Abstract. Introduction

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1 Analysis of BEM and FEM codes for the modelling of microwave passive dielectric devices C. Di Nallo, F. Frezza, A. Galli Department ofelectrical Engineering, "La Saprienza " University of Rome, Via Eudossiana 18, Roma, Italy Abstract In this work, studies and comparisons have been carried out on the performances presented by numerical codes for the analysis and design of compact, light-weight, and high-performance microwave guided-wave devices that employ dielectric components. The features of original numerical codes based on boundary elements (BEM) have been compared with versatile canonical codes, such as those using finite elements (FEM). The BEM implementations have expressly been conceived for the study of a variety of electromagnetic problems in dielectric structures, and show notable advantageous properties as concerns accuracy and efficiency. The relevant basic formulation is oriented to the modal analysis of arbitrary dielectric waveguides, but it can be applied to characterize important classes of dielectric resonators as well. Extensions of the BEM approach allow us to analyze also the coupling problems for dielectric components, and in particular between dielectric guides and resonators, but in such cases the analysis with FEM is clearly more flexible. Based on the application of such BEM and FEM codes, fundamental theoretical information is then usable for the design of microwave and millimeter-wave passive devices, such as advanced solutions of filters. Introduction Dielectric structures are increasingly employed in up-to-date devices for electronic and communication systems, where light-weight and compact circuits are required at ever higher frequencies. Both in the upper part of microwave spectrum (millimeter and sub-millimeter waves) and in optics, basic components such as dielectric guides and resonators, optical fibers, etc. [1-3], have a prominent importance, particularly for their use in guided-wave passive components.

2 378 Software for Electrical Engineering The electromagnetic (EM) characterization and modeling of such types of devices is a fundamental requisite in the design procedures. To this aim, a large quantity of different numerical methods have been proposed in the literature to solve many practical problems [4,5]. Among the various codes and tools for microwave design, CAD packages are surely increasing their popularity: in connection with constant improvements of the computing resources, the current trend is to employ commercial codes suited for accurate EM characterization of a wide number of problems, thus avoiding types of software based on approximate or empirical methods of analysis. In this area, packages based on thefiniteelement method (FEM) [6,7] result particularly favorable for their properties of precision and wide versatility (e.g., some of these codes for microwave passive components have recently been extended to open structures as well). In contrast with the generality of these FEM tools, the notable required amount both of temporary storage memory and of computing time can often represent a serious limit of applicability, thus affecting the accuracy degree as well; therefore, FEM requirements for CPU and RAM may still result exceedingly heavy in many practical cases. Obviously, the optimal numerical technique is intrinsically connected to the kind of the investigated EM problem (topology of the structure, etc.), so that different specific numerical approaches may be requested to strongly improve efficiency and economy without affecting accuracy. The purpose of this work is to present new computational tools, alternative to standard approaches, which can suitably model a class of problems for guiding-wave dielectric structures, with a specific interest for advanced applications in microwave and millimeter-wave passive circuitry. The basic software under investigation, using a novel formulation of the boundary element method (BEM) [8], has a wide generality since it can be employed for characterizing the modal properties of microwave dielectric guiding structures (i.e., waveguides and classes of resonators with arbitrary geometry), but it can by the same be applied to other ranges, such as in optics. The knowledge achievable through this code can suitably be extended and applied for problems concerning a specific topology, the "nonradiative dielectric" (NRD) waveguide, which has recently been proposed for high-performance applications specifically at millimeter waves [9]. Our attention has particularly been focused on the analysis and design of composite guided-wave passive components, such as various classes offiltersfor satellite and mobile communications. The results of these implementations are compared with other techniques presented in the literature and specifically with FEM codes. Characterization of guided-wave dielectric structures Since many years, the characterization of the modal properties for dielectric waveguides with various shapes has got great interest in the literature, in conjunction with the relevant applications to microwaves and optics. Among several approaches, the boundary element method (BEM) [1,5] presents many attractive features for the rigorous analysis of arbitrary dielectric geometries: even though a large amount of preprocessing is typically required, its basic advantage is related to the reduction of complexity by one dimension, which provides very useful economy in memory storage and computing time; furthermore, BEM appears well suited for problems in open structures.

