3D Analytical Calculation of Forces Between Linear Halbach-Type Permanent Magnet Arrays Hicham Allag, Jean-Paul Yonnet, Mohamed E

3D Analytical Calculation of Forces between Linear Halbach-Type Permanent Magnet Arrays Hicham Allag, Jean-Paul Yonnet, Mohamed E. H. Latreche

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Hicham Allag, Jean-Paul Yonnet, Mohamed E. H. Latreche. 3D Analytical Calculation of Forces between Linear Halbach-Type Permanent Magnet Arrays. ELECTROMOTION, Jul 2009, Lille, France. hal-00402515

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3D Analytical Calculation of Forces between Linear Halbach-Type Permanent Magnet Arrays

H. Allag 1,2 , J-P. Yonnet 1 and M. E. H. Latreche 2 1- Laboratoire de Génie Electrique de Grenoble, G2E Lab (UMR 5269 CNRS/INPG-UJF), Institut Polytechnique de Grenoble, ENSE3, BP46, St Martin d’Hères cedex, France 2- Université de Constantine, Labo. L.E.C., Dept. d’Electrotechnique, 25000 Constantine, Algérie Email: [email protected] , [email protected]

Abstract - Usely, in analytical calculation of magnetic and unique property of producing a strong periodic magnetic field mechanical quantities of Halbach systems, the authors use the on one side of the array with a minimal field on the opposite Fourier series approximation because the exact calculations are side of the array. The Halbach array has been applied in particle more difficult. In this work the interaction forces between linear accelerators or wigglers, magnet bearings [12], electrical Halbach arrays are analytically calculated thanks to our recent machines [1, 2, 7, 8], and Maglev designs (Indutrack method) development 3D exact calculation of forces between two cuboïdal magnets with parallel and perpendicular magnetization. We [3, 4, 6]. essentially describe the way to separately calculate the forces For example, wigglers can be build using permanent between two magnets, between one magnet and a Halbach array magnets or electromagnets. Permanent magnet wigglers are and between two Halbach systems made using rare-earth materials such as SmCo or NdFeB. There magnet configurations is often referred to as the Halbach Keywords— Halbach array, Permanent magnet, Magnetic configuration [5]. Forces II. ANALYTICAL CALCULATION BACKGROUND I. INTRODUCTION Since the discovery and the development of Samarium The analytical calculation is a very powerful and a very fast Cobalt magnets in the 70 years and recently the development of method to compute magnetic interactions. It is why the rare-earth materials such as SmCo or NdFeB, the designers can analytical expressions of all the interactions, energy, forces and use magnets owning a really rigid magnetization. They are the torques between two cuboïdal magnets are very important magnets which can be used in repulsion without any risk of results. Many problems can be solved by the addition of demagnetization. Their magnetization can be easily modelled element interactions. The simpler shape of elementary volume by magnetic charges on the poles, or by equivalent currents. is the parallelepiped, with its cuboïdal volume. It is why many One of the first applications of Samarium Cobalt magnets were 3D calculations can be made by the way of 3D interactions the magnetic bearings. Due to the circular symmetry, the between two elementary magnets of cuboïdal shape. Up to now, calculation can be made in 2D. The first 2D analytical only the force components between two magnets with the expressions of the forces between magnets were given by magnetization direction parallel to one edge of the Marina Marinescu [9] and Jean-Paul Yonnet [10]. The stiffness parallelepipeds have been analytically solved. We have of a magnetic bearing can be easily calculated by analytical succeeded in a new result of first importance for the analytical expressions [11]. The magnetic couplings are another calculation: the force when the magnetization directions are application working by interaction forces between permanent perpendicular. Consequently, by combining parallel and magnets. When their length is long in comparison with the perpendicular magnetization directions, the analytical airgap dimensions, the 2D analytical calculation can be used expressions can be written for any magnetization direction. [12]. Otherwise it is necessary to use 3D calculation. The 3D Thanks to all this new result, the interaction energy and all the analytical calculation is obviously more difficult than the 2D. It components of the forces can be calculated by fully analytical is not 2 but 4 successive analytical integrations which must be expressions, for any magnetization direction, and for any calculated, and the difficulty fastly increases with the number relative position between the two magnets. of integrations. Many persons thought that the last analytical The only two hypotheses are that the magnets own a cuboïdal integration was not possible, and must be made by a numerical shape, and they are uniformly magnetized. These results are way. We had worked on this problem, and we have succeeded very interesting to understand the mechanical forces acting on in solving the calculation. At that time all the integrations had permanent magnets. For those reasons we are interesting in been made by brain-work, with a pen and a piece of paper. Halbach permanent magnet. The Halbach array is a special The first 3D forces analytical expressions were published in array of high-field permanent magnets in which the orientation 1984 by Gilles Akoun and Jean-Paul Yonnet [13]. The forces of the magnetic poles of each magnet in the array has the were analytically calculated for two cuboïdal magnets with

