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Variable-speed switched reluctance motorsP.J.Lawrenson, D.Sc, Fel. I.E.E.E., C.Eng., F.I.E.E., J.M.Stephenson, &Sc,Ph.D., C.Eng., M.I.E .E., P.T.BIenkinsop, B .Sc, Ph.D., J.Corda, Dipl. El. Ing.,Ph.D., and N.N.Fu lton, B.Sc, Ph.D., C.Eng., M.I.E.E.

Indexing term: Reluctance motorsAbstract: The paper explores the theory and potential of a family of doubly salient electronically-switchedreluctance motors. It is demonstrated that the machine provides the basis for fully-controllable variable-speed systems, which are shown to be superior to conventional systems in many respects. The motor retainsall the advantages normally associated with induction motors and brings significant economy in the driveelectronics. The basic modes of operation, analysis, design considerations and experimental results from arange of prototype motors up to 15 kW at 750 rev/min are described. The most recent prototype has achieveda continuous rating which is 1-4 times that of the equivalent induction motor.

List of principal symbolsdERfSiikLLNPQRr,sTVW 1|30\pV

= rotor diameter= energy ratio (see Section 2.2.3)= frequency= interpolar airgap= instantaneous current= coefficient of inductance overlap= self-inductance of stator circuit= pole num ber= output power= number of phases= resistance of stator circuit= subscripts denoting rotor and stator, respectively= electromagnetic torqu e= applied voltage= co-energy= pole arc= rotor pole pitch= flux linkage= rotor speed= maximum value= minimum value

1 IntroductionThis paper lays general foundations for the practical designof a family of switched reluctance motors, and, further, itdemonstrates that machines in this family are capable ofextremely high levels of performance, can be controlled inexceptionally simple and flexible ways, are simple andcheap to manufacture and can offer important operationaladvantages in both industrial and domestic applications.Because of these qualities, and because of many unfamiliarfeatures, mainly arising from the highly nonlinear nature ofmost aspects of their operation, an attempt is made here,with due reference to related work, to provide a compre-hensive basic treatment of switched reluctance (s.r.) motors.Reluctance motors are most familiar with conventional,cylindrical stators and distributed, 3-phase windings and, inthis form, have been developed over the last two decades toPaper 795 B, received 2nd January and in revised form 29t h April1980Prof. Lawrenson, Dr. Stephenson and Dr. Fulton are with, andDr. Blenkinsop and Dr. Corda were formerly with, the Departmentof Electrical and Electronic Engineering, University of Leeds, LeedsLS2 9 JT, England. Dr. B lenkinsop is now with Patscentre Inter-national, Melbourn, Royston, Herts. SG8 6DP and Dr. Corda is withthe Faculty of Electrical Engineering, University of Sarajevo, 71113Sarajevo, Yugoslavia

provide high efficiencies and high specific outputscomparable with those achievable from induction motors.1They are thought of as orthodox members of the a.c.motor family, of use when true synchronous speed (orposition synchronisation) is needed and when minimumfirst cost is not an overriding consideration, though theyare also used with variable-frequency inverters to providevariable-speed drives.

By contrast, the reluctance machines discussed here areof inherently variable-speed type. They may be thought ofas 'brushless' machines having parallels with other suchmachines, either of the d.c. type, but without field excita-tion, or of the a.c. self-synchronous type, again withoutfield excitation. It will be shown that the absence of fieldexcitation does not lead to inferior performance, as mightbe imagined, but brings important advantages in costsavings with the additional advantage of the elimination ofa commutator.The motor has salient poles on both stator and rotor(i.e. it is doubly salient), the windings on the stator are ofparticularly simple form and there are no windings of anykind on the roto r. Currents in the stator circuits are switchedon and off in accordance with the rotor position and, withthis simplest form of control, the mo tor inherently developsthe to rqu e speed characteristics typica l of a series-connectedd.c. machine. However, using appropriate strategies, it isboth easy and extremely cheap to give the motor a widerange of different characteristics.The use of reluctance mo tors with roto r position switch-ing has a long history which has now come full circleWithin the last 15 years or so it has appeared frequently inthe guise of advanced, closed-loop control schemes forstepping motors and also for various special applications;prior to that it was used with mechanical contactors in low-cost single-phase versions, as in shaver motors, for example;more than 100 years ago, it was employed in the original'electromagnetic engines' with which modern variable-reluctance stepping motors have much in common. Morerecently, a number of workers have directed their attentionto various particular design developments of steppingmotors aimed at considerably increasing the power levels.

Of this work, perhaps the most straightforward extensionof stepping-motor strategies has been that by Bausch andRieke. They have briefly described 2 a 4-phase, double-stackmotor using an earlier drive system 3 to give low-speedrunning in the manner familiar in small-angle steppingmotors. They were particularly interested to apply the

