Modelling the performance of the vernier hybrid machine

Modelling the performance of the vernier hybridmachine

M.A. Mueller and N.J. Baker

Abstract: Variable reluctance permanent magnet machines exhibit shear stresses higher thanconventional machine topologies. However, analysis and design of these machine topologies is notstraightforward owing to the nature of the flux paths in the machine. In a transverse flux machineflux flows in all three dimensions and the machine structure is non-conventional. The vernierhybrid machine is another member of this family, but the flux paths are restricted to twodimensions, and can therefore be constructed using laminations. Significant leakage flux paths existin the vernier hybrid machine, which must be modelled accurately during machine design. The useof flux-linkage maps to determine the performance of a linear vernier hybrid machine is described.Experimental results are presented to verify the model. The technique can be applied to anyvariable reluctance permanent magnet machine topology.

List of symbols

F forceV voltageE no-load induced emfL inductanceI,i currentB flux densityR resistancelm magnet lengthAm magnet cross-sectional areaBr remanent flux densityW co-energyx displacementC flux linkagef fluxm0 permeability of free spacemr permanent magnet recoil permeability

1 Introduction

The vernier hybrid machine (VHM) is a member of thevariable reluctance permanent magnet (VRPM) machinefamily. There are many variations in the topology ofmachines making up this family, some of which have beenpublished by various authors [1–5]. The basic principle ofoperation in these machines is similar: permanent magnetsof short pole pitch (o50mm) move with respect to a slottedsurface of similar pitch. The slotted surface provides thevariation in reluctance and produces rapid flux reversal of

the permanent magnet (PM) field. There is therefore a highrate of change of energy in the airgap resulting in airgapshear stresses orders of magnitude greater than in conven-tional machines. Weh et al. reported a shear stress in theregion of 200kN/m2 for the transverse flux machine (TFM)[1].The potential of the TFM in terms of shear stress has

been widely reported, but manufacture of the TFM isdifficult because of its non-conventional structure. Spoonerand Haydock applied the conformal mapping technique toderive expressions for shear stress and power factor of theVHM [5]. Results in the paper showed that the machinetopology was capable of high shear stresses (less than theTFM), but also had low power factor. However, themagnetic and physical structure is more similar to aconventional machine, and hence more desirable in termsof manufacture. The analysis of Spooner and Haydock wasused to demonstrate the concept and not investigate theperformance of this machine topology. In this paper theauthors present and verify a model for investigatingmachine performance. Modelling of VRPM machines isnot straightforward, because of complex leakage flux pathsand in the case of the TFM three-dimensional flux paths. Inorder to take into account saturation, leakage paths andarmature reaction effects a complete picture of the fluxdistribution with position and armature current is required.This so-called flux-linkage map provides a completedescription of the machine enabling performance to bedetermined. In previous work the technique has beenapplied to PM machines with large pole pitch and toswitched reluctance machines [6–8]. However, the magnetpole pitch in the VHM is smaller than in conventional PMmachines and magnets sit side by side, which influences theforce calculation. The effect of a magnet pair must beincluded in any flux-linkage map. In conventional PMmachines only one magnet is modelled in the flux-linkagemap, but in this paper the effect of a pair of magnets isincluded in the model. Electromagnetic finite elementmodelling can be used to determine the flux-linkage map.It has the advantage of being able to model complexgeometries and saturation. More classical techniques could

The authors are with the School of Engineering, University of Durham, SouthRoad, Durham, UK

r IEE, 2003

IEE Proceedings online no. 20030642

doi:10.1049/ip-epa:20030642

Paper first received 6th November 2002 and in revised form 1st May 2003.Originally published online: 15th August 2003

