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Finite-element solutions for devices with permanent magnets

The magnetic field of a complicated magnetic circuit consisting of both electromagnets and permanent magnets situated in air can be derived from the solution of a boundary value problem. Such a problem can be formulated so as to find a potential function giving a stationary value to an appropriate functional. A numerical solution can then be found by applying the finite-element method. Expressions of the functional for materials with different magnetic properties and the derivation of the numerical solution of the boundary value problem by the finite-element method are given with special emphasis on systems with permanent magnets. In one of the two examples illustrating the application of the method, design considerations of the magnetic device are related to the solution.