Hybrid analytical modeling : fourier modeling combined withMesh-based magnetic equivalent circuitsCitation for published version (APA):Pluk, K. J. W., Jansen, J. W., & Lomonova, E. A. (2015). Hybrid analytical modeling : fourier modeling combinedwith Mesh-based magnetic equivalent circuits. IEEE Transactions on Magnetics, 51(8), [8106812].https://doi.org/10.1109/TMAG.2015.2419197

DOI:10.1109/TMAG.2015.2419197

Document status and date:Published: 01/01/2015

Document Version:Accepted manuscript including changes made at the peer-review stage

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1

Hybrid Analytical Modeling:Fourier Modeling Combined with Mesh-Based Magnetic Equivalent

Circuits

K.J.W. Pluk, J.W. Jansen, and E.A. Lomonova

Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

This paper presents a hybrid analytical modeling method which integrates a mesh-based magnetic equivalent circuit modelwith the Fourier modeling approach. This hybrid analytical modeling is capable of correctly predicting the electromagnetic fielddistributions for various two-dimensional geometries. The generalized approach of the presented hybrid modeling concept, makesthe modeling technique applicable to a wide range of electromagnetic devices. By only meshing the parts of the domain in thevicinity of the high permeable materials, the increase in computational effort is limited compared to the sole use of Fourier modeling.The proposed hybrid analytical modeling method predicts the force in the geometry with ≥ 97 [%] accuracy with respect to finiteelement analysis.

Index Terms—Analytical models, electromagnetic modeling, Fourier analysis, magnetic equivalent circuit, reluctance network

LIST OF SYMBOLS⇀

A Magnetic vector potential[Wb m−1

]⇀

B Magnetic flux density vector [T]⇀

H Magnetic field strength vector[A m−1

]⇀

J Current density[A m−2

]⇀

M0 Magnetization vector[A m−1

]e Unit vector [−]Br Remanent flux density [T]µ0 Permeability of vacuum 4π ·10−7

[H m−1

]µr Relative permeability [−]i Arbitrary Fourier region [−]m Arbitrary MEC-region [−]k Arbitrary MEC-element number [−]ψ Magnetic scalar potential [A]ϕ Magnetic flux [Wb]

ωin Spatial frequency

[m−1

]n Harmonic number [−]N Total number of harmonics [−]xp Width of the periodicity [m]

I. INTRODUCTION

ELECTROMAGNETIC field modeling is a crucial stepin the design process of electromagnetic devices. Very

extensive magnetic field modeling is necessary to achievean optimal design. A lot of research and development isconducted in methods to predict the magnetic field distributioninside electromagnetic structures. Especially during the designof an electromagnetic device, the computational efforts shouldbe low to enable a broad exploratory investigation of thedesign space. Therefore, the numerical methods (such as FiniteElement Analysis [1]) are less convenient, and if possible thereduction from a three-dimensional (3-D) geometry to a two-dimensional (2-D) model is made.

Corresponding author: K.J.W. Pluk (e-mail: K.J.W.Pluk@tue.nl)

Analytical or semi-analytical modeling methods are lesstime consuming than the numerical methods, however, usuallythe accuracy that can be obtained with these methods is lower.During the design, the accuracy of the magnetic field calcu-lations is less important which makes the (semi-)analyticalmodeling methods a suitable way to predict the magneticfield. In the literature, many (semi-)analytical methods arediscussed, including

• Magnetic Equivalent Circuit (MEC) modeling [2, 3];• (surface) charge modeling (CM) [4–7];• Fourier modeling [8–11].

For applications where the flux paths are largely pre-definedby the high-permeable materials, the MEC modeling is usuallythe appropriate choice. This pre-defined path enables the usageof flux-tubes to simplify the geometry, and therewith, enablesthe calculation the magnetic field. To apply the MEC modelingto more complex geometries, the modeling can be extendedwith the Tooth Contour Method [12] or a conformal mappingtechnique [13]. The extension enables the calculation of thenon-arbitrary couplings between, for instance, the stator andthe rotor.

For unbounded problems where no coils and no high-permeable material is present, the surface charge modelingis one of the preferable options. The surface charge modelingis extended by applying images [14, 15] or by including thesecondary magnetic field [16] to mimic the high-permeablematerials in the vicinity.

Geometries with repeating geometrical patterns are a typicalexample for a spatial harmonic description such as Fouriermodeling. Using the Fourier modeling, the different materialsare incorporated as regions, where in each region a Fourierbased expression for the magnetic field is used. However,this definition is limited to regions, and therefore materials,of equal width (in the periodic direction). Extending theFourier modeling with mode-matching, regions (i.e. material

2

blocks) with a different width can be incorporated [9, 17].Unfortunately, an additional physically non-existing boundaryis necessary to obtain a solution. Furthermore, in situationswhere small features are important, the number of harmonicscan be limited due to the high fundamental spatial frequency,while a large number of harmonics is required to achievesufficient accuracy.

This paper proposes a semi-analytical modeling methodcapable of incorporating high permeable materials with smallfeatures (such as saliency, tooth tips, slot shapes and smallslots), while the presence of both coils and magnets is pos-sible. The hybrid analytical modeling method discussed inthis paper combines Fourier modeling with a mesh-basedmagnetic equivalent circuit model. The Fourier modeling isextensively discussed by Gysen [9], while the mesh-basedMEC modeling is known from [18–20]. The combination of amesh-based MEC modeling with harmonic modeling methodsis proposed in [21–24]. Where Ouagued [21], Barakat [22]and Laoubi [23] only connect the mesh-based modeling tothe harmonic modeling on one side, this paper applies a bi-directional coupling on both sides of the MEC-region.

In this paper, the hybrid analytical modeling method is de-scribed in a generalized manner for two-dimensional Cartesiansituations. First, the focus is on the model formulation beforeintroducing the full modeling methodology and the boundaryconditions. Afterwards a validation of the modeling method isperformed using a comparison with 2-D finite element modelsfor two different geometries.

II. MODEL FORMULATION

A. Model Assumptions

To apply the hybrid modeling to a certain situation orgeometry, the following assumptions are made:

1) the problem is formulated in 2-D2) the problem is time independent (i.e. quasi-static)3) periodicity is present in the x-direction4) only homogeneous, isotropic, linear materials are used5) magnetic field sources (magnets and coils) are homoge-

neous and invariant in the z-direction.To enable a Fourier based description of the magnetic field,

periodicity is implied, where in this model the periodic direc-tion is chosen along the x-direction. If no actual periodicityis present, the periodicity should be chosen such that theinfluence of the adjacent periods on the observed geometryis limited. Choosing a very large period seems ideal for thesecases, however, this will directly result in a large number ofharmonics required, and therefore, a large computational price.

