slotless permanent-magnet machines: general analytical ... · pfister and perriard: slotless...

IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 6, JUNE 2011 1739

Slotless Permanent-Magnet Machines: General AnalyticalMagnetic Field CalculationPierre-Daniel Pfister�� and Yves Perriard�

Moving Magnet Technologies SA (MMT), 25000 Besançon, FranceLaboratory of Integrated Actuator (LAI), Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland

This paper presents a general analytical model for predicting the magnetic field of slotless permanent-magnet machines. The modeltakes into account the effect of eddy currents in conductive regions and notably in conductive permanent magnets without neglectingtheir remanent field. The modeling of this effect is important for the design of very high speed slotless permanent-magnet machines, asthe power losses are linked with the frequency of the field. The model takes into account any number of layers. It implies that, for one,the fields can easily be calculated in a design including a permanent magnet and a conductive retaining sleeve. The model is applicableboth to internal rotor and external rotor permanent-magnet machines. The effect of the relative permeability and of the conductivity ofthe permanent magnet or of the yoke on the magnetic fields is also taken into account. Any magnetization can be taken into account, inparticular a Halbach type permanent magnet, or a radially magnetized permanent magnet can be considered.

Index Terms—Analytical magnetic fields solutions, permanent-magnet machines, slotless motors.

I. INTRODUCTION

S LOTLESS permanent-magnet (PM) machines are increas-ingly used for very high speed (VHS) applications. The

analytical modeling of the magnetic field is important for thedesign and optimization of such applications. Also, analyticalmodels of the fields calculated for slotless structures are widelyused in the design of slotted structures using conformal map-ping or Carter coefficients.

Many papers (see Section I-C) have already been writtenabout the analytical calculation of magnetic fields in severalparticular cases. The objective of the present paper is to showa general model (see Table I), applicable to a very large familyof slotless machines.

A. Considered Structure

The structure considered in this paper is the following. Itis made of -concentric contiguous hollow cylinders. Everycylinder is characterized by:

• its inner and outer radius;• its permeability and conductivity;• its spatial and temporal applied current density harmonics;• its remanent field spatial harmonics;• its rotational speed.At the interface between two contiguous hollow cylinders,

the interface is characterized by its spatial and temporal appliedsurface current density harmonics.

Each hollow cylinder is called a “layer.” The model is calledan -layer model. Every layer in which the magnetic field iscalculated, and every boundary is numbered starting with theinnermost one as shown in Fig. 1.

In the presented model any spatial magnetization harmonicscan be taken into consideration. Allowing models of differentkinds of magnetization and in particular the ideal Halbach mag-netization as it is defined in [1] and [2] to be created. Radialmagnetization can be approximated by this model.

Manuscript received June 25, 2010; revised September 15, 2010; acceptedDecember 06, 2010. Date of publication February 17, 2011; date of current ver-sion May 25, 2011. Corresponding author: P.-D. Pfister (e-mail: [email protected]).

Digital Object Identifier 10.1109/TMAG.2011.2113396

Fig. 1. Section of the contiguous hollow cylinders: the example of a 3-layerstructure, with infinite permeability in the center and in the exterior. The struc-ture is defined to be general. The model is able to deal with any number of layers.It is able to deal with PM motors, PM generators, eddy-current brake, and otherkind of electromechanical structures.

B. Which Differential Equation?

As it is shown in Section II, starting from Maxwell’s equa-tions and in the constitutive equations of the materials, a diffu-sion equation can be derived in its generalized form: it includesthe effect of magnetic remanent fields, eddy currents, and ap-plied currents. In this paper, it is called: a generalized diffusionequation (GDE).

Depending on the assumptions which govern the physics ofthe structure, the governing equation can take different formsthat are named differently (Fig. 2). These different forms can bededuced from the GDE.

Different authors have used these different forms of the GDE:Laplace equation [3], Poisson equation [4], the diffusion equa-tion [5], and the generalized form of the diffusion equation [6].

C. Literature Review

The resolution of a differential equation in order to obtainthe magnetic field in an electrical machine is not recent. Withthe exception of some papers where the 3-D analytic magneticfield solution is presented [7], [8], most publications involve

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1740 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 6, JUNE 2011

Fig. 2. The differential equation used for the calculation of the magnetic vectorpotential.

2-D solutions. Here are some important contributions to thistheory.

