Practical method for analyzing the effect of induced eddy currents on the vertical instability of a tokamak plasma

The effect of induced eddy currents on the vertical instability of a noncircular tokamak plasma is studied in the formalism of circuit equations. Assuming a rigid plasma and an arbitrary yet equatorially symmetric configuration we show that through diagonalization and the use of a positivity condition following from Lenz's principle the dispersion relation can always be cast into the form of the in-hour equation in fission reactor kinetic theory. The graphical representation of this dispersion relation shows that: (a) an unstable equilibrium cannot be stabilized; (b) there is only one positive time period and all the other solutions are damped modes; and (c) only the most damped mode may be oscillatory. In analogy to the one delayed neutron group approximation to the in-hour equation, a one pair of eddy currents (OPEC) approximation is introduced, which unambiguously defines an effective resistance and effective inductance of the eddy current carriers, these two parameters being mathematically similar to the averaged delayed neutron lifetime and the total delayed neutron population. The OPEC approximation reduces the dispersion relation to a cubic equation, greatly simplifying the computation