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The design parameters of the motor are as follows:
topological configuration shown in Fig. 1total air gap = 1/8 inblade thickness = 3/8 innumber of turns on the core = 600coil resistance = 0-25 Q
The motor, as described above, is not self-starting and hasto be started manually. In the above, the 6-blade 6-coil motorsystem is purely arbitrary. In fact, for the maximum utilisationof space, the maximum number of blades and coils should beused, because the total torque produced by the motor is the
sum of the torques produced by each coil-blade set.
Test resultsA 6-blade 6-coil motor was built, based on the above
principles. The approximate blade shape is shown in Fig. la,and they move in C shaped core. The motor runs on n o loadand on load, and a typical speed/torque curve is shown inFig. 4. The speed/torque curve was determined by the brake
28
24
20
bt 12
7 0 80 90 100 110 120 130speed, rev/mln
Fig. 4Torquelspeed characteristic of the motor
test. Tests were performed at 12 V d .c , with a current of 8 Aflowing through each coil. More tests will have to be done todetermine the possible applications of the motor.
Future work and conclusionsThe motor described in the preceding sections is not
self-starting. As an immediate next step toward further work,a self-starting mechanism has to be devised. The overall per-formance of the motor has to be improved. This could,perhaps, be done by increasing the number of coils and thenumber of blades, and by obtaining an optimum matchbetween the switching circuit and motor driving coil. Possiblea.c. switching has to be investigated.
Based on the work reported here, it is concluded that it ispracticable to build d.c. (and universal) motors without anycommutators or other moving contacts. It also seems to befeasible to develop linear, d.c.-switched reluctance motorsbased on the principles described here.
AcknowledgmentThe work reported here was supported by US National
Science Foundation grant GY-3942.
30th January 1969 S. A. NASAR
Department of Electrical EngineeringUniversity of KentuckyLexingtonKy. 40506, USA
References1 NASAR, s. A. : 'Solid-state switchingin rotating electrical machines',
Proc. 1EE, 1966, 113, (2), pp. 338-339PROC. IEE, Vol. 116, No. 6, JUNE 1969
2 BATES, j .J.: 'Thyristor-assisted commutationin electrical machines'ibid., 1968, 115, (6), pp. 791-801
3 ALEXANDERSON,E. F. w., and MiTTAG,A. H. : 'The thyratron motor 'Elect. Engng., 1934, 53, pp . 1517-1523
4 SAY, M. c: 'Elec t r ica l des ign manual ' (Chapman& Ha'i \Qf>2\5 GENTRY, F. E., GUTZWILLER, F. W. , HOLONYOK N..JON., and VON
ZASTROW, E. E. : 'Semiconductor controlled rectifiers' (Prentice-Hall,1964)
6 GUTZWILLER, F.w. (Ed.): 'SCR manual' (General Electric,1967)
AVERAGE ASYNCHRONOUS TORQUEOFSYNCHRONOUS MACHINES, WITHPARTICULAR REFERENCETORELUCTANCE MACHINES
Synchronous motors with noncylindrical rotors developaverage asynchronous torques which are negative at zeroslip. This phenomenon, which is very dependent on the ratioXd \X q and is particularly marked in modern reluctancemachines, is explained in terms of conventional 2-axis theo ry.Additionally, some points of clarification concerning theterm 'load angle' are made.
In connection with our work on 3-phase reluctance motors,measurements and calculations have been made of asyn-chronous torque/slip curves. As with any noncylindricalrotor machine, the torque is not constant but pulsates as thetravelling field moves with respect to the 'saliency'. In modernreluctance machines the ratio Xd \X q is frequently high (sixwould not be an uncommon value), and the torque pulsationsare correspondingly large. Conventional analysis assumesthat the slip is constant for any given loading. For experi-mental purposes, the variations in slip can be kept small byusing a large coupled inertia, and the average load torqueand average slip can be recorded and plotted.
Curve a of Fig. 1 shows an experimental characteristic fora typical machine. The most striking feature of this curve isthat it cuts the slip axis below the synchronous speed andshows a negative torque at zero slip. (The friction and windageloss for this machine is a negligible 5 W). Torque/slip curveshave been published for salient-pole machines many timesbefore, but they have never, to our knowledge, included thisfeature of negative torque at zero slip. As will be seen, this
inaccuracy is not likely to be serious for conventionalmachines, because of their relatively low Xd \X q ratio (andusually low stator resistance also), but it was not justified inFig. 13 of Reference 3, in which one of the curves was extra-polated at low slips and assumed to pass through zero. Thisletter is primarily intended to explain this negative averagetorque at small and zero slip, in terms of the conventional2-axis theory, but the opportunity is also taken to make someminor points concerning the 'load angle' of a reluctancemachine.
