Research Article Dominant Mode Wave Impedance of Regular Polygonal Waveguides

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1 Microwave Sciece ad Techology, Article ID , 4 pages Research Article Domiat Mode Wave Impedace of Regular Polygoal Waveguides Vyacheslav V. Komarov Istitute of Electroic ad Mechaical Egieerig, Yuri Gagari State Techical Uiversity of Saratov, Saratov , Russia Correspodece should be addressed to Vyacheslav V. Komarov; vyacheslav.komarov@gmail.com Received 25 October 2013; Revised 26 December 2013; Accepted 30 December 2013; Published 6 February 2014 Academic Editor: Chie-Je Wag Copyright 2014 Vyacheslav V. Komarov. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Polygoal metal waveguides are aalyzed aalytically ad umerically. Classical equatio for the wave impedace of arbitrary shaped waveguides is completed with approximate expressio for the cutoff wavelegth of the domiat mode. Proposed approach is tested with the help of 3D fiite differece time domai models of microwave waveguides juctios. Obtaied data are used for computer-aided desig of microwave trasitio from coaxial lie to cylidrical waveguide. 1. Itroductio Regular polygoal waveguides (RPW) with differet umber of side walls () fid applicatio i microwave egieerig as basic uits of ateas [1], orthomode trasducers [2], T- juctios [3], mode trasformers [4], heatig devices [5], ad power combiers [6]. Distributio of the TE- ad TM-modes isuchwaveguidescabeobtaiedfromthehelmholtz equatio solutio with Neuma ad Dirichlet boudary coditios o metal walls. But the exact aalytic solutio of this problem is possible oly for the triagular ( = 3) [7] ad square(=4) waveguides. Approximate aalytical approaches based, for example, o perturbatio method [8], method of aalytical regularizatio [9], method of aalogy [10], or some others [11] are also used for calculatio of the eigevalues ad eigefields of RPW. Some of these methods are restricted either by the waveguide shape [7] or by the mode type [11]. Numerical techiques implemeted i commercial packages, HFSS, CST MWS, ad MATLAB, are the alterative tools of modelig electromagetic fields i RPW [9, 10, 12]. Most of publicatios metioed above are devoted to the computatio of the cutoff waveumbers of hollow RPW with 3 8. Closed-form expressios for approximate calculatio of the characteristic impedace, atteuatio, ad phase costat have bee derived i [10], where a very simplified equatio for wave impedace without ay examples of its applicatio is represeted. The objective of the preset study was to check the applicability of metal waveguides theory to the modelig of wave impedace of RPW. 2. Wave Impedace of RPW Electromagetic characteristics of ay RPW deped o the metal wall size (a ), which ca be determied usig aother geometrical parameter outer radius (R ): a =2R si ( π ). (1) As it is kow, a >R,for3 5; a 6 = R 6, while for the rest waveguides a <R whe 7(Figure 1). The wave impedace of the lowest TE-mode of ay shaped metal waveguide: Z TE = kη β = 120π 1 (λ/λ c ) 2, (2) where k is the waveumber; η is the free space wave impedace; β is the phase costat; λ is the wavelegth i free space; λ c is the cutoff wavelegth of the domiat

