2200 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 9, SEPTEMBER 2011

3-D Analytical and Numerical Modeling of Tubular ActuatorsWith Skewed Permanent Magnets

B. L. J. Gysen, K. J. Meessen, J. J. H. Paulides, and E. A. Lomonova

Department of Electrical Engineering Electromechanics and Power Electronics Eindhoven University of Technology,Eindhoven 5600, MB, The Netherlands

This paper considers analytical and numerical techniques to model the magnetic field distribution in a tubular actuator with skewedpermanent magnets (PMs). A fast 3-D analytical model based on Fourier analysis is developed for calculation of the various field com-ponents resulting from the skewed PMs for various skewing topologies. This techniques provides means for validating the assumptionsof 2.5-D multilayer methods. Furthermore, a 2.5-D analytical multilayer model is derived for calculation of the cogging force due to theslot openings including skewed PMs. The analytical methods are validated by means of 3-D finite element analysis.

Index Terms—Boundary value problem, Fourier analysis, permanent magnet, skewing, 3-D modeling, tubular actuators.

I. INTRODUCTION

T UBULAR actuators are ever more used in the industrydue to their high force density, excellent servo characteris-

tics, and no need for end windings due to their cylindrical struc-ture [1]–[3]. However, tubular actuators inhibit a relatively largeforce ripple which is caused by three effects:

• the finite length of the stator and/or translator causesfringing fields at the ends which leads to a force ripplerelative to the ratio of the finite length of stator and/ortranslator and the pole pitch and/or slot pitch;

• the slot openings of the stator cause fringing fields leadingto a force ripple dependent on the slot/pole number;

• the winding distribution together with the form of excita-tion (ac or dc) cause an additional force ripple due to theapparent magnetomotive force harmonics and higher har-monics of the magnetic field distribution in the air gap;

where the latter is only apparent under excitation. These forceripples are undesired in most applications, especially when ahigh position accuracy is necessary.

Reduction of these force ripples has been the subject of manyresearchers and many techniques exists to reduce the aforemen-tioned list. Optimizing the finite length can reduce the forceripple due to the finite length of the stator and/or translator[4]. Choosing the proper slot/pole number [5], using soft-mag-netic wedges [6], changing the magnet to pole pitch ratio [7],slot tip width or even alter the magnet and/or tooth shape cansignificantly reduce the force ripples due to the slot openings.Finally, designing for sinusoidal induced electromotive force(EMF) waveforms and using ac excitation will reduce the forceripple due to excitation [8].

In general, these methods only reduce one of the threeapparent force ripples. An alternative effective method for re-ducing one or more forms of force ripples at ones is skewing ofthe stator or translator, as shown in Fig. 1 for translator skewing,

Manuscript received December 23, 2010; accepted March 28, 2011. Date ofpublication April 07, 2011; date of current version August 24, 2011. Corre-sponding author: B. L. J. Gysen (e-mail: b.l.j.gysen@tue.nl).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2011.2139220

Fig. 1. Three-phase slotted tubular actuator with triangular skewed permanentmagnets.

[9], [10]. Skewing of the translator requires special permanentmagnet shapes which is more expensive, however, the costcan be reduced for large numbers. Stator skewing requirescomplex ferromagnetic tooth shapes and coil shapes which ismore expensive but also more complicated from a fabricationand assembly point of view, especially when a laminated statoris considered. Furthermore, skewing of the stator can lead toadditional undesired forces or torques [8], [9].

A few publications consider the analysis of skewed tubularactuators. In [4], skewing of the ends of the stator of a tubularactuator was investigated in order to reduce the cogging forcedue to the finite stator length. A 2-D analytical solution of asingle end is used and a 2.5-D multilayer model is created bymeans of extrusion of the 2-D solution. Successful verificationwas obtained by 3-D finite element analysis (FEA) and measure-ments. Combined minimization of the force ripple due to the fi-nite length and the slot openings has been investigated in [9] bymeans of multilayered 2-D FEA where a reduction of 90% isobtained for the total cogging force. In [11], various 3-D mul-tislice finite element methods were discussed for calculation ofthe force ripple due to the slot openings in tubular motors withskewed permanent magnets. In [8] a 2-D analytical model hasbeen derived for the magnetic field distribution neglecting theslot openings and finite length, hence only the force ripple dueto the winding distribution was considered. Based upon this 2-Dsolution, a 2.5-D analytical model was constructed by means

0018-9464/$26.00 © 2011 IEEE

GYSEN et al.: 3-D ANALYTICAL AND NUMERICAL MODELING OF TUBULAR ACTUATORS WITH SKEWED PERMANENT MAGNETS 2201

of extrusion for various PM skewing topologies. The reductionof the force ripple due to the winding distribution is calculatedwhere triangular skewing offered the highest reduction in ripplefor the lowest reduction in mean force.