IEE Proc.-Electr. Power Appl., Vol. 150, No. 6, November 2003 647

be adopted, such as a magnetic circuit approach, butknowledge of the flux paths is required before developingsuch models.The high shear stress capability of the VHM makes it

attractive in cases where a conventional machine topologywould be physically very large and heavy, for example lowspeed generators in direct drive wind turbines or direct drivewave energy converters. In the case of wind turbines largecranes are required to lift the generator to the top of a towerthat could be in excess of 80m. As turbines move offshorethis becomes even more of an issue. Any gains in mass fordirect drive systems will be of benefit. In terms of motoring,power dense machines have been proposed to replaceconventional synchronous machines in marine applica-tions [9]. The more conventional structure of the VHMcompared to the TFMmakes it a suitable candidate for thisapplication.In this paper the authors demonstrate the use of flux-

linkage maps to determine the performance of VHMmachines, with particular reference to a linear version.There will be a brief summary of the machine operation.Flux plots generated from finite element modelling arepresented to highlight leakage and armature excitationeffects. Results from 2D magnetostatic finite elementanalysis are applied to modelling techniques to calculatethe force, no-load and on-load performance of the machine.

2 The vernier hybrid machine (VHM)

2.1 Principles of operationA linear version of the VHM was chosen to explain theoperation and investigate performance modelling (Fig. 1). Alinear toothed translator constructed from iron laminationsmoves between two C-cores, with a coil wound on each poleand magnets mounted on the pole face. The translatortooth and slot width are similar in dimension to the magnetpitch. When the teeth are fully aligned with magnets, fluxflows as shown by the dotted line in Fig. 1a. As thetranslator moves this flux decays to zero in the unalignedposition and then reverses polarity. In this way the coil seesa rapid change in flux over a short distance. The electricalfrequency of the flux pulsations is greater than thefrequency of the translator, producing a magnetic gearingeffect.Figure 1b, shows the mechanism of force production.

The PMs are replaced by equivalent coils and the field isproduced by armature excitation. A higher flux density (B1)is observed in the gap under a tooth compared to the field inthe slot region (B2). Peak force is produced when theinterface between two magnets is aligned with the centre ofa tooth. If the field is in the direction shown, a force F1 is

produced under a tooth and a smaller force, F2, is producedin the slot region acting in the opposite direction.In a three-phase machine there would be two other

similar units, but displaced such that the flux pulsations are120 electrical degrees out of phase. Physically this isachieved by separating adjacent phases by a multiple oftwo thirds of a rotor pitch. Each phase is then magneticallyisolated from the other and the machine can be built up interms of modules. Adopting this modular approachprovides a high degree of flexibility in the design and choiceof power ratings.Table 1 gives the dimensions of a prototype machine

designed and built to verify the modelling techniques usedin the paper. Fig. 2 shows a photo of the built machine. Theanalysis derived in [5] was used to design the prototype andfurther details can be found in [10].

Mecrow and Jack proposed a similar topology, but withthe magnets on the moving part, and a castellated statorpole face [2]. In the VHM the magnets are mounted on thestator so that magnet weight and cost are reduced, and the

core back

slot region

translator

coils

magnets

flux path

B1 > B2

B2 B1

F2 F1

tooth slot

PMs withequivalent coilrepresentation

net force = F1 − F2

a bFig. 1a One phase module of the linear VHMb Force production in a VRPM machine

Table 1: Main dimensions of the prototype

Magnet pitch, mm 12

Magnet thickness, mm 4

Airgap, mm 1

Core length, mm 100

Magnets per pole 6

Rotor pole pitch, mm 24

Rotor slot depth, mm 10

C-core slot depth, mm 100

C-core slot width, mm 144

Core back, mm 50

Fig. 2 VHM test rig

648 IEE Proc.-Electr. Power Appl., Vol. 150, No. 6, November 2003

moving member is simply constructed from iron lamina-tions.