1) Already in 1929, B. Hague wrote a book about 2-D solu-tions of Poisson’s equation [9]. He calculated the fields dueto currents in a cylindrical geometry.

2) In 1977, A. Hughes and T. J. E. Miller [10] presented amodel of the field created by a conducting sheet in a 5-layerstructure.

3) Based on the work of B. Hague, N. Boules wrote a paper in1985 [11] entitled “Prediction of no-load flux density dis-tribution in permanent magnet machines,” where the PM’smagnetization is replaced by equivalent currents.

4) In 1993, Z. Q. Zhu and D. Howe wrote an excellent seriesof four papers [12]–[15] on the calculation of the magneticfield in electrical machines based again on the magneticscalar potential. The fields due to the PM are directly cal-culated using the PM magnetization.

5) In 1995, Z. J. Liu et al. used the magnetic vector potential tocalculate the fields and the eddy currents in the stator yoke[16]. They divided the space into three concentric layersand expressed the analytical field solution in the layers.

6) In 1997 and 1998, after discussion with N. Boules, F. Dengwrote two papers [17], [18] about eddy-current power lossesdue to the commutation of the PM machine. She was able tosolve the differential equation not only in different layers,but also in different sectors in some layers. In these cal-culations, the remanent field of the PMs is neglected. Theapplied current in the coils is approximated by a surfacecurrent density between two layers. This model enables thecalculation of the eddy-current power losses in the motorteeth due to the pulsewidth modulation [19].

Several improvements and contributions have been made onthis topic in the last 10 years. Amongst these, the following pub-lications can be cited.

1) In 2003, S. R. Holm calculated the magnetic fields in acylindrical structure. He included in his calculations thefields due to an applied current density in a layer [4].

2) In 2007, Shah et al. solved a 6-layer structure in cartesiancoordinates [20], based on a previous work [21].

3) In 2007, A. Chebak determined the solution of the differen-tial equation for a 4-layer structure, -pole-pair PM, withmagnetization harmonics and calculated the eddy currentsin the stator [22].

4) In 2008, M. Markovic calculated the eddy-currentpower losses in a one-pole-pair PM [5], consideringits magnetization.

This timeline is not exhaustive, but shows some importantmilestones on the path to the resolution of the differential equa-tions for obtaining the fields. Many other papers could be cited:[6], [23], [24], [27], [28]. A brief overview of some papers which

solve the differential equation is presented in Table I. Some re-marks about these two tables are as follows.

• A “Yes” in the “Eddy currents” row means that the eddy cur-rents, if they are calculated, are a direct solution of the differ-ential equation. If the geometry is more complex than a con-centric contiguous hollow cylinder, it may be better to makesimplifying assumptions than to directly solve the generaldifferential equation in order to derive eddy currents.

• Concerning the “Innermost boundary” row:— “ ” means that the material inside the boundary

1 is considered to be of infinite permeability. No mag-netic field is calculated in this region. Since the layeris defined as a region in which the magnetic fields arecalculated, in this case the center is not considered as alayer, as in Fig. 1.

— “ ” means that the radius of the boundary 1 tendsto zero.

• Concerning the “Outermost boundary” row:— “ ” means that the exterior of the boundary

is considered to be of infinite permeability.— “ ” means that the outermost layer covers the

whole space: the radius of the boundary tends toinfinity.

For more information about the “Innermost boundary” andthe “Outermost boundary,” see Section IV-B.

• If a “(1)” stands in the “Magnetic potential” row, it meansthat instead of solving a differential equation involving amagnetic scalar or vector potential, the authors solved adifferential equation directly involving the current density.

• If a “(2)” stands in the “Eddy currents in the PM(s)” row,it means that the PM remanent field and its harmonics areneglected in the calculation of the eddy currents. In thesecases the eddy currents are due to the excitation current.The interaction of the PM’s magnetization with the eddycurrents is also neglected.

• As is shown in Section III, the form of the diffusion equa-tion in a PM where eddy currents are considered is differentand implies a much simpler solution when the PM is paral-lelly magnetized than when higher harmonics of remanentfield are considered. If a “(3)” stands in the “Eddy currentsin the PM(s)” row, it means that the calculation is valid onlyfor a 1-pole-pair parallelly magnetized PM.

There are some limitations to the different resolutions ofthe differential equation in cylindrical coordinates presented inTable I.

