In this paper, a fast and accurate full-wave analysis tool of passive structures based on Electromagnetic-Bandgap (EBG) waveguides periodically loaded with metal ridges is proposed. For this purpose, a very efficient Integral Equation (IE) technique is followed to model the planar steps involving arbitrary waveguides. The well-known Boundary Integral - Resonant Mode Expansion (BI-RME) method is used to obtain the modal chart of the ridged waveguides. In order to show the advantages of this tool, a periodically loaded E-plane filter with improved stopband performance is analyzed and compared to standard implementation. Dispersion relations are also derived and used as guidelines for designing an EBG fifth-order low-pass filter

Boria Esbert, Vicente Enrique

Coves Soler, Ángela María

Gimeno Martínez, Benito

Marini, Stephan

Taroncher Calduch, Máriam

Full-wave analysis and applications of EBG waveguides periodically loaded with metal ridges

Repositorio Institucional de la Universidad de Alicante

2009 IEEE MTT-S International Microwave Workshop Series onSignal Integrity and High-Speed Interconnects (IMWS2009-R9)Full-Wave Analysis and Applications of EBG WaveguidesPeriodically Loaded with Metal RidgesStephan Marini*, Angela Covest, Mairiam Taroncher+, Vicente E. Boria+, Benito Gimeno&,* Department of Physics, Systems Engineering and Signal Theory, University of Alicante, Spain,Email:smarini @dfists.ua.es, vboria@dcom.upv.es, benito.gimeno @uv.esAbstract- In this paper, a fast and accurate full-waveanalysis tool of passive structures based on Electromagnetic-Bandgap (EBG) waveguides periodically loaded with metalridges is proposed. For this purpose, a very efficient IntegralEquation (IE) technique is followed to model the planar stepsinvolving arbitrary waveguides. The well-known BoundaryIntegral - Resonant Mode Expansion (BI-RME) method isused to obtain the modal chart of the ridged waveguides.In order to show the advantages of this tool, a periodicallyloaded E-plane filter with improved stopband performanceis analyzed and compared to standard implementation. Dis-persion relations are also derived and used as guidelines fordesigning an EBG fifth-order low-pass filter.Index Terms- Modal analysis, Integral equations, Electro-magnetic bandgap (EBG), periodic ridged waveguides.I. INTRODUCTIONPeriodic structures are currently receiving a renewedinterest due to their potential use in the microwave andmillimeter-wave frequency bands. For filtering applica-tions, such structures are reported to offer reduced physicalsize and improved stopband performance [1], [2]. Thesefeatures are direct consequence of the slow waves propa-gating inside periodic structures, which present a reducedphase velocity and wavelength. Therefore, the physicallengths of half-wavelength resonators, when periodicallyloaded, are considerably reduced. Furthermore, due to thedispersion behavior of slow waves, an improved stopbandperformance can be achieved. The E-plane waveguidetechnology, periodically loaded with reactive obstacles inform of metal ridges, offers a convenient way for realizingEBG passive waveguide structures (see [2] and [3]).This paper proposes a novel method for the fast andrigorous full-wave analysis of passive structures based onElectromagnetic-Bandgap (EBG) waveguides periodicallyloaded with metal ridges. For such purpose, an IntegralEquation (IE)-based method, originally outlined in [4] forrectangular waveguides, is updated to cope with arbitrarilyshaped geometries. This method makes use of the modalchart of ridged waveguides, which is determined followinga revisited version (see [5]) of the classical and well-knownBI-RME technique.In order to validate the new analysis tool proposed inthis paper, we have considered two practical applicationsof EBG ridged waveguides. First, an E-plane filter periodi-cally loaded with metal ridges is successfully analyzed, aswell as compared in terms of physical size and out-of-bandresponse to a standard E-plane topology. Then, an EBGlow-pass ridge filter, designed from the pass- and stop-bands derived from the corresponding dispersion diagram,is discussed.II. THEORYAn example of a periodically loaded structure consid-ered in this paper, in particular an E-plane filter withperiodic metal ridges, can be seen in Fig. 1.taFig. 