3 Software for Electrical Engineering 379 Referring to the analysis of arbitrarily-shaped dielectric guides, a new BEM approach is here considered, which exhibits sensible improvements in both efficiency and accuracy with respect to the other formulations [1,5]. Here we do not intend to go into details about the specific theoretical aspects of this formulation, some of which can be found in [8]: in the following subsections we prefer to stress our attention on the main distinctive features and possible practical applications. BEM code for the analysis of arbitrary dielectric waveguides Our BEM software is aimed to describe completely the modal properties (wavenumbers, fields, etc.) of arbitrarily-shaped dielectric guiding structures as schematized in Fig. 1. Figure 1: Cylindrical dielectric waveguiding structure of arbitrary cross section in a rree-space region. The local rectangular coordinate system on the boundary s, and the electromagnetic and geometrical parameters, useful for the BEM analysis, are also indicated. In the preprocessing phase, we have employed suitable analytical manipu aliens to reduce the singularity problems related to the integral equations' dyadic kernels, which can give serious troubles for numerical convergence and stability. In our formulation the derivative operations on the unknowns have intentionally been avoided, thus allowing, for instance, field expansions with piecewise-constant basis functions [8]. Consequently, it has been possible to achieve a significant enlargement of the class of discretization methods for the numerical solution of the integral equations' set. On this basis the implementation of an unconventional computational procedure, based on quadrature formulas (Nystrom's method) instead of canonical moment methods has given extremely speed, economic, and precise results

4 380 Software for Electrical Engineering Tests and comparisons concerning a variety of applications (particularly those ones that are quite delicate from a computational viewpoint) prove that, due to such desirable features, this BEM code is absolutely competitive with respect to other well-stated numerical techniques. Our approach results very stable in terms of the input physical parameters (characteristics of the mediums, geometries of the structures, frequency ranges, and so forth). Then, the number of discretization points for the investigated structures can drastically be reduced, with profitable consequences on the memory storage and computing time as well (e.g., the computation of the modal parameters with FEM codes [5,6] requires far larger CPU and RAM occupations). It should be noted that the implementation permits us to derive all the modal solutions, and therefore is very useful for full-wave analysis; in addition, spurious solutions are never present with this formulation. Some representative results are now shown, particularly for specific applications where EM characterization is rather difficult. Applications to dielectric guides and other components To check the accuracy and reliability of the present BEM implementation, many different applications have been considered concerning dielectric waveguiding structures for which data are available in the literature. In addition to the dispersion properties of guided modes, interesting comparisons may be derived considering also dielectric resonators (DRs) [2] in the NRD topology [10]. Very high degrees of accuracy have first been obtained for standard shapes. Comparisons of data for typical dielectric structures, such as rectangular guides, image lines, etc., have always given very accurate results. Also curved geometries have been analyzed, which have been approximated with proper polygonal contours for the discretization process. NRD disk DR Rescmances 1,1,1 (HEM) 0,1,1 (TE) 0,1,2 (TM) 2,1.1 (HEM) 1,2,1 (HEM) 1,3,1 (HEM) fres [GHz] Error % ~ 0.0 < +0.1 < +0.1 < +0.1 < +0.2 < +0.1 Figure 2: Test on BEM solutions: table of the resonance frequencies (left) for an NRD circular dielectric resonator of radius R and height a (right)', modal indices are referred to angular, radial, and longitudinal variations. Comparisons are presented with respect to exact analytical data. Parameters: e^lo; R=5 mm; a=10 mm. Comparisons with rigorous reference data achievable e.g. for circularsection structures have always shown very satisfactory precision. An example of accuracy is shown in Fig. 2 for data of resonance frequencies (left) referring to a circular-section NRD DR (right); comparisons are referred to exact analytical data [10]. It should be noted that some commercial FEM codes can analyze only 'realistic' practical structures while cannot calculate eigensolutions for isolated structures, such as DRs; in any case, the amount of computing resources with FEM for such type of problems is clearly unfavorable.