The following well known expression, 1 based on 2-axistheory, gives the average asynchronous torque at constantslip, for a salient-pole machine:
T =
2 r\\Y d \*Y q + \Y q \*Y d )} . . (I)
where a = 1 2s
D = a - r 2 Yd Yq - sr(Y d + Yq )
Although derived in per-unit form in Reference 1, thisequation gives the torque, in synchronous watts, for a star-connected machine, when dimensional parameters are usedand V is the r.m.s. line, voltage. The expression is arrived atby considering the axis currents and voltages as phasors ofconstant slip frequency, and it gives the average torque perslip cycle.
Calculations using eqn. 1 show that the shape of thetorque/slip curve varies very much with changes in the
relative resistances and reactances of the rotor windings andwith the degree of saliency. Curves b and c of Fig. 1 showthe effects of changing the direct-axis rotor resistance, for atypical machine; the two curves assume the same leakagereactance for both rotor windings, but the direct-axis rotorresistance is half that of the quadrature-axis rotor resistancefor curve b, whereas it is double that of the quadrature-axis
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rotor resistance for curve c. Note that both curves exhibitthe negative motoring torque at positive slip, indicated bythe experimental curve a, an d that, while th e point of zero
*1O 0 7 5 0-5 0-25normalised slip
Fig. 1Characteristics showing the average asynchronous torque at constantslip as a function of slipa Measured characteristicfor a typical reluctance motorb Calculated characteristicfor a typical machine with direct-axis rotor resistance
half the quadrature-axis rotor resistancec Calculated characteristicfor the same machine as in b, bu t with direct-axis
rotor resistance twice thatof the quadrature axis
torque occurs at a larger slip for the machine with the higherresistance, both have the same negative torque at zero slip.
Letting the slip in eqn. 1 tend to zero,
(7%=0 = - 2{X d Xq + r 2 ))2 (2)
This is the average torque per slip cycle as the slip tends tozero, and it is always negative, except when the stator resis-tance or saliency is zero, when it also becomes zero. Evi-dently, it is necessary to supply shaft torque to keep themachine running at small positive slips, th e power beingrequired to supply the losses due to the pulsations of currentcaused by the saliency. Th e interpretation of the negativetorque at zero slip, and the reconcilation of this with the theoryof synchronous operation, calls for further consideration.
In th e 2-axis theory of synchronous machines, th e slip is
equal to the time rate of change of load angle ~ . 8 V is the
load angle between the quadrature-axis and the supply-voltage phasors, and it is shown in the phasor diagram (Fig. 2),for a synchronous reluctance motor. For any constant supplyfrequency, the voltage phasor has a constant angular velocity,and so is the electrical load angle which is observed undersynchronous conditions, if the shaft is illuminated by astroboscope triggered from the supply voltage.
The synchronous torque of a star-connected reluctancemachine can be expressed in terms of 2-axis theory as
_ V \X d -X q )2{X dXq + r
2 ) 2,X q - r
2) sin 28 v
+r(X d + X q )cos28 v -r(X d -X q )} . . (3)
where V is the r.m.s. line voltage. In this expression, 8 V hasbeen defined as positive for motoring and negative forgenerating operation. Curve a of Fig. 3shows the synchronoustorque as a function of 8 V, for both motoring and generatingregions for a typical machine. (The machine parameters arethe same as those used to calculate the curves of Fig. 1).
The significance of the negative torque at zero slip (Fig. 1)1050
can now be seen. Eqn. 1 gives the average torque with respect
to time, over a cycle of constant slip. However, since " isdt
equal to the (constant) slip, the time-averaged torque over a
q axis4
JXqlq
d axis
Fig. 2
Phasor diagram for a synchronous reluctance motor
1-5
1O
0-5
I '0 5
F -10
-1-5
- 2 0
- 2 - 5 l I I L _-9 0 -45 0 45
load angle,degFig. 3Curves showing the variation of the synchronous torques developed bya typical reluctance machine as functions of load angle(a) S v(b) 8 l-S ttMotoring torqueis taken as positive
slip cycle is the same as the average torque obtained byintegration with respect to 8 V over a cycle of 8 V. Thus, in thelimit, as the slip tends to zero, the average torque, as givenby eqn. 2, is also obtained by the integration of eqn. 3 withrespect to 8 V from 0 to TT. Because of the stator resistance, theaverage torque for the generating half cycle is always greaterthan for the motoring half cycle, and so the average torqueper slip cycle at zero slip is always negative for a motor.
