Three-Dimensional Analytical Model for an Axial-Field Magnetic Coupling

Transcription

1 Three-Dimensional Model for an Aial-Field Magnetic Coupling Bastien Dolis, Thierr Lubin, Smail Mezani, Jean Lévêque To cite this version: Bastien Dolis, Thierr Lubin, Smail Mezani, Jean Lévêque. Three-Dimensional Model for an Aial-Field Magnetic Coupling. Progress in Electromagnetics Research M, 214, 35, pp <1.2528/PIERM143145>. <hal > HAL Id: hal Submitted on 24 Jun 214 HAL is a multi-disciplinar open access archive for the deposit and dissemination of scientific research documents, whether the are published or not. The documents ma come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Progress In Electromagnetics Research M, Vol. 35, , 214 Three-Dimensional Model for an Aial-Field Magnetic Coupling Bastien Dolis *, Thierr Lubin, Smail Mezani, and Jean Lévêque Abstract In this paper, we propose an analtical method for modeling a permanent magnets aial field magnetic coupling. The three-dimensional model takes into account the radial fringing effects of the coupler. The analtical solution requires resolving the Laplace equation in low permeabilit subdomains. The magnetic field calculation allows the determination of global quantities like aial force and torque. 3D finite element computations as well as measurements validate the proposed model. 1. INTRODUCTION Magnetic couplings are used to transmit torque from a prime mover to its load without mechanical contact. This is useful in applications requiring hermetic isolation between two rotar parts. In addition, the absence of mechanical contact eliminates matter s wear. This snchronous tpe coupling also provides intrinsic protection against overloads. Magnetic coupling h e h Magnets From motor To load R in R out Shaft Soft-Iron oke Figure 1. Structure of a magnetic coupling (p = 6). The studied magnetic coupling (Figure 1) is composed of two identical rotors equipped with aiall magnetized permanent magnets (PMs). The PMs are glued on an iron oke to form a north-south alternating polarit. The two rotors are separated b an air gap. The torque is created when the magnetic fields due to the two rotors are shifted b an angle ϕ, called the load angle. The aim of this paper is to develop an eact 3D analtical model which allows the computation of the magnetic field distribution, the aial force and the torque. In the literature, there are two tpes of analtical modeling for magnetic actuator: Received 14 March 214, Accepted 4 April 214, Scheduled 6 April 214 * Corresponding author: Bastien Dolis (bastien.dolis@univ-lorraine.fr). The authors are with the Laboratoire GREEN, Université de Lorraine, Faculté des Sciences et Technologies, BP 239, 5456, France

3 174 Dolis et al. - 3D Modeling is based on Amperien model using Biot-Savart law or on Coulombian model using charge surfaces [1 16]. These methods can be used onl in free space without ferromagnetic material. To take into account the magnetic oke, the method of images is used. - Fourier analsis based on the resolution of Mawell s equations b the method of separation of variables [17 24]. This method assumes an infinite permeabilit of the ferromagnetic materials so iron faces appear as homogeneous boundar conditions. Periodicities can also be taken into account. modeling based on Fourier method is widel used. Most of the computations are done in 2D so neglecting edge effects [19, 2]. To consider 3D effects in a 2D model, a multi-slice approach can be used so better results can be obtained [21, 22]. Three-dimensional models [23,24] are also developed for electrical machines analsis but the slotting effects are taken into account using the Carter coefficient. In this paper, we present a 3D analtical approach based on the resolution of Mawell s equations. The 3D model takes into account the edges effects. The analtical model is validated through 3D FE computations and eperiments. 2. PROBLEM DESCRIPTION AND ASSUMPTIONS The magnetic coupling is onl composed of permanent magnets, ferromagnetic okes and air. Hence, a magnetic scalar formulation is used. The magnetic field H and the scalar potential V are related b H = grad(v ) (1) In the different media, the flu densit is In the air: B = µ H (2) In the magnet: B ( ) = µ µ r H + M (3) where µ and µ r are the vacuum permeabilit and the relative permeabilit of the magnets and M is the magnetization. When clindrical coordinates are use, the solutions arising from the resolution of the PDE contain Bessel s functions. This leads to some numerical difficulties and a lack of accurac since onl few harmonics can be used. These problems decrease the reliabilit of the design tool. A linearization assumption at the mean radius (Figure 2) is used here to overcome the problems encountered using clindrical coordinates. This assumption is valid if the curvature effects are ignored. Of course, this is true for most electromechanical devices because the air gap thickness is usuall much lower than the other dimensions of the sstem. This issue will be discussed in Section 4.2. π/p D l a l a R mean la l D=Rout -Rmean D D a la V( = D ) = -V( = -D ) D >> l a for B = +_ D = θ -D -D D V( = D ) = -V( = -D ) Figure 2. Linearization at the mean radius. Figure 3. Anti-periodic conditions in the sstem.

