3-D Analytical Model of Axial-Flux Permanent Magnet Machine With Segmented Multipole-Halbach Array

This paper presents a 3-D analytical model of an axial-flux permanent magnet (AFPM) machine with a segmented multipole-Halbach PM array. Closed-form solutions are self-consistently derived in terms of modified Bessel functions of the first- and the second-kind by solving analytically Laplace and Poisson equations by the method of magnetic scalar potential subject to the appropriate boundary conditions. In the preceding studies, their formulations are based on a 2-D or quasi 3-D geometry, and their discussions are often limited to the magnetic fields with low-poles of the regular PM. The proposed model successfully provides more rigorous and widely applicable expressions for magnetic fields, back-electromotive force, Lorentz torque and torque constant without limitations on the number of poles and the arrangements of the PM. Behavior of the torque constant is then shown against the number of poles ranging widely from low-poles to high-poles of the regular PM, the standard-Halbach PM and the multipole-Halbach PM for changeable geometrical parameters. The obtained results are of much use in understanding intrinsically the performance characteristics of the AFPM.


I. INTRODUCTION
The axial-flux permanent magnet (AFPM) machine has potentially attractive features of high torque density, high efficiency and high magnetic poles due to its flat geometry in comparison to the ordinal radial flux permanent magnet (RFPM) machine. The AFPM machine has thus recently attracted great attention as a powerful candidate for a next generation driving source in many applications, e.g. robotics [1], [2], [3], [4], electric vehicles [5], [6], [7], [8] and power generation [9], [10], [11], [12]. In particular, industrial robots involving actuators have been researched and developed in the world by academic institutes [13], [14] and leading companies [15], [16], [17]. Thanks to them, man-powered heavy works have increasingly been replaced and automated by the robots. In multi-jointed robots, the The associate editor coordinating the review of this manuscript and approving it for publication was Xiaodong Liang . performance required for servomotors ranges widely from high-revolution to high-torque, depending on the role of each joint. Specifically, a quick response is demanded for the joint close to the point of action, where low-pole and high-revolution RFPM would be suitable. On the other hand, a large load is inevitably applied on the joint corresponding to the point of effort, where multipole and high-torque AFPM would be effective. In either case, the basic performance of the PM machines is almost characterized by the torque constant. It is therefore very important to understand characteristics of the torque constant of the AFPM as well as the RFPM over a global range of the various designing parameters.
In order to generate high torque in electric machines, a Halbach PM array has so far been suggested and studied in various areas by many researchers [18], [19], [20], [21], [22], [23], [24]. The Halbach array is superior to ordinal one in several ways; First, the Halbach PM produces an about twice stronger magnetic flux density than ordinal one. Second, the magnetic field distribution is approximately sinusoidal. Third, the leakage flux from back side is almost negligible. These advantages are expected to be enhanced for a multipole-Halbach array with higher poles by the interaction of magnetic fields in the adjacent magnets. However, characteristics of the AFPM with a multipole-Halbach PM has not yet sufficiently been studied in comparison with those of the RFPM, and most of previous studies have been confined to the low number of poles because of the facility in modeling, designing and manufacturing.
In designing high performance machines, it is of intrinsic importance to study magnetic field structures in the air-gap and their Fourier spectra, since the fundamental amplitude of the fields produces the net Lorentz force by interacting synchronously with the alternating electric current of the coil, while the harmonics results in the ineffective force such as cogging torque and/or torque ripple. Most preceding works utilize commercially available simulation programs, in which electromagnetic (EM) fields are numerically solved by, e.g. the finite element method (FEM) [25], [26], [27]. However, such a numerical method requires enormous computation time for a 3-D structure of the AFPM, and makes the principle governing underlying physics unclear. Furthermore, the FEM analysis forces us to remodel every time the geometrical parameters, e.g. the number of poles and/or the arrangements of the PM change, and the computation accuracy becomes unstable for higher poles. For these reasons, the FEM is not suitable for a global parametrical study on the 3-D AFPM model. This work therefore adopts an analytical approach, which makes it possible to investigate EM problems not only quickly, but also flexibly, and then to provide deep physical insight through mathematical formulations.
