Analytical Surface Charge Method for Rotated Permanent Magnets: Boundary Element Method Comparison and Experimental Validation J.R.M. van Dam, J.J.H. Paulides, E.A. Lomonova, and M. Dhaens DI-07 Analytical Surface Charge Method for Rotated PMs: Boundary Element Method Comparison and Experimental Validation Wednesday, 2:42 pm 1:30 pm - 3:18 pm DI New Applications: Magneto-Elastic, Magneto-Electric and Other Devices Aqua 310 (3rd Floor) 12 minutes per presentation

Contents Introduction Boundary element method Surface charge method Results for rotated permanent magnets Conclusions

Introduction Energy harvesting application Energy harvesting based on permanent magnets An optimal harvester design starts with an accurate permanent magnet ineteraction model Non-magnetic Permanent magnet

Introduction Permanent magnets Interaction between permanent magnets 𝜽 PM2 PM2 z y x 𝜃 PM1 45° 10° 90°

Introduction Model requirements The PM interaction model requirements consist of Accurate magnetic field description Direct force calculation Allow for rotated PMs Preferably analytical

Introduction Magnetic models Methods for modeling magnetic structures Method DOMAIN SOLUTION TYPE RELATIVE ROTATIONS BETWEEN PMs FEM Bounded to infinite box Bounded to infinite box Numerical Numerical Yes Yes BEM Unbounded Unbounded Numerical Numerical Yes Yes Fourier Bounded to Dirichlet boundary condition Bounded to Dirichlet boundary condition Analytical Analytical No No The design process of the electromagnetic applications on the previous slide, would benefit from a model which: Provide an analytical solution (the equations are solved in the entire domain, albeit for a limited number of cases in the charge model); Are meshfree (solution speed); Provide the required physical quantities such as the fields, forces, torques, and energies; And are able to do this for all dimensions. The method of choice in this paper, is the surface charge method. Charge model Unbounded Unbounded Analytical Analytical Around x- or y-axis Around x- or y-axis

Boundary Element Method (BEM) In comparison to FEM * FEM Numerical methods Geometry discretization FEM: Full geometry BEM: Boundaries BEM is suitable for unbounded problems BEM requires linear and homogeneous material properties BEM In the BEM, the geomertry discretization results in Problem dimension reduced by one As each point p in the domain is expressed in terms of the boundary values, once all boundary values are known ANY potential value within the domain can now be found, which makes the BEM a powerful method. * http://www.conceptanalyst.com/technical.html

Surface charge method General model Magnet field description Charge density replaces magnet ++++++++ - - - - - - - -

Results for rotated permanent magnets Simulation results Analytically, all positions along the path are calculated The BEM matches closely to ANA, at the expense of setup and computation time The FEM shows deviations due to numerical noise Numerical noise

Results for rotated permanent magnets Experimental validation Close agreement between simulations and experiments has been found An additional advantage of the surface charge method is its ability to directly calculate torques and stiffnesses Logging PC dSPACE PM2 BEM is an interesting alternative as it allows for fast calculation in simple problems, and it allows for multiple rotations, whereas the surface charge method does not allow for rotations around the z-axis. However, for more complex problems, the non-symmetrical matrix drastically increases computation time. Unlike the surface charge method, the BEM assumes linear and homogeneous material properties. This is an important drawback in the design and analysis of electrical machines. Deviation between ANA and EXP at 10 degrees is attributed to inaccuracies in the test setup, which are composed of deviations in the remanent magnetization and magnetization angle of the PMs, 3-D printer manufacturing tolerances, and deviations in the alignment of the test setup with respect to the xy-plane of the load cell. An additional advantage of the surface charge method is its ability to, directly from magnetic fields and forces, calculate torques and stiffnesses. PM1 Load cell 3-D printed test setup

Conclusions The progress in the analytical surface charge method for permanent magnets (PMs) has been addressed. The method has been derived, and the expressions for the interaction force between PMs with an (anti-)parallel, perpendicular, and rotated magnetization have been provided. The surface charge method has been applied to rotated PMs, the analytical results of which have been experimentally validated. In the boundary element method, similarly accurate results have been achieved compared to the surface charge method, at the expense of simulation setup time. Moreover, the surface charge method allows for direct calculation of torques and stiffnesses. Currently, 2-D force expressions can be obtained if PMs are rotated around non-magnetized axes. Future work will consider additional rotations around magnetized axes.