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inher ent series characteris tic to a vehicle drive. Develop-ments closely related to those discussed in this paper havebeen presented by Unnewehr and Koch,4 Koch,5 and Byrneand Lacy.6 Unnewehr and Koch described a multiple-disc-rotor machine using a separate set of laminations for eachstator phase, but the mechanical complexities and cost ofthis form of construction seem to have held back itsdevelopment. Subsequently, Koch presented a theoreticaltrea tm ent of a reversible 3-phase single-stack motor of thekind discussed in the present paper, but his treatment wasseriously incomplete in that, not only was it restrictedthrough using a linear model, but it failed to recognise thefundamental effects of the negative torque developedduring part of the operating cycle. Furthermore, no con-sideration was given to the possibility of switching currentinto a winding before the onset of the period of increasingwinding in duct ance wh ich, as discussed Mow, is anessential feature of a practical system. The motor describedby Byrne and Lacy was also of single-stack type but wasrestricted to a 4-pole 2-phase stator and 2-pole rotor, adesign which is inherently unidirectional. The discussion,as in the underlying patent specification,7 gave particularatten tion to th e creation of abnorm al levels of saturationwith the objective of increasing the torque output asproposed by Jarret.8 By contrast, the present authors donot follow this method of increasing the saturation to'unnecessarily' high levels (e.g. by reducing the'packing factor' of the iron) and it will be demonstratedbelow that very high levels of specific torque are developedwell in line with those foreshadowed for stepping motors,albeit of small frame size, in the paper by Harris et al.9The work described here was initiated in rudimentaryform at Leeds University more than 10 years ago. Theobjectives were to build on the existing experience w,.hhigh-performance reluctance motors and to exploit the factthat the reluctance moto r (unlike conventional i' ..ction orsynchronous motors) can be operated with unidirectionalphase currents and, hence, with the minimum number ofswitches. (This economy in switches has, of course, beenrecognised by several of the authors referred to above andby Ray and Davis.10) Transistor or thyristor switches can beused but, in either case, the system costs are minimised.These costs can be expected to be significantly below thoseof equivalent indu ction mo tor systems and, in a number ofcases, can already be seen to be below those of equivalentd.c. motor systems.

Following early work on 1-phase mo tors , detailed studieshave been made since 1972 of lower-power motors (from10W to over lkW at various speeds up to 10 000 rev/min)using mainly transistor drive circuits. Since 1975 a substan-tial programme, in close collaboration with colleagues fromNottingham University, has been in progress, directedtowards the development of a 50 kW drive (using thyristors)for a battery-powered road vehicle. The part of this pro-gramme at Nottingham has been oncerned with electroniccircuits for use with s.r. moto rs and, in their recent paper,10Ray and Davis have presented a study, using a linear model,of switching device rating and of control techniques. Thatpaper and this one were prepared essentially as companionpapers. Discussion of circuits here is accordingly limited tothe minimum of principles necessary to discuss the opera-tion of the motor.In the following Section the basic principles of operationof the motor (including the patterns of energy flow) aresummarised, and a concept appropriate in place of power

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factor is introduced. Section 3 describes fundamentaldesign considerations raised by the motor. This is followedin Section 4 by a brief discussion of the performancecharacteristics, both the inherent one and the manyachievable ones, and of the methods of controlling and s.e.characteristics. The highly nonlinear nature of the machineand some of the important implications of this for practicalrealisations of the system are discussed in Section 5. InSection 6, experimental results for different machine sizesand types demonstrate the considerable attractiveness ofthe s.r. motor system for variable-speed applications. Theseresults give particular attention to the specially demandingcharacteristics required for an electric-vehicle drive,including a constant-power capability over a wide speedrange.2 Basic principlesFig. 1 shows the basic elements of a doubly-salient reluc-tance motor, where it is to be understood that only two ofa larger number of stator poles are included. The motorcomprises a single stack and both stator and rotor areconstructed from laminations. Most simply, diametricallyopposite stator poles carry coils connected in series to givea 2-pole field pattern. In this Section, the principles ofoperation are explained and the parameters fundamental tothe operation are L entified.2.1 Inductance variation and torque productionTorque is developed by the tendency for the magneticcircuit to adopt a configuration of minimum reluctance,i.e. for the rotor to move into line with the stator poles andto maximise the inductance of the excited coils. Note thatthe torque is inae lendent of the direction of current flow,so that unidirectional currents can be used, permitting asimplification of the electronic driving circu its. In general,because of magnetic nonlinearities, the torque T mustbe calculated in terms if co-energy W 'as

bW (0 , / )30 (1 )

Fig. 1 Elements of a doubly-salient reluctance motorIEEPROC , Vol. 127, Pt. B, No. 4, JULY 1980


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where 0 is the angle describing the rotor position (see Fig.l)and / is the current in the coils. Note that changes in co-energy depend both on the angular position of the rotorand on the instantaneous value of the current. Properallowance for both of these factors must be made inpredicting performance, but it is helpful to consider first asimplified model in which magnetic nonlinearity is neglectedand eqn. 1 can be simplified toi2 dL2 d0 (2)

where L is the self inductan ce of the circuit at any value of 0.Fig. 2a shows the variation of inductance with rotorposition for the pair of sta tor poles shown in Fig. 1, idealisedin that magnetic saturation and the 'rounding'effect of thefringing fields are neglected. The number of cycles ofinductance variation per revolution is proportional to thenumber of rotor pole pairs, and the 'length' of the cycle isequal to the rotor pole pitch 0. The physical significance ofthe different regions R of the variation needs to berecognised for later discussion:R01 : at 0O the 'leading' edges of rotor poles meet theedges of stator poles and the inductance starts a linearincrease with rotation, continuing until the poles are fullyoverlapped at Q x, when the inductance reaches itsmaximum value Lmax.Ri2 ' from Qx to 02 the inductance remains constant atLmax, through the region of complete overlap. This regionis generally known as the 'dead zone'.K23 : from 02 to 03 the inductance decreases linearly 'to the minimum value, Lmjn ./? 3 4 : from 03 to 04 the stator and rotor poles are notoverlapped and the inductance remains constant at Lmin.The associated variation of torque for a constant coilcurrent follows from eqn. 2 and is shown in Fig. 2b . Thetorque can be controlled to give a resultant which is

e 0 e, e 2 e3

e22TTrotor angle, 9

Fig. 2 a Variation of inductance of one stator circuit as a func-tion of rotor positionb Variation of torque with cons tan t cu rren t as a functionof rotor position

