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