1. Three-dimensional layout of a periodically loaded E-plane filter.For analysis purposes, it is first required to know accu-rately the complete modal chart of the ridged waveguidespresent in the structure of Fig. 1. Following the well-known BI-RME method, the electric field at a genericobservation point r inside the arbitrary (ridged) waveguidecan be obtained by:E (r) =-jr1k j Ge (r, s', k) * J, (1') dl' (1)where the meaning of all involved variables, dyadic andvector functions can be seen in [5]. Splitting Ge and J,into its transversal and longitudinal components, and im-posing the corresponding boundary conditions (i.e. Et (r).t (1) = 0, and E, (r) = 0) on or, two integral equationsare obtained for the transversal electric field Et (TEmodes) and the longitudinal electric field E, (TM modes).Then, both integral equations are solved via the Galerkinversion of the MoM, where the chosen basis and testingfunctions are overlapping piece-wise parabolic splines.Applying the Galerkin approach, a generalized/standardeigenvalue problem is obtained for the TE/TM case. The978-1-4244-2743-7/09/$25.00 C2009 IEEE 87 Guadalajara, Mexico, Feb. 19-20, 2009Authorized licensed use limited to: UNIVERSIDAD DE ALICANTE. Downloaded on May 11, 2009 at 06:16 from IEEE Xplore.  Restrictions apply.2009 IEEE MTT-S International Microwave Workshop Series onSignal Integrity and High-Speed Interconnects (IMWS2009-R9)solutions for these problems provide as eigenvalues thecutoff wavenumbers, and as eigenvectors the modal expan-sion coefficients and the amplitudes of the transversal andlongitudinal components of the unknown current density.An advantage of the BI-RME technique is that, withouthardly additional computational effort, the coupling coef-ficients between the ridged waveguide and the standardrectangular contour enclosing the ridged profile can beeasily computed (see [6]). The coupling integrals betweenthe modes of the ridged waveguide and the ones of therectangular waveguide are defined as follows:Ipq j e() e(2)dS (2)where e(1) and e(2) are, respectively, the normalizedelectric modal vectors of the rectangular and ridged waveg-uides, being S the cross-section of the smaller (ridged)guide.In order to obtain a full-wave characterization of theplanar junction between two waveguides, a very efficientmethod based on an integral equation technique, originallydescribed in [4] for dealing with rectangular waveguides,has been properly updated. The objective of this techniqueis to obtain a Generalized Impedance Matrix (GIM) repre-sentation of each planar waveguide junction. A remarkablecontribution of this method is the distinction made betweenaccessible and localized modes: accessible modes are thoseused to connect transitions, while localized modes are onlyused to describe the electromagnetic fields in the junction.Imposing the boundary condition at the junction plane, andafter some mathematical manipulations (see details in [4]),it is possible to obtain the following integral equation:(3)h(6) (s) = MJM6) (s')iK(s, s') ds'cs (2)where s and s' are the observation and source points,respectively, K(s, s') is the kernel of the integral equation,M$6)(s') are unknown vector functions which define themagnetic field in the junction, and (6) = (1), (2) indicatesthe suitable region of the junction. Finally, the solutionof the integral equation can be performed using theGalerkin procedure, where the unknown vector functionsare expanded in terms of the smallest waveguide modes.Proceeding in this way, we can compute the impedanceparameters of the generalized Z-matrix as follows:+( ff)- J (M$<)(sl) h(6)*(s) ds'ics(2)M(a) f(nf- h 2) h(6)* dslq=l cs(2)(4)On the other hand, once the generalized Z matriz isobtained, the full-wave two-port ABCD parameters ofthe unit cell of the periodic structure can be numericallyderived:p A B j (5)where:A = Zll . Z21-B = Zl Z211 * Z22 -Z12C = Z21D =Z2.-* Z22(6)(7)(8)(9)In a periodic structure of period p, a generic componentF(z) of the electromagnetic field must satisfy the Floquetcondition [7]:F(z + p) = e-PF(z) Vz (10)where -' is the propagation constant of the Floquet mode.Imposing the Floquet condition at the unit cell, we obtain:(I,) (-2) (-2) (11)The right-hand side of equation (11) can now be recog-nized as the classical canonical form:[P] [x] = A [I] [x] (12)where the eigenvalues and, consequently, the propagationconstants, can be straightforwardly determined using well-established routines. At a given frequency, once the eigen-values A have been obtained, the real and imaginary partsof the Floquet propagation constant = a+j/3 are directlyderived by:ln 