Representation of axisymmetric magnetic fields in computer programs

The standard expression for the off-axis fields in terms of the axial values and their differentials is examined. It is shown that if terms higher than the second are used, the accuracy becomes worse instead of better, unless the input data are mathematically perfect. It is not possible to remedy this by smoothing experimental data with a polynomial curve-fitting routine. Alternative expressions of the fields in terms of Legendre polynomials and of elliptic integrals are considered. It is shown that the elliptic integral method is the most convenient in practical cases, and a fast computer routine is given for the required K and E. In this method, the fields are defined by the parameters of a set of current loops, and the problem of finding these parameters is discussed. No general solution can be given, but a considerable class of cases is reduced to an explicit solution of a cubic equation by Cardan's method. It is shown that hybrid methods can also be used; the data is first represented by a set of current loops. A precise set of axial values is calculated from these loops, and the differential routine can then be used with acceptable accuracy, including terms up to the sixth order.