1 ELECTROMOTION 2009 parallel magnetization directions. The calculus was made by the From the analytical expression of the scalar potential, the way of interaction energy determination. The forces magneticr field H can be easily calculated by derivation. expressions were obtained by derivation of the energy H = −gr ad (V ), it can be expressed as: expression. Until now, all the analytical force calculation has 1 1 σ ' + been made for cuboïdal magnets which magnetization is H = ()−1 i j ε(Ui ,Vj ,W) (6) πµ ∑∑ parallel to one of the edge of the magnet. It means that the 4 0 i=0 j =0 magnetic poles are only on two rectangular faces of the magnet − UV ε = ln( r −V ) ε = ln( r −U) ε = tg 1( ) [14]. Recently, we have worked again on the 1984 x y z Wr formulations, and we succeed in the analytical interaction By adding the field created by the two surfaces, we obtain the energy calculation when the magnetization directions of the two field created by a cuboïdal magnet (Fig. 2). cuboïdal magnets are perpendicular. From the expression of M x energy, the forces can be easily calculated by derivation. It is an z y important step, because its opens the way to the calculation of the interactions when the magnetization directions are in every z direction. This result will be used easily for linear Halbach 2a + + array magnetic field and forces calculations which have been + O y studied until now, using different approaches. - - y 2c III. THE BASIC MATHEMATICAL MODEL - - -

A. Magnetic induction created by a magnet x 2b Let us consider a rectangular surface 2a x 2b, wearing a Figure 2 – Magnetic induction created by a magnet σ uniform pole density (Fig. 1). We will calculate the scalar The geometric parameters are shown on Fig. 2. Intermediary potential in the point P, which coordinates are (X, Y, Z). variables are: The scalar potential V is given by: U = x − (− )1 i a ; V = y − (− )1 j b and W = −(− )1 k c σ ⋅ i j k = 1 dS (1) V ∫∫ r r = 2 + 2 + 2 (7) πµ − r U i V j W k 4 0 S r r' By using the cartesian coordonates, it is equivalent to: The magnetic field and induction components are given by : σ 1 1 1 σ b a 1 = − i+ j+k ε (8) = (2) H ∑∑∑( )1 (Ui ,Vj ,Wk ) V dy ' dx ' 4πµ = = = πµ ∫ ∫ 2 2 2 0 k0 i0 j 0 4 0 −b −a (x'−x) + (y'−y) + z σ 1 1 1 B = µ H = (− )1 i+ j+k ε(U ,V ,W ) (9) After two analytical integrations, we obtain: 0 π ∑∑∑ i j k 1 1 4 k=0 i =0 j =0 σ + V = ()−1 i jφ(Ui ,Vj ,W) (3) πµ ∑∑ With ε , ε , ε and ε are the same in equation (6) 4 0 i=0 j =0 x y z The function φ is given by:

− UV  φ(U,V,W, r) = −U ln( r −V) −V ln( r −U) −W ⋅tg 1   (4) B. Interaction energy calculation for parallel magnetization  rW  directions Intermediary variables are: We study the interactions between two parallelepipedic = − − i = − − j = Ui x ( )1 a , V j y ( )1 b and W z magnets. Their edges are respectively parallel (Fig. 3). The magnetizations J and J’ are supposed to be rigid and uniform in = 2 + 2 + 2 (5) r U i V j W each magnet. The dimensions of the first magnet are 2a x 2b x 2c, and its polarization is J. Its center is O, the origin of the axes Oxyz. For the second magnet, the dimensions are 2A x 2B x 2C, its polarization is J’, and the coordinates of its center O’ M x z are ( α, β, γ). The side 2a is parallel to the side 2A, and so on. y The magnet dimensions are given on Table 1. z

