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Application Limits of the Airgap Maxwell Tensor

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Application Limits of the Airgap Maxwell Tensor Application Limits of the Airgap Maxwell Tensor Raphaël Pile, Guillaume Parent, Emile Devillers, Thomas Henneron, Yvonnick Le Menach, Jean Le Besnerais, Jean-Philippe Lecointe To cite this version: Raphaël Pile, Guillaume Parent, Emile Devillers, Thomas Henneron, Yvonnick Le Menach, et al.. Application Limits of the Airgap Maxwell Tensor. 2019. hal-02169268 HAL Id: hal-02169268 https://hal.archives-ouvertes.fr/hal-02169268 Preprint submitted on 1 Jul 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Application Limits of the Airgap Maxwell Tensor Raphael¨ Pile1,2, Guillaume Parent3, Emile Devillers1,2, Thomas Henneron2, Yvonnick Le Menach2, Jean Le Besnerais1, and Jean-Philippe Lecointe3 1EOMYS ENGINEERING, Lille-Hellemmes 59260, France 2Univ. Lille, Arts et Metiers ParisTech, Centrale Lille, HEI, EA 2697 - L2EP - Laboratoire d’Electrotechnique et d’Electronique de Puissance, F-59000 Lille, France 3Univ. Artois, EA 4025, Laboratoire Systemes` Electrotechniques´ et Environnement (LSEE), F-62400 Bethune,´ France Airgap Maxwell tensor is widely used in numerical simulations to accurately compute global magnetic forces and torque but also to estimate magnetic surface force waves, for instance when evaluating magnetic stress harmonics responsible for electromagnetic vibrations and acoustic noise in electrical machines. This article shows that airgap surface forces based on Maxwell tensor with a circular contour depend on the radius of application. The dependence of surface forces on radius is analytically explained using an academic case of a slotless electric machine. The corrective coefficients that should be applied to transfer the airgap surface forces to stator (or rotor) bore radius are presented. The coefficients depend on the airgap geometry and surface forces wavenumber. The use of transfer coefficients is recommended to quantify the deviation due to the radius of application in the airgap. In addition, the transfer coefficients could be applied to correct the radial surface force component on the tip of the teeth assuming infinite permeability. Index Terms—Maxwell tensor, Magnetic stress, Electrical machines, Magneto-mechanical, Vibroacoustic. I. INTRODUCTION application of airgap surface force based on MT and the surface forces at stator bore radius. Due to numerous numerical AXWELL tensor (MT) is used to accurately compute artifacts in practical electrical machines, only a slotless stator M overall forces on a given body. In particular, electrical and rotor with constant permeability is studied in this article. It machines designers are used to apply MT technique in the allows to obtain the analytic expression of the airgap potential middle of the airgap to calculate the electromagnetic torque and resulting surface forces at any radius. Thus the surface experienced by the rotor. This technique has been naturally force transfer deviation is quantified by analytically-derived extended to the study of magnetic force waves experienced by transfer coefficients (20) and (24) which depend on airgap the outer structure - generally the stator - for vibroacoustic geometry and magnetic surface force wavenumber. analysis. The method is generally based on a circular path in the airgap to ease physical interpretations with the Fourier II. PROBLEM DEFINITION transform [1,2]. In that case, one may look for a circular path as close as possible from the stator bore radius to accurately A. Maxwell Tensor estimate forces experienced by the stator. Indeed, for magnetic According to [3,7], for an incompressible linearly linear media [3], the theoretical magnetic surface force density magnetizable media the magnetic flux density B is related to at the air-ferromagnetic interface is exactly given by MT the magnetic field H 8x 2 Ω by B(x) = µ(x;B)H(x) such expression. that the magnetic stress tensor reduces to the following form: However, the radius of the circular path has a strong H · H T = −µ I + µHH (1) influence on calculated stress harmonics [4]–[6]. In Finite m 2 Element Analysis (FEA) simulations, the mesh is generally I the finest in the airgap so that it is recommended to evaluate with the identity tensor. In particular, the Vacuum Maxwell µ = µ forces in the middle of the airgap. Besides, 2D FEA based stress tensor is obtained for 0. The theoretic application on scalar potential cannot ensure high accuracy of both radial of this magnetic stress tensor for the computation of magnetic P and tangential flux density at the air-ferromagnetic interface, surface force density on a contour which exactly corresponds which are both involved in MT expression. Then the numerical to the air-ferromagnetic interface (defined by its local normal n application of the MT at the interface is source of numerical vector ) leads to [3,5]: errors [5]. 