A Method for Calculating Active Feedback System to Control Vertical Position of Plasma in a Tokamak

In designing tokamaks, the maintenance of vertical stability of plasma is one of the most important problems. For this purpose systems of passive and active feedbacks are applied. The role of passive system consisting of a vacuum vessel and passive stabilizer plates is to suppress fast MHD (magnetohydrodynamic) instabilities. The active feedback system is applied to control slow motions of plasma. The objective of this paper is to investigate three successive problems the solution of which will allow to determine the possibility to control plasma motions. The first problem is the vertical stability problem under the assumption of ideal conductivity of plasma and passive stabilizing elements. The problem is solved analytically and on the basis of the obtained solution a criterion of MHD-stability is formulated. The second problem is the vertical stability when finite conductivity of stabilizing elements is taken into account. The dispersion equation relative to instability growth rate is obtained and analyzed. For practical values of the parameters it is shown that there is a unique root with positive real part, which presents the growth rate of only unstable mode. The third problem is connected with the control of plasma vertical position with application of active feedback system. The problem of calculation of feedback control parameters is formulated as an optimization problem and its approximate solving method is suggested.