Improved finite element computations of torque inbrushless permanent magnet motors

M. Popescu, D.M. Ionel, T.J.E. Miller, S.J. Dellinger and M.I. McGilp

Abstract: Two new finite element (FE) methods for improved electromagnetic torque computationare proposed in the paper. The first method uses an analytical filter applied to the Maxwell stresstensor calculation. The second method is based on the virtual work principle and directly providesthe separation of torque in its average and pulsating components. The FE computational methodsare validated on a brushless PM motor having a rotor with surface mounted ferrite arcs. Therelative merits of each method are discussed, together with computational issues, such as theinfluence of the meshing and the convergence of the calculation.

1 Introduction

Other authors have previously studied the difficultiesassociated with the accurate computation of the electro-magnetic torque from FE solutions, and some improvedprocedures have been proposed [1–8]. The study presentedin this paper brings further original contributions tothe subject, by introducing two new formulations: onelocal, based on the Maxwell stress method; and one global,based on the magnetic co-energy variation (virtual work)method.

2 Analytical filter for Maxwell’s stress tensormethod

According to Maxwell’s stress tensor method, the electro-magnetic torque in the motor airgap, on a radial closedsurface of radius r, is calculated as a surface integral, whichfor a two-dimensional electromagnetic field model of aradial-flux rotating machine, is reduced to a circularintegral:

Te ¼lfem0

Z 2p

0

r2BrBydy ð1Þ

where Br and By are the radial and tangential componentsof the flux density B and lfe is the active axial length of themachine. Only the flux density components are thereforerequired for torque computation, which would, in principle,allow for a simple and fast calculation.

In common practice, first-order triangular elements areemployed, mainly to ensure the highest computationalspeed. In this case, satisfactory accuracy is achieved forthe magnetic vector potential A. However, the distributionof the flux density B is one order less accurate, since

it is obtained by differentiating the trial functions of A.While A is described by a linear function over eachfirst-order triangular element, B is piece-wise constantover each element and significant errors can arise in itstangential component, especially in the airgap elementsadjacent to boundaries between materials with differentpermeability.

One method of improving the accuracy of the Maxwellstress torque computation is based on the direct derivationof the flux density components from an analyticalexpression of the magnetic vector potential in the airgapof the motor. In cylindrical co-ordinates, we define twoconcentric circles of radii R1 and R2 inside the motor airgapand set on them nonhomogenous Dirichlet boundaryconditions, with the values determined by the FE magneticvector potential solution. Inside the cylindrical shell(R1oroR2) the magnetic vector potential can be analyti-cally described as in [9]

A r; yð Þ ¼X1n¼1

cnrn þ dnr�nð Þ gn cos nyð Þ þ hn sin nyð Þð Þ½ �ð2Þ

The magnetic vector potential on the two circularboundaries can be expressed in Fourier series:

A R1; yð Þ ¼ a01 þX1n¼1

an1 cos nyð Þ þ bn1 sin nyð Þ½ � ð3Þ

A R2; yð Þ ¼ a02 þX1n¼1

an2 cos nyð Þ þ bn2 sin nyð Þ½ � ð4Þ

The magnetic flux density components are calculated byderivation of the magnetic vector potential:

Br ¼@Ar@y

; By ¼ �@A@r

ð5Þ

From the previous equations, a new computational formulais derived for the electromagnetic torque:

Te ¼P2

� �2plfem0

X1n¼1

n2

R1

R2

� �n

� R2

R1

� �n� � an2bn1 � an1bn2ð Þ

ð6Þ

M. Popescu, T.J.E. Miller and M.I. McGilp are with the SPEED Laboratory,Glasgow University, G12 8LT, UK

D.M. Ionel is with the AO Smith Corporate Technology Center, 12100W. ParkPlace, Milwaukee, WI 53224, USA

S.J. Dellinger is with the AO Smith Electric Products Company, 531 N. FourthStreet, Tipp City, OH, 45371, USA

r IEE, 2005

IEE Proceedings online no. 20055107

doi:10.1049/ip-epa:20055107

Paper first received 29th July 2004. Originally published online: 24th January2005

IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 271

This equation clearly illustrates the proportionality oftorque with the motor polarity P and the torque productionmechanism by the interaction of the same-order airgap fieldharmonics. From a computational point of view, (6) isadvantageous because of its reduced sensitivity to the finiteelement meshing, especially in the airgap. On the otherhand, the analytical solution based on the Fourier series issensitive to the maximum order of the harmonicsconsidered in the summation.

