Keywords: Torque transmission, axial magnetic coupling, analytical model, axial force

In this paper, a theoretical analysis of an axial magnetic coupling is presented, leading to new closed-form expressions for the magnetic axial-force and torque. These expressions are obtained by using a two-dimensional (2-D) approximation of the magnetic coupling geometry (mean radius model). The analytical method is based on the solution of Laplace's and Poisson's equations by the separation of variables method. The influence of geometrical parameters such as number of pole pairs and air-gap length is studied. Magnetic field distribution, axial force and torque computed with the proposed 2-D analytical model are compared with those obtained from 3-D finite elements simulations and experimental results.

I. INTRODUCTION

AGNETIC couplings are of great interest in many industrial applications. They can transmit a torque from a primary driver to a follower without mechanical contact. As the torque could be transmitted across a separation wall, axial field magnetic couplings are well suited for use in isolated systems such as vacuum or high pressure vessels. Moreover, they present a maximum transmissible torque (pull-out torque) giving an intrinsic overload protection.

Axial magnetic couplings consist of two opposing discs equipped with rare earth permanent magnets as shown in Fig. 1. The magnets are magnetized in the axial direction. They are arranged to obtain alternately north and south poles. The flux is closed by soft-iron yokes. The torque applied to one disc is transferred through an air-gap to the other disc. The angular shift between the two discs depends on the transmitted torque value. The main drawback of axial-type magnetic couplings is the significant value of the axial attractive force between the two discs.

An accurate knowledge of the magnetic field distribution is necessary for predicting the torque and the axial force. The magnetic field can be evaluated by analytical methods or by numerical techniques like finite elements [START_REF] Ferreira | Torque analysis of permanent magnet coupling using 2D and 3D finite elements methods[END_REF][START_REF] Wu | Analysis and design optimisation of magnetic couplings using 3D finite element modelling[END_REF][START_REF] Wang | Design and analysis of a permanent magnet axial coupling using 3D finite element field computations[END_REF][START_REF] Chan | Performance of an Axial-Flux Permanent Magnet Synchronous Generator From3-D Finite-Element Analysis[END_REF].

Finite elements simulations give accurate results considering three dimensional (3-D) effects and nonlinearity of magnetic materials. However, this method is computer time consuming Manuscript received October 31, 2011. T. Lubin, S. Mezani and A. Rezzoug are with the Groupe de Recherche en Electrotechnique et Electronique de Nancy (GREEN), Université Henri Poincaré, 54506 Nancy, France (e-mail: thierry.lubin@green.uhp-nancy.fr). Analytical methods are, in general, less computational time consuming than numerical ones and can provide closed-form solutions giving physical insight for designers. So, they are useful tools for first evaluations of magnetic couplings performances and for the first step of design optimization.

Three-dimensional analytical models for ironless permanent magnet couplings have been proposed in the literature [START_REF] Yonnet | Permanent magnet bearings and couplings[END_REF][START_REF] Furlani | Formulas for the force and torque of axial couplings[END_REF][START_REF] Yonnet | Analytical calculation of permanent magnet couplings[END_REF][START_REF] Furlani | A three-dimensional field solution for radially polarized cylinders[END_REF][START_REF] Furlani | A three-dimensional model for computing the torque of radial couplings[END_REF][START_REF] Yao | Theoretical computations for the torque of magnetic coupling[END_REF][START_REF] Waring | An investigation of face type magnetic couplers[END_REF][START_REF] Furlani | Analysis and optimization of synchronous couplings[END_REF][START_REF] Furlani | A three-dimensional field solution for permanent-magnet axial-field motors[END_REF][START_REF] Elies | Analytical optimization of the torque of a permanent-magnet coaxial synchronous coupling[END_REF][START_REF] Charpentier | Optimal design of cylindrical air-gap synchronous permanent magnet couplings[END_REF][START_REF] Charpentier | Study of ironless permanent magnet devices being both a coupling and an axial bearing for naval propulsion[END_REF][START_REF] Rakotoarison | Using Coulombian approach for modelling scalar potential and magnetic field of a permanent magnet with radial polarization[END_REF][START_REF] Ravaud | Comparison of the Coulombian and Amperian current models for calculating the magnetic field produced by arc-shaped permanent magnets radially magnetized[END_REF][START_REF] Ravaud | Permanent magnet couplings: Field and torque three-dimensional expressions based on the Coulombian model[END_REF][START_REF] Ravaud | Analytical design of permanent magnet radial couplings[END_REF]. The proposed models are developed for axial magnetic couplings with parallelepiped magnets or cylindrical tile magnets. As the magnets are in free space (with no other magnetic materials present), analysis is based either on the amperian model with Biot-Savart law or on the coulombian method with equivalent surface charges. Although these methods give very accurate results, they are not suitable for the study of magnetic couplings with iron-core structures.

An alternative analytical method to compute the torque for magnetic couplings with iron yokes is based on boundary value problems with Fourier analysis. This method consists in solving directly the Maxwell's equations in the different regions (air-gap, magnets....) by the separation of variables method [START_REF] Gysen | General formulation of the electromagnetic field distribution in machines and devices using Fourier analysis[END_REF], [START_REF] Lubin | Exact analytical method for magnetic field computation in the air-gap of cylindrical electrical machines considering slotting effects[END_REF]. The magnetic field distribution is obtained in each region by using boundary and interface conditions. The torque and the force are then computed by using the Maxwell stress tensor. In [START_REF] Hornreich | Optimal design of synchronous torque couplers[END_REF] and [START_REF] Fontchastagner | Some co-axial magnetic couplings designed using an analytical model and an exact global optimization code[END_REF], two-dimensional (2-D) analytical models for radial-type magnetic couplings were developed and closed-form expressions for the torque was given and used for design optimization. In [START_REF] Parviainen | Modeling of axial flux permanent-magnet machines[END_REF] and [START_REF] Azzouzi | Quasi-3-D analytical modeling of the magnetic field of an axial flux permanent-magnet synchronous machine[END_REF], quasi 3-D analytical models are proposed to compute the performances of axial-flux permanent magnets machines. A Simple Analytical Expressions for the Force and Torque of Axial Magnetic Couplings Thierry Lubin, Smail Mezani, and Abderrezak Rezzoug M modulation function is defined to take into account the radial dependence of the magnetic field.

In this paper, we propose new formulas for the torque and the axial force of an axial-type magnetic coupling with iron yokes (fig. 1). The analytical study is based on the solution of 2-D Laplace's and Poisson's equations in air-gap and permanent magnets regions by using the separation of variables method. The torque expression is used to study the influence of geometrical parameters (number of pole pairs and air-gap length). In order to study the accuracy of the proposed formulas, the results are compared with those obtained from 3-D finite elements simulations and experimental results.

II. PROBLEM DESCRIPTION AND ASSUMPTIONS

As shown in Fig. 1, the geometrical parameters of the studied magnetic coupling are the inner and outer radii of the magnets R 1 and R 2 , the air gap length e, and the magnets thickness h. The pole-arc to pole-pitch ratio of the permanent magnets is α. The number of pole-pairs is p.

Analytical study of axial magnetic couplings is complicated because of the three-dimensional nature of the magnetic field distribution. However, in order to simplify the analysis and to carry out closed-form expressions for the axial force and torque, the 3-D problem is reduced to a 2-D one by introducing a cylindrical cutting surface at the mean radius of the magnets R e =(R 1 +R 2 )/2 at which the magnetic field will be computed [START_REF] Parviainen | Modeling of axial flux permanent-magnet machines[END_REF], [START_REF] Azzouzi | Quasi-3-D analytical modeling of the magnetic field of an axial flux permanent-magnet synchronous machine[END_REF].

Fig. 2 shows the resulting 2-D model by considering the unrolled cylindrical cutting surface. With this approach, we neglect the radial component of the magnetic field and we consider that the axial and tangential components do not depend on the r-coordinate. Moreover, for simplicity, we adopt the following assumptions:

1) The iron yokes have infinite magnetic permeability, 2) The magnets are axially magnetized with relative recoil permeability 1 = r µ . As shown in Fig. 2, the whole domain is divided into three regions: the PMs regions (regions I and III) and the air-gap region (region II). The magnets of region III are shifted by an angle δ (torque angle) from the magnets of region I. Due to the periodicity of the magnetic field distribution, the studied domain is limited by 0 ≤ θ ≤ 2π/p.