• The number of layers is limited.• The number of simplifying hypotheses is high.• The models of eddy currents in the PMs neglect the effect

of the remanent field. The only known exception is onecase examined in the literature which is a one-pole-paircentral PM diametrically magnetized [5].

• Each model is the result of laborious calculations.

D. Objective

The aim of the present paper is to show a model which hasthe following advantages.

1) The procedure which gives the analytical solution is fast.2) The procedure gives a model for any number of layers.3) As the magnetization of the PM is defined by its harmonics,

any magnetization can be set, including the ideal Halbachmagnetization.

PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES 1741

TABLE ICOMPARISON OF THE SOLUTIONS OF THE DIFFERENTIAL EQUATIONS. THE REMARKS ARE IN THE TEXT

4) The eddy currents in the PM are determined, with anynumber of pole pairs, magnetization harmonics, appliedcurrent density harmonics and applied surface current den-sity harmonics.

5) The model can handle for the innermost boundary: zeroradius or nonzero radius with infinite permeability insidethe boundary.

6) The model can handle for the outermost boundary: infi-nite radius or finite radius with a material of infinite per-meability outside it.

7) The model can handle at the same time applied surfacecurrent densities and applied current densities.

E. Outline of the Paper

Maxwell’s equations and two equations which describe ma-terials are used to obtain the GDE. The GDE is considered in theFourier space and solutions are found. Boundary conditions aredescribed and finally a general solution is found for a multilayersystem.

F. Assumptions

1) The permeability and the conductivity are isotropic.2) Until (22), the system is considered to be 3-D. Afterwards,

3-D effects are neglected and the system is considered tobe 2-D.


3) For the 2-D, no lamination of any materials is considered.4) The materials are considered to be linear with no

saturation.5) The PMs do not demagnetize.6) Each layer is cylindrical.7) The wavelengths of all time-varying fields are large com-

pared with the physical dimensions of the device.

II. FROM THE MAXWELL’S EQUATIONS TO THE GENERALIZED

DIFFUSION EQUATION

A. Maxwell’s Equations

Maxwell’s equations are written in the following form:

(1)

(2)

(3)

(4)

where is the electric displacement, is the resistivity, isthe electric field strength, is the magnetic flux density, isthe magnetic field strength, and is the current density.

B. The Constitutive Materials Equations

The constitutive equation of a PM is

(5)

where is the remanent field and is the permeability. Sincethe permeability is isotropic, is scalar. Since the materials areassumed to be linear without saturation, is constant. In softferromagnetic materials, the same equation is used, but with

.The current density in a moving conductor with relative ve-

locity is generated by the Lorentz force and is given by [29]:

(6)

is the conductivity which is assumed to be constant.

C. Vector Potential

As the divergence of is equal to zero , a mag-netic vector potential such that

(7)

is taken into consideration. Equations (2) and (7) are combinedto obtain

(8)

which gives after integration

(9)

where is a electric scalar potential.

is not uniquely defined. Let be an arbitrary scalar func-tion, and and be defined as

(10)

(11)

It implies that

(12)

(13)

Therefore, the potentials defined by (10) and (11) give the samefield. To define uniquely, the Coulomb gauge [30] has beenchosen:

(14)

D. Generalized Diffusion Equation

Since the permeability is assumed to be isotropic and con-stant, (4), (5), (6), and (7) are combined to obtain

(15)

Using the Coulomb gauge and the identity

(16)

(15) is rewritten as

(17)

Equation (7) is the most general form of the GDE presentedin this paper. It can be simplified, depending on material’s prop-erties and other hypotheses:

• It can be reduced in PMs with no applied currents, wherethe eddy currents are considered. A GDE type equation isobtained:

(18)

• It can be reduced in a conductive media, where no currentis applied where the eddy currents are considered. A diffu-sion type equation is obtained:

(19)• It can be reduced in the air. A Laplace type equation is

obtained:

(20)

• It can be reduced in a static media, where the eddy currentsare not considered, but where current is applied. A Poissontype equation is obtained:

(21)


where is the applied current density. For (21), theapplied current density is deduced from the scalar potentialusing (6):

(22)

By hypothesis, the complete system is 2-D. The cylin-drical coordinate system is used. The magneticvector potential and the current density are hence along

. and are in theplane. The angular velocity of the material is so .Equation (17) is expressed along :

(23)

III. THE GENERALIZED DIFFUSION EQUATION SOLUTIONS

Now that the GDE is formulated, its solutions need to befound. Since the structure is made of cylinders, there is a spa-tial periodicity of any quantity, but in particular of the magneticvector potential. Moreover, as the transient states are not consid-ered, there is a time periodicity of period which correspondsto the period of the applied current. The angular frequency isdefined by

(24)

For any quantity , the following periodicity conditionapplies:

(25)

(26)

The easiest way to find solutions is to consider the differentvariables in the Fourier space.