In references [4, 10], formulae, equivalent in principle to(6), have been deduced using a different approach based onthe coefficients cn, dn, gn,hn from (2). We believe that (6),beyond its advantage of a much simpler formulation andfaster implementation, clearly shows the influence of thecomputational parameters, such as R1 and R2.

3 Virtual work method with segregation ofaverage and pulsating torque components

The method of energy functional variation, which uses aderivative of the magnetic energy with respect to position atconstant flux linkages or the derivative of magnetic co-energy with respect to position at constant current, tocompute the electromagnetic torque, is generally acceptedas a method well suited for FE [1, 9]. This is mainly becauseof its global formulation, which is less dependent on localerrors, caused for example by meshing.

The electromagnetic torque may be found as a derivativeof the magnetic energy with respect to position at constantflux linkages or the derivative of magnetic co-energy withrespect to position at constant current:

Te ¼ �@Wmag

@y

����c¼ct¼@W 0

mag

@y

����i¼ct

ð7Þ

One approach developed by Coulomb and Mennier [3],is mostly employed to compute the electromagnetictorque using a local functional energy or coenergyderivative for each rotor position.

The total magnetic field energy produced by a system of ncoils having the flux-linkages c1;c2; . . . ;cn and thecurrents i1; i2; . . . ; in is given by:

Wen ¼Xn

k¼1ikck ð8Þ

In a conservative (lossless) system this energy equals thesum between the magnetostatic energy of the system (i.e.motor) and the mechanical work. The magnetostatic energystored in the motor is calculated as the surface integral ofthe energy density w(x) in the motor cross-section, times theactive length:

Wmag ¼ lfe

ZwmagðxÞdxdy ð9Þ

The energy density has components in three different typeof materials: nonmagnetic (e.g. air), soft-magnetic materials(i.e. steel laminations) and hard-magnetic materials (i.e.permanent magnets):

wmagðxÞ ¼ waðxÞ þ wfðxÞ þ wmðxÞ ð10ÞA similar expression can be used to compute the system’sstored co-energy ðW 0

magÞ.The energy density wa in nonmagnetic media with mr¼ 1

is equal to the co-energy w0a and is calculated as:

wa ¼ w0a ¼1

2BH ð11Þ

In soft-magnetic materials with the hysteresis loopneglected, the energy wf and co-energy density w0f are givenby (Fig. 1a):

wf ¼Z BðxÞ

0

H � dB

w0f ¼ B � H �Z BðxÞ

0

H � dB

ð12Þ

For isotropic permanents magnets [11–12] the energy andco-energy density are defined by the hashed areas in Fig. 1bas wm and, respectively, w0m.

The input energy variation in between two time steps,each with a different rotor position and set of statorcurrents, is:

DWen ¼ Wenðkþ1Þ � WenðkÞ

¼Xn

k¼1ik þ Dikð Þ ck þ Dckð Þ �

Xn

k¼1ikck

" #ð13Þ

Based on the energy conservation law in a losslesssystem, a variation of the input energy will produce avariation of the mechanical work and of the storedmagnetic energy:

DWen ¼Xn

k¼1ikDck þ

Xn

k¼1ckDik

!

¼ DWmag þ dL ¼ DXn

k¼1ikck

!� DW 0

mag þ dL

ð14Þ

Substituting (14) in (13) and considering that rotor move-ment can be regarded as realised at constant current orconstant flux-linkage, the electromagnetic torque can becomputed from the equation:

Te ¼dLDy

����i;c¼ct¼

DWen � DWmag

� �Dy

����i;c¼ct

¼

Pnk¼1

ikDck � DWmag

� �����i¼ct

Dy¼�DWmag

��c¼ct

Dy

ð15Þ

wf

w ′

BBr

B

H −H

a b

f

w ′m

wm

Fig. 1 Magnetic energy and co-energy for a Soft magneticmaterial (lamination steel) and a permanent magnet with a rigiddemagnetisation curvea Soft magnetb permanent magnet