The semi-analytical modeling given in this paper is onlyvalid under the assumption that all materials used are ho-mogeneous, isotropic and linear. In the initial design of anelectromagnetic device, it is sufficient to know the value of themagnetic flux density inside the materials to indicate whetherthe material might saturate, therefore, the assumed linearityof the materials is not an issue. The influence of the possiblenon-homogeneity and anisotropy of the materials is marginaland, therefore, negligible.

O

Even Periodic Boundaries

yx

z

xc,vcb xvcmxvcb

xc,vcm

xg

xp

xt

xmz xmx

zg

zs1

zs2

zvc

zm

zb,vc

Halbach Array

Shield

Voice-Coil

Fig. 1. Shielding geometry, a locally thickened magnetic shield with a holebetween a permanent magnet Halbach array and voice-coil actuator.

The magnetic field sources, i.e. permanent magnets andcoils, should be homogeneous in their amplitude and novariations in the z-direction are allowed. The invariancy in thez-direction in the modeling is ensured by a proper division ofthe geometry in regions, while the homogeneity of the sourcesis a necessity for the source description in the model. Non-homogeneity of a source in the x-direction can be incorporatedby placing multiple sources with different amplitudes adjacentto each other, i.e. a discrete description of variation in thesource. Furthermore, the permanent magnets in the problemare magnetized in the x or the z-direction, while only currentsources in the y-direction are included, due to the assumed2-D description.

B. Electromagnetic examples

The proposed hybrid analytical modeling method is appliedon two electromagnetic examples, one concerning shielding,and one showing the capabilities of the modeling method inmotor design.

The geometry shown in Fig. 1 consists of a permanentmagnet Halbach array, above which a voice coil actuator, witha permanent magnet, is located. The permanent magnet ofthe voice coil actuator experiences an undesired force causedby the electromagnetic fields originating from the Halbacharray. To reduce the magnetic field of the permanent magnetarray, a magnetic shielding plate is introduced between thepermanent magnet array and the voice coil actuator. Sincea solid ferromagnetic shield would give a large reluctanceforce on the permanent magnet of the voice coil actuator, ahole in the magnetic shield is located, directly underneath thepermanent magnet. Around the hole, the magnetic shield, islocally thickened. The dimensions and material parameters forthe shielding geometry are given in Table I.

To show the possibilities of the modeling method forthe machine design, a multi-tooth flux-switching permanentmagnet linear motor, with double saliency, is modeled. Dueto the capability of the proposed modeling method to includesmall features with a high-permeability, the hybrid analyticalmodeling method is suited for analyzing design features suchas tooth tips, slot shapes and saliency. The chosen multi-tooth flux-switching permanent magnet linear motor geometryand its dimensions are based on the motor presented in [25].

3

TABLE IGEOMETRICAL AND MATERIAL PARAMETERS FOR SHIELDING GEOMETRY

Parameter Value Descriptionxp 100 [mm] Width of the periodicityxmz 20 [mm] Width of magnets in z-directionxmx 5 [mm] Width of magnets in x-directionxg 14 [mm] Width of the hole in the shieldxt 4 [mm] Width of the thickening of the shieldxvcm 10 [mm] Width of the voice coil magnetxvcb 10 [mm] Width of the voice coil bundlexc,vcb 29 [mm] Center of the voice coil bundlexc,vcm 43 [mm] Center of the voice coil magnetzm 20 [mm] Height of the magnetszg1 10 [mm] Gap between magnets and shieldzs1 1 [mm] Thickness of the shieldzs2 2 [mm] Thickness of the shield with thickeningzvc 20 [mm] Height of the voice coilzb,vc 35 [mm] Bottom coordinate of the voice coilBr 1.4 [T] Remanence of the permanent magnetsJ 5

[A/mm2

]Current density in voice coil

µr,m 1.05 [−] Relative permeability of magnetsµr,s 1500 [−] Relative permeability of the shield

yx

z

O

zsb

zstzag

zmt1

zmt2

zmb

xst xsg

xmg

xmt1

xmm

xmt2

xp

zmm

Even Periodic Boundaries

stator

mover

Fig. 2. A multi-tooth flux-switching permanent magnet linear motor geometry.

The geometry is shown in Fig. 2, while the geometrical andmaterial parameters are specified in Table II.

C. Division in Regions

Observing the geometries shown in Fig. 1 and 2, thecoordinate system is defined such that the present periodicityis aligned with the x-axis and the invariant direction is chosenin the y-direction. The z-direction is the direction with thechanging materials. For the hybrid analytical modeling, thegeometry should be divided in regions where each regioncontains only one material, and has a width equal to theperiodicity. The regions are numbered ascending in the z-direction.

A permanent magnet is incorporated in the modeling as aregion containing the magnetic material while the permanentmagnet is represented by a block-shaped magnetization func-tion. Equivalently, the coil bundles are modeled by a regionwith µr = 1 [−] and a block-shaped current density function(see Region 6 in Fig. 3).

In general, in the hybrid analytical modeling all high-permeable materials (µr >> 10) that do not have a widthequal to the periodicity, are modeled using a MEC-region.

TABLE IIGEOMETRICAL AND MATERIAL PARAMETERS FOR MULTI-TOOTH MOTOR

GEOMETRY

Parameter Value Descriptionxp 48 [mm] Width of the periodicityxst 3.6 [mm] Width stator teethxsg 4.4 [mm] Width of gap between stator teethxmt1 3 [mm] Width of small mover teethxmg 3 [mm] Width of gap between small mover teethxmm 3 [mm] Width of mover magnetsxmt2 5.5 [mm] Width of large mover teethzsb 10 [mm] Height of stator backironzst 7.5 [mm] Height of stator teethzag 0.5 [mm] Height of airgapzmt1 3 [mm] Height of small mover teethzmt2 6.5 [mm] Height of large mover teethzmb 4 [mm] Height of mover backironzmm 17.5 [mm] Height of mover magnetBr 1.2 [T] Remanence of the permanent magnetsµr,m 1 [−] Relative permeability of magnetsµr,i 750 [−] Relative permeability of the iron

O

Even Periodic Boundaries

yx

z

1

2

34

5

6

7

MEC

Fig. 3. The division in regions for the shielding geometry shown in Fig. 1.

Inside this MEC-region all materials (high-permeable, but alsoair) and sources are meshed. Therefore, in the geometry ofFig. 1 the magnetic shield with its thickened parts are modeledusing the mesh-based magnetic equivalent circuit modeling.

