total MMF, regardless of whether it happens to be tightly or loosely wound. Of course it does make sense for the coils to be wound as tightly as is practicable, since this not only minimises the resistance of the coil (and thereby reduces the heat loss) but also makes it easier for the heat generated to be conducted away to the frame of the machine. The air-gap In motors, we intend to use the high Xux density to develop force on current-carrying conductors. We have now seen how to create a high Xuxdensityinsidetheironpartsofamagneticcircuit,but,ofcourse,itis12 Electric Motors and Drives physicallyimpossibletoputcurrent-carryingconductorsinsidetheiron. Wethereforearrangeforanair-gapinthemagneticcircuit,asshownin Figure1.7.Wewillseeshortlythattheconductorsonwhichtheforceis to be produced will be placed in this air-gap region. If the air-gap is relatively small, as in motors, we Wnd that the Xux jumpsacrosstheair-gapasshowninFigure1.7,withverylittletendency toballoonoutintothesurroundingair.WithmostoftheXuxlinesgoing straight across the air-gap, the Xux density in the gap region has the same high value as it does inside the iron. Inthemajorityofmagneticcircuitsconsistingofironpartsandoneor moreair-gaps,thereluctanceoftheironpartsisverymuchlessthanthe reluctance of the gaps. At Wrst sight this can seem surprising, since the distanceacrossthegapissomuchlessthantherestofthepaththrough the iron. The fact that the air-gap dominates the reluctance is simply a reXection of how poor air is as a magnetic medium, compared to iron. To put the comparison in perspective, if we calculate the reluctances of twopathsofequallengthandcross-sectionalarea,onebeinginironand theotherinair,thereluctanceoftheairpathwilltypicallybe1000times greater than the reluctance of the iron path. (The calculation of reluc- tance will be discussed in Section 1.3.4.) Returning to the analogy with the electric circuit, the role of the iron parts of the magnetic circuit can be likened to that of the copper wires in the electric circuit. Both oVer little opposition to Xow (so that a negligible fraction of the driving force (MMF or EMF) is wasted in conveying the Xow to where it is usefully exploited) and both can be shaped to guide the Xow to its destination. There is one importantdiVerence,however.Intheelectriccircuit,nocurrentwillXow until the circuit is completed, after which all the current is conWned inside the wires. With an iron magnetic circuit, some Xux can Xow (in the surrounding air) even before the iron is installed. And although most of the Xux will subsequently take the easy route through the iron, some will still leak into the air, as shown in Figure 1.7. We will not pursue leakageXux here, though it is sometimes important, as will be seen later. Reluctance and air-gap flux densities If we neglect the reluctance of the iron parts of a magnetic circuit, it is easy to estimate the Xux density in the air-gap. Since the iron parts are thenineVect‘perfectconductors’ofXux,noneofthesourceMMF(NI) isusedindrivingtheXuxthroughtheironparts,andallofitisavailable to push theXux across the air-gap. The situation depicted in Figure 1.7Electric Motors 13 MMF =NI g AreaA Figure 1.8 Air-gap region, with MMF acting across opposing pole-faces therefore reduces to that shown in Figure 1.8, where an MMF of NI is applied directly across an air-gap of length g. To determine how much Xux will cross the gap, we need to know its reluctance.Asmightbeexpected,thereluctanceofanypartofthemag- netic circuit depends on its dimensions, and on its magnetic properties, andthereluctanceofarectangular‘prism’ofair,ofcross-sectionalarea AandlengthgasinFigure1.8isgivenby g R ¼ (1:5) g Am 0 wherem istheso-called‘primarymagneticconstant’or‘permeabilityof 0 free space’. Strictly, as its name implies, m quantiWes the magnetic 0 properties of a vacuum, but for all engineering purposes the permeabil- ity of air is alsom . The value of the primary magnetic constant (m)in 0 o 7 theSIsystemis410 H/m;rathersurprisingly,thereisnonamefor the unit of reluctance. Inpassing,weshouldnotethatifwewanttoincludethereluctanceof the iron part of the magnetic circuit in our calculation, its reluctance would be given by l fe R ¼ fe Am fe andwewouldhavetoaddthistothereluctanceoftheair-gaptoobtain the total reluctance. However, because the permeability of iron (m )is fe so much higher than  , the iron reluctance will be very much less than 0 the gap reluctance, despite the path length l being considerably longer than the path length (g) in the air. Equation 1.5 reveals the expected result that doubling the air-gap would double the reluctance (because the Xux has twice as far to go),14 Electric Motors and Drives whiledoublingtheareawouldhalvethereluctance(becausetheXuxhas twoequallyappealingpathsinparallel).TocalculatetheXux,F,weuse the magnetic Ohm’s law (equation 1.4), which gives MMF