positive (i.e. motor action) or is negative (i.e. generatoraction) simply by switching the current in the coil on andoff at appropriate instants during the inductance cycle.There seems to have been an assumption in many earlierpublications that maximum operating torque and power areachieved b y switching the current on only when th e rotor isin the region R0l. It is important to recognise that this isnot so. For example, to maximise motoring torque, currentmust be switched on when the rotor is in the region R34 or2.2 Energy flow and current and flux waveforms2.2.1 Switching circuits and energy flow: Fig. 3a shows asimple form of switching circuit for controlling the currentin the stator coils which, though simple, is sufficientlygeneral to study the possible patterns of energy flow in themotor. At this stage it remains sufficient to consider only asingle circuit on one pair of stator poles (along with onepair of rotor poles). When the switch S is closed and therotor is stationary, current builds up in the winding andenergy is taken from the d.c. source 1. When S is opened,the current continues to flow, but now through the diodeso that the stored magnetic energy is transferred to d.c.

d.c.source 1motor

motor

F ig. 3 a Essen ial elemen ts of a general switching circuitb Circuit with two switches per coil pairc Grcuit with one switch and bifilar-wound coilsIEE PROC , Vol. 127, Pt. B, No. 4, JUL Y 1980 255


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source 2. The general equation governing the flow of statorcurrent may be writtendi//dt (3 )

where v is the voltage (of appropriate polarity) appliedacross the winding and \JJ is the flux linking the coil. Againit is helpful to consider first a simplified model which againassumes magnetic linearity and also negligible resistance.On this basis eqn. 3 may be rewritten

di dLv = L + i codt dd (4 )where co (=d0/dr) is the speed of rotation. The rate of flowof energy is given by

vi= \Lvdt\2.2 dL+*-2 dd co (5 )

This equa t ion show s tha t , w hen r unning as a m o t o r , th einput e lectr ical power goes par t ly to increase th e s tor edmagne t ic ene r gy ( ] Li2) an d par t ly to pr ovide mechanica l

o u t p u t p o w e r i2 dL co,) the la t ter being associated

w i t h th e ' m o t i o n a l e.m.f.' in the s tator c ircui t . Thus , withthe sw i tch S closed dur ing th e region of r is ing inductanceR01, par t of the energy f rom source 1 is conver ted in tom e c h a n i c a l o u t p u t an d par t is s tor ed magne t ica l ly ; but w i t hS open dur ing R01, s tored magnetic energy is par t ly con-ver ted to m e c h a n i c a l o u t p u t and par t ly t ransfer red tosour ce 2. If cur r en t is still flowing during th e ( c o n s t a n t )m a x i m u m i n d u c t a n c e p e r i o d , th e energy is s imply trans-ferred to sour ce 2. F i n a l l y , if current f lows during th edecr eas ing induc tance pe r iod , ene r gy is t ransfer red to thesour ce not only f rom that s tored magnetical ly but a l so ,t h r o u g h th e gener a t ion of negative torque, f rom a m e c h -an ica l sour ce , w hich is the c o n d i t i o n of regenerat iveo p e r a t i o n .

The poss ibi l i ty of f u l l f our - quadr an t oper a t ion , cont r o l leds imply th r ough the ins tan t s at which circui ts are sw i tched( w i t h o u t th e famil iar inconveniences of c o n t a c t o r s an dr ever s ing a r r angements ) , is an impor tan t f ea tur e of thepr esen t sys tem.

F igs . 3b and c show dif ferent forms of switching circui t .In th e circui t of Fig. 3b, clos ing Sa and Sb al lows energy tof low into th e w inding and opening b o th sw i tches causes th ec u r r e n t to f low through D a and D b , so t h a t the s tor edener gy is r e t u r n e d to the ( s ingle) supply. In the circui t ofFig . 3c it has been poss ible to halve th e n u m b e r of switchesand d iodes by the i n t r o d u c t i o n of a bif i lar winding of thes t a t o r (i.e. a pr imar y an d secondar y c i r cu i t w ound toge the ron each pole for max imu m coupl ing) . W hen S is c losed ,cur rent f lows into the p r i m a r y {Lx, Rx), and w h e n S iso p e n e d th e pr imary cur rent fa l ls abruptly to zero but acor r esponding cur r en t is es tab l i shed in the secondar y(L 2, R2) so as to maintain constant f lux l inkages ( s toredmagn e t ic ene r gy) . This s econdar y cur r en t , f lowing th r oug hD , r e tur ns ener gy to the supply. This c ircui t of Fig. 3c isthe mos t economica l f o r m, r equi r ing on ly one e lec t r on icsw i tch pe r coil pair , but a pr ice for this great s implici ty has

t o be paid in tha t the voltage to be w i t h s t o o d by the switchis twice th e supply voltage (assuming an equa l number oft u r n s on pr imary and secondar y) and, of c o u r s e , an extraw inding is required on the p o l e .

Results are pr esen ted in Sec t ion 5 for machines oper a tedwith circuits basically of the t y p e s h o w n in Figs . 3b and c.2.2.2 Current and flux waveforms: All cur r en t and fluxw avef or ms are w hol ly nons inuso ida l and those of cur r en tvary widely with operat ing condit ions . Fig. 4 show s dia-grammatical ly one cycle of the w avef or ms of cur r en t andf lux (in accordance with eqn. 4), typ ica l of the i m p o r t a n tmotor ing condi t ion in w h i c h th e w inding is switched on atsome angle 0., in advance of the o n s e t of the risinginductance region (Sect ion 2.1). The ef fect ive inductanceof th e circuit is Lmin, ini t ia l ly, so al lowing th e cur r en t tobuild up rapidly to its maximum va lue (and to maximise itsto r que- pr oduc ing e f f ec t ) . Subsequent ly , the r is ing induct-ance and the m o t i o n a l e.m.f. cause th e cur r en t to fall untilthe switch is opened at some angle dx, ( typical ly) beforethe maximum induc tance is r eached . Ther ea f te r , thecurrent fa l ls more rapidly because an oppos ing po la r i ty isapplied to the w inding by vir tue of the current f lowing intothe supply . N ote , in accor dance w i th th e i n t r o d u c t o r yc o m m e n t s , t h a t the di r ec t ion of cur r en t dur ing the cyclenever reverses. The angle be tw ee n dt an d 6X is referred to asthe conduction angle 6C, and is of cons ide r ab le impor tancefo r th e cont r o l of the machine ( Sec t ion 4) .

