FORCE CALCULATIONS WITH 3-DIMENSIONAL SCALAR POTENTIAL BASED FINITE-ELEMENT PROGRAMS

In the design of many electromagnetomechanical devices one of the prime parameters of interest is the mechanical force of torque produced by the device. The generated forces are of interest in two forms. The first is the distribution, which might be used to indicate a particular loading on some part of the structure; the second is a global value which appears at the mechanical interface. It is with the calculation of the latter that this paper is concerned. Much has been published concerning force calculations in two-dimensional solutions and it is the intention of this paper to discuss the extension of these methods to three-dimensional, scalar potential based analyses and to give examples of computed and measured results on several different devices. In general, the methods that are used in two dimensions are applicable in three dimensions. These are ususally classified as single- and multiple-solution techniques. Into the first category fall Maxwell stresses and the Coulomb implementation of virtual work; into the second falls the classical virtual work approach. The Maxwell stress approach requires a surface integral of a product of flux density (B) components. In two dimensions this is a relatively simple task; it reduces to defining a line in the plane and computing B as the differential of A (the vector potential). In three dimensions, the definition of a surface may not be simple and, if a scalar potential is used for the solution, the computation of flux density components is far from trivial. When these requirements are compared with the classical virtual work approach, the reduction in computing effort by using a single-solution technique many not seem so significant. A second factor influencing the use of a multiple-solution approach is fundamental to the finite-element method; in general, the method produces accurate values for global quantities such as coenergy, with coarser discretizations than those required for local accuracy. Single-solution methods tend to require high-accuracy local fields and thus require considerably more computer effort per solution. On problems ranging from a disk drive actuator to a permanent magnet generator accuracies within 3% of measured have been computed with virtual work. The equivalent two-dimensional solutions did not do better than 10%.