At higher frequencies, metal loss effects in passive waveguide components become more pronounced and hazardous. In this paper, we propose two integral equation techniques, based on the generalized admittance and impedance matrices, for the accurate consideration of losses in the metal walls of waveguide junctions. Both techniques have been evaluated in terms of accuracy and numerical efficiency, and conclusions are drawn regarding the best properties of the admittance parameter formulation. Finally, combining such technique with a classical perturbative method for considering propagation losses, we have successfully predicted all loss effects in two real waveguide filters used for commercial applications.
 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. , is no longer valid.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].
 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 . 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. .
 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 , 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. . 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 , Itoh , and Sorrentino et al. ) 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 , but such an approach cannot presently deal with waveguide junctions.
 In this context, the objective of this work is to extend both admittance and impedance integral equation formulations, originally proposed by Gerini et al.  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. , 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.
 In this section we present the theory developed for the accurate consideration of metal losses at waveguide junctions. After a brief review of the integral equation technique leading to the admittance matrix representation, we derive the novel expressions for the impedance matrix case. Next, the efficient implementation of both techniques is fully described.
 Let us consider a general lossy junction between two waveguides of arbitrary cross section as shown in Figure 1. According to Figure 1, the cross sections of the waveguides on each region (δ = 1, 2) are defined as CS(1) = A ∪ W and CS(2) = A. The tangential electric and magnetic fields at each waveguide region are
where Vn and In are the modal voltages and currents, respectively, whereas en and hn are the waveguide modal vector functions [see Marcuvitz, 1986].
2.1. Admittance Integral Equation Formulation
 For a good conductor, the boundary conditions of the tangential electric field are
where Js(r) is the surface current density and Zs is the surface impedance, which for good conductors can be approximated as follows [see Balanis, 1989]
As it is well known, in (4) ω represents the angular frequency, μ the magnetic permeability of the medium, and σ the metal conductivity.
 The original boundary conditions (2) and (3) can be rewritten as a single equation
where ξ(r) is an auxiliary function defined in the following way
 Proceeding now as indicated by Hueso et al. , we easily obtain the next two sets of Fredholm integral equations of the second kind
where Nn(δ) are the unknown basis functions to be solved, whereas K(1)(s, s′) and K(2)(s, s′) are the kernels defined as
 In (9) and (10), N(δ) means the number of accessible modes considered in each waveguide, whose concept was already introduced by Rozzi and Meelenbrauker . On the other hand, Zn(δ) is the modal impedance [see Marcuvitz, 1986], and n(δ) denotes its asymptotic behavior as defined by Hueso et al. . For the accelerated evaluation of both kernels, it is recommended to follow the technique described in Appendix A.
 Once the two integral equations are solved as detailed in section 2.3, we can easily compute the admittance matrix elements of the lossy junction as follows
and the auxiliary voltages n are defined in Figure 2.
2.2. Impedance Integral Equation Formulation
 A completely dual procedure to the one just described before can be followed to formulate a generalized impedance matrix representation of the lossy waveguide junction. In this case, the starting equations are obtained from forcing the boundary conditions for the tangential magnetic field of a good conductor
where Zs is the surface impedance previously defined in (4), and equation (14) has been obtained taking vectorial products in both sides of equation (3)
 It must be noted that equations (14) and (3) are completely dual if (14) is rewritten in terms of the metal surface admittance Ys. For very good conductors, which is the most typical situation, it is well known that such admittance tends to infinity, thus requiring more terms in the series appearing in (14) to satisfy the equation. Therefore it can be expected that the impedance formulation will behave rather worse than the admittance one in terms of convergence speed.
 Now we can rewrite (13) and (14) as a single equation
where the following auxiliary function has been introduced
Then, following a dual procedure to the one detailed by Hueso et al. , we finally obtain the next two sets of Fredholm integral equations of the second kind
where Mn(δ) are now the unknown basis functions. In this case, the two integral equation kernels K(1)(s, s′) and K(2)(s, s′) have dual expressions to the ones shown in (9) and (10) for the admittance case and can be simply obtained by replacing in such two equations m(δ), Zm(δ) and em(δ) by m(δ), Ym(δ) and hm(δ), respectively. Therefore the acceleration technique described in Appendix A for the efficient computation of both kernels can also be followed.
 Once the unknown Mn(δ) basis functions are solved as shown in section 2.3, the impedance matrix elements of the lossy junction can be easily computed as
In this case, the generalized impedance matrix representation of the lossy junction is shown in Figure 3, where the presence of asymptotic modal admittances (n(δ)) in a shunt configuration can be seen.