A magnetic vector potential formulation is used in 2D cylindrical coordinates to describe the problem. According to the adopted assumptions, the magnetic vector potential in each region has only one component along the r-direction and only depends on the θ and z-coordinates. The electromagnetic equations in each region expressed in term of the magnetic vector potential are

µ  ∇ = -∇ ×   ∇ =   A M A (1)
with 0 r z B M µ = = ± z z M e e ( 2
)
where M is the magnetization vector, B r the remanence of the magnets, e z the unit vector along the axial direction and ± indicates the magnetization direction.

III. 2-D ANALYTICAL MODEL

By using the separation of variables method, we now consider the solution of Poisson's equations for PMs regions and Laplace's equation for the air-gap region.

A. Solution of Poisson's Equation in the PMs Regions (Regions I and III)

Poisson's equation in the magnets region (region III) can be written in a cylindrical coordinates system as

+ ≤ ≤ +   ≤ ≤  ( 3
)
where µ 0 is the permeability of the vacuum and M z is the axial magnetization of the magnets. Knowing that the tangential component of magnetic field at 2 z h e = + is null (soft-iron yoke with infinite permeability) and considering the continuity of the axial component of the flux density at z h e = + , we obtain the following boundary

conditions 2 0 III z h e A z = + ∂ = ∂ (4) ( , ) ( , ) III II A h e A h e θ θ + = + (5)
where ( , )

II A z
θ is the magnetic vector potential in the air-gap region.

The distribution of the axial magnetization M z is plotted in Fig. 3, δ is the relative angular position between the magnets of region I and region III. The axial magnetization can be expressed in Fourier's series and replaced in (3) ( )

( ) 1 ( ) sin z k k M M kp θ θ δ ∞ = = - ∑ (6)
( )

0 4 cos 1 with 1,3,5, 7.... 2 r k B M k k k π α πµ   = - =     (7)
Taking into account the boundary conditions ( 4) and ( 5), the general solution of the magnetic vector potential in Region III can be written as

( ) ( ) ( ) ( ) 1 1 ( , ) 2 ( cos( )) cos 2 ( sin(
)) sin

III e III k k k e e III k k k e A z kp ch z h e R a K kp kp kp ch h R kp ch z h e R c K kp kp kp ch h R θ δ θ δ θ ∞ = ∞ = =   --     +         --     + +       ∑ ∑ (8) with 0 2 e k k R K M kp µ = (9)
The integration constants III k a and III k c are determined using a Fourier series expansion of ( , )

II A h e θ + over the interval [0, 2π/p] 2 / 0 2 cos( ) ( , ) cos( ) 2 p III k k II p a K kp A h e kp d π δ θ θ θ π + = + ∫ (10) 2 / 0 2 sin( ) ( , )sin( ) 2 p III k k II p c K kp A h e kp d π δ θ θ θ π + = + ∫ (11)
The expressions of the coefficients III k a and III k c are given in the appendix.

One can apply the same procedure for region I by considering a zero value for δ. This leads to the following expression for the magnetic vector potential

( ) ( ) 1 1 ( , ) ( ) cos sin e I I k k k e e I k k e kp ch z R A z a K kp kp ch h R kp ch z R c kp kp ch h R θ θ θ ∞ = ∞ =       = +             +       ∑ ∑ (12)
The integration constants I

B. Solution of Laplace's Equation in the Air-Gap Region (Region II)

Laplace's equation in the air-gap region can be written in a cylindrical coordinates system as 1 0

II II e A A R z θ ∂ ∂ + = ∂ ∂ for 0 2 / h z h e p θ π ≤ ≤ +   ≤ ≤  ( 15
)
The continuity of the tangential component of the magnetic field at z h = and at z h e = + leads to the following boundary conditions

II I z h z h A A z z = = ∂ ∂ = ∂ ∂ and II III z h e z h e A A z z = + = + ∂ ∂ = ∂ ∂ (16)
By taking into account the boundary conditions ( 16), the general solution of the magnetic vector potential in the air-gap can be written as

( ) ( ) ( ) ( ) ( ) ( ) 1 1 ( , ) ( ) cos
∞ = ∞ = =     -- -         - +                 -- -         + - +             ∑ ∑ (17)
The integration constants

p II I k h A p a kp d z π θ θ π ∂ = ∂ ∫ (18) 2 / 0 2 cos( ) 2 p II III k h e A p b kp d z π θ θ π + ∂ = ∂ ∫ (19) 2 / 0 2 sin( ) 2 p II I k h A p c kp d z π θ θ π ∂ = ∂ ∫ (20) 2 / 0 2 sin( ) 2 p II III k h e A p d kp d z π θ θ π + ∂ = ∂ ∫ (21)
The expressions of these coefficients are developed in the appendix.

The axial and tangential components of the magnetic flux density in the air-gap can be deduced from the magnetic vector potential by

1 II IIz e A B R θ ∂ = - ∂ II II A B z θ ∂ = ∂ (22)

IV. AXIAL-FORCE AND TORQUE EXPRESSIONS

A. Electromagnetic torque

The electromagnetic torque is obtained using the Maxwell stress tensor. A line at

[ ]

, z h h e ζ = ∈
+ in the air-gap region is taken as the integration path so the electromagnetic torque is expressed as follows

2 3 3 2 1 0 0 ( , ) ( , ) 3 e II IIz R R T B B d π θ θ ζ θ ζ θ µ - = ∫ (23)
Incorporating ( 22) into [START_REF] Ferreira | Torque analysis of permanent magnet coupling using 2D and 3D finite elements methods[END_REF], the analytical expression for the electromagnetic torque becomes ( )

3 3 2 1 0 1 ( ) 3 e k k k k k R R T W X Y Z π µ ∞ = - = + ∑ ( 24
)
where the coefficients W k , X k , Y k and Z k are given in the appendix.

The torque can be computed with a good precision by considering only the fundamental components of the the flux density distribution in the air-gap (k = 1). This is especially true for large number of PM pole-pairs and/or large air-gap.

Considering the first harmonic approximation, we can derive a closed-form expression for the electromagnetic torque which depends directly on the geometrical parameters.

( ) ( ) ( )

3 2 2 3 2 1 2 0 2 16 1 sin 3 2 2 1 r e B R sh a T R sin p sh a R π α δ π µ ν         = -        +       (25) with e h a p R = and 2
e h ν = (26)
As expected, the torque presents a sinusoidal characteristic with the relative angular position δ. Its maximum value (pullout torque) is obtained at the angle δ=π/2p.

B. Axial-Force

Axial magnetic force is an important parameter for the design of an axial magnetic coupling. This attractive force must be known because it affects directly the rotor structure and bearings. Indeed, the bearing lifetime depends on the bearing load. By using the Maxwell stress tensor, the axial force expression is ( )

2 2 2 2 2 2 1 0 0 ( , ) ( , ) 4
IIz II R R F B B d π θ θ ζ θ ζ θ µ - = - ∫ (27)
Substituting ( 22) into (27), the analytical expression for the axial force becomes

( ) ( ) ( ) ( ) 2 2 2 1 2 2 0 1 4 k k k k k R R F Z X W Y π µ ∞ = - = + - + ∑ (28)
Considering only the fundamental component of the magnetic field in the air-gap (k = 1), we can derive a closedform expression for the axial force

( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 1 2 2 0 2 8 1 sin 2 2(1 ) cos 2(1 ) 1 r sh a B R F R R sh a p ch a π α π µ ν δ ν         = -        +       × + + (29)
From ( 25) and (29), we can see that the torque and the axial force dependence on the design parameters are explicit. For engineering purpose, it is important to have simple relations to study rapidly the effects of the geometrical parameters on the coupling performances. This is developed in the following subsection.

V. RESULTS OBTAINED WITH 2-D ANALYTICAL MODEL

In this section, we use the proposed 2-D analytical model to compute the magnetic field distribution in the air-gap for different angular position between the two discs. For each position, the torque and the axial force are calculated by respectively using ( 25) and (29). Then, the influence of some geometrical parameters on the coupling performances is investigated (particularly the air-gap length and the pole-pairs number). The geometrical parameters of the studied device are given in Table I. These parameters correspond to the one which give a pull-out torque of around 90 Nm (obtained using ( 25)) when we consider an air-gap length of 3 mm and a 6 pole-pairs.