Pioneering analytical studies on the AFPM can be found in the literature [28], [29], in which magnetic fields in the air-gap have been investigated by integrating the free-space Green function. Magnetic fields of the disk-type PM have also been analyzed by taking the Hankel transformation [30]. However, these preceding papers are limited to the study only on the magnetic field structure itself of the regular PM array with low-poles. 2-D analytical studies of the AFPM have been reported in [31] and [32], which provide the relevant solutions formulated in terms of a vector potential by introducing an artificial correction function in the radial direction. Such a 2-D approximate model would be valid only when the inner radius is close to the outer one. 3-D analytical models of the ironless AFPM machine have been suggested in [33], [34], in which magnetic fields are studied only for the regular PM array with low-poles, and torque characteristics is not reported. In a similar paper [35], magnetic fields and the relevant torque have also been analyzed on the basis of a 3-D AFPM model with a standard-Halbach PM, but high-order space harmonics are neglected in their torque formula, and the results are limited to a single case of low-poles. A complete 3-D model of the AFPM is therefore highly required in order to investigate systematically magnetic fields, back electromotive force(EMF) and torque characteristics without limitation on the regular PM with low-poles.
From a theoretical point of view, in the existing papers, e.g. [33] and [34] basic solutions have been given in the single form of the first-kind Bessel function under the open-circuit condition in order to avoid complicated formulations. Their formulae actually include only one unknown quantity to be determined by the boundary condition. This work is different from the preceding ones in that general solutions have more rigorously been derived and expressed in the combination of the first-and the second-kind modified Bessel functions by solving straightforwardly Laplace and Poisson equations in the air-gap and the magnet regions, respectively. Our formulae thus involve six unknowns to be determined by the appropriate boundary conditions. The novelty of this work lies in the fact that the proposed model successfully provides more rigorous and widely applicable mathematical expressions for magnetic fields, back-EMF, Lorentz torque and torque constant without limitations on the number of poles and the arrangements of the PM. To the best of the authors' knowledge, such a rigorous formula for the torque constant based on the 3-D multipole-Halbach AFPM has not been found in the preceding papers. The objective of this paper is to derive analytically an expression for the torque constant, and to show its dependence on the number of poles ranging widely from low-poles to highpoles for geometrical parameters of the multipole-Halbach AFPM. In addition, the proposed model enables us to study rapidly EM characteristics of the AFPM, incorporating the effect of high-order space harmonics without sacrificing a degree of accuracy, even if the number of poles is as high as ∼ 100.
This paper is organized as follows. In Section II, geometries of the proposed model are presented. In Section III, basic equations are self-consistently formulated, and then the rigorous expressions for magnetic fields, back-EMF, Lorentz torque and torque constant are successfully derived. Section IV focuses on the validity, results and discussion. Finally, Section V summarizes this work. Fig.1a) presents the analytical 3-D model of the AFPM machine developed in this work, where R o and R s are inner and outer radii of the stator core, R r and R m are inner and outer radii of the rotor magnet, respectively, g is a mechanical airgap, z m is a thickness of the magnet in the axial direction, is an angle spread by a segment arc, θ c is an angle spread by a coil arc, and θ s is a coil-pitch. This model comprises of four assembly: a rotor magnet, a back yoke, three-phase (U,V,W) coils, and a slotless stator.

