Romain Ravaud

Valerie Lemarquand
email: guy.lemarquand@ieee.org

Ironless Permanent Magnet Motors: Three-Dimensional Analytical Calculation

Keywords: Ironless motor, analytical calculations, torque, magnetic field, PM Synchronous motors !

We present a three-dimensional expression of the magnetic torque exerted between a tile permanent magnet radially magnetized and a winding in ironless structures. Such an expression can be used for calculating the magnetic torque transmitted between the stator and the rotor of an ironless permanent magnet motor. The calculations are carried out without using any simplifying assumptions. Consequently, the expression of the torque obtained is exact whatever the magnet or winding dimensions. The ironless structure we consider in this paper is commonly used for high speed rotation devices in which eddy currents must be avoided. Indeed, it is noted that more and more structures are ironless because they offer promising perspectives.

INTRODUCTION

I RONLESS electrical machines are generally used when high-speed rotations are required. These ironless motors [START_REF] Boules | Prediction of no-load flux density distribution in permanent magnet machines[END_REF] [START_REF] De Boeij | Modeling ironless permanent-magnet planar actuator structures[END_REF] differ from iron-core motors [START_REF] Muljadi | Axial-flux modular permanent-magnet generator with a toroidal winding for wind-turbine applications[END_REF] [START_REF] Kamper | Analysis and performance of axial flux permanent-magnet machine with air-cored nonoverlapping concentrated stator windings[END_REF] because the last ones are used up to a few hundred of hertz. Even though ironless motors are less efficient than iron-core motors, they constitue technological solutions [START_REF] Baran | Synchronous couplings with sm co5 magnets[END_REF] [START_REF] Jang | Design and experimental evaluation of synchronous machine without iron loss using double-sided halbach magnetized pm rotor in high power fess[END_REF] for avoiding the drawbacks of eddy currents that appear in iron-core motors. Many studies on ironless structures have been performed by several authors [START_REF] Madawala | Magnetic field analysis of an ironless brushless dc machine[END_REF] [START_REF] Ragot | Optimization of electric motor for a solar airplane application[END_REF] with analytical or semi-analytical approaches. In addition, some studies dealing with iron-core electric machines have been carried out with different analytical calculations [START_REF] Zhu | Comparison of halbach magnetized brushless machines based on discrete magnet segments or a single ring magnet[END_REF] [START_REF] Zhu | Analytical prediction of the cogging torque in radial-field permanent magnet brushless motors[END_REF]. Striclty speaking, iron-core motors using permanent magnets and windings are composed of iron parts whose aim is to confine the magnetic field produced by the permanent magnets. However, for high speed rotation devices, the presence of iron is problematic in the structure because eddy currents appear. Nevertheless, it is well known that the presence of these eddy currents generates a decrease in the efficiency of the torque transmitted between the stator and the rotor of electrical machines. This is why we can say that two kinds of applications must be identified and depend on both the input frequency and the rotation speed required. In addition, the magnet topology is also of great importance in high-speed motors and must be clearly optimized [START_REF] El-Hasan | Magnet topology optimization to reduce harmonics in high-speed axial flux generators[END_REF].

Several methods can be used for studying analytically ironless motors [START_REF] Babic | An improvement in the calculation of the magnetic field for an arbitrary geometry coil with rectangular cross section[END_REF]- [START_REF] Ravaud | Discussion about the analytical calculation of the magnetic field created by permanent magnets[END_REF]. These methods generally use the three-dimensional magnetic potential created by permanent magnets [START_REF] Ravaud | Torque in permanent magnet couplings: comparison of uniform and radial magnetization[END_REF] and allow us to optimize easily such structures [START_REF] Lemarquand | Ironless loudspeakers[END_REF]- [START_REF] Ravaud | The three exact components of the magnetic field created by a radially magnetized tile permanent magnet[END_REF].

Consequently, high speed rotation devices are only composed of ironless parts with samarium-cobalt magnets that are not conductor of electricity. The efficiency of such ironless structures is weaker than iron electrical machines but they can reach very high rotating speeds without the drawbacks of the eddy current effects. The accurate knowledge of the torque in such ironless structures is required for optimization purposes. It is noted that a semi-analytical calculation of this torque is possible because the structure considered is ironless. For classical electrical machines, such calculations are more difficult and finiteelement methods are appropriate.

We present in this paper an analytical calculation of the torque transmitted between the stator and the rotor of an ironless motor. We use the analogy between the coulombian model and the amperian current model for calculating the torque transmitted between a tile permanent magnet radially magnetized and a winding in the stator. Then, by using the principle of superposition, this calculation is used for calculating the torque transmitted between th stator and the rotor.

NOTATION AND GEOMETRY

The geometry considered and the related parameters are shown in Fig 1 . We consider both an outer tile permanent magnet and an inner winding. Such a configuration corresponds to an element of an ironless permanent magnet motor. Both the tile permanent magnet and the winding can be modeled by using the coulombian model. Indeed, by using the analogy between the coulombian and the amperian current models, the winding can be represented by using the fictive magnetic pole densities that are located on both the inner and outer surfaces delimited by the winding. The radially magnetized tile permanent magnet is also represented by fictitious magnetic pole surface densities [START_REF] Jang | Design and experimental evaluation of synchronous machine without iron loss using double-sided halbach magnetized pm rotor in high power fess[END_REF] that are located on its inner and outer faces as well as in its volume. We denote H θ , the azimuthal component of the magnetic field created by the tile permanent magnet. Its expression is given by

H θ = H(r, θ, z). u θ (1)
where

H(r, θ, z) = -∇ (Φ(r, θ, z)) (2)
and Φ(r, θ, z) is the scalar magnetic potential produced by the tile permanent magnet radially magnetized. This scalar magnetic potential 1. Geometry considered: an outer tile permanent magnet radially magnetized (polarization J) and an inner winding with a current, I, flowing through it. The inner radius of the outer tile permanent magnet is r in1 and its outer radius is r out1 . Its height is z b -z a and its angular width is θ 2 -θ 1 . Its magnetic polarization is denoted J. The inner radius of the winding is r in2 and its outer radius is r out2 . Its height is h 2 -h 1 and its angular width is θ 4 -θ 3 .

is expressed as follows:

Φ(r, θ, z) = 1 4π S in J r -r r in1 dθ 1 dz 1 - 1 4π S out J r -r r out1 dθ 1 dz 1 + 1 4π V J r -r dr 1 dθ 1 dz 1 (3)
where r -r is the distance between the observation point and a point located on the charge distribution, S in is the inner surface of the tile permanent magnet, S out is the outer surface of the tile permanent magnet and V is its volume.

In addition, K, the equivalent charge densities of the winding, is defined as follows:

K = µ 0 N I (r out2 -r in2 ) (4)
where µ 0 is the permeability of the vacuum, I is the intensity flowing in the winding and N is the number of current loops. In the numerical simulation carried out in this paper, we have taken N I = 240 At and K=0.1 T.

The magnetic torque can be determined as

Parameters Expressions T1(a, b) -a 2 b 2 JK (g(θ2) -g(θ1)) /4πµ0 T 2 (a, b) -ab 2 JK (g(θ 2 ) -g(θ 1 )) /4πµ 0 T3(a, b) -a 2 bJK (g(θ2) -g(θ1)) /4πµ0 T 4 (a, b)
-abJK (g(θ 2 ) -g(θ 1 )) /4πµ 0 TABLE 1 Parameters used for calculating the torque transmitted between a permanent magnet and a winding follows:

T = S in2 H θ Kr in2 d Sin2 - S out2 H θ Kr out2 d Sout2 + V H θ Kd Ṽ ( 5
)
where V is the volume of the winding, S in2 and S out2 are its inner and outer faces. The azimuthal component H θ of the magnetic field created by the outer tile permanent magnet can be determined either by using the scalar magnetic potential, the vector potential or the three-dimensional Green's function [START_REF] Baran | Synchronous couplings with sm co5 magnets[END_REF]. Without any simplifying assumptions, there are up to six integrations necessary for calculating the magnetic torque. However, this expression can also be reduced to the following form:

