Optimal design of feedback coils for the control of external modes in tokamaks

A formalism is developed for optimizing the design of feedback coils placed around a tokamak plasma in order to control the resistive shell mode. It is found that feedback schemes for controlling the resistive shell mode fail whenever the distortion of the mode structure by the currents circulating in the feedback coils becomes too strong, in which case the mode escapes through the gaps between the coils, or through the centers of the coils. The main aim of the optimization process is to reduce this distortion by minimizing the coupling of different Fourier harmonics due to the feedback currents. It is possible to define a quantity α0 which parametrizes the strength of the coupling. Feedback fails for α0⩾1. The optimization procedure consists of minimizing α0 subject to practical constraints. If there are very many evenly spaced feedback coils surrounding the plasma in the poloidal direction then the optimization can be performed analytically. Otherwise, the optimization must be performed numerically. The optimal configuration is to have many, large, overlapping coils in the poloidal direction.