4 Progress In Electromagnetics Research M, Vol. 35, h t z r > 1 h -D I II r > 1 Domain II Domain I D Figure 4. Resolution domain in the (, z) plan. Figure 3 shows the studied domain in the plan together with the adopted periodicit conditions. Onl one pole is considered with anti-periodic boundar conditions along. The anti-periodicit along the ais is a fictitious one adopted to facilitate the computation. B setting D l a, this condition is equivalent to a more phsical condition which is B = at = ±D. The permeabilit of ferromagnetic parts (okes) is considered infinite (µ r 1). Hence, at the interface between the iron and the air, we impose the orthogonalit of the flu lines b n H = (4) where n is the outward normal to the considered surface. The permeabilit of the permanent magnets (NdFeB) is considered equal to that of air (µ r = 1). The smooth geometr of the coupling (no slotting effects) allows us to calculate onl the field created b one rotor (the magnets on the other rotor are switched off). The field due to the second rotor is then deduced from the computed solution which is shifted b the angle ϕ. The total field is obtained b superposing the two solutions. Figure 4 shows the whole domain of resolution taking into account all the assumptions and the equations to solve. Domain I ( z h) corresponds to the first rotor magnet region whose magnetization is noted M. Domain II (h z h t ) is an air region composed of the airgap and the second rotor magnet whose magnetization is set to zero. The geometrical parameters gathered from Figure 2 and Figure 4, are: 2 D : Length along the -ais, 2 D : Length along the -ais, 2 l a : Length of the magnet along the -ais, 2 l a : Length of the magnet along the -ais, h: Height of the magnet, h t : Total height of the sstem. 3. MAGNETIC FIELD CALCULATION The method of separation of variables is used to solve the PDEs in the two studied subdomains Domain I ( z h) In the domain I, we have to solve the following equation 2 V I V I V ( ) I z 2 = div M for { D D D D (5)

5 176 Dolis et al. The magnetization vector M has onl a component in the z-direction ( M = M z (,) u z ). In this case, the divergence term is zero so (5) reduces to a Laplace equation 2 V I V I V I z 2 = (6) The anti-periodic boundar conditions at = ±D and = ±D lead to V I ( = D ) = V I ( = D ) V I ( = D ) = V I ( = D ) B using the method of separation of variables, the solution of (6) is [ V I (,, z) = A I e k z + B I e k z] (C I cos(w n ) + D I sin(w n )) n=1 m=1 (E I cos(w m ) + F I sin(w m )) with w n = n π ; w m = m π ; k = wn D D 2 + wm 2 where the coefficients A I, B I, C I, D I, E I and F I are the constants to be determined for each harmonic orders (n and m). Because of the smmetries along and, the aial component of the flu densit B z is zero at = ±D and = ±D. B z =±D&=±D = (9) Then, the solution given b (8) simplifies to [ V I (,,z) = A I e k z + B I e k z] cos(w n ) cos(w m ) (1) n=1 m= Domain II (h z h t ) The Laplace equation has to be solved in this air domain 2 V II V II V II z 2 = for (7) (8) { D D D D (11) Because of the smmetries described above, the solution of (11) reduces to [ V II (,, z) = A II e k z + B II e k z] cos(w n ) cos(w m ) (12) n=1 m=1 where the coefficients A II and B II are constants to be determined for each harmonic orders n and m Determination of A I, B I, A II and B II These coefficients are obtained using the interface and boundar conditions. The solutions given b (1) and (12) require the determination of 4 N M unknowns, where N and M are the number of harmonics used to solve this problem. For this purpose, the following conditions are used: Boundar Condition in Domain I at z = We replace (1) into (4) to obtain: H = V I H = V I = z= = z= (13)