For the geometry in Fig. 2, the stator teeth are modeled asa MEC-region, while the full mover is modeled as anotherMEC-region. The obtained MEC-regions have a width equalto the periodicity, while the mesh inside the MEC-regions ischosen such that the material changes are incorporated.

The resulting division in regions for the shielding geometryis given in Fig. 3 where Region 4 is the MEC-region. For themulti-tooth motor, Regions 3 and 5 are MEC-regions as shownin Fig. 4, where the division in regions is shown. Throughoutmost of this manuscript a general definition of the regions isused, where i is an arbitrary Fourier region, while m is usedfor the MEC-region.

D. Definition of MEC-element

As previously indicated, inside the MEC-region a mesh-based discretization is used. The MEC-region is meshed in thex, z-plane with rectangular elements. Even though the meshingin both the x− and the z-direction can be non-uniform, in theused formulation, all elements in the z-direction with a certain

4

Even Periodic Boundaries

yx

z

O1

2

4

5

6

3

MEC

MEC

Fig. 4. The division in regions for the multi-tooth flux-switching permanentmagnet linear motor of Fig. 2.

(a)

1 2 3 L

K

(b)

Fig. 5. The meshing principle illustrated on a simplified system where(a) shows the geometry and (b) the obtained meshing including numberingand locally increased mesh-density. (The dashed (vertical) lines indicate theperiodic boundaries.)

x-coordinate will have an equal width in the x-direction andvice-versa. The meshing of the MEC-region is performed suchthat the material boundaries of the geometry are coincidingwith the edges of the MEC-elements. The number of MEC-elements inside the MEC-region can be increased for a moreaccurate solution. Due to the possibility of a non-uniform meshsize, local increase of mesh-density is possible. However, inthe used formulation a local increase of the mesh in the x-direction will be visible throughout the full z-direction of theMEC-region as is shown in Fig. 5 (where the numbering ofthe mesh-elements is indicated as well).

With the described meshing method, the resulting MEC-elements will contain only one material (i.e. iron, air ofmagnet). In each MEC-element, a potential node is definedin its geometrical center. From this potential node towardsall four sides of the MEC-element a magnetic reluctanceis present. In both the positive and negative x-direction, areluctance is assumed, which covers half of the width ofthe MEC-element. While the reluctances in the positive andnegative z-direction each cover half of the height of the MEC-element. The reluctances used are calculated according to

Rkx+ = Rk

x− =lkx

2µ0µkrS

kyz

(1)

Rkz+ = Rk

z− =lkz

2µ0µkrS

kxy

(2)

where lkx and lkz are the sizes of MEC-element k in the x−

Rx−

Rz−

Rz+

Rx+

Fig. 6. Illustration of a mesh element in the mesh-based MEC modeling. Thethick red solid-line square indicates the actual MEC-element. Centered in theelement, the red circular marker, indicates the potential node. The reluctanceelements are indicated by the Rx+, Rx−, Rz+ and Rz−.

and z-direction, respectively, where µ0 = 4π ·10−7 [H/m] isthe permeability of vacuum, µk

r is the relative permeability ofthe material in this MEC-element. Furthermore, Sk

xy and Skyz

are the cross-sectional areas of MEC-element k in the x, y-and y, z-plane, respectively. Due to the assumption of a 2-Dgeometry, a unity size of a MEC-element in the y-direction ischosen.

The reluctances in each MEC-element are illustratedin Fig. 6, where the thick (red) solid-line rectangle indicatesthe actual mesh element, and the red point gives the potentialnode. Furthermore, the solid, dashed, dotted and dash-dottedlines illustrate the flux-tubes corresponding with the reluctanceelements.

With the reluctances, the center node is coupled to theadjacent MEC-elements on the top, bottom, left and right, tocomplete the MEC-region. The periodicity inside the MEC-region is taken into account by assuming that the last elementin the x-direction is adjacent to the first element in the x-direction and vice-versa.

III. MODELING METHODOLOGY

A. Fourier Modeling Description

In order to solve the magnetostatic field distribution ina region, the magnetic flux density,

⇀

B, can be written interms of the magnetic vector potential,

⇀

A as explained in [9].By assuming a 2-D geometry with the invariancy in the y-direction, where the magnetization is only possible in thex- and z-direction and is independent of z, and where onlya current density in the y-direction exists, a second-orderdifferential equation is obtained as given by

∇2Ay (x, z) = −µ0µrJy (x) + µ0∂Mz (x)

∂x(3)

where µr is the relative permeability of the material in theregion, Jy is the current density in the y-direction and Mz isthe residual magnetization in the z-direction.

In each region the solution for Aiy , where i is an arbitrary

region with a Fourier description, is chosen such that the Pois-son equation (3) is fulfilled, while concerning the periodicity

5

in the x-direction. The chosen general solution for Ay is givenby

Aiy (x, z) = Ai

y0 (z) +

N∑n=1

[Ai

ycn (z) cos(ωinx)

+Aiysn (z) sin

(ωinx)]

(4)

where N is the total number of harmonics taken into account,and where Ai

y0 (z), Aiycn (z) and Ai

ysn (z) are given by

Aiy0 (z) = −µ0M

ix0z (5)

Aiycn (z) = − 1

ωin

qin eω

inz − 1

ωin

rin e−ω

inz

− µ0

ωin

M izsn −

µ0µr

(ωin)

2 Jiycn (6)

Aiysn (z) = − 1

ωin

sin eω

inz − 1

ωin

tin e−ωinz

+µ0

ωin

M izcn −

µ0µr

(ωin)

2 Jiysn (7)

where qin, ri

n, sin and tin are the a-priori unknown coefficients

which have to be solved using boundary conditions, and whereM i

zsn, M izcn, J i

ysn and J iycn are the sine and cosine terms of

the Fourier description of the magnetization in the z-directionand the current density in the y-direction, respectively. Fur-thermore, M i

x0 is the DC-term in the magnetization in the x-direction which will occur in the boundary description of thetangential magnetic field strength (later on), and ωi

n is n timesthe fundamental spatial frequency ω0 which is determinedfrom the periodicity according to

ωin = nω0 = n

2π

xp(8)

where xp is the width of the periodicity.With the given definition of the magnetic vector potential,

the magnetic flux density description can be derived accordingto

⇀

Bi (x, z) = −∂Ai

y (x, z)

∂zex +

∂Aiy (x, z)

∂xez. (9)

The magnetic flux density inside an arbitrary Fourier regionis given by

⇀

Bi (x, z) = Bix (x, z) ex +Bi

z (x, z) ez (10)

Bix (x, z) = µ0M

ix0 +

N∑n=1

[Bi

xsn (z) sin(ωinx)

+Bixcn (z) cos

(ωinx)]

(11)

Biz (x, z) =

N∑n=1

[Bi

zsn (z) sin(ωinx)

+Bizcn (z) cos

(ωinx)]

(12)

where Bixsn, Bi

xcn, Bizsn and Bi

zcn are given by

Bixsn (z) = si

n eωinz − tin e−ω

inz (13)

Bixcn (z) = qi

n eωinz − ri

n e−ωinz (14)

Bizsn (z) = qi

n eωinz + ri

n e−ωinz + µ0M

izsn +

µ0µr

ωin

J iycn

(15)

Bizcn (z) = −si

n eωinz − tin e−ω

inz + µ0M

izcn −

µ0µr

ωin

J iysn.

