In this paper, a fast and accurate Computer-Aided Design (CAD) tool of periodically loaded E-plane filters is proposed. The tool is based on a very efficient Integral Equation (IE) technique that provides a full-wave modal analysis of discontinuities between arbitrarily shaped waveguides, i.e. ridge waveguides, and rectangular ones. For solving the complete filter, the Boundary Integral - Resonant Mode Expansion (BI-RME) method is also employed. In order to show the advantages of this CAD tool, a periodically loaded E-plane filter with improved stop-band performance is analyzed and compared to standard solutions. For validation purposes, numerical and experimental results are successfully compared

Boria Esbert, Vicente Enrique

Coves Soler, Ángela María

Gimeno Martínez, Benito

Goussetis, George

Marini, Stephan

Taroncher Calduch, Máriam

Repositorio Institucional de la Universidad de Alicante

FAST AND ACCURATE CAD TOOL OF PERIODICALLY LOADED E-PLANE FILTERSS. Marini(1), M. Taroncher(2), V.E. Boria(2), B. Gimeno(3), A. Coves(4), G. Goussetis(5)(1)Dpt. Fı´sica, Ing. Sistemas y Teorı´a de Sen˜al, Universidad de Alicante,Campus de San Vicent del Raspeig, Ap. 99, E-03080 Alicante, SPAIN,Phone: +34 965903400 (ext. 2204), Email: smarini@ua.es(2)Dpt. Comunicaciones, Univ. Polite´cnica de Valencia, Campus Vera, E-46022 Valencia, SPAIN,Phone: +34 963879718, Fax: +34 96 3877309, Email: vboria@dcom.upv.es(3)Dpt. Fı´sica Aplicada - I.C.M.U.V., Univ. de Valencia, Campus Burjassot, Valencia, SPAIN(4)Dpt. Fı´sica y Arquitectura Computadores, Univ. Miguel Herna´ndez Elche, Alicante, SPAIN(5)Dpt. of Electronic & Electrical Eng., Loughborough Univ., Loughborough, United KingdomAbstractIn this paper, a fast and accurate Computer-Aided Design (CAD) tool of periodically loaded E-plane filters isproposed. The tool is based on a very efficient Integral Equation (IE) technique that provides a full-wave modalanalysis of discontinuities between arbitrarily shaped waveguides, i.e. ridge waveguides, and rectangular ones. Forsolving the complete filter, the Boundary Integral - Resonant Mode Expansion (BI-RME) method is also employed.In order to show the advantages of this CAD tool, a periodically loaded E-plane filter with improved stop-bandperformance is analyzed and compared to standard solutions. For validation purposes, numerical and experimentalresults are successfully compared.Index TermsE-plane Filters, Periodic Structures, Computer-Aided Design (CAD), Integral Equation, Moments MethodI. INTRODUCTIONPeriodic structures are currently object of great interest in the microwave area for their potentialapplication in the microwave and millimeter-wave domains. In filtering applications, periodic structureshave been reported to offer reduced physical size and improved stop-band performance, due to the well-known slow-wave effect caused by periodicity [1]-[3]. The effective wavelength in a periodically loadedstructure is significantly reduced relative to the same parameters of a wave propagating in a comparablehomogeneous line. Therefore, the half wavelength of a resonator in a periodic structure is considerablyreduced. Furthermore, due to the dispersion relation of slow waves, an improved stop-band performancecan also be achieved in such a case.E-plane technology including ridge waveguides offers a very convenient way for realizing periodicstructures, through the loading of the waveguide with reactive obstacles in form of ridges. In this highdemanding scenario, the usage of flexible CAD tools, based on accurate and efficient electromagneticsimulators, reveals absolutely necessary. This paper revisits a novel hybrid method for the fast and rigorousanalysis of periodically loaded E-plane filters composed of cascaded ridge waveguides. For such purposes,an IE-based modal method, originally described in [4] for rectangular waveguides, has been updated to copewith arbitrarily shaped geometries. This method makes use of the modal chart of waveguides with arbitrarycross-section, which is determined following an improved version (see [5] and [6]) of the classical andwell-known BI-RME technique [7]. A comprehensive example, which fully validates the right functioningand the efficiency of the new CAD tool developed, is included.II. THEORYPeriodically loaded E-plane filters involve the cascaded connection of ridge and rectangular waveguides.For analysis