In most of the aforementioned literature skewing is mod-eled by means of the 2.5-D multilayered technique or time con-suming 3-D FEA. The multilayer technique neglects the angularfield component, however this assumption might be incorrect forrelatively large skewing angles. In general, time consuming 3-DFEA verifies this assumption and therefore, this paper proposesa fast 3-D analytical model to investigate the effect of the an-gular component which was to date never considered. This 3-Danalytical modeling technique is based on Fourier analysis andseparation of variables and the solution is obtained by solvinga boundary value problem. Only translator skewing is consid-ered for various magnetization patterns and skewing topologieswhich allows one to investigate the assumption of the multilayermethod without using 3-D FEA.

Besides the 3-D analytical model, a 2-D analytical model willbe given which includes the slot openings and allows for cal-culation of the resulting fringing fields and force ripple due tothese slot openings, [12], [13]. Modeling of slotted actuators isgenerally performed using the Carter factor [14] to account forthe reduction in the mean force however does not allow for cal-culation of the force ripple. The proposed method on the otherhand exactly describes the fringing fields and resulting coggingforce. The proposed 2-D analytical model will then be used tocreate a 2.5-D analytical model by means of extrusion (multi-layer method). This model allows for fast calculation of the re-duction in force ripple dependent on the skewing angle.

The benchmark topology of the tubular permanent magnetactuator for this paper will be given in Section II together withthe considered skewing topologies. Section III describes the 3-DFEA model which will be used to verify the analytical methods.The 3-D analytical model will be discussed in Section IV wherean analysis of the angular component on triangular and sinu-soidal skewing will be presented. Finally, the 2.5-D multilayertechnique will be shown and verified with 3-D FEA for trian-gular skewing in Section V followed by conclusions.

II. TOPOLOGY

The benchmark topology consists of a slotted stator on theoutside, and a permanent magnet array with translator back ironon the inside, see Fig. 1. Only the force ripple due to the slotopenings is considered and therefore no armature reaction is in-cluded. End effects are excluded and only one periodic section istaken into account, shown in Fig. 2. A topology with 3 slots perpole-pair is investigated, however the analysis can be repeatedfor different slot-pole combinations. The relative displacementbetween stator and translator will be denoted as . The per-manent magnets are magnetized in the positive and negative ra-dial direction alternatively where skewing is considered in theaxial direction dependent on the angular position. The amountof skewing as a function of the angular position, , for trian-gular skewing is given by

(1)

Fig. 2. One periodic section of the slotted tubular PM actuator with skewedPMs including geometric parameters.

Fig. 3. Skewing functions for (a) triangular and (b) sinusoidal skewing.

with the peak to peak value of skewing in meters, seeFig. 3(a). When, for example, sinusoidal skewing is considered,the dependency is given by [Fig. 3(b)]

(2)

All skewing functions are chosen to have even symmetry aroundwhich simplifies the analysis.

The skewing angle, , in electrical degrees is defined as theratio of the skewing amplitude with respect to the pole pitch

(3)

Similar analysis can be applied for different magnetizationpatterns or skewing patterns, although triangular skewing is con-sidered the most appropriate one in terms of effectiveness [8]and production costs of the permanent magnets. The various ge-ometrical parameters and material properties of the benchmarktopology as indicated in Fig. 2 are listed in Table I.

III. 3-D FINITE ELEMENT MODEL

The 2.5-D and 3-D analytical models discussed in the fol-lowing sections will be verified with 3-D finite element anal-ysis. The finite element model is constructed with Flux3D from

2202 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 9, SEPTEMBER 2011

TABLE ILIST OF GEOMETRIC PARAMETERS AND MATERIAL PROPERTIES

Fig. 4. Mesh distribution of the 3-D FE model.