A. Flux density distribution and torque calculation for e = 3mm and p=6

Figs. 4a and4b show respectively the flux lines (for two pole pitches) and the axial component of the flux density in the middle of the air-gap under no-load condition (δ = 0°). The magnetic flux density distribution along the air-gap is computed by using ( 17) and [START_REF] Azzouzi | Quasi-3-D analytical modeling of the magnetic field of an axial flux permanent-magnet synchronous machine[END_REF]. We can observe that the flux lines are almost axial along the air-gap (the tangential component of the flux density is null in the middle of the airgap). For this position, the torque is then equal to zero and the axial force is attractive and reaches its maximum value. By using (29), we obtain F ≈ 2500N as shown in Fig. 8. 5b and Fig. 5c show respectively the axial and the tangential components of the flux density in the middle of the air-gap. For this angular position, the torque reaches its maximum value T e = 94 Nm (pull-out torque) as indicated in fig. 7 and the axial force is still attractive (F = 423N) as shown in fig. 8. As it can be observed in fig. 8, the axial force reaches a null value for an angle slightly higher to half the pole pitch (around 17°). Fig. 7 and Fig. 8 summarize the variation of torque and axial force as a function of the angular displacement δ. As shown previously, the maximum torque occurs at an angular shifting of half pole pitch angle. We can observe that the first harmonic approximation gives accurate results (the error is less than 5%) compared to the ones obtained by taking into account 10 harmonic terms in ( 24) and ( 28).

B. Influence of the air-gap length

The length of the air-gap has a significant influence on the characteristics of the axial magnetic coupling. Fig. 9 and Fig. 10 show respectively the pull-out torque and the maximal axial force as a function of the air-gap length. The results have been computed by using ( 25) and (29). The geometrical parameters are the ones given in table I and we have considered a number of pole pairs p = 6. As shown in fig. 9, the pull-out torque of the magnetic coupling decreases quickly as the distance between the magnets increases. The maximum torque is almost divided by two when the air-gap is increased from 2mm to 7mm. In the same way, the maximal axial force is reduced when the air-gap length increases (fig. 10).

C. Influence of the number of pole pairs

The variation of pull-out torque and maximal axial force versus the number of pole pairs are respectively shown in fig. 11 and fig. 12. The results have been computed by using ( 25) and (29). For the study, we have considered several air-gap lengths. The other geometrical parameters are those given in Table I. Fig. 11 shows that all the curves present a maximum which depends on the air-gap length. The optimum value of the number of pole pairs is shifted to the right when the air-gap is reduced. This result is well known for magnetic couplings. For the studied coupling (Table I), the optimal number of pole pairs is p = 6 if we consider an air-gap length e = 5mm. We can observe in Fig. 12 that the maximum axial force decreases when the number of pole pairs increases.

We have shown here that the torque formula (25) can predict the effects of the geometrical parameters on the coupling performances and from fig. 11, we can choose rapidly the optimum value of the number of pole pairs when the other geometrical parameters are given.

In the next subsection, we investigate the precision of the 2-D approximation [START_REF] Wang | Design and analysis of a permanent magnet axial coupling using 3D finite element field computations[END_REF], by comparing the previous analytical results with 3-D FEM simulations and experimental results.

VI. 3-D FEM SIMULATIONS AND EXPERIMENTAL RESULTS

In order to show the limits of the formulas ( 25) and (29), the analytical results have been compared to 3-D FEM simulations in one hand and to experimental results in another hand. For the 3-D finite element simulations, we have used COMSOL multiphysics software.

For the experimental validation, we have manufactured an axial magnetic coupling prototype using sector type NdFeB magnets glued on iron yokes. The geometrical parameters of the prototype are those of Table I. We choose a number of pole pairs p = 6 that corresponds practically to the optimum value for an air-gap value e = 5mm. Fig. 13 shows the axial magnetic coupling placed on the test bench. The axial coupling is inserted between two electrical machines. In fig. 13, the air-gap value is e = 9.5mm.

A. 3-D FEM results

Fig. 14 and Fig. 15 show respectively the pull-out torque and the maximal axial force as a function of the air-gap length obtained with 3-D finite elements analysis and with the 2-D analytical model. The number of harmonic terms used in [START_REF] Wu | Analysis and design optimisation of magnetic couplings using 3D finite element modelling[END_REF] and ( 28) is N = 10. The geometrical parameters are those given in Table I. For this study, the pole pair number is fixed to p = 6. As expected for this type of device, the 2-D analytical prediction gives higher values for both pull-out torque and maximal axial force as compared to 3-D FE analysis. This is mainly due to the 3-D effects which are not taken into account in the proposed model (the radial dependence of the magnetic field is not considered). The error on the pull-out torque prediction ranges from 22% for e=2mm to 37% for e=12mm. The error on the maximal axial force prediction is less important and ranges from 8% for e=2mm to 34% for e=12mm.

Figs. [START_REF] Ravaud | Analytical design of permanent magnet radial couplings[END_REF] show the pull-out torque versus the number of pole pairs computed with 3 methods (( 24), (25) and3-D FEM). The results are given for three values of the air-gap lengths. As it can be observed in Figs. [START_REF] Ravaud | Analytical design of permanent magnet radial couplings[END_REF], although the analytical formula [START_REF] Wang | Design and analysis of a permanent magnet axial coupling using 3D finite element field computations[END_REF] predicts higher torque values, the number of pole pairs which corresponds to the maximal value of the torque is almost the same for the 3 methods in use. Fig. 17 presents the synthesis of Fig. 16 and gives the optimal value of the pole-pair number versus air-gap lengths. This is an important result since we can observe in fig. 17 that the analytical formula [START_REF] Wang | Design and analysis of a permanent magnet axial coupling using 3D finite element field computations[END_REF] is suitable in the determination of the optimum value of the pole-pair number with the air-gap value when the other geometrical parameters are fixed.

B. Experimental results

Fig. 18 compares the measured values of the axial flux density and the ones obtained with the proposed 2D analytical model for no load condition (δ=0). The measurements are made along the θ coordinate at the mean radius of the magnets R e = 45 mm. For this test, the air-gap is fixed at e = 9.5mm. A Hall probe placed on a XY table is used to measure the magnetic field distribution. As the magnetic flux density is measured at the mean radius R e , we can observe very good agreement between experimental results and the ones obtained with the 2-D analytical model.

To show the limits of the 2-D analytical model, we have measured the radial dependence of the axial flux density at a center line of a pole for no-load condition (δ=0). The air-gap is fixed at e = 9.5mm. The results are shown in fig. 19. As it can be observed, the axial flux density shows large variations along the radial expanse of the magnet. This is due to the large value of the air-gap. This result can not be predicted by the 2-D analytical model which neglects the radial dependence of the magnetic field. We can note a good agreement between 3-D FEM simulations and experimental results. Although the proposed 2D analytical model shows some lack of accuracy compared to 3D finite-element simulations and experimental results (error of around 30% on the pull-out torque prediction), we have shown that it can be used to determine rapidly the optimal value of the pole-pair number when the other geometrical parameters are given.

Moreover, the proposed analytical formulas can be useful tools for the first step of design optimization since continuous derivatives issued from the analytical expressions are of great importance in most optimization methods.

APPENDIX

• The development of [START_REF] Elies | Analytical optimization of the torque of a permanent-magnet coaxial synchronous coupling[END_REF] and [START_REF] Charpentier | Optimal design of cylindrical air-gap synchronous permanent magnet couplings[END_REF] The development of ( 13) and ( 14) gives

    -- -         = - +                 -- -         = -               -- -     = - +       ( ) ( ) (
                -- -         = - +             (A.6)

Fig. 1 .