purposes, it is required to know the complete modal chart of such arbitrary shaped waveguidesin a very accurate way. To reach this aim, the well-known BI-RME method (see [7]) is proposed to beused. Following this classical method, the electric field at a generic observation point r inside the ridgewaveguide can be obtained byE (r) = −jηk∫σGe (r, s′, k) · Jσ (l′) dl′ (1)where s′ indicates a source point on σ (the arbitrary ridge contour), η =√µ/ is the characteristicimpedance and k = ω√µ is the wavenumber, Ge is the two-dimensional dyadic Green’s function of theelectric type for the two-dimensional resonator of rectangular cross section and Jσ is the current density.Splitting Ge and Jσ into its transversal and longitudinal components, and imposing the correspondingboundary conditions (Et (r) · tˆ (l) = 0, and Ez (r) = 0) on σ, we obtain the integral equations for thetransversal electric field Et (TE modes) and for the longitudinal electric field Ez (TM modes).The next step in the BI-RME method is the segmentation of the problem, in order to solve the integralequations via the Galerkin version of the MoM [8], where the basis and testing functions are chosen tobe overlapping piece-wise parabolic splines. Following the Galerkin approach, a generalized eigenvalueproblem is obtained for the TE case, whereas a standard eigenvalue problem is got for the TM modes(see [5]). The solution of both problems provides as eigenvalues the TE and TM cutoff wavenumbers, andas eigenvectors the modal expansion coefficients and the amplitudes of the transversal and longitudinalcomponents of the unknown current density.An advantage of the BI-RME technique is that, without hardly additional CPU effort, the couplingcoefficients between the arbitrarily-shaped waveguide and the standard rectangular contour enclosing thearbitrary profile can be easily computed (see [9]). The coupling integrals between the modes of the ridgewaveguide and those of the rectangular one are defined as followsIpq =∫Sep · e♦q dS (2)where ep and e♦q are, respectively, the normalized modal vectors of the rectangular and ridge waveguides.In order to obtain a full-wave characterization of any planar junction between two waveguides, a veryefficient method based on an integral equation technique, originally described in [4] for dealing withrectangular waveguides, has been properly updated. The objective of this technique is the representationof each planar waveguide junction in terms of a Generalized Impedance Matrix (GIM). A remarkablecontribution of this method is the distinction made between accessible and localized modes: accessiblemodes are those used to connect transitions, while localized modes are only used to describe the elec-tromagnetic fields in the junction (the number of localized modes is always greater than the number ofaccessible ones).The first step of this technique consists in imposing the boundary condition at the junction, where thetotal transverse magnetic field in each waveguide region can be expressed in terms of the waveguidemodes (see [10]) as followsH(δ)t =∑mI(δ)m h(δ)m (3)and (δ) = (1), (2) indicates the suitable region of the junction, h(δ)m is the magnetic vector mode functionrelated to the m-th mode of region (δ). After some mathematical manipulations (see details in [4]), it ispossible to obtain the following integral equationh(δ)n (s) =∫∫CS(2)M(δ)n (s′)K(s, s′) ds′ (4)where s and s′ are the observation and source points, respectively, K(s, s′) is the kernel of the integralequation, and M(δ)n (s′) are unknown vector functions which define the magnetic field at the junction.tba(a)tab(b)Fig. 1. Three-dimensional layout of the 2 resonator filter: the standard E-plane filter in (a) and the periodically loaded E-plane filter in (b).The common dimensions of the two filters are a = 22.86 mm, b = 10.16 mm and t = 0.10 mm.Finally, the solution of the integral equation can be carried out using the Galerkin’s procedure, wherethe unknown vector functions are expanded in a set of basis functions used in (δ) region. Proceeding inthis way, we can easily compute the impedance parameters of the generalized Z-matrix representationthrough the following equationZ(δ,γ)m,n =∫∫cs(2)M(γ)n (s′) · h(δ)∗m (s′) ds′ =M(γ)∑q=1α(n,γ)q∫∫cs(2)h(2)q · h(δ)∗m ds′ (5)III. RESULTSIn order to demonstrate the improvements introduced by the proposed structure in terms of stop-bandperformance, the