Axis Ox Oy Oz 2a + σ + + First Magnet (J) 2a 2b 2c + y ’ y’+ dy y Second Magnet (J’) 2A 2B 2C + , x’ Second Magnet Position O’ α β γ x’+ dx Table 1 : Magnet dimensions and position 2b The magnetization directions shown on Figure 3 correspond Figure 1 – Geometrical disposition for the potential and the field calculation to the case when the polarizations J and J’ have the same direction, parallel to the side 2c. Note that the calculation stay valid when they are in opposite direction; only the expression sign is reversed. The polarizations J and J’ are supposed to be

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1 1 1 1 1 1 rigid and uniform. They can be replaced by distributions of J ⋅J' + + + + + = − i j k l p q φ (14) magnetic charges on the poles. It is the coulombian F ∑∑∑∑∑∑( )1 (. Uij ,Vkl ,Wpq ,r) 4πµ ======representation of the magnetization. 0 i0 j0 k0 l0 p0 q 0 with z Z (V 2 −W2 )  UV  1 2B φ (U,V,W,r) = ln ()r −U +UV ln( r −V)+VW ⋅tg 1-  + U⋅r γ x 2 W⋅r  2 J’ 2C O’ (U2 −W2)  UV  1 Y φ (U,V,W,r) = ln ()r −V +UV ln( r −U) +UW ⋅ tg 1-   + V ⋅ r 2A y 2 W ⋅ r  2

 UV  φ (U,V,W,r) = −UW ln ()r −U −VW ln( r −V) +UW ⋅ tg 1-   −W ⋅ r 2a z W ⋅ r  O J β y 2c α C. Interaction energy calculation for perpendicular

x 2b magnetization directions Figure 3 – Magnet configuration With applying the same procedure to magnets with r r perpendicular magnetization Fig. 4, we obtain for energy: Their density σ is defined by σ = J ⋅ n .On the example of J .J ' 1 1 C A b a 1 Figure 3, since J is perpendicular to the surfaces 2a x 2b and E = ∑ ∑ dZ dX dy dx (15) σ πµ ∫ ∫ ∫ ∫ oriented to the top, these faces wear the density = +J on the 4 0 l=0 p = 0 − C − A −b − a r upper face (North Pole), and σ = -J on the lower face (South z Z Pole). All the analytical calculations have been made by 2B successive integrals. We have determined the scalar potential created by one charged surface. From this scalar potential, the γ 2C induction components can be obtained by derivation. For the O’ whole magnets, we have calculated the interaction energy. The forces and the torques can be deduced by linear and angular J’ 2A derivation. The most difficult analytical calculation is the 2a interaction energy in 3D. It is made by four successive integrations. The first one gives a logarithm function. In the J O β y second one, you have two logarithm and two arc-tangent 2c α functions. The last one owns many complex functions based on logarithm and arc-tangent functions. The interaction energy in a 2b system of two magnets with parallel magnetization directions x (Fig. 3) is given by: Figure 4 – Magnet configuration with perpendicular magnetization direction The obtained expressions of the interaction energy are: 1 1 C A b a 1 1 1 1 1 1 J.J' + 1 J⋅J' + + + + + E = (− )1 p q dY dX dy dx (10) E = (− )1 i j k l p q.ψ(U ,V ,W ,r)(16) πµ ∑∑ ∫ ∫ ∫ ∫ πµ ∑∑∑∑∑∑ ij kl pq 4 0 p=0 q =0 −C −A −b −a r 4 0 i=0 j =0 k =0 l =0 p =0 q =0 with r = (α+ X −x)2 +(β +Y − y)2 +(γ +(− )1 l C−(− )1 pc)2 with V(V 2 −3U 2 ) W(W 2 −3U 2 ) The obtained expressions of the interaction energy are: ψ(U,V,W,r) = ln( W + r)+ ln( V + r) 6 6 (17) ⋅ 1 1 1 1 1 1 J J' i+j+k+l+p+q   = − ψ (11) U 2 −1UW  2 −1 UV  2 −1 VW  E ∑∑∑∑∑∑( )1 . (Uij ,Vkl ,Wpq r), + 3V tg  +3W tg  +U tg   πµ 6 V ⋅r  W ⋅r  U ⋅r  4 0 i=0 j =0 k =0 l =0 p =0 q =0   V ⋅W ⋅r with +UVW .ln( −U + r)+ U (V 2 − W 2 ) V (U 2 − W 2 ) 3 ψ (U ,V ,W , r) = ln( r − U ) + ln( r − V ) (12) 2 2 The variables U,V,W,r are the same than (Equation (13)) From the gradient of energy the force components are: −1UV  r 2 2 2 + UVW ⋅ tg   + ()U + V − 2W 1 1 1 1 1 1 J ⋅J' + + + + +  rW  6 F = (− )1 i j k l p q φ(. U ,V ,W ,r)(18) πµ ∑∑∑∑∑∑ ij kl pq The secondary variables are: 4 0 i=0 j =0 k =0 l =0 p =0 q =0 = α + − j − − i U ij ( )1 A ( )1 a with = β + − l − − k φ (U ,V ,W , r) = −VW ln( r − U ) + VU ln (r + W ) + WU ln( r + V ) Vkl ( )1 B ( )1 b (13) x 2 2 2 = γ + − q − − p U 1-  VW  V 1-  UW  W 1-  UV  Wpq ( )1 C ( )1 c − tg   − tg   − tg   2 U ⋅ r  2 V ⋅ r  2 W ⋅ r  2 2 2 with r = U + V + W 2 2 ij kl pq (U −V ) UW  1 φ (U,V,W,r) = ln ()r +W −UW ln( r −U) −UV ⋅ tg 1-   − W ⋅r From the interaction energy, the force components can be y ⋅ r r 2 V r  2 obtained by = − . Consequently the force ( 2 − 2 ) F gr ad E φ = U W ()+ − − − ⋅ 1-  UV − 1 ⋅ z (U,V,W,r) ln r V UV ln( r U) UW tg   V r components are: 2 W ⋅r  2