T = θ 4 θ 3 (T 1 (r out1 , r out2 ) -T 1 (r in1 , r out2 )) dθ i + θ 4 θ 3 (T 1 (r in1 , r in2 ) -T 1 (r out1 , r in2 )) dθ i + θ 4 θ 3 r out1 r in1 (T 2 (r 1 , r in2 )) dr 1 dθ i - θ 4 θ 3 r out1 r in1 (T 2 (r 1 , r out2 )) dr 1 dθ i + θ 4 θ 3 r out2 r in2 (T 3 (r 2 , r in1 )) dr 2 dθ i - θ 4 θ 3 r out2 r in2 (T 3 (r 2 , r out1 )) dr 2 dθ i + θ 4 θ 3 r out1 r in1 r out2 r in2 T 4 (r 1 , r 2 )dθ i dr 1 dr 2 (6)
The parameters used in ( 6) are defined in Table 1. Furthermore, the functions used in Table 1 are defined as follows:

g(θ j ) = y(a, b, h 2 , z a ) -y(a, b, h 1 , z a ) +y(a, b, h 1 , z b ) -y(a, b, h 2 , z b ) (7)
with

y(a, b, z 1 , z 2 ) = - η ab arctan ξ η + ξ ab ( 8
)
and

ξ = a 2 + b 2 + (z 2 -z 1 ) 2 -2ab cos(θ i -θ j ) (9) η = -(z 1 -z 2 ) 2 (10)
The expression obtained can be used for realizing easily parametric studies. It is emphasized here that Eq. ( 2) contains three types of interactions that correspond to the interactions between the surface and volume densities of the tile and of the winding. In addition, the numerical integrations of Eq. ( 4) have a robust convergence. As a consequence, the determination of Eq. ( 2) is both fast and accurate and allows us to study the magnetic torque transmitted by several radially magnetized tile permanent magnets on several windings.

APPLICATION: MAGNETIC TORQUE IN AN IRONLESS PERMANENT MAGNET MO-TOR

We discuss now the interest of using such an analytical expression for calculating the magnetic torque transmitted in an ironless permanent magnet motor. For this purpose, let us consider a motor whose rotor bears n radially magnetized permanent magnets and whose stator bears k windings. The total magnetic torque can be expressed as follows:

T total = n i=1 k j=1 α (i,j) T i,j (11)
where i represents the i th magnet of the rotor, j represents the j th winding of the stator and T i,j represents the magnetic torque transmitted between the i th magnet and the j th winding. The parameter α (i,j) is 1 if the magnet and winding polarizations are in the same sense and -1 if Fig. 2. Ironless permanent magnet motor with 8 radially magnetized permanent magnets in its outer rotor and 4 windings in its inner stator. their polarizations are opposite. We illustrate it with the following set of parameters: n = 8, k = 4, as shown in Fig 2 . The total magnetic torque transmitted by the stator on the rotor is represented versus the angular shift in Fig. 2 in which we give the numerical results given by the three-dimensional expression of the torque transmitted between a rotor with 8 radially magnetized tile permanent magnets and a stator with 4 windings. In addition, it is to be noted that no simplifying assumption is done and that the computational cost is nevertheless very low (less than 250s). This computational cost can still be dropped by neglecting the term owing three numerical integrals in Eq. ( 2). Such simplifications can safely done because the interactions between the fictious volume charges have not a great influence on the torque calculation between the winding and the tile permanent magnet.

INFLUENCE OF THE POLE RADIAL WIDTHS ON THE TORQUE TRANSMITTED BETWEEN THE STATOR AND THE ROTOR

In our configuration, the number of poles and the permanent magnet and winding dimensions have a great influence on the torque transmitted between the stator and the rotor. By using the analytical calculations established in this paper, it is possible to optimize the permanent magnet dimensions in order to obtain the greatest torque. We illustrate in Fig 4 how varies the torque transmitted between one permanent magnet at the rotor and one winding in the stator when their radial widths increase by keeping the same equivalent charge density K = 0.1 T. Figure 4 shows that the more the radial widths are important, the more the torque transmitted between the permanent magnet and the winding is. However, other parameters must be taken into account as the magnet volume [START_REF] Charpentier | Calculation of ironless permanent magnet coupling using seminumerical magnetic pole theory method[END_REF] and the structure dimensions [START_REF] Lemarquand | Nonsinusoidal torque of permanent-magnet couplings[END_REF]. Such optimizations are possible with analytical calculations as they are fast to carry out.

CONCLUSION

This paper presents an exact three-dimensional analytical method for calculating the torque transmitted between the stator and the rotor of ironless permanent magnet motors with radially magnetized tile permanent magnets and windings. Such structures are generally used when high-speed rotations are required. It is emphasized that the calculations have been carried out without using any simplifying assumptions. The expression of the torque is obtained in a semi-analytical form that can be still simplified by neglecting the interactions between the charge volume densities. Indeed, this interaction has a very low influence of the torque value. In addition, the computational cost of such an expression being very low, this approach is a good alternative to the finiteelement methods for carrying out parametric studies in ironless structures. However, it must be emphasized that such calculations can be carry out analytically possible because the structure considered is ironless.

  Fig.1. Geometry considered: an outer tile permanent magnet radially magnetized (polarization J) and an inner winding with a current, I, flowing through it. The inner radius of the outer tile permanent magnet is r in1 and its outer radius is r out1 . Its height is z b -z a and its angular width is θ 2 -θ 1 . Its magnetic polarization is denoted J. The inner radius of the winding is r in2 and its outer radius is r out2 . Its height is h 2 -h 1 and its angular width is θ 4 -θ 3 .

Fig. 3 .

 3 Fig. 3. Representation of the torque T versus the angular shift between the stator and the rotor, rin1 = 0.025 m, rout1 = 0.028 m, rin2 = 0.021 m, rout2=0.024 m, J=1 T, K=0.1 T for the three representations and all the tiles have the same height h=0.003 m.

Fig. 4 .

 4 Fig. 4. Representation of the torque T versus the angular shift between one permanent magnet and a winding, J= 1 T, K=0.1 T and all the tiles have the same height h= 0.003m.


Rémi Douvenot
email: remi.douvenot@enac.fr

SSW-2D: some open-source propagation software introducing split-step wavelet and wavelet-to-wavelet propagation techniques

Keywords: Atmospheric propagation, parabolic wave equation, wavelet transforms, open-source software

In this paper, the open-source software SSW-2D for propagation simulation is introduced. The software includes the classical split-step Fourier, as well as the more recent splitstep wavelet (SSW) and wavelet-to-wavelet propagation (WWP) techniques. SSW and WWP benefit from the properties of wavelet decompositions. Therefore, they are less memory-intensive and present a lower complexity than SSF. However, the coding is more demanding. It requires additional algorithmic steps -including the computation of wavelet propagators, which asks for a fine understanding of the decompositions.

This free and open-source Python software is proposed for encouraging the community to handle and improve SSW and WWP. The interaction with the computation core can be made through CSV files or with a dedicated GUI. Note that the final field is saved with respect to altitude, whereas only the wavelet decomposition of the total field is saved. This emphasizes the memory gain due to the sparse wavelet decompositions.

I. INTRODUCTION

The modelling of the propagation of radiowaves on large distances is a major issue for sizing many surveillance or telecommunication systems. When dealing with open spaces like suburban, rural, maritime, or even space environments, deterministic tools are often preferred. In this goal, the splitstep Fourier (SSF) resolution of the parabolic equation is a widely used technique for the modelling of radio waves [START_REF] Levy | Parabolic Equation Methods for Electromagnetic Wave Propagation[END_REF] in slowly-varying environments, as well as for acoustic waves [START_REF] Jensen | ser. Modern Acoustics and Signal Processing[END_REF].

SSF is based on a resolution of the propagation equation in two steps: a propagation step in the Fourier domain and an environment step -to account for atmosphere and relief -in the space domain. Here, we introduce a tool based on SSW [START_REF] Zhou | Modeling the long-range wave propagation by a split-step wavelet method[END_REF] and WWP [START_REF] Zhou | A fast wavelet-to-wavelet propagation method for the simulation of long-range propagation in low troposphere[END_REF]. SSW mimics SSF, except that the propagation steps are performed in the wavelet domain. Basically, this brings two main advantages and one drawback. On one hand, the wavelet decomposition is faster than the Fourier transform and the wavelet representation of the signal allows for sparse representations. On the other hand, unlike in the Fourier domain, the propagator is not diagonal in the wavelet domain, which complicates this step.

In WWP, all the propagation is performed in the wavelet domain, which implies a faster computation and additional hypotheses.

The goal of this paper is to comprehensively introduce and illustrate the SSW-2D propagation tool and the techniques within. It is proposed for research and/or educational purposes first. Note that this software is available online [START_REF] Douvenot | SSW-2D[END_REF].

In Section II, the SSW and WWP methods are introduced. The current capacities of the SSW-2D software are presented in Section III. Then, Section IV is dedicated to a discussion on SSW and WWP, including some research leads for the computational physics community.

