1. 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

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

3. 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

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

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

6. 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

7. 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. *

8. Surface charge method General model
Magnet field description
Charge density replaces magnet

9. 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

10. 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

11. 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.

13. Question slides

14. 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

15. 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.

16. 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.

17. 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)

18. 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

19. 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:
Temperature dependency
PM manufacturing

20. 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

21. 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.

22. 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. *

23. 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. *

24. 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. *