Interactions between polyhedral permanent magnets

Abstract

With the current trend toward industrial automation, efficient energy generation, and electric motor vehicles, permanent magnets are seeing more widespread use than ever before. They permeate our world, enabling sound generation through loudspeakers, mass data storage in the server farms keeping us online, and even the vibration motors in our pockets notifying us of new messages. Never before have permanent magnets seen such widespread use, and thus it is paramount to understand the interactions between them. The primary aim of this thesis is to investigate and model the magnetic fields produced by generalised polyhedral permanent magnets, and the forces and torques between them. To achieve this aim, two main objectives were identified. The first objective was to analytically solve the magnetic charge model field equations for arbitrary polyhedral permanent magnets with a relative permeability of unity. This was performed using two unique approaches, leading to two unique but equivalent sets of field solutions, with the first being more effective when the field is calculated at few points, and the second being more effective when the field is calculated at many points. These field solutions were implemented in MATLAB code with a focus on computation efficiency, thus reducing calculation time. The solutions may also be used to numerically integrate over the surface of another magnet to accurately estimate the force and torque imparted. The second objective was to derive a methodology to model the field due to a polyhedral permanent magnet with non-unity relative permeability. This was done by applying a surface mesh to a magnet, and allowing the ‘magnetic charge’ on each surface element to vary based on the permeability and magnetic field passing through the element. This was derived in such a way that the field is calculated only once, with no iteration required. Rather, a matrix equation is solved to give the surface charge distribution, leading to calculations of the magnetic fields, forces, and torques based on the previous objective. This was again implemented in MATLAB code with focus on computation efficiency, leading to fast calculations. This thesis begins with a short prologue, giving a brief historical overview of the development of magnetism as a physical science. Chapter 1 follows, outlining the theory used for modelling magnets and giving a review of relevant literature. Chapters 2 and 3 outline two new methods for calculating the magnetic field produced by general ideal polyhedral permanent magnets, each with benefits and drawbacks over the other. In addition, Chapter 2 found that a pair of pyramid frustum magnets produce a larger mutual force than a pair of cuboidal magnets, suggesting further investigation into frustum magnets. Chapter 4 applies the methodology from Chapter 3 to a planar array of frustum magnets, finding no significant benefit over traditional cuboidal planar arrays. Chapter 5 explores magnetic permeability, deriving a methodology to calculate magnetic fields, forces, and torques imparted by linear magnetic materials of polyhedral geometry. Finally, the thesis is concluded in Chapter 6, summarising the preceding chapters and outlining potential future work to follow this thesis. The primary outcome of this thesis is the development of a new methodology which can accurately and quickly compute the magnetic fields, forces, and torques imparted by magnetic materials of polyhedral geometry. The methodology allows for materials with constant non-unity relative permeability, more accurately reflecting permanent magnet materials and magnetic behaviour. Moreover, other geometries may be accurately approximated by polyhedra and the methodology applied, allowing the fast and accurate approximation of any current-free magnetostatic system.