Measurement of plasma current distribution using magnetic probes

Abstract

Magnetic probes are commonly used for measuring current distributions in plasmas. Unfortunately they must be placed inside the plasma, perturbing the current distribution under measurement. Other authors have studied the influence of a magnetic probe on a current sheet for both thin and thick current distributions. For the case of thick current sheets (as happens in plasma focus discharges), the measured voltage can be written as a first kind convolution of a Volterra equation. In order to get the real current distribution, an inversion of this equation is needed. First kind Volterra equations (Abel integrals included) are in some sense not a well-posed problem and, for numerical inversion, the input data must be smoothed. Further, the magnetic probe problem represents a particular case among first kind Volterra equations because the kernel is singular and the illness of the problem becomes more severe. On the other hand, we are interested in inverting experimental data with errors, so it is mandatory to evaluate the error in the inverted profile. For Volterra integral equations, error propagation has not been widely discussed in the literature. In this work a direct method, the matrix method plus an explicit filter, are used to obtain the solution. One of the main advantages of the method is the easy calculation of the error in the inverted signal from the errors of the parameters and the input data. In order to minimize error magnification, some limitations to the geometrical configuration of the probe are found. Also, it is shown that this method is a good compromise between accuracy and computational speed (it can be used on line working with thousands of data points).