II. MODEL
This figure is illustrated for changeable parameters p = 2, Q = 6 and l = 2 as one of the examples, where p is the number of pole-pairs, Q is the number of coils, and l is the number of segments per pole. Throughout the paper, the machine is driven by three-phase sinusoidally alternating electric currents, and the ratio of the pole-pairs and the number of coils is chosen to be the most popular combination Q/(2p) = 3/2. All of the magnet pieces are sequentially assigned with number i = 1, 2, · · · , 2pl, and all of the coils are similarly numbered as j = 1, 2, · · · , Q in the order of U, V and W phases. The analysis regions are divided into three regions, 1 , 2 and 3 , as shown in the figure. In this model, l = 1 corresponds to an ordinal PM, l = 2 to a standard-Halbach PM, l ≥ 3 to segmented multipole-Halbach PMs, and l 1 approaches an ideal Halbach PM, as shown in Fig.1b).
This work is based on the following assumptions; First, permeability of the stator yoke and the magnet yoke is infinite, in other words magnetic saturation in the yoke is negligible. Second, thickness of the coil is neglected, that is to say, delta-function coil is adopted. Third, effects of the eddy current, e.g. Joule heat in the yoke and/or magnetic coupling between the rotor and the stator, are not incorporated. Fourth, secondary magnetic fields coming from the electric current of the coil is much weaker than those of the magnet. Fifth, the PM has a linear magnetization characteristics with a constant remanence and a constant recoil permeability. The effect of demagnetization is thus not considered in the model. It is also important to study such nonlinear complicated effects in a practical stage of designing and/or manufacturing, but is beyond the scope of this paper.

A. MAGNETIC FIELDS
The governing equations for magnetostatic fields B and H are self-consistently formulated as where M is a magnetization vector of the PM, µ 0 = 4π × 10 −7 H/m is a permeability of the vacuum, and µ r is a relative recoil permeability of the PM. As shown in Fig.2, the magnetization vector M i for the i-th segment of the PM with an array variable l can be expressed in a cylindrical coordinates system (r, θ, z) as where and ϕ i ≡ π(i − 1)/l + π/2 is an elevation angle of the magnetization vector, ϑ i ≡ π(i − 1)/(pl) is an azimuthal angle measured from the x-axis to the line extended radially from the origin O through the center G i , e r , e θ and e z are unit vectors in the radial, azimuthal and axial direction, respectively. It is worth noting that magnetization pattern of the PM can exactly be realized through the elevation angle ϕ i and the azimuthal angle ϑ i by substituting integers into the array parameter l. Magnetic scalar potential φ = φ(r, θ, z) can now be introduced because of the curl-free relation (2) as Taking divergence of (3) and (4) in combination with (1) provides Laplace and Poisson equations, respectively General solution for the Laplace equation (10) can be expanded in the form of double-Fourier series in the manner of separation of variables as where ν ≡ np, λ ≡ 2kπ/τ , τ ≡ 2z s , and X nk are the specific functions to be determined below in each region in terms of the normalized radial coordinate, x ≡ λr.
Components of the magnetization vector can similarly be expressed as where and with α q ≡ z m /z s and δ nm = 1 (n = m) or 0 (n = m) a Kronecker delta notation. The radial component M r can safely be dropped because it does not contribute to the net Lorentz torque. The Laplace and Poisson equations (10) and (11) can therefore be rewritten in terms of the specific functions X nk (x) as The general solutions in the regions 1 , 2 and 3 can now respectively be expressed by where I ν (x) and K ν (x) are modified Bessel functions of the first-and the second-kind for the ν-th order, respectively, and other notations are defined bỹ and with x r ≡ λR r . See also Appendix A for the relevant derivations. Six unknown coefficients A nk , B nk , C nk , D nk , E nk and F nk have to be determined subject to the following boundary conditions: The boundary condition (34) is intrinsically equivalent to (32), and (37) to (35) likewise. The number of equations to be solved therefore amounts to six. It should also be noted that the relations (31) and (38) where with x i ≡ λ k R i (i = o,r,m,s). Other coefficients are given by Practical technique for numerically computing the coefficients C nk and D nk is given in Appendix B.

B. BACK-EMF
When the PM rotor revolves around a z-axis with a uniform angular frequency ω, the axial component B z of the magnetic field permeating a coil surface located at z = z s is given by As shown in Fig.3, the linkage flux j for the j-th coil can straightforwardly be obtained by taking the surface integral for B z over the area S j of the coil as where θ ± j ≡ θ s (j − 1) + π/(2p) ± θ c /2 correspond to both edges oriented in the radial direction of the j-th coil, and the specific integral I nk defined by reflects the magnetic field structure in the radial direction. The back-EMF E j in the j-th coil is now induced by the time-varying flux according to the Lenz's law where N c is the number of turns in a coil, and hereafter N c = 1 for simplicity. The relevant line back-EMF between the Uand V-phase is then obtained by taking difference between the phase back-EMFs with j = 1 and 2 where q(= 3) represents the number of phases.

