In this paper, the very accurate and efficient modal analysis of arbitrarily shaped waveguides, whose cross-section is defined by a combination of straight, circular and/or elliptical arcs, is solved. An extension of the well-known Boundary Integral-Resonant Mode Expansion (BI-RME) method is proposed. Successful comparisons with available numerical and experimental data fully demonstrates the validity of the improved BI-RME method proposed.This work has been supported by European Commission under the Research and Training Networks Programme, Contract No. HPRN-CT-2000-00043

Boria Esbert, Vicente Enrique

Cogollos Borras, Santiago

Esteban González, Héctor

Gimeno Martínez, Benito

Marini, Stephan

Morro Ros, José Vicente

Soto Pacheco, Pablo

Repositorio Institucional de la Universidad de Alicante

IFTH-7 Modal Computation of Arbitrary Waveguides Composed of Linear, Circular and Elliptical Arcs Santiago Cogollos*, Stephan Marini*, Pablo Soto', Hector Esteban*, Jose V. Mono*, Vicente E. Bona* and Benito Gimenol *Grupo de Aplicaciones de las Microondas Departamento de Comunicaciones, Universidad Politbcnica de Valencia, Camino de Vera s/n, E-46022 Valencia, SPAIN Email: vboria@dcom.upv.es b p t o .  Fisica Aplicada - I.C.M.U.V. Universidad de Valencia, Dr. Moliner 50, E-46100 Burjassot, Valencia, SPAIN Email: benito.gimeno@uv.es Absbcrci-In this paper, the very accurate and efficient modal analysis of arbitrarily sbaped wavgnides, whose cross-section is defined hy a combination of straight, cimrlar and/or ellip- tical arcs, is solved. An extension of the well-known Bound- ary Integral-Resonant Mode Expansion (BI-RME) method is proposed. Successful comparisons with available numerical and experimental data fully demonstrates the validity of the improved BI-RME method proposed. I. INTRODUCTION Arbitrary waveguides with a cross-section composed of Iin- ear, circular andor elliptical arcs (see fig. 1) are usually present in many passive waveguide devices [I]. For instance, ridge rectangular or circular waveguides, as well as cross-shaped irises, are frequently used in dual mode empty or dielectric loaded resonator filters. In the same way, the presence of rounded comers in rectangular waveguide passive components, mainly due to the manufacturing techniques presently em- ployed, is already being under investigation. Another example of great practical use is the elliptical waveguide, which has found increasing application in the design of many microwave smctures, such as dual and triple mode filters, circular waveguide polarizers, twists and mode launchers, radiators, resonators, and corrugated horns. Nowadays, most of the CAD tools used in the design of such components require to know, in a vely accurate way, the full-wave modal spectrum of the arbitrarily shaped waveguides considered in this paper. Furthermore, the coupling integrals between these modes and those of the standard rectangular waveguide that fully encloses the arbitrary cross-section (see 12 in fig. 1) must also be known. Among the many methods published in the literature to solve this problem, the BI-RME method originally proposed in [2], [3] has become one of the most popular ones, mainly due to its high speed and accuracy. The BI-RME methods is based on the solution of an Integral Equation by the Method of Moments, whose practical implementation (see details in [2]) requires to subdivide the boundary contour of the arhitrary waveguide (D in fig. 1). Even though many efforts have k e n  devoted to improve the BI-RME technique in the last years (see [l], [41), all Y T  Fig. 1. Arbitrmy waveguide with cross section S embedded in a standard rectangular waveguide R. Vector t is tangent to (r and I is a suitable abscisa taken on the line. the practical implementations of such technique subdivide the arbitrary contour using only straight segments. In many practical applications, e.g. when circular and/or elliptical arcs are present, the aforementioned subdivision approach based only on straight segments can lead to some errors in the results provided by the BI-RME method. Furthermore, as indicated in 151, such errors are even higher for the lower order modes of the arbitrary waveguides, thus degrading the accuracy of the modal analysis tools. In this paper, an extension of the BI-RME method, based on a subdivision of the arbitrary contours in rectangular, circular andor elliptical arcs, is proposed. Several practical results confirming the improved accuracy related to this new technique are offered. 11. IMPROVING THE TM PROBLEM In order to compute the TM modes of an arbitrarily shaped waveguide using the original BI-RME method described in [2], ~ the following matrix elements must be computed m: -. G3 = J, J, "z(l)g(s,S')2L3(l')dzdl' (1) where the functions U% and 7-13 are piecewise parabolic basis and weighting functions related to the MOM. The domain of 0-7803-7695-l/03/$17.00 0 2003 IEEE 2065 2003 IEEE MlTS Digest Authorized licensed use limited to: UNIVERSIDAD DE ALICANTE. Downloaded on March 26,2010 at 07:27:07 EDT from IEEE Xplore.  