The s implici ty of the waveform of f lux linking thew inding is par t icular ly interes t ing; recal l ing that , for themoment , cons ide r a t ion is being res tr ic ted to the case ofzero res is tance, as long as a posi t ive constant vol tage isappl ied , th e flux increases at a cons tan t r a te (see eqn. 3)and, conversely, when a constant negat ive voltage is appliedthe flux decreases also uni form ly. The maximum f luxalways occurs at the ins tan t of switch-of f def ined by 6 .

F ig. 4 Current and flux waveforms derived using a linear variationof inductancecurrentfluxinductance

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2.2.3 Phase relationship and 'power factor': The abovewaveforms clearly indicate fundamental differencesbetween the switched reluctance motor and conventionala.c. machines. Two other distinctive differences should betaken up at this stage: the first is the straightforward oneof the relationship between currents in different statorcircuits; the second is the more subtle one as to whatconcept is appropriate in place of power factor.In connection with the first, it has simply to be notedthat the stator circuits are not 'phases' in the conventionalsense, there being no automatically fixed relationshipbetween the time when one 'phase' is switched off andanother is switched on. The conduction time of a 'phase'winding is chosen in relation to the inductance variation ofthat winding and to the required operating conditions. Theconduction periods of adjacent phases may or may notoverlap. The operation of the multiphase motor can bedetermined by combining the current and torque (and flux)functions as shown in Figs. 2 and 4 for the individualphases. Appropriate allowance must be made for theangular displacement between adjacent pairs of stator polesand, of course, for the magnetic nonlinearity of the

machine.Turning to the matter of the 'quality' of performancewhich, for a.c. machines is measured by power factor, itseems simplest and most helpful to seek a direct measure interms of the useful energy flow as a proportion of thetotal energy flow (which includes, of course, the circulatingmagnetic field energy). This leads to the concept of an'energy ratio' defined, over a cycle, astotal energy supplied to machine -energy returned to supplyER total energy supplied (6 )

The significance of this, as with power factor in a.c. systems,is particularly in relation to device ratings and costs. (Notethat, of course, device ratings depend on several factors inaddition to ER/PF, e.g. voltage rating, current waveform,duty cycle.) The ER values, as will be seen from Section 6,are generally very satisfactory and lead to total deviceratings which are less than those of induction-motor-basedsystems. (The detailed discussion of electronic circuitdesign will be the subject of a future publication by theauthors' colleagues at Nottingham University.)3 Fundam ental design considerationsThe discussion so far has been restricted to basic aspects ofbehaviour in the context of the simplest model of an s.r.motor. It is necessary now to explore the questions ofdesign strategy which immediately arise when a practicalsystem is contemplated. The number of questions is verylarge because many of the constraints which 'simplify'a.c. machines do not apply: the number of phases to beemployed is open to choice between one and many; theratio of phase number to stator pole number is not fixed;the ratio of rotor pole number to stator pole number isopen to a wide variety of choices; the 'best' values of rotorpole arc and stato r pole arc have to be considered; at tentionhas to be paid to starting capability; matters concernedwith core losses, switching frequencies and varying fluxdistrib ution s in different parts of the magnetic circuit haveto be studied. It is not possible in the context of this paper

to cover all these questions adequately, nor can the impli-cations of different applications be explo red, but several aredealt with below in broad terms. Atten tion is given only t oself-starting reversible designs.A basic observation is fust made concerning mutualinductance. It is known from stepping motor studies that itis desirable, from the point of view of maximising machineoutput, to eliminate mutual inductance between phases.Careful consideration will show that mutu al inductance dueto main (airgap) flux will indeed be zero for the features ofthe model assumed so far, namely:

(a ) the iron is effectively infinitely permeable(b ) the stator poles are excited in diametricallyopposite pairs(c) the rotor has an even number of poles.3.1 Necessary conditions on pole n umbers and pole arcsConsiderable detailed argument is involved in establishingthe conditions which must be satisfied by the numbers andthe arcs of the stator and rotor poles. However, taking intoaccount:

(i) the above point about mu tual inductance(ii) the possible basic patterns of 'repeatability'between relative stator and rotor pole dispositions(iii) the need to minimise the permen ance associatedwith Lmin(iv) the requirement for self-starting capability in eitherdirection from any rotor position(v) the desirability of minimising switching frequ ency,it can be shown, for pole num bers, thatVv,Ny) = qN y (7 )

andLCM (N s,N r) >Ns>Nr (8 )where N s and N r are even, the number of phases q is greaterthan 2 and the symbol LCM denotes the lowest commonmultiple. It can also be shown, for pole arcs, that

(9 )and

2TT (10)

3.2 Preferred values of pole numbers and pole arcsEqns. 710 define necessary conditions to be satisfied byN s, N r, (5S and fir, but further consideration is necessary todetermine the values of these parameters to give a 'good'design. First, consider (3 S and the constraints on its pos-sible values. When ps > 0r, $r = 2n/qNr, so that