5 Software for Electrical Engineering 381 Irregular geometries have then been tested, and particularly those ones which are usually rather delicate from a numerical viewpoint, such as structures with sharp edges. On this subject, a particular practical interest is furnished by slightly-perturbed geometries (e.g., circular or square sections with cuts or notches), which are further complicated by the presence of very close (quasidegenerate or quasi-dual) eigensolutions. In this sense, specific applications have been considered for notched dielectric resonators that can suitably be coupled in integrated NRD circuitry for compact filters of dual mode type [11]. It should be reminded that, for similar devices, the canonical way of achieving double poles with a single element is by employing metallic perturbations (such as screws or posts) close to the resonator, so that its rotational symmetry is perturbed: this EM modeling is very complicated, and empirical procedures have often to be used, which can result rather inadequate and expensive. The alternative way of altering the resonator geometry with small cuts or notches is more promising, since it allows a much more accurate and predictable analysis. In particular, with the BEM code, the exact location of quasi-dual resonance frequencies for NRD DRs can accurately be determined as a function of any geometrical perturbations and of the other parameters. An example of such behavior is shown in Fig. 3a (right), for a square-section NRD resonator (left) An increasing splitting of the dual resonance frequencies occurs as the geometrical perturbation d (notch amplitude) enlarges. The presence of quasidegenerate modes is clearly shown through the two different configurations of the resonance modes that are almost orthogonal, as depicted in Fig. 3b (it can be seen that the behavior of such notched square DRs cannot be derived as a simple slight perturbation of a degenerate rectangular-section DR, as the modal fields are really different). The accurate knowledge of these resonances is also a basic requisite for deriving the mutual coupling between dual modes, when such types of DRs are used as selective components for filter of dual-mode type [111- for the case presented in Fig. 3a, the theoretical behavior of the mutual coupling between dual resonances is given in Fig. 3c. Analysis of novel topologies of dielectric filters For all such cases, BEM has shown a high reliability and precision, together with a dramatic saving in computation time with respect to FEM. We can hence u "S*!?* B^M implementation is a powerful and convenient tool as far as the EM modeling of the basic dielectric components is concerned. Actually the same approach can be extended for instance to characterize coupled dielectric guides. Anyway, for more involved problems, the advantages of BEM versus FEM are no longer prominent, because the amount of analytical preprocessing required may become prohibitive, and the class of solvable problems is reduced To this aim, the involved problem of a rigorous quantification of the coupling phenomena between dielectric resonators and guide has also been considered referring to the NRD topology, in order to get all the information for microwave filter design. The basic lines of a BEM solution to such problem can be found in [12], where the approach was particularized to cylindrical dielectric resonators, set at a certain distance apart from the NRD guide. It results that with BEM, the presence of boundaries of different shape (the straight ones of the guide and the circular one of a resonator) may require evaluations of involved testing integrals and sums of slowly-convergent series.

6 382 Software for Electrical Engineering t I V \ < ^ \ K. f ^ ^ k + v ^ b) ^ / V a) 11.0 Coupling Coefficient K Resonance Frequencies [GHz] Notch d [mm] Notch d [mm] Figure 3: a) A notched square NRD DR (left), and the theoretically-calculated location of the 'quasi-dual' resonant frequencies/;,^ as a function of the notch amplitude on a corner d (right)', b) Top views of the 'quasi-dual' electric-field configurations for a notched square DR, which show prevalent diagonal field lines at the two close resonances; c) Relevant behavior of the mutual dual-mode coupling coefficient K vs. notch d. Parameters of NRD DR for an X-band simulation: permittivity,.=2.53, height a=12.3 mm, sides b=l=10 mm. On the other hand, the FEM analysis for the same types of coupled structures can be performed without significant analytical complications with respect to the single components. In this case, the physical device under investigation is discretized through suitable meshes, and its EM behavior is described in its whole through global parameters: therefore, FEM codes can work well as analysis tool, while the synthesis procedures need anyway a deep understanding of the EM interaction phenomena. Tests here proposed concern the study of interacting structures for filters, such bandpass and bandstopfilterswith DRs coupled to NRD guide (Fig. 4a). As illustrated in the topology schematized in Fig. 4b, using a notched DR, a double-pole behavior is achieved through a single element, with resonance modes accessed simultaneously. The quantification of the coupling amount between guide and resonator acting on two different modes is a very involved matter, depending on DR's orientation with respect to the line too [11]. In fact, while the notch acts basically on the separation in frequency of the peaks, their location is influenced by the external coupling with the feeding line; the tuning is adjustable through the rotation a of the DR around its axis. These characteristics can be analyzed globally by FEM through the scattering parameters of the filtering structure.

