Reduction of cogging force in PM linear motors bypole-shifting

N. Bianchi, S. Bolognani and A.D.F. Cappello

Abstract: A technique is described to reduce cogging force in permanent-magnet linear motors dueto the interaction between the permanent magnets (PMs) and the stator teeth. It consists of aproper shift of the PMs so that this effect is compensated for among the poles of the machine. Thistechnique, generally adopted for surface-mounted PM motors only, is also applied to interior PMmotors. The rules to arrange the magnets appropriately are illustrated and the technique is verifiedby finite-element analysis and some experimental tests.

1 Introduction

Permanent magnet linear (PML) motors are more andmore used in factory automation and numerical controlsystems. One of the main reasons is the possibility to have alinear motion without the need for a rotary-to-linearconversion, allowing a great or positional accuracy to beachieved. There are various typologies of PML motors.They may be classified according to:

� the armature (or primary), which can be slotted or slotless;in this paper we focus on the slotted primary, which is themain origin of cogging force

� the configuration of the PM secondary, which can be withinterior permanent magnets (IPM) or with surface-mountedpermanent magnets (SPM)

� the shape, which can be flat (see Fig. 1a) or tubular (seeFig. 1b);

� the core length, which can be with the secondary longerthan the primary (see Fig. 2a), called configuration X orshorter than the primary (see Fig. 2b), called configurationY; in this paper the PM secondary is considered to beshorter than the armature (i.e. configuration Y), and innerin case of tubular machine.

In spite of their advantages, the PML motors exhibit somedrawbacks: one of them is the presence of cogging forcewhich can introduce a disturbance in positioning precision.The cogging force, in PML motors is caused by twophenomena. The first one arises from the interactionbetween the PMs and the finite length of the armaturecore, and is often called ‘end effect’. It can be minimisedadopting a suitable stator length [1–3] or modifying theextremity shape of the shorter part [4]. This phenomenon isprevalent in PML motors of configuration X, but it will notconsidered in the following. The second phenomenon is dueto the interaction between PMs and the armature slots. It is

called ‘slotting effect’ and is present in slotted PML motorsof both configurations X and Y.

Different techniques may be adopted to reduce this effect,which are also employed in rotary PM machines [5, 6]. Aclassical method is PM skewing [7] but it requires a morecomplex PM shape, especially for tubular configurations [8].Another method is to vary the PMwidth [4, 10] or thickness[11], again yielding to very complex PM shape.

An alternative to these methods is the technique of pole-shifting, first presented in [12], which is often employed inthe rotary machines. It consists of a proper shift of the PMs,in order to compensate for the cogging force between thepoles of the motor. An application of such a technique onthe SPM linear motor is reported in [13], illustrating a shiftof half slot of alternate PMs. Unlike PM skewing, thistechnique does not introduce difficulty in the shape of PMs,

a b

Fig. 1 Flat and tubular configurations of PML motora Flatb Tubular

a b

Fig. 2 Motor configurationsConfiguration X: PM secondary longer than primaryConfiguration Y: PM secondary shorter than primary

The authors are with the Department of Electrical Engineering, University ofPadova, via Gradenigo 6/A, Padova I-35131, Italy

E-mail: bianchi@die.unipd.it

r IEE, 2005

IEE Proceedings online no. 20045082

doi:10.1049/ip-epa:20045082

Paper first received 1st July and in revised form 11th November 2004. Originallypublished online: 8th April 2005

IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 703

which remain rectangular in the flat configuration or simplerings in the tubular configuration, making this methodavailable for mass production.

Pole-shifting is commonly used in SPM motors, whilethere is no literature for IPM motors. Anyway such atechnique can be indifferently adopted with both SPM andIPM configurations. This paper generalises the theory ofpole-shifting, extending the discussion to the IPM motor.The rules to calculate the appropriate PM shift aredescribed. The method has been verified by finite-element(FE) analysis and confirmed by experimental measurementson some PML motor prototypes.

2 Analytical expression for the cogging force

The cogging force due to the interaction between the PMsand the armature slotting is essentially independent of thearmature current. The anisotropy of the armature, due tothe presence of teeth and slots, causes a variation ofmagnetic energy with the cursor position (that moves in thez-axis direction) and then an oscillating force.