Cedrat [15]. Since this paper only considers the cogging com-ponent due to the slot openings, only one periodic section ofthe stator and translator is modeled with cyclic boundary con-ditions at and , hence end-effects are neglected.Furthermore, a zero flux boundary is applied atmm, where is the outer radius of the stator. A sliding sur-face exists at the center of the air gap where a non-conformingmesh is authorized. The nodal values on both sides of the slidingsurface are linearly interpolated. Remeshing during movementbecomes unnecessary which saves computation time. A secondorder mesh is generated where the number of nodes is around530 000 depending on the skewing amplitude . The mesh dis-tribution is shown in Fig. 4 and a closer view at the air gap isvisible in Fig. 5.

The radial, angular and axial component of the flux densitysolution in the center of the air gap, , are shownin Figs. 6, 7, and 8, respectively, for a skewing amplitude of

mm or a skewing angle of electrical degreesand a relative distance of mm.

It can be observed that the angular and axial component arequite noisy, even using a dense second order mesh distribution.The noise is particulary present at the corners of the slot open-ings. Two effects are clearly visible, the fringing at the slot open-ings and the effect of skewing of the PMs. For the axial com-ponent, , it can be observed in Fig. 8 that the fringing effect

Fig. 5. Closer view of the mesh distribution of the 3-D FE model.

Fig. 6. Radial component of the flux density in the center of the air gap of the3-D FE model for triangular skewing.

Fig. 7. Angular component of the flux density in the center of the air gap of the3-D FE model for triangular skewing.

is larger than the skewing effect. However, for the angular com-ponent in Fig. 7, , the skewing effect is dominant and at theposition of the slot openings, the angular component appearsin the form of noise since the field component at the positions

GYSEN et al.: 3-D ANALYTICAL AND NUMERICAL MODELING OF TUBULAR ACTUATORS WITH SKEWED PERMANENT MAGNETS 2203

Fig. 8. Axial component of the flux density in the center of the air gap of the3-D FE model for triangular skewing.

Fig. 9. Angular component of the flux density in the air gap of the unskewed3-D FE model.

of the slots changes direction within one mesh element (meshsize around 0.5 mm). In order to proof the noise is a result ofmesh discretization, an unskewed 3-D FEA with equal parame-ters is performed and the angular component is shown in Fig. 9where again a noisy solution is obtained with equivalent ampli-tude. Since an unskewed topology by definition has no angularcomponent, the noisy solution should be a result of mesh dis-cretization. This indicates that the slot openings have no signifi-cant influence on the angular field component as expected sincethe fringing flux at the slot openings is axial directed. These re-sults will be used to compare the correctness of the analyticalmodels described in the following sections.

IV. 3-D ANALYTICAL MODEL

In this section, a 3-D analytical model is developed in order tocalculate all the field components with regard to the skewing ef-fect. Although the developed cogging force only depends on theradial and axial component, skewing of the permanent magnetsintroduces an angular component, and consequently alters theradial and axial component in form and amplitude. The previous

Fig. 10. Boundary value problem of the 3-D analytical model.

described 3-D finite element model has the ability to calculatethese components, however considering the computational time,complexity and inaccuracy due to mesh discretization, an an-alytical model is preferred especially in case of a parametricsearch. A 3-D analytical model is presented without inclusion ofthe slot openings, since they have a very low influence on the an-gular component as was shown in the previous section. This sig-nificantly simplifies the analysis and implementation. Althoughthis model does not allow for cogging force calculations, it is ex-tremely useful to investigate the assumptions of the multilayeredmethod for different skewing topologies and varying geomet-rical parameters such as pole pitch, magnet pitch, magnet heightand air gap length. The 3-D analytical model is based on Fourieranalysis and separation of variables where the total geometry isdivided into concentric regions, permanent magnets (region I)and the air gap (region II), [16] as shown in Fig. 10. The ironis assumed infinite permeable, hence the reluctance of the teeth,stator and translator back iron are considered to be zero. Instead,a boundary condition will be applied at the stator bore and innerradius of the PMs as indicated in Fig. 10. The permanent mag-nets are modeled having a linear second quadrant with remanentflux density and relative permeability . Finally, end-ef-fects are excluded and only one pole-pair is considered with pe-riodic boundary conditions at and .