 1 Fig. 1. Geometry of the studied axial-type magnetic coupling (p = 6) and poorly flexible for the first step of design stage.Analytical methods are, in general, less computational time consuming than numerical ones and can provide closed-form solutions giving physical insight for designers. So, they are useful tools for first evaluations of magnetic couplings performances and for the first step of design optimization.Three-dimensional analytical models for ironless permanent magnet couplings have been proposed in the literature[START_REF] Yonnet | Permanent magnet bearings and couplings[END_REF][START_REF] Furlani | Formulas for the force and torque of axial couplings[END_REF][START_REF] Yonnet | Analytical calculation of permanent magnet couplings[END_REF][START_REF] Furlani | A three-dimensional field solution for radially polarized cylinders[END_REF][START_REF] Furlani | A three-dimensional model for computing the torque of radial couplings[END_REF][START_REF] Yao | Theoretical computations for the torque of magnetic coupling[END_REF][START_REF] Waring | An investigation of face type magnetic couplers[END_REF][START_REF] Furlani | Analysis and optimization of synchronous couplings[END_REF][START_REF] Furlani | A three-dimensional field solution for permanent-magnet axial-field motors[END_REF][START_REF] Elies | Analytical optimization of the torque of a permanent-magnet coaxial synchronous coupling[END_REF][START_REF] Charpentier | Optimal design of cylindrical air-gap synchronous permanent magnet couplings[END_REF][START_REF] Charpentier | Study of ironless permanent magnet devices being both a coupling and an axial bearing for naval propulsion[END_REF][START_REF] Rakotoarison | Using Coulombian approach for modelling scalar potential and magnetic field of a permanent magnet with radial polarization[END_REF][START_REF] Ravaud | Comparison of the Coulombian and Amperian current models for calculating the magnetic field produced by arc-shaped permanent magnets radially magnetized[END_REF][START_REF] Ravaud | Permanent magnet couplings: Field and torque three-dimensional expressions based on the Coulombian model[END_REF][START_REF] Ravaud | Analytical design of permanent magnet radial couplings[END_REF]. The proposed models are developed for axial magnetic couplings with parallelepiped magnets or cylindrical tile magnets. As the magnets are in free space (with no other magnetic materials present), analysis is based either on the amperian model with Biot-Savart law or on the coulombian method with equivalent surface charges. Although these methods give very accurate results, they are not suitable for the study of magnetic couplings with iron-core structures.An alternative analytical method to compute the torque for magnetic couplings with iron yokes is based on boundary value problems with Fourier analysis. This method consists in solving directly the Maxwell's equations in the different regions (air-gap, magnets....) by the separation of variables method[START_REF] Gysen | General formulation of the electromagnetic field distribution in machines and devices using Fourier analysis[END_REF],[START_REF] Lubin | Exact analytical method for magnetic field computation in the air-gap of cylindrical electrical machines considering slotting effects[END_REF]. The magnetic field distribution is obtained in each region by using boundary and interface conditions. The torque and the force are then computed by using the Maxwell stress tensor. In[START_REF] Hornreich | Optimal design of synchronous torque couplers[END_REF] and[START_REF] Fontchastagner | Some co-axial magnetic couplings designed using an analytical model and an exact global optimization code[END_REF], two-dimensional (2-D) analytical models for radial-type magnetic couplings were developed and closed-form expressions for the torque was given and used for design optimization. In[START_REF] Parviainen | Modeling of axial flux permanent-magnet machines[END_REF] and[START_REF] Azzouzi | Quasi-3-D analytical modeling of the magnetic field of an axial flux permanent-magnet synchronous machine[END_REF], quasi 3-D analytical models are proposed to compute the performances of axial-flux permanent magnets machines. A

2 0 2 in

 22 Regions I and III (PMs) 0 in Region II (air-gap)

Fig. 2 . 2 -

 22 Fig. 2. 2-D model of the axial magnetic coupling at the mean radius of the magnets Re =(R1+R2)/2.

Fig. 3 .

 3 Fig. 3. Magnetization distribution along θ-direction (region III).

Fig. 4 .

 4 No load condition (δ = 0°): (a) magnetic flux lines, (b) axial component of the flux density in the middle of the air-gap.

Fig. 5

 5 Fig. 5 corresponds to the full load condition (δ = 15°). We can observe clearly on Fig. 5a the distortion of the flux lines due to the angular displacement of the upper magnets. Fig. 5b and Fig. 5c show respectively the axial and the tangential components of the flux density in the middle of the air-gap.For this angular position, the torque reaches its maximum value T e = 94 Nm (pull-out torque) as indicated in fig.7and the axial force is still attractive (F = 423N) as shown in fig.8. As it can be observed in fig.8, the axial force reaches a null value for an angle slightly higher to half the pole pitch (around 17°).

Fig. 5 .

 5 Fig. 5. Full load condition (δ = 15°): (a) magnetic flux lines, (b) axial component of the flux density, (c) tangential component of the flux density.

Fig. 6

 6 Fig. 6 corresponds to an angular displacement δ = 30° (unstable position). In this position, the magnets of the two discs are in opposed direction and the flux lines repel each other. The flux density presents only a tangential component in the middle of the air-gap. The torque is then equal to zero and the axial force is now repulsive as shown in fig. 8. The axial force value computed with (29) gives F = -1628 N.Fig.7and Fig.8summarize the variation of torque and axial force as a function of the angular displacement δ. As shown previously, the maximum torque occurs at an angular shifting of half pole pitch angle. We can observe that the first harmonic approximation gives accurate results (the error is less than 5%) compared to the ones obtained by taking into account 10 harmonic terms in (24) and (28).

Fig. 6 .

 6 Fig. 6. Magnets in opposed direction (δ = 30°): (a) magnetic flux lines, (b) tangential component of the flux density in the middle of the air-gap.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Torque versus the angular displacement δ for e = 3mm and p = 6.

Fig. 9 .

 9 Fig. 9. Pull-out torque versus air-gap length for p = 6.

Fig. 10 .

 10 Fig. 10. Maximum axial-force versus air-gap length for p = 6.

Fig. 12 .

 12 Fig. 12. Maximum axial force versus the number of pole pairs for several airgap values.

Fig. 13 .

 13 Fig. 13. Axial magnetic coupling prototype placed on the test bench (e = 9.5mm).

Fig. 14 .

 14 Fig. 14. Pull-out torque versus the air-gap length for p = 6: 3-D FEM and 2-D analytical results.

Fig. 15 .Fig. 16 .

 1516 Fig. 15. Maximum axial-force versus the air-gap length for p = 6, 3-D FEM and 2-D analytical results.

Fig. 17 .

 17 Fig. 17. Optimal value of the pole-pair number versus air-gap dimension computed with 3-D FEM and 2-D analytical model (25).

Fig. 18 .

 18 Fig. 18. Measured and computed (2D analytical model) axial flux density in the middle of the air-gap at the mean radius Re =(R1+R2)/2 for e = 9.5 mm.

Fig. 19 .

 19 Fig. 19. Measured and computed axial flux density along a radial lines in the middle of the air-gap at a center line of a pole for e = 9.5 mm.

Figs. 20

 20 Figs.[START_REF] Fontchastagner | Some co-axial magnetic couplings designed using an analytical model and an exact global optimization code[END_REF] show the comparison between the measured values of the static torque and the calculated ones by using the 2-D analytical model (25) and 3-D FEM. The static torque was measured by suspending weights from a wire attached to a rod (a rotor is locked and the other one can rotate). The relative angular position δ was measured using an encoder with a resolution of 4096 steps per revolution (0.088 degree). Two values of the air-gap dimension were considered (e = 4mm and e = 9.5mm). It can be noticed that the experimental measurements are in good agreement with the results obtained with 3-D FE simulations. As shown previously, analytical formula[START_REF] Wang | Design and analysis of a permanent magnet axial coupling using 3D finite element field computations[END_REF] gives higher values of around 30% for the pullout torque.

Fig. 20 .

 20 Measured and computed static torque versus the angular displacement δ for p = 6: (a) e = 4mm; (b) e = 9.5mm VII. CONCLUSION In this paper, we have proposed new simple analytical expressions for computing axial force and torque of an axial magnetic coupling. These expressions are determined by the solution of 2-D Laplace's and Poisson's equations (mean radius model) in the different regions (air-gap and magnets).

  gives

TABLE I PARAMETERS

 I

		OF THE STUDIED AXIAL COUPLING
	Symbol	Quantity	value
	R1	Inner radius of the magnets	30 mm
	R2	Outer radius of the magnets	60 mm
	h	Magnets thickness	7 mm

e Air-gap length variable α PMs pole-arc to pole-pitch ratio 0.9 p Pole-pairs number variable Br Remanence of the permanent magnets 1.25 T

  Expression of the coefficients W k , X k , Y k and Z k given in (24) and (28)

			I k a K +	k	= -	II k a	e R kp	ch sh	     	e e kp R kp R	e e	     	+	II k b	e R kp	sh	  	1 kp e R	e	  	(A.3)
			I k c	= -	II k c	e R kp	ch sh	     	e e kp R kp R	e e	     	+	d	II k	e R kp	sh	  	1 kp e R	e	  	(A.4)
	• Expressions of the coefficients II k a , II k b , II k c and II k d for the
	air-gap region
	The development of (18) to (21) gives
	a	II k	=	I k a	e kp R	th	  	e kp R	h	  	II k b	= -	III k a	e kp R	th	  	e kp R	h	  
	II k c	=	I k c	e kp R	th	  	e kp R	h	  	d	II k	= -	III k c	e kp R	th	  	e kp R	h	  	(A.5)
	• ( e II k k kp sh R W a sh	e kp R ζ	h e e	)	II k b	sh	e kp sh R	( R e kp ζ	e	h	)
	X	k	II k c	ch		( sh e kp R	e kp R ζ	h e e	)	d	II e k kp R	ch	e kp R sh	( R e kp ζ	e	h	)
	k Y			II k c	sh		( sh e kp R	e R kp ζ	h e e	)	d	II e k kp R	sh	e kp R	ζ