Assuming a periodicity equal to one slot pitch, thecogging force due to each PM can be expressed by means ofa Fourier series expansion. Referring to the j-th pole, thecogging force is

Fcog;jðzÞ ¼X1k¼1

Fk;j sin k2pzc

tsþ jk;j

� �ð1Þ

where ts is the slot-pitch and zc defines the cursor positionalong the z-axis. Fk,j is the amplitude of the kth harmonic;however, its calculation is not important for the followinganalysis. Each force harmonic is characterised by a differentinitial phase angle. The angle jk,j defines the initial phaseangle of the kth harmonic of the jth pole. It is useful toexpress the angles jk,j of the generic jth pole as a function ofthe initial phase angles of a reference pole, e.g. the polelabelled by j¼ 0, as

jk;j ¼ jk;0 þ j2pqp ð2Þ

where qp is the number of slots per pole.The total cogging force of the motor is expressed as the

sum of the individual contributions of each pole given by(1), obtaining

Fcog ¼X2p�1

j¼0

X1k¼1

Fk;j sin k2pzc

tsþ jk;j

� �ð3Þ

In (2) two cases may occur, according to whether qp isinteger or not. If qp is an integer number, the angles jk,j arethe same for each jth pole making the total cogging force 2ptimes higher than the cogging force of a single pole. If qp isnot an integer number, the force waves of the various polesare not in phase, and then the resultant cogging force isgenerally lower.

Equation (3) is based on the key assumption thatsuperposition can be applied to the forces produced byeach PM of the secondary. Actually superposition holds forthe flux density components, and the application ofsuperposition of the forces requires special care. In theSPM motor, it is possible to assume that each PM edge hasno coupling effect with other PM edges, even after pole-shifting. On the contrary, in the IPM motor, the widths ofthe iron rings are modified, so that flux density amplitudeand distribution change across each pole, as will be shownlater. Therefore a mutual coupling among the PMs exists,and the assumption made about the superposition of forcesmight be not satisfied.

In any case, (3) highlights that cogging force can bereduced by pole-shifting, and suggests a rule to estimate afavorable pole displacement.

3 Technique of pole-shifting

The technique of pole-shifting consists of shifting each jthPM by a quantity Dzj so that the components of the coggingforce of the various poles expressed in (1) are no more inphase, even with qp integer. Thus the harmonic contents ofthe total cogging force expressed by (3) are stronglyreduced. Figure 3 illustrates the basic idea of the pole-shifting technique, considering a two-pole PML motor. InFig. 3a, the two magnets are arranged symmetrically withrespect to the armature teeth, so that the contributions ofthe two poles to the total cogging force are in phase. Thetotal cogging force is therefore two times the single polecogging force. Figure 3b shows that one pole has beenshifted. After such pole-shifting, the two main componentsof the cogging force are 1801 out of phase; thus the totalcogging force is reduced [12].

It is possible to obtain a general expression of the pole-shifting for any number of poles of the motor. For instance,let us consider a generic 2p-pole PML motor. At first, eachPM is numbered as j¼ 0,2,y , (2p�1). A suitable shift Dzj

is computed to distribute the components of cogging forceof each pole along the motor length. It follows that

Dzj ¼ jts

2pð4Þ

so that a sort of stepped skewing along the motor length isobtained. Of course the technique is more effective inmotors with a higher number of poles, since the shift can bebetter distributed.

If qp is not integer, the quantity Dzj can be lower thanthat presented in (4) since there are more periods of coggingforce during a slot-pitch displacement [14]. The number ofFcog periods during a slot-pitch displacement is computed as

Np ¼2p

GCD½Q; 2p� ð5Þ

where Q and 2p are the number of slots and motor poles,respectively, and the denominator is the greatest commondivisor between Q and 2p [5].

With the assumption made for achieving (3), the numberj used in the numeration of the PMs is not significant. Thismeans that different configurations may be obtained.However, two considerations should be made. First,problems of space between adjacent PMs may suggestconsideration only of some suitable combinations. This is

τ τ

ττ

τs

τa

b

fcog, tot

fcog, tot

fcog,1 = fcog,2

fcog,1 fcog,2

Fig. 3 Pole-shifting effect on cogging force (t is the pole-pitch andts is the slot-pitch)

704 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005

the case for the SPM motor with very large PM width.Secondly, magnetic coupling between adjacent poles mayexist, so that differences in the resulting force can be foundadopting different numbering sequences for the PMs. Thisis the case again for the SPM motor with large PM width,but also for the IPM motor in which the pole-shiftingmodifies the flux density amplitude of the poles.