A. Complex Fourier Series

The complex Fourier series of any quantity is defined as

(27)

where is the real part of .The GDE (23) is expressed for the th harmonic:

(28)

which can be simplified as

(29)

with

(30)

The solution of (29) depends on the material properties andon many parameters. Two cases need to be considered.

1) The case where eddy currents can possibly occur, if thelayer of the system is defined by the right harmonics, ro-tation speed and material conductivity. This case is calledECPO.

2) The case where the hypothesis is made that no eddycurrents occur or where they are neglected. This case iscalled NECO.

Some examples of both cases follow.1) The ECPO case:

• A PM layer with a given conductivity rotating in a fieldcreated by different applied current harmonics.

• A cylinder of copper rotating in a synchronous field. Noeddy currents occur, nevertheless, if the harmonic con-tent of the field is enriched, eddy currents would occur.

2) The NECO case:• Any nonconductive media.• Laminated iron in which eddy currents are neglected. If

the eddy currents are not neglected in laminated iron,the problem is intrinsically 3-D and cannot be solvedby the present model. For laminated iron, the simplestapproach is to solve the equation without eddy currentsfirst, and then use an a posteriori model of the powerlosses due to the eddy currents as a function of the field.

• A coil layer. The insulation between the wires impliesthat the NECO case is a better approximation than theECPO case.

B. The Generalized Diffusion Equation in the “Eddy CurrentsCan Possibly Occur” Case

Good examples of conductive media would be: titanium orcopper cylinders, a conductive PM, and iron. In the ECPO case,the following hypotheses are made.

1) and are constant. This assumption is al-ways true for any ideal Halbach PM. This assumption is agood approximation in PMs used for electrical machines.

2) There is no external current: .3) The material is conductive: .4) The field is synchronous: , with the number of

poles pairs. In the moving part: . In the standstillpart: .

In that case (29) becomes

(31)

In the resolution of (31), two cases need to be separated.1) The first case is implied by the following conditions:

• For a part which belongs to the rotor: or.

• For a part which belongs to the stator: or .If one of the above conditions is fulfilled, the solution is

(32)


with the particular solution of the differential equation:

(33)

It is important to notice that in a material which has noremanent field . Also if the remanent field isparallel .

2) The second case is implied by the following conditions:• For a part which belongs to the rotor: and

.• For a part which belongs to the stator: and .The solution is

(34)

where is the Bessel function of the first kind (seeAppendix E) and is the Bessel function of the secondkind (see Appendix F).

The particular solution is the following for even :

(35)

where is the Struve function (see Appendix G),is a generalized hypergeometric function (see

Appendix H), and

is the generalized Meijer G function (see Appendix I).With odd, no general formula for was found. Here

is the formula for :

(36)

From (31), it can be simply deduced that when the magnetiza-tion of the part is parallel or when there is no remanent magne-tization: .

Fig. 3. Diagram of an electromagnetic �-layer structure. Every layer is charac-terized by its constitutive material properties, by its rotational speed and by theexternal current applied. This structure is general, it can represent, for example,eddy-current brakes and PM slotless motors. In this representation, the layer �contains a coil defined by the angles � and � .

C. The Generalized Diffusion Equation Solutions in the “NoEddy Currents Occur” Case

With respect to the solution of the GDE in the domain of elec-trical machines, magnets with low conductivity, air, litz wire,can be related to or approximated by the NECO case. In a non-conductive media, the following assumptions are made.

1) As in the ECPO case: and are constant.2) The field is synchronous .The following expression of the GDE can be solved when the

material is not conductive, or when the eddy currents can beneglected:

(37)

In the resolution of (37), different cases are separated de-pending on

(38)

with

(39)

In the case of a balanced three-phase machine, the harmonicsare given by

(40)

where and are the boundary angle of the coil, isthe time harmonics of the current density. The two anglesand are shown in Fig. 3.


Fig. 4. Boundary conditions.