1Ap-= -ZAp(13)In the canonical problem (12), the number of eigenval-ues (A) obtained is equal to the order of the matrix Pin the left-hand side. These eigenvalues are, in general,complex and appear as pairs corresponding to forwardand backward directions of propagation of the same mode.Pass-bands and stop-bands are characterized, respectively,by the condition as to whether or not the periodic structuresupports at least one propagating Floquet mode within theconsidered frequency range.III. RESULTSIn order to demonstrate the practical application of anEBG ridged waveguide structure, a two-resonator peri-odic filter (proposed in [2]) has been first considered,as well as compared with an equivalent standard E-planetopology. The layout and dimensions of these two filters(which are implemented with standard WR-90 input/outputwaveguides, i.e. a = 22.86 mm and b 10.16 mm, andconsidering a metal insert thickness t 0.10 mm) areshown in Fig. 1 and Fig. 2.978-1-4244-2743-7/09/$25.00 C2009 IEEE 88 Guadalajara, Mexico, Feb. 19-20, 2009Authorized licensed use limited to: UNIVERSIDAD DE ALICANTE. Downloaded on May 11, 2009 at 06:16 from IEEE Xplore.  Restrictions apply.2009 IEEE MTT-S International Microwave Workshop Series onSignal Integrity and High-Speed Interconnects (IMWS2009-R9)bbLs, Lr, Ls2 Lr2 Ls,< >< >< >< 2 <AII(a)Ls, Lr, Ls2 Lr2 Ls3< >< >< >< >< >< p>(b)Fig. 2. Views of the metal inserts for the standard E-plane filter in(a) and the periodically loaded E-plane filter in (b). The dimensions forthe standard E-plane filter are: septa lengths L,j = Ls3 = 0.31 mmand L,2 = 1.68 mm, resonator length L,j = L,2 = 20.10 mm, totallength of filter 42.50 mm. The dimensions for the periodically loadedE-plane filter are: septa lengths L,j = Ls3 = 1.20 mm and L,2 =4.20 mm, resonator length L,j = L,2 = 8.0 mm, ridge dimensionssI = b 10.16 mm and s2 = 1.0 mm , ridge lengths II = 1.0 mmand 12 0.5 mm (period p = 1.5 mm), total length of periodicallyE-plane filter 22.40 mm.The homogeneous rectangular waveguides acting asresonators of the standard E-plane filter (see Fig. 2 (a)) arereplaced with periodic structures consisting of a cascade ofridged waveguides (see Fig. 2 (b)). The periodically loadedfilter is very similar to the standard E-plane topology, butinstead of having a homogeneous waveguide of length Lrbetween the two septa of length Ls (see Fig. 2 (a)), acascade of ridged waveguides of equal length are placedinside the resonator section (see Fig. 2 (b)).In Fig. 3 (a), the E-plane standard filter responseobtained with our full-wave software package is wellcompared with Ansoft HFSS vlO.0 results. Exploitingthe symmetry of the structure, the analysis method hasrequired to consider 80 accessible modes, 250 basis func-tions and 400 kernel terms. The complete simulation of theelectrical response has only taken a CPU effort of 0.11 sper frequency pointIn Fig. 3 (b), the periodically loaded E-plane filterresponse obtained with our full-wave software packageis successfully compared again with Ansoft HFSS vlO.0data, as well as with measurements of a manufacturedfilter prototype from [2]. Exploiting again the symmetryof the structure, in this case our full-wave analysis methodhas required to consider 80 accessible modes, 250 basisfunctions and 500 kernel terms. The complete simulationof the electrical response has only taken a CPU effort of0.35 s per frequency point (177 s in total). As it can beseen in both figures, the periodic loaded filter has betterselectivity and wider stop-band response than its standard'All the CPU times reported in this work were obtained with a PentiumIV platform at 3.2 GHz with 512-MB RAM.(a)10 11 12Frequency (GHz)14(b)Fig. 3. Electrical responses of the E-plane standard filter in a), and ofthe periodically loaded E-plane filter in b).E-plane counterpart. Furthermore, the periodic filter is47% shorter than the standard filter, because the lengthsof the periodically loaded resonators are near halved.Fig. 4 compares the dispersion relation for the TE10mode of the standard WR-90 guide and for the EBG ridgedwaveguide with dimensions shown in Fig. 2 (b). As itcan be seen, due to the slow wave effect, the dispersivebehavior of the EBG waveguide is significantly more linearin the 8-9 GHz bandwidth than for the standard case.Additionally, the dispersion diagram that can be ob-tained with the theory described in the previous sectionis useful for the practical design of waveguide filters.Analyzing the pass- and stop-bands (gaps) in the disper-sion diagram of a unit cell, it is possible to generate afinite structure with a very similar electrical response. Forinstance, the layout of an EBG fifth-order low-pass filter,originally proposed in [3], can be seen in Fig. 5. This filteris composed of a cascade of five ridged waveguides. Thedimensions of the unit cell have been selected in orderto obtain a stop-band from 11 GHz, which corresponds978-1-4244-2743-7/09/$25.00 C2009 IEEE 89 Guadalajara, Mexico, Feb. 19-20, 2009Authorized licensed use limited to: UNIVERSIDAD DE ALICANTE. Downloaded on May 11, 2009 at 06:16 from IEEE Xplore.  