4 Finite-element results

The proposed technique has been verified by means of two-dimensional FE analysis, comparing the results without andwith the proposed shift. A four-pole, a six-pole and aneight-pole tubular PML motor have been considered,characterised by an outer primary containing the armaturewinding, and by an inner secondary containing the PMs.

The cogging force has been computed at no load, varyingthe position of the cursor, integrating the Maxwell’s stresstensor along a surface in the middle of the airgap. Table 1gives motor geometry data, which are the same for the four-pole, six-pole and eight-pole PML motors. The results forthe six-pole SPM and IPM motors are reported later.

4.1 Six-pole SPM PML motorFirst a six-pole tubular PML motor is considered. Figure 4shows a section of the motor, highlighting the six PMs(PM1, PM2,y , PM6). In the same figure, PM numeratingused for pole-shifting is shown. The value of j of the variousPMs is given in Table 2. The slot pitch ts of the motor is11mm, corresponding to three slots per pole. Each jth PMis shifted by a quantity Dzj, as also shown in Table 2. Theresulting PM positions are those of Fig. 4b.

However, to obtain a better flux distribution in the yoke,the configuration shown in Fig. 5 is more convenient. In thelatter, the two end PMs are halved, but they could beconsidered as a unique PM in the numeration. Then theshifting rule is applied in the same way as for the previousconfiguration.

Figure 6 shows the airgap flux density distributionwithout and with the pole-shifting of the secondaryconfiguration of Fig. 4. With the aim of highlighting theeffect of pole-shifting, such a distribution refers to a slotlessprimary. In this way, the distorting effect of the slot openingis avoided.

The results of the FE analysis at no load are shown inFig. 7a, according to the geometrical data given in Tables 1and 2 and secondary configuration of Fig. 5. Without pole-shifting, the peak-to-peak cogging force is equal to 165N.After the pole-shifting, the peak-to-peak cogging force hasbeen reduced to 13N, that is a 92% decrease. Simulation ofthe force under load is shown in Fig. 7b. After pole-shifting,the ripple force is reduced by 86.6% (from 175 to 24N).

As for the stepped skewing, the adopted technique reducesthe average force. The reduction of the average force underload is evident in Fig. 7b. Such a reduction corresponds to5.5% (from 301 to 284N).

Table 1: Tubular PML motor data

Outer cylinder (containing armature winding) Inner cylinder (containing PM)

SPM IPM

Outer diameter 105.4mm PM outer diameter 52mm 52mm

Inner diameter 53.4mm PM inner diameter 44mm 30mm

Slot bottom diameter 93.4mm PM height 4mm 11mm

Wedge height 1.5mm PM width 25mm 5mm

Slot width 7mm Pole-pitch 33mm 33mm

Tooth width 4mm Shaft diameter 30mm 30mm

Number of turns/coil 100 turns

PM3 PM4 PM5 PM6

PM3PM2PM1

PM2PM1

PM4 PM5 PM6

j =0

j =0 j =2 j =4 j =5 j =3 j =1

j =2 j =4 j =5 j =3 j =1

a

baxis of symmetry

axis of symmetry

Fig. 4 Section of six-pole SPM linear motor, configuration 1, withPM numeration used for pole-shiftinga Symmetrical PM distributionb Unsymmetrical PM distribution

Table 2: Six-pole motor: PM numeration and correspondingshift

PM PM numeration PM shift

PM1 j¼ 0 Dz0¼ 0.00mm

PM2 j¼ 2 Dz2¼ 3.66mm

PM3 j¼ 4 Dz4¼ 7.32mm

PM4 j¼ 5 Dz5¼ 9.15mm

PM5 j¼ 3 Dz3¼ 5.49mm

PM6 j¼ 1 Dz1¼ 1.83mm

PM1 PM2 PM3 PM4 PM5 PM6 PM1

PM1 PM2 PM3 PM4 PM5 PM6 PM1

j = 0 j = 2 j = 4 j = 5 j = 3 j = 1 j = 0

j = 0 j = 2 j = 4 j = 5 j = 3 j = 1 j = 0

axis of symmetry

axis of symmetrya

b

Fig. 5 Section of six-pole SPM linear motor, configuration 2, withPM numeration used for pole-shiftinga Symmetrical PM distributionb Unsymmetrical PM distribution

IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 705

4.2 Six-pole IPM PML motorAlthough the technique of pole-shifting is applied to SPMmotors, there are no applications for IPMmotors. With theaim of extending the technique of pole-shifting to the IPMconfiguration, let us consider a six-pole IPM motor, asshown in Fig. 8.

