Axisymmetric Magnetic Field Calculation with Zonal Harmonic Expansion

The magnetic field of an axially symmetric coil or magnetic material system can be computed by expansion of the central and remote zonal harmonics, using the Legendre polynomials. This method can be 100–1000 times faster than the more widely known elliptic integral method and is more general than the similar radial series expansion. We present the zonal harmonic method for field, scalar and vector potential calculation of circular current loops, of general axisymmetric coils and magnetic materials, and of special coils with rectangular cross section, with various source representations: currents, magnetic dipoles and equivalent magnetic charges. We discuss in detail the convergence properties of the zonal harmonic expansions, and we show the generalization of the method for special three-dimensional magnetic systems.