7 Software for Electrical Engineering 383 For the bandstop topology of Fig. 4b, examples of transmission scattering parameters (db) vs. frequency (GHz) are reported in Fig. 4c: the tuning effect is shown to be strongly sensitive to three different values of DR's orientation a. The analyses based on FEM permit advantageous multimodal investigations, from which other essential aspects related to these structures can be enlightened. E.g., for such NRD filters, fundamental information has also been achieved on the undesirable effects of power conversions between operating and parasitic modes [11]. In all these cases, comparisons with experimental results as well give quite satisfactory agreement with the theoretical predictions. a) GHZ GHz io.75 xt ^ e basic geometry of an NRD guide; b) A typical topology of an NRD bandstopfilterconfiguration using a square DR with a notch (top view without plates); c) Examples of theoretically-calculated dependence of the guide/ resonator coupling on the orientation a of the DR around its axis, measured through the transmission coefficient (db) as a function of frequency (GHzV up a=45 ; left down, a=0 ; right down, a=90. Parameters as in Fig 3

8 384 Software for Electrical Engineering Conclusion The modeling of components employing dielectrics has a remarkable interest in electromagnetics for its theoretical and practical implications, and a lot of different software has been proposed in the literature for such type of analysis, based on all the most common numerical methods (finite elements, finite differences, mode matching, integral equations, etc.). The novel BEM implementation that has firstly been developed for the characterization of arbitrarily-shaped dielectric guides has shown excellent advantageous features as regards efficiency, economy, and precision, even compared with the most employed numerical techniques. The amount of preprocessing is considerable, but can become very heavy when composite structures are studied, as in filters where interaction occurs between guide and resonators. In this problems, FEM tools are quite satisfactory tools of analysis. Based on these BEM and FEM codes, examples of practical devices, till now explored only experimentally, have been discussed and tested. Both the accurate quantification of the basic parameters and the comprehension of disregarded electromagnetic phenomena are achieved for novel solutions of passive components for microwaves and millimeter waves. References [1] R. E. Collin, Field theory of guided waves. New York: IEEE Press, [2] D. Kajfez and P. Guillon, Eds., Dielectric resonators. Norwood: Artech, [3] D. Marcuse, Theory of dielectric optical waveguides. Orlando: Academic Press, [4] T. Itoh, Ed., Numerical techniques for microwave and millimeter-wave passive structures. New York: Wiley, [5] R. Sorrentino, Ed., Numerical methods for passive microwave and millimeter wave structures. New York: IEEE Press, Part 5, [6] Ansoft Co., HP 85180A High-Frequency Structure Simulator - User's reference. Santa Rosa: Hewlett Packard Ed., May [7] B. E. MacNeal, Ed., MSC/EMAS user's manual. Los Angeles: MacNeal- Schwendler Co., [8] C. Di Nallo, F. Frezza, and A. Galli, "Full-wave modal analysis of arbitrarily-shaped dielectric waveguides through an efficient boundary-elementmethod formulation," Digest MTT-S Int. Microwave Symp., pp , May * [9] T. Yoneyama, "Nonradiative dielectric waveguide," in K. J. Button, Ed., Infrared and millimeter-waves, New York: Academic Press, vol. 11, pp ** [10] F. Frezza, A. Galli, G. Gerosa, and P. Lampariello, "Characterization of resonant and coupling parameters of dielectric resonators for NRD filtering devices," Digest MTT-S Int. Microwave Symp., pp , June [11] C. Di Nallo, F. Frezza, A. Galli, G. Gerosa, M. Guglielmi, and P. Lampariello, "Experimental investigation on NRD-guide dual-mode filters," Digest MTT-S Int. Microwave Symp., pp , May [12] C. Di Nallo, F. Frezza, A. Galli, G. Gerosa, and P. Lampariello, "A Boundary-Element-Method formulation for the electromagnetic coupling between dielectric waveguides and resonators," Computational Mechanics, vol. 13, n. 1/2, pp , Nov