  )

														h
										sh	e kp R	e
	Z	k	II k a	ch	e kp R sh	e kp R ζ	h e e	II k b	ch	e kp sh R	e R kp ζ	e	h


Thierry Lubin

Smail Mezani

Abderrezak Rezzoug

2D Analytical Calculation of Magnetic Field and Electromagnetic Torque for Surface-Inset Permanent Magnet Motors

Keywords: Analytical calculation, Magnetic field distribution, Surface-inset permanent magnet machine, Reluctance torque

The paper deals with a subdomain model for predicting the magnetic field distribution in surface-inset permanent-magnet motors with semi-closed slots under load conditions. Due to the presence of electrical current in the stator slots, a magnetic vector potential formulation is used. The magnetic vector potential distribution in each subdomain (air-gap, stator slots, and rotor slots with inset PMs) is obtained by solving two-dimensional Laplace's and Poisson's equations by the separation of variables method. One of the main contributions of the paper concerns the magnetic vector potential expression in the PMs region. Indeed, PMs are inset into the rotor iron which leads to solve Laplace's equation with nonhomogeneous Neumann boundary conditions. Magnetic field distribution and electromagnetic torque computed with the proposed analytical method are verified with those obtained from finite element analyses.

I. INTRODUCTION

NALYTICAL models for electrical machines and actuators are based on simplified geometries and often consider infinite permeability for the soft-magnetic material. However, they present less computational time consuming compared to numerical models and provide closed-form solutions giving physical insight for designers. Analytical models are very flexible to compare rapidly different machine topologies and can be useful tools in the first step of design optimization process.

Several models were proposed recently for computing the air-gap magnetic field distribution in PMs motors considering slotting effects. Two analytical methods are mainly developed. The first one is based on Schwarz-Christoffel conformal mapping and provides a 2-D relative permeance function to account for slotting effect [START_REF] Zhu | Instantaneous magnetic-field distribution in brushless permanent-magnet dc motor, part III: Effect of slotting[END_REF]- [START_REF] Boughrara | Magnetic field analysis of inset and surface-mounted permanentmagnet synchronous motor using Schwarz-Christoffel transformation[END_REF]. The actual air-gap flux density is calculated by multiplying the relative permeance function by the radial flux density of a slotless motor. The second one, called the subdomain method, consists in solving directly the Maxwell's equations in the different subdomains i.e. air-gap, stator slots and magnets, by the separation of variables method [START_REF] Gu | Effect of slotting in PM electrical machines[END_REF]- [START_REF] Zhu | An accurate subdomain model for magnetic field computation in slotted surface-mounted permanent magnet machines[END_REF]. The magnetic field distribution is obtained in each subdomain by using boundary and interface conditions.

In this paper, we present a subdomain model to calculate the magnetic field distribution in surface-inset PMs motors with semi-closed slots under load conditions (Fig. 1). The proposed model can be used for PMs motors with any pole and slot number combinations including fractional-slot machines. Due to the presence of electrical current in the stator slots, a magnetic vector potential formulation is used.

Although some papers were published on the analytical calculation of the magnetic field in surface-inset PMs motors, none was concerned with the analysis of the magnetic field under load conditions. Previous analytical models deal only with no-load conditions. They are all based on a magnetic scalar potential formulation [START_REF] Sargos | Analytical calculation of airgap magnetic field produced by inset permanent magnet rotor machine[END_REF]- [START_REF] Rahideh | Analytical magnetic field distribution of slotless brushless machines with inset permanent magnets[END_REF].

This paper extends the model that was presented recently by the authors [START_REF] Lubin | 2-D Exact analytical model for surface-mounted permanent-magnet motors with semi-closed slots[END_REF] to a machine with a surface mounted permanent magnet rotor topology. Therefore, we focus essentially on the determination of the magnetic vector potential expression in the rotor PMs region (region k of Fig. 1). Indeed, the magnetic vector potential formulation leads to solve Laplace's equation with nonhomogeneous Neumann boundary conditions when the PMs are inset into the rotor iron. Digital Object Identifier inserted by IEEE slots. The geometrical parameters are, the inside radius of the permanent magnets R 1 , the radius of the rotor R 2 , the inner and outer radii of the slot-opening R 3 and R 4 respectively, and R 5 is the radius of the stator slot bottom. The opening angle of the PMs is α, and p is the number of pole pairs (p = 4 in fig. 1).

II. MOTOR GEOMETRY AND ASSUMPTIONS

The stator presents Q semi-closed slots (Q = 6 in fig. 1) with current density J j in each slot. The slot-opening angle (region j in Fig. 1) is δ and the slot isthmus angle is β (region i in Fig. 1).

The analytical model is based on the following assumptions:

• end effects are neglected,

• stator and rotor iron cores are infinitely permeable,

• the PMs are radially magnetized and present a relative recoil permeability µ r ,

• all the boundary surfaces are defined by a constant angle and/or a constant radius.

As shown in Fig. 1, the whole domain is divided into four types of subdomains: the 2p rotor PMs subdomains (regions k, k=1,2,….,2p), the air-gap subdomain (region II), the Q stator slots-opening subdomains (regions i, i=1,2,….,Q) and the Q stator slots subdomains (regions j, j=1,2,….,Q).

The problem is solved in 2D polar coordinates. According to the adopted assumptions, the magnetic vector potential has only one component along the z-direction and only depends on the r and θ coordinates. The notations used in the paper are ( , ) where z e is the unit vector in the z-direction. Analytical solutions for magnetic vector potential expressions in the air-gap (region II) and in the stator slots subdomains (regions i and j) were presented recently by the authors for a surface-mounted PMs motor [START_REF] Lubin | 2-D Exact analytical model for surface-mounted permanent-magnet motors with semi-closed slots[END_REF]. Compared to [START_REF] Lubin | 2-D Exact analytical model for surface-mounted permanent-magnet motors with semi-closed slots[END_REF], only the rotor topology is changed and this introduces more difficulties. Hence, in the next section, we focus only on the magnetic vector potential calculation in the PMs rotor subdomain (region k).

III. ANALYTICAL SOLUTION OF MAGNETIC FIELD IN THE PMS

REGION

The k-th permanent magnet subdomain is shown in Fig. 2. It is delimited by the radii R 1 and R 2 and by the angles θ k and θ k +α. In a frame linked to the stator, the angular position of the k-th PMs region is defined as

2 k k p α π θ ∆ = -+ + with 1, 2,....2 k p = ( 1
)
where ∆ is the mechanical position of the rotor.

A. Problem Formulation

The magnetic field in a permanent magnet satisfies the basic equations of magnetostatics

0 k B ∇ ⋅ = (2) 0 k H ∇ × = (3)
where k B and k H are respectively the magnetic flux density and the magnetic field vectors in the region k.

The magnetic flux density can be written in terms of the magnetic vector potential k A as

k k B A = ∇ × (4)
Modern permanent magnet materials (NdFeB or SmCo) present a linear second quadrant characteristic such as

0 0 k r k k B H M µ µ µ = + (5)
where µ 0 is the permeability of air, µ r is the relative recoil permeability, and k M is the residual magnetization vector of region k. Equation (3) and ( 5) are combined to give

0 k k B M µ ∇ × = ∇ × (6)
Using (4) and choosing the Coulomb gauge ( 0

k A ∇ ⋅ = ), (6) leads to the Poisson equation 2 0 k k A M µ ∇ = -∇ × (7)
If we consider that the permanent magnets are radially magnetized with a uniform residual magnetization, the residual magnetization vector in region k can be written as

0 ( 1) k r k r B M e µ = - with k = 1,2,….,2p (8)
where B r is the remanence of the magnets and r e the unit vector in the r-direction. This reduces the Poisson equation [START_REF] Ackermann | Analytical modeling of the cogging torque in permanent magnet motors[END_REF] to a Laplace equation

2 0 k A ∇ = (9)
The magnetic vector potential has an unique component in the z-direction, then (9) becomes

k k k A A A r r r r θ ∂ ∂ ∂ + + = ∂ ∂ ∂ (10)

B. Boundary conditions

The tangential component of the magnetic field along the lateral sides and the bottom of the magnets (iron boundaries) has to be zero because of the infinite permeability for the rotor iron core.