A. Magnetization Vector

The skewed magnetization profile has only a component inthe radial direction dependent on and written as

(4)

with the unit vector in the radial direction. Since the radialmagnetization component, , is periodic in the -direc-tion over and in the -direction over it can be written asa double Fourier series given by

(5)

2204 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 9, SEPTEMBER 2011

with and the spatial frequencies in the angular andaxial direction, respectively, with the harmonic number in the

-direction. A maximum number of harmonics, and , areincluded for the angular and axial direction, respectively. Thecoefficients and are calculated as

(6)

(7)

when and as

(8)

(9)

for . For triangular skewing, the coefficients are given by

(10)

(11)

where , is the magnet to pole pitch ratio and

(12)

The magnetization profile is shown in Fig. 11 forand with a skewing amplitude of mm and rel-ative position of mm. When other skewing topolo-gies need to be considered, for example in case of sinusoidalskewing, the integrals (6) to (9) can be evaluated numericallyresulting in the magnetization profile shown in Fig. 12. Thefollowing analysis for the field calculation is identical for anyskewing topology with even symmetry around .

B. Field Solution

A scalar potential formulation of the boundary value problemis used, [17]. Hence the magnetostatic field equations for regionI (permanent magnets) and II (air gap) can be written as

(13)

(14)

where the scalar potential, , is defined as

(15)

Fig. 11. Skewed magnetization profile � as a function of � and � for trian-gular skewing (� � ��� � � ��).

Fig. 12. Skewed magnetization profile � as a function of � and � for trian-gular skewing �� � ���� � ���.

with the magnetic field strength vector. Using separation ofvariables, the solution for the scalar potential of region I can bewritten as a double Fourier series

(16)

with

(17)

(18)

GYSEN et al.: 3-D ANALYTICAL AND NUMERICAL MODELING OF TUBULAR ACTUATORS WITH SKEWED PERMANENT MAGNETS 2205

with and the modified Bessel functions ofthe first and second kind, respectively, of the th order withargument and and coefficients forevery and . The function is given by

(19)

The solution of the scalar potential in the air gap (region II), ,is equal to the one of region I by setting andand changing all indexes from I to II. The functionhas therefore not the be considered for the air gap region. Themagnetic field strength for for region I, , is obtained with (15),and can be written as

(20)

with and the radial, angular and axial componentand and the unit vector in the angular and axial direction,respectively. The radial component for region I, , isgiven by

(21)

with

(22)

(23)

where the function is given by

(24)

The angular component for region I, , is given by

(25)

with

(26)

(27)

Finally, the axial component for region I, , is givenby

(28)

with

(29)

(30)

Again, the magnetic field strength for the air gap (region II),, is equal to the solution of the magnetic field strength in

region I by setting and to zero and changing all indexesfrom I to II. The magnetic flux density solution can be obtainedfrom the magnetic field strength by means of the constitutiverelations

(31)

(32)

C. Boundary Conditions

The field equations for region I (permanent magnets) and re-gion II (air gap) are derived but the coefficients

and for both regions ( I, II) are obtained bysolving a set of boundary conditions. The tangential magnetic

2206 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 9, SEPTEMBER 2011

field strength needs to be zero at the inner radius of the perma-nent magnets, and the stator bore, . Further-more, the normal magnetic flux density and tangential magneticfield strength needs to be continuous between region I and II at

. These boundary conditions can be listed as

(33)

Only the conditions (33) (a)–(d) are the independent ones sincethe three conditions (33) (e)–(f) lead to the same equations asthe first three conditions (33) (a)–(c). Furthermore, the coeffi-cients and can be solved independent fromand . The conditions for the coefficients andresulting from the four conditions (33) (a)–(d) can be written as

(34)

(35)

(36)

(37)

The set of equations for and are identical, onlyreplacing by by and by . Hence,in total eight times equations are necessary to solve theset of unknowns.

