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A cavity resonator technique for the measurement of isotropic homogeneous nonmagnetic dielectric materials is addressed. The materials to be measured are placed in a well-defined position inside a circular cylindrical cavity resonator through a filling hole in the head end cap of the resonator. The measurement procedure solves the inverse problem based on a variational formulation in terms of the magnetic field in the resonator. The effects of lossy walls and of the filling hole are taken into account by using inhomogeneous surface impedance boundary conditions. In order to reduce the problem to matrix form, the magnetic field in the variational formulation is expanded in terms of undamped eigenmodes of the lossless cavity. The resulting equation is a generalized eigenvalue problem for the unknown material parameters. It is solved by using standard techniques such as the generalized Schur decomposition. The proposed method is tested against experimental data, including measurements at various temperatures, in order to show its capabilities as well as to see the effect of the filling hole on the results.