Confusion can arise over th e term 'load angle', because
PROC. IEE, Vol. 116, No. 6, JUNE 1969
I
1Y/
/
rAV\\
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another load angle,in addition to 8V, has been usedin thedevelopment of the theory of reluctance machines(e.g.Reference 2). Thisis defined,as the electrical angle by whichthe direct axisis ahead of the axis of phase a, in whichthecurrent isa positive maximum,at / = 0. This load angle can,alternatively, be defined as the angle by which the directaxis leads the axis of the stator po tential.In Reference 2, thiselectrical load anglewas designated 8 e , to differentiate itfrom its mechanical equivalent8, but here it will be denotedby 8,-, in recognition of the fact thatit is one of two electricalload angles, beingthe one which relates the stator-currentphasor to the direct axis.It is negative in the motoring con-
dition, and it is shown in the phasor diagram (Fig. 2). Thisload angle is particularly useful in the calculation of thereactancesof machines in termsof their physical geometry.
The equivalent expression for eqn . 3,in terms of 8,-, is-2V\Xd-Xq)sm28i
[{lr-{Xd-Xq) sin 28 i}2+{(Xd+Xq)+(Xd-Xq) cos 28,}2](4)
(Eqn. 4 is derived in Reference 2, although the initial minussign was unfortunately omitted). Curveb of Fig. 3 shows thetorque given by eqn. 4 plotted against8,-. The very different
LAWRENSON, p. j . , an d AGU,L. A. : 'Theory an d performance ofpolyphase reluctance machines',Proc. IEE, 1964, 111, (8), pp. 1435-1445LAWRENSON, p. j . , an d GUPTA, s. K. : 'Developments in the per-formance and theory of segmental-rotor reluctance motors',ibid.1967, 114, (5), pp. 645-653
GENERALISED THEORY OF INDUCTIONMOTORS WITH ASYMMETRICALPRIMARY WINDINGS
In the paper by Butler and Wallace[Proc. IEE, 1968,
115, (5), pp. 685-694], the generalised analysishas beendevelopedfor induction motors having any numberof asym-metrical primary windings,and assuming a symmetricalorasymmetrical power supply. The method, dueto Campbell,31Butterworth32 and Calvaer,33 of transforming mutual leakagereactance into an equivalent self-leakage reactancewasapplied with great advantage. The simplicity achievedis ofgreat interest. The generalityof the approach is such thatithas been easily appliedto shaded-pole motors having morethan one shading winding.
SastryA has conducted experimentson a single-phaseinduction motor with its starting winding notin quadrature,
180 r
6 0 -
oa
VOo>$o
-
N*^X X
1 1
X
i
5
X
i
o"~ " o
y
1
1200
8 0 0
4 0 0
A A
12r
oa
1 1
0
X
1
0
1
0
x
0
X
10-2 0-4 0-6 0-8
4)
E2 2
1 0slip
0-2 0-4 0-6 0-8 10
Fig. AComparison of experimental and computed performances of an 8-pole single-phase induction motorApplied voltage = MOVCapacitance across phase B = 267|i.FShift between two phases = 60 (for fundamental)
Calculated: Measured:phase A x phase Aphase B o phase Bline A line
1 torque torqueCalculated values include all spatial harmonics, including subharmonics
shapes of the two curvesin Fig. 3 emphasises the following:, . d8 v d8: , , . d8= .(a) 7-7^ -7 - and therefore -1 is not constant at constant
at at atslip*, so that integrationof eqn. 4 does not lead to theaverage torque per slip cycle at constant slip;and (b ) 8 Vand 8; are, in general, very differentfor any given torque,and therefore a stroboscope whichis triggered fromthesupply voltage will no t indicate 8,- even approximately.8,- can, of course, be observed by triggering the strobo scopefrom the supply current.13th December 1968 J. M. STEPHENSON
P. J. LAWRENSONDepartment of Electrical& Electronic Engineering
Universityof LeedsLeeds 2, England
References1 ADKINS, B .: 'The general theory of electrical machines (C hapman
& Hall, 1957)* At finite constant slip, the stator phase current can only be represented by aphasor whose magnitudeand phase, relative to the supply voltage, vary perio-dically with the slip frequency
PROC. IEE, Vol. 116, No. 6, JUNE 1969
and an analysis has been made to take into account any spaceangle, and any number of space harm onics, including sub-harmonics.This was developedas an extensionof the workof Puchstein and Lloyd,8 and the recent workof Buchanancon the double-revolving-fieldtheory. In this analysis, the slotmutual leakage is accounted for by the calcu lation of the slotleakage factor.0 The results of this investigationare sum-marised in Fig. A.
In Appendix 11.3, the au thors have presenteda method ofarrivingat the effective value of the air gap in thesalient-poleconstructions.This derivation seemsto be valid only for thefundamental component. Sincethe permeance distributionsare different for different harmonics,E a different constantvalue of th e air gap, equivalent to the varying one, has to betaken for each harmonic.This is not clearly app arent.
With a view toavoidingany error in sucha choice,and alsoto determinethe variatipn that would otherwise existas aresult of neglecting the m utual leakage, we have conductedtests in the following way. A conventional single-phaseinduction motor, with its starting winding shifted fromthemain winding by 40 (electrical), was chosen. The startingwinding was short-circuited througha condenser,and the
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