NI Am 0 F¼ ¼ (1:6) R g We are usually interested in the Xux density in the gap, rather than the totalXux, so we use equation 1.1 to yield F m NI 0 B¼ ¼ (1:7) A g Equation1.7 isdelightfully simple,andfrom itwe cancalculatetheair- gapXuxdensityonceweknowtheMMFofthecoil(NI)andthelength ofthegap(g).Wedonotneedtoknowthedetailsofthecoil-windingas long as we know the product of the turns and the current, nor do we needtoknowthecross-sectionalareaofthemagneticcircuitinorderto obtain theXux density (thoughwe do ifwe want to knowthe totalXux, see equation 1.6). For example, suppose the magnetising coil has 250 turns, the current is 2 A, and the gap is 1 mm. The Xux density is then given by 7 4p10 2502 B¼ ¼ 0:63 tesla 3 110 (We could of course obtain the same result using an exciting coil of 50 turnscarryingacurrentof10 A,oranyothercombinationofturnsand current giving an MMF of 500 ampere-turns.) Ifthecross-sectionalareaoftheironwasconstantatallpoints,theXux density would be 0.63 T everywhere. Sometimes, as has already been mentioned,thecross-sectionoftheironreducesatpointsawayfromthe air-gap, as shown for example in Figure 1.3. Because the Xux is com- pressedinthenarrowersections,theXuxdensityishigher,andinFigure 1.3iftheXuxdensityattheair-gapandintheadjacentpole-facesisonce againtakentobe0.63 T,thenatthesectionaa’(wheretheareaisonlyhalf thatattheair-gap)theXuxdensitywillbe20:63¼1:26T. Saturation It would be reasonable to ask whether there is any limit to the Xux density at which the iron can be operated. We can anticipate that there must be a limit, or else it would be possible to squash the Xux into aElectric Motors 15 Effective reluctance 0 0 12 Flux density (tesla) Figure 1.9 Sketch showing how the eVective reluctance of iron increases rapidly as the Xux density approaches saturation vanishingly small cross-section, which we know from experience is not the case. In fact there is a limit, though not a very sharply deWned one. Earlierwenotedthattheironhasalmostnoreluctance,atleastnotin comparison with air. Unfortunately this happy state of aVairs is only true as long as the Xux density remains below about 1.6 – 1.8 T, depending on the particular steel in question. If we try to work the iron at higher Xux densities, it begins to exhibit signiWcant reluctance, and no longer behaves like an ideal conductor of Xux. At these higher Xux densities, a signiWcant proportion of the source MMF is used in drivingtheXuxthroughtheiron.Thissituationisobviouslyundesirable, sincelessMMFremainstodrivetheXuxacrosstheair-gap.Sojustaswe wouldnotrecommendtheuseofhigh-resistancesupplyleadstotheload in an electric circuit, we must avoid overloading the iron parts of the magnetic circuit. The emergence of signiWcant reluctance as theXux density is raised is illustrated qualitatively in Figure 1.9. When the reluctance begins to be appreciable, the iron is said to be beginning to ‘saturate’. The term is apt, because if we continue increas- ingthe MMF, or reducing the area of the iron, we will eventually reach an almost constant Xux density, typically around 2 T. To avoid the undesirable eVects of saturation, the size of the iron parts of the mag- netic circuit are usually chosen so that the Xux density does not exceed about1.5 T.AtthislevelofXuxdensity,thereluctanceoftheironparts will be small in comparison with the air-gap. Magnetic circuits in motors The reader may be wondering why so much attention has been focused on the gapped C-core magnetic circuit, when it appears to bear little16 Electric Motors and Drives Figure 1.10 Evolution of d.c. motor magnetic circuit from gapped C-core resemblance to the magnetic circuits found in motors. We will now see that it is actually a short step from the C-core to a magnetic motor circuit, and that no fundamentally new ideas are involved. The evolution from C-core to motor geometry is shown in Figure 1.10, which should be largely self-explanatory, and relates to the Weld system of a d.c. motor. We note that the Wrst stage of evolution (Figure 1.10, left) results in theoriginalsinglegapoflengthgbeingsplitintotwogapsoflengthg/2, reXecting the requirement for the rotor to be able to turn. At the same time the single magnetising coil is split into two to preserve symmetry. (Relocating the magnetising coil at a diVerent position around the magnetic circuit is of course in order, just as a battery can be placed anywhere in an electric circuit.) Next, (Figure 1.10, centre) the single magnetic path is split into two parallel paths of half the original cross- section, each of which carries half of the Xux: and Wnally (Figure 1.10, right), the Xux paths and pole-faces are curved to match the rotor. The coil now has several layers in order to Wt the available space, but as discussed earlier this has no adverse eVect on the MMF. The air-gap is still small, so theXux crosses radially to the rotor. TORQUE