## 1. Introduction

[2] The modeling of losses due to nonperfect conducting materials used in the fabrication of microwave components is a fundamental key for an accurate characterization of passive waveguide devices, such as filters, couplers, polarizers, orthomode transducers, diplexers and multiplexers [see *Matthaei et al.*, 1980; *Conciauro et al.*, 2000]. These components are key elements in most present and future wireless and/or space communication systems operating at higher frequencies (microwave and millimeter-wave bands), where the conventional approach of neglecting losses or considering them as a second-order effect, as followed by *Alvarez et al.* [1996], is no longer valid. In this high-demanding scenario, an accurate consideration of the metal loss effects within the present computer-aided design (CAD) tools of passive waveguide devices is required [see *Melloni and Gentilli*, 1995].

[3] The effect of losses in microwave waveguides and cavities has been under consideration in the technical literature during recent decades. A first approximation for the analysis of the metal losses only considers the attenuation of the power flow in the direction of propagation due to the finite conductivity of the waveguide metallic ones [see, e.g., *Balanis*, 1989; *Collin*, 1991]. In this perturbation method it is assumed that the electromagnetic fields for a given mode in the presence of finite conductivity walls are essentially the same as when the conductivity is infinite. As a consequence, the surface currents flowing on the guide walls can be directly calculated. Such approach permits to evaluate the losses in the walls from the known surface current density of the unperturbed mode, which directly leads to the calculation of the modal attenuation constant related to the propagation direction. Another very early contribution to the general properties of degenerated modes in lossy waveguides, which are treated by employing a variational method, can be found in work by *Gustincic* [1963]. However, all these perturbative approaches do not take into account the losses in the transversal metallic walls of the waveguide junctions, which can be important as proved by *Hueso et al.* [2004].

[4] Therefore, in order to increase the accuracy of modern CAD tools, the rigorous consideration of the losses due to the transversal metallic wall of a junction between two waveguides with different cross sections should be included. Up to now, very few techniques have been developed in order to solve accurately junctions of waveguides with finite wall conductivity. For instance, an *E* field mode-matching method combined with the conservation of complex power technique was first proposed by *Wade and McPhie* [1990], whereas a pure mode-matching technique formulated in terms of electric and magnetic currents, and leading to the generalized scattering matrix (GSM) representation, can be found in work by *Trinchero et al.* [1997]. Both works show very interesting preliminary results, but information about numerical efficiency and convergence issues (i.e., the relative convergence phenomenon widely documented by *Leroy* [1983], *Itoh* [1989], and *Sorrentino et al.* [1991]) are missed. More recently, the longitudinal and transversal attenuation constants for a lossy waveguide (also including the presence of lossy dielectrics) have been evaluated by means of a nonperturbative technique by *Mattes and Mosig* [1999], but such an approach cannot presently deal with waveguide junctions.

[5] In this context, the objective of this work is to extend both admittance and impedance integral equation formulations, originally proposed by *Gerini et al.* [1998] for the lossless case, to the accurate consideration of metal losses in waveguide junctions. The basics of this theory for the generalized admittance matrix (GAM) formulation can be found in work by *Hueso et al.* [2004], where it was successfully applied to a few simple examples. In this paper, the theory extension for solving the same problem in terms of a generalized impedance matrix (GIM), as well as the efficient solution of both integral equation formulations, are detailed. Then a comparative study of GAM and GIM techniques in terms of numerical robustness and efficiency is presented, thus revealing the best performance of the admittance parameter formulation. Finally, combining the GAM technique with the classical perturbation method cited before [see *Balanis*, 1989; *Collin*, 1991], we have successfully predicted all loss effects in two real application prototypes: an inductive band-pass filter at 28 GHz for local multipoint distribution systems (LMDS) and a commercial filter composed of a band-pass and a low-pass section for a C band (6 GHz) communication satellite.