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We present a development on an existing model for the electromagnetic interaction of a coil with a conductive cylinder. We can now calculate analytically both the electromagnetic field and the impedance of arbitrarily positioned probe coils. This is possible by using an appropriate expansion of cylindrical eigenfunctions in terms of eigenfunctions of the Cartesian coordinate system. The extended model is useful in simulating eddy current inspections of cylindrical structures under conditions of probe tilt as well as inspections with special coil orientations like the rotating probe coil.