The modeling of eddy current (EC) problems is not only helpful for the probe design and the interpretation of measurement data, but also essential to the corresponding inverse problems. In order to deal with high conductivity and low operating frequency, the conventional pure numerical methods require large memory and long computing time to accurately solve EC problems. In this dissertation, semi-analytical approaches are presented to efficiently solve a series of EC problems in cylindrical polar coordinates.

By applying the truncation in an unbounded domain, the fields are expanded into a linear combination of the eigenfunctions. Each eigenfunction belongs to different eigenvalue. The expansion coefficients are determined by the continuity conditions at the interface between different regions. For a given coil source, the semi-analytic solution of the fields in an unflawed conductor can be obtained using this method.

The electromagnetic fields are expressed in terms of transverse electric and transverse magnetic modes. At a circularly cylindrical interface, these two modes are coupled with each other. The coupling is considered in the borehole and tube problems. Then the concept is further extended into the borehole opening problem, in which the coupling between different eigenmodes is added.

A well-developed crack model is used to solve the crack problems and estimate the impedance change of an induction coil due to a thin crack in conductors. The thin crack is equivalent to an electric current dipole source distributed over the crack region. An electric integral equation is constructed to solve this equivalent source, and the coil impedance change is predicted. The dyadic Green's kernel, which is required in the integral equation, is derived for different structures.

Special numerical implementations are proposed to deal with the issues arising in our program to make sure the calculation is accurate and the computational cost is small. The convergence of the series expansion is discussed. Methods used to compute the eigenvalues are also investigated. Predictions of the probe signals show a good agreement with experimental measurements in several examples. Compared to numerical methods, the approach is much faster to get the results with same accuracy.