272 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005

or if we use the co-energy,

Te ¼dLDy

����i;c¼ct

¼DWen þ DW 0

mag � DPnk¼1

ikck

� �� �����i;c¼ct

Dy

¼�Pnk¼1

ckDik þ DW 0mag

� �����c¼ct

Dy

¼DW 0

mag

���i¼ct

Dy

ð16Þ

In the actual rotor movement, usually both currents andflux-linkages vary. Thus, by adding (15) and (16) one mayobtain:

Te ¼1

2

Pnk¼1

ikDck

����c¼ct

Dy� 1

2

Pnk¼1

ckDik

����i¼ct

Dy

þDW 0

mag

���c¼ct�DWmag

����i¼ct

Dy

ð17Þ

By integrating over an electrical cycle we can obtain theaverage electromagnetic torque:

Tavg ¼1

2TDy

Z T

0

Xn

k¼1ikDck �

Xn

k¼1ckDik

!dy

þ 1

TDy

Z T

0

DW 0mag � DWmag

� dy

ð18Þ

Note that the second integral in (18) is equal to zero.The electromagnetic torque can be expressed as the sum

of its average and pulsating components:

Te ¼ Tavg þ Tpls ð19ÞIf the currents are considered constant during incrementalmoving from (15) and (19), one can obtain:

Tpls ¼ �DWmag

Dy

����i¼ctþ 1

Dy

Xn

k¼1ikDckð Þji¼ct

" #� Tavg ð20Þ

The average electromagnetic torque (18) is equivalent tothe torque predicted by the i-psi diagram [13] or the flux-MMF diagram [8]. The right hand side of the pulsatingtorque equation (20) contains a first term, which representsthe torque caused by the magnetic circuit permeancevariation. The difference between the second term and theaverage torque on the right hand side of the pulsatingtorque equation (20) corresponds to the ripple torque,which includes the effect of space and time stator MMFharmonics. This separation of torque components clearlyindicates the sources of parasitic torque harmonics and canprove useful in electrical machine design optimisation.

4 Experimental validation

The proposed methods have been implemented in thescripting language of an FEA software [14] and have beenused to compute the performance of a 3-phase 36-slot, 12-pole PM motor, which has been especially built as anexperimental validation tool. The stator has been wound ina 12-pole fully pitched distributed winding arrangement. Norelative axial skew has been provided in between the statorand the rotor. Such a motor design, with a surface mounted

PM rotor and a one slot per pole and phase stator with adistributed winding, is known for its relatively high coggingand ripple torque.

An example mesh, with 1272 nodes and 2488 first orderfinite elements for one pole of the motor cross-section, isshown in Fig. 2, and a detail of the airgap mesh, which has4 circular layers of elements, is illustrated in Fig. 3. The fluxlines for brushless DC operation, i.e. square-wave currentswith two phases switched on and the third switched off, areplotted in Fig. 4 for a value of the phase current of 6A. Theagreement between the measured and computed open-circuit back emf indicates that the special sine-wavemagnetisation of the prototype rotor is satisfactorilymodelled. This is particularly important because the shapeof rotor magnetisation influences the cogging and rippletorque, as shown by other authors [15].

The cogging torque has been measured at standstill(statically) using an in-line torque transducer and aprecision torque table, by which the rotor was moved andheld at incremental positions. The experimental challengesin precise cogging measurements are widely accepted, andtherefore repeated measurements over full clockwise andcounter-clockwise rotor rotations have been averaged to

Fig. 2 FE mesh of brushless PM motor built for experimentalvalidation

Fig. 3 Detail of airgap mesh of Fig. 2

Fig. 4 Flux lines in motor cross-section for brushless DC loadoperation with 6A square-wave phase currents

IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 273

provide the values plotted in Fig. 5 for a half period ofcogging torque. The same test bed and method has beenused for electromagnetic torque measurements with twomotor phases connected in anti-series and the third leftunconnected (Figs. 6 and 7). The resultant torque againstrotor position waveform is relevant to the BLDC motoroperation with square-wave currents. In such a case andwith reference to Fig. 6, the motor would be dynamicallyoperated with a similar combination of on-off phases inbetween 60 and 120 electrical degrees.

