An efficient integral equation technique for the analysis of arbitrarily shaped capacitive waveguide circuits

Transcription

1 RADIO SCIENCE, VOL. 46,, doi: /2010rs004458, 2011 An efficient integrl eqution technique for the nlysis of rbitrrily shped cpcitive wveguide circuits F. D. Quesd Pereir, 1 P. Ver Cstejón, 1 A. Álvrez Melcón, 1 B. Gimeno, 2 nd V. E. Bori Esbert 3 Received 15 June 2010; revised 15 November 2010; ccepted 24 Jnury 2011; published 21 April [1] In this contribution new nd efficient integrl eqution formultion is presented for the nlysis of rbitrrily shped cpcitive wveguide devices. The technique benefits from the symmetry of the structure in order to reduce the dimensions of the problem from three to two dimensions. For the first time, this technique formultes the wveguide cpcitive discontinuity problem s 2 D scttering problem with oblique incidence, combined with n efficient clcultion of the prllel plte Green s functions. The numericl method llows the efficient evlution of the electromgnetic fields inside the nlyzed structures. Results for different prcticl cpcitive wveguide devices re successfully compred with commercil softwre tools for vlidtion of the proposed theory. Finlly, novel low pss filter implementtion bsed on circulr conducting posts hs been proposed. The field contour lines in the criticl gps of the new structure re curved due to the use of rounded posts. This could result in improved power hndling cpbilities with respect to stndrd corrugted low pss filters. Cittion: Quesd Pereir, F. D., P. Ver Cstejón, A. Álvrez Melcón, B. Gimeno, nd V. E. Bori Esbert (2011), An efficient integrl eqution technique for the nlysis of rbitrrily shped cpcitive wveguide circuits, Rdio Sci., 46,, doi: /2010rs Informtion nd Signl Theory Deprtment, Technicl University of Crtgen, Crtgen, Spin. 2 Applied Physics Deprtment, University of Vlenci, Vlenci, Spin. 3 Communictions Deprtment, Technicl University of Vlenci, Vlenci, Spin. Copyright 2011 by the Americn Geophysicl Union /11/2010RS Introduction [2] Cpcitive rectngulr wveguide structures re typiclly used for the design of low pss filters s proposed by Levy [1973], impednce converters [see Young, 1962], nd mtching networks. In the pst, severl numericl techniques hve been used for computing the electricl response of this kind of devices, such s the finite elements method [see Slzr Plm et l., 1998] employed by widely used commercil softwre pckges like HFSS, modl nlysis techniques s described by Guglielmi nd Gheri [1994], or the boundry integrl resonnt mode expnsion (BI RME) used by Arcioni et l. [1996] for the brod bnd nlysis of cpcitive microwve devices. Although generl purpose finite elements codes cn del properly with E plne (cpcitive) structures, their min drwbck is the lck of efficiency, since one hs to setup three dimensionl model of the circuits. On the other hnd, modl nlysis s employed by Guglielmi nd Gheri [1994] re very efficient for the study of E plne devices composed of cnonicl rectngulr wveguide sections, but cnnot esily hndle complex geometries such s rounded corners introduced during mechnicl mnufcturing procedures, or other useful elements such s conducting rounded posts. This limittion is due to the need of the computtion of the modl chrt of the bsic wveguide sections defining the whole structure. [3] Moreover, the method of nlyticl regulriztion (MAR) [Nosich, 1999] hs successfully been employed for the nlysis of E plne nd H plne rectngulr wveguide microwve devices [Kirilenko et l., 1994, 1996; Lypin et l., 1996]. This technique llows for the nlysis of circulr posts inside rectngulr wveguides following procedure similr to tht proposed by Twersky [1962]. The min dvntge of MAR with respect to stndrd integrl eqution technique solved by the method of moments (MOM), is the formultion in terms of Fredholm mtrix equtions of the second kind which gurntee numericl convergence. However, MAR cnnot del with rbitrrily shped cpcitive geometries, other thn circulr posts. [4] This drwbck hs been overcome by other techniques llowing the efficient evlution of modl chrts in complex geometries, such s the BI RME method. Alterntively, the nlysis of E plne rbitrrily shped microwve components cn lso be performed by computing the dmittnce mtrix of the structure with the 2 D BI RME technique, s presented by Arcioni et l. [1996]. The technique is bsed on solving n eigenvlue problem where the whole structure is surrounded by n uxiliry rectngulr cvity resontor, nd the input nd output ports re short circuited. The dynmic vrition of the fields inside the structure re expnded in terms of the resonnt modes of the surrounding cvity used s reference. Once the eigenvlue problem is solved, the brodbnd dmittnce 1of11

