Off-axis expansion solution of Laplace's equation: Application to accurate and rapid calculation of coil magnetic fields

A flexible algorithm for the accurate computation of off-axis magnetic fields of coils in cylindrical geometry is presented. The method employs a partial power series decomposition of Laplace's equation about the symmetry axis where the series coefficients are derivatives of the field along the axis. A method for computing high order analytic derivatives for four "basic" coil types (loop, annular disk, thin solenoid, and full coil) will be demonstrated. Utilizing these derivatives, highly accurate off-axis fields can be calculated for the basic coil types. For ideal current loops, field errors of less than 0.1% of the exact elliptic integral solution can be obtained out to approximately 70% of the loop radius. Accuracy improves substantially near the symmetry axis and is higher than normally achievable with mesh-based or integral solvers. The simplicity, compactness and speed of this method make it a good adjunct to other techniques and ideal as a module for incorporation into more general programs.