(16)

B. Source Term Description

In the magnetic flux density description, the magnetizationand current density terms are derived as a Fourier descrip-tion. In general, a permanent magnet will be replaced bya rectangular shaped magnetization function. Equivalently, arectangular shaped current density function is assumed. Thesemagnetization and current density functions are described as

M ix (x) = M i

x0 +

N∑n=1

[M i

xsn sin(ωinx)

+M ixcn cos

(ωinx)]

(17)

M iz (x) = M i

z0 +

N∑n=1

[M i

zsn sin(ωinx)

+M izcn cos

(ωinx)](18)

J iy (x) = J i

y0 +

N∑n=1

[J iysn sin

(ωinx)

+ J iycn cos

(ωinx)].

(19)

Except for Mx0, all DC-terms can be neglected since onlyMx0 is required in the Fourier description. The sine and cosineterms of the magnetization in the x-direction are not directlyvisible in the magnetic flux density description, however theseterms are used in the boundary conditions.

According to the general Fourier theory [9], the source terms(i.e. Mx0, Mxsn, Mxcn, Mz0, Mzsn, Mzcn, Jy0, Jysn andJycn), when considering the rectangular shaped definition ofthe source curves, are given by

M ix0 =

1

xp

Six∑

six=1

Msix

x

(xsix

1 − xsix

0

)(20)

M ixsn =

2

xp

Six∑

six=1

Msix

x

ωin

(cos(ωinx

six

0

)− cos

(ωinx

six

1

))(21)

M ixcn =

2

xp

Six∑

six=1

Msix

x

ωin

(sin(ωinx

six

1

)− sin

(ωinx

six

0

))(22)

where six is a counter through the magnets magnetized in the

x-direction in region i, and x0 and x1 are the x-coordinate ofthe left and right sides of the magnet, respectively, and whereM

six

x is the amplitude of the magnetization of this magnet.Equivalently, the source terms for Mz and Jy are found.

A permanent magnet for which the magnetization is notaligned with either the x− or z-direction, is modeled by

6

Rkz+

Rkz−

Rkx+R

kx−

ψk

ψk+1ψk−1

ψk+L

ψk−L

ϕkz+

ϕkz−

ϕkx−

ϕkx+

Fig. 7. Illustration of the magnetic flux, ϕ, reluctance, R, and magneticpotential, ψ, involved in the magnetic equivalent of Kirchhoff’s currentlaw (23).

superimposing a magnet magnetized in x−, and a magnetmagnetized in the z-direction. The strength of these magnetsis obtained by using the vector projection on the axis of theoriginal magnetization.

C. MEC Modeling Description

The harmonic modeling method described above is limitedto describing situations where the high permeable materialspans the full period of the model. The inclusion of smallhigh-permeable features (such as tooth-tips, slot shapes andsmall slots) is not directly possible. Therefore, the mesh-basedMagnetic Equivalent Circuit (MEC) modeling technique isapplied to model the high permeable features that are smallerthan the period of the Fourier model.

As previously indicated, inside each MEC-element apotential-node is defined. With the magnetic equivalence ofKirchhoff’s current law, the magnetic scalar potential distri-bution in the center nodes of the MEC-elements in the MEC-region is found. The magnetic equivalence of Kirchhoff’scurrent law applied to MEC-element k (shown in Fig. 7) gives∑

ϕin =∑

ϕout ⇒ ϕx− + ϕz− − ϕx+ − ϕz+ = 0 (23)

where ϕ is the magnetic flux in a branch, calculated accordingto

ϕkx− =

ψk-1 − ψk

Rk-1x+ +Rk

x−(24)

ϕkx+ =

ψk − ψk+1

Rk+1x− +Rk

x+

(25)

ϕkz− =

ψk-L − ψk

Rk-Lz+ +Rk

z−(26)

ϕkz+ =

ψk − ψk+L

Rk+Lz− +Rk

z+

(27)

where ψk is the magnetic scalar potential at the center node ofMEC-element k, and where L is the number of MEC-elementsin a layer of the MEC-region (i.e. all elements in the MEC-region with an equal z-coordinate). Therefore, MEC-elementk+L is directly on top of element k.

To include the periodicity in the MEC-layer, for the far leftand far right MEC-elements, their adjacent MEC-elements in

Rkz+

Rkz−

Rkx+R

kx−

ψk

ψk+1ψk−1

ψk+L

ϕkz+

ϕkz−

ϕkx−

ϕkx+

(a)

Rkz+

Rkz−

Rkx+R

kx−

ψk

ψk+1ψk−1

ψk−L

ϕkz+

ϕkz−

ϕkx−

ϕkx+

(b)

Fig. 8. Illustration of the coupling between the MEC-region and the Fourierregion, (a) at the bottom, and (b) at the top.

the x-direction are replaced according to

k-1 = k-1+L for k = 1 + qL (28)k+1 = k+1-L for k = L + qL (29)

where q = 0, 1, . . . ,K/L−1, and where K is the total numberof MEC-elements in the region. The top and bottom layerof the MEC-region are linked to the Fourier regions that areadjacent at the top or bottom. From the MEC-region, theFourier regions are considered to be a flux source, therefore,these sources are coupled as shown in Fig. 8. The amplitudeof the flux sources is calculated by

ϕkz− =

¨Bm-1

z (x, z)∣∣z=hb

dSkxy (30)

which is the total amount of flux entering through the cross-sectional area at the bottom of a MEC-element in the bottomlayer of the MEC-region (region m), where hb is the z-coordinate of the boundary between the involved regions(i.e. the MEC-region and the Fourier region underneath).Evaluating the integrations for a unit depth in the y-direction,the strength of the flux source is given by

ϕkz− =

N∑n=1

[−1

ωm-1n

(cos(ωm-1n xk

1

)− cos

(ωm-1n xk

0

))Bm-1

zsn (hb)

+1

ωm-1n

(sin(ωm-1n xk

1

)− sin

(ωm-1n xk

0

))Bm-1

zcn (hb)

](31)

where Bzsn and Bzcn are given by (15) and (16), respectively,and xk

0 and xk1 are the left and right x-coordinate of MEC-

element k, respectively, and k is bounded by 1 ≤ k ≤ L, whichare all MEC-elements in the bottom layer. Equivalently, theflux at the top of a MEC-element in the top layer of the MEC-region is calculated, with the Fourier description in region m+1and for MEC-elements K− L + 1 ≤ k ≤ K.