In the case of the IPM motor, the same considerationscan be repeated. The PMs are numbered and then shifted inthe same manner, without considering the iron core ringsplaced among the magnets. The latter have to be adapted tothe new geometry of the cursor.

Figure 8a shows the six PMs (PM1, ..., PM6) that arealternated by the iron core rings. They are numbered as thePMs of the SPM configuration. The PMs are shifted by aquantity Dzj computed by (4) as given in Table 2, where theslot pitch ts¼ 11mm. The resulting PM position is that ofFig. 8b.

Figure 9 show the airgap flux density distribution withoutand with pole-shifting. As for the SPM motor, a slotlessprimary has been considered to highlight the effect of pole-shifting in such a distribution, so that the slot opening effectis avoided. Note that, in this case, the iron core rings havedifferent width, so that the pole width and the pole positionare modified. Therefore the airgap flux density exhibits adifferent amplitude from one pole to another. As mentionedabove, in this case a different numbering sequence of thePMs yields different values of flux density over the poles,confirming the interaction between the displacement of theadjacent PMs.

As shown in Fig. 10a, at no load, the peak-to-peakcogging force of the initial configuration is equal to 64N.With pole-shifting the peak-to-peak cogging force isreduced to 11N, that is a reduction by 83.2%.

0 50 100 150 200

− 1.0

− 0.8

− 0.6

− 0.4

− 0.2

0

0.2

0.4

0.6

0.8

1

without shifting with shifting

flux

dens

ity ,T

position, mm

Fig. 6 Airgap flux-density distribution of the six-pole SPM linearmotor, configuration 1 (referring to a slotless primary), without andwith pole-shifting

0

50

100

150

200

250

300

350

400

−20 −15 −10 −5 0 5 10 15 20

−20 −15 −10 −5 0 5 10 15 20−120

−100

−80

−60

−40

−20

0

20

40

60

80

100

forc

e, N

b

a

position, mm

forc

e, N

without shifting

position, mm

without shiftingwith shifting

with shifting

Fig. 7 Force behaviour with and without pole-shifting for asix-pole SPM motor, configuration 2a No loadb Under load

PM1 PM2 PM3 PM4 PM5 PM6

PM1 PM2 PM3 PM4 PM5 PM6

j = 0 j = 2 j = 4 j = 5 j = 3 j = 1

j = 0 j = 2 j = 4 j = 5 j = 3 j = 1

axis of symmetry

axis of symmetry

b

a

Fig. 8 Section of six-pole IPM linear motor with PM numerationused for pole-shiftinga Symmetrical PM distributionb Unsymmetrical PM distribution

0 50 100 150 200−1.0

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

flux

dens

ity, T

without shifting

with shifting

position, mm

Fig. 9 Airgap flux-density distribution for six-pole IPM linearmotor (the analysis has been carried out using a slotless primary)

706 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005

Simulations of the force under load is shown in Fig. 10b.After pole-shifting, the ripple force is reduced by 78% (from109 to 24N) and the average force by 1.5% (from 252 to248N). Once again the benefit of the pole-shifting is evident.

The numeration of the PMs in the IPM motor influencesthe result a little. In fact, if the poles are narrow the airgapflux density due to the PMs is higher, and vice versa.Different numerations have been tested in order to choosethe most profitable. With the numeration reported in Table2, a satisfactory result is obtained.

4.3 Effect of number of polesThe decrement of cogging force is higher in PML motorswith a higher number of poles. This is because the higher thenumber of poles, the better distributed the cogging forcecomponents. Conversely, the reduction of the average forceis essentially the same, since the number of slots per pole isthe same. Thus, as for the stepped skewing, the correspond-ing shift does not vary between four-pole, six-pole to eight-pole motors. A lower reduction of the average force couldbe obtained with a higher number of slots per pole and witha non integer qp: in both cases the required shift is lower.Some results are given in Table 3. This shows that thereduction of both cogging and ripple force is more effectivewith a higher number of poles. The reduction of the averageforce is also shown in Table 3. In the SPM motor such areduction always decreases with the number of poles. In the

IPM motor such a reduction is not always as expected. Thereason for that can be attributed to the interaction betweenthe adjacent poles after pole-shifting. It is worth noting that,after pole-shifting, the position of the ‘average’ polar axis (ord-axis) is modified. The advance displacement is given by

Dzd ¼1

2p

X2p�1

j¼0Dzj ð6Þ

Then the currents feeding the motor have to be modifiedconsequently to achieve the maximum force density.