2 [6] This contribution presents the bsic formultion nd the theory underlying the developed integrl eqution. The technique is demonstrted in three E plne prcticl wveguide devices, including simultion dt from commercil softwre pckges for vlidtion. A novel low pss filter bsed on circulr cpcitive steps is proposed, to show the cpbilities of the new technique to del with complex geometries. The electromgnetic fields distributions re obtined for the two lst exmples. Results show tht in the criticl gps the field contour lines re curved due to the use of rounded posts. This could result into beneficil effect for power hndling cpbilities issues with respect to stndrd corrugted low pss filters presented by Cmeron et l. [2007]. Figure 1. A cpcitive step inside rectngulr wveguide of width nd height b. The coordinte xis used in this pper is lso shown. prmeters of the structure re obtined by opening the input nd output ports. [5] On the other hnd, integrl eqution techniques re very populr for the nlysis of useful microwve devices [see Aud nd Hrrington, 1984; Bunger nd Arndt, 2000], nd 2 D/3 D scttering problems s described by Hrrington [1968] nd Ro et l. [1982]. An interesting exmple is the study of inductive microwve components in rectngulr wveguides [see Estebn et l., 2002]. This problem ws lso solved by Quesd Pereir et l. [2007] nd Pérez Soler et l. [2007] using 2 D integrl eqution formultion under norml plne wve excittion. In ddition, 2 D integrl eqution formultions with oblique incident ngle hve been used in generl scttering problems, involving both metllic nd dielectric objects s described by Peterson et l. [1998] nd Rojs [1988]. However, to the uthors knowledge, such methods hve never been pplied to the study of prcticl wveguide microwve components. In this pper we solve for the first time the wveguide cpcitive discontinuity problem with n oblique incidence integrl eqution formultion. The technique uses the prllel plte Green s functions of line sources, efficiently computed with the theory previously reported by Quesd Pereir et l. [2007]. Therefore, the clcultion of modl chrts in complex wveguide structures is not needed for the tretment of complex cpcitive wveguide geometries. The proposed technique provides full wve nlysis of prcticl cpcitive wveguide circuits. Unlike other modl bsed techniques, it cn esily del with rbitrrily shped cpcitive discontinuities nd posts inside rectngulr wveguides. The technique is lso efficient for the nlysis of these devices, since it exploits the symmetry of the structure to reduce the nlysis to 2 D problem. All these resons mke of the technique proposed very ttrctive nd useful strtegy for the design of cpcitive wveguide devices. 2. Theory [7] The formultion of the proposed method is setup considering rectngulr wveguide where ll the discontinuities re invrint long the x xis, s shown in Figure 1. In Figure 1, the relevnt dimensions (width nd height b) of the wveguide re indicted, together with the orienttion of the coordinte xes. [8] The originl structure shown in Figure 1 is replced, fter the ppliction of the surfce equivlent principle [see Blnis, 1989], by induced electric current densities on the boundries of the conducting discontinuities. In this wy, the rectngulr wveguide is reduced to prllel plte wveguide due to the invrint geometry of the problem long the x xis. After these considertions, n electric field integrl eqution (EFIE) [see Peterson et l., 1998] is employed for computing the response of the equivlent two dimensionl problem shown in Figure 2. [9] The excittion