Restrictions apply. Yt  y s  Fig. 2. Arbimrily oriented straight segment with length Al. Fig. 3. Arbitrarily oriented circular arc with radius r and length rAq. these functions are the elements i and j of U that can be straight segments, circular arcs andor elliptical arcs. In (U, the double integral is computed numerically with a  in^ is s&aightforward. Gauss quadrature rule, but this is not possible for the diagonal elements of the L' matrix (i.e. when i = j )  because of the singularity of g (the scalar Green function of a cylindrical waveguide). A rapidly convergent expression for g is A. Straight segments For straight segments, the integration of the singular term arbitrary oriented straight segment can be defined as follows (see fig, 2) = 2o + (t + 0 . 5 ) ~ ~  caseg = yo + ( t  + 0.5)ALsinO (9) l m  TAOT;~ 47r g = -  In- T;OTA1 m=-w where (3) (4) 7r Tz9 = 1 - 2eTms cos (y - (-1)qy') + 7r Tmp = - 15 - (-1)*2' + 2" b The singularity is introduced by the In Tto term when the field point ( q y )  approaches the source point (dry ' ) .  This term tends to infinity Like In R. In order to treat this singularity, g must be expanded as a Taylor series, and taking into account the following limit where t E [-0.5,0.5] and the expression of InR is very simple InRZ = In [(z - 2') + (y - y')] = In [AL2(t - t')'] (IO) (2) and the singular integral can be solved immediately be- cause the integration of parabolic functions U, multiplied by In AZ2(t - t')' is analytically known. B. Circular arcs Following the same procedure, a circular arc (see fig. 3) can be defined as follows 2 = 20 t rcosp(t) y = yo + rsinp(t)  (11) (5) where T p  2 2 = (i) lim P Y '  z-z' (. - 2')' + (y - y') (12) d t )  = pi + Ap(t +0.5) then, the green function can be split in the following way Ap = ' ~ z - p i  (13) 1 277 g = g1 - -In R where 1 Tto 91 = go--lu- AT R2 where t E [-0.5,0.5] and T is the constant radius of the circular arc. Here, taking into account the Taylor series of R2, InR' can be decomposed in two parts (6) In R2 = In Ap2(t R2 - t')' +In  [Ap'(t - t')'] (14) (7) Y-Y' and the functions go and 91 are regular everywhere. Then, the singular part of the diagonal elements of L' matrix is analytically performed. Moreover, this analytic integration is always done in the same limits (in the parameter interval defined by t E [-0.5,0.5]) to fully exploit the code. With this last result, the regular term in (14) can be added to the regular part of the Green function to be integrated numerically, and the singular term of (14) can be integrated analytically because it is similar to (IO). 2066 Authorized licensed use limited to: UNIVERSIDAD DE ALICANTE. Downloaded on March 26,2010 at 07:27:07 EDT from IEEE Xplore.  Restrictions apply. ’4 , , (x, , ,v.)  f l u , :  .............. ,,i ....... Fig. 4. Arbitrarily oriented elliptic arc. C. Elliptical arcs Again, following the same procedure of the previous sec- tions, an elliptic arc is defined. In this case, a local coordinate system (U, v) has been chosen (see fig. 4), thus obtaining U = acosq(t)  v = bsinq(t) (16) where q( t )  = VI + Aq(t +0.5) AV = 02 ~ 71 (17) with t E [-0.5,0.5]. Using the Cartesian coordinate system ( x , ? ~ )  we obtain In the ‘E case, an additional problem appears when the dyadic GSt is expanded into a Taylor series, where the following two terms are obtained which are not singular but discontinuous. This means that the limit when the source point approaches the field point depends on the approximation path. If we have three kinds of segments, these limits must be properly calculated for each case. A. Straight segments If we use (Y), the limits of (22)  and (23) are easily calculated lim = cos2$ (24) 2-2‘ 1I-Y‘ lim V = -sin(26’) z-z’ 2 U-Y’ (25) 1 B. Circular arcs In this case, using ( I  I), the limits of Q and I are Iim Q = sin2[p(t)] (26) 2-B‘ YWY’ lim I = -1 sin[ap(t)l (27) 5-21 2 Y-Y‘ C. Elliptical arcs In this case, taking (16), the limits of Q and I are (28) [-a cos a sin q(t)  - bsin a cos q(t)12 (;) = (20) YO + (cp,. s ina  - s t n o ) .  