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IV. VALIDATION WITH FINITE ELEMENT METHOD In order to validate our analytical model, we chose two From the last results, we remark the great resemblance of magnets which dimensions are presented in Fig. 5 : two cubes curves, but the analytical method presents some others of 1 cm edge, at a distance of 1 cm. advantages resumed in the facility of implementation and the rapidity of execution – 1minut for the analytical model versus 20 minuts for the FEM, for this example-. Direction of second 0,01m V. HALBACH MAGNET SYSTEM STUDY magnet 0,01m The ideal linear Halbach array has pure sinus magnetic Z Distance D=0,01m profile, on the enhanced side of the array, while canceling the Y 0,01m field on the other side [2]. Figure 8.a illustrates an ideal X Halbach array, but it is impractical to fabricate. Instead, an array of rectangular or square permanent magnets (PM) is actually used. The practical (non-ideal) four- and eight-piece Figure 5 – Geometrical disposition of the magnets Halbach arrays are shown respectively in Fig. 8.b and Fig. 8.c. These non-ideal Halbach arrays are not able to provide the zero Using 3D scalar magnetic potential finite element method for magnetic fields on the cancelled side and the purely sinusoidal equal values of magnetization (1T for each permanent magnet), magnetic field on the enhanced side. However, they still we obtain the potential from Flux3D ® such as presented in provide better performance characteristics than simply using an Fig.6 array composed of alternating polarity magnets.

Figure 8 – Linear Halbach arrays permanent magnets configurations For presenting the manner of interaction force calculation, let

us considering a system composed of two eight-piece Halbach Figure 6 – Scalar potential presented for the two magnets arrays Fig. 9, designed for a wiggler.