IV. BOUNDARY CONDITIONS

Inside each Fourier and MEC-region, the description of themagnetic field is now defined. To apply the Maxwell’s equa-tions throughout the whole domain, these equations shouldhold on the boundaries between the regions as well. ApplyingMaxwell’s equations to the boundary results in the continuityof both the magnetic flux density normal to the boundary andthe magnetic field strength,

⇀

H , tangential with the boundary,

7

at the z-coordinate of this boundary (z = hb). Therefore, theboundary conditions are given by

Biz (x, z)

∣∣z=hb

= Bi+1z (x, z)

∣∣z=hb

(32)

H ix (x, z)

∣∣z=hb

= H i+1x (x, z)

∣∣z=hb

(33)

Using the constitutive relation,⇀

B = µ0µr⇀

H+µ0⇀

M0, with⇀

M0

the residual magnetization, (33) becomes

µi+1r

µir

Bix (x, hb)−

µ0µi+1r

µir

M ix (x) = Bi+1

x (x, hb)− µ0Mi+1x (x) .

(34)

A. Boundary conditions between Fourier regions

When both regions i and i+1 are Fourier regions, the con-tinuity of the normal magnetic flux density and the tangentialmagnetic field strength is included by means of the separationof variables. Since both regions i and i+1 have an equal widthof the periodicity, their fundamental frequencies are equal.This implies that a separation of the sine and cosine functionsis valid, and therefore, for every harmonic n, equation (32) isgiven by

Bizsn (hb) = Bi+1

zsn (hb) (35)

Bizcn (hb) = Bi+1

zcn (hb) (36)

The tangential magnetic field boundary condition (34) forevery harmonic n is given by

µi+1r

µir

Bixsn (hb)−

µ0µi+1r

µir

M ixsn = Bi+1

xsn (hb)− µ0Mi+1xsn (37)

µi+1r

µir

Bixcn (hb)−

µ0µi+1r

µir

M ixcn = Bi+1

xcn (hb)− µ0Mi+1xcn (38)

With these equations, 4N boundary equations are obtainedto solve, together with the other boundary conditions, the 4Nunknowns inside a Fourier region (i.e. qn, rn, sn and tn).

B. Boundary conditions between Fourier and MEC-region

The continuity of the normal magnetic flux density and thetangential magnetic field strength should hold on the boundarybetween a Fourier region and a MEC-region as well. Since themagnetic flux density in the z-direction is used as a flux sourcein the MEC-elements connected to the bottom and top layerof the MEC-region, the continuity of the normal magnetic fluxdensity is already covered.

To ensure the continuity of the tangential magnetic fieldstrength, (34) is rewritten for the MEC-region, given by

µkr

µm-1r

Bm-1x (x, hb)−

µ0µkr

µm-1r

Mm-1x (x) = Bk

x (x, hb)− µ0Mkx (x)

(39)

where k is the MEC-element for the observed x-coordinate,and where m-1 is the Fourier region underneath the MEC-region. The magnetic flux density in the x-direction for MEC-element k is found by dividing the average flux in the x-direction by its cross-sectional area,

Bkx (x, hb) =

1

2Skyz

(ϕkx− + ϕk

x+

)(40)

where ϕx− and ϕx+ are given in (24) and (25), respectively.Substituting the general expression (11) for the magnetic fluxdensity in the x-direction for a Fourier region, the tangentialmagnetic field boundary condition, which should hold for allMEC-elements at the boundary, is given by

N∑n=1

[(Bm-1

xsn (hb)− µ0Mm-1xsn

)sin(ωm-1n x

)+(

Bm-1xcn (hb)− µ0M

m-1xcn

)cos(ωm-1n x

)]=

µm-1r

2µkrS

kyz

(ψk-1 − ψk

Rk-1x+ +Rk

x−+

ψk − ψk+1

Rkx+ +Rk+1

x−

)− µ0µ

m-1r

µkr

M kx

(41)

where Bm-1xsn (hb) and Bm-1

xcn (hb) are given by (13) and (14)with assuming that z = hb, the z-coordinate of the boundarybetween the MEC-region and the Fourier region below.

Inside the MEC-region, the distribution of the tangentialmagnetic flux density is a staircase-shaped function, due tothe constant magnetic flux density assumed inside the MEC-elements. This staircase-shaped function is described as aFourier series. Due to the separation of the sine and cosineterms, the tangential magnetic field equations are given by

sm-1n eω

m-1n hb − tm-1

n e−ωm-1n hb − µ0M

m-1xsn+

2µm-1r

xpωm-1n

L∑k=1

[{1

2µkrSyz

(ψk-1 − ψk

Rk-1x+ +Rk

x−+

ψk − ψk+1

Rkx+ +Rk+1

x−

)+

−µ0

µkr

M kx

}(cos(ωm-1n xk

1

)− cos

(ωm-1n xk

0

))]= 0 (42)

qm-1n eω

m-1n hb − rm-1

n e−ωm-1n hb − µ0M

m-1xcn+

−2µm-1r

xpωm-1n

L∑k=1

[{1

2µkrSyz

(ψk-1 − ψk

Rk-1x+ +Rk

x−+

ψk − ψk+1

Rkx+ +Rk+1

x−

)+

−µ0

µkr

M kx

}(sin(ωm-1n xk

1

)− sin

(ωm-1n xk

0

))]= 0 (43)

where the summation over all L elements in the bottomlayer of the MEC-region is performed. By taking the averagemagnetic flux in MEC-element k, the magnetic scalar potentialof three different MEC-elements (i.e. MEC-element k and itsadjacent elements in the x-direction) is used in each term ofthe summation. Since the summation is cyclic over the bottomlayer (see (28) and (29)) the equations can be rewritten towards

sm-1n eω

m-1n hb − tm-1

n e−ωm-1n hb +

µm-1r

2Syz

2

xpωm-1n

L∑k=1

[1

µk-1r

−1

Rk-1x+ +Rk

x−cos(ωm-1n x

)∣∣x=xk-11

x=xk-10

+

1

µkr

(−1

Rk-1x+ +Rk

x−+

1

Rkx+ +Rk+1

x−

)cos(ωm-1n x

)∣∣x=xk1

x=xk0

+

1

µk+1r

1

Rkx+ +Rk+1

x−cos(ωm-1n x

)∣∣x=xk+11

x=xk+10

]ψk = µ0M

m-1xsn+

L∑k=1

[2µm-1

r

xpωm-1n µk

r

M kx

(cos(ωm-1n xk

1

)− cos

(ωm-1n xk

0

))](44)

