## 1. Introduction

[2] The need to develop techniques which allow for the prediction of the electrical behavior of shielded circuits and cavity backed antennas has triggered the growth of studies in this area. In order to make an accurate analysis of this kind of circuits, pure numerical techniques such as finite elements, finite differences or the transmission line matrix method can be used [see, e.g., *Jin and Volakis*, 1991; *Omiya et al.*, 1998; *Hoefer*, 1985].

[3] Moreover, the integral equation technique combined with the Method of Moments [see *Harrington*, 1968] has become very popular, because of its efficiency and moderate computational cost for circuits with electrical size in the order of few wavelengths [*Gentili et al.*, 1997; *Livernois and Katehi*, 1989]. The formulation of the integral equation requires an algorithm which leads to the computation of the electromagnetic scalar and vector potentials of the problem, i.e. the so called mixed potentials Green's functions. In the literature, several formulations can be found for the case of shielded structures, where the enclosure influence has to be included inside the Green's functions [see *Eleftheriades et al.*, 2004; *Gentili et al.*, 1997]. In this case, the numerical treatment of the problem is reduced to the printed circuits itself, and therefore, the computational cost decreases considerably.

[4] The Green's functions for rectangular enclosures have been extensively studied in the past [see *Railton and Meade*, 1992; *Karen and Atsuki*, 1995; *Park and Nam*, 1997]. The first studies expressed the Green's functions in terms of spectral domain slowly convergent series of vector modal functions inside rectangular cavities [see *Park and Nam*, 1997; *Dunleavy and Katehi*, 1988]. Recently, the computation of the Green's functions have been performed by using spatial domain formulations [see *Melcón et al.*, 1999], expressing them as slowly convergent series of spatial images. However, in order to evaluate efficiently the Green's functions it is necessary to use special acceleration algorithms [see *Park and Nam*, 1997; *Melcón and Mosig*, 2000; *Eleftheriades et al.*, 2004; *Hashemi-Yeganeh*, 1995], in both spectral and spatial domains.

[5] Because of this complex mathematical treatment, the circular waveguide has been less studied. Green's functions inside circular geometries can be formulated by using spectral domain techniques, which express them as vector modal series of Bessel functions [see *Leung and Chow*, 1996; *Zavosh and Aberle*, 1994]. These methods are strongly dependent on the chosen numerical approach, since the higher order Bessel functions are not easily computed with enough accuracy.

[6] On the other hand, a new spatial domain method for the Green's functions computation inside circular cylindrical cavities was recently proposed by *Vera-Castejón et al.* [2004] and *Quesada-Pereira et al.* [2005]. The technique uses image theory to enforce the proper boundary conditions for the fields. The numerical evaluation of the Green's functions under electric current excitation inside an empty circular cylindrical cavity was described by *Vera-Castejón et al.* [2004], whereas in the work of *Quesada-Pereira et al.* [2005] Green's functions under magnetic currents were studied. Besides, the technique was extended by combining it with the potentials of a stratified medium formulated in the spatial domain with the Sommerfeld integral [see *Mosig*, 1989], which allows the analysis of practical multilayer printed circuits.

[7] In this framework, this paper presents an extension of the original image theory that permits its application to arbitrarily shaped cylindrical geometries. The new formulation to compute both the electric scalar potential and the magnetic vector potential dyadic Green's functions produced by electric currents inside arbitrarily shaped cylindrical cavities is shown. This formulation computes the Green's functions when source and observation points belong to the same transverse plane. However, a multiring approach, similar to the one presented by *Pereira et al.* [2005] for circular cavities, can be used in order to evaluate observation points located at a different height than the source point. Geometries, such as rectangular, triangular and cross shaped are analyzed, while fast convergence behavior is exhibited.

[8] A new technique for the evaluation of the accuracy of the Green's functions is introduced, whereas two methods to increase the precision are proposed. The first is based on properly locate the spatial images, by optimizing its distance from the cavity wall in order to obtain the lowest possible error. The second is based on a gradient technique which optimizes either the complex value of the images or their location. Some useful results are obtained, showing the effectiveness of both techniques for reducing the computational error during the calculation of the Green's functions in several cavities.

[9] In addition, the novel image technique has been applied successfully for the calculation of the resonant frequencies inside arbitrarily shaped cavities, and several results are given. Furthermore, a comparison between the distribution of the potentials and the electric field components (provided by HFSS^{©}) is presented. It is shown that they have the same distribution inside the cavity, because they satisfy the same boundary conditions. Finally, a practical 4-poles band-pass filter based on coupled lines is analyzed, showing the practical value of the method proposed.