C. LORENTZ FORCE AND TORQUE
Let us now formulate Lorentz force and torque. In this paper, the Lorentz torque is defined as the moment of force that purely derives from the Lorentz force oriented in the circumferential (θ) direction. As shown in Fig.3, the Lorentz force df j exerted on an infinitesimal current element I j dr of the j-th coil is given by where B ± j ≡ B z (r, θ ± j , z = z s , t) are axial magnetic fields at both edges of the j-coil, and I j ≡ I 0 cos p(ωt − δ j ) is an electric current in the j-th coil with I 0 a peak current and δ j ≡ 2π(j − 1)/(pq) a phase shift. The Lorentz force df per pole-pair is driven by a set of the U-, V-and W-phase coils, and is expressed as Noticing the explicit relation one obtains with θ j ≡ θ s (j − 1). Taking the following identity of the summation in terms of j where φ n (t) ≡ (1+n)pωt and ψ n,j ≡ 2π(1+n)(j−1)/q. The Lorentz torque T can finally be obtained by the integration with respect to dT ≡ rdf as where we have used the relation Eigen modes and eigen frequencies are summarized in Table 1, where nontrivial modes are indicated by bold letters. Trivial modes do not generate the net Lorentz force. The Lorentz torque can therefore be expanded in terms of the nontrivial modes as follows 1  where n ≡ νθ c /2. The first term represents a uniform Lorentz torque independent of time, the second, third and fourth terms correspond to torque ripples of the 6th, 12th and 18th order harmonics, respectively. In a compact form, one can rewrite (69) as The torque constant K t is defined as the uniform Lorentz torque T 0 per current as where T 0 corresponds to the first term of eq.(70).

A. VALIDITY OF ANALYTICAL MODEL
Main parameters adopted in this model are listed in Table 2.
In order to validate the model, analytical results of the magnetic field distributions in the air-gap, back-EMF waveform and torque waveform are calculated by the use of the derived formulae (55), (60) and (69), respectively, and then compared with those of the FEM. The axial components B z of the magnetic flux density at the central radius r = R c ≡ (R r + R m )/2 and just below the upper yoke z = z s are plotted as a function of the azimuthal angle θ for some typical geometries of p = 2, 3 and 4 in Figs.4a), b) and c), respectively. The corresponding Fourier spectra are also drawn in Figs.4a ), b ) and c ), respectively. The magnetic field amplitudes B z,n are defined as B z,n ≡ k (−1) k+1 λX 2,nk (x c ) with x c ≡ λR c . It is shown that magnetic field distributions of l = 1 are almost the same irrespective of the number of poles, while those of l = 2, 3, 4 on the d-axis corresponding to θ = π/p gradually increase with an increase in poles. It is also found from the Fourier spectra that the fundamental amplitudes of the fields become strong with an increase in l except for the low-pole case a ), and that the space harmonics with n = 2kl + 1(k = 1, 2, · · · ) are excited for all cases. Specifically, the third harmonics appears only for the regular PM array. It would be expected from the spectra that the multipole-Halbach PM with l = 4 delivers the most superior performance in this study since the fundamental amplitude is the highest, and the harmonics is the lowest. The results of the analytical model for the magnetic field distributions and their Fourier spectra are in excellent agreement with those of the FEM. In Fig.5, the axial components B z in the air-gap on the d-axis corresponding to θ = 0 are now plotted against the radial coordinate r for the same geometrical parameters as above. The magnetic field amplitudes increase in the order of l = 1, 2, 3 and 4, as is expected. More interestingly, their amplification of the fields becomes significant for higher poles. It should also be noted that the curves have a negative (right down) slope for l ≥ 2 since the interaction of the magnetic fields in the Halbach PM is enhanced in the interior region. The analytical results agree well with the FEM ones except for some small deviations in the region close to the outer edge of the PM.
Temporal waveforms of the line back-EMF E u − E v between the U-and V-phase and the relevant Lorentz torque T for the same parameters as above are shown in Figs.6 and 7, respectively. As l increases, the waveforms of the back-EMF approach sinusoidal irrespective of the number of poles. Simultaneously, the waveforms of the Lorentz torque shift upward decreasing their fluctuation, i.e torque ripple, for p ≥ 2. However, amplitudes of the torque ripple increase with increasing the number of poles for all l. The observed improvement in the torque performance is consistent with the behavior of the Fourier spectra discussed above. It can be confirmed once again that the results of the analytical model agree well with those of the FEM.   Furthermore, the validity of the proposed model can well reasonably be explained by showing the fact that the mechanical output power T ω is identical with the electrical input power E u I u + E v I v + E w I w . Namely, energy conservation law holds at anytime in the system of equations. It can be demonstrated not only numerically but also analytically that  the expression for the Lorentz torque T given in (70) is consistently reproduced by calculating straightforwardly the formula (E u I u + E v I v + E w I w )/ω with the aid of (59). Mathematical proof of the energy conservation law is presented in the Appendix C.
In the calculations of the FEM, commercial EM simulation software, JMAG-designer (x64) ver.21.0 [36] has been utilized on a high-end workstation equipped with a hexa-core processor. In this environment it takes 5 to 6 hours for the FEM, while only a few minutes for the analytical model to obtain a single waveform of the back-EMF or Lorentz torque.