electrical responses of a two-resonator periodic filter (originally proposed in [3]) andof a standard E-plane solution with the same in-band response have been compared. The layout and thedimensions of such two resonator filters are shown on Fig. 1 and Fig. 2. The homogeneous rectangularwaveguide sections acting as the resonators of the standard E-plane filter (see Fig. 2 (a)) are replacedwith periodic structures consisting of a cascade of ridge waveguides (see Fig. 2 (b)).bLs1Ls2Ls3Lr1 Lr2(a)b s1s2l2 l1 s1 s2 l2 l1Ls1Ls2Ls3Lr1 Lr2(b)Fig. 2. The dimensions for the standard E-plane filter (a) are: septa lengths Ls1 = Ls3 = 0.31 mm and Ls2 = 1.68 mm, resonatorlengths Lr1 = Lr2 = 20.10 mm, total length of 42.50 mm. The dimensions for the periodically loaded E-plane filter (b) are: septa lengthsLs1 = Ls3 = 1.20 mm and Ls2 = 4.0 mm, resonator length Lr1 = Lr2 = 8.0 mm, ridge dimensions s1 = b = 10.16 mm and s2 = 1.0 mm,ridge lengths l1 = 1.0 mm and l2 = 0.5 mm, total length of 22.40 mm.In Fig. 3 (a), the E-plane standard filter response obtained with our CAD software package is successfullycompared with simulations performed with Ansoft HFSS1. Exploiting the symmetry of the structure, ouranalysis method has required to consider 80 accessible modes, 250 basis functions and 400 kernel terms.In our case, the complete electrical response simulation has only taken a CPU effort of 0.11 sec. perfrequency value, whereas HFSS has needed 1.20 sec. per each point. Both software packages have beenrun in a Pentium IV platform at 3.2 GHz with 1-GB RAM.In Fig. 3 (b), the periodically loaded E-plane filter response obtained with our CAD software packageis well compared with Ansoft HFSS v10.0 simulations, and also with measurements of a filter prototype1HFSS v.10.0, Ansoft Corporation, Pittsburgh, PA, 2005.fabricated and measured (see [3]). An excellent agreement between all results can be observed. As it isshown in the figures, the periodic loaded filter has better selectivity and wider stop-band response thanits standard E-plane counterpart. Furthermore, the overall periodic filter is 47% shorter than the standardfilter since the resonator lengths have been widely reduced, as it is expected from the well-known slowwave effect in periodic structures. In the periodic case, by exploitation of the symmetry of the structure,the full-wave analysis method has only required to consider 80 accessible modes, 250 basis functions and500 kernel terms. Our software tool has only involved a CPU effort of 0.35 sec. per frequency point. Onthe contrary, the HFSS results have required a CPU effort of 1.20 sec. for each analysis point.8 9 10 11 12 13 14−60−50−40−30−20−100Frequency (GHz)Scattering parameters (dB)|S11||S21|BI−RMEHFSS(a)8 9 10 11 12 13 14−60−50−40−30−20−100Frequency (GHz)Scattering parameters (dB)|S11| |S21|BI−RMEHFSSMeasurements(b)Fig. 3. Electrical responses of the E-plane standard filter in a), and of the periodically loaded E-plane filter in b).IV. CONCLUSIONSA very efficient and accurate CAD tool of periodically loaded E-plane filters has been described. Forthe fast and rigorous analysis of such filters, an advanced hybrid method combining the BI-RME and theIE techniques has been proposed. This novel CAD tool has been successfully applied to the analysis ofa compact E-plane filter, composed of periodic ridge waveguides, with an improved stop-band response.Numerical and experimental results are presented to fully validate these arguments.REFERENCES[1] G. Goussetis and D. Budimir, “Waveguide filters with improved stopband performance,” Proc. 30th Eur. Microwave Conf., Oct. 2000.[2] J. Sor, Y. Quian, and T. Itoh, “Miniature low loss CPW periodic structure for filter applications,” IEEE Trans. Microwave Theory Tech.,vol. 49, pp. 2336–2341, Dec. 2001.[3] G. Goussetis and D. Budimir, “Novel periodically loaded e-plane filters,” IEEE Microwave and Wireless Comp. Letters, vol. 13, no. 6,pp. 193–195, June 2003.[4] G. Gerini, M. Guglielmi, and G. Lastoria, “Efficient integral formulations for admittance or impedance representation of planarwaveguide junctions,” IEEE MTT-S Int. Microwave Symp. Dig., vol. 3, pp. 1747–1750, June 1998.[5] S. Cogollos, S. Marini, V. E. Boria, P. 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Fast and accurate CAD tool of periodically loaded E-plane filters

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