A new technique for calculating toroidal harmonic coefficients for curl-free magnetic fields in helical toroidal devices

Abstract

In a curl-free toroidal magnetic system, the vector magnetic field may be derived from a scalar magnetic potential expressed as a series of toroidal harmonic functions. A new method of computing the coefficients of the toroidal harmonics of the magnetic scalar potential is presented in this paper. The method uses Fourier integrals performed over the poloidal and toroidal angles on a closed surface. The integrand includes the component of the magnetic field normal to the toroidal surface and a multiplying factor chosen to create a set of simple equations which are linear in the coefficients. This set of equations is then inverted to obtain the toroidal harmonic coefficients. The magnetic fields calculated using the magnetic scalar potential with these coefficients are found to agree well with the original magnetic field even for a small number of coefficients. For a small number of coefficients used in the scalar potential, the time required to calculate the magnetic field is reduced by up to a factor of five from standard Biot-Savart methods.