D. Results and Analysis

The flux density solution for triangular skewing is calculatedin the center of the air gap (region II), , andthe radial, , angular, , and axial, , component areshown in Figs. 13, 14, and 15, respectively for mm,

mm, and . It can be observed thatthe same levels and distribution are obtained as with the FEA ofSection III despite of the fringing at the slot openings. Further-more, a mesh-free solution is obtained and the effect of skewingon the various field components can be investigated more accu-rately and computational efficient. In Figs. 16, 17, and 18, theflux density solution is shown for sinusoidal skewing for thesame parameters as for triangular skewing, showing the abilityof the analytical model to handle complex skewing patterns. Theuse of these complex skewing patterns does not influence thecomputation time nor the complexity which is in clear contrastto the FEA.

Fig. 13. Radial component of the flux density in the air gap of the 3-D analyticalmodel for triangular skewing.

Fig. 14. Angular component of the flux density in the air gap of the 3-D ana-lytical model for triangular skewing.

Fig. 15. Axial component of the flux density in the air gap of the 3-D analyticalmodel for triangular skewing.

With this 3-D analytical model, one is able to calculate thethree field components in a very short computational time

GYSEN et al.: 3-D ANALYTICAL AND NUMERICAL MODELING OF TUBULAR ACTUATORS WITH SKEWED PERMANENT MAGNETS 2207

Fig. 16. Radial component of the flux density in the air gap of the 3-D analyticalmodel for sinusoidal skewing.

Fig. 17. Angular component of the flux density in the air gap of the 3-D ana-lytical model for sinusoidal skewing.

Fig. 18. Axial component of the flux density in the air gap of the 3-D analyticalmodel for sinusoidal skewing.

and investigate the level of the angular component comparedto the radial or axial component. In Fig. 19, the ratio in per-centage between the maximum angular, , and maximum

Fig. 19. Ratio between maximum angular and maximum radial component ofthe flux density in the air gap of the 3-D analytical model for triangular skewing.

Fig. 20. Ratio between maximum angular and maximum axial component ofthe flux density in the air gap of the 3-D analytical model for triangular skewing.

radial component, , as a function of the skewing angle isshown for various air gap lengths. It can be observed that theratio increases linearly with the skewing angle and the slopeincreases quadratically with the air gap length. Increasing theair gap length reduces the radial component and simultaneouslyincreases the leakage flux from one magnet to the consecutivemagnets, hereby increasing the angular component. However,the percentage is relatively low, since for slotted actuators, theair gap length is in the order of 1 mm and skewing amplitudesare typically in the order of 60 electrical degrees, hence theinfluence of the -component is in the order of 2.5%.

The ratio between the maximum angular, , and maximumaxial component, , shown in Fig. 20, again increases lin-early with the skewing angle, , however the slope is inde-pendent of the air gap length, . Increasing the air gap lengthincreases the leakage flux in both the angular and axial direc-tion. For common skewing angles, this ratio is in the order of10%. This analysis indicates that the assumption of the com-monly used 2.5-D multilayered method for skewed permanentmagnets is valid for the benchmark topology and reasonableamounts of skewing. An equivalent analysis can be conductedfor sinusoidal skewing, where the ratio between the maximum

2208 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 9, SEPTEMBER 2011

Fig. 21. Ratio between maximum angular and maximum radial component ofthe flux density in the air gap of the 3-D analytical model for sinusoidal skewing.

Fig. 22. Ratio between maximum angular and maximum axial component ofthe flux density in the air gap of the 3-D analytical model for sinusoidal skewing.

angular and maximum radial flux density in percentage is shownin Fig. 21 and the ratio between the maximum angular and max-imum axial flux density in percentage is shown in Fig. 22. Thesame dependency is visible as for triangular skewing, however,the percentages are a factor 1.5 higher, indicating that the influ-ence of the angular component might become important, leadingto an inaccurate cogging force prediction if a 2.5-D analyticalmodel would be used for extreme values of skewing angles. Fur-thermore, considering different sizing, material properties andskewing topologies, the influence might become significant. Theproposed 3-D analytical model provides a means for investiga-tion of this influence.

V. 2.5-D ANALYTICAL MODEL

This section describes the 2.5-D multilayer analytical methodto calculate the force ripple due to the slot openings for skewedactuators. The 2.5-D analytical model is based on a 2-D axisym-metric model, where the skewing is modeled by an extrusion ofthis 2-D model where the magnets are shifted in the -direc-tion according to the skewing dependency . This was donein [8], however, the slot openings were not included in the 2-D

Fig. 23. Geometry and division in regions for the 2.5-D analytical model.

axisymmetric model. Hence, only the force ripple due to thewinding distribution and excitation could be investigated.