II. THE PROPAGATION ALGORITHMS

A. Overview of the split-step techniques

We work on a 2D scenario in cylindrical coordinates (r, θ, z), assuming an invariance along θ. In these conditions, one must solve the propagation equation in harmonic mode

∂ 2 u ∂r 2 -2jk 0 ∂u ∂r + ∂ 2 u ∂z 2 + k 2 0 (n 2 -1)u = 0, (1)
where u = Ee jk0r is the reduced complex electric field, k 0 is the wavenumber in the vacuum and n is the refractive index of the medium. By neglecting the backward propagation, and using the approximation

√ 1 + A + B ≈ √ 1 + A + √ 1 + B -1 for A, B ≪ 1,
this equation can be reduced to a one-order differential equation with r, denoted as the wide-angle parabolic equation [START_REF] Dockery | An improved impedance-boundary algorithm for Fourier split-step solutions of the parabolic wave equation[END_REF]. Its iterative solution is

u(r + ∆r, z) = e -jk0(n-1)-j k 2 0 + ∂ 2 ∂z 2 -k0 ∆r u(r, z).
(2) The classical SSF algorithm numerically solves this equation by passing the second term in the Fourier domain, which yields u(r + ∆r, z) = e -jk0(n-1)∆r F -1 e -j(kr -k0)∆r F [u(r, z)] ,

where F denotes for the Fourier transform and k r is the horizontal wavenumber. The term e -j(kr -k0)∆r is the freespace propagator in the sense that it transforms the plane wave spectrum at ∆r into the plane wave spectrum at r + ∆r considering free space between them.

B. Split-step wavelet

Similarly to SSF, the numerical solution of (2) with SSW is [START_REF] Zhou | Modeling the long-range wave propagation by a split-step wavelet method[END_REF]

u(r + ∆r, z) = e -jk0(n-1)∆r W -1 [PW [u(r, z)]] , (4)
where W denotes for the wavelet decomposition (and compression), W -1 is the wavelet recomposition, and P is the free-space propagator in the wavelet domain detailed in Section II-D.

The wavelet decomposition leads to a multiscale representation of a discrete signal of size N z given by

u[p z ] = Nz/2 L p=1 a L,p φ L,p [p z ] + L l=1 Nz/2 l p=1 d l,p ψ l,p [p z ], (5)
where ψ l,p denotes a wavelet of level l and φ L,p is the scaling function. The translations p are necessary for the wavelets to cover the entire vertical domain. The fast wavelet transform (FWT) [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] is used to rapidly obtain the coefficients a L,p and d l,p , with a linear complexity with N z . Most of the coefficients can be set to 0 afterwards with a minimal and controlled loss of information via a compression step [START_REF] Cohen | Numerical Analysis of Wavelet Methods[END_REF].

The details and validations of the SSW algorithm are given in [START_REF] Zhou | Modeling the long-range wave propagation by a split-step wavelet method[END_REF]. We have especially demonstrated that the algorithm has a linear time complexity whereas SSF complexity follows O(N z log N z ). Moreover, the compression steps in SSW imply an error that can be managed by choosing appropriate thresholds [START_REF] Bonnafont | Determination of the thresholds in the split-step wavelet method to assess accuracy for longrange propagation[END_REF].

An example of a SSW result is plotted in Figure 1a. The propagation factor is displayed with respect to distance and altitude. It is calculated in the presence of a triangular relief (in black) and a trilinear atmospheric duct. Frequency is 500 MHz and the source is a complex source point placed 50 m before the first vertical of computation and at 50 m of altitude. Apodisation (absorbing layer) is performed above the dotted horizontal line with a Hanning window.

In Figure 1b, the wavelet decomposition of the vertical field at 35 km on L = 3 levels is displayed. The white spaces correspond to null coefficients. One can see that most of the field is carried by the scaling functions φ L . The detail coefficients corresponding to wavelets of lower levels are nonzero only where the field is divergent, from 300 m to 500 m of altitude. That is consistent with the fact that the lower wavelets carry the higher part of the signal spectrum.

C. Wavelet-to-wavelet propagation

In the WWP algorithm, the main idea is to perform all the propagation in the wavelet domain. Therefore, a step of propagation is given by

U (r + ∆r) = N PU (r), (6)
where U is the wavelet transform of u, and N is the application of the refraction in the wavelet domain. This refraction is applied on the center position of each wavelet as detailed in [START_REF] Zhou | A fast wavelet-to-wavelet propagation method for the simulation of long-range propagation in low troposphere[END_REF]. The apodisation step is performed in the wavelet domain too. Therefore, to apply SSW, a first wavelet transform is necessary to initiate the algorithm, and an inverse wavelet transform is applied on the last vertical to obtain the final field. In other words, the actual field is not computed during the propagation, only its wavelet transform. Since the propagation is performed in the wavelet domain, the ground composition and the relief cannot be accounted. The solution is to hybridize SSW with WWP. SSW is used to calculate the field at the lower altitudes whereas WWP is used for higher altitudes. This hybridization technique, denoted as WWP-H, is also detailed in [START_REF] Zhou | A fast wavelet-to-wavelet propagation method for the simulation of long-range propagation in low troposphere[END_REF].

The same simulation as in the previous section is performed with WWP-H. Only the relief is not present because it is not available yet in the software.

One can see in Fig. 2a that both refractivity and apodisation are efficiently accounted in the wavelet domain. However, the quantifying of this model error is still an open question. The wavelet coefficients at 35 km plotted in Fig. 2a show that in the absence of relief, all the energy is carried by the scaling functions. Therefore, the compression rate is high (83.7 % on this vertical).

In the next section, the generation and use of the free-space propagator in the wavelet domain are presented.

D. The wavelet-domain propagator

The tricky part of the SSW and WWP algorithms is the generation and use of the wavelet-to-wavelet propagator.

From our knowledge, the propagation of a wavelet on a step ∆r has no closed-form expression. Therefore, the propagator is numerically calculated beforehand using SSF on one freespace step then stored.

What we call the free-space propagator in the wavelet domain is actually a dictionary of local propagators for each wavelet. Since all the wavelets on one level of decomposition are identical, their propagated counterparts are also identical, and the final number of required propagators (2 L ) is kept small. The way to rapidly calculate these propagators, including the associated pseudo-code, is given by Bonnafont et al. [START_REF]A local split-step wavelet method for the long range propagation simulation in 2D[END_REF]. Fig. 3 gives two examples of the local propagators used to generate Fig. 1a. They are associated to the scaling function and to a wavelet of level 2, respectively.

With these propagators, the propagation operator P in the wavelet domain in (4) consists in applying the proper propagator to each wavelet coefficient of u(r, z) to obtain the wavelet decomposition of u(r + ∆r, z) propagated in free space on one step ∆r. The inverse wavelet transform is finally applied to obtain the very field.

Note that if geometrical parameters are modified during the propagation, the propagators must be updated. This operation is fast and can be performed on the fly [START_REF]A local split-step wavelet method for the long range propagation simulation in 2D[END_REF].

A. Source computation

The source module is dedicated to the initial field computation. For the time being, only the complex source point [START_REF] Deschamps | Gaussian beam as a bundle of complex rays[END_REF] is available. This module reads the source parameters, calculates the associated field, and writes the corresponding electric field in a CSV file.

Any other source could be considered at minimum cost, including a field exported from an antenna simulation software.

B. Terrain generation

The terrain module is used to generate the relief. Triangular and realistic random reliefs are available. As for the initial field, this module writes this output in a dedicated file.

Similarly to the source input, other source generations can be considered as long as the output is in the proper format.

C. Propagation simulation

The propagation module is the core of the software. After the calculation of the dictionary of propagators, this function propagates the initial field with the selected technique. Up to now, available techniques are split-step Fourier (SSF), splitstep wavelet (SSW), wavelet-to-wavelet propagation (WWP), and hybridized SSW-WWP (WWP-H).

Three variables are saved at the end of the calculation: the dictionary of propagators, the final field, and the wavelet coefficients all along the propagation (this requires an additional step for SSF). Note that the storage of the wavelet coefficients is much less demanding than the equivalent electric field.

D. Post processing

The post_processing module is mainly dedicated to the results display. At this time, it allows for plotting the dictionary of propagators, the total field from the saved wavelet coefficient or the corresponding propagation factor, and the wavelet decomposition of the field at a given distance from the initial plane. This last function also gives the compression rate obtained after the wavelet decomposition the field.

E. Graphic user interface

Finally, the module GUI is used to launch the graphic user interface (GUI). In this case, the inputs are directly filled in on the GUI, that automatically updates the CSV files read by the other modules. Press buttons launch the aforementioned modules source, terrain, and propagation, respectively. A display of the GUI is given is Fig. 4.