0 k H n × = ( 11
)
where n is the unit vector normal to the iron boundaries.

From the constitutive relations ( 5) and ( 8), the boundary conditions for the k-th PMs subdomain written in terms of magnetic vector potential are given by ( 1)

k k k r A r B θ θ θ = ∂ = - ∂ (12)
( 1)

k k k r A r B θ θ α θ = + ∂ = - ∂ ( 13
) 1 0 k r R A r = ∂ = ∂ (14)
Note that the Neumann boundary conditions [START_REF] Kumar | Improved analytical model of a permanentmagnet brushless DC motor[END_REF] and ( 13) are not homogeneous (not equal to zero) and depend on the r coordinate.

The continuity condition between the k-th PMs region and the air-gap region leads to ( , ) ( , )

k II A R A R θ θ = ( 15
)
where A II (r,θ) is the magnetic vector potentials in the air-gap which is given by [START_REF] Meeker | Finite Element Method Magnetics[END_REF] in [START_REF] Lubin | 2-D Exact analytical model for surface-mounted permanent-magnet motors with semi-closed slots[END_REF].

C. Solution of Laplace's Equation with Nonhomogeneous Neumann Boundary Conditions

The k-th PMs subdomain with its associated boundary conditions is shown in Fig. 3. We have to solve the Laplace equation [START_REF] Li | Analytical solution of air-gap field in permanent magnet motors taking into account the effect of pole transition over slots[END_REF] with nonhomogenous Neumann boundary conditions [START_REF] Kumar | Improved analytical model of a permanentmagnet brushless DC motor[END_REF] and [START_REF] Bellara | Two-dimensional exact analytical solution of armature reaction field in slotted surface mounted PM radial flux synchronous machines[END_REF]. In the separation of variables method, homogeneous boundary conditions are necessary to determine the eigenvalues [START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF]. Hence, problems with nonhomogeneous boundary conditions need to be transformed to Poisson's equation, with homogeneous boundary conditions.

Using the principle of superposition, the magnetic vector potential A k (r,θ) can be expressed as a sum of two functions

( , ) ( , ) ( , )
k k k A r U r V r θ θ θ = + (16)
The function U k (r,θ) is chosen to satisfy the nonhomogeneous boundary conditions ( 12) and ( 13)

( , ) ( 1)
k k r U r r B θ θ = - (17)
From ( 16) and ( 17), Laplace's equation (10) becomes

2 2 2 2 2 1 1 1 ( 1) k k k k r V V V B r r r r r θ θ ∂ ∂ ∂ + + = -- ∂ ∂ ∂ (18)
with homogeneous boundary conditions

0 k k V θ θ θ = ∂ = ∂ 0 k k V θ θ α θ = + ∂ = ∂ (19)
It is now possible to use the classical method of separation of variables to solve [START_REF] Zhu | An accurate subdomain model for magnetic field computation in slotted surface-mounted permanent magnet machines[END_REF] with homogeneous boundary conditions [START_REF] Sargos | Analytical calculation of airgap magnetic field produced by inset permanent magnet rotor machine[END_REF]. Eigenvalues and eigenfunctions of the homogeneous problem are given by [START_REF] Lubin | Exact analytical method for magnetic field computation in the air-gap of cylindrical electrical machines considering slotting effects[END_REF] 2

m mπ λ α   =     with 0,1, 2,3,..... m = (20)
( ) cos ( )

km k mπ Θ θ θ θ α   = -     (21)
The analytic series solution can be written as ( )

( ) 0 0 1 ( 1) ( , ) ln 2 2 cos k k k r k k m m k k k m m m k m B V r a b r r m a r b r rK π π α α θ θ α π θ θ α ∞ - = - = + - +       + + - -         ∑ (22
( ) 2 2 2 ( 1) 1 ( 1) ( ) 1 k m r k m B K m m α π π α - -- =     -           (23)
Substituting ( 17) and ( 22) into ( 16) and taking into account the boundary conditions ( 14) and the interface condition [START_REF] Dubas | Analytical solution of the magnetic field in permanent-magnet motors taking into account slotting effect: no-load vector potential and flux density calculation[END_REF], the general solution of the magnetic vector potential in region k can be written as ( )

( ) 0 / 1 / 2 1 1 1 1 1 ( , ) ( 1) 2 ( , ) cos ( , )
cos

k k k r k k m m k m m m k m k m A r c r B r R m c R R m R r m K r R π α π α π α α θ θ θ Ρ π θ θ Ρ α π π θ θ α α ∞ = - ∞ =   = + - --       + -             - + -               ∑ ∑ (24)
where 0 k c and k m c are new integration constants and

1 1 1 ( , ) m m m r r r R R R π π α α π α Ρ -     = +         (25)
The integration constants 0 k c and k m c are determined using a Fourier series expansion of the air-gap magnetic vector potential ( ) 24) in [START_REF] Lubin | 2-D Exact analytical model for surface-mounted permanent-magnet motors with semi-closed slots[END_REF]) over the interval

2 , II A R θ (equation (
[θ k , θ k +α ]. 0 2 1 ( , ) k k k II c A R d θ α θ θ θ α + = ∫ (26) 2 1 1 2 2 2 2 ( , ) cos ( ) k k m k k m m II k R m R m c K R R m A R d π α θ α θ π π α α π θ θ θ θ α α +         = +                 + -     ∫ (
r k k A B r θ ∂ = ∂ k k A B r θ ∂ = -∂ (29)
Their expressions are ( )

( ) / 1 / 2 1 1 1 1 1 ( , ) ( 1) ( , ) sin ( , ) sin r k k r k m m k m m m k m k m B r B r R m m c r R R m R m r m K r r R π α π α π α θ Ρ π π θ θ α Ρ α π π π θ θ α α α ∞ = - ∞ = = -   - -             + + -               ∑ ∑ (30) ( ) ( ) / 1 / 2 1 1 2 1 1 1 ( , ) ( 1) 2 ( , ) cos ( , ) 1 cos
k k r k k m m k m m m k m k m B r B r R m m c r R R R m r m K r R θ π α π α π α α θ θ θ Ε π π θ θ α Ρ α π π θ θ α α ∞ = - ∞ =   = -- --       - -               + - -                   ∑ ∑ (31) where 1 1 1 ( , ) m m m r r r R R R π π α α π α Ε -     = -         (32)

D. Integration Constants for the Air-gap Region

Compared to the problem linked to the machine with surface-mounted permanent-magnets [START_REF] Lubin | 2-D Exact analytical model for surface-mounted permanent-magnet motors with semi-closed slots[END_REF]

2 1 2 1 cos( ) 2 k k p II k n r k r R A a n d r θ α θ θ θ π µ + = = ∂ = ∂ ∑ ∫ (33) 2 2 1 2 1 sin( ) 2 k k p II k n r k r R A c n d r θ α θ θ θ π µ + = = ∂ = ∂ ∑ ∫ (34)
where µ r is the relative permeability of the magnets. The

IV. ANALYTICAL RESULTS AND COMPARISON WITH FINITE ELEMENT CALCULATION

In order to validate the proposed model, the analytical results are compared with those obtained from 2D FEM simulations [START_REF] Meeker | Finite Element Method Magnetics[END_REF] when the same assumptions are used. The mesh in the different regions is refined until convergent results are obtained. The flux density in the air-gap, in the PMs, and in the slot-opening and slots regions are computed with a finite number of harmonic terms N I and N i,j,k as it is indicated in Table I.

As an example of surface-inset PMs machine, we investigate here the performances of a three-phase fractionalslot PMs motor. This machine presents 4-pole/15-slot corresponding to a number of slot per pole and per phase equal to q = 1.25. The geometrical parameters used in the simulation studies are given in Table I.

This fractional slot machine has a double-layer winding with a coil pitch equals to 3 slots. Fig. 4 shows the windings distribution in the stator slots.