2.2 Modelling the VHMAt first sight it appears that this machine could be modelledusing the conventional PM synchronous machine equiva-lent circuit, viz. a no-load emf behind a fixed synchronousreactance and winding resistance. However, initial calcula-tions showed that this circuit did not accurately representthe machine on-load.The voltage equation for one phase of the VHM is:

V ¼ iRþ dðCarm þCPMÞdt

ð1Þ

where Carm is the flux linkage due to armature excitationand CPM is the flux linkage due to PM excitation.The rate of change of flux linkage due to the PMs is the

no-load induced emf (E), and the flux linkage due to thearmature excitation is equal to the product of inductance(L) and current (i).

V ¼ iRþ E þ Ldidtþ i

dLdt

ð2Þ

To solve the current from (2) accurate models of the no-load emf and the inductance variation are required. Some ofthe problems faced when modelling these machines can beconsidered by analysing flux plots generated from 2D finiteelement modelling software [11]. A finite element model ofthe geometry shown in Fig. 1a with dimensions given inTable 1 has been created.The flux paths due to PM excitation only are shown in

the region of a pole face in Fig. 3. There is a significantamount of flux flowing between neighbouring magnets

reducing the amount of flux linking the armature coils.Moving from the left to the right-hand side of the statorpole face, each aligned magnet contributes less flux. Thefinal two magnets on the right-hand side make a negligiblecontribution to the magnetising flux. When the translatorhas moved one pole pitch the two left-hand magnets willmake negligible contribution. This fact holds true for anyeven number of magnets on the pole face. This effect wouldnot occur if the magnets were on the rotor (as is the case inthe general VRPM topology), as there would always be aneighbouring magnet for the flux to flow into. In thecalculation of no-load induced emf it is therefore importantto take into account edge effects at the c-core poles and fluxflow between neighbouring magnets.On-load, flux flows due to both PM and armature

excitation. Flux due to the latter is affected by reluctancevariation provided by the slotted structure and the opposingPM mmfs. The latter modifies the flux distribution

depending upon the armature excitation polarity. Figure 4shows two flux plots in the pole face region with thetranslator in the same position, but with current flowing inopposite directions in Figs. 4a and 4b. In Fig. 4a the net fluxflow sees a larger reluctance than in Fig. 4b due to theopposing mmfs of the coil and the aligned PMs. The totalflux flowing in the machine depends upon the translatorposition and on the magnitude and polarity of the armatureexcitation. For example in the position shown in Fig. 4 theflux linkage in Fig. 4a is 3.51Wb turns and in Fig. 4b it is5.24Wb turns.

Flux paths are complex in this machine, when with onlyPM excitation and both PM and armature excitation. Theflux-linkage map is the best way of modelling the machinebecause it provides a complete description of the flux atdifferent positions and levels of excitation.

2.3 Electromagnetic force calculationThrust force in a linear machine is determined from the rateof change of co-energy with position. The latter is calculatedfrom the flux-linkage map. A typical C�I trajectory isshown in Fig. 5. The instantaneous total flux linkage isplotted against mmf for different positions of the translatorover one electrical cycle. Both of these quantities are afunction of rotor position and coil current, and for a givenelectrical loading provide a closed trajectory per electricalcycle. The average force is proportional to the total co-energy converted in one electrical cycle, which is equal tothe shaded area enclosed in Fig. 5. The instantaneous co-energy is equal to the area bounded by two finitely spacedpositions and their magnetic curves. Instantaneous force is

Fig. 3 2D flux plot for a displacement of 3 mm from the fullyaligned position

Fig. 4a Flux plot with coil current¼�10 Ab Flux plot with coil current¼ 10 A


then calculated from the rate of change of co-energy, (3).This technique is well published and further informationcan be found in [7].

F ¼ @W 0ði; xÞ@x i¼constant

ð3Þ

A 2D finite element model of the geometry in Fig. 1a withdimensions in Table 1 has been created. The flux-linkageagainst current against position map was obtained fordifferent coil excitations at different translator positionsover one electrical cycle. Fig. 6 shows the results obtainedfrom the finite element model.