The use of the GUI is recommended to get familiar with SSW-2D and to test parameter values. Inputs through CSV files are more fitted for expert users willing to modify and/or improve the software.

On this GUI, the initial field and the environment (atmosphere and relief) are also plotted. After the propagation computation, the computed field and the final vertical cut are plotted. The type of output (electric field, propagation factor or Poynting vector) and the dynamic can be tuned as well. The values above the black dotted lines are in the absorbing layer.

Now that both the method and the software have been presented, the conclusion is dedicated to a discussion about the future of the research based on SSW and the usefulness of SSW-2D in this scope.

IV. CONCLUSION AND FUTURE LEADS

First of all, SSW and WWP techniques are from now on mature techniques to model radiowave propagation. They are shown to be faster than SSF (specially WWP, having a much better complexity) and they allow for a sparse representation of the propagated field. The limitations are similar than for SSF, i.e., the paraxial cone of validity and the neglected backward propagation. The improved efficiency of SSW and SSW is at the cost of compressions of both the signal and the propagator that are acceptable.

In practice, SSW and WWP are particularly interesting on large domains and/or with very high frequencies for which the number of vertical points is large. For instance, it is obviously relevant in a radio occultation context [START_REF] Douvenot | Modelling the radiowave propagation with a split-step wavelet method for radio occultation[END_REF].

As for WWP, the additional errors due to apodisation and atmosphere in the wavelet domain are not yet quantified. However, first tests show that these errors seems easily acceptable. This could not be the case in fast varying atmosphere, in the presence of turbulence notably.

As is, SSW-2D is a research code that does not claim to compete with the commercial SSF codes like AREPS or TEMPER. A significant amount of work would be necessary to optimise the algorithms and translate them in a compiled language. That is also the reason why computation times are not mentioned in this work. Also, many different features are planned during the next months of development: new sources, ground roughness, surface wave, new reliefs, other wavelet families. But the main objectives now are to track down bugs and propose a detailed documentation. The GIT deposit is the opportunity for the community to test and improve this software.

In addition to these simple features, some leads for future improvements still require research efforts. SSW-2D is promising for applications that demand lots of resources. The straightforward extension is 3D propagation. This has already been proposed [START_REF] Bonnafont | Split-step wavelet with local operators for the 3D long-range propagation[END_REF] but relief and non-homogeneous impedance grounds are not properly accounted yet.

Apodisation may be improved using the wavelet decomposition. The residual field on top on the domain (see Fig. 1a) is due to the periodic boundary conditions for the wavelet decomposition that ensure the orthonormality of the basis. However, this could be improved as well.

Studies could also focus on the very propagators. Today, they are generated with SSF. A more appropriate technique could be derived. For example, a closed-form expression based on the wavelet connection coefficients could improve the overall method.

Last, these techniques can be straightforwardly adapted for acoustic or optic propagation. SSW-2D could be useful in these domains as acoustics deals with large 3D environments and optics with very small meshes.

Fig. 1 :

 1 Fig. 1: Propagation factor (dB) obtained by SSW and the corresponding wavelet coefficients of the field in dB at 35 km.

Fig. 2 :

 2 Fig. 2: Propagation factor (dB) obtained by WWP and the corresponding wavelet coefficients of the field in dB at 35 km.

(a) φ 3 (b) ψ 2 Fig. 3 :

 323 Fig. 3: Two propagators in the wavelet domain: for the scaling function and for one wavelet.

ACKNOWLEDGEMENTS

The author wishes to thank his colleagues and students from ENAC, and especially Dr. Zeina El-Ahdab and Victor Darchy for the discussions, suggestions and tests of SSW-2D.


Benoit Merit
email: benoit.merit.etu@univ-lemans.fr

Guy Lemarquand

Bernard Nemoff
email: bernard.nemoff@orkidia-audio.com

An ironless low frequency subwoofer functioning under its resonance frequency

Keywords:

HAL is

INTRODUCTION

In sound reproducing systems, the bass range is important because it contributes more to the feeling of power and quality than any other part of the sound spectrum: when the bass is solid and realistic the resultant sound is usually perceived as being very good to excellent. However problems to reproduce tight, clean, and realistic sound in the bass range are not trivial. This paper presents a loudspeaker bound to produce low and very low frequencies, the particularity coming from the fact that the proposed loudspeaker works under its resonance frequency. This type of subwoofer has more demands regarding its designing than traditional ones, with the aim of being very accurate and efficient.

WORKING UNDER THE RESONANCE FRE-QUENCY

When looking at its pressure response, a loudspeaker acts like a high-pass driver (Fig. 1). Since every frequencies have to be reproduced ideally with the same level, a classical loudspeaker works above its critical resonance frequency f 0 where its pressure response is relatively flat. Thus, the major problem when manufacturing subwoofers is to decrease their mechanical resonance frequency to extend their bandwidth towards the low frequencies. When mounted in a closed enclosure, the resonance frequency of the loudspeaker is proportional to the square root of the total mechanical stiffness of the mounted loudspeaker over the total moving mass of the loudspeaker. Therefore to extend the bandwidth of a subwoofer towards low frequencies, ones has to decrease the stiffness while increasing the moving mass, at the expense of the efficiency of the subwoofer. Since a suspension whose stiffness tends to be null does not exist, the widening of this bandwidth also gives rise to large closed enclosures. It is also met subwoofers used on a frequency range that comprises their mechanical resonance, despising all spoiling risks and phase reversals.

Numerous theories and ideas have led to designs which have significantly improved low frequency response of such loudspeakers [START_REF] Stahl | Loudspeaker Lower Bass Response Using Negative Resistance and Impedance Loading[END_REF], [START_REF] Thuras | Sound Translating Device[END_REF], [START_REF] Olson | Loudspeaker and Method Of Propagating Sound[END_REF], [START_REF] Geddes | Passively Assisted Loudspeakers[END_REF]. Unfortunately these methods deteriorate the phase response and the consequent smearing of the arrival times of these now audible low frequency sound components. However phase distortion is well known to be a critical criterion for the quality and accuracy of the response of a loudspeaker [START_REF] Suzuki | On the Perception of Phase Distortion[END_REF].

In this paper a subwoofer working under its resonance frequency, in the bandwidth 10Hz-100Hz, is investigated. The first advantage is that the behavior of the loudspeaker is really well known and well modeled in this frequency range: supplying tension, current and displacement of the diaphragm are in phase. The phase of the diaphragm velocity is almost constant which is an important point when searching for coherent response of the loudspeaker. Let's remind that under the resonance frequency, the pressure response of the loudspeaker falls at a rate of 12dB per octave as the frequency is decreased. This means that an embedded electronics using a double integrator is necessary to raise this slope and to flatten the pressure response of the loudspeaker under its resonance frequency. The integrator is also used to perfectly correct phase variations of the pressure response of the loudspeaker, at the same time as to filter the highest frequencies, in contemplation of a good agreement with a bass or bass-medium loudspeaker. Long et al. have also worked on a necessary electronics to protect each piece of the loudspeaker from large displacements at very low frequencies [START_REF] Long | Method and Apparatus for Operating a Loudspeaker Below Resonant Frequency[END_REF]. And it appears that a custom-built loudspeaker is necessary to well fit the functioning under the resonance frequency.

Therefore, with regard to the subwoofer itself, a coherent work leads to make a light moving mass to increase the bandwidth of the subwoofer at the same time as to increase its efficiency. Efficiency of this subwoofer also represents an important work to compensate for the drawback of the 12dB/octave slope under the resonance frequency, and to make this subwoofer be very dynamic with sharper time response.

STRUCTURE OF THE LOUDSPEAKER

The structure of the loudspeaker designed for a better adaptation to the functioning under its resonance frequency is shown Figure 2. It is made of a flat diaphragm, only centered thanks to the surround suspension. The former coil and the voicecoil are located in the airgap between two sets of three annular radially magnetized permanent magnets. The following presents the design proposed to make the subwoofer be very performant at low and very low frequencies with really reduced nonlinearities.

