SLOTLESS permanent-magnet (PM) machines are increasingly

IEEE TRANSACTIONS ON MAGNETICS, VOL 47, NO 6, JUNE 2011 1739 Slotless Permanent-Magnet Machines: General Analytical Magnetic Field Calculation Pierre-Daniel Pfister 1;2 and Yves Perriard 2 Moving Magnet Technologies SA (MMT), 25000 Besançon, France Laboratory 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 model takes into account the effect of eddy currents in conductive regions and notably in conductive permanent magnets without neglecting their remanent field The modeling of this effect is important for the design of very high speed slotless permanent-magnet machines, as the 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 applicable both to internal rotor and external rotor permanent-magnet machines The effect of the relative permeability and of the conductivity of the permanent magnet or of the yoke on the magnetic fields is also taken into account Any magnetization can be taken into account, in particular 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 SLOTLESS permanent-magnet (PM) machines are increasingly used for very high speed (VHS) applications The analytical modeling of the magnetic field is important for the design and optimization of such applications Also, analytical models of the fields calculated for slotless structures are widely used in the design of slotted structures using conformal mapping or Carter coefficients Many papers (see Section I-C) have already been written about the analytical calculation of magnetic fields in several particular cases The objective of the present paper is to show a general model (see Table I), applicable to a very large family of slotless machines A Considered Structure The structure considered in this paper is the following It is made of -concentric contiguous hollow cylinders Every cylinder 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 applied surface current density harmonics Each hollow cylinder is called a layer The model is called an -layer model Every layer in which the magnetic field is calculated, and every boundary is numbered starting with the innermost one as shown in Fig 1 In the presented model any spatial magnetization harmonics can be taken into consideration Allowing models of different kinds of magnetization and in particular the ideal Halbach magnetization as it is defined in [1] and [2] to be created Radial magnetization can be approximated by this model Manuscript received June 25, 2010; revised September 15, 2010; accepted December 06, 2010 Date of publication February 17, 2011; date of current version May 25, 2011 Corresponding author: P-D Pfister (e-mail: pierredaniel pfisterpublic@gmailcom) Digital Object Identifier 101109/TMAG20112113396 Fig 1 Section of the contiguous hollow cylinders: the example of a 3-layer structure, with infinite permeability in the center and in the exterior The structure 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 other kind of electromechanical structures B Which Differential Equation? As it is shown in Section II, starting from Maxwell s equations and in the constitutive equations of the materials, a diffusion equation can be derived in its generalized form: it includes the effect of magnetic remanent fields, eddy currents, and applied currents In this paper, it is called: a generalized diffusion equation (GDE) Depending on the assumptions which govern the physics of the structure, the governing equation can take different forms that are named differently (Fig 2) These different forms can be deduced from the GDE Different authors have used these different forms of the GDE: Laplace equation [3], Poisson equation [4], the diffusion equation [5], and the generalized form of the diffusion equation [6] C Literature Review The resolution of a differential equation in order to obtain the magnetic field in an electrical machine is not recent With the exception of some papers where the 3-D analytic magnetic field solution is presented [7], [8], most publications involve 0018-9464/$2600 2011 IEEE