For the enclosed numerical examples, the Maxwell stressand the analytically filtered Maxwell method have beenapplied using 1001 points equidistantly distributed in theairgap along two circular contours, through the middle ofthe two central FE layers, respectively (see Fig. 3). Thevalues reported for the Maxwell stress method werecomputed by averaging the results for these two contours.The two contours defined the cylindrical shell ðR1oroR2Þas per (6), which was used for the analytically filteredMaxwell method. The two inner FE airgap layers have beenlinked to the rotor and the two outer FE airgap layers havebeen linked to the stator. Discrete rotor positions have beensimulated maintaining the same meshes for the stator androtor, moving the rotor and re-meshing the airgap layers.The comparative graphics between static measurements andcomputations from Figs. 5 and 6 illustrate the numericalnoise present in the conventional Maxwell stress calcula-tions and, to some extent, also in the results of the virtualwork method. Such numerical noise is caused by thedifferentiation, discrete mesh and rotor position variation.By increasing the number of rotor positions used in thevirtual work computations, the numerical noise from Fig. 5has been reduced in Figs. 6 and 7.

Most notably, the analytical method derived from theMaxwell stress ‘filters’ the numerical noise and appears toprovide the most realistic torque waveforms. The accuracyof the electromagnetic torque computations is, of course,dependent on many numerical factors, such as the meshdensity, which are discussed in the following Section.

5 Numerical study

The example results from Figs. 8 and 9 indicate thatincreasing the overall mesh density does not necessarilyguarantee the convergence of the FE torque solutioncalculated with the conventional Maxwell stress method.Such a finding is in line with the results reported by otherauthors, who have studied the intricate relationship betweenthe mesh quality and/or density and the Maxwell stress [6].On the other hand, the analytically filtered Maxwell stressmethod and the virtual work method, which are both basedon global variables, are less sensitive to meshing, although asignificantly larger mesh is recommended for coggingcalculations.

The torque values from Figs. 8 and 9 have beennormalised with reference to the value calculated with theanalytically filtered Maxwell method and the largest mesh.This approach was preferred based on the numerical and

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0

−0.05

cogg

ing

torg

ue, N

m

0 1 2 3 4 5

rotor position, mech. deg.

virtual work

analytical filtermeasured

Maxwell stress

Fig. 5 Experimental and FE computed cogging torque

measuredanalytical filterMaxwell stressvirtual work

elec

trom

agne

tic to

rque

, Nm

1

2

3

4

5

rotor position, el. deg.

1801501209060300

0

Fig. 6 Torque against rotor position for static experiment with 3 Adc supply

00

30 60 90 120 150 180

rotor position, el. deg.

10

8

6

4

2

elec

trom

agne

tic to

rque

, Nm

virtual workMaxwell stressanalytical filtermeasured

Fig. 7 Torque against rotor position for static experiment with 6 Adc supply

0 5000 10000 15000total no. of nodes in FE mesh

170

160

150

140

130

120

110

100

90

norm

alis

ed c

oggi

ng to

rque

, %

analytical filterMaxwell stress

virtual work

Fig. 8 Example of FE mesh influence on convergence of coggingtorque calculation at 2.5 mech. deg. rotor position

274 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005

experimental results discussed in the preceding Section. Theanalytically filtered Maxwell stress results are also influ-enced by the number of harmonics considered in thesummation of (6). The examples provided in Table 1indicate that a larger number of harmonics is required forthe cogging torque calculation.

Based on the virtual work method with the segregation ofcomponents, the average torque can be computed with (19)and (20) and requires only the current distribution data andthe flux linkage results. Therefore, in principle, a coarsermesh can be used resulting in computational-time savings.However, because of the differentiation with respect torotor position, special care is required in selecting therotation steps to minimise the numerical noise. An i-psi

diagram [13] or a flux-mmf diagram [8] provides an efficientway of plotting and integrating the data. The area of the i-psi diagram from Fig. 10 is equal to the average torque ofthe prototype motor for an ideal BLDC square-wavecurrent excitation (two phases on and the third off) at 3Aand 6A phase current, respectively. The example diagramsalso illustrate a linear relationship between average torqueand phase current, indicating virtually no influence of thesaturation on the prototype motor operation in the loadrange considered.