2 2 Microwave Sciece ad Techology R 8 R 3 a 3 (a) a 8 (b) Figure 1: Triagular (a) ad octagoal (b) waveguides. mode. The eigevalues of RPW ca be calculated usig the results of umerical modelig obtaied with the help of the fiite elemet method (FEM) implemeted i PDE Toolbox of MATLAB software [13]. Logitudial formulatio of 2D Helmholtz equatio with Neuma boudary coditios was utilized i fiite elemet modelig of the lowest TE-mode of each RPW. Automated mesh geerator ad adaptive mesh refier based o Roseberg-Steger scheme are realized i PDE Toolbox for flexible triagulatio of arbitrary shaped 2D domais. Electromagetic field compoets were approximated by the first-order polyomial fuctios ad geeralized eigevalue problem was solved by Aroldi algorithm. Number of triagular elemets i the mesh for all RPW did ot exceed 40,000. Prelimiary testig of umerical solutio o examples of square ( = 4) ad cylidrical ( 32) waveguides has show a good agreemet with aalytical solutios for these classical trasmissio lies available from the literature. Additioal details of RPW simulatios by meas of the FEM are described i [12, 13]. Curve-fittig procedures of MATLAB ad MS Excel have bee applied to derive a closed-form expressios usig umerically obtaied depedecies λ c (), where3 100: λ c = ( ) a, 3 8, (3) λ c = ( ) R =qr, 3 8, (4) λ c = ( ) R, (5) Determiatio coefficiet for (3) (5)wasR Numerical Verificatio Numerical simulatio tool commercial software Quick- Wave 3D [14] based o the fiite differece time domai (FDTD) method has bee utilized i this study for checkig (4). Comparative aalysis of two umerical techiques, FEM ad FDTD, ad aalytical approach described i [10] are giveitable 1. Table 1: Normalized cutoff wavelegths of the lowest TE-mode of RPW. λ c /R FEM FDTD Aal. [10] Juctios of RPW ca be successfully employed for desig of microwave trasitios from widely available coaxial lie to RPW. Oe of such approaches is aalyzed i preset study. Let us cosider stadard cylidrical waveguide (CW) with radius R = cm ad the TE 11 -mode propagatig at ISM-frequecy 2.45 GHz. Matchig of this waveguide with RPW is achieved whe S 11 = TE Z Z mi, (6) Z+Z TE where S 11 is the reflectio coefficiet; Z is the wave impedace of the TE 11 -mode of CW at 2.45GHz ad accordig to (2), Z = Ω. Coditio (6)willbesatisfiedifZ TE Z. The, i order to fid sizes of RPW which correspod to (6)wecarewrite (2)as λ c = λ 1 (120π/ Z) 2. Ad ow employig (4)we obtai R = λ q 1 (120π/ Z) 2. (7) (8)

Microwave Sciece ad Techology 3 D d Adapter t s w Y Z SQW CW Figure 2: Juctio of cylidrical ad hexagoal waveguides. Juctio Figure 3: Cofiguratio of microwave trasitio uder study.

3 Microwave Sciece ad Techology 3 D d Adapter t s w Y Z SQW CW Figure 2: Juctio of cylidrical ad hexagoal waveguides. Juctio Figure 3: Cofiguratio of microwave trasitio uder study. Table 2: Reflectio coefficiet of waveguide juctios R, cm 5,488 5,045 4,732 4,563 4,459 4,390 S Six 3D FDTD models of CW-RPW juctios for 3 8 have bee built applyig QuickWave 3D ad (8). Oe of such models is show i Figure 2. Iput sigal i the form of a pulse of spectrum from 2.3 GHz to 2.6 GHz ad TE 11 -mode was selected for CW. Aalogous sigal but for arbitrary shaped trasmissio lie was used i the output port. Simulatio results: absolute values of reflectio coefficiet S 11 at 2.45 GHz are listed i Table 2. Obtaied data demostrate that proposed approach to the RPW wave impedace defiitio works well whe 4. Forthe = 3, S 11 value is higher tha expected. Additioal calculatios have show that matchig coditio (6) is satisfied for R 3 =4.9cmad S 11 = Example of Microwave Trasitio Desig Aalytical approach described i preset study has bee used for buildig the model of coaxial waveguide trasitio show i Figure 3. Trasitio is cosisted of two sectios: coaxial square waveguide (SQW) adapter ad SQW-CW juctio. CoaxialliewithsizesD =7mmadd = 3.04 mm excites TE 10 mode i SQW 100 mm legth with side wall size a 4 = mm. The domiat mode of SQW is trasformed i TE 11 -modeofcw15mmlegthad83.62mmdiameteri waveguide juctio. Cylidrical metal probe t = 2mm coected to the coaxial feeder is placed at the distace w = 30mm from short circuited wall of SQW. Optimizatio problem solutio has allowed determiig the probe height value s=25mm. Results of umerical modelig: absolute values of reflectio coefficiet i frequecy rage 2.2 f, GHz 2.7 are represeted i Figure 4. FDTDmodelhasbee verifiedby 3D FEM model developed with the help of COMSOL software Reflectio coefficiet Y X Z FDTD FEM Frequecy (GHz) Figure 4: Absolute values of reflectio coefficiet for microwave coaxial SQW adapter (1) ad complete trasitio coaxial SQW-CW (2). [15]. Both techiques agree well ear the ISM-frequecy 2.45 GHz. Ad ow takig ito accout results i Table 2,RPWwith ayumberofsidewallscabeusedisteadofcwtodesig trasitio from coaxial lie to the selected RPW. 5. Coclusio It has bee proved that the classical approach to the wave impedace defiitio previously applied for rectagular ad cylidrical waveguides is also applicable for RPW. Exceptio observed for the triagular waveguide should be a subject of a separate study. Obtaied data ca be employed i desig of various microwave compoets, for example, trasitios, o RPW. Coflict of Iterests The author declares that there is o coflict of iterests regardig the publicatio of this paper.