1740 IEEE TRANSACTIONS ON MAGNETICS, VOL 47, NO 6, JUNE 2011 Fig 2 The differential equation used for the calculation of the magnetic vector potential 2-D solutions Here are some important contributions to this theory 1) Already in 1929, B Hague wrote a book about 2-D solutions of Poisson s equation [9] He calculated the fields due to currents in a cylindrical geometry 2) In 1977, A Hughes and T J E Miller [10] presented a model of the field created by a conducting sheet in a 5-layer structure 3) Based on the work of B Hague, N Boules wrote a paper in 1985 [11] entitled Prediction of no-load flux density distribution in permanent magnet machines, where the PM s magnetization is replaced by equivalent currents 4) In 1993, Z Q Zhu and D Howe wrote an excellent series of four papers [12] [15] on the calculation of the magnetic field in electrical machines based again on the magnetic scalar potential The fields due to the PM are directly calculated using the PM magnetization 5) In 1995, Z J Liu et al used the magnetic vector potential to calculate the fields and the eddy currents in the stator yoke [16] They divided the space into three concentric layers and expressed the analytical field solution in the layers 6) In 1997 and 1998, after discussion with N Boules, F Deng wrote two papers [17], [18] about eddy-current power losses due to the commutation of the PM machine She was able to solve the differential equation not only in different layers, but also in different sectors in some layers In these calculations, the remanent field of the PMs is neglected The applied current in the coils is approximated by a surface current density between two layers This model enables the calculation of the eddy-current power losses in the motor teeth due to the pulsewidth modulation [19] Several improvements and contributions have been made on this topic in the last 10 years Amongst these, the following publications can be cited 1) In 2003, S R Holm calculated the magnetic fields in a cylindrical structure He included in his calculations the fields due to an applied current density in a layer [4] 2) In 2007, Shah et al solved a 6-layer structure in cartesian coordinates [20], based on a previous work [21] 3) In 2007, A Chebak determined the solution of the differential equation for a 4-layer structure, -pole-pair PM, with magnetization harmonics and calculated the eddy currents in the stator [22] 4) In 2008, M Markovic calculated the eddy-current power losses in a one-pole-pair PM [5], considering its magnetization This timeline is not exhaustive, but shows some important milestones on the path to the resolution of the differential equations 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 remarks about these two tables are as follows A Yes in the Eddy currents row means that the eddy currents, if they are calculated, are a direct solution of the differential equation If the geometry is more complex than a concentric contiguous hollow cylinder, it may be better to make simplifying assumptions than to directly solve the general differential 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 magnetic field is calculated in this region Since the layer is defined as a region in which the magnetic fields are calculated, in this case the center is not considered as a layer, as in Fig 1 means that the radius of the boundary 1 tends to 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 to infinity For more information about the Innermost boundary and the Outermost boundary, see Section IV-B If a (1) stands in the Magnetic potential row, it means that instead of solving a differential equation involving a magnetic scalar or vector potential, the authors solved a differential 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 are neglected in the calculation of the eddy currents In these cases the eddy currents are due to the excitation current The interaction of the PM s magnetization with the eddy currents is also neglected As is shown in Section III, the form of the diffusion equation in a PM where eddy currents are considered is different and implies a much simpler solution when the PM is parallelly magnetized than when higher harmonics of remanent field are considered If a (3) stands in the Eddy currents in the PM(s) row, it means that the calculation is valid only for a 1-pole-pair parallelly magnetized PM There are some limitations to the different resolutions of the differential equation in cylindrical coordinates presented in Table 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 one case examined in the literature which is a one-pole-pair central 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 has the 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 Halbach magnetization

PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES 1741 TABLE I COMPARISON OF THE SOLUTIONS OF THE DIFFERENTIAL EQUATIONS THE REMARKS ARE IN THE TEXT 4) The eddy currents in the PM are determined, with any number of pole pairs, magnetization harmonics, applied current density harmonics and applied surface current density harmonics 5) The model can handle for the innermost boundary: zero radius or nonzero radius with infinite permeability inside the boundary 6) The model can handle for the outermost boundary: infinite radius or finite radius with a material of infinite permeability outside it 7) The model can handle at the same time applied surface current densities and applied current densities E Outline of the Paper Maxwell s equations and two equations which describe materials are used to obtain the GDE The GDE is considered in the Fourier space and solutions are found Boundary conditions are described and finally a general solution is found for a multilayer system 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 to be 2-D

