In this paper exact analytical expressions for the impedance of a cylindrical air-core coil above a layered metal structure whose conductivity varies continuously with depth are presented. Although the model is general, attention is focused on three conductivity profiles: the linear, the quadratic and the exponential. The derived expressions for the impedance change for each profile could provide a useful tool for the solution of the inverse problem: that of determining the conductivity from variable frequency measurements of the impedance. Furthermore, the obtained final formulas contain elegant mathematical functions and show a substantially higher computational efficiency with respect to existing methods.
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