2.3. Efficient Solution of the Integral Equations
 To solve the integral equations previously derived for the admittance and impedance matrix formulations, the well-known Galerkin version of the method of moments (MOM) described by Harrington  will be employed. A different expansion for the basis functions, with respect to the one proposed by Gerini et al. , has been used in this work. In our case, we have chosen modal solutions belonging only to region (1) (bigger waveguide), which has allowed us to achieve a complete set of modal functions for expanding the unknown vector functions Nn(δ) (Mn(δ)) for the admittance (impedance) formulation. Therefore
where fq(1) will be eq(1) for the admittance case and hq(1) for the impedance formulation.
 Then the application of Galerkin's procedure provides a simple linear problem. The integral equations for both formulations can be handled in terms of a suitable defined linear operator ℒ as shown next
where p = 1, 2, …, M(δ), thus obtaining M(δ) equations with the same number of unknowns, and 〈·,·〉 means the usual inner product defined in the domain A + W.
 For the admittance formulation, the operator ℒ acts in the following way
whereas for the impedance case it is defined as follows
 From this point, the classical Galerkin approach can be easily followed. The operator ℒ should be applied to each mode eq(1) (admittance case) or hq(1) (impedance case), and the expressions can be easily written in terms of coupling integrals among the modal vector functions of the waveguides involved in the lossy junction. For the efficient evaluation of such coupling integrals, the reader is referred to Appendix B.
 To validate the theory presented in this paper, we have compared the loss effects predicted by our model with the numerical results provided by Ansoft HFSS v.8.5, as well as with experimental measurements. In this study, for comparative purposes, a Pentium IV platform running at 2.8 GHz with 1 GB DDRAM has been used for computing our results and those provided by HFSS.
 This section is organized as follows. First, a convergence study has been carried out in order to compare the admittance and impedance formulations. For that purpose, the losses introduced by the frontal walls of a simple waveguide step have been analyzed. Once the metal losses are accurately considered in the waveguide junctions, the propagation losses in the waveguide sections connecting such junctions in real passive devices have been included following a classical perturbation approach. Then we have considered two real devices whose experimental electrical responses are available. The measured S parameters of an inductive filter for LMDS applications and of a satellite commercial filter are successfully compared with the simulated results provided by the proposed technique.
3.1. Convergence Study
 To start the validation of both formulations, we have chosen a simple waveguide step, shown in Figure 4, where the shaded area has been first considered as a perfect conductor. For this limit case, the comparison is performed with the classical impedance integral equation (Z-IE) technique which does not consider losses in the device, already validated by Gerini et al.  and therefore used as a reference.
 To get convergent results (see Figure 5), the reference method has required only one accessible mode, 70 basis functions for the Galerkin MOM, and 250 terms in the kernels. Both admittance and impedance matrix techniques results (considering a very high value for the metal conductivity) are also represented in Figure 5. We can see that the results provided by the admittance matrix formulation are very accurate, and they have been reached using only one accessible mode, 250 basis functions for the Galerkin MOM, and 400 terms in the kernels. On the other hand, in order to recover the less accurate results provided by the impedance matrix formulation (see Figure 5), one accessible mode, 2500 basis functions for the Galerkin MOM, and 8000 terms in the kernels have been required. Therefore the convergence is reached at a very high CPU cost in the case of the impedance formulation, as was already predicted theoretically in section 2.2. It is interesting to remark that this behavior is completely opposite to what happens in the lossless case, where impedance IE outperforms admittance IE, that is, in terms of convergence [see Gerini et al., 1998].
 Once the two proposed formulations have been initially validated with a lossless case, we have studied the convergence behavior of both techniques for predicting the losses of the waveguide step shown in Figure 4 when the shaded area is made of copper (σCu = 5.8 × 107 S/m). In this case, the results provided by HFSS for such a lossy junction are selected as a reference.
 For comparative purposes, all the results for this lossy junction are shown together in Figure 6. As happened for the lossless case, the Y-IE formulation converges faster and better to accurate results than the Z-IE technique. In fact, to get the excellent results provided by the Y-IE algorithm, only 12 s were needed for solving 250 frequency points, whereas HFSS and Z-IE algorithm did require 60 and 1000 min, respectively. As a consequence, the admittance matrix formulation has been selected for the accurate analysis of real waveguide devices considered next.
3.2. LMDS Filter at 28 GHz
 Next, we have studied all loss effects in a four-pole filter used at the input front end of an LMDS receiver operating in the Ka band. A prototype of such filter has been made (see Figure 7) employing an aluminum alloy of conductivity equal to 1.2 × 107 S/m. This inductive and symmetric LMDS filter has been designed (see dimensions of the structure in the caption of Figure 7) to have a band-pass response of 800 MHz bandwidth centered at 28 GHz.