The corresponding connecting matrix that represents the stator windings distribution in the slots is A/mm²) Figure 5 shows the magnetic flux lines in the machine under no-load condition. The radial and tangential components of the flux density in the middle of the air-gap (at 4.05 r cm = ) are shown in Fig. 6. Excellent agreement is obtained between analytical and FE results. One can see clearly the distortion of the radial component of the flux density at the location of the slot-opening placed in front of the magnets. For no-load condition, we can observe that the air-gap flux density is null in front of the rotor saliency.

2 1 0 1 2 0 0 1 1 0 0 1 1 0 0 [ ] 0 1 1 0 0 2 1 0 1 2 0 0 1 1 0 0 0 1 1 0 0 1 1 0 0 2 1 0 1 2 C -- --     = -- --     -- --   (35)
Using (30) and (31), the radial and tangential components of the flux density in the 1 st PMs region (k=1; at 3.6 r cm = and 22.5°< θ <67.5°) are calculated and plotted in Fig. 7. In accordance with the boundary conditions ( 12) and ( 13) used in the analytical model, one can observe that the radial component of the magnetic field at the sides of the magnet is equal to the remanence B r . Due to the demagnetization effect (air-gap and slots), one can observe on Fig. 7a that the radial component of the flux density is somewhat lower than the remanence of the magnet. Fig. 7b shows that it is difficult to predict accurately the tangential component of the flux density with FEM [START_REF] Meeker | Finite Element Method Magnetics[END_REF] when PMs are inset into the rotor iron, despite using a very fine mesh size.

The analytical determination of the back-EMF waveform as a function of the rotor position for n turn =1 is presented in Fig. 8. The computation is done for a rotating speed Ω = 1500 rpm.

The maximum value of the back-EMF is around 10V. The analytical and numerical predictions are close to less than 1%.

The cogging torque is given in Fig. 9. The torque is evaluated at consecutive rotor positions using Maxwell stress tensor method. The angular period of the cogging torque corresponds to the Least Common Multiple of 2p and Q giving 360°/LCM(Q,2p)=6°. It can be seen that the proposed analytical model can predict the cogging torque with an excellent precision. 2) Armature Reaction Field (B r = 0T and J rms =4A/mm²) Figure 10 shows the flux distribution in the machine caused by the armature reaction acting alone. The magnets are removed and replaced by air. The three-phase stator windings are fed with electrical current such as I a = I and I b = I c =-I/2 corresponding to AC operation. Fig. 11 and Fig. 12 compare the FE and analytically predicted flux density waveforms in the air-gap and PM regions. Fig. 12a shows that the radial component of the flux density is now equal to zero at the iron boundaries.

Due to its rotor saliency (the reactance X d is smaller than X q ), this motor develops a reluctance torque. Fig. 13 shows the comparison between analytical and FE predictions for the static reluctance torque. As expected, the reluctance torque reaches its maximum value for an angle of about 22.5°. Fig. 14 shows the electromagnetic torque waveform versus rotor position for a control angle giving the maximum torque. At each rotor position, the current values in the different slots are updated to have a sinusoidal current waveform. It can be seen that the studied machine produces an average reluctance torque of about 5.5 Nm. The proposed analytical model can predict the reluctance torque with a very good precision. 3) Results for Load Condition (B r =1.2T,J rms =4A/mm²) Figure 15 shows the flux distribution due to both PMs and armature current (load condition). The three-phase stator windings are fed so that I a = I and I b = I c =-I/2 corresponding to AC operation (J rms = 4A/mm 2 ). The radial and tangential components of the flux density in the air-gap and magnet regions are respectively shown in Fig. 16 and Fig. 17. The influence of the armature reaction on both radial and tangential components of the flux density waveforms is noticeable in comparison with the no-load results of Fig. 6 and Fig. 7.

The static torque versus mechanical rotor position is presented in Fig. 18. Maximum torque occurs at an angle of about 60°. It is well-known that for a surface-inset PMs motor, the maximum torque occurs at an angle greater than 45°. Surface-inset PMs motors produce not only permanent magnets torque component (for which the maximum torque value occurs at an angle of 45°) but also reluctance torque component as shown in Fig. 13. Figure 19 shows the electromagnetic torque waveform versus rotor position. At each rotor position, the current values in the different slots are updated to have a sinusoidal current waveform. It can be seen that the studied machine produces an average torque of about 26 Nm. The torque ripples are due to cogging torque (Fig. 9) and reluctance torque (Fig. 14). If we compare Fig. 14 and Fig. 19, one can observe that the mean value of the reluctance torque is about 20% of the total torque. Once again, the analytical and the FEM results are in good accordance.

In order to have a good precision in the analytical torque evaluation, the number of harmonic terms in each region should be chosen carefully. In our case, a number of harmonic terms equal to N I = 30 for the air-gap region and N i,j,k =10 for the slots, slot-opening and PMs regions gives a good precision. For a given rotor position, the computation time is about 70 ms with the analytical model whereas the linear FEM takes about 3 s for a mesh of 33500 elements. The analytical computations being much faster, the presented model can advantageously be used in a preliminary design of surfaceinset PMs motors.

4) Model Limitations due to Magnetic Saturation

In order to study the model limitations due to magnetic saturation of the iron parts, we have compared the maximum torque value obtained with the analytical model (which considers infinite permeability of the iron parts) with the one obtained by nonlinear finite elements analysis [START_REF] Meeker | Finite Element Method Magnetics[END_REF]. The B(H) characteristic used in FE simulations is shown in Fig. 20 (knee point of around 1.6T). This curve corresponds to M-27 steel given in the FEMM software [START_REF] Meeker | Finite Element Method Magnetics[END_REF].

Figure 21 shows the maximum torque as a function of the current density obtained with the analytical model and with FE simulations (the geometrical parameters are those given in Table I). In order to show the effect of the magnetic saturation, we have considered different values for the current density in the slots (J rms = 2 A/mm 2 to J rms = 12 A/mm 2 ). As expected, the maximum torque obtained with nonlinear FE analysis is below the analytical prediction especially for large value of the current density.

Nevertheless, as it can be observed in Fig. 21, the infinite permeability assumption of the stator/rotor back iron has a low impact on the accuracy of the torque determination for the studied machine working in normal conditions (error less than 4% for for J rms ≤ 6 A/mm 2 ). This is due to the fact that PM machines usually have a large air-gap (1mm for the studied machine) and that the relative permeability of magnets is very close to unity (µ r = 1.05). Therefore the reluctance of the effective air-gap is considerably higher compared to that of the stator and rotor back-iron. So, the stator teeth are not highly saturated, except for large values of the current density.

V. CONCLUSION

In this paper, an analytical approach to calculate the magnetic field distribution in surface-inset PM motors, including both stator slotting effects (semi-closed slots) and rotor saliency, was developed. The proposed model is sufficiently general to be used for any poles and slots combination including fractional slot winding machines. Flux density distribution, back-EMF and electromagnetic torque computations for no-load and load conditions are favorably compared to those issued from finite element simulations.

As the analytical model is less computational time consuming than FEM, it can advantageously be used in the initial design and optimization of surface-inset PMs motors.

APPENDIX

For the determination of the integration coefficients in the different regions, we have to calculate integrals of the form

∫ + ⋅         - ⋅ = β θ θ θ θ θ β π θ i i i d k n i n k f ) ( cos ) cos( ) , , ( (A.1) ∫ + ⋅         - ⋅ = β θ θ θ θ θ β π θ i i i d k n i n k g ) ( cos ) sin( ) , , ( (A.2)
∫ + ⋅ = β θ θ θ θ i i d n i n r ) cos( ) , ( (A.3) ∫ + ⋅ = β θ θ θ θ i i d n i n s ) sin( ) , ( (A.4) ∫ +         -       - - - = β θ θ θ θ θ β π δ β θ θ δ π i i d k m k m F i i ) ( cos )) ( 2 1 ( cos ) , ( (A.5)
The calculations of the previous integrals lead to the following results: -for

β π n k ≠ ( ) 2 2 2 2 2 ) sin( ) ( sin ) 1 ( ) , , ( β π θ θ β β n k n n n i n k f i i k - - + - - = (A.6) ( ) 2
)

) cos( ) cos( 1 ) , ( i i n n n n i n s θ β θ + + - = (A.11)
The development of (A.5) gives -for The development of (33) and (34) gives ( )