PRODUCTION Having designed the magnetic circuit to give a high Xux density under the poles, we must obtain maximum beneWt from it. We therefore need toarrangeasetofconductors,Wxedtotherotor,asshowninFigure1.11, andtoensurethatconductorsunderaN-pole(atthetopofFigure1.11) carrypositivecurrent(intothepaper),whilethoseundertheS-polecarry negative current. The tangential electromagnetic (‘BIl’) force (see equa- tion1.2)onallthepositiveconductorswillbetotheleft,whiletheforce on the negative ones will be to the right. A nett couple, or torque will thereforebeexertedontherotor,whichwillbecausedtorotate. (Theobservantreaderspottingthatsomeoftheconductorsappearto have no current in them will Wnd the explanation later, in Chapter 3.)Electric Motors 17 Figure 1.11 Current-carrying conductors on rotor, positioned to maximise torque. (The source of the magneticXux lines (arrowed) is not shown.) At this point we should pause and address three questions that often cropupwhentheseideasarebeingdeveloped.TheWrstistoaskwhywe have made no reference to the magnetic Weld produced by the current- carrying conductors on the rotor. Surely they too will produce a mag- netic Weld, which will presumably interfere with the original Weld in the air-gap,inwhichcaseperhapstheexpressionusedtocalculatetheforce on the conductor will no longer be valid. The answer to this very perceptive question is that theWeld produced bythecurrent-carryingconductorsontherotorcertainlywillmodifythe originalWeld(i.e.theWeldthatwaspresentwhentherewasnocurrentin the rotor conductors.) But in the majority of motors, the force on the conductor can be calculated correctly from the product of the current and the ‘original’ Weld. This is very fortunate from the point of view of calculating the force, but also has a logical feel to it. For example in Figure 1.1, we would not expect any force on the current-carrying conductor if there was no externally applied Weld, even though the current in the conductor will produce its own Weld (upwards on one side of the conductor and downwards on the other). So it seems right thatsinceweonlyobtainaforcewhenthereisanexternalWeld,allofthe force must be due to thatWeld alone. The second question arises when we think about the action and reac- tionprinciple.Whenthereisatorqueontherotor,thereispresumablyan equalandoppositetorqueonthestator;andthereforewemightwonderif the mechanism of torque production could be pictured using the same ideasasweusedforobtainingtherotortorque.Theanswerisyes;thereis always an equal and opposite torque on the stator, which is why it is usuallyimportanttoboltamotordownsecurely.Insomemachines(e.g. theinductionmotor)itiseasytoseethattorqueisproducedonthestator18 Electric Motors and Drives by the interaction of the air-gap Xux density and the stator currents, in exactlythesamewaythattheXuxdensityinteractswiththerotorcurrents toproducetorqueontherotor.Inothermotors,(e.g.thed.c.motorwe havebeenlookingat),thereisnosimplephysicalargumentwhichcanbe advanced to derive the torque on the stator, but nevertheless it is equal andoppositetothetorqueontherotor. TheWnalquestionrelatestothesimilaritybetweentheset-upshownin Figure 1.10 and the Weld patterns produced for example by the electro- magnets used to lift car bodies in a scrap yard. From what we know of thelargeforceofattractionthatliftingmagnetscanproduce,mightnot we expectalarge radialforce betweenthestatorpoleandtheironbody of the rotor? And if there is, what is to prevent the rotor from being pulled across to the stator? AgaintheaYrmativeansweristhatthereisindeedaradialforcedueto magneticattraction,exactlyasinaliftingmagnetorrelay,althoughthe mechanismwherebythemagneticWeldexertsapullasitentersironorsteel isentirelydiVerentfromthe‘BIl’forcewehavebeenlookingatsofar. It turns out that the force of attraction per unit area of pole-face is proportionaltothesquareoftheradialXuxdensity,andwithtypicalair- gapXuxdensitiesofupto1 Tinmotors,theforceperunitareaofrotor 2 surface works out to be about 40N=cm . This indicates that the total radial force can be very large: for example the force of attraction on a smallpole-faceofonly510cmis2000 N,orabout200 Kg.Thisforce contributes nothingtothetorqueofthemotor,andismerely anunwel- come by-productof the‘BIl’ mechanismwe employto produce tangen- tial force on the rotor conductors. Inmostmachinestheradialmagneticforceundereachpoleisactually agooddealbiggerthanthetangentialelectromagneticforceontherotor conductors, and as the question implies, it tends to pull the rotor onto thepole. However,themajority ofmotors areconstructedwithan even numberofpolesequallyspacedaroundtherotor,andtheXuxdensityin eachpoleisthesame,sothatintheoryatleasttheresultantforceon the complete rotor is zero. In practice, even a small eccentricity will cause the Weld to be stronger under the poles where the air-gap is smaller, and this