Question slides

Surface charge method Development of charge model for cuboidal magnets * * z y x z y x z y x 3-D: B, F, W Fy, Fz, Ty, Tz 3-D: F, K, T Surface charge model (Anti-)Parallel magnetization Rotated ( 𝜃 𝑥 ) magnetization Perpendicular magnetization Variable charge density * * 1984 1999 & 2013 2009 2014 present μr =1 (analytical) μr,PM =1.05 (analytical) * μr,steel (semi-analytical) * Charpentier (1999) IEEE Trans. Magn. Robertson (2013) PhD Dissertation * Akoun & Yonnet (1984) IEEE Trans. Magn. * Van Casteren (2014) IEEE Trans. Magn. * Kremers (2013) IEEE Trans. Magn. * Jansen (2009) Sensor Letters

Surface charge method Mathematical correctness The net charge of a magnet should equal to zero, to satisfy Gauss’ law for magnetism Relative permeability corrresponds with measurements (Kremers (2013) IEEE Trans. Magn.) Field solution Outside of the magnet: Correct field solution. Magnet surfaces: Improved accuracy by using a variable charge density. Inside of the magnet: Incorrect field solution. ++++++++ - - - - - - - - Van Casteren (2014) IEEE Trans. Magn.

Surface charge method Equations Magnetostatic Maxwell equations for current-free regions 𝛁× 𝑯 =𝟎 and 𝛁⋅ 𝑩 =𝟎 Introduce the magnetic scalar potential, 𝝋 𝒎 𝛁× 𝛁 𝝋 𝒎 =𝟎 and 𝑯 =−𝛁 𝝋 𝒎 Substitute the constitutive relation 𝑩 = 𝝁 𝟎 ( 𝑯 + 𝑴 ) to obtain 𝛁 𝟐 𝝋 𝒎 =𝛁⋅ 𝑴 . Here, 𝑴 is the magnetization vector of the permanent magnet, assumed bounded by the PM volume, hence, 𝝋 𝒎 𝒙 = 𝟏 𝟒𝝅 𝑽 𝝆 𝒎 ( 𝒙 ′ ) | 𝒙 − 𝒙 ′| 𝐝𝒗′ + 𝟏 𝟒𝝅 𝑺 𝝈 𝒎 ( 𝒙 ′) | 𝒙 − 𝒙 ′| 𝐝𝒔′ . For a uniformly magnetized PM, 𝝆 𝒎 =𝟎.

Design tool for machines and actuators State-of-the-art and extensions Design of a transverse flux machine Highly permeable steel is incorporated in a (semi-)analytical charge model Non-cuboidal inidividual PM shapes in charge method Charge distribution * Charge model The present surface charge model is suitable for interactions between similarly shaped PMs, but differentely shaped PMs, soft-magnetic materials, multiphysical and manufacturing effects have not yet been taken into account. Cylinder Arc Ring Triangle Pyramidal frustum * M.F.J. Kremers and D.T.E.H. van Casteren (2015)

Design tool for machines and actuators Hard– versus soft–magnetic materials In order for the surface charge method to replace the FEM Operating region PMs Linear 𝝁 𝒓 Operating region soft-magnetics Nonlinear 𝝁 𝒓 Analytically (Semi-)analytically

Design tool for machines and actuators Different PM shapes and multiphysical effects Eddy currents in PM Brittle nature Interactions The required model extensions start with the inclusion of the forces between cylindrical PMs with rotation around one axis. Subsequently, the forces and torques between pairs of cuboidal and pairs of cylindrical PMs with arbitrary rotation should be considered. Then, expressions for the forces and torques between pairs of spherical, triangular, and, finally, differently-shaped permanent magnets should be developed. A latter example is the force between spherical and triangular PMs. PM manufacturing process: neorem.fi Brittle PM: http://theodoregray.com/periodictable/Elements/060/index.s14.html Temperature dependency PM manufacturing

Finite Element Method Infinite box Using an infinite box, the boundary condition translates to the magnetic potential at infinity equating to null. PM2 AIR PM1 INFINITE BOX

BEM versus FEM Active volume versus source volume BEM: only sources FEM: very large volume (air) does not contain sources, so very much unused mesh nodes and elements, which makes it rather slow.

Boundary conditions Dirichlet: Specifies the value of the function on a surface, . Neumann: Specifies the normal derivative of the function on a surface, . Robin: The sum of a Dirichlet and a Neumann boundary condition. * http://mathworld.wolfram.com/BoundaryConditions.html

Dirichlet boundary condition When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take on along the boundary of the domain. The question of finding solutions to such equations is known as the Dirichlet problem. In engineering applications, a Dirichlet boundary condition may also be referred to as a fixed boundary condition. * https://en.wikipedia.org/wiki/Dirichlet_boundary_condition

Boundary Element Method (BEM) In comparison to FEM Resulting flux lines * FEM Numerical methods Geometry discretization FEM: Full geometry BEM: Boundaries BEM is suitable for unbounded problems BEM requires linear and homogeneous material properties BEM In the BEM, the geomertry discretization results in Problem dimension reduced by one As each point p in the domain is expressed in terms of the boundary values, once all boundary values are known ANY potential value within the domain can now be found, which makes the BEM a powerful method. * http://www.conceptanalyst.com/technical.html