At each value of current the flux-linkage distribution issymmetrical over one electrical cycle (two pole pitches) asexpected. However symmetry does not exist over thecurrent range at a fixed position. For example at the fullyaligned position, x¼ 0 or x¼ 24mm, the flux linkage is�6Wb turns for an excitation of �15 A whereas it is7.5Wb turns for 15 A. This asymmetry can be explainedwith reference to Figs. 4a and 4b.

2.4 Cogging force calculationTheC–I map in the form shown in Fig. 6 only provides thetotal electromagnetic force. A different approach is requiredto calculate the cogging force, the tendency of the PMs toalign with the teeth on the translator. To calculate a value ofcogging force the data need to be converted into the flux-mmf diagram for a permanent magnet [12, 13]. The flux

emanating from a magnet corresponds to the mmf withinthat magnet according to (4). This is also the load line forthe magnetic circuit.

mmf ¼ lmm0mr

fAm

� Br

� �ð4Þ

The intersection of the load line and the magnet’s de-magnetisation curve is the operating point, which willchange with position as shown in Fig. 7. Over one coggingforce cycle the operating point moves up and down the PMdemagnetisation curve. The area swept out by this linerepresents the change in co-energy contained within themagnet.

This method has been applied successfully to PMmachines with large magnet pole pitch. In the VHMhowever, the pole pitch is of the order 10 to 20mm andmagnets sit side by side. The cogging force is produced bythe interaction of a tooth on the translator and a pair ofmagnets of opposite polarity on the stator. The effect of apair of magnets must therefore be considered.Each magnet will have its own operating point on the

demagnetisation curve. When a rotor tooth overlaps 2 PMsequally, the so-called unaligned position (Fig. 8a), the

co-energy

position, x

mmf, A turns

�, Wb turns

Fig. 5 Typical C�I trajectory for a PM machine

8

6

4

2

0

−2

−4

−6

−825

2015 15

−155

5−510

10−100

0

current, Aposition, mm

flux

linka

ge, W

b tu

rns

Fig. 6 Flux-linkage map for the linear VHM prototype

change in co-energyover small

displacement

flux-mmf curvefor a PM during onecogging torque cycle

demagnetisationcurve of PM

mmf, A turns

effect ofrotation

flux,

Wb

Fig. 7 Flux-mmf diagram for a PM

N S

N S

unalignedposition

PM demagnetisationcurve

flux linkage

flux linkage

load line forboth N & S

PMs

load line for N PM

load line for S PM

mmf

mmf

displacementtowards

neighbouringPM

a

b

Fig. 8a Coincidental operating points in the unaligned positionb Operating points diverge as the translator moves towards fullalignment


operating points of the two magnets coincide. However, asthe rotor tooth moves towards the N-pole PM thereluctance of the S-pole PM flux increases and thereluctance of the N-pole PM flux decreases. The operatingpoint for the N-pole therefore moves up the demagnetisa-tion curve, and the S-pole operating point moves down thecurve. The area enclosed by the airgap lines for each magnetand the demagnetisation curve is equal to the co-energy,and hence the instantaneous cogging force is equal to therate of change of this co-energy with displacement.Figs. 8a and 8b show load lines with different positions

for a pair of magnets. When the translator is in the fullyaligned position the operating points are at maximumdivergence and the area between the two load lines and thedemagnetisation curve is equal to the total energy that hasto be overcome in order to move the translator.

2.5 Current CalculationEquation (1) defines the voltage relationship for one phase.This equation has been used to calculate the current flow inorder to overcome the problems of separating the magnetflux from the armature flux. As shown in Fig. 4 thearmature flux paths are influenced by the relative polarity ofthe armature flux compared to the polarity of the PM flux.The rate of change of flux linkage includes the no-load emfand the effect of armature reaction. In generating mode therate of change of flux linkage will be greater than theterminal voltage, V:

V ¼ iRþ dCdt

ð5Þ

Assuming an initial current of 0A, the rate of change of fluxlinkage is equal to the terminal voltage. The flux linkage isdetermined by numerically integrating (5) using the Runge-Kutta algorithm. The current can then be found from theC�I map and the translator position. This new currentvalue is then used to calculate the new rate of change of fluxlinkage, and the steps are repeated.