2n (11)

When(12)

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The range of B. is therefore277 (13)

and similarly it can be shown that the range of fi r is- v 2TT / 2

(14)Thus, both /3r and B s have the same range (increasing withincreasing q and decreasing with increasing N r), althoughthey are at the same time governed by

2TT 05)These relationships may be shown diagrammatically as inFig. 5, where the limits of combinations of the values ofBr and Bs are denoted by the sides of the triangle XYZ. Atpoint X (where the machine would contain the leastamo unt of iron) the inductance variation has no dead zone(since B s = B r) and is of the form shown in Fig. 6a . As thegeom etry changes from the po int represented by X alongXY or XZ to Y or Z, a dead zone appears and the lengthof the minimum inductance period diminishes accordinglyuntil, at Y or Z, the inductance has the form shown inFig. 6b . Point Z corresponds to a machine with maximumwinding space (as at X) and point Y to a machine withzero winding space (since stator pole arc = stator polepitch when Bs = B~ s). The latter is not physically realisable,but it should be noted that keeping Br towards \ implieslow rotor inertia.A special case should be noted at point W, where B s= Br =0/2. At this point, both the period of minimum inductanceof Fig. 6a and the dead zone of Fig. 6b have vanished, asshown in Fig. 6c . Any geometry corresponding to a pointinside the triangle has an inductance pattern which exhibitsfeatures of all three patterns shown, depending on theposition of the point. The line of Bs = Br divides the triangleas shown into regions fy > Br and B s Bs and there-fore lie in the minor triangle of XWZ. It can be noted inpassing that any such practical design will have a corres-ponding cou nterpart in the minor triangle XWY (where itis represented by a point symmetrical about the line XW)and will have the rotor and stator pole arc values inter-changed.Attention is turned now to the number of rotor poles.As the torque at any point is proportional to the slope ofthe inductance, it follows th at, to increase torq ue, thepermeance corresponding to Lmin should be made as smallas possible. It can be shown that if the airgap due to theminor rotor diameter (dimension #; in Fig. 1) is greater thand{(p-Br)j2, the inductance mainly depends on the rotorinterpolar arc (0-/3,.) since fringing effects become pre-dominant in the value of inductance. Hence,(


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num ber of difficult quest ions when detailed motor designand system performance have to be considered. Fig. 7illustrates, by way of example, the voltage and flux wave-forms in various parts of the magnetic circuit for a 3-phase6/4 pole machine (in accordance with eqn. 3 with R = 0).The voltage switching frequency of any phase f h is (seeFig. la):

and of the supply is(17)

(18)The period and form of the flux variations in the iron varybetween the d ifferent parts of the magnetic circuit, andmay be deduced by superposition of separate variations, ofthe type shown in Fig. 4, corresponding to the voltagesapplied to the individual phases. It will be seen that thestator poles experience unidirectional flux pulsations (thesense of which depends on the sense of the winding connec-tion) of frequency fph .

The flux patterns in the stator back iron are illustratedby considering the back iron divided into sections boundedby pole centre lines. Assuming that the winding connectionsare 1, 2, 3 on adjacent poles, followed by 1 ', 2 ', 3' onopposite poles, then the sections between the centre linesof poles 1 and 2, 2 and 3 , l' and 2 ', 2' and 3' will have fluxwaveforms of the type shown in Fig. 7c. The waveform ismade up of a triangular component of frequency f h and ad.c. component, the sense of which depends on theparticular section. The remaining sections, i.e. between thecentre lines of poles 3 and l', 3' and 1 have the type ofwaveform shown in Fig. Id , which has a d.c. componentwith a 'ripple' component of frequency fappliedphasevoltages

statorpole flux(phase 1)

statorcore flux (1)

nmmnmmnmmnmm

rotorpole flux

rotorcore flux

Fig. 7 Voltage and flux waveform s for a 3-ph ase 6/4 pole s.r.motor

The poles of the rotor experience bidirectional pulseswhich have an overall frequenc y of co/2vr as shown inFig. le , and the core of the rotor experiences bidirectionalpulses of flux as shown in Fig. 7/, which again have anoverall frequency of cofl-n, although the waveform isconsiderably more complex and contains higher-orderfrequencies. It is because the rotor sees a minimum fre-quency CJ/27T that it has to be laminated.Any stator with N s teeth will have (Int (NJ4)+1) differentwaveforms occurring in the core (this is derivable from thesymmetry of the magnetic circuit) and, therefore, a 4-phasemachine has 3 different waveforms, compared to 2 wave-forms for the 3-phase case. In the same way as noted forthe 3-phase case, those waveforms with d.c. bias occursymmetrically about the core with opposite bias. Insummary, the following can be stated:(a) The switching frequency is very dependent on thethe num ber of phases (e.g. a change from 3-phase to 4-phasedoubles the frequency).(b) The core loss associated with these frequencies willbe exceedingly complex (and may defy realistic prediction).(c) Losses in the driver unit will increase with rising-numbers of phases.

If efficiency is imp ortan t in a particu lar appli catio n, e.g. ina battery traction drive, then the above relationships implythat for normal motor speeds the number of rotor polesand the number of phases must be kept as low as possible.(Reduction of the number of phases may well also minimisethe cost of the associated power-switching devices.)However, if a low-speed application is being considered, thelimita tions impo sed by frequency will be less one rou s, andthe designer has much greater freedom in his choice ofnumbers of poles and phases.