The three forces components are calculated by the two models – analytical and by FEM- and presented in Fig. 7. T he upper magnet moves in translation along the Ox axis Z above the lower fixed magnet. The distance between the X Y two magnets (airgap when the upper magnet is above the fixed magnet) is 10 mm (Fig. 5). Figure 9 – Eight-piece Halbach wiggler system

1.5 By applying our model and superposition after, we can treat Fx Analytical Force [N] Fy Analytical 1 Fz Analytical the problem at three essential cases, as follow: Fx Flux3D Fy Flux3D Fz Flux3D 0.5 A. Parallel case

0 b) a) -0.5 Z -1 Y X -1.5 Figure 10 – Parallel case configuration -2 For the first parallel case Fig. 10.a the forces component are

-2.5 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 given from the equation (14). The same results are obtained for X displacement [m] the second configuration Fig. 10.b with considering the

following permutations (V →W equation (13)) and (F y→ Fz Figure 7 – Three force components calculated from analytical model and equation (14)) finite element method

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B. Perpendicular case

8 4 Force Force 1-1 Force1-1 Force Fz [N] 6 Force 1-2 3 Force 2-1 Fy [N] Z Force 1-3 Force 3-1 4 Force 1-4 2 Force 4-1 Force 1-5 Force 5-1 Y 2 X 1 Figure 11– Parallel case configuration 0 0 -2 -1 In perpendicular case, the final expressions of force are -4 -2 -6 similar – see equation (18)- by considering simply the -3 -8 -4 following changing in axis (V →W and W →-V) -0.04 -0.02 0 0.02 0.04 0.06 0.08 -0.04 -0.02 0 0.02 0.04 0.06 0.08 Y [m] Y [mm] C. Inclined case Figure 15 – Fz and Fy forces exerted on the upper magnet When one of the two magnetization directions is inclined (Z magnetization direction) Fig.12, it can be decomposed as the sum of parallel and Fig. 15 show the calculated forces (Fz and Fy) exerted on the perpendicular case. The whole energy and the forces can be upper magnet by all magnets of the lower Halbach array. For calculated by addition of the interactions between the example the curve in blue present the interaction force between magnetization components. the two first magnet noted 1-1 (when β=0 the force Fz is J// J┴ J┴ J maximal and the corresponding Fy is equal to zero). By the J J// same consideration we can show all interaction forces (from 1

Z to 5).

Y If we repeat the same procedure for the second magnet, X Fig.16, we obtain also the interaction forces between all Figure 12 – Inclined case configuration magnets Fig. 17, but the difference is the translation of all curves by a quantity 2B=0.01m ( 2B is the Y width of the upper VI. APPLICATION magnet, see Figure 3 and 4) . It’s logical because the centre of all magnets in Y arrangement is the multiple of the Y width A simple example has been chosen fig. 13. (n(2B)) Z

0.01 m Y

0.01 m Y displacement

z X D=0.005 m Z

y 0.01 m Y x X Figure 16 – Second magnet above the four-piece Halbach array Figure 13 – Four-piece Halbach wiggler system

4 4 Force 2-1 It’s composed of two four-piece Halbach array, one upper Force 2-1 Force Force 3 Force 2-2 Fz [N] 3 Force 2-2 Fy [N] the other. The magnetizations parallel to z axis are inversed for Force 2-3 Force 2-3 2 Force 2-4 2 Force 2-4 Force 2-5 Force 2-5 the two arrangements. The 3D axes are considered at the centre 1 1 of the first magnets. All the magnetizations are equal and their 0 0 value is 1T. -1 -1

For the first result we consider only the first magnet of the -2 -2 -3 upper four-piece Halbach array Fig. 14, this magnet moves -3 -4 -4 -0.04 -0.02 0 0.02 0.04 0.06 0.08 -0.04 -0.02 0 0.02 0.04 0.06 0.08 linearly in Y direction (from -0,04m to 0.08m) – in our model Y [m] Y [m] calculation this displacement is considered by imposing the Figure 17 – Fz and Fy forces exerted on the upper magnet desired value at β parameter (see section III-a and III-b)) – (Y magnetization direction) Now, it is interesting to calculate the forces in the three

Y directions which act in the centre of the upper system. For

displacement realize this, it is necessary to sum all the forces resulting from

Z the first configuration Halbach and which are exerted on all the magnets of the second. Y X The total Fz and Fy are presented in Fig. 18 for β=-0.08 to Figure 14 – First magnet above the four-piece Halbach array 0.08m For the wiggler system Fig. 13, the global force exerted between the two parts is only a vertical force of 50 N.