6 Progress In Electromagnetics Research M, Vol. 35, These conditions state that the tangential magnetic field components are zero at z =. The development of these equations gives us N M equations linking the coefficients in the domain I. A I + B I = (14) Boundar Conditions in Domain II at z = h t The conditions are the same as in (13), so we can deduce N M equations linking the coefficients of the domain II Interface Conditions between Domains I and II (z = h) A II e k ht + B II e k ht = (15) Domains I and II have the same magnetic permeabilit (µ r = 1), so the normal induction (B z ) and the tangential magnetic field (H and H ) of the domain I and II will be equal at z = h. B Iz z=h = B IIz z=h H I z=h = H II z=h H I z=h = H II z=h (16) Substituting (1), (2) and (3) into (16), ields to the following equations: V I = V II V I = V II V I z + M z = V II z M z is obtained b epanding the magnetization into a double Fourier series along the and -directions in the domain I (Figure 5). The magnetization is epressed as M z (,) = M cos(w n ) cos(w m ) n=1 m=1 (18) 16 B r with M = µ n m π 2 sin(w n l a ) sin(w m l a ) B r is the remanent flu densit of the permanent magnets. Finall, b replacing (1), (12) and (18) in (16), we obtain the 2 N M missing equations. A I e k ha + B I e k ha A II e k ha B II e k ha = k A I e k ha k B I e k ha k A II e k ha + k B II e k ha = M (17) (19) M () z Br/µ -D -la M () z la D Br/µ -D -la la D Figure 5. Magnetization M z as a function of and.

7 178 Dolis et al. The coefficients A I, B I, A II and B II are calculated b solving, for each pair of harmonics n and m, an algebraic sstem of linear equations arising from (14), (15) and (19). The presented methodolog concerns the determination of the solution due to a magnet in one rotor (the magnetization of the magnets on the second rotor is set to zero). The solution due to the second rotor magnets is obtained in the same wa. Finall, the complete solution is obtained b superposing the two results. If one notes B 1 the flu densit due to the magnets on the first rotor, the flu densit B 2 due to the magnets on the second rotor with the spatial offset X i = R mean ϕ, is B 2 (,, z) = B 1 (,, z ) with = X i ; z = (z h t ). Then, the overall solution is obtained b superposition as B (,, z) = B 1 (,, z) + B 2 (,, z) (2) 4. RESULTS AND DISCUSSIONS In this section, we compare the results issued from the 3D analtical model to those obtained from a finite element analsis using COMSOL Multiphsics R. The results are also compared to some measurements Validation of the Linearized Model The geometrical parameters of the linearized coupling are given in Table 1. Figure 6 shows the magnetic flu densities B, B and B z along the -direction in the middle of the air gap (z = h t /2) and = with a magnet offset X i = D. The analtical and the numerical results are in ver good agreement. Figure 7 shows the electromagnetic static forces vs. X i applied to one rotor pole. These forces are derived from the integration of the Mawell stress tensor on a surface in the middle of the air gap. Unlike the F force, the aial force (F z ) has an non nil average value due to the attraction of the iron okes. It is important to note that the computation time of the analtical model for one position of the magnet is about 1 millisecond whereas the numerical calculation (3D mesh and resolution) takes several minutes Curvature Effects As stated in the introduction of this paper, the linearized model ignores the curvature effects. Hence, we have to analze the error introduced b the linearized model when evaluating the performances of a clindrical aial coupling, for a given air gap, this error depends on the radial ecursion (R out R in ) defined in Figure 1 and on the mean pole pitch which is equal to (R out + R in )π/2p. Table 1. Dimension of the linearized magnetic coupling. Parameter Description Unit Value 2 D Length of the domain along mm 1 2 D = 8 D Length of the domain along mm 4 2 l a Length of the magnet along the -ais mm 8 2 l a Length of the magnet along the -ais mm 8 h Height of the magnet mm 15 e Thickness of the air gap mm 5 h t = 2h + e Total height of the domain along z-ais mm 35 B r Remanent flu densit of permanent magnets T 1.2 N Number of harmonics along the -ais - 3 M Number of harmonics along the -ais - 3

8 Progress In Electromagnetics Research M, Vol. 35, Flu densit B (T) z FEM (m) (a).25.2 Flu densit B (T) FEM (m) FEM (b) Flu densit B (T) (m) Figure 6. Flu densit distribution under 1 pole in the middle of the air-gap (z = h t /2) at = l/2: (a) z component, (b) component and (c) component. (c) F (N) FEM Xi (m) (a) Aial force F (N) z FEM Xi (m) (b) Figure 7. (a) Static forces F and (b) aial force F z. To analze the influence of these parameters, we introduce a dimensionless number λ which allows comparing the pole pitch and the radial etrusion λ = p 1 β 1 + β with β = R in R out (21) Two computations are performed: the first one evaluates, b 3D FEM, the torque of the clindrical coupling and the second one uses the analtical model to calculate the torque of the linearized clindrical coupling. For a magnet height h = 1 mm, we var R out, R in and p for two values of air gap (e = 5 mm