8

qm-1n eω

m-1n hb − rm-1

n e−ωm-1n hb +

−µm-1r

2Syz

2

xpωm-1n

L∑k=1

[1

µk-1r

−1

Rk-1x+ +Rk

x−sin(ωm-1n xk-1

1

)∣∣x=xk-11

x=xk-10

+

1

µkr

(−1

Rk-1x+ +Rk

x−+

1

Rkx+ +Rk+1

x−

)sin(ωm-1n xk

1

)∣∣x=xk1

x=xk0

+

1

µk+1r

1

Rkx+ +Rk+1

x−sin(ωm-1n xk+1

1

)∣∣x=xk+11

x=xk+10

]ψk = µ0M

m-1xcn+

L∑k=1

[−2µm-1

r

xpωm-1n µk

r

M kx

(sin(ωm-1n xk

1

)− sin

(ωm-1n xk

0

))]. (45)

For the continuity of the tangential magnetic field at the topof the MEC-region, equations (44) and (45) are used for MEC-elements K−L + 1 ≤ k ≤ K and for the Fourier quantities ofregion m+1.

C. Non-continuous boundary conditions

Besides the continuous boundaries, other boundary condi-tions are applicable as well. For a region, which has one of itsboundaries at z = ±∞, the magnetic field will vanish towardsthe boundary. For a Fourier region, this boundary condition isgiven by

Bix (x, z)

∣∣z=±∞ = 0 (46)

Biz (x, z)

∣∣z=±∞ = 0. (47)

Since no sources can exist which have an infinite size, theresults from boundary condition derived from Bx and Bz

become equal. The boundary conditions are, therefore, givenby

Bixsn (z)

∣∣z=±∞ = 0 (48)

Bixcn (z)

∣∣z=±∞ = 0 (49)

Substituting (13) and (14), this is simplified to

r1n = 0 & t1n = 0 if z = −∞ (50)

qIn = 0 & sI

n = 0 if z =∞ (51)

where I depicts the last region of the geometry.Inside a MEC-region, the same kind of boundary condition

can hold if it is assumed that no magnetic field can escapethe MEC-region at one of the sides (i.e. the magnetic field istangential with the boundary). For these situations, the flux inthe z-direction on one of the boundary interfaces is assumedto be zero, therefore, this boundary condition is implementedby assuming that the flux towards this boundary is zero or,equivalently, the reluctance (in the z-direction) towards thisboundary is infinite, given by

If m = 1, Rkz− =∞ for 1 ≤ k ≤ L (52)

If m = I, Rkz+ =∞ for K− L + 1 ≤ k ≤ K. (53)

Finally, the condition that the magnetic field is fully per-pendicular to the boundary, is concerned. In fact this boundarycondition is equivalent with assuming that the material locatedon the other side of the boundary has an infinite permeabil-ity. This means that the magnetic flux density parallel to

the boundary interface, Bx, is zero, therefore the followingboundary equation is used

Bix (x, z)

∣∣z=hb

= 0 (54)

where hb is the z-coordinate of the boundary plane. For aFourier region, (54) can directly be implemented, by

Bixsn (hb) = 0 (55)

Bixcn (hb) = 0 (56)

For a MEC-region, the boundary condition of (54) is mim-icked by assuming that the reluctances in the x-direction insidethe layer of MEC-elements that are the closest to the boundaryinterface are infinite, as given by

Rkx− =∞ & Rk

x+ =∞ (57)

where the boundary elements k are denoted by

1 ≤k ≤ L if m = 1 (58)K− L + 1 ≤k ≤ K if m = I. (59)

Due to implying an infinite x-reluctance throughout the fulllayer of MEC-elements that are closest to the boundary, nomagnetic flux in the x-direction is assumed in this layer ofMEC-elements. Therefore, to increase the accuracy of thesolution, the size in the z-direction of the elements adjacentto the boundary should be taken small.

V. CALCULATING THE COEFFICIENTS

All boundary conditions are known, therefore, the unknowncoefficients are solvable. For each Fourier region, 4N a-priori unknown coefficients are used (i.e. qn, rn, sn and tn).Furthermore, both at the top and bottom of each Fourier re-gion, N equations concerning the sine-terms and N equationsconcerning the cosine-terms are obtained, independent of thetype of boundary conditions used as depicted in Table III.While for the MEC-region, there are K unknown coefficients(i.e. ψk), for which K independent equations are given (basedon Kirchhoff’s current law (23)).

To solve all unknowns, the boundary conditions and coef-ficients are collected in one matrix equation, given by

E1b

E1t

E2b

...Em-1

b

Em-1MEC

EKCL

Em+1MEC

Em+1t

Em+2b...EI

t

︸ ︷︷ ︸

Etot

Q1

Q2

...Qm-1

Ψ

Qm+1

Qm+2

...QI

︸ ︷︷ ︸

X

=

Y 1b

Y 1t

Y 2b...

Y m-1b

Y m-1MEC

YKCL

Y m+1MEC

Y m+1t

Y m+2b...Y It

︸ ︷︷ ︸

Ytot

(60)

where E are the sine and cosine equations of the boundaryconditions of the Fourier region (according to Table III)

9

TABLE IIIEQUATIONS USED FOR BOUNDARY CONDITIONS OF A FOURIER REGION

Type Continuous MEC Zero-flux µ = ∞Top Bottom

Sine (35) (37) (44) (48) (55)Cosine (36) (38) (45) (49) (56)

with trailing and/or leading zeros to ensure the connectionto the correct coefficients, Y are the terms in the boundaryconditions that are not dependent on the unknown coefficients(i.e. the source terms), Q = [ qn, rn, sn, rn]

T representsthe column vector with Fourier unknowns, and Ψ is a columnvector with the magnetic potential inside all MEC-elements.Furthermore, the subscripts b and t represent the bottom andtop of the region, while EMEC is given by (44) and (45), andEkcl is given by (23), and where I indicates the last regionconsidered.

Before solving this set of matrix equations, the coefficientsthat are directly found by a boundary condition (i.e. (50)and (51)) should be removed from the set of equations,together with the corresponding boundary conditions. Af-terwards, the matrix equation (60) is solved using an LU-decomposition, and all the (a-priori) unknown coefficients arefound.

