Stellarator magnetic field configurations are analysed using a harmonic expansion of the scalar potential. Harmonic functions for fields due both to external coils (vacuum fields) and to plasma currents are derived from those for a straight stellarator, using a perturbation technique to obtain first-order toroidal corrections. In the case of the vacuum fields, the coefficients, or generalized multipole moments, are determined by a least-squares fit to the 'exact' fields, which are calculated from the coil geometry using the Biot-Savart method. The mode structure of the Advanced Toroidal Facility (ATF) torsatron, in particular, is basically that of a puresystem, augmented by an asymmetric broadening in the poloidal mode number spectrum because of toroidicity. Configuration control in ATF is accomplished by varying the axisymmetric moments: the magnetic axis shift is most sensitive to the m = 1 moment and the central rotational transform is most sensitive to the m = 2 moment. These moments, in turn, are expressible simply as linear combinations of the coil currents, which allows them to be determined readily from experimental data.
Optimization techniques for modular stellarator coils