5 Experimental results

Some experimental tests have been performed to verify theresults presented above. The prototype used in the tests isshown in Fig. 11: it is a tubular PML motor designed tooperate with both SPM and IPM cursors. The geometricaldata are given in Table 1. The two cursors have been builtwith four poles.

The two configurations of the motor differ from eachother in the cursor assembly only, while the same externalcylinder and winding have been used. In the SPM motor,the magnets are radially magnetized and mounted on theouter surface of the cursor, alternated by aluminum rings,

−20 −15 −10 −5 0 5 10 15 20

−20 −15 −10 −5 0 5 10 15 20

−50

−40

−30

−20

−10

0

10

20

30

40

50

0

50

100

150

200

250

300

forc

e, N

fo

rce,

N

without shifting

with shifting

position, mm a

position, mm

b

without shifting

with shifting

Fig. 10 Force behaviour with and without pole-shifting of asix-pole IPM motora No loadb Under load

Table 3: Effect of pole-shifting on motors with differentnumber of poles

SPM motor, configuration 1

Four poles Six poles Eight poles

cogging force �18% �78% �90%

average force �7.8% �4.7% �4.5%

ripple force �35% �70% �82%

SPM motor, configuration 2

Four poles Six poles Eight poles

cogging force �92.2% �92.0% �95.7%

average force �5.6% �5.5% �4.4%

ripple force �87.9% �86.6% �91.6%

IPM motor

Four poles Six poles Sight poles

cogging force �71% �83.2% �88%

average force �1.0% �1.5% �1.4%

ripple force �73% �78% �81%

Fig. 11 Photograph of PML motor

IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 707

as in Fig. 12a. In the IPM motor, the magnets are axiallymagnetised rings alternated by iron rings (see Fig. 12b).Starting from the initial configuration, with the magnetsequally spaced, both the SPM and the IPM cursors havebeen modified, changing the distribution of the PMs asshown in the previous Section.

In the SPM cursor, the position of the PMs has beencomputed according to (4). In the new configuration, thedistances between the PMs are different compared withthose of the initial one; thus the aluminum rings have beenreshaped. The final SPM cursor is shown in Fig. 12a. In thesame way, in the IPM cursor, the position of the PMs has

been computed by (4); the new distances between the PMsrequired the iron rings to be reshaped. The final IPM cursoris that of Fig. 12b. However, it is worth noticing that inboth the configurations, no difficulties are introduced in thecursor assembly.

Figures 13 and 14 and show the results of stationarymeasurements at different positions of the cursor, obtainedafter locking the cursor by means of a clamping screw.Some difficulties in measurement were met, because of thefriction between the secondary and the brasses. Themeasurement was repeated several times and the averagevalue is reported in the Figures.

The cogging force of the SPM and IPMmotors, obtainedwith the initial PM configuration, shows three completeoscillations per pole-pitch (t¼ 25mm in the prototype),equal to the number of slot per pole (qp¼ 3). This result isthe consequence of adopting a Y configuration: in such acase, the end effect is practically negligible and the coggingforce is mainly caused by the interaction between themagnets and the stator teeth.

After pole-shifting, a considerable reduction of coggingforce has been verified as shown in Figs. 13 and 14. Theperiod of the cogging force changed, confirming that thelower order harmonics have been cancelled [5, 12]. In spiteof the effort in measuring, it is possible to note a non-perfect

magnets

aluminum rings

magnets

iron cores

a

b

Fig. 12 Cursors with shifted polesa SPM motorb IPM motor

100 105 110 115 120 125−50

−25

0

25

50

forc

e, N

without shifting with shifting

position, mm

Fig. 13 Measured cogging force in four-pole SPM motor withoutand with pole-shifting (pole pitch equal to 25 mm)

100 105 110 115 120 125−50

−25

0

25

50

without shifting with shifting

position, mm

forc

e, N

Fig. 14 Measured cogging force in four-pole IPM motor withoutand with pole-shifting (pole pitch equal to 25 mm)

0

10

20

30

40

50

60

70

80without shifting

with shifting

SPM motor IPM motor

cogg

ing

forc

e, N

Fig. 15 Peak-to-peak cogging force without and with pole-shifting

708 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005

equality among the periods, and it is difficult to individua-lise the periodicity of the solutions with shifted magnets.

In conclusion, the modifications to the cursor allow thecogging force to be reduced by 55% in the SPMconfiguration and 65% in the IPM configuration. In thefirst case it is lower than expected, while in the second case itis as expected (see Table 3).