The loudspeaker motor

The drawbacks due to the iron in classical loudspeakers are really well known [START_REF] Vanderkooy | A Model of Loudspeaker Driver Impedance Incorporating Eddy Cur-rents in the Pole Structure[END_REF], [START_REF] Klippel | Loudspeaker Non Linearities -Symptoms, Parameters, Causes[END_REF], [START_REF] Lemarquand | Ironless Loudspeakers[END_REF]. Iron is the source of Eddy currents and reluctant effects, especially at low frequencies, and is the origin of a magnetic field that is a function of the axial position in the airgap. Some solutions have been proposed based on specific design of ironless motor structures [START_REF] Ohashi | Magnetic Circuit and Speaker[END_REF], [START_REF] House | Transducer Motor Assembly[END_REF]. However, since their created magnetic induction highly depends on the position in the airgap, these assemblies are only suitable to small voice-coil displacement and thus to medium or high frequency loudspeakers. Indeed bass loudspeakers also require the most uniform magnetic induction possible over the whole displacement of the voice-coil. When only using magnets, analytical calculations of the magnetic field created by different magnetic structures are now usable [START_REF] Bancel | Three Dimensional Analytical Optimization of Permanent Magnets Alternated Structure[END_REF], [START_REF] Ravaud | Analytical Calculation of Magnetic Field Created by Permanent Magnet Rings[END_REF]. The knowledge of analytical solutions makes possible the fast optimization of conceivable structures made of magnets, and searching for linear motors, leads finally to the magnetic structure of the presented loudspeaker. It uses two sets of three stacked annular radially magnetized permanent magnets. In one set, the two external rings are used to make the radial component of the magnetic induction created by the central ring become constant along the total height of this central ring (Fig. 3). For eco- nomical reasons, the chosen dimensions and magnetizations of each ring of the presented structure are not optimal. Nevertheless Figure 3 shows that the created induction is really quite uniform. In an underhung configuration, this structure makes possible a constant force factor along the entire displacement of the voice-coil up around ±20mm. First attempts lead to a constant force factor of 35N.m along the 40mm height of the central rings, for an electrical resistance of 5Ω. By optimizing the magnets and the wire used in the voice-coil, it may be reached more significant values of this force factor.

Thus, a very linear motor has been designed, even for large displacements of the voice-coil. We are nevertheless conscious that this structure is subject to magnetic flux leakages. The use of the three radially magnetized rings represents a considerable reduction of the total height of the magnetic structure, in prevision of reduced dimensions of the former coil to increase its resonance frequencies.

The moving part of the loudspeaker

In addition to be ironless, the loudspeaker has the characteristic to have a flat circular diaphragm. This choice enables to fit as well as possible the approximation of circular piston used in the models, and is reinforced by the conclusions of Quaegebeur , s work [START_REF] Quaegebeur | Influence of Material and Shape on Sound Reproduction by an Electrodynamic Loudspeaker[END_REF]. Interferences, diffraction phenomena and cavity effects caused by the conical shape usually given to the loudspeaker diaphragm are thus removed [START_REF] Sakamoto | Loudspeaker with Honeycomb Disk Diaphragm[END_REF]. The use of a flat diaphragm intrinsically makes possible to use a voice-coil whose diameter is larger than usual. The former coil diameter is calculated with the aim of pushing back the first resonances of the diaphragm towards the high frequencies. Figure 4 shows that the optimal diameter of the voice-coil is 0.7 times the diameter of the diaphragm, when the coil and its support are considered rigid. The 20" diameter flat diaphragm posses two resonance frequencies in its wanted working bandwidth, measured at 38Hz and 61Hz. In presence of the former coil, the first resonance of the diaphragm occurs at 83Hz. Moreover, thanks to the previously described motor, the total height of the moving part of the loudspeaker is low compared to the radius of the diaphragm; the gravity centre of the moving element is thus rather close to the diaphragm so that only the external suspension is necessary to centre and control the motion of the moving part. The removal of the spider represents a consequent decrease of the excursion non-linearities. Since a high mechanical resonance frequency is reached, a low volume box is used to enclose the loudspeaker and provides 50% of the total stiffness. The total needed stiffness implies the use of a material with high modulus for the surround suspension: the suspension brings a non-negligible boundary condition. Note that in the case of a clamped boundary condition, first resonance of the diaphragm with the former coil is predicted at 135Hz; with the surround suspension, the first resonance frequency is then expected up to 100Hz.

Conclusion

An ironless loudspeaker well suitable to a functioning under its mechanical resonance frequency has been described. First calculations and measurements are quite promising for its conception and its improvement. The perfectly constant force factor, the removal of the iron and the flat diaphragm are the basis of a really expected accurate, clear and dynamic low frequency subwoofer.

Figure 1 :

 1 Figure 1: Behaviour of the pressure response of a loudspeaker vs. frequency.

Figure 2 :

 2 Figure 2: Composition of the ironless subwoofer.

Figure 3 :

 3 Figure 3: Magnetic structure used in the subwoofer and its created radial magnetic induction, Br(T ), along the observation axis.

Figure 4 :

 4 Figure 4: Evolution of the resonance frequencies of an arbitrary circular plate (diameter a), when fixed to a circular support whose diameter b varies.


R Ravaud

Senior IEEE G Lemarquand
email: guy.lemarquand@ieee.org

V Lemarquand

Magnetic Field Created by Tile Permanent Magnets

Keywords: Analytical model, magnetic field, permanent magnet, axial magnetization, radial magnetization

This paper presents the analytical calculation of the three components of the magnetic field created by tile permanent magnets whose magnetization is either radial or axial. The calculations are based on the coulombian model of permanent magnets. The magnetic field is directly calculated, without the magnetic potential. Both axial and radial magnetization of the tiles are considered. The expressions obtained give the magnetic field in all the space. Such analytical expressions are very useful for the design and optimization of many industrial applications.

I. INTRODUCTION

A FTER the parallelepiped, the most common shape for a permanent magnet in electrical engineering is certainly the tile, which can also be described as a ring sector. Such tiles can be either axially or radially magnetized. Therefore, the calculation of the magnetic field created by tile magnets is of great utility and numerous approaches exist.

Numerical approaches do not allow one to perform numerous parametric studies quickly and have generally a high computational cost. Consequently, authors are looking for alternative solutions. Their approaches are often semianalytical ones [START_REF] Azzerboni | Three-dimensional calculation of the magnetic field created by current-carrying massive disks[END_REF], and they represent important steps toward the ideal analytical ones. In fact, different points of view can be adopted, which all look for the same thing: the calculation of the magnetic field and the magnetic forces. Kim et al. [START_REF] Kim | Restoring force between two noncoaxial circular coils[END_REF][3] calculate the magnetic force from the vector potential, with complete elliptic integrals and a mesh-matrix method. Kwon et al. [START_REF] Kwon | Analysis of the far field of permanent magnet motors and effects of geometric asymmetries and unbalance in magnet design[END_REF] work with spherical coordinates and multipole expansions to calculate the far field of permanent magnet motors. Selvaggi et al. [START_REF] Selvaggi | Calculating the external magnetic field from permanent magnets in permanent-magnet motors -an alternative method[END_REF] propose an approach with cylindrical coordinates and the use of Green's function and Fourier series expansion to characterize the field from a set of permanent magnets. Babic and Akyel [START_REF] Babic | Improvement in the analytical calculation of the magnetic field produced by permanent magnet rings[END_REF]- [START_REF] Babic | Magnetic force calculation between thin coaxial circular coils in air[END_REF] calculate the force between coils thanks to Heuman's Lambda function. Toroidal harmonics yield interesting solutions to formulate the magnetic field created by permanent magnet cylinders [START_REF] Selvaggi | Computation of the three-dimensional magnetic field from solid permanent-magnet bipolar cylinders by employing toroidal harmonics[END_REF]. Conway uses Bessel's function to calculate the inductance of coils using the vector potential [START_REF] Conway | Inductance calculations for noncoaxial coils using bessel functions[END_REF] and proposes also a direct calculation of the magnetic field -without the vector potential-to reach the same goal [START_REF] Conway | Noncoaxial inductance calculations without the vector potential for axisymmetric coils and planar coils[END_REF].

While these studies show variety in the starting point for the problem they want to solve, many others are more specific and describe methods to calculate the magnetic field created by toroidal magnets. Perigo et al. [START_REF] Perigo | General expressions for the magnetic flux density produced by axially magnetized toroidal permanent magnets[END_REF] present analytical-integral expressions for the axial and radial magnetic flux density components of axially magnetized toroidal magnets, either on-or off-axis. The application considered is an electron beam focusing system. Zhilichev [START_REF] Zhilichev | Calculation of magnetic field of tubular permanent magnet assemblies in cylindrical bipolar coordinates[END_REF] works with cylindrical coordinates and uses separation of variables to evaluate the magnetic field from the scalar potential for tubular linear permanent magnet machines. He gives 2D and 3D approximations. Furlani et al. propose solutions based on the vector potential for radially polarized multipole cylindrical magnets [START_REF] Furlani | A three-dimensonal field solution for radially polarized cylinders[END_REF] as well as for axially polarized magnets for axial field motors [15][16]. Rakotoarison et al. [START_REF] Rakotoarison | Using coulombian approach for modeling scalar potential and magnetic field of a permanent magnet with radial polarization[END_REF] give a semianalytical method to calculate the field created by radially magnetized tile magnets from the scalar potential. They use a coulombian model of the magnets and take the volume charge density into account.