β π δ π k m ≠             - +       + - ×         -       = δ β δ π δ β δ π β π δ π δ π ( 2 sin ( 2 sin ) 1 ( ) , ( 2 2 m m k m m k m F k (A.12) -for β π δ π k m =         - = ) ( 2 cos 2 ) , ( δ β β π β k k m F (A.
2 1 2 / 1 2 2 / 1 2 1 1 1 2 2 1 2 1 1 ( 1) 1 sin( ( )) sin( ) ( , ) 2 ( , ) 1 ( , , ) ( , ) 1
( , , )

p k II r n k k r k p k m m r m k m m p k m r k m B a n n s n k n n R R m c f m n k R R R K R m f m n k R π α π α π α α θ α θ µ π Ε µ α Ρ π µ π α = ∞ = =   + ∞     = = -   = + + -     -         - -               ∑ ∑∑ ∑∑ (A.14) ( ) 2 1 2 / 1 2 2 / 1 2 1 1 1 2 2 1 2 1 1 ( 1) 1 cos( ( )) cos( ) ( , ) 2 ( , ) 1 ( , , ) ( , ) 1 ( , ,
)

p k II r n k k r k p k m m r m k m m p k m r k m B c n n r n k n n R R m c g m n k R R R K R m g m n k R π α π α π α α θ α θ µ π Ε µ α Ρ π µ π α = ∞ = =   + ∞     = = -   = - + + +     -         - -               ∑ ∑∑ ∑∑ (A.15)
where k m K is given by [START_REF] Lubin | 2-D Exact analytical model for surface-mounted permanent-magnet motors with semi-closed slots[END_REF].

The coefficient II n b and II n c defined in [START_REF] Lubin | 2-D Exact analytical model for surface-mounted permanent-magnet motors with semi-closed slots[END_REF] are given by 0

3 1 / 3 4 3 / 3 4 1 1 3 / 3 4 1 1 ( , ) ( , ) ( , , ) ( , ) 2 ( , , ) ( , )
Q i II n i Q k i k k i k Q i k k i k b b r n i R R R k a f k n i R R R k b f k n i R R R π β π β π β π Ρ β Ε β Ε = ∞ = = ∞ = = = + - ∑ ∑∑ ∑∑ (A.16) 0 3 1 / 3 4 3 / 3 4 1 1 3 / 3 4 1 1 ( , ) ( , ) ( , , ) ( , ) 2 ( , , ) ( , ) Q i II n i Q k i k k i k Q i k k i k b d s n i R R R k a g k n i R R R k b g k n i R R R π β π β π β π Ρ β Ε β Ε = ∞ = = ∞ = = = + - ∑ ∑∑ ∑∑ (A.17)
where Q is the number of stator slots.

• Expressions of coefficients 0 k c and k m c for the PMs subdomain :

The calculation of ( 26) and ( 27) yields to the following linear relations between the integration constants in the PMs and subdmains where J j is the current density in the slot j.

a b R R R R R a b r n i n R R n R R R R R R c d s n i n R R n R R Ρ β Ε β Ε Ρ β Ε β Ε ∞ = ∞ = + = + + + ∑ ∑ (A.
j i i j m j m m m R a b R a J R R R R R m m a m R R π β π β µ Ρ δ πβ π β π Ε δ ∞ =   + = + - +                         ∑ (A.
j i m k i i k k k k k m m a b m R R R k a b F m k R R R R R π β π β π β πβ π π δ Ρ π Ε Ε δβ ∞ =     = +           + -       ∑ (A.25)
We have to solve a system of linear equations whose dimension is equal to the number of unknowns. By rewriting the above equations in matrix and vectors format, a numerical solution can be found thanks to mathematical software (Matlab).

Fig. 1 .

 1 Fig. 1. Surface-inset permanent magnet motor with semi-closed slots (p = 2 and Q = 6).

Fig. 1

 1 Fig.1shows a surface-inset PMs motor with semi-closed

Fig. 2 .

 2 Fig. 2. k-th permanent magnet subdomain.

Fig. 3 .

 3 Fig. 3. PMs subdomain (region k) with its boundary conditions.

  27) Expressions of the coefficients 0 k c , and k m c are given in the appendix. The radial and tangential flux densities r k B and k B θ in the PMs regions are deduced from the magnetic vector potential expression (

  , if the boundary condition for r = R 3 are the same and lead to the same coefficient expressions II n b and II n d , the ones on r = R 2 must be totally reconsidered in the case of the present machine. The constants II n a and II n c have to be re-written as

2

 2

  coefficients II n a and II n c are developed in the appendix.

Fig. 4 .

 4 Fig. 4. Double-layer windings distribution for a three phases machine with 15 slots and p = 2 (q = 1.25).

Fig. 5 .Fig. 6 .

 56 Fig. 5. Magnetic flux distribution for no-load condition obtained with FEM.

Fig. 8 .Fig. 7 .

 87 Fig. 8. Per turn phase back-EMF waveform.

Fig. 9 .

 9 Fig. 9. Cogging torque waveform.

Fig. 11 .

 11 Flux density distribution for radial (a) and tangential (b) components of armature reaction field in the middle of the air-gap for Jrms=4A/mm², Ia= I and Ib = Ic =-I/2.

Fig. 10 .Fig. 12 .

 1012 Fig. 10. Armature reaction magnetic flux distribution obtained with FEM for Jrms=4A/mm² (the magnets are removed).

Fig. 13 .

 13 Fig. 13. Static torque versus rotor position due to slots currents alone (reluctance torque) for Jrms=4 A/mm².

Fig. 14 .

 14 Fig. 14. Electromagnetic torque versus rotor position due to slots currents alone (reluctance torque) for Jrms=4A/mm².

Fig. 15 .

 15 Fig. 15. Magnetic flux distribution for load condition.

Fig. 17 .

 17 Flux density distribution for radial (a) and tangential (b) component in the 1 st PMs region (k=1) at r=(R1+R2)/2 for load condition.

Fig. 18 .

 18 Fig. 18. Static torque versus rotor position for Jrms=4A/mm².

Fig. 19 .

 19 Fig. 19. Electromagnetic torque versus rotor position for Jrms=4A/mm².

Fig. 20 .Fig. 21 .

 2021 Fig. 20. B(H) characteristic of the stator/rotor back iron used in nonlinear FE simulations.

Thierry

  Lubin was born in Sedan, France, in 1970. He received the M.S. degree from the University of Paris 6, France in 1994 and the Ph.D. degree from the University Henri Poincaré, Nancy, France, in 2003.He is currently a lecturer of Electrical Engineering at the University of Nancy at the Groupe de Recherche en Electrotechnique et Electronique de Nancy. His interests include modeling and control of electrical machines and applied superconductivity in electrical devices.Smail Mezani was born in Algiers, Algeria, in 1974. He received the engineer diploma and the magister degree from the University of Sciences and Technology Houari Boumediene, Algiers, Algeria in 1996 and 1999 respectively. He obtained the Ph.D. degree from the Institut National Polytechnique de Lorraine, France, in 2004.He is currently a lecturer at the University Henri Poincaré of Nancy, France, at the Groupe de Recherche en Electrotechnique et Electronique de Nancy where his research interests include the applications of superconductors in electromechanical devices.Abderrezak Rezzoug received the electrical engineer degree from ENSEM INPL, Nancy, France in 1972, and the Dr. Ing. diploma and the Ph.D. degree from INPL, in 1979 and 1987 respectively. After working at the INPL as an assistant Professor until 1991, he is currently a Professor of Electrical Engineering at the University Henri Poincaré, Nancy, France. As a member of the Groupe de Recherche en Electrotechnique et Electronique de Nancy, his main subjects of research concern superconducting applications to electrical devices, and the control and diagnosis of electrical machines.

TABLE I PARAMETERS

 I OF THE SURFACE-INSET PMS MACHINE

	Symbol	Quantity	value
	R1	Inner radius of the PMs	3.2 cm
	R2	Radius of the rotor surface	4 cm
	R3	Stator bore radius	4.1 cm
	R4	Outer radius of the slot-opening	4.3 cm
	R5	Outer radius of the slot	5.3 cm
	L	Axial length	20 cm
	δ	Slot-opening angle	12°
	β	Slot isthmus angle	6°
	α	PMs opening angle	45°
	Br	Remanence of the permanent magnets	1.2 T
	µr	Relative recoil permeability	1.05
	p	Pole-pairs number	2
	Q	Number of stator slots	15
	Jrms	RMS current density	4 A/mm²
	NI	Number of harmonics used for magnetic field
		calculation in the air-gap domain

100 Ni,j,k Number of harmonics used for magnetic field calculation in the slot, slot-opening and PMS domains 50 1) Results for No-Load Condition (B r = 1.2T and J rms =0

( , )