For unsaturated motors, the on-load ‘cogging’ (i.e. thepulsating torque component due to the magnetic circuitpermeance variation) can be estimated using the coggingvalues computed at open-circuit operation (Fig. 5). How-ever, this is not necessarily the case in a highly saturatedmotor. To illustrate the point, two additional computa-tional models have been analysed. One is identical to theprototype motor with the exception of the rotor magnets,which are made of NdFeB, with a remanence approxi-mately three times higher than the original ferrite. Anothermodel also employs NdFeBmagnets but considers twice theelectric load, i.e. 12A per phase. The load operation of theoriginal and the two additional models has been simulatedby FEA and the pulsating torque computed with (20). Theon-load cogging torque, computed as the differential of themagnetic energy with respect to rotor position at constantcurrent, and normalised by the value of the peak-to-peaktotal torque pulsations for the three FE models, is plottedin Fig. 11.

0 5000 10000 15000total no. of nodes in FE mesh

analytical filterMaxwell stress

virtual work

90

95

100

105

norm

alis

ed e

lect

rom

agne

tic to

rque

, %

Fig. 9 Example of FE mesh influence on convergence ofelectromagnetic torque calculation at 6A load and 100 el. deg. rotorposition

Table 1: Influence of number of harmonics on cogging andload torque calculated by the analytically filtered Maxwellmethod

No. ofharm.

Cogging at 2.5mech. deg., %

Torque at 90el. deg., %

Torque at 120el. deg., %

1 65.99 111.95 94.28

3 126.90 111.58 94.52

5 183.76 97.29 105.25

7 157.36 101.97 101.17

9 120.81 101.85 100.82

11 121.83 100.00 99.88

13 120.30 99.75 100.23

15 114.21 99.75 100.23

17 110.66 100.00 100.35

19 113.20 99.63 100.23

21 114.21 99.63 100.12

31 96.45 99.75 99.88

41 99.49 100.00 100.35

51 104.57 99.88 100.35

71 95.94 100.00 100.00

91 158.38 100.00 100.12

101 100.00 100.00 100.00

121 99.49 100.00 100.00

131 100.51 100.00 100.00

191 100.00 100.00 100.00

201 100.00 100.00 100.00

−8−250

−200

−150

−100

−50

−6 −4 −2 0 2 4 6 8

current, A

3A6A

0

50

100

150

200

250

flux

linka

ge, W

b

Fig. 10 i-psi diagrams for prototype motor with square-wavecurrents

00

1 2 3 4 5rotor position, mech.deg.

100

80

60

40

20

cogg

ing/

peak

to p

eak

puls

atin

g to

rque

, %

ferrite 6ANdFeB 6ANdFeB 12A

Fig. 11 Contribution of cogging torque to load ripple for differentmotor design

IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 275

The non-linear effect of the magnetic saturationcauses the model with NdFeB magnets and the original6A load to exhibit the highest contribution of the on-loadcogging to the total torque pulsation. The on-load coggingeffect in the fully saturated model with NdFeB and 12Aload is comparable with the effect in the prototype motormodel.

6 Conclusions

The numerical and experimental studies performed haveindicated that, out of the two new methods presented, theanalytically filtered Maxwell stress method is less sensitiveto the variation of computational FE parameters andexhibits reduced numerical noise. This, together with itsapparent improved accuracy, makes it a recommendedchoice for sensitive calculations such as cogging torque orfor final design checks and calibration against experimentaldata.

The method of virtual work with segregation of torquecomponents has to its advantage the fact that being a globalmethod a coarser mesh could be used in principle, at leastfor the average torque calculation. The method is thereforesuitable to be used in an industrial design environment,which requires fast computational tools. Furthermore, thismethod provides the theoretical background for separatingthe cogging and the ripple torque component from the totalon-load pulsating torque and can be used therefore tosupport the decision process on the methods to be used forthe mitigation of parasitic torques for a given motor designand performance specification.

7 Acknowledgments

The authors would like to thank Prof. T.J.E. Miller and M.McGilp, who are with SPEED Laboratory, University ofGlasgow, and to S. Dellinger, who is with AO SmithElectric Products Company, for the technical insightsprovided and for their contributions in preparing the paper.

Funding for this work was provided by the companies fromSPEED Consortium.

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