Magnetic torque or force calculation by direct differentiation of finite element coenergy

A vector-potential finite-element method for calculating magnetic force or torque by direct analytic differentiation of coenergy is derived and applied to two- and three-dimensional actuators. The results are shown to agree well with calculations using the conventional difference between coenergies at two positions and with experimental measurements. Another method of force calculation, Maxwell's stress tensor, is also derived by differentiating the coenergy, resulting in a formula that is applicable on the surface of nonlinear materials.<>