All the studies dedicated to the calculation of the magnetic field created by toroidal or tile magnets are very useful: they constitute tools for the design and the optimization of devices which use such magnets and enable to meet specific requirements. For example, tile magnets with rotating magnetizations are used to make discrete Halbach cylinders [START_REF] Halbach | Design of permanent multiple magnets with oriented rec material[END_REF] which have various applications, such as electrodynamic wheels [START_REF] Bird | Characteristics of an electrodynamic wheel using a 2-d steady-state model[END_REF] for maglev devices, the creation of homogeneous fields or tailored gradient fields [START_REF] Hilton | Halbach cylinders with improved field homogeneity and tailored gradient fields[END_REF] [START_REF] Blache | High magnetic field gradients in flux confining permanent magnet structures[END_REF], the magnetization of brushless machines to obtain sinusoidal field variation in the airgap [START_REF] Zhu | Comparison of halbach magnetized brushless machines based on discrete magnet segments or a single ring magnet[END_REF]. Tile magnets are used with peculiar profiles to reduce the cogging torque in axial flux machines [START_REF] Aydin | Minimization of cogging torque in axial-flux permanent-magnet machines: design concepts[END_REF] as well as in radial flux ones [START_REF] Li | Optimum design of magnet shape in permanent-magnet synchronous motors[END_REF], but also to control the torque in permanent magnet couplings [START_REF] Quellec | New permanent magnet couplings for screwing devices[END_REF]- [START_REF] Charpentier | Calculation of ironless permanent magnet coupling using semi-numerical magnetic pole theory method[END_REF] and gears [START_REF] Charpentier | Mechanical behavior of axially magnetized permanent-magnet gears[END_REF]. They are constitutive parts of displacement sensors [START_REF] Blache | New structures for linear displacement sensor with hight magnetic field gradient[END_REF] and also ironless loudspeakers [START_REF] Berkouk | Analytical calculation of ironless loudspeaker motors[END_REF]- [START_REF] Ravaud | Ironless loudspeakers with ferrofluid seals[END_REF] demonstrating their widespread use.

This paper presents analytical expressions for the three components of the magnetic field created by tile permanent magnets. The magnetic field is directly calculated, without a previous calculation of the scalar or vector potential [START_REF] Ravaud | The three exact components of the magnetic field created by a radially magnetized tile permanent magnet[END_REF] [START_REF] Ravaud | Discussion about the analytical calculation of the magnetic field created by permanent magnets[END_REF]. The expressions given are for a ring sector, but remain valid when the sector is extended to the whole ring. The axial magnetization of the tiles is considered in the first section below and the radial magnetization in the second.

II. AXIALLY MAGNETIZED TILES

A. Notation and Geometry

The geometry which is considered is a tile permanent magnet and the related parameters are shown in Fig. 1. The tile inner radius is r in , the tile outer one is r out and its height is h. The angular width of the tile is θ 2 -θ 1 . The axis z is an axis of symmetry. The coulombian model of a permanent magnet is used. Consequently, the tile permanent magnet is represented by two curved planes which correspond to the upper (z = h) and lower (z = 0) faces of the ring sector. The upper one is charged with a surface magnetic pole density +σ * ; the lower one is charged with the opposite surface magnetic pole density -σ * . All the illustrative calculations are done with σ * = J. n = 1T . In order to simplify the calculations, the upper face only is taken into account to determine the three magnetic field components. However, the total magnetic field can be calculated by the application of the linear superposition principle to both faces.

Let us consider a point P on the ring sector upper face. The magnetic field H created by the source point P (r, θ s , z) at any observation point M (r, θ, z) of the space is given by [START_REF] Azzerboni | Three-dimensional calculation of the magnetic field created by current-carrying massive disks[END_REF].

H(r, θ, z) = σ * 4πµ 0 θ2 θ1 rout rin --→ P M --→ P M 3 r 1 dr 1 dθ s (1)
where µ 0 is the vacuum magnetic permeability (µ 0 = 4π.10 -7 SI) and σ * is the fictitious magnetic pole surface density in tesla. Equation ( 1) can be written as follows:

H(r, θ, z) = σ * 4πµ 0 θ2 θ1 rout rin (r -r 1 cos(θ -θ s )) u r -r 1 sin(θ s -θ) u θ + (z -z 1 ) u z (r 2 1 + r 2 + (z -z 1 ) 2 -2r 1 r cos(θ -θ s )) 3 2 r 1 dr 1 dθ s (2)

B. Components along the three directions

u r , u θ , u z
The integration of (2) leads to the magnetic field components along the three axes defined

H r (r, θ, z), H θ (r, θ, z), H z (r, θ, z). 1) Azimuthal component H θ (r, θ, z):
The magnetic field azimuthal component H θ (r, θ, z) created by the upper face is given by [START_REF] Kim | Mutual inductance of noncoaxial circular coils with constant current density[END_REF].

H θ (r, θ, z) = σ * 4πµ 0 (η(θ, θ 1 ) -η(θ, θ 2 )) (3)
with

η(θ, θ i ) = r 2 + r 2 in + (z -h) 2 -2rr in cos(θ i -θ) r + cos(θ i -θ) log r in -r cos(θ i -θ) + r 2 + r 2 in + (z -h) 2 -2rr in cos(θ i -θ) - r 2 + r 2 out + (z -h) 2 -2rr out cos(θ i -θ) r -cos(θ i -θ) log r out -r cos(θ i -θ) + r 2 + r 2 out + (z -h) 2 -2rr out cos(θ i -θ) (4)
Equation ( 3) is valid for any observation point M (r, θ, z) with 0 ≤ θ ≤ 2π.

h = 3 mm, θ 1 = 0 rad, θ 2 = π 2 rad, θ = 0 rad, r in = 25 mm, r out = 28 mm. 2) Radial component H r (r, θ, z):
The field radial component H r (r, θ, z) created by the upper face is given by [START_REF] Selvaggi | Calculating the external magnetic field from permanent magnets in permanent-magnet motors -an alternative method[END_REF].

H r (r, θ, z) = σ * 4πµ 0 (α(u 2 , r, z) -α(u 1 , r, z)) (5)
; h = 3mm, r in = 25mm, rout = 28mm, θ 1 = 0, θ 2 = π 2 , θ = π 2 where u i = cos(θ -θ i ) (6)
and

α(u i , r, z) = F 1 (u i , r, z) G 1 (u i , r, z)E * arcsin[ r 2 + r 2 out -2rr out u i + (z -h) 2 (r + r out ) 2 + (z -h) 2 ], (r + r out ) 2 + (z -h) 2 (r -r out ) 2 + (z -h) 2 +F 1 (u i , r, z)Π * arcsin[ r 2 + r 2 out -2rr out u i + (z -h) 2 (r + r out ) 2 + (z -h) 2 ], (r + r out ) 2 + (z -h) 2 (r -r out ) 2 + (z -h) 2 -F 2 (u i , r, z) G 2 (u i , r, z)E * arcsin[ r 2 + r 2 in -2rr in u i + (z -h) 2 (r + r in ) 2 + (z -h) 2 ], (r + r in ) 2 + (z -h) 2 (r -r in ) 2 + (z -h) 2 -F 2 (u i , r, z)Π * arcsin[ r 2 + r 2 in -2rr in u i + (z -h) 2 (r + r in ) 2 + (z -h) 2 ], (r + r in ) 2 + (z -h) 2 (r -r in ) 2 + (z -h) 2 + (u 2 i -1) 1 -u 2 i log r out -ru i + r 2 + r 2 out -2rr out u i + (z -h) 2 - (u 2 i -1) 1 -u 2 i log r in -ru i + r 2 + r 2 in -2rr in u i + (z -h) 2 (7)
with :

