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A new eigenmode algorithm based on the method of lines (MoL) is presented for the analysis of different kinds of discontinuities in microstrip and waveguide circuts. The basic procedure known from the integrated optics as MoL-BPM is adapted and extended for the application to microwave circuits by including metallizations and taking into account reflections at all discontinuities. The easy mathematical description and a general formulation allow an investigation of a great variety of complex and technical interesting structures. The fact that all eigenmodes are considered in the analysis of the structures, the analytical calculation in one direction and the matching of the field components instead of the modes at the discontinuities guarantee a great accuracy of the calculations. The validity of the new procedure is proved by a comparison of first numerical results for an impedance step and a microstrip bend. Especially the improvement on the previous results with the MoL is made evident.