Restrictions apply.2009 IEEE MTT-S International Microwave Workshop Series onSignal Integrity and High-Speed Interconnects (IMWS2009-R9)12T1058Q6420N15GX 100 0.1 0.2 0.3 0.4(rad/mm)0 0.5 1 1.5p (rad)2 2.5 3Fig. 4. Dispersion relation for the rectangular WR-90 waveguide (dashedline) and E-plane EBG waveguide (solid line) in the same housing oflength p = 1.5 mm, with 82 = 1.0 mm,11 = 1.0 mm and t = 0.1 mmLw LrFig. 5. Layout of the EBG fifth-order low-pass filter. The dimensions are:L, = 6.0 mm, L,j = L,5 = 1.0 mm, L,2 = L,3 = L,4 = 2.0 mm,ridge dimensions b= 10.16 mm 82 = 1.0 mm, and t = 0.1 mm.(as shown in Fig. 6) to the starting frequency of thefirst band gap of the periodic structure. The number ofunit cells of the finite structure defines the order of thefilter, the attenuation in the stop-band and the out-of-bandresponse. It should be noted that the finite structure is notcompletely periodic, since the first and last ridge loadingshave been slightly modified in order to match the EBGwaveguide impedance to the TE10-mode impedance of theinput waveguide. In Fig. 7, the EBG fifth-order low-passfilter response provided by our analyis tool is successfullycompared with Ansoft HFSS vlO.0 results.IV. CONCLUSIONAn efficient and accurate tool for the full-wave analysisof passive structures based on Electromagnetic-Bandgap(EBG) ridged waveguides has been presented. For the fastand rigorous analysis of these filters, an advanced modalmethod combining BI-RME and IE techniques has beenoutlined. This novel tool has been successfully appliedfirst to the analysis of a compact periodically loaded(with metal ridges) E-plane filter with an improved stop-band response. Then, the dispersion behavior of EBGwaveguides are derived and used for the practical designof an EBG low-pass filter.Fig. 6. Dispersion behavior (3-f) for EBG periodically loaded waveg-uides. The dimensions of the unit cell are p 8 mm, l= 2 mm, and82 1 mm. The circles are taken from [3].8 10 -2 14+H 16--20'-40j__ISI21+ HFSS-808 10 12 14 16Frequency (GHz)Fig. 7. Electrical responses of the EBG fifth-order low-pass filter.REFERENCES[1] J. Sor, Y Quian and T. Itoh, "Miniature Low loss CPW periodicstructure for filter applications," IEEE Trans. Microwave Theory &Tech., vol. 49, no. 12, pp. 2336-2341-1504, December 2001.[2] G. Goussetis, and D. Budimir, "Novel periodically loaded E-planefilters", IEEE Microwave and Wireless Comp. Lett., vol. 13, no. 6,pp. 193-195, June 2003.[3] G. Goussetis, A. Faresidis and P. Kosmas, "Efficient analysis, design,and filter applications of EBG waveguide with periodic resonantloads", IEEE Trans. Microwave Theory & Tech., vol. 54, no. 11,pp. 3885-3892, November 2006.[4] G. Gerini, M. Guglielmi and G. Lastoria, "Efficient integral for-mulations for admittance or impedance representation of planarwaveguide junctions", IEEE MIT-S Int. Microwave Symp. Dig., vol.3, pp. 1747-1750, June 1998.[5] S. Cogollos, S. Marini, V. E. Boria, P. Soto, A. Vidal, H. Esteban, J.V. Morro, and B. Gimeno , "Efficient modal analysis of arbitrarilyshaped waveguides composed of linear, circular and elliptical arcsusing the BI-RME method," IEEE Trans. Microwave Theory & Tech.,vol. 51, no. 12, pp. 2378-2390, December 2003.[6] P. Arcioni, "Fast evaluation of modal coupling coefficients of waveg-uide step discontinuities", IEEE Microwave Guided Wave Lett., vol.6, no. 6, pp. 232-234, June 1996.[7] R. E. Collin, Field Theory of Guided Waves, New York: McGraw-Hill, 1960.978-1-4244-2743-7/09/$25.00 C2009 IEEE, ' - --TE |/0,' ~EBG -I90 Guadalajara, Mexico, Feb. 19-20, 2009Authorized licensed use limited to: UNIVERSIDAD DE ALICANTE. Downloaded on May 11, 2009 at 06:16 from IEEE Xplore.  Restrictions apply.

Full-wave analysis and applications of EBG waveguides periodically loaded with metal ridges

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