VI. FORCE CALCULATION

With all coefficients defined, the magnetic flux densitythroughout the whole domain is known, therefore, for calcu-lation of the force on a part of the geometry, the Maxwellstress tensor is applied. By defining the integration domain asa closed box around the part of the domain for which the forcedesired (including air), an accurate calculation of the force isobtained with a numerical integration. As derived in [26], theforce density per unit depth on a 2-D geometry is given by

Fx =1

µrµ0

−ˆ

Cb

BxBz dx−ˆ

Cl

(1

2B2

x −1

2B2

z

)dz

+

ˆ

Ct

BxBz dx+

ˆ

Cr

(1

2B2

x −1

2B2

z

)dz

(61)

Fz =1

µrµ0

−ˆ

Cb

(1

2B2

z −1

2B2

x

)dx−

ˆ

Cl

BzBx dz

+

ˆ

Ct

(1

2B2

z −1

2B2

x

)dx+

ˆ

Cr

BzBx dz

(62)

where Cb, Cl, Ct and Cr are the bottom, left, top and rightedges of the integration domain, respectively.

VII. MODEL VALIDATION

The proposed hybrid analytical modeling (HAM) method isvalidated using 2-D Finite Element Analysis (FEA) models forthe shielding and multi-tooth motor geometry given in Fig. 1and 2, respectively. The 2-D FEA models are implemented in

(a)

(b)

Fig. 9. The magnetic flux density for the shielding geometry of Fig. 1, with(a) the absolute value obtained with the proposed HAM, and (b) the differencebetween FEA and HAM, ∆ |B| = |BFEA|−|BHAM |. N = 100 harmonicsare used in all Fourier regions and an x × z-discretization of 265 × 22 isused in Region 4.

FLUX2D [27] and are densely meshed, while the periodicity,and all material properties and geometrical parameters areequal to the implemented hybrid analytical models.

For the shielding geometry shown in Fig. 1, the magnitudeof the magnetic flux density obtained with the hybrid analyticalmodeling is shown in Fig. 9(a). As it is visible, a smooth mag-netic flux density pattern is obtained with the proposed hybridmethod. In Fig. 9(b), the difference between the FEA resultsand the HAM results is shown as, ∆ |B| = |BFEA|−|BHAM |.In the regions where the permanent magnets are located, theharmonic description of the permanent magnets is clearlyvisible (i.e. the Gibbs phenomenon [28]). For the permanentmagnet of the voice coil, a deviation is visible that is extendingfurther in the x-direction than inside the magnet array. Fur-thermore, some deviations are shown inside the MEC-region.These deviations are mainly due to the discretization of theMEC-region, where the magnetic flux density contains discretesteps instead of the desired smooth pattern.

A more thorough comparison between the results obtained,with finite element analysis and the hybrid analytical model,for the shielding geometry is given in Fig. 10, where themagnetic flux density on a path in the middle of the shield

10

−5

0

5

10

Bx[T

]

(a)

FEA

HAM

−2

−1

0

1

Bz[T]

(b)FEA

HAM

0 25 32 36 50 54 75 100−0.2

0

0.2

0.4

x [mm]

∆B

[T]

(c)

Bx

Bz

Fig. 10. The magnetic flux density in the middle of the shield (z =30.5 [mm]) for the FEA and HAM. (a) the magnetic flux density in thex-direction, (b) the magnetic flux density in the z-direction, and (c) thedifference between FEA and HAM, ∆ |B| = |BFEA| − |BHAM |.

(z = 30.5 [mm]) is shown. Observing Bx and Bz , the HAMclosely matches the FEA results. However, the differences,shown in Fig. 10(c), clearly reveal the deviations. For the mag-netic flux density in the x-direction a discretization noise isvisible, the amplitude of the noise increases with the derivativeof Bx. Furthermore, for both Bx and Bz , there are deviationsvisible around the edges of the shield (at x = 36 [mm] andx = 50 [mm]). The remaining large differences are locatedaround the edges of the thickening of the shield.

For a numerical comparison, the force per unit depth in thex− and z-direction on the permanent magnet of the voice coilactuator obtained from FEA and HAM is given in Table IV.Besides the calculated force, the difference in force betweenthe FEA and HAM are given as well, where ∆Fx << 1 [%]and ∆Fz ≈ 3 [%].

For the multi-tooth motor geometry shown in Fig. 2, theabsolute value of the magnetic flux density obtained withHAM is shown in Fig. 11(a). Again a rather smooth magneticflux density pattern is obtained with the proposed hybridmethod. In Fig. 11(b), the difference between the FEA resultsand the HAM results is shown, ∆ |B| = |BFEA| − |BHAM |.Besides the edges of the high permeable material, only smalldeviations are visible.

A more thorough comparison between the finite elementand hybrid analytical results for the multi-tooth motor is givenin Fig. 12, where the magnetic flux density in the centerof the airgap (z = 17.75 [mm]) is shown. The HAM isvisually similar with the FEA results when observing the Bx

and Bz . The difference between the FEA and HAM, shownin Fig. 12(c), clearly shows a deviation between the models.Most of these deviations, for both Bx and Bz , occur around theedges of the teeth of either the stator or the mover. Note that itis well known that an accurate prediction of the magnetic fluxdensity obtained with FEA is not ensured around the edges ofa material.

For a numerical comparison, the force per unit depth onthe mover in the x− and z-direction calculated from FEA and

(a)

(b)

Fig. 11. The magnetic flux density for the multi-tooth motor geometryof Fig. 2, with (a) the absolute value obtained with the proposed HAM, and(b) the difference between FEA and HAM, ∆ |B| = |BFEA| − |BHAM |.N = 100 harmonics are used in all Fourier regions, and an x×z-discretizationof 127× 33 and 225× 73 is used in Regions 3 and 5, respectively.

−0.5

0

0.5

Bx[T]

(a)

FEA

HAM

−2

0

2

Bz[T]

(b)FEA

HAM

0 8 16 24 32 40 48−0.1

0

0.1

x [mm]

∆B

[T]

(c)

Bx

Bz

Fig. 12. The magnetic flux density in the middle of the airgap (z =17.75 [mm]) of Fig. 2 for FEA and HAM. (a) The magnetic flux density inthe x-direction, (b) the magnetic flux density in the z-direction, and (c) thedifference between (FEA) and HAM, ∆ |B| = |BFEA| − |BHAM |.

11

TABLE IVTHE FORCE ON THE PERMANENT MAGNET OF THE VOICE-COIL ACTUATOR

AND THE MOVER OF THE MULTI-TOOTH MOTOR.