of the problem (~E i, ~H i ) is the dominnt mode of the rectngulr wveguide TE z 10. Although the geometry of the cpcitive device is invrint long the x xis, this fundmentl mode exhibits hlf period trigonometric vrition of sine type long the wveguide width, s cn be observed from its bsic expression: ~E i ¼ A 10 " 0 sin x e jkzz^y where k z is the propgtion constnt long the wveguide z xis, nd A 10 is n rbitrry mplitude constnt [Blnis, 1989]. This vrition of the exciting field prevents to formulte the problem s simple scttering problem of norml incidence contined in the (y, z) plne, s it hs been done in the works of Quesd Pereir et l. [2007], Aud nd Hrrington [1984], Conciuro et l. [1996], nd Estebn et l. [2002]. [10] Fortuntely, the previous vrition of the field long the x xis is known nd does not chnge due to the symmetry of the problem. After pplying the Euler s formul to eqution (1), the incident electric field cn be considered s tht produced by two plne wves with n oblique incident ngle with respect to the x xis ( ~ k 1 = ^x + k z^z nd ~ k 2 = ^x + k z^z, see Figure 3): ~E i ¼ A 10 e j x e j x " 0 2j e jkzz^y ð1þ ð2þ 2of11

3 Figure 2. Equivlent prllel plte wveguide problem considered for the evlution of the electromgnetic response of the device. The constitutive prmeters of the medium inside the wveguide re the sme s those corresponding to vcuum (" 0, m 0 ). The electromgnetic fields exciting the structure (~E i, ~H i ) correspond to the fundmentl mode of the rectngulr wveguide TE z 10 ;(^c) is unit vector tngent to the contour, wheres (^n) is the norml unit vector. The loction of the input (z = 1 ) nd output (z = 2 ) ports is lso represented in the plot. [11] This decomposition of the exciting field in two plne wves, which re propgting t n oblique ngle with respect to the x xis, llows to formulte the problem s 2 D scttering problem of oblique incidence. Following this theory, the electromgnetic fields scttered by the conducting cpcitive discontinuities inside the rectngulr wveguide cn be computed s the sum of the contributions of the two previous plne wves. A very efficient wy for solving this problem is bsed on writing the EFIE in the dul sptilfrequency domin (spectrl vrible k x =± )[Peterson et l., 1998], resulting: h ^n ~E ~ i þ ~E ~ i s ¼ 0; on the metllic contourðþ c ð3þ h ^n ~E ~ i h i ¼ ^n j! ~A ~ i þ ~rf e ; on the metllic contourðþ c ð3bþ [12] In equtions (4) nd (4b), ll the relevnt mgnitudes re written in mixed sptil spectrl domin (sptil vribles of the contour (c) in the (y, z) plne nd the spectrl vrible k x ). [13] Another importnt difference with respect to n inductive problem [see Quesd Pereir et l., 2007; Aud nd Hrrington, 1984; Estebn et l., 2002], is tht the unknown induced electric current ~ ~J(c ) = ~J x (c )^x + ~J c (c )^c presents two components, one long the longitudinl x xis, nd the other long the contour (c) of the cpcitive problem. On the other hnd, G A (~, ~ ) is digonl dydic Green s function corresponding to the mgnetic vector potentil, wheres G V (~, ~ ) is the electric sclr potentil Green s function.the mthemticl forms of these Green s functions for the geometry under considertion re described in section 2.1. [14] Finlly, the integrl eqution (equtions (3) (3c)) hs been solved by mens of the method of moments expnding ~r ¼ jk x^x þ d dc ^c ð3cþ where mixed potentil representtion of the electric field is ssumed, nd the (~) denotes Fourier trnsformtion with respect to the x coordinte. Using the Green s functions formlism, the explicit form of the integrl eqution is written s ~~E i ðþ c ¼ j! G A ð~;~ Þ ~J x ðc Þ^x þ ~J c ðc Þ^c dc tn c h i þ jk x^x þ d jk x ~J x ðc Þþ d ~J c ðc Þ dc ^c dc c j! G V ð~;~ Þdc ð4þ tn ~ ¼ y^y þ z^z ð4bþ z Figure 3. The excittion of the problem, TE 10 mode, is the sum of two different plne wves. The propgtion constnts of the plne wves re ~ k 1 = ^x + k z^z nd ~ k 2 = ^x + k z^z. 3of11