IV. THE -LAYER PROBLEM

The vector potential obtained by solving the GDE is deter-mined in the previous sections. The constants andremain to be determined. These two constants are defined bythe boundary conditions. The considered structure is an -layerconcentric structure as shown in Fig. 3. The two boundary con-ditions for each interface are deduced in Section IV-A. As thereare layers, there are interfaces and hence condi-tions. In Section IV-B, the innermost and outermost boundariesgive two more conditions. The total number of conditions is .The conditions give the equations that are needed to de-fine the constants.

The interior boundary of the th layer is called . Its perme-ability is called . is the coefficient associated withthe th layer for the harmonic and . represents thecoefficient of the th layer for . repre-sents the function of the layer for .

The condition is defined to be true for layer if:• For a layer which belongs to the rotor:

and .• For a layer which belongs to the stator: and

.Otherwise it is defined to be false.

represents the condition which expresses the factthat the harmonic creates eddy currents in the layer .

A. Two Kinds of Boundary Conditions

The boundary defined by between layerand layer is taken into consideration. implies

that . By the definition of the magneticvector potential, in the 2-D case it follows:

(41)

which can be expressed as a Fourier series:

(42)

This is the first boundary condition.implies that

(43)

where is by definition the surface current density at theboundary between layer and layer . Fig. 4 shows the dif-ferent vectors at the boundary.

Using , the following expression is obtained:

(44)

which gives

(45)

in the 2-D approximation. The second boundary condition forthe vector potential is obtained:

(46)

The last expression can be expressed as a Fourier series:

(47)

B. Three Kinds of Innermost and Outermost BoundaryConditions

The innermost and outermost boundary conditions are at theinner side of the innermost layer and at the outer side of the out-ermost layer of the -layer structure. The boundary conditionsare deduced in the following three cases:

1) Center: If the innermost boundary is defined in ,this boundary is called “center.”

needs to be defined in . It implies that ifis true, (34) gives that

(48)

(49)

and if is false, (32) and (38) imply that

(50)

(51)

2) Infinite Permeability: If inside from the innermostboundary the permeability is considered to be infinite: ,(47) gives

(52)

If the exterior of the outermost boundary is considered to beinfinite, (47) gives

(53)


3) Infinite Radius: In this case, for the layer , only a non-conductive material is considered. The assumption is made thatthere is no flux in . This implies that the magneticvector potential needs to be zero in . Equation (38)gives

(54)

(55)

C. Matrix

The magnetic vector potential given by a single wire is nottaken into consideration. In the problem, for any current densitywhich flows in one direction, there is always a current densitywhich flows in the other direction. Therefore, the amplitude ofthe vector potential’s harmonic is assumed to be equal tozero.

The coefficients , with and needto be calculated to obtain the magnetic vector potential. In orderto calculate them, the following vector is defined:

...

(56)

Using the boundary conditions of Sections IV-A and IV-B,the following system is found:

(57)

can be expressed as

��

��

��

� �

�

��

� �

� �

� �

� �

� �

� ��

�

��

� �

� �

� �

� �

......

. . ....

......

......

.... . .

......

......

� �

� �

� �

� �

� �

� �

�

��

� � � � � � � � � � � � ��

with and being (1 2) matrices, and andbeing (2 2) matrices. The expression of these matrices

is given in Appendix A.The vector has elements:

...(58)

The variables are defined in Appendix B.

Fig. 5. Representation of the magnetic field calculated analytically consideringeddy currents in the outer yoke. The different layers from the center to the ex-terior are: the rotor (iron yoke, 2-pole-pair ideal Halbach PM), the air gap, thestator (iron yoke), air.

D. Solution

Equation (57) is inverted to obtain the formula for eachconstant:

(59)

V. FIELD REPRESENTATION

A. Magnetic Flux Density

The magnetic flux density is given by

(60)

which gives in polar coordinates

(61)

Now can be represented as a function of the different har-monics of the magnetic vector potential:

(62)

B. Some Illustrations

The magnetic field in different configurations is represented,as an illustration of the power of the model. The following rep-resentations are the result of the fully analytical model.

• Fig. 5 represents a five-layer model of the following struc-ture, from the center to the exterior: the rotor (iron yoke,2-pole-pair ideal Halbach PM), the air gap, the stator (ironyoke), air. The innermost boundary: “center,” and the out-ermost boundary: “infinite radius.” The figure shows thedeformation of the magnetic flux density field lines due tothe eddy currents in the stator.