1 1 1 2 µ0 − µ 2 P = − Bn − Ht n (2) Therefore, it is interesting to know how to transfer surface 2 µ0 µ 2 forces based on MT calculated in the airgap to another radius, where Bn and Ht are the magnetic flux density locally normal in particular at tooth tip radius. Such a transformation would (resp. magnetic field locally tangential) to the interface. also be useful when using analytical electromagnetic models such as permeance magneto-motive force models, which focus on the accurate calculation of the flux density distribution at B. Airgap Surface Force the airgap radius. One of the most used method to compute magnetic surface Therefore, this article compares the deviation between the forces based on MT is to use Stoke’s theorem [8] along a Stator the magnetic potential z-component Az is imposed at radius Γ Rag such that 8θ 2 [0; 2π]: ∗ Rag Az(Rag; θ) = βn sin(nθ + n); n 2 N (5) The continuity of tangential magnetic field can be expressed R s with the magnetic potential at the interface between the air Rotor (permeability µ0) and ferromagnetic media (permeability µ) 8θ 2 [0; 2π]: Fig. 1. Slotless electrical machine used to compare magnetic surface force on the stator and in the airgap 1 @Az − 1 @Az + (Rs ; θ) = (Rs ; θ) (6) µ0 @r µ @r Thus the boundary condition at r = R with µ ! 1 becomes: circular closed boundary Γ in the airgap at a given radius Rag s r as illustrated in Fig 1. The result gives the total magnetic force @Az (Rs; θ) = 0 (7) Fm acting on either the stator or rotor [1,5]: @r I For these boundary conditions, Poisson’s equation is solved µ0 2 Fm = − H n + µ0HnH dΓ (3) for the 2D magnetic vector potential in polar coordinates 8θ 2 Γ 2 [0; 2π] and 8r 2 [Rag; Rs]: Then the airgap MT approximation is to assume the term 1 @ 1 @A @2A under the integral to be the magnetic surface force density in z + z = 0 (8) the polar referential: r @r r @r @θ2 This partial derivative problem can be solved 1 2 µ0 2 Pr(Rag) ≈ Br(Rag) − Hθ(Rag) analytically [9]. Thus the magnetic potential, flux density and 2µ0 2 (4) field can be analytically computed in the studied domain. The P (R ) ≈ B (R )H (R ) θ ag r ag θ ag next section provides the calculation steps which lead to the with Pr the radial magnetic surface force density, Pθ the analytical expression of flux density (15) in the slotless case. tangential magnetic surface force density, Br the radial magnetic flux density, and Hθ the tangential magnetic field. D. Analytic Magnetic Solving in Slotless Case Neglecting the ferromagnetic permeability µ contribution A solution exists and is unique for the previous system into (2) gives an expression very similar to (4) justifying the composed of (8), (7) and (5) according to [8]. Then a method use of airgap surface force approximation. consists in stating a function Az and to check if it fulfills The application of (2) at the air-ferromagnetic interface the boundary conditions: a solution similar to [9] is searched remains source of numerical errors [5]: in the case of 8θ 2 [0; 2π] and 8r 2 [Rag; Rs]: numerical model based on FEA, the continuity of both normal γ E (r; R ) + α E (R ; r) magnetic flux Bn density and tangential magnetic field Ht n n s n n ag Az(r; θ) = sin(nθ + n) (9) cannot be ensured on the interface between materials. As a En(Rag; Rs) consequence, publications often use airgap approximation (4) with En a polynomial function defined by: for the vibroacoustic design of electrical machines because 2 n n the middle of the airgap is better meshed, it is less En :(x; y) 2 R ! (x=y) − (y=x) (10) sensitive to variational formulation, and it allows the physical Under this form, the vector Az is satisfying Poisson’s understanding of each magnetic force wave (for instance with equation (8). Then the coefficient γn, αn, have to be analytical models) [1,2]. determined in order to satisfy the boundary conditions. If a correct set of these coefficients is found, then Az would be the C. Studied Slotless Case unique solution of the problem. First, satisfying the boundary condition (5) in the airgap leads to: Understanding the sources of airgap surface force (4) variations is a difficult task because of numerous artifacts such γn = βn (11) as slotting effect, sharp geometries, interference between the The second boundary condition (7) leads to: wavenumbers, etc. To avoid these main artifacts, a specific case is defined: αn = 2βn=Fn(Rag; Rs) (12) • Slotless stator and rotor to avoid sharp geometries and with Fn a polynomial function defined by: slotting effect.
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