cosa (;) (18) lim @ = Y-Y’ and finally 5-2’ a2 sin2 q(t)  + b2 cos2 q(1) (29) l x ( t ) s in2a -abs in2qcos2a  lim P = ~ v-v‘ R2 = a2 [cosq(t) - c o ~ q ( t ’ ) ] ~  + b2 [sinq(t) ~ sinq(t’)]’ (1% 5-5’ 2 a2 sin2 q(t)  + b2 cos2 q(t)  Now, taking into account the Taylor series of R2, InR2 can be decomposed in two parts where ~ ( t )  = a2 sin2 q(t)  - b2 cos2 q(t)  (30) In R2 = In A$(t R2 - t’)2 +In [A$@ - t’)’] (20) IV. RESULTS This section presents some results that fully validate the extension of the BI-RME method iust described. Successful because the first term of the r.h.s part of (20) is regular Y-Y’ A. n2e circular waveguide In this case, the singular term of (20) needs a further treatment in order to obtain an analytical solution. . ~n this example, we compute the cutoff frequencies of the TM modes of a circular waveguide, which is obtained 111. IMPROVING THE TE PROBLEM from a standard square contour (6 in fig. 1) perturbed by a tubular sheet (U in fis. 1). Table I comDares the relative - ,  For the TE problem, the singularities appear in the static part of the bidimensional Green’s dyadic GSt,  whose four components are detailed in [2], and are of the same kind solved before. Therefore, the same procedure described in the previous section can he followed. enors of such cutoff frequencies for the classical BI-RME approach (using only straight segments) and the new extension proposed (considering circular arcs). This benchmark test fully demonstrates the accuracy improvement introduced by the BI- RME extension proposed. 2067 Authorized licensed use limited to: UNIVERSIDAD DE ALICANTE. Downloaded on March 26,2010 at 07:27:07 EDT from IEEE Xplore.  Restrictions apply. TABLE I RELATIVE RROR IN THE CUTOFF FREQUENCIES OF THE BI-RME METHOD USING STRAIOHT SEGMENTS AND CIRCULAR ARCS. 1 Mode Enor BI-RME I Error BI-RME I (TM) (straight segments) (circular segments) ~~ T M i i  TMoz TMii TM12 0.206 % 0.009 % 0.207 % 0.004 % 0.320 % 0.011 % 0.439 % 0.013 % TABLE II CUTOFF WAVENUMBERS USINO BI-RME WITH ELLIPTICAL ARCS VERSUS THE METHOD OUTLINED IN [61 ..TMiz ~ Mode W M )  TE TE TM TE TE TM __ 0.585 % 0.015 % 0.422472 0.422540 0.519355 0.519450 0.644532 0.644830 0.658631 0.659039 Relative emx 0.015 0.016 0.018 0.04 B. The elliptical waveguide Next, an elliptical waveguide defined within a rectangular WR-90 waveguide has been considered. Here, U is an ellipse of eccentricity equal to 0.5 and semi-major axis equal to 5". Table II successfully compares the cutoff wavenumbers provided by this method with results from [6]. C. Filter analysis As a final example of practical use, a four-pole inductively coupled rectangular waveguide filter in with rounded corners has been designed, manufactured and measured. The rounded corners that usually appear in the manufacturing process have been accurately considered using the extended BI-RME method proposed in this paper. As can be seen in fig. 5, this filter is composed of several pieces that can be manufactured using low cost milling. Figure 6 shows a very good agreement between simulated and measured'results. V. CONCLUSION In this paper, an extension of the BI-Rh4E method that can handle straight, circular and elliptical segments in the dis- cretization of arbitrary cross section waveguides is presented. The validation of this extended technique has been carried out with numerical and experimental results. Finally, this method has been used in the design process of a rectangular waveguide filter with rounded corners. The simulated results have been successfully compared with measurements of a low cost manufactured prototype. Fig. 5. Pieces of the four pole inductive filter with rounded comers Fig. 6. Simulated and measured S-parameten of the four pole inductive filter with rounded comers. ACKNOWLEDGMENT This work has been supported by European Commission un- der the Research and Training Networks Programme, Contract No. HPRN-CT-2000-00043. REFERENCES Ill G. Conciauro, M. Guglielmi. and M. Somentino, Advanced Modal Anoly- sis . CAD Techniquesfor Woveguide Components Md Filters. Chichester: John Wiley & Sons, Ltd., 2000. [Z] G. Conciauro, M. Bressan, and C. Zuffda, "Waveguide modes via an integral equation leading to B linear mauix eigenvalue problem," IEEE Trans. Micmwove Weory Tech., vol. 32, no. 11, pp. 1495-1504, 131 P. h i a n i ,  "Fat evaluation of modal coupling coefficients of waveguide step discontinuities:' IEEE Micmwave wireless Compon. Len., vol. 6, no. 6, pp. 232-234, June 1996. [4] M. Bozzi, G. Conciauro, and L. Perreprini, "On the evaluation of * modal coupling coefficients by contour integrals:' lEEE Trans. Micmwave Theory Tech., vol. 50, no. 7, pp. 1853-1855, July 2CQ2. [5l R. Courant and D. Hilbert. Methods ofMarhemaricnl Physics. ser. Wiley Classics Libmy. New York John Wiley & Sons Wiley-lntencience. 1989, vol. I. [6] B. Gimeno and M. Guglielmi. "Full wave network representation for rectangular, circular, and elliplical to elliptical waveguide juctians:' IEEE Trans. Micmwave Theory Tech.. vol. 45, no. 3, pp. 376-384. March 1997. November 1984. 2068 Authorized licensed use limited to: UNIVERSIDAD DE ALICANTE. Downloaded on March 26,2010 at 07:27:07 EDT from IEEE Xplore.  Restrictions apply. 

Modal computation of arbitrary waveguides composed of linear, circular and elliptical arcs

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