• Fig. 6 represents a two-layer model. The inner layer is anideal Halbach 3-pole-pair PM, and the exterior layer is air.


Fig. 6. Representation of the magnetic field calculated analytically due to athree-pole-pair ideal Halbach PM. The innermost and outermost boundarieshave the condition of “infinite permeability.” The two layers from the centerto the exterior are: the PM and air.

Fig. 7. Representation of the magnetic field calculated analytically due to aone-pole-pair PM. The PM is magnetized with a first and second harmonic. Itmakes it asymmetric. The five different layers from the center to the exteriorare: a rotor yoke, a PM, air, a stator yoke, and air.

The innermost and outermost boundaries have the condi-tion of “infinite permeability.”

• Fig. 7 represents the possibilities of considering differentmagnetization harmonics.

• Fig. 8 represents the difference between an ideal HalbachPM and a radially magnetized PM. The radially magne-tized PM is created taking into consideration the harmonics1,3,5,7 of the radial remanent field, and the tangential re-manent field harmonics are equal to zero.

VI. MODEL VALIDATION

A. Very High Speed Permanent-Magnet Machine

As it is not possible to validate the model in its generalityusing finite-element methods, we present here different caseswhich show the model’s validity. The first illustration is avery high speed PM machine whose specifications are givenin Table II. A section of the machine is shown in Fig. 10.This machine is slotless and reaches more than 200 000 rpmand more than 2 kW of output power [35]. As the machine is

Fig. 8. Representation of the magnetic field calculated analytically due to atwo-pole-pair ideal Halbach PM (left), and due to a radially magnetized PM(right).

TABLE IIPROTOTYPE SPECIFICATIONS

Fig. 9. Geometry of a coil of the VHS PM machine.

slotless, the coils shown in Fig. 9 are in the air gap between thestator yoke and the rotor.

The field calculated using the analytical model (10 spaceharmonics) and using finite-element methods is represented inFig. 10. In each figure, a current density of 10.4 10 A/m isapplied to the coil which is on the upper right side. No currentis applied to the two other coils. The value of permeability andremanence are the ones given in Table III. The permeability ofthe outer yoke is assumed to be . Fig. 10 showsthat the agreement is excellent.

Another good physical property that can be calculated tocompare the model to finite-element methods, is the total fluxpassing through one coil. The same hypotheses are consideredas the one used for the calculation of Fig. 10 except that noapplied current is inside the coils. The total flux passing through


Fig. 10. Comparison of the magnetic field given by the analytical model (top)and the finite-element methods (bottom) during the calculation of the torquecreated by one phase. The current in set in coil 1. In a+, the current densityis 10.4 A/mm and in a- the current density is �� A/mm . In b and c thecurrent density is equal to 0.

TABLE IIIPROTOTYPE MATERIALS

a coil is calculated as a function of the angle. The dots in Fig. 11represent the finite-element method calculations, the contin-uous line represents the analytical model. As for Fig. 10, theagreement is excellent. The comparison between the analyticalmodel and the finite-element method gives a difference of lessthan %.

B. Eddy-Current Brake

The dynamic electromagnetic model is validated using a2-pole-pair eddy-current brake structure. The structure made ofconcentric cylinders is the following, starting from the center.

Fig. 11. Total flux passing through one coil as a function of the angular positionof the rotor. The dots represent calculation using the finite-element methods, thecontinuous line represents the calculation using the analytical model.

Fig. 12. Magnetic field in the eddy-current brake obtained by finite-elementmethods.

1) A 2-pole-pair ideal Halbach type PM in the center: the re-manent field is T and the relative permeabilityis 1.03. The outer radius is 5 mm. The PM is rotating at aspeed of rad/s.

2) A layer of air.3) A yoke: the inner radius is 5.5 mm, the outer radius is

7 mm, its relative permeability is 2000, is conductivity2.4 10 S.

4) A layer of air.The magnetic fields calculated using the analytical model is

represented in Fig. 13 and the one obtained using the finite-elements methods is represented in Fig. 12. Fig. 14 representsthe radial magnetic field in the conductive yoke at different radii.We see a good agreement between the model and the finite-element methods.

VII. CONCLUSION

Starting with Maxwell’s equations, the formalism developedin this paper allows the obtention of the analytical expressionof the vector potential and the magnetic field at any point of a-layer cylindrical system, whereas in the literature only special

cases of this model have been found.The presented model is successful in the following aspects:

each layer can rotate, be conductive, have a remanent magnetic


Fig. 13. Magnetic field in the eddy-current brake calculated using the analyticalmodel.