3 Results

3.1 Electromagnetic and cogging forceThe static force with only one phase excited was measuredfor the prototype linear VHM shown in Fig. 2. In order tomeasure the static force the drive mechanism was replacedwith a screw thread mounted load cell, which enabled easypositioning of the translator. The position of the rotor wasselected and subsequently held in place by the cell. The celldeforms less than 0.4mm at the rated full load 1000kg andthe rotor can therefore be assumed stationary as current isaltered within the coils. However, static friction exists in thebearing arrangement, which consists of pairs of bearingsattached to the uprights pinching the upper and lowersurfaces of the runner (see Fig. 9). In order to take this intoaccount static force measurements were taken at fixedpoints over an electrical cycle in the push and pull directionsresulting in two waveforms, the average of which was takenas the measured force. The force was measured for anumber of different coil excitations (�20 to 20A) and at1mm intervals over one electrical cycle (two pole pitches or24mm in this case). Cogging force was also measuredtaking the same precautions for no coil excitation. Acomparison of the average experimental and predicted forceis shown in Fig. 10 for an excitation of 16A. In the modelzero displacement corresponds to the fully aligned position,but in the experimental work the start position was notalways the same. There is, however, excellent correlationbetween the nature of the force distributions and the peak

experimental and predicted values are within 6.7% of eachother.The flux-linkage map for a pair of magnets over one

electrical cycle was determined using 2D finite elementanalysis. There are three pairs of magnets per pole face andfour pole faces per phase. Fig. 11 shows the total cogging

Bearingpositions

Load cell

TranslatorBearing runner

Fig. 9 Position of load cell

4000

3000

2000

1000

0

−1000

−2000

−3000

−4000

forc

e, N

0 5 10 15 20 25 30

position, mm

average exppredicted

Fig. 10 Force per phase against position at I¼ 16 A

800

600

400

200

−200

−400

−600

−800

0

forc

e, N

−5 0 5 10 15 20 25

position, mm

experiment 1experiment 2predicted

Fig. 11 Experimental and predicted cogging forces


force per phase. Experimental results were measured overone electrical cycle in both directions illustrating the effectof static friction. For displacements up to 6mm the resultsdo not compare very well. This problem may arise if therunners are not smooth or excessive dirt and grease arepresent over this portion of the displacement. Under suchconditions the influence of static friction will be differentcompared to the remainder of the displacement. Thecalculated results follow the same trend and are of thesame order of magnitude as the measured values, providingsome confidence in the method.Both experimental and predicted results show two pairs

of different peaks. This is entirely as expected and dependsupon the number of rotor teeth overlapping the stator pole.During one part of the cycle only two rotor teeth are in theunaligned peak force position producing the lower peakforce, whereas the larger peak is produced by three rotorteeth in the unaligned peak force position.

3.2 No-load induced emfFig. 12 shows the open circuit emf results calculated fromthe flux-linkage map (Fig. 6) plotted on the same axis as theexperimentally measured results. The correlation betweenthe two is good. As, by definition, no current flows in anopen circuit test, the predicted results were obtained simplyby differentiating the predicted flux-linkage time profilefrom the experimental time displacement results.

The predicted waveform is more regular and slightlylarger than the experimental one. Each phase module has tobe adjusted manually, and it is likely that the airgapbetween the translator and the pole faces differs betweenpole faces and across each pole face. Simulated results alsoappear to move slightly out of phase during the time periodshown. The model was run with the exact analoguedisplacement data obtained from the transducer. Super-imposed on this signal was a high-frequency noise signal,produced by the inverter used to drive the test rig. As thedisplacement signal has to be differentiated into velocity,this noise becomes significant and hence requires filteringout. A 200 point moving average method was used, and it islikely that this will account for the gradual loss of agreementbetween used and actual displacement results. It should bepointed out that the predicted waveform has been phaseshifted so that it lies on top of the experimental results. Thisis perfectly valid because the start position of theexperimental results is not precisely known.

