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In this study, the efficient full-wave analysis of passive devices composed of circular and arbitrarily-shaped waveguides is considered. For this purpose, the well-known boundary integral-resonant mode expansion (BI-RME) method has been properly extended. Circular waveguides are used for resonant mode expansion, whereas the arbitrary contour is defined by any combination of straight, circular and elliptical segments, thus allowing the exact representation of the most widely used geometries. The proposed algorithm extends previous implementations of the BI-RME method based on circular waveguides by considering circular and elliptical arcs for defining arbitrary geometries. Similarly, it allows the efficient analysis of passive devices based on circular waveguides with arbitrary perturbations, thus providing more accurate results with less computational efforts than a rectangular waveguide-based BI-RME approach. The extended method has been successfully tested with several practical application examples, having compared its performance with the BI-RME method based on rectangular waveguides.