On the use of the total scalar potential on the numerical solution of fields problems in electromagnetics

SUMMARY The paper summarizes the formulation of a set of computer algorithms for the solution of the three-dimensional non-linear Poisson field problem. Results are presented that were obtained by applying algorithms to the analysis of two-dimensional magnetostatic fields. Scalar and vector potentials were used, and it is shown that the convenient single valued scalar potential associated with the induced sources gives severe accuracy problems in permeable regions. The results become as good as those obtained using vector potential if the scalar potential associated with the total field is used for permeable regions. The combination of two scalar potentials has a significant advantage for three-dimensional problems. The non-linear Poisson equation occurs in many areas of physics and engineering. The equation is relatively easy to solve compared to other defining equations, but for many applications the solutions must be very accurate. In magnetostatic problems, for example, the geometry of boundaries and surfaces separating differing media is often complicated and field accuracies of the order of 0.1 per cent and higher are essential. These conditions are frequently encountered in the wide range of electromagnets associated with the design of charged particle accelerators, spectrometers, detectors, focusing devices and plasma containment experiments used in physics and also in the broad spectrum of machines, transformers, etc. used in electrical engineering. Although the methods discussed in this paper are of general applicability they are looked at with particular reference to electromagnetics. Both differential and integral operator formulations have been used to solve the magnetostatic problem. Many well established programs solve the two-dimensional cases using differential formulations based directly on the defining equation usually in terms of the single component vector potentiaI. 1 - 3 These programs are capable of giving high accuracy although the position of the far field boundary can have a significant effect on the results. This latter difficulty has been overcome at the expense of increased computational cost by solving the integral form of the equations in terms of field components directly. An added advantage of this approach is that only regions containing material media (e.g., iron) are discretised. This is very useful in extending to three dimensions where the need to have a mesh of elements connecting many different regions of complex shape is seen as a limitation of the differential approach. This difficulty is avoided in programs based on integral formulations 4-6 and at the present time these provide a general technique for solving non-linear three-dimensional magnetostatic problems providing high accuracy is not required.