3.4 Choice of number of phasesThe choice of phase number is influenced in a major wayby the required starting torque (and hence the effectivevalues of dL/dd). To ensure adequate starting torque at allrotor angles there must be adequate 'overlap' between theL{6) variations of 'adjacent' phases. The adequacy of theoverlap can depend significantly on the effects of fluxfringing which lead to rounding of the corners of the L{6)function and to reduced values of dL/dd (and hence torque)at either end of the rising inductance period R0l (seeSection 5). However, useful guidance about the effects ofoverlap can be inferred from the ideal L(d) variations asfollows:If kL is defined as the ratio of inductance overlap of twoadjacent phases to the angle over which the inductan ce ischanging, then, from Fig. 2a,

" Bo) ~min(fl)

2TT

(19)

(20)

The variation of kL is best described by a diagram, sincevalues of A and /^ are, to some extent, dependent on q . Itis shown in Fig. 8 as a function of (5S, where /35 variesIEEPRO C, Vol. 127, Pt. B, No. 4, JULY 1980 259


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between the m aximu m and minim um values allowed by acombination of q an d N r. Curve A corresponds to q = 3 ,N r = 4 and curve B corresponds to q =4 , N r = 6. It will beseen that higher values of kL are achieved at relativelylow values of (i s for the 4-phase case. For example,M l and M2 mark the midpoints of the two ranges of /3S,and it will be seen that the 4-phase machine has 20% moreoverlap. Put another way, for a specified amount of over-lap, the 4-phase machine can work much closer to ,a condition which has already been demonstrated to bebeneficial. To this extent, the 4-phase machine allowsgreater flexibility in design and will have the better startingperformance.4 Con trol and performance characteristicsIt is helpful in discussing the control of the s.r. motor tobegin by considering its 'natur al' or 'inherent ' characteristics,i.e. that which occurs under conditions of fixed supplyvoltage and fixed switching angles. Linear analysis showsthat the torque/sp eed curve is then the same as that of ad.c. series motor.11 '12 This can readily be seen to followfrom the fact that, as the motor slows down, the timeduring which a phase winding is excited increases inverselywith the fall in speed and, therefore, so also does the flux.The torq ue, ho wever, as in all such devices, is propo rtionalto the square of the flux and so the resulting torque/speedcurve is defined by

kT=. c o 2

and the power is given bykP= CO

(21)

(22)The analogy with the d.c. series machine immediatelypoints to the possibility of control through terminal voltageor supply current, and both of these can be readily imple-mented in the overall system. Additionally, however, thereare two further important parameters available to thedesigner. These are the switch-on angle 6{ and the switch-offangle dx (or its equivalent, the conduction angle 0 C).Control of these angles is easily and economically achieved,

1 0

0 8

0 6

0

0 2

T I / 6 -TT/3 TT/2

involving only the appropriate conditioning of timing pulses,and makes available a very wide range of performancecharacteristics and control possibilities. In practice, ofcourse, control parameters are chosen so as to optimiseoverall system performance (e.g. to minimise currents or tomaximise efficiency), as well as to achieve particularcharacteristics. It is impossible to set out here the fulldependence of performance on the control parameters, butbasic modes of operation are explained briefly below.The first mode is the natural one with fixed supplyvoltage and fixed switching angles. There is, of course, afamily of series characteristics for varying supply voltages(at a given speed the flux is proportional to the voltage V,and the torque varies as V2), with an upper limitingcharacteristic set by maximum rated voltage. As the speedfalls, the flux rises and 'base speed' coft is defined as thatwhich corresponds to maximum flux (and current) atmaximum voltage. Base speed is the lowest speed at whichmaximum power can be obtained and the highest speed formaximum torque.

The second important mode of operation involves thecontrol of speed below co b. By analogy with the d.c. seriesmotor, this can be achieved by varying the effective appliedvoltage either literally, using a variable d.c. link, or bymodulation or chopping using the main switching devices.As in the d.c. machine, a constant to rque characteristic canbe obtained in this mode by 'current limit' current chop-ping, or the torque can be varied by contro l of the choppinglevel. An alternative possible means of controlling speedbelow cj b with the s.r. motor is to reduce the conductionangle 6C at constant voltage, but this entails an increasingpeak flux for constant average torque.The third important mode of operation is that ofcontrolled speed above co b. Although, as explained above,torque falls 'naturally' as co~2, the natural fall in torquewith increasing speed can be offset by increasing the con-duction angle proportionately. The conduction time doesnot then fall as co" 1, nor does the flux, and a variety ofcharacteristics can be obtained. A characteristic of particularinterest (especially for traction applications) is that ofconstan t power over a range of speeds. This requires theflux to fall as to', so that the torque falls as co"1. As theconduction angle is increased, there comes a point at whichthe 'switched o n' period for a phase winding is equal to the'switched off period (d c = 0/2) and, for an ideal case ofzero resistance, if 6C becomes larger than this, the flux levelwould tend towards infinity (because a net d.c. voltagewould be applied to a purely inductive circuit). This maybe taken as defining the upper limit of the third mode ofcontrol, but it will be seen in Section 6 that, within thislimit, a wide speed range of constant power operation canbe achieved with excellent overall performance. At speedsabove the highest speed obtained at constant power in thethird control mode, the motor reverts to its 'natural'characteristic appropriate to 6C = 0/2.