This paper focuses on the force ripple due to the slot open-ings by considering the slot openings as separate regions [12],[13] and the 2-D axisymmetric model of [8] is extended to theone shown in Fig. 23 without considering the armature reac-tion. In this way, the force ripple due to the slot openings canbe calculated together with the effect of skewing of the per-manent magnets. The slot width is assumed equal to the slotopening which is valid for the cogging force calcu-lation since the fringing fields are predominantly apparent at theslot tips. This assumption is not necessary but reduces the com-plexity since less regions need to be considered. Consequently,the height of the adapted slot has become the height of the slotopening plus the slot depth, . The 2-D analytical modelis extended to a 2.5-D analytical model by including skewing ofthe permanent magnets. Skewing of the permanent magnets at aparticular angle is modeled as an additional relative displace-ment in the axial direction given by the skewing function .Hence the total relative displacement of the translator with re-spect to the stator is given by , see Fig. 23. Note thatfor this 2.5-D multilayer model, has become a parameter in-stead of a variable. Again, the soft-magnetic material is assumedto have infinite permeability and the permanent magnets havinga linear second quadrant with remanent flux density andrelative permeability . Therefore, considering radial magne-tization and including the slot openings, three separate regionsare now considered:

• Region I: permanent magnets,• Region II: air gap,• Region IIIs: slot openings .

A. Magnetization Vector

The magnetization vector of region I has again only a ra-dial component which is know described by a 1-D Fourier seriesin the -direction

(38)

(39)

GYSEN et al.: 3-D ANALYTICAL AND NUMERICAL MODELING OF TUBULAR ACTUATORS WITH SKEWED PERMANENT MAGNETS 2209

Fig. 24. Magnetization profile of the 2.5-D analytical model for triangularskewing.

The coefficients and are calculated as

(40)

(41)

For any skewing function, , the coefficients andcan be written as

(42)

(43)

The magnetization vector for triangular skewing is calculatedfor harmonics and for in 100 stepsand shown in Fig. 24, where mm and mm.

B. Field Solution

The field solution is again obtained by solving the magneto-static Maxwell equations in terms of the magnetic scalar poten-tial for every region which can be written as

(44)

(45)

(46)

Note that the scalar potential is now only dependent on andsince is a parameter. Using separation of variables, the solu-tion of the scalar potential for region I, , can be writtenas

(47)

with

(48)

(49)

The solution for region II is obtained by setting andto zero and changing the indexes from I to II. The solution foreach slot region III, is given by

(50)

where the spatial frequencies, , are chosen such that atthe left and the right side of the slot opening the scalar potential,

, is zero. The offset, , of each slot number ,is the left side of each slot given by (see Fig. 23)

(51)

From the solution of the scalar potential, the magnetic fieldstrength is again obtained using (15)

(52)

Note that now only the radial and axial components are ob-tained. The radial component of region I, , can bewritten as a 1-D Fourier series as

(53)

with

(54)

(55)

where the function is given by

(56)

2210 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 9, SEPTEMBER 2011

The axial component for region I, , is given by

(57)

with

(58)

(59)

For the field solution in the air gap, the solution of region Ican be used by setting and and replacingall indexes I by II. The radial and axial magnetic field strengthsolution in the slot region has the form of

(60)

(61)

Finally, the magnetic flux density, , is obtained using theconstitutive relations

(62)

(63)

(64)

which is left for the reader.

C. Boundary Conditions

The coefficients for every region are solved by a set ofboundary conditions which are similar to the ones given in the3-D analytical model, however, since we have slot openingsat the stator bore, the tangential magnetic field strength has tobe continuous at the slot openings but zero elsewhere. Further-more, the tangential magnetic field strength has to be zero atthe outer radius of the slot opening, . The total list ofboundary conditions is given by

(65)

(66)

(67)

(68)

Fig. 25. Radial component of the flux density in the air gap of the 2.5-D ana-lytical model for triangular skewing.

Fig. 26. Axial component of the flux density in the air gap of the 2.5-D analyt-ical model for triangular skewing.