4 4 Microwave Sciece ad Techology Refereces [1] J. M. Simeoi ad M. Jofre, Equilateral triagular waveguide atea a spectral domai aalysis, IET Microwaves, Ateas ad Propagatio, vol. 4, o. 3, Article ID IMAPCH , pp , [2] J.-H. Hwag ad Y. Oh, Compact orthomode trasducer usig sigle-ridged triagular waveguides, IEEE Microwave adwirelesscompoetsletters,vol.21,o.8,pp ,2011. [3]Y.Tao,Z.She,adG.Liu, Closed-formexpressiosfor the equivalet circuit model of square-waveguide T-juctios ad its applicatio i ortho-mode trasducer desig, IEEE Trasactios o Microwave Theory ad Techiques, vol.58,o. 5, pp , [4]A.Mediavilla,J.L.Cao,adK.Cepero, Otheoctave badwidth properties of octagoal-shaped waveguide mode trasformers, IEEE Trasactios o Microwave Theory ad Techiques,vol.59,o.10,pp ,2011. [5] V. N. Tra, A applicator desig for processig large quatities of dielectric materials, i Proceedigs of the Symposium o Microwave: Theory ad Applicatios i Material Processig, pp , Ciciati, Ohio, USA, [6] V. S. Il i, S. N. Il i, ad D. A. Usaov, Properties of a orthogoal-turstile juctio of three square waveguides ad a hexagoal waveguide, Commuicatios Techology ad Electroics,vol.42,o.2,pp ,1997. [7] C. Y. Wag, Exact solutio of equilateral triagular waveguide, Electroics Letters,vol.46,o.13,pp ,2010. [8] C. Y. Wag, Cutoff frequecies of perturbed circular waveguides ad polygoal waveguides, Electromagetic Waves ad Applicatios,vol.16,o.2,pp ,2002. [9] M. Lucido, G. Paariello, ad F. Schettio, Full wave aalysis of arbitrary polygoal sectio waveguides, i Proceedigs of the IEEE MTT-S Iteratioal Microwave Symposium (IMS 07),pp , Hoolulu, Hawaii, USA, Jue [10]B.RaveloadA.K.Jastrzebski, EMparametersofregular polygoal waveguide, i Proceedigs of the 14th Europea Microwave Coferece (EuMC 11), pp , Machester, UK, October [11] Q. Zheg, l. Ma, ad J. Li, A simple estimate for cutoff umber of lowest order TM mode of hollow metallic waveguide of arbitrary cross sectio, RF ad Microwave Computer-Aided Egieerig,vol.10,o.3,pp , [12] V. V. Komarov, Eigemodes of regular polygoal waveguides, Ifrared, Millimeter, ad Terahertz Waves,vol.32,o. 1, pp , [13] MathWorks, PDE Tootbox,MATLAB,Natick,Mass,USA,2013, [14] QWED, QuickWave-3D, QWED,Warsaw,Polad,2013, [15] COMSOL Multiphysics V.4.3,2013,

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