Fig. 14. Comparison between the radial magnetic field obtained by finite-ele-ment methods or calculated analytically in the conductive part of the eddy-cur-rent brake.

field and be subject to an applied current density, each boundarycan be characterized by the presence of a current surface density.The calculation of the fields in multipolar conductive magnetsis also a contribution of this paper.

The study of different designs shows a good agreement be-tween the general analytical model and finite-element methods.

APPENDIX A

A. The Expression of

can be expressed as

��

��

��

� �

�

��

� �

� �

� �

� �

� �

� ��

�

��

� �

� �

� �

� �

......

. . ....

......

......

.... . .

......

......

� �

� �

� �

� �

� �

� �

�

��

� � � � � � � � � � � � ��

with and being (1 2) matrices, and andbeing (2 2) matrices.

The first column of :

(63)The second column of :

(64)The first column of :

(65)

The second column of :

(66)

and depend on the side boundary conditions, theyare defined using considerations in Section IV-B.

is considered first. If the interior side boundary is a“center”:

(67)

If the interior side boundary is “infinite permeability”, the firstcolumn of is

(68)


The second column of :

(69)

is now taken into consideration. By hypothesis, inthe case of an exterior side boundary which is an “infinite ra-dius” only a nonconductive material is taken into account:

If the exterior side boundary is “infinite permeability,” the firstcolumn of is

(70)The second column of :

(71)

B. The expression of

The vector has elements:

...(72)

depends on the innermost boundary condition:

(73)

and depends on the outermost boundary condition:

(74)For :

(75)

For :

and

C. Gamma Function

The function [36] is defined by

(76)

This implies that for any :

(77)

D. Pochhammer Function

The general definition of the Pochhammer function [36] is

(78)

It can be simplified when is a positive integer:

(79)

E. Bessel Function of the First Kind

The Bessel function of the first kind [36], , satisfies thefollowing differential equation:

(80)

It is defined as

(81)


F. Bessel Function of the Second Kind

The Bessel function of the second kind [36], , also sat-isfies the following differential equation:

(82)

If , it is defined as

(83)

If , it is defined as

(84)

G. Struve Function

The Struve function [36] is defined as

(85)

H. Generalized Hypergeometric Function

For positive Generalized hypergeometric function [37],, is defined as

(86)

where is the Pochhammer function.

I. Generalized Meijer G Function

The Generalized Meijer G function can be defined in termsof the Fox H function:

��

��

� ��

��

��

��

��

�

��

��

�

��

��

��

(87)

with and . The infinite contourof integration separates the poles of at

from the poles of at. Such a contour always exists in the cases

.Any good mathematical software can calculate directly such

a function. Many books and papers give more detail about thisfunction [37].

ACKNOWLEDGMENT

The authors want to thank Moving Magnet Technologies SAand Sonceboz SA for their support for the research accomplishedon very high-speed machines [26]. The present paper is resultingfrom this research.

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Pierre-Daniel Pfister was born in Bienne, Switzerland, in 1980. He received theM.Sc. degree in physics in 2005 and the Ph.D. degree in 2010 from the SwissFederal Institute of Technology-Lausanne (EPFL). He studied for one year atthe University of Waterloo, Canada.

After receiving the Ph.D. degree, he continued to serve as a developmentengineer for Sonceboz SA (Switzerland)/Moving Magnet Technologies SA(France). His research interests are in the field of permanent magnet machines,very high speed machines, and analytical optimization.

Yves Perriard was born in Lausanne, Switzerland, in 1965. He received theM.Sc. degree in microengineering from the Swiss Federal Institute of Tech-nology-Lausanne (EPFL) in 1989 and the Ph.D. degree in 1992.

Co-founder of Micro-Beam SA, Yverdon, Switzerland, he was CEO of thiscompany involved in high precision electric drives. He was a Senior Lecturerfrom 1998 and has been a Professor since 2003. He is currently director of theIntegrated Actuator Laboratory and vice-director of the Microengineering In-stitute at EPFL. His research interests are in the field of new actuator designand associated electronic devices. He is author and co-author of more than 80publications and patents.

slotless permanent-magnet machines: general analytical ... · pfister and perriard: slotless...

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