1742 IEEE TRANSACTIONS ON MAGNETICS, VOL 47, NO 6, JUNE 2011 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 compared with the physical dimensions of the device II FROM THE MAXWELL S EQUATIONS TO THE GENERALIZED DIFFUSION EQUATION be an arbitrary scalar func- is not uniquely defined Let tion, and and be defined as It implies that (10) (11) (12) 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, is the electric field strength, is the magnetic flux density, is the magnetic field strength, and is the current density B The Constitutive Materials Equations The constitutive equation of a PM is where is the remanent field and is the permeability Since the permeability is isotropic, is scalar Since the materials are assumed to be linear without saturation, is constant In soft ferromagnetic materials, the same equation is used, but with The current density in a moving conductor with relative velocity is generated by the Lorentz force and is given by [29]: is the conductivity which is assumed to be constant C Vector Potential As the divergence of is equal to zero, a magnetic vector potential such that is taken into consideration Equations (2) and (7) are combined to obtain which gives after integration (5) (6) (7) (8) (9) (13) Therefore, the potentials defined by (10) and (11) give the same field To define uniquely, the Coulomb gauge [30] has been chosen: (14) D Generalized Diffusion Equation Since the permeability is assumed to be isotropic and constant, (4), (5), (6), and (7) are combined to obtain Using the Coulomb gauge and the identity (15) is rewritten as (15) (16) (17) Equation (7) is the most general form of the GDE presented in this paper It can be simplified, depending on material s properties and other hypotheses: It can be reduced in PMs with no applied currents, where the eddy currents are considered A GDE type equation is obtained: (18) It can be reduced in a conductive media, where no current is applied where the eddy currents are considered A diffusion 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 currents are not considered, but where current is applied A Poisson type equation is obtained: where is a electric scalar potential (21)

PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES 1743 where is the applied current density For (21), the applied current density is deduced from the scalar potential using (6): with (30) (22) By hypothesis, the complete system is 2-D The cylindrical coordinate system is used The magnetic vector potential and the current density are hence along and are in the plane 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 be found Since the structure is made of cylinders, there is a spatial periodicity of any quantity, but in particular of the magnetic vector potential Moreover, as the transient states are not considered, there is a time periodicity of period which corresponds to the period of the applied current The angular frequency is defined by For any quantity applies: (24), the following periodicity condition (25) (26) The easiest way to find solutions is to consider the different variables in the Fourier space A Complex Fourier Series The complex Fourier series of any quantity where is the real part of The GDE (23) is expressed for the is defined as th harmonic: (27) The solution of (29) depends on the material properties and on many parameters Two cases need to be considered 1) The case where eddy currents can possibly occur, if the layer of the system is defined by the right harmonics, rotation speed and material conductivity This case is called ECPO 2) The case where the hypothesis is made that no eddy currents occur or where they are neglected This case is called NECO Some examples of both cases follow 1) The ECPO case: A PM layer with a given conductivity rotating in a field created by different applied current harmonics A cylinder of copper rotating in a synchronous field No eddy currents occur, nevertheless, if the harmonic content 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 solved by the present model For laminated iron, the simplest approach is to solve the equation without eddy currents first, and then use an a posteriori model of the power losses due to the eddy currents as a function of the field A coil layer The insulation between the wires implies that the NECO case is a better approximation than the ECPO case B The Generalized Diffusion Equation in the Eddy Currents Can Possibly Occur Case Good examples of conductive media would be: titanium or copper cylinders, a conductive PM, and iron In the ECPO case, the following hypotheses are made 1) and are constant This assumption is always true for any ideal Halbach PM This assumption is a good 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 standstill part: In that case (29) becomes which can be simplified as (28) (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 (29) (32)

1744 IEEE TRANSACTIONS ON MAGNETICS, VOL 47, NO 6, JUNE 2011 with the particular solution of the differential equation: (33) It is important to notice that in a material which has no remanent field Also if the remanent field is parallel 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 (see Appendix E) and is the Bessel function of the second kind (see Appendix F) The particular solution is the following for even : Fig 3 Diagram of an electromagnetic s-layer structure Every layer is characterized by its constitutive material properties, by its rotational speed and by the external current applied This structure is general, it can represent, for example, eddy-current brakes and PM slotless motors In this representation, the layer i contains a coil defined by the angles and C The Generalized Diffusion Equation Solutions in the No Eddy Currents Occur Case With respect to the solution of the GDE in the domain of electrical machines, magnets with low conductivity, air, litz wire, can be related to or approximated by the NECO case In a nonconductive 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 be neglected: (37) (35) where is the Struve function (see Appendix G), is a generalized hypergeometric function (see Appendix H), and In the resolution of (37), different cases are separated depending on is the generalized Meijer G function (see Appendix I) With odd, no general formula for was found Here is the formula for : with (38) (39) In the case of a balanced three-phase machine, the harmonics are given by (36) From (31), it can be simply deduced that when the magnetization of the part is parallel or when there is no remanent magnetization: (40) where and are the boundary angle of the coil, is the time harmonics of the current density The two angles and are shown in Fig 3

PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES 1745 Using, the following expression is obtained: (44) Fig 4 Boundary conditions which gives IV THE -LAYER PROBLEM The vector potential obtained by solving the GDE is determined in the previous sections The constants and remain to be determined These two constants are defined by the boundary conditions The considered structure is an -layer concentric structure as shown in Fig 3 The two boundary conditions for each interface are deduced in Section IV-A As there are layers, there are interfaces and hence conditions In Section IV-B, the innermost and outermost boundaries give two more conditions The total number of conditions is The conditions give the equations that are needed to define the constants The interior boundary of the th layer is called Its permeability is called is the coefficient associated with the th layer for the harmonic and represents the coefficient of the th layer for represents 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 fact that the harmonic creates eddy currents in the layer A Two Kinds of Boundary Conditions The boundary defined by between layer and layer is taken into consideration implies that By the definition of the magnetic vector potential, in the 2-D case it follows: which can be expressed as a Fourier series: This is the first boundary condition implies that (41) (42) (43) where is by definition the surface current density at the boundary between layer and layer Fig 4 shows the different vectors at the boundary (45) in the 2-D approximation The second boundary condition for the vector potential is obtained: The last expression can be expressed as a Fourier series: (46) (47) B Three Kinds of Innermost and Outermost Boundary Conditions The innermost and outermost boundary conditions are at the inner side of the innermost layer and at the outer side of the outermost layer of the -layer structure The boundary conditions are 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 if is true, (34) gives that and if is false, (32) and (38) imply that (48) (49) (50) (51) 2) Infinite Permeability: If inside from the innermost boundary the permeability is considered to be infinite:, (47) gives (52) If the exterior of the outermost boundary is considered to be infinite, (47) gives (53)

1746 IEEE TRANSACTIONS ON MAGNETICS, VOL 47, NO 6, JUNE 2011 3) Infinite Radius: In this case, for the layer, only a nonconductive material is considered The assumption is made that there is no flux in This implies that the magnetic vector potential needs to be zero in Equation (38) gives (54) (55) C Matrix The magnetic vector potential given by a single wire is not taken into consideration In the problem, for any current density which flows in one direction, there is always a current density which flows in the other direction Therefore, the amplitude of the vector potential s harmonic is assumed to be equal to zero The coefficients, with and need to be calculated to obtain the magnetic vector potential In order to calculate them, the following vector is defined: Fig 5 Representation of the magnetic field calculated analytically considering eddy currents in the outer yoke The different layers from the center to the exterior are: the rotor (iron yoke, 2-pole-pair ideal Halbach PM), the air gap, the stator (iron yoke), air D Solution Equation (57) is inverted to obtain the formula for each constant: (59) (56) V FIELD REPRESENTATION A Magnetic Flux Density The magnetic flux density is given by Using the boundary conditions of Sections IV-A and IV-B, the following system is found: X mn = can be expressed as B 1mn 0 0 0 0 0 111 0 0 0 0 L 2mn R 2mn 111 0 0 0 0 0 0 0 L 3mn R 3mn 111 0 0 0 0 0 0 0 111 L smn R smn 111 0 0 B (s+1)mn (57) with and being (1 2) matrices, and and being (2 2) matrices The expression of these matrices is given in Appendix A The vector has elements: The variables are defined in Appendix B (58) which gives in polar coordinates (60) (61) Now can be represented as a function of the different harmonics 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 representations are the result of the fully analytical model Fig 5 represents a five-layer model of the following structure, from the center to the exterior: the rotor (iron yoke, 2-pole-pair ideal Halbach PM), the air gap, the stator (iron yoke), air The innermost boundary: center, and the outermost boundary: infinite radius The figure shows the deformation of the magnetic flux density field lines due to the eddy currents in the stator Fig 6 represents a two-layer model The inner layer is an ideal Halbach 3-pole-pair PM, and the exterior layer is air

PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES 1747 Fig 6 Representation of the magnetic field calculated analytically due to a three-pole-pair ideal Halbach PM The innermost and outermost boundaries have the condition of infinite permeability The two layers from the center to the exterior are: the PM and air Fig 8 Representation of the magnetic field calculated analytically due to a two-pole-pair ideal Halbach PM (left), and due to a radially magnetized PM (right) TABLE II PROTOTYPE SPECIFICATIONS Fig 7 Representation of the magnetic field calculated analytically due to a one-pole-pair PM The PM is magnetized with a first and second harmonic It makes it asymmetric The five different layers from the center to the exterior are: a rotor yoke, a PM, air, a stator yoke, and air The innermost and outermost boundaries have the condition of infinite permeability Fig 7 represents the possibilities of considering different magnetization harmonics Fig 8 represents the difference between an ideal Halbach PM and a radially magnetized PM The radially magnetized PM is created taking into consideration the harmonics 1,3,5,7 of the radial remanent field, and the tangential remanent 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 generality using finite-element methods, we present here different cases which show the model s validity The first illustration is a very high speed PM machine whose specifications are given in Table II A section of the machine is shown in Fig 10 This machine is slotless and reaches more than 200 000 rpm and more than 2 kw of output power [35] As the machine is Fig 9 Geometry of a coil of the VHS PM machine slotless, the coils shown in Fig 9 are in the air gap between the stator yoke and the rotor The field calculated using the analytical model (10 space harmonics) and using finite-element methods is represented in Fig 10 In each figure, a current density of 104 10 A/m is applied to the coil which is on the upper right side No current is applied to the two other coils The value of permeability and remanence are the ones given in Table III The permeability of the outer yoke is assumed to be Fig 10 shows that the agreement is excellent Another good physical property that can be calculated to compare the model to finite-element methods, is the total flux passing through one coil The same hypotheses are considered as the one used for the calculation of Fig 10 except that no applied current is inside the coils The total flux passing through

1748 IEEE TRANSACTIONS ON MAGNETICS, VOL 47, NO 6, JUNE 2011 Fig 11 Total flux passing through one coil as a function of the angular position of the rotor The dots represent calculation using the finite-element methods, the continuous line represents the calculation using the analytical model Fig 10 Comparison of the magnetic field given by the analytical model (top) and the finite-element methods (bottom) during the calculation of the torque created by one phase The current in set in coil 1 In a+, the current density is 104 A/mm and in a- the current density is 010:4 A/mm In b and c the current density is equal to 0 TABLE III PROTOTYPE MATERIALS a coil is calculated as a function of the angle The dots in Fig 11 represent the finite-element method calculations, the continuous line represents the analytical model As for Fig 10, the agreement is excellent The comparison between the analytical model and the finite-element method gives a difference of less than % B Eddy-Current Brake The dynamic electromagnetic model is validated using a 2-pole-pair eddy-current brake structure The structure made of concentric cylinders is the following, starting from the center Fig 12 Magnetic field in the eddy-current brake obtained by finite-element methods 1) A 2-pole-pair ideal Halbach type PM in the center: the remanent field is T and the relative permeability is 103 The outer radius is 5 mm The PM is rotating at a speed of rad/s 2) A layer of air 3) A yoke: the inner radius is 55 mm, the outer radius is 7 mm, its relative permeability is 2000, is conductivity 24 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 finiteelements methods is represented in Fig 12 Fig 14 represents the radial magnetic field in the conductive yoke at different radii We see a good agreement between the model and the finiteelement methods VII CONCLUSION Starting with Maxwell s equations, the formalism developed in this paper allows the obtention of the analytical expression of 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

PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES 1749 with and being (1 2) matrices, and and being (2 2) matrices The first column of : The second column of : (63) Fig 13 model Magnetic field in the eddy-current brake calculated using the analytical The first column of : (64) (65) Fig 14 Comparison between the radial magnetic field obtained by finite-element methods or calculated analytically in the conductive part of the eddy-current brake The second column of : field and be subject to an applied current density, each boundary can be characterized by the presence of a current surface density The calculation of the fields in multipolar conductive magnets is also a contribution of this paper The study of different designs shows a good agreement between the general analytical model and finite-element methods A The Expression of can be expressed as X mn = APPENDIX A B 1mn 0 0 0 0 0 111 0 0 0 0 L 2mn R 2mn 111 0 0 0 0 0 0 0 L 3mn R 3mn 111 0 0 0 0 0 0 0 111 L smn R smn 111 0 0 B (s+1)mn (66) and depend on the side boundary conditions, they are 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 first column of is (68)

1750 IEEE TRANSACTIONS ON MAGNETICS, VOL 47, NO 6, JUNE 2011 The second column of : For : (69) is now taken into consideration By hypothesis, in the case of an exterior side boundary which is an infinite radius only a nonconductive material is taken into account: If the exterior side boundary is infinite permeability, the first column of is and The second column of : (70) C Gamma Function The function [36] is defined by (76) (71) This implies that for any : B The expression of The vector has elements: (77) (72) D Pochhammer Function The general definition of the Pochhammer function [36] is depends on the innermost boundary condition: (78) It can be simplified when is a positive integer: (73) (79) and depends on the outermost boundary condition: E Bessel Function of the First Kind The Bessel function of the first kind [36], following differential equation:, satisfies the (80) For : (74) (75) It is defined as (81)

PFISTER AND PERRIARD: SLOTLESS PERMANENT-MAGNET MACHINES 1751, also sat- F Bessel Function of the Second Kind The Bessel function of the second kind [36], isfies the following differential equation: If If, it is defined as, it is defined as G Struve Function The Struve function [36] is defined as (82) (83) (84) (85) H Generalized Hypergeometric Function For positive Generalized hypergeometric function [37],, is defined as where is the Pochhammer function (86) I Generalized Meijer G Function The Generalized Meijer G function can be defined in terms of the Fox H function: G j;k u;v z; r a 1; ;a k ;a k+1; ;a u b1; ;b j;b j+1; ;b v = rh j;k u;v z (a1;r); ; (a k;r); (a k+1;r); ; (a u;r) (b1;r); ; (b j ;r); (b j+1;r); ; (b v ;r) = r 2{ L u k j d=1 0(1 0 a d 0 rs) d=1 0(b d + rs) v d=j+1 0(1 0 b d 0 rs) z0s ds d=k+1 0(a d + rs) (87) with and The infinite contour of 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 this function [37] ACKNOWLEDGMENT The authors want to thank Moving Magnet Technologies SA and Sonceboz SA for their support for the research accomplished on very high-speed machines [26] The present paper is resulting from this research REFERENCES [1] K Atallah, D Howe, and P H Mellor, Design and analysis of multi-pole Halbach (self-shielding) cylinder brushless permanent magnet machines, in Eighth Int Conf Electrical Machines and Drives, Cambridge, UK, Sep 1997, pp 376 380 [2] Z Q Zhu and D Howe, Halbach permanent magnet machines and applications: a review, in IEE Proc-Elect Power Appl, Jul 2001, vol 148, pp 299 308 [3] Z Q Zhu, K Ng, N Schofield, and D Howe, Improved analytical modelling of rotor eddy current loss in brushless machines equipped with surface-mounted permanent magnets, IEE Proc Elect Power Appl, vol 151, no 6, pp 641 650, Nov 2004 [4] S R Holm, Modelling and Optimization of a 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