Eddy current method is widely used in practice for quality testing of conducting materials (examples include determination of electrical conductivity, thickness of metal coatings, identification of flaws in a conducting medium). In the present paper a semi-analytical method for solution of direct eddy current problems for the case of a conducting medium of finite size is considered. The method is applied to several eddy current problems with cylindrical symmetry. The following problem is analyzed in detail. Consider a coil with alternating current located above a conducting medium in the form of a circular cylinder (such a model can be used for design of coin validators which are based on the estimation of electrical conductivity of a coin). We assume that the electromagnetic field is exactly zero at a sufficiently large distance from the coil (the distance can be chosen on the basis of the required accuracy of the solution). The solution is constructed using the method of separation of variables which includes two steps where numerical calculations are necessary: (a) computation of complex eigenvalues without good inital guess for the roots and (b) solution of a system of linear algebraic equations. Computations of the change in impedance of the coil for different frequencies with the semi-analytical method are in good agreement with experimental data and results of numerical simulation with finite element method. Solution of other problems with cylindrical symmetry is also discussed (a flaw in the form of a circular cylinder in a conducting half-space or a plate). Such models can be used for the analysis of quality of spot welding (in case of a volumetric flaw) and estimation of the effect of corrosion (for surface flaws).