Calibration of the high-frequency magnetic fluctuation diagnostic in plasma devices

The increasing reservoirs of energetic particles which drive high-frequency modes, together with advances in the understanding of magnetohydrodynamics, have led to a need for higher-frequency (50 kHz to >20 MHz) measurements of magnetic field fluctuations in magnetic fusion devices such as tokamaks. This article uses transmission line equations to derive the voltage response of a Mirnov coil at the digitizer end of a transmission line of length l. It is shown that, depending on the terminations of the line, resonances can occur even for lλ≪1, with λ the wavelength of a fluctuation in the transmission line. A lumped-circuit model based on the approach of Heeter [R. F. Heeter, A. F. Fasoli, S. Ali-Arshad, and J. M. Moret. Rev. Sci. Instrum. 71, 4092 (2000)] is extended to enable the inclusion simultaneously of both serial resistance and parallel conductance elements. As originally proposed by Heeter the lumped-circuit model offers the advantage of remote calibration; this may be of particular value when upgrading existing systems to operate at frequencies above the original design specification. It is formally shown that the transmission line equations for the transfer function and measured impedance reduce to those of the lumped circuit model of Heeter under specific conditions. The result extends the use of the lumped-circuit model of Heeter, which can be used to extract the transfer function from measurement of the impedance, beyond the case of an open-circuit termination. Although the numerical procedure does exhibit some problems associated with non-uniqueness, it provides a simple calibration method for systems that are not well defined. Using typical parameters for a high-frequency Mirnov coil installed on the Joint European Torus (JET) tokamak, the lumped-circuit approximation agrees with the steady-state transmission line model to within 0.015° in phase and 22% in amplitude for frequencies up to 1 MHz. A matched termination, though eliminating line resonances and reducing the length of time for the system to reach steady state, is inappropriate for the JET-type coils which exhibit significant temperature-dependent resistance. Finally, for fluctuations of finite duration, a method of computing the discrepancy due to the simplifying assumption of Fourier-stationary conditions is described.