F 1 (u i , r, z) = 1 1 -u 2 i 2r out (1 + u i ) rrout(ui-1) (r-r out ) 2 +(z-h) 2 r 2 +r 2 out -2rr out u i +(z-h) 2 (r+r out ) 2 +(z-h) 2 rr out (1+u i ) (r+rout) 2 +(z-h) 2 r 2 + r 2 out -2rr out u i + (z -h) 2 (8)
F 2 (u i , r, z) = 1 1 -u 2 i 2r in (1 + u i ) rrin(ui-1) (r-r in ) 2 +(z-h) 2 r 2 +r 2 in -2rr in u i +(z-h) 2 (r+r in ) 2 +(z-h) 2 rr in (1+u i ) (r+rin) 2 +(z-h) 2 r 2 + r 2 in -2rr in u i + (z -h) 2 (9
r out = 28mm, θ 1 = 0 rad, θ 2 = π 2 rad, θ = π 2 rad G 1 (r, z) = (r -r in ) 2 + (z -h) 2 2rr in ( 10
)
G 2 (r, z) = (r -r out ) 2 + (z -h) 2 2rr out (11) E * [k] = φ= π 2 0 1 -k 2 sin(θ) 2 dθ (12)
Equation ( 5) is valid for any observation point M (r, θ, z) with θ = θ i and 0 ≤ θ < 2π. This expression remains valid for ring permanent magnets, for which the angular width is 2π (θ 2 -θ 1 = 2π). It leads to the expression of the radial component already given by the authors for ring magnets [START_REF] Ravaud | Analytical calculation of the magnetic field created by permanent-magnet rings[END_REF]. Figure 3 represents the field radial component H r (r, θ, z) versus the radial distance r. The used parameters are h = 3 mm, θ 1 = 0 rad, θ 2 = π 2 rad, θ = 0 rad, r in = 25 mm, r out = 28 mm. 3) Axial component H z (r, θ, z): The field axial component H z (r, θ, z) created by the upper face is given by [START_REF] Zhilichev | Calculation of magnetic field of tubular permanent magnet assemblies in cylindrical bipolar coordinates[END_REF].

H z (r, θ, z) = σ * 4πµ 0 (γ(θ, θ 2 ) -γ(θ, θ 1 )) (13)
with [START_REF] Zhilichev | Calculation of magnetic field of tubular permanent magnet assemblies in cylindrical bipolar coordinates[END_REF] with

γ(θ, θ i ) = h 1 η 1 (θ, θ i )Π * 2(c 1 + d 1 )f 2c 1 f - √ 2 d 2 1 f (-e + f ) , i sinh -1 -1 c 1 + d 1 c 1 -d 1 cos(θ -θ i ) , c 1 + d 1 c 1 -d 1 +h 2 η 1 (θ, θ i )Π * 2(c 1 + d 1 )f 2c 1 f + √ 2 d 2 1 f (-e + f ) , i sinh -1 -1 c 1 + d 1 c 1 -d 1 cos(θ -θ i ) , c 1 + d 1 c 1 -d 1 -h 3 η 2 (θ, θ i )Π * 2(c 2 + d 2 )f 2c 2 f - √ 2 d 2 2 f (-e + f ) , i sinh -1 -1 c 2 + d 2 c 2 -d 2 cos(θ -θ i ) , c 2 + d 2 c 2 -d 2 -h 4 η 2 (θ, θ i )Π * 2(c 2 + d 2 )f 2c 2 f + √ 2 d 2 2 f (-e + f ) , i sinh -1 -1 c 2 + d 2 c 2 -d 2 cos(θ -θ i ) , c 2 + d 2 c 2 -d 2 (14)
Parameters a 2(z -h)(r 2 + (z -h) 2 ) b 1 2(z -h)rr out c 1 r 2 + r 2 out + (z -h) 2 d 1 -2rr out e -r 2 -2(z -h) 2 f r 2 b 2 2(z -h)rr in c 2 r 2 + r 2 in + (z -h) 2 d 2 -2rr in
η 1 (θ, θ i ) = -i d 1 (-1+cos(θ-θ i )) c1-d1 d 1 (1+cos(θ-θ i )) c1+d1 1 cos(θ-θi) 2 -1 c 1 +d 1 d 2 1 f (-e + d 1 )(d 2 1 (e -f )) + 2c 2 1 f (15) η 2 (θ, θ i ) = -i d2(-1+cos(θ-θi)) c 2 -d 2 d2(1+cos(θ-θi)) c 2 +d 2 1 cos(θ-θ i ) 2 -1 c 2 +d 2 d 2 2 f (-e + d 2 )(d 2 2 (e -f )) + 2c 2 2 f ( 16
)
h 1 = 2ad 1 ( √ 2c 1 f + d 2 1 f (-e + f )) + b 1 ( √ 2d 2 1 (e -f ) -2c 1 d 2 1 f (-e + f )) (17)
h 2 = 2ad 1 (- √ 2c 1 f + d 2 1 f (-e + f )) + b 1 ( √ 2d 2 1 (-e + f ) -2c 1 d 2 1 f (-e + f )) (18)
h 3 = 2ad 2 ( √ 2c 2 f + d 2 2 f (-e + f )) + b 2 ( √ 2d 2 2 (e -f ) -2c 2 d 2 2 f (-e + f )) (19)
h 4 = 2ad 2 (- √ 2c 2 f + d 2 2 f (-e + f )) + b 2 ( √ 2d 2 2 (-e + f ) -2c 2 d 2 2 f (-e + f )) (20)
where Π * [n, φ, m] is given in terms of the incomplete elliptic integral of the third kind by [START_REF] Blache | High magnetic field gradients in flux confining permanent magnet structures[END_REF].

Π * [n, φ, m] = φ 0 1 (1 -n sin(θ) 2 ) 1 -m sin(θ) 2 dθ ( 21
)
Although the result H z (r, θ, z) is a real number, equation ( 14) contains the imaginary number i (i 2 = -1) because we did not succeed in obtaining a real expression for the axial component H z (r, θ, z). The parameters used in ( 14) are defined in Table I. However, as the imaginary part is the consequence of numerical noise and nearly equals zero, when the expression ( 14) is used in symbolic mathematical tools such as Mathematica or Maple, the real part of Hz(r, θ, z) only has to be considered. Equation ( 13) is valid for any observation point M (r, θ, z) with θ = θ i and 0 ≤ θ < 2π. Here again, this expression remains valid for ring permanent magnets, i.e. when the angular width is 2π (θ 2 -θ 1 = 2π). It also

r out = 28mm, θ 1 = 0 rad, θ 2 = π 2 rad, θ = π 2 rad
leads to the expression of the axial component already given by the authors for ring magnets [START_REF] Ravaud | Analytical calculation of the magnetic field created by permanent-magnet rings[END_REF]. Figure 4 represents the axial component H z (r, θ, z) versus the radial distance r of the observation point. The parameter values are h = 3 mm, θ 1 = 0 rad, θ 2 = π 2 rad, θ = 0 rad, r in = 25 mm, r out = 28 mm.

III. RADIALLY MAGNETIZED TILES

A. Notation and geometry

The geometry and its parameters are shown in Fig. [START_REF] Selvaggi | Calculating the external magnetic field from permanent magnets in permanent-magnet motors -an alternative method[END_REF]. The axis z is an axis of symmetry. Again, the coulombian model of permanent magnets is used. The permanent magnet ring sector is thus represented by two curved planes which correspond here to the inner and outer faces of the ring. The inner face is charged with a magnetic pole surface density +σ * ; the outer one is charged with the opposite magnetic pole surface density -σ * . We only consider the inner face to simplify the analytical calculation. As stated previously, the total magnetic field can be calculated by the application of the linear superposition principle to both faces.

The magnetic pole volume density is not taken into account in this paper. This means that the total sum of all the charges in the model does not equal zero. Indeed, as the magnetization is radial, the magnetic pole surface density of the curved planes is uniform. The charge on the outer plane is thus greater than the charge on the inner plane, as the surfaces of these planes. The volume charge density, linked to the magnetization divergence, appears in fact to set the global charge to zero. If the radial width of the tile is small, which also means that the tile is thin, then the difference between the inner and outer plane surface is small, and so is the magnetic pole volume density: its neglecting is an acceptable approximation. This approximation becomes less and less valid when the thickness of the tile increases. This paper presents expressions for radially magnetized thin tiles.

Let us consider a point P on the tile inner face. The magnetic field H created by the source point P (r, θ s , z) at any observation point M (r, θ, z) of the space is given by [START_REF] Zhu | Comparison of halbach magnetized brushless machines based on discrete magnet segments or a single ring magnet[END_REF].