3.3 Short-circuit testIn the short-circuit test the load voltage, V in (5), is zero(Fig. 13). The results from this model are reasonable andhave an error of similar magnitude to those for the open-circuit results.

3.4 On-load resultsA parallel R-C load was connected across each phase of thegenerator. A capacitive load was chosen to compensate forthe high generator inductance. The voltage across the R-Cload was measured and calculated using the current modeldescribed earlier. Fig. 14 shows the experimental andcalculated results of the load voltage.

4 Discussion

The flux-linkage map is a powerful method of determiningthe performance of any electrical machine. If finite elementanalysis is used to determine the map, second-order effectssuch as saturation and leakage fields are taken into account.However, to generate the map using finite elements anumber of solutions are required which can be time-consuming even on a modern PC. In total 724 solutionswere generated to obtain the C�I map in Fig. 6. However,inspection of the flux-linkage map in Fig. 6 shows that auniform distribution exists. For example the flux-linkagedistribution with position at a fixed current is sinusoidal in

150

100

50

0

−50

−100

−1500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

time, s

volta

ge, V

predictedexperimental

Fig. 12 Open-circuit emf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

time, s

2.5

2.0

1.5

1.0

0.5

−0.5

0

−1.0

−1.5

−2.0

−2.5

curr

ent,

A


Fig. 13 Experimental and predicted short-circuit current

800

600

400

200

−200

−400

−600

−800

volta

ge, V

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

time, s


Fig. 14 Voltage across an R-C load, R¼ 600O and C¼ 150mF


nature. As the current varies this distribution is shiftedlinearly depending upon the excitation. Only three flux-linkage values would be required: in the fully alignedposition at an excitation level equal to 7 the maximumcurrent and in the unaligned position with maximumpositive (or negative) current excitation. If it is assumed thatthe PM-only flux linkage varies sinusoidally over oneelectrical cycle a similar simplification can be made tocalculate the PM C-I map for cogging torque calculations.With this simplified method for determining the flux-linkagemap the method can be more readily adopted in the designoffice, where rapid solutions are required, but stillincorporating the benefits of finite element modelling.Iron losses and PM eddy-current losses cannot be

calculated from the C�I map, but flux density informationcan be extracted from the map, which could then be fed intoclassical expressions to provide the designer with anindication of the likely losses. Although the method hasbeen applied to a linear version of the VHM, it appliesequally to rotary machines, and to other topologies ofvariable-reluctance permanent magnet machines.The no-load, short-circuit and load results are very

similar in distribution, but the calculated values are 20 to30% greater than measured. The phase inductance wasmeasured and found to be 10% greater than the valuecalculated from a 2D finite element model. The use ofconcentrated coils on a wide stator pole results in asignificant endwinding, which should not be neglected.Throughout the calculations it was assumed that the airgapwas 1mm along the length of the machine. However, noaccount was taken for the increase in gap due to the glueused in attaching the magnets, and adjustment of the airgapwas difficult with some cores slightly skewed with respect tothe translator. In addition the bearing guideways were nottrue resulting in an increase in airgap during motion. Toinvestigate the effect of an increased airgap the model wasrerun with an airgap of 1.2mm. The results of open-circuitemf and load voltage are shown in Figs. 15 and 16.Compared to the original simulated results the errorbetween experimental and calculated results is muchreduced, but this is expected if the airgap is increased.The model relies on mechanical parameters and theseresults highlight how important the mechanical set-up is,with respect to its effect on the electrical system. It is theauthors’ opinion that the comparison between experimentaland calculated results in Figs. 12 to 16 does provide someconfidence in the model: the distribution is correct and the

error is due to the limitations of the mechanical systemused.