 In Figure 8, we can see a detailed view of the LMDS filter insertion losses. Figure 8 includes the simulated results using the Y-IE formulation and HFSS considering the conductivity value of the employed aluminum alloy, as well as a comparison with the experimental measurements of the prototype. As can be seen, our results agree with those given by the commercial software. However, the two simulated results are slightly different from measurements, which can be attributed to several reasons. First, the real conductivity of a material can be altered from its nominal value by environment temperature and humidity. Besides, additional insertion losses can be introduced by each specific manufacturing technique. Finally, conductivity values are typically provided at zero frequency (direct current (DC). However, at higher frequencies (microwaves and millimeter waves), such values can be affected by the surface properties, since the induced electrical current flows in smaller layers.
 To alleviate such small differences, or even for predicting conductivity values for unknown materials, we can make use of our accurate simulation tool in order to find an equivalent conductivity which allows us to recover experimental results. This process obviously needs previous measurements in order to calibrate the equivalent conductivity value. Proceeding in this way, the loss effects in new manufactured filters using the same materials and fabrication techniques could be accurately predicted.
 The complete reflection and transmission responses of this LMDS filter are shown in Figure 9. To compute these results, our simulation tool only takes 14 min for 250 frequency points (3.4 s per frequency value), whereas HFSS requires about 140 min.
3.3. C Band Communication Satellite Filter
 Finally, a commercial communications satellite filter has been fully characterized. This device consists of the cascade of an eight-pole band-pass filter, a six-pole low-pass filter, and two double plane step transformers, and it has been designed for operating in the C band (see dimensions in Figure 10). A prototype was built using two aluminum blocks with subsequent silver plating and was measured experimentally by Alvarez and Guglielmi .
 In this example, since we did not know a nominal value for the conductivity of the silver-plated aluminum, we found an equivalent conductivity value for such real material following the calibration procedure just described before (σEq = 3 × 107 S/m). As should be expected, this equivalent value is higher than the conductivity of the aluminum alloy used in the previous example (1.2 × 107 S/m) and smaller than pure silver (6.7 × 107 S/m). Using the predicted equivalent value, we computed the transmission and reflection parameters shown in Figure 11. A detailed view of the band-pass insertion losses is shown in Figure 12, which are approximately equal to 0.2 dB. The complete analysis of this device has only taken 52 min for 200 frequency points (i.e., 15.6 s per frequency value).
 Two extended integral equation techniques for the accurate consideration of losses in waveguide junctions are presented and evaluated. From this comparative study, we conclude that the impedance integral equation formulation does not offer good convergence results. On the contrary, the admittance formulation provides accurate loss characterization with very low CPU cost. The theory proposed for the lossy junctions has been combined with a classical perturbative approach for considering all loss effects present in passive waveguide structures. Successful results for two real filters used in LMDS and space applications have proved the accuracy and numerical efficiency of the proposed theory.
Appendix A:: Efficient Computation of Integral Equation Kernels
 For this purpose, we have extended a procedure originally proposed by Boria and Guglielmi  and already applied successfully to the lossless case by Gerini et al. . Proceeding in this way, the kernels of our two integral equation formulations can be evaluated as
where (δ) and 2r(δ) have the following expressions
and R is usually set to 4, since it has been verified that it provides very accurate results in negligible computation time.
 In (A2) and (A3), Xm(δ) and fm(δ) mean Zm(δ) and em(δ) for the admittance case and Ym(δ) and hm(δ) for the impedance formulation, respectively. In the admittance matrix formulation, we have for the TE case
and for the TM case
 For the impedance matrix formulation, because of duality, the expressions of the B2r coefficients for the TE (TM) modes are the ones corresponding to the TM (TE) modes of the admittance case.
 Then, all of the infinite summations involved in the computation of the integral equation kernels are now frequency independent and can therefore be evaluated only once outside the frequency loop.
Appendix B:: Efficient Evaluation of Coupling Integrals
 It should be pointed out that, in our case, the following new terms involving coupling integrals in the W region appear
 Obviously, the Am,n coupling integral would be δm,n if the integration domain was A ∪ W, because of the well-known orthonormalization condition of standard modes [see Marcuvitz, 1986]. Using such normalization property, these terms can be easily computed in the following way
 In the previous expression, we have transformed an integration on the metallic wall W into an integration on the small aperture A (whose implementation is typically easier). In any case, the Am,n terms are frequency independent, and therefore they can also be computed outside the frequency loop. Following this procedure, the loss effects in passive waveguide components can be considered very efficiently.
 This work has been supported by research projects TIC2000-0591-C03-01 and TIC2000-0591-C03-03 and special action ESP2001-4547-PE. The authors would like to thank Alejandro Alvarez-Melcon, from Universidad Politcnica de Cartagena, Spain, for valuable discussions during the preparation of this manuscript.