H(r, θ, z) = σ * 4πµ 0 θ 2 θ 1 h 0 --→ P M --→ P M 3 r 1 dz 1 dθ s ( 22
)
where µ 0 is the magnetic permeability of the vacuum (µ 0 = 4π.10 -7 SI) and σ * is the fictitious magnetic pole surface density given in tesla . Equation ( 22) can be written as follows:

H(r, θ, z) = σ * 4πµ 0 θ 2 θ 1 h 0 (r -r 1 cos(θ -θ s )) u r -r 1 sin(θ s -θ) u θ + (z -z 1 ) u z (r 2 1 + r 2 + (z -z 1 ) 2 -2r 1 r cos(θ -θ s )) 3 2 r 1 dz 1 dθ s (23) B. Components along the three directions u r , u θ , u z
The integration of ( 23) leads to the magnetic field components created by the inner face along the three axes defined H r (r, θ, z), H θ (r, θ, z), H z (r, θ, z).

1) Azimuthal component H θ (r, θ, z): The field azimuthal component H θ (r, θ, z) is given by [START_REF] Li | Optimum design of magnet shape in permanent-magnet synchronous motors[END_REF].

H θ (r, θ, z) = σ 4πµ 0 (β(θ, θ 1 ) -β(θ, θ 2 )) (24)
with

β(θ, θ i ) = r in (b -z) r -(b -z) 2 arctan r 2 + r 2 in + (b -z) 2 -2rr in cos(θ i -θ) -(b -z) 2 - r in r tanh -1 r 2 + r 2 in + z 2 -2rr in cos(θ i -θ) z (25)
Equation ( 24) is valid for any observation point M (r, θ, z) with 0 ≤ θ ≤ 2π. Figure 6 represents the azimuthal component H θ (r, θ, z) versus the radial distance r of the observation point. The parameters values are h = 3mm,

θ 1 = 0, θ 2 = π 2 , θ = 0, r in = 25mm, r out = 28mm. 2) Radial component H r (r, θ, z):
The radial component of the field H r (r, θ, z) is given by [START_REF] Lemarquand | Nonsinusoidal torque of permanent-magnet couplings[END_REF].

H r (r, θ, z) = σ * 4πµ 0 (β(u 1 ) -β(u 2 )) (26)
u i = cos(θ -θ i ) (27)
and

β(u i ) =    2i(1 + u i ) d(-1+u i ) c+e1+dui (-(a 1 d + b 1 (c + e 1 ))) F * i sinh -1 [ √ -c+d-e1 √ c+e 1 +du i ], c+d+e1 c-d+e1 d √ -c + d -e 1 e 1 d(1+ui) c+e 1 +du i 1 -u 2 i    +    2i(1 + u i ) d(-1+u i ) c+e1+dui (b 1 c -a 1 d)Π * e1 c-d+e1 , i sinh -1 [ √ -c+d+e1 c+e1+dui ], c+d+e1 c-d+e1 d √ -c + d -e 1 e 1 d(1+ui) c+e 1 +du i 1 -u 2 i    +    2i(1 + u i ) d(-1+u i ) c+e2+dui (-(a 2 d + b 2 (c + e 2 ))) F * i sinh -1 [ √ -c+d-e2 √ c+e 2 +du i ], c+d+e2 c-d+e2 d √ -c + d -e 2 e 2 d(1+ui) c+e 2 +du i 1 -u 2 i    +    2i(1 + u i ) d(-1+u i ) c+e 2 +du i (b 2 c -a 2 d)Π * e2 c-d+e 2 , i sinh -1 [ √ -c+d+e2 c+e 2 +du i ], c+d+e2 c-d+e 2 d √ -c + d -e 2 e 2 d(1+ui) c+e 2 +du i 1 -u 2 i    (28)
where F * [φ, m] is defined by [START_REF] Lemarquand | Ironless loudspeakers[END_REF] and Π * [n, φ, m] is given by [START_REF] Blache | High magnetic field gradients in flux confining permanent magnet structures[END_REF]. Here again, Eq.( 28) contains the imaginary number i (i 2 = -1) although the result H r (r, z) is a real number because we did not succeed in obtaining a real expression for the radial component H r (r, θ, z). The parameters used in [START_REF] Charpentier | Mechanical behavior of axially magnetized permanent-magnet gears[END_REF] are defined in Table II. Of course, when using the expression (28) in tools like Mathematica or Mapple, the imaginary part of H r (r, θ, z) has to be dropped, as it only corresponds to numerical noise and nearly equals zero. Equation ( 26) can be used to calculate the radial component of the magnetic field at any observation point M (r, θ, z) with θ = θ i and 0 ≤ θ < 2π. When the tile angular width becomes 2π (θ 2 -θ 1 = 2π), so for ring magnets, the expression becomes the one already given in a previous paper [START_REF] Ravaud | Analytical calculation of the magnetic field created by permanent-magnet rings[END_REF]. Figure 7 represents the field radial component H r (r, θ, z) versus the radial distance r of the observation point. The parameter values are h = 3 mm, θ 1 = 0 rad, θ 2 = π 2 rad, θ = 0 rad, r in = 25 mm, r out = 28 mm. 3) Axial component H z (r, θ, z) : The field axial component H z (r, θ, z) is given by ( 29):

Parameters

a 1 r in rz b 1 -r 2 in z c r 2 + r 2 in d -2rr in e 1 z 2 a 2 -r in r(z -h) b 2 r 2 in (z -h) e 2 (z -h) 2
H z (r, θ, z) = σ * 4πµ 0 (α(θ, θ 2 ) -α(θ, θ 1 )] (29)
with

α(θ, θ i ) = 2r in (r -r in ) 2 + (z -h) 2 F * θ -θ i 2 , - 4rr in (r -r in ) 2 + (z -h) 2 - 2r in (r -r in ) 2 + z 2 F * θ -θ i 2 , - 4rr in (r -r in ) 2 + z 2 (30)
where F * [φ, m] is given in terms of the elliptic integral of the first kind by [START_REF] Lemarquand | Ironless loudspeakers[END_REF].

F * [φ|m] = θ=φ θ=0 1 1 -m sin(θ) 2 dθ ( 31
)
Equation ( 29) is valid for any observation point M (r, θ, z) with 0 ≤ θ ≤ 2π.

IV. CONCLUSION

This paper gives the analytical expressions of the three components of the magnetic field created by tile permanent magnets whose magnetization is either radial or axial. When the tiles are axially magnetized, the magnetic field can be calculated using these expressions at any point in space. For radially magnetized tiles, the expressions given correspond to the field created by thin tiles, whose inner and outer radii are not too different, and can be used at any point of the space as well. The magnetic field is directly calculated, without the help of the magnetic potentials. The utility of such analytical expressions for the three components of the magnetic field created by tile permanent magnets lies in the fact that the related calculations have a low computational cost, especialy with regard to methods using finite elements or finite differences, and that they allow efficient parametric optimization studies of devices, which is very important for all applications. The Mathematica files containing the analytical expressions used to calculate the three components of the magnetic field for axial and radial magnetizations are available line [36].

Fig. 1 .

 1 Fig. 1. Axially magnetized tile permanent magnet : parameter definition.

Figure 2

 2 represents the azimuthal component H θ (r, θ, z) versus the radial distance r of the obervation point. The parameter values are

Fig. 2 .

 2 Fig. 2. Field azimuthal component H θ (r, θ, z) versus the radial distance r of the observation point;h = 3mm, r in = 25mm, rout = 28mm, θ 1 = 0, θ 2 = π 2 , θ = π

Fig. 3 .

 3 Fig. 3. Field radial component Hr(r, θ, z) versus the radial distance r of the observation point; h = 3mm, r in = 25mm, r out = 28mm, θ 1 = 0 rad, θ 2 = π 2 rad, θ = π 2 rad

Fig. 4 .

 4 Fig. 4. Field axial component Hz(r, θ, z) versus the radial distance r of the observation point; h = 3mm, r in = 25mm, r out = 28mm, θ 1 = 0 rad, θ 2 = π 2 rad, θ = π 2 rad

Fig. 5 .

 5 Fig. 5. Radially magnetized permanent magnet tile: parameter definition

Fig. 6 .

 6 Fig. 6. Field azimuthal component H θ (r, θ, z) versus the radial distance r of the observation point; h = 3 mm, r in = 25 mm, rout = 28 mm, θ 1 = 0 rad, θ 2 = π 2 rad, θ = 0 rad

Fig. 7 .

 7 Fig. 7. Field radial component Hr(r, θ, z) versus the radial distance r of the observation point; h = 3 mm, r in = 25 mm, r out = 28 mm, θ 1 = 0 rad, θ 2 = π 2 rad, θ = π 2 rad

Fig. 8 .

 8 Fig. 8. Field axial component H z (r, θ, z) versus the radial distance r of the observation point; h = 3 mm, r in = 25 mm, rout = 28 mm, θ 1 = 0 rad, θ 2 = π 2 rad, θ = π 2 rad