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REFERENCES [1] Jang, S., and Sung, L.: ‘Comparison of two types of PM linear synchronous servo and miniature motor with air cored film coil’,IEEE Trans. Magn., 2002, 38, (5), pp. 3264–3266 40 50 [2] Halbach, K.: ‘Application of permanent magnets in accelerators and Force Force 30 Fz [N] 40 Fy [N] electron storage rings (invited)’, J. Appl. Phys., 1985, 57, (8),pp. 3605–3608 30 20 [3] Kratz, R., and Post, R.F.: ‘A null-current electro-dynamic levitation 20 10 system’, IEEE Trans. Appl. Supercond., 2002, 12, (1), pp. 930–932 10 0 [4] Kartz, R., and Post, R.F.: ‘Halbach arrays for maglev 0 -10 -10 applications’.Proc. 6th Int. Symp. on Magnetic Suspension Technology, -20 -20 Turin, Italy, 2001

-30 -30 [5] E. P. Furlani, “Permanent Magnet and Electromechanical Devices; -40 -40 Materials, Analysis and Applications ”, Academic,Elsevier, New York, 2001. -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 Y [m] Y [m] [6] P. Friend “Magnetic Levitation train technology”, progress report, Figure 18 – The total Fz and Fy forces exerted on the centre of the upper Bradley university, 2004 four-piece Halbach magnet [7] Trumper, D., Williams, M., and Nguyen, T.: ‘Arrays for synchronous machines’. Proc. IEEE Industry Applications Society AnnualMeeting, 1993, We can easily study another simple planar array, it’s Vol. 1, pp. 9–18 composed only of parallel magnetization alternatively oriented, [8] Zhu, Z., and Howe, D.: ‘Halbach permanent magnet machines and this planar array was used as motor by Asawaka [15]. The applications: a review’, IEE Proc. Electr. Power Appl., 2001, 148, (4),pp. Asakawa motor is the first kind of permanent magnet planar 299–308 [9] M. Marinescu, "Analytische Berechnungen und Modellvorstellungen motors. It is characterized by a large number of permanent für Systeme mit Dauermagneten und Eisen", Ph. D., TU Braunschweig, magnet cubes, upper the magnets move a rectangular coil and April 1980 the available forces are varied with the relative position [10] J-P. Yonnet, "Etude des Paliers Magnétiques Passifs", Ph.D., INP between the coils and the matrix magnet [15-16]. Grenoble, July 1980 [11] J-P. Yonnet "Analytical Calculation of Magnetic Bearings", Proc of We can study such model, by considering the analogy the 5 th Int Workshop on Rare Earth Cobalt Permanent Magnets and their between the coil and a permanent magnet. A general algorithm Applications, Roanoke, Virginia (USA), June 1981, p. 199 – 216 is realized for this possibility. An example of (5*6) permanent [12] J-P. Yonnet, S. Hemmerlin, E. Rulliere, G. Lemarquand, "Analytical magnets with alternatively opposite (Z) magnetizations calculation of permanent magnet couplings", IEEE Trans. on Magnetics, Vol. 29, Nov. 1993, p 2932-2934. directions is shown on Fig. 19. The coil can be considered like [13] G. Akoun, J-P. Yonnet, "3D analytical calculation of the forces exerted a big magnet upper the planar array. between two cuboïdal magnets", IEEE Trans. Magnetics, MAG 20, n° 5, Sept. 1984, p. 1962-1964. [14] J-P. Yonnet, "Magneto-mechanical devices", Chap. 9 of "Rare Earth Iron Permanent Magnet" J.M.D. COEY Ed., Oxford University Press, Sept. 1996. [15] T.Asakawa. “Two Dimensional Precise Positioning Devices for use in semiconductor apparatus”. U.S. Patent Office, Patent # 4 535 278, August 1985. [16] Kim W.j.: «High Precision Planar Magnetic Levitation. Ph.D. thesis, Massachusetts Institute of Technology (1997).

Figure 19 – geometric presentation of (5*6) patchwork array

VII. CONCLUSION In this paper, The Coulombian approach for 3D interaction forces calculation is shown for cuboïdal permanent magnets with parallel, perpendicular and for any magnetization direction. This approach is perfectly adapted for linear Halbach systems study. The results obtained for the 4-pieces Halbach wiggler system prove that we can calculate the forces exerted on each magnet separately and also determinate the global force on all the Halbach arrays. By our method the calculation of many magnet systems is possible if we always consider that magnet interactions can be obtained by the superposition of interactions between cuboïdal elements.