Voice-coil Multi-tooth motorFx [N/m] Fz [N/m] Fx [N/m] Fz [N/m]

FEA 121.6 327.5 238 −15.9 ·103

HAM 121.6 317.7 235 −16.0 ·103

∆F [N/m] −0.05 9.83 3.02 141∆F [%] −0.05 3.09 2.24 −0.89

TABLE VTHE SIMULATION PARAMETERS FOR THE VOICE-COIL ACTUATOR AND

THE MULTI-TOOTH MOTOR FOR HAM. N IS THE NUMBER OF HARMONICSAND ”TIME” IS THE SIMULATION TIME INCLUDING MESHING, SOLVING

AND CALCULATING B-FIELD AND FORCE.

Voice-coil Multi-tooth motorHAM HAM

N 100 100

mesh-size 265× 2297× 33169× 73

Time [min] 1.25 12

HAM is given in Table IV. Besides of the calculated force,the difference in force between the FEA and HAM are givenas well, where ∆Fx < 1.5 [%] and ∆Fz < 1 [%].

In Table V, the simulation parameters for the HAM aregiven. For the MEC-regions, the number of mesh elementsis included by ”number of mesh-elements in x × number ofmesh-elements in z”. The simulation time includes meshing,solving and calculation of the B-field and force.

VIII. DISCUSSION

In the previous section, it has been shown that the magneticflux density distribution found with the proposed Hybrid 2-Dmodel with respect to 2-D FEA gives accurate results. In theproposed HAM method, the benefits of two analytical model-ing methods are combined, i.e. Fourier modeling and mesh-based MEC modeling. Both analytical modeling methods havesome drawbacks which have been highlighted in the results ofthe previous section.

The differences in the predicted magnetic flux density insidethe regions with a permanent magnet clearly reveals theharmonic description (i.e. Gibbs phenomenon). The deviationsoccurring are due to the description of a rectangular shapedmagnetization function with a number of harmonics. A morenoticeable difference is visible in Region 6 when compared toRegion 2 of Fig. 9(b). The deviations between the modelsextends further in the x-direction in Region 6 due to theassumption of a full region with one single permeability. In theHAM, the full region has the permeability of the permanentmagnet, while the FEA only uses this permeability inside thepermanent magnet itself.

Since the HAM method uses a mesh-based MEC-approach,the mesh density is an important factor in the HAM. Asespecially visible in Fig. 10(c), the meshing of the MEC-regionresults in an discretization error. The discretization noise iscaused by the fact that the coordinates, on which the magneticflux density is extracted from the HAM, are not the points

on which the magnetic scalar potential is calculated duringsolving. The mesh-based MEC modeling assumes an equalmagnetic flux density inside each of the flux tubes illustratedin Fig. 6, therefore the most accurate result is obtained in thepotential node. Between two potential nodes, the magnetic fluxdensity is assumed constant, which results in a ”staircase”-shaped flux density. To reduce this discretization error, thenumber of MEC-elements inside the MEC-region could beincreased.

Another deviation caused by the discretization is visiblearound the edges of the materials, clearly shown in Fig. 11(b).On the edges of the materials, the magnetic permeability willchange rapidly. Due to the calculation of the magnetic fluxaccording to equations (24) to (27), the changing permeabilitywill be visible between the two potential nodes located aroundthe material edge. By decreasing the size of the MEC-elementsaround the edges of the materials, the deviation of the magneticflux density around these edges is decreased.

Only linear materials are assumed in the shown formulation,which results in the very large magnetic flux density occurringinside the magnetic shield of Fig. 9(a). These magnetic fluxdensity values will most definitely cause saturation inside themagnetic shield. Saturation effects are not included in themodeling, however, based on an iterative process the magneticpermeability of the materials inside the MEC-layer can bechanged. Therefore, local saturation effects can be included inthe MEC-region of the HAM, while only global saturationeffects can be included in most other analytical modelingmethods.

As previously indicated, the accuracy of the magnetic fluxdensity results increases when the number of harmonics takeninto account is increased. However, due to the large differencein values in the E-matrix (60), which are caused by the pos-itive and negative exponential values in the Fourier modeling,singularity in the matrix might occur. This singularity limitsthe number of harmonics since the harmonic number n is usedin the exponential terms.

An increase in the number of MEC-elements gives a moreaccurate solution, however, increasing the mesh-density, inequality with FEA, will result in a larger computational effort.From computational point of view, the proposed HAM is lesscomputational intensive with respect to FEA, especially sinceonly a part of the domain is meshed. A large, additionalreduction of the computational efforts is possible in theHAM, since the current implementation is not optimized forcalculation time. For both the FEA and the HAM, a trade-off should be made between the desired mesh-density andthe resulting calculation times. However, since the HAM onlymeshes the regions with changing material properties, theairgap of a motor is not meshed, while especially in the airgapof a machine, a high mesh-density in FEA is required for anaccurate result.

A significant reduction in calculation time of the magneticfield and force is possible for certain variations. Since allsource terms (i.e. the magnetization and current density terms)are collected in the Ytot-matrix, variations of the sourceamplitude and movement the source in the x-direction isincluded by only recalculating Ytot. A movement of a source

12

in the z-direction or movement of (and movement within) theMEC-region requires recalculation of the full system of linearequations.

The proposed modeling method is not only suited for ge-ometries in cartesian coordinate systems. With the adaption ofthe harmonic description according to [9], polar and cylindricalcoordinate systems are possible as well.

IX. CONCLUSION

The principle of combining Fourier modeling with mesh-based Magnetic Equivalent Circuit (MEC) modeling is demon-strated on two electromagnetic examples. It is shown thatthe Hybrid Analytical Modeling (HAM) technique can beapplied to any geometry where a periodicity is present orcan be assumed in cartesian coordinate systems. With theinclusion of the mesh-based MEC modeling, the possibilitiesand applicability of the modeling for geometries with smallfeatures of high-permeable material, such as saliency, tooth-shoes and slot shapes, is increased. Since HAM describes theairgap region of a machine by means of a Fourier series,it requires a less dense meshing compared to finite elementanalysis.

For the electromagnetic example structures shown in thispaper, an excellent agreement is obtained with respect tofinite element analysis, which proves the applicability of thismodeling method for a wide class of electromagnetic devices.For the forces calculated with the hybrid analytical model, lessthan 3 [%] deviation with respect to finite element analysis isobtained.

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[2] H. C. Roters, Electromagnetic Devices. New York: Wiley, 1941.[3] V. Ostovic, Dynamics of Saturated Electric Machines. New York:

Springer-Verlag, 1989.[4] E. P. Furlani, Permanent Magnet and Electromechanical Devices,

I. Mayergoyz, Ed. Elsevier, 2001.[5] J. L. G. Janssen, B. L. J. Gysen, J. J. H. Paulides, and E. A. Lomonova,

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