5 Conclusions

This paper has described methods of calculating forces andpredicting the performance of the VHM using flux-linkage,current and position data: the so-called C�I map.Experimental results on a three-phase linear VHM havebeen presented to verify the methods described. Eventhough finite element analysis has been used to generate theC�I map, the technique could be adapted for design officeuse by making assumptions about the symmetry of theC�Imap, leading to fewer finite element computations.

6 Acknowledgments

The authors would like to thank Newage AVK SEG andthe Engineering and Physical Sciences Research Council forproviding financial assistance to Mr. Baker, and to theUniversity of Durham for providing facilities. This workwas completed as part of EPSRC grant N38299. Finally theauthors would like to acknowledge the contributions oftechnicians David Jones and Ted Jones.

7 References

1 Weh, H., Hoffman, H., and Landrath, J.: ‘New permanent magnetexcited synchronous machine with high efficiency at low speeds’.Proc. Int. Conf. on Electrical machines, Pisa, Italy, Sept. 1988, pp.35–40

2 Mecrow, B.C., and Jack, A.G.: ‘A new high torque density permanentmagnet machine configuration’. Proc. Int. Conf. on Electricalmachines, Cambridge Mass., USA, Sept 1990, pp. 1046–1052

3 Harris, M.R., and Mecrow, B.C.: ‘Variable reluctance permanentmagnet motors for high specific output’. Proc. IEE Conf. on Electricalmachines & drives, Oxford, 1993, pp. 437–442

4 Iwabuchi, N., Kawahara, A., Kume, T., Kabashima, T., andNagasaka, N.: ‘A novel high-torque reluctance motor with rare-earthmagnet’, IEEE Trans. Ind. Appl., 1994, 30, (3), pp. 609–614

5 Spooner, E., and Haydock, L.: ‘Vernier hybrid machines’, Submittedto IEE Proc., Electr. Power Appl

6 Staton, D.A., Deodhar, R.P., Soong, W.L., and Miller, T.J.E.:‘Torque prediction using the flux-MMF diagram in AC, DC andreluctance motors’, IEEE Trans. Ind. Appl., 1996, 32, pp. 180–188

7 Miller, T.J.E.: ‘Brushless permanent-magnet and reluctance motordrives’, (Clarendon Press, Oxford, 1989)

8 Deodhar, R., Anderson, S., Boldea, I., and Miller, T.J.E.: ‘The fluxreversal machine: a new brushless doubly salient permanent magnetmachine’, IEEE Trans. Ind. Appl., 1997, 33, (4), pp. 925–934

9 Hodge, C., and Mattick, D.: ‘Direct drive naval propulsion motors’.Presented at UK Magnetics Society Seminar, New Topologies forElectrical Machines, De Montfort University, Leicester, 25th Septem-ber 2002

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volta

ge, V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

time, s


Fig. 15 Open-circuit emf with an airgap of 1.2 mm in the model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

time, s

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Fig. 16 Voltage across an R-C load, R¼ 600O and C¼ 150mF,with an airgap of 1.2 mm in the model


10 Mueller, M.A., and Baker, N.J.: ‘A low speed reciprocating perma-nent magnet generator for direct drive wave energy converters’.Proc. IEE Power electronics, machines & drives Conf., Bath, UK,April 2002, pp. 468–473

11 PC-OPERA user guides (Vector Fields Ltd., Oxford, UK)

12 Campbell, P.: ‘Comments on energy stored in permanent magnets’,IEEE Trans. Magn., 2000, 36, pp. 402–403

13 Deodhar, R.P., Staton, D.A., Jahns, T.M., and Miller, T.J.E:‘Prediction of cogging torque using the flux-MMFdiagram technique’,IEEE Trans. Ind. Appl., 1996, 32, pp. 569–575


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Modelling the performance of the vernier hybrid machine