B. RESULTS OF ANALYTICAL MODEL
Hereafter, discussion concentrates on the results obtained by the analytical model for globally parametrical study. In Fig.8, the axial components B   PMs of l = 2, 3, and 4 increase inwardly irrespective of the values of the inner radii. While in c) for middle-poles, the fields are formed almost uniformly in a wide range of the radius for all l. In d) for high-poles, all of the fields increase outwardly contrary to the case a). It is found that the magnetic field structure in the radial direction strongly depends on the number of poles due to the convergence effect of the Halbach PM. It should be mentioned that the artificial correction function exploited in some preceding works would potentially overestimate or underestimate the magnetic field strength for low-poles and/or high-poles especially when the geometry of the AFPM is radially thick. The obtained results can be recognized by the fact that reducing R r is intrinsically equivalent to increasing p and/or l because the distance between the neighboring PM segments close to the inner edge is estimated to be R r π/(pl).
In general, the torque constant of the RFPM machine is proportional to the product of the amplitude of the fundamental magnetic fields and the number of poles, i.e. K t ∝ pB (1) ρ , e.g., [37], where B ρ is a radial component of the magnetic fields. On the other hand, the torque constant of the AFPM depends not only on the number of poles, but also on the radial structure of the magnetic fields through the integral term I 1k in (71). This is because the magnetic field in the air-gap of the RFPM is almost independent of the axial coordinate, while that of the AFPM is strongly dependent on the radial coordinate, as already shown in Fig.8. So now, Fig.9 shows the torque constants against the number of pole-pairs for some typical geometries of the AFPM with (R r , R m )=(60,100) mm for a) g = 0.5 mm, b) g = 1.0 mm, c) g = 1.5 mm and d) g = 2.0 mm. It is found that in the narrow air-gaps of a) and b) the torque constants monotonically increase with increasing p in the global range of 2 ≤ p ≤ 100, while that in the large gaps of c) and d), maximum turning points clearly appear for all l.
In Fig.10, the torque constants are then plotted against the inner radius R r of the PM with the fixed air-gap g = 0.5 mm and R m = 100 mm for a) p = 5, b) p = 10, c) p = 20 and d) p = 50. For the low number of poles, significant difference among array-variables l cannot be found. However, for the high-poles p ≥ 10 enhancement in the torque constant becomes more apparent because of the focusing effect of the multipole-Halbach PM as the number of poles increases, and/or as the inner radius of the PM decreases.
The torque constant per unit volume of the PM can be defined asK t ≡ K t /{π(R 2 m − R 2 r )z m }. Hereafter,K t is referred to as the torque constant density. Fig.11 presents the torque constant densities against the inner radius of the PM with g = 0.5 mm and R m = 100 mm for a) p = 5, b) p = 10, c) p = 20 and d) p = 50. As shown in the figure, the torque constant densities increase with an increase in the number of poles, and become higher in the order of l = 1, 2, 3 and 4 except for the low-poles. Fig.11a) shows that the torque constant densities of the standard-Halbach PM are weaker than that of the regular PM since the focusing effect of the magnetic fields would be insufficient in almost the whole range of R r .
For the middle-poles of b) and c), difference between the regular and multipole-Halbach PM becomes significant. For the high-poles of d), the torque constant densities keep high, but moderately decrease with decreasing inner radius for all l.