Figure 15 shows a final comparison of the measureddata.

6 Conclusions

The technique of pole-shifting to reduce the cogging force inlinear PM motors has been presented. Such a techniqueallows a considerable reduction of cogging force to beobtained with a moderated effect on the average force. Therules to apply this technique are described. To verify theeffectiveness of such a technique, FE analysis andmeasurements have been carried out on a tubular PMLmotor. Such a technique has been employed not only on anSPM linear motor but also on an IPMmotor, showing thatit is effective for such a configuration as well. A reduction ofthe cogging force by 60–65% has been measured in a four-pole motor, while a reduction by 80 and 90% has beenobtained by FE simulations in a six-pole and eight-polemotor, respectively.

7 Acknowledgment

The measurements were carried out on a tubular linearmotor provided by Magnetic S.p.a. in MontebelloVicentino (VI), Italy, which the authors are gratefullyacknowledge. A special thanks toM. Bellomi andM. Trovafor their help during the tests.

8 References

1 Hor, P.J., Zhu, Z.Q., Howe, D., and Rees-Jones, J.: ‘Minimization ofcogging force in a linear permanent magnet motor’, IEEE Trans.Magn., 1998, 34, (5), pp. 3544–3547

2 Zhu, Z.Q., Hor, P.J., Howe, D., and Rees-Jones, J.: ‘Calculation ofcogging force in novel slotted linear tubular brushless permanentmagnet motor’, IEEE Trans. Magn., 1997, 33, (5), pp. 4098–4100

3 Zhu, Z.Q., Xia, Z.P., Howe, D., and Mellor, P.H.: ‘Reduction ofcogging force in slotless linear permanent magnet motors’, IEEEProc., Electr. Power Appl., 1997, 144, (4), pp. 277–282

4 Jung, S.Y., and Jung, H.K.: ‘Reduction of force ripples in permanentmagnet linear synchronous motor’. Proc. Int. Conf. on ElectricMachines (ICEM), Bruges, Belgium, 2002, (CD–ROM)

5 Bianchi, N., and Bolognani, S.: ‘Design techniques for reducing thecogging torque in surface-mounted PM motors’, IEEE Trans. Ind.Appl., 2002, 38, (5), pp. 1259–1265

6 Jahns, T., and Soong, W.: ‘Pulsating torque minimization techniquesfor permanent magnet AC motor drives: a review’, IEEE Trans. Ind.Electron., 1996, 43, (2), pp. 321–330

7 Jung, I.S., Hur, J., and Hyun, D.S.: ‘Performance analysis of skewedPM linear synchronous motor according to various design para-meters,’, IEEE Trans. Magn., 2001, 37, (5), pp. 3653–3657

8 Bianchi, N., Bolognani, S., Corte, D., and Tonel, F.: ‘Tubular linearpermanent magnets motors: an overall comparison’, IEEE Trans. Ind.Appl., 2003, 39, (2), pp. 466–475

9 Wakiwaka, H., Senoh, S., Yajima, H., and Yamada, H.: ‘Smootherthrust on multi-polar type linear DC motor’, IEEE Trans. Magn.,1997, 33, (5), pp. 3877–3879

10 Cruise, R.J., and Landy, C.F.: ‘Reduction of detent forces in linearsynchronous motors’. Proc. Int. Symp. on Linear Drives for IndustryApplications (LDIA), Tokyo, Japan, 1998, pp. 412–415

11 Lee, J., Lee, H.W., Chun, Y.D., Sunwoo, M., and Hong, J.P.: ‘Theperformance prediction of controlled-PM LSM in various designschemes by FEM’, IEEE Trans. Magn., 2000, 36, (4), pp. 1902–1905

12 Li, T., and Slemon, G.: ‘Reduction of cogging torque in permanentmagnet motors’, IEEE Trans. Magn., 1988, 24, (6), pp. 2901–2903

13 Lim, K.C., Woo, J.K., Kang, G.H., Hong, J.P., and Kim,G.T.: ‘Detent force minimization techniques in permanent magnetlinear synchronous motors’, IEEE Trans. Magn., 2000, 38, (2),pp. 1157–1160

14 Yoshimura, T., Watada, M., Torii, S., and Ebihara, D.: ‘Study on thereduction of detent force of permanent magnet linear synchronousmotor’. Proc. Int. Symp. on Linear Drives for Industry Applications(LDIA), Nagasaki, Japan, 1995, pp. 207–210

IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 709