It will be appreciated that changes in dc as discussed inthe context of constant torque and constant power opera-tion can be achieved by associated changes in 6( and 8X, and,the proper choice has to be made in relation to applicationand performance optimisation.Fig. 8 Variation of inductance overlap coefficient k^ with statorpole arcA q = 3,N r = 4B q = 4, N r = 6

5 Nonlinearities and performance predictionIn predicting the performance of conventional machines itis usually satisfactory to use a model in which the par-26 0 IEEPROC , Vol. 127, Pt. B, No. 4, JULY 1980


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ameters are constant, provided that these constant valuesare also 'effective' values allowing appropriately formagnetic nonline arities and o ther influences, such as skineffect etc. (Transient performance computations continue,of course, to involve nonlinear equations, because of thespeed dependence, though they can be handled withvarying degrees of simplification depending upon the infor-mation required.) With the s.r. motor, however, behaviourdepends so much on the nonlinear 'magnetic' parametersthat no corresponding simplification using constant'effective' param eters is acceptab le. It becomes essential forall serious design work to recognise the dependence ofinductance on instantaneous current as well as position.Consideration is given, first, to the nonlinear nature of theflux-linkage, inductance and static torque characteristicsand, secondly, to performance predictions using thesecharacteristics. The basic data presented were measuredusing a representative 4-phase motor built into a D90induction-motor frame.5.1 Static characte ristics: flux-linkage inductance andtorqueTypical variations of flux-linkage with current, inductancewith angle and torque with angle are shown in normalisedterms in Fig. 9a, b and c, respectively. The very significantdivergence of these characteristics from the ideal ones (asin Fig. 2 for L(d) and T(6)) is immediately apparent. Onlyat very low values of excitation or for rotor positions nearthe middle of the minimum inductance region R 34 do theyapproximate to the ideal: for currents above 0-15/ the \p/irelationship ceases to be linear; the rising section of theinductance curve ROi becomes very rounded at both ends;and the torque curve quickly loses its rectangular shape.The differences are owing, of course, mainly to sa turati oneffects in the magnetic circuit, but near the ends of regionROi the influence of flux fringing is significant.Two distinct effects of saturation must be recognised.The first is the 'bu lk' effect on the magnetic circuit as awhole when the excitation is raised to practical levels, andthis is similar to the effect in other types of machine. Thesecond is the effect of the intense 'local' saturation in thepole tips when rotor and stator poles are only partiallyoverlappped. It is particularly severe when the degree ofoverlap is small and, as known from experience withstepping motors,0 requires careful consideration.13 Ingeneral, both local and bulk effects are present and interact,but their effects can be isolated by observations at particularrotor angles in Fig. 9a .

Bulk saturation effects can be observed best for positionswith full overlap between stator and rotor poles (whenthere is no influence from local saturation). From Fig. 9a itcan be seen, considering, say, the curve for 0 = 0-380, thatbulk saturation occurs at all flux levels in excess of 0-51//;from Fig, 9c the associated large loss of torque at largeoverlap and high current is apparent. Clearly, local satura-tion can only be separated from bulk saturation below thisflux level of 0-5^, and is best observed at small amounts ofoverlap. It is most evident in the nonlinearity of the i//-zcurves for OO50


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bifilar-wound type s, with power ou tpu ts ranging from 10Wat 10 000 rev/min to 25kW at 750 rev/m in. For ease ofconstruction, and to permit unambiguous comparison withconventional motors and systems, several of these havebeen built into commercially available induction-motorframes. They have also employed airgaps which are con-sistent with conventional techniques of induction-motormanufacture. The smaller motors have employed tran-sistor convertors only up to a rating of 1 -5kW, although,with the transistors now available, this type of convertor

could be used at much higher powers. Alternatively, gate-turn-off thyristors could be used. The larger motors (lOkWand above) have been developed specifically for battery-powered traction applications, and have employed th yristorconvertors developed by colleagues at the University ofNottingham. This Section presents representative results ofthe authors' experience.Fig. 11 shows the measured performance of a 3-phasenonbifilar wound motor in a D90 frame, having N r = Aand Af = 6. The core size was identical with that of the

0 -25 0-50n o r m a l i s e d c u r r e n t , i / i0 7 5

1 00- 0 07,

normal ised angle , 9 /


11/13

standard ind uction moto r in the frame, and the same airgapdiameter was retained (to facilitate comparisons), eventhough these dimensional choices are unfavourable to thes.r. motor. The Figure shows the measured power/speedcharacteristics and values of measured efficiency atimportant load points. The characteristics show regions ofconstant torque, constant power and series-motor roll-off,2 0 r

- 0 18 0 0 13normalised angle, 9/


12/13

It is to be no ted th at, although only motoring character-istics have been presented here for clarity, regeneration hasbeen achieved equally successfully over the whole torque/speed range. These machines used digital control of switch-ing angles, a method which is very convenient for researchand development purposes. The method of rotor-positiondetectio n is extremely simple; a pair of optical switches andan appropriately slotted disc clamped to the rotor shaft arethe only components required on the motor, and from theresulting signals appropriate firing pulses are generated bythe electronic controller.7 ConclusionsA family of variable-speed drives has been described whichuses doubly salient rotor-position-switched reluctancemotors in conjunction with various forms of electronicpower convertor. Particular attention has been paid tovarious highly nonlinear features, which do not occur inconventional a.c. or d.c. machines, with a view to laying ageneral foundation for the practical design of s.r. mot ors.