9 18 Dolis et al. and 1 mm) in the following intervals: R out = [.1 m to.3 m] with a step of.1 m (three values); R in = [.2 R out to.8 R out ] with a step of.1 R out (seven values); p = [2 to 8] with a step of 1 (seven values); This corresponds to 294 combinations. Figure 8 shows the relative difference between the torque evaluations for different values of λ. It can be seen that the error is less than 4% for an airgap of 5 mm. As epected, this error rises to 6% when the airgap value is 1 mm. This demonstrates the abilit of the analtical model to accuratel evaluate the torque. Hence, the developed model can be used as an efficient tool for quickl sizing clindrical magnetic couplings Eperimental Checking In this section, we compare the analtical results and some measurements carried out on a constructed magnetic coupling [19]. The dimensions of the magnetic coupling are given in Table 2. A photograph of the prototpe is shown Figure 9. This coupling has a value λ = 2. Figured 1 and 11 compare the measured and the computed aial flu densities in the middle of the air gap. The measurements have been taken at no load (ϕ = ) using a Hall effect probe driven b a XY displacement table. The air gap thickness is set to e = 9.5mm. The calculated and the measured aial flu densities B z are almost identical. The small difference when evaluating B z along the radial coordinate is due to the curvature effects. Figure 12 shows the static torque vs. load angle for air gap values e = 9.5mm and e = 4 mm. 6 5 e=5 mm e=1 mm 4 Error (%) λ Figure 8. Error on the maimum torque when using the (linearized-analtical) and the (clindrical-numerical) calculations. Hall probe Figure 9. Demonstrator aial magnetic coupling (p = 6). Flu densit B (T) z Anaticall Eperimental Circumferential angle (degree) Figure 1. Aial flu densit (B z ) vs. angular position in the middle of the air gap (z = h t /2) and mean radius R mean for e = 9.5 mm. Flu densit B (T) z Eperimental Radial coordinate (mm) Figure 11. Aial flu densit (B z ) vs. radial coordinate in the middle of the air gap (z = h t /2) at the center of the pole for e = 9.5mm.

10 Progress In Electromagnetics Research M, Vol. 35, Table 2. Dimension of the magnetic coupling. Parameter Description Unit Value R out Outer radius mm 6 R in Inner radius mm 3 h Magnet thickness mm 7 e Thickness of the air gap mm variable α Magnet opening to pole-pitch ratio -.9 p Number of pole pairs - 6 B r Remanence of permanent magnets T Torque (N.m) Eperimental Torque (N.m) Eperimental Angular displacement (degree) (a) Angular displacement (degree) Figure 12. Static torque vs. load angle for two values of the air gap: (a) e = 9.6mm and (b) e = 4 mm. (b) The relative difference between the analtical results and the measurements is lower than 3% for both air gap values. The model slightl overestimates the torque. 5. CONCLUSIONS In this article, we have presented an analtical model that can quickl predict the flu densit distribution, the forces and the torque of PMs aial magnetic couplings. It has been shown that linearizing the actual clindrical topolog doesn t affect the accurac of the model. In fact, several numerical and eperimental checking show the abilit of the developed analtical model to accuratel evaluate the magnetic field distribution and the torque. This powerful tool can then be used for fast and accurate design of PMs magnetic couplings or to be integrated within evolutionar optimization procedures. ACKNOWLEDGMENT The authors would like to thank the DGA (Direction Générale de l Armement) for funding this work. REFERENCES 1. Yonnet, J. P., Permanent magnet bearings and couplings, IEEE Trans. Magn., Vol. 17, No. 1, , Furlani, E. P., Formulas for the force and torque of aial couplings, IEEE Trans. Magn., Vol. 29, No. 5, , Sep Yonnet, J. P., S. Hemmerlin, E. Rulliere, and G. Lemarquand, calculation of permanent magnet couplings, IEEE Trans. Magn., Vol. 29, No. 6, , Nov Furlani, E., S. Reznik, and A. Kroll, A three-dimensional field solution for radiall polarized clinders, IEEE Trans. Magn., Vol. 31, No. 1, , Jan

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