![]() This manuscript version is made available under the CC-BY-NC-ND 4.0 license These equations are significantly faster than finite element simulations, and can therefore be used for efficient optimisation of magnet geometry or topology, real-time simulations, and for approximation of curved surfaces of permanent magnets. These equations were implemented in Matlab code and validated using finite element simulations and literature. This paper presents a new set of simplified and exact equations describing the magnetic field produced by an arbitrarily-shaped polyhedral permanent magnet with constant uniform magnetisation and a relative permeability of unity. Some authors have derived magnetic field equations for polyhedral magnets, allowing more general magnet shapes, but these are either not fully simplified or computationally inefficient. In recent decades, researchers have derived equations describing magnetic fields, forces, and torques, but these are usually limited to cuboid or ring-shaped magnets. For example, the analytical expression can be included in optimization software allowing to directly obtaining the shape optimization by a fast way.ĭue to their wide use in industrial and commercial devices, it is important to accurately and effectively model permanent magnets, leading to better magnet designs and more desirable magnetic characteristics. The 3-D analytical expressions are difficult to obtain, but the expressions of energy, force, and torque are very simple to use. Analytical calculation owns many advantages in comparison with other calculation methods. ![]() The results have been verified and validated by comparison with finite-element calculation. The three components of the torque are written with functions based on logarithm and arc-tangent. The torque is calculated for rotational movement of the second magnet around its center. The analytical calculation is made by replacing magnetizations by distributions of magnetic charges on the magnet poles. ![]() ![]() The only hypothesis is that the magnetizations J and J ' are supposed to be rigid and uniform in each magnet. We have solved it, so that now all the interactions (energy, three forces components, and three torque components) can be expressed by analytical formulations. Until now, the torque components have never been analytically calculated because of the angular derivation. It was published for the first time in 1984. By 3-D fully analytical calculation, up to now, only the force components problem has been solved. Each elementary magnet is submitted to a force and a torque. Most of the systems working by magnet interactions can be calculated by superposition of the interactions between parallelepiped elementary magnets. ![]()
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