The motor retains all the advantages of robustness,cheapness, safety and minimal maintenance associated withthe cage induction motor, and permits a power-convertordesign which uses only half the number of main switchingdevices needed in the normal inverter for induction motoruse. In terms of cost, therefore, the new system is signifi-cantly cheaper than equivalent, a.c. motor based, variable-speed systems, and it is expected to retain this advantage asrelative costs of machine and electronic components vary.It is also cheaper than many d.c. motor based systems and,with the high proportions of the cost of such systems beingin the d.c. motor, the s.r. motor system should gain pro-gressively. One example of this is in the application toelectric-vehicle drives which has received attention inassociation with colleagues at Nottingham University.The performances already obtained from s.r. systemsshows them to be attractive over a wide range of sizes formany applications. Ratings and efficiencies of total s.r.systems have been obtained which better those of a standardinduction motor in the same frame, even when fed from afixed-frequency sinusoidal sup ply. Indee d, recent resultshave been obtained in which the continuously-rated specific

output of a 15kW s.r. motor is some 1 -3 - 1-4 times higherthan that of the equivalent induction motor.It is also appropriate to emphasise the range and con-venience of control which s.r. systems permit; they arefully reversible, regenerative and provide very flexibletorque/speed characteristics merely through appropriatetiming of the switching angles, giving full 4-quadrant opera-tion with no additional convertor comp onents. Additionally,of course, they have the sought-after operational advantages

Fig. 13 Current waveforms for the same motor as Fig. 12, lOkWoutput:a 750rev/min ER = 0-72b 1500 rev/min ER = 0-84C 2250 rev/min ER = 0-87X-scale: 1 ms/div; .y-scale: 20A/div

Fig. 14 Measured flux waveforms of a 4-phase s.r. motora Waveform corresponding to the 3-phase waveform of Fig. 7c& Waveform corresponding to the 3-phase waveform of Fig. id

264 IEEPROC , Vol. 127, Pt. B, No. 4, JULY 1980


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of being 'brushless' designs. It seems entirely reasonable toconclude that, with fuller development either in general orin relation to specific applications, the new system willprove commercially and technically very attractive.

8 AcknowledgmentsAcknowledgment is due to the UK Science ResearchCouncil and the University of Leeds for the financing ofDr. P.T. Blenkinsop and Dr. J. Corda, respectively, to enablethem to work on the theory and design of switched reluc-tance motors.

The development of the traction system, to whichreference has been made in the paper, involved extremelyclose collaboration between those responsible for themotor. The authors wish to acknowledge the part played inthe success of this development by Mr. Davis and Mr. Ray,and more recently Mr. Blake of the University ofNottingham. They also wish to thank Chloride TechnicalLtd. for financing the combined project, and in particularDr. M.F. Mangan for many constructive discussions. Theauthors are indebted to Brook Crompton Parkinson Ltd.for their assistance, particularly in the provision of data oninduction machines.

} References1 LAWRENSON, P.J.: 'Synthesis and performance of improvedreluctance motors'. Proceedings of the international conferenceon electrical machines, London, 1974, pp. C3-1 - C3-102 BAUSCH, H., and RIEKE, B.: 'Performance of thyristor-fedelectric car reluctance machines'. Proceedings of the internationalconference on electrical machines, Brussels, 1978, pp. E4/2 -1 -E4/2-103 BAUSCH, H., and RIEKE, B.: 'Speed and torque control of

thyristor-fed reluctance motors'. Proceedings of the internationalconference on electrical machines, Vienna, 1976, Part I, pp.128-1 - 128-10

4 UNNEWEHR, L.E., and KOCH, W.H.: 'An axial air-gap reluctancemotor for variable-speed applications', IEEE Trans, 1974,PAS-93, pp.367 - 3765 KOCH, W.H.: Thyristor controlled pulsating field reluctancemotor system', Electric Machines and Electromechanics, 1977,1, pp. 201-2156 BYRNE, J.V., and LACY, J.G.: 'Characteristics of saturablestepper and reluctance motors', in 'Small electrical machines'.IEE Conf. Publ. 136,1976, pp. 93-967 BYRNE, J.V., and LACY, J.G.: UK Patent 132111 0, 19738 JARRET, J.: 'Machines electriques a reluctance variable et adents saturees', Tech. Mod., 1976, 2, pp.78-809 HARRIS, M.R., ANDJARGHOLI, V., LAWRENSON, P.J.,HUGHES, A., and ERTAN, B.: 'Unifying approach to the statictorque of stepping-motor structures', Proc. IEE, 1977, 124, (12),pp . 1215-122410 RAY, W.F., and DAVIS, R.M.: 'Inverter drive for doubly salientreluctance motor', IEE. J. Electr. Power AppL, 1979, 2, (6),pp . 185-19311 BLENKINSOP, P.T.: 'A novel, self-commutating, singly-excitedmotor'. Ph.D. thesis, University of Leeds, 197612 CORDA, J.: 'Switched reluctance machine as a variable-speeddrive'. P h.D. thesis, U niversity of Leeds, 197913 ERTAN, H.B., HUGHES, A., LAWRENSON, P.J., and HARRIS,M.R.: 'A new approach to the prediction of the static torquecurve of saturated VR stepping motors'. Proceedings of the 8thincremental motion control systems and devices symposium,University of Illinois, May 197914 SIMKIN, J., and TROWBRIDGE, C.W.: 'Three dimensionalcomputer program (TOSCA) for nonlinear electromagneticfields'. Paper RL-79-097, SRC Rutherford Laboratory, Dec.197915 CORDA, J., and STEPHENSON, J.M.: 'An analytical estimationof the minimum and maximum inductances of a double-salientmotor'. Proceedings of the international conference on steppingmotors and systems, Leeds, 1979, pp.50-5916 HUGHES, A., LAWRENSON, P.J., STEELE, M.E., andSTEPHENSON, J.M.: 'Prediction of stepping motor perform-ance'. Proceedings of the international conference on steppingmotors and systems, Leeds, .1974, pp.67-7617 STEPHENSON, J.M., and CORDA, J.: 'Computation of torqueand current in doubly salient reluctance motors from nonlinearmagnetisation data', Proc. IEE, 1979, 126, (5), pp.393-396

18 CREIGHTON, G.K., SMITH, I.R., and MERGEN, A.F.: 'Lossminimisation in 3-phase induction motors with p.w.m. invertersupplies', IEEJ. Electr. Power Appl, 1979, 2, (5), pp. 167-173

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