will give rise to an unbalanced pull, resulting in noisy running and rapid bearing wear. Magnitude of torque Returningtoouroriginaldiscussion,theforceoneachconductorisgiven byequation1.2,anditfollowsthatthetotaltangentialforceFdependson theXuxdensityproducedbytheWeldwinding,thenumberofconductorsElectric Motors 19 ontherotor,thecurrentineach,andthelengthoftherotor.Theresultant 1 torqueorcouple (T)dependsontheradiusoftherotor(r),andisgivenby T ¼ Fr (1:8) WewilldevelopthisfurtherinSection1.5,afterweexaminetheremark- able beneWts gained by putting the conductors into slots. The beauty of slotting If the conductors were mounted on the surface of the rotor iron, as in Figure 1.11, the air-gap would have to be at least equal to the wire diameter, and the conductors would have to be secured to the rotor in ordertotransmittheirturningforcetoit.Theearliestmotorsweremade like this, with string or tape to bind the conductors to the rotor. Unfortunately,alargeair-gapresultsinanunwelcomehigh-reluctance inthemagneticcircuit,andtheWeldwindingthereforeneedsmanyturns andahighcurrenttoproducethedesiredXuxdensityintheair-gap.This means that theWeld winding becomes very bulky and consumes a lot of power.Theearly(Nineteenth-century)pioneerssoonhitupontheideaof partially sinking the conductors on the rotor into grooves machined paralleltotheshaft,theintentionbeingtoallowtheair-gaptobereduced sothattheexcitingwindingscouldbesmaller.Thisworkedextremelywell asitalsoprovidedamorepositivelocationfortherotorconductors,and thusallowedtheforceonthemtobetransmittedtothebodyoftherotor. Before long the conductors began to be recessed into ever deeper slots until Wnally (see Figure 1.12) they no longer stood proud of the rotor surfaceandtheair-gapcouldbemadeassmallaswasconsistentwiththe needformechanicalclearancesbetweentherotorandthestator.Thenew ‘slotted’ machines worked very well, and their pragmatic makers were unconcernedbyrumblingsofdiscontentfromscepticaltheorists. Figure1.12 InXuence onXuxpaths when the rotor is slotted to accommodate conductors 1 OlderreaderswillprobablyhavelearnedthetermsCoupleandMoment(ofaforce)longbefore realising that they mean the same as torque.20 Electric Motors and Drives The theorists of the time accepted that sinking conductors into slots allowed the air-gap to be made small, but argued that, as can be seen fromFigure1.12,almostalltheXuxwouldnowpassdowntheattractive low-reluctance path through the teeth, leaving the conductors exposed totheverylowleakageXuxdensityintheslots.Surely,theyargued,little or no ‘BIl’ force would be developed on the conductors, since they would only be exposed to a very low Xux density. ThescepticswererightinthattheXuxdoesindeedXowdowntheteeth; but there was no denying that motors with slotted rotors produced the same torque as those with the conductors in the air-gap, provided that theaverageXuxdensitiesattherotorsurfacewerethesame.Sowhatcould explainthisseeminglytoogoodtobetruesituation? The search for an explanation preoccupied some of the leading thinkers long after slotting became the norm, but Wnally it became possible to verify theoretically that the total force remains the same as itwouldhavebeeniftheconductorswereactuallyintheXux,butalmost allofthetangentialforcenowactsontherotorteeth,ratherthanonthe conductors themselves. This is remarkably good news. By putting the conductors in slots, we simultaneously enable the reluctance of the magnetic circuit to be re- duced,andtransfertheforcefromtheconductorsthemselvestothesides oftheironteeth,whicharerobustandwellabletotransmittheresulting torque to the shaft. A further beneWt is that the insulation around the conductors no longer has to transmit the tangential forces to the rotor, anditsmechanicalpropertiesarethuslesscritical.Seldomcantentative experiments with one aim have yielded rewarding outcomes in almost every other relevant direction. Therearesomesnags,however.Tomaximisethetorque,wewillwant as much current as possible in the rotor conductors. Naturally we will work the copper at the highest practicable current density (typically 2 between 2 and 8A=mm ), but we will also want to maximise the cross- sectional area of the slots to accommodate as much copper as possible. This will push us in the direction of wide slots, and hence narrow teeth. But we recall that the Xux has to pass radially down the teeth, so if we maketheteethtoonarrow,theironintheteethwillsaturate,andleadto a poor magnetic circuit. There is also the possibility of increasing the depthoftheslots,butthiscannotbetakentoofarorthecentreregionof therotorironwhichhastocarrytheXuxfromonepoletoanother will become so depleted that it too will saturate. Inthenextsectionwelookatwhatdeterminesthetorquethatcanbe obtainedfromarotorofagivensize,andseehowspeedplaysakeyrole in determining the power output.

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