V. SUMMARY
The novel 3-D analytical model of the AFPM machine with a segmented multipole-Halbach PM array has been proposed in this paper. The mathematical expressions for magnetic fields, back-EMF, Lorentz torque and torque constant have successfully been derived in the explicit closed form, incorporating the effect of the space harmonics of the fields. Validity of the model can reasonably be demonstrated by the fact that the results obtained by the analytical model are in good agreement with those by the FEM, and that the law of energy conservation holds for the input-and output-power in the system of equations. On the basis of this model, the torque constant of the AFPM has been investigated in a wide range of the designing parameters, involving the number of poles, the number of segments per pole, air-gap and radius of the PM. The derived formulae and the obtained results are of much use in understanding the performance characteristics of the AFPM, and are available especially in early-designing of the electric machines, e.g. servomotors in multi-jointed robots or torque-controllable in-wheel AFPM machines on-board electric vehicles.

APPENDIX A: SOLUTION OF THE POISSON EQUATION
Poisson equation in the region 2 is given in the form Because independent solutions for the homogeneous differential equation with R = 0 are ϕ ν ≡ I ν (x) and ψ ν ≡ K ν (x), general solution can be expressed as While special solution for the inhomogeneous equation with R = 0 can be written is a Wronskians. Therefore, general solution for the inhomogeneous equation is given by

APPENDIX B: FOURIER COEFFICIENTS
In order to avoid overflow in the numerical integration involving Bessel's functions, the relevant coefficients C nk and D nk can respectively be modified to and

APPENDIX C: ENERGY CONSERVATION
It can mathematically be demonstrated on the basis of the derived formulae for the back-EMF and the torque that the law of energy conservation T ω = E u I u + E v I v + E w I w holds at anytime in the system. Electrical input power P e in can be expressed by with δ j = 2π(j − 1)/(pq) and θ j = θ s (j − 1) + π/(2p). Using the trigonometric identity, one obtains with ± nj ≡ 2π(n±1)(j−1)/3. The terms of summation with respect to j can be dropped or simplified into the form q j=1 sin ± nj = 0 (C-5) q j=1 cos + nj = qδ n,6m−1 (m = 1, 2, 3, · · · ) (C-6) q j=1 cos − nj = qδ n,6m +1 (m = 0, 1, 2, · · · ) (C-7) where δ nm = 1 (n = m) or 0 (n = m) is a Kronecker delta notation. Substitution of (C.2) and (C.3) into (C.1) with the help of (C.5)-(C.7) leads to the following formula λ I nk sin n cos p(n − 1)ωtδ n,6m+1 . (C-9) The electrical input power can finally be given in the following expression This formula exactly coincides with the mechanical output power P m out ≡ T ω. HISAKO HARADA received the M.E. degree in engineering from the Graduate School of Science and Technology, Chiba University, Japan, in 2002. Since 2002, she has been working as a Research Engineer with Seiko Epson Corporation. Currently, she is a Senior Researcher with the Research and Development Department, where she is engaged in the electromagnetic field simulation of electric machines. VOLUME 11, 2023