(69)

(70)

These boundary conditions result in a set of equa-tions for the coefficients and

which is extensively described in[12] and are listed in the Appendix. Note that for this boundaryvalue problem, all the coefficients have to be solved in oncesince the relative position of the slots causes coupling betweenthe sine and cosine terms of the field solution.

D. Results and Analysis

Calculating the field distribution in the center of the air gapwith the 2.5-D analytical model for

triangular skewing with a skewing amplitude of mmand relative displacement of mm gives the solutionshown in Fig. 25 for the radial component and in Fig. 26 for theaxial component. Note again that the angular component is notincluded, but the fringing effect at the slot openings is obtained

GYSEN et al.: 3-D ANALYTICAL AND NUMERICAL MODELING OF TUBULAR ACTUATORS WITH SKEWED PERMANENT MAGNETS 2211

Fig. 27. Cogging force waveform for triangular skewing of the 2.5-D analyticalmodel for various skewing amplitudes,� , verified with 3-D FEA.

as is clearly visible in Figs. 25 and 26. The number of harmonicsincluded in region I and II is and for region III is .The flux densities are calculated for in 100steps (100 layers) and for every value of , the set of boundaryconditions has to be solved again which makes this method quitetime consuming for investigation of the field distribution. Theforce calculation can be much more computational efficient aswill be shown. The flux density results are in very good agree-ment with the FEA of Section III.

The axial force calculation or cogging force per periodic sec-tion for the unskewed actuator, , as a function of therelative displacement, , is obtained by means of inte-gration of the Maxwell tensor in the air gap

(71)

evaluating the integral gives

(72)

The axial force calculation for the skewed actuator,, can be obtained from the unskewed force

profile with the multilayer method

(73)

with the number of layers considered. The force is calcu-lated as a function of the relative displacement, , for trian-gular skewing and various skewing amplitudes and shownin Fig. 27 verified with 3-D FEA. It can be observed that a very

Fig. 28. Cogging force reduction in percentage as function of the skewing anglefor triangular skewing calculated with the 2.5-D analytical model and 3-D FEA.

good agreement is obtained. Note that only half of the profileneeds to be calculated since the waveform inhibits odd sym-metry. The calculation of each of those profiles takes about 18hours for the 3-D finite element model. The calculation of eachforce profile for the 2.5-D analytical multilayer method is withina minute. Now this 2.5-D analytical model can be used to cal-culate the minimization of the cogging force component de-pending on the skewing angle . The calculation is performedusing harmonics in the air gap, harmonics for theslot openings, and layers around the circumference. Thecogging force is calculated for triangular skewing with varyingskewing angle, . In Fig. 28 the decrement in cogging force re-lated to the unskewed actuator in percentage is shown as well asthe verification with 3-D FEA. It can be observed that for a verygood agreement is obtained even for relatively large skewingangles. However, care should be taken using this method sincealtering sizes, skewing topology and material properties mightresult in an incorrect force prediction for this method and the3-D analytical method should provide as means for validation.

VI. CONCLUSION

To date, analytical modeling of skewed tubular actuators isperformed using the 2.5-D multilayer method as an approxima-tion for this 3-D problem. By means of 3-D FEA, it has beenshown that the slot openings do not affect the angular field com-ponent resulting from skewing. Therefore, it is possible to derivea 3-D analytical model based on Fourier analysis and separationof variables while neglecting the slot openings for investigationof the angular field component. The 3-D analytical model pro-vides a means for fast calculation of the magnetic field distribu-tion for different skewing topologies depending on the variousgeometry parameters and material properties which validates ifthe assumption of the 2.5-D multilayer method is valid. Further-more, a 2.5-D analytical multilayer method is proposed for cal-culation of the cogging force component resulting from the slotopenings which is validated by means of 3-D FEA. It is shownthat for triangular skewing with radial magnetization, modelingof relatively large skewing angles with the 2.5-D multilayermethod is a valid approach for the benchmark topology. All

2212 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 9, SEPTEMBER 2011

the aforementioned analysis can easily be extended for differentskewing topologies, actuator sizes, and material properties.

APPENDIX

The set of equations resulting from the boundary conditionsfor the 2-D analytical model are listed as

(74)

(75)

(76)

(77)

(78)

(79)

(80)

with the correlation functionsand written as

(81)

(82)

(83)

(84)

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