4 the unknown electric current with tringulr subsectionl bsis functions, nd using the sme set of functions for testing (Glerkin procedure) ~~J ðc Þ ¼ ~J x ðc Þ^x þ ~J c ðc Þ^c ¼ XNx n f nx ðc Þ k x ^x þ XNc b n f nc ðc Þ k x ^c ¼ XNx n f nx ðc Þ k x þ ^x þ XNc b n f nc ðc Þ k x þ ^c ð5þ [15] These functions re defined on the liner segments used in the discretiztion of the contour of the conducting discontinuities inside the prllel plte wveguide. In the expnsion of the currents, N x nd N c re the number of subsectionl tringulr bsis functions long the longitudinl ( f nx ) nd trnsversl ( f nc ) directions, respectively. On the other hnd d (k x ) is spectrl Dirc s delt function defined t the sptil frequencies (k x =± ) corresponding to the oblique incident ngle of the plne wves exciting the rectngulr wveguide (see eqution (2) nd Figure 3). The excittion of the problem cn be written in the mixed sptil frequency domin s: ~~E i ¼ A 10 " 0 k x kx þ 2j e jkzz ^y [16] The electric current in equtions (4) nd (4b) is replced by its expnsion (5). After tht, the resulting expression is tested with functions oriented long x xis ( f mx ) nd long the trnsverl (y, z) plne ( f mc ). Finlly, system of liner equtions is obtined nd cn be written in mtrix form for ech one of the involved sptil frequencies (k x =± ), s follows xx mn xc mn n ð n Þ b n ðb n Þ cx mn cc mn ¼ where ( n, b n ) re the unknown expnsion coefficients under plne wve excittion (k x =+ ), nd ( n, b n ) re the corresponding coefficients under the excittion (k x = ). After some mthemticl mnipultions, the different submtrices tke the form mn c xx ¼ j! f mx ðþ c G xx A ð~;~ Þf nxðc Þdc dc m c n jð= Þ2 f mx ðþ c G V ð~;~ Þf nx ðc Þdc dc ð8þ! c m c n mn xc ¼= f mx ðþ c G V! ð ~;~ Þ dfnc c ð Þ dc dc ð8bþ c m c n dc mn cx ¼= df mc ðþ c G V ð~;~ Þf nx ðc Þdc dc ð8cþ! c m dc c n mn c cc ¼ f mc ðþ c c y c y G yy A ð~;~ Þþc zc G zz z A ð~;~ Þ fnc ðc Þdc dc m c j df mc ðþ c G V ð~;~ Þ df ncðc Þ dc dc ð8dþ! c m dc c dc e m ¼ A 10 c m " 0 0 e m e jkzz f mc ðþc c y dc 2j ð6þ ð7þ ð8eþ The tngent unit vectors to the conducting posts in equtions (8) (8e) re defined s ^c = c y^y + c z^z nd ^c = c y ^y + c z ^z for the observtion nd the source cells, respectively. [17] Since we re employing the sme set of functions for the expnsion of the electric current density nd for testing, following Glerkin procedure, the reltion xc mn = cx mn is stisfied. Moreover, the following reltion holds between the current expnsion coefficients under the two plne wve excittions: n ¼ n b n ¼ b n ð9þ ð9bþ [18] The previous reltions llow to solve the system of liner eqution (7) only once for computing the totl induced electric current density on the conducting posts. Using these reltions, the totl electric current in the sptil domin cn be expressed s ~J ðc Þ ¼ XNx n f nx ðc Þe j x ^x þ XNx n f nx ðc Þe j x ^x þ XNc b n f nc ðc Þe j x ^c XNc b n f nc ðc Þe j x ^c X Nx ¼ 2 cos x n f nx ðc Þ^x þ 2j sin x XNc b n f nc ðc Þ^c ð10þ [19] Once the unknown current hs been found, the scttering prmeters cn directly be computed by evluting the rtio between the incident nd the scttered fields on the ports of the device (z = 1 nd z = 2 shown in Figure 2), s described by Levitn et l. [1983]. The computtion of the electromgnetic fields is briefly described in section 2.2. It is importnt to stress tht, lthough the excittion of the cpcitive problem is split into two different plne wves, it is only necessry to solve the lgebric problem for one of them, leding to very efficient formultion. [20] It cn be observed tht the proposed technique implements n electric field integrl eqution (EFIE) for the nlysis of cpcitive discontinuities inside rectngulr wveguides. As it is known, this kind of integrl eqution my fil t certin frequencies for closed structures. For the prcticl circuits treted in this pper this is not problem due to the electriclly smll size of the conducting posts inside the rectngulr wveguides considered. Nevertheless, the presented formultion cn be dpted to combined field integrl eqution (CFIE) for fr out of bnd nlysis. In this cse, the posts could become electriclly lrge enough to trigger defect frequencies when using n EFIE. It is known tht the CFIE is free from these defect frequencies nd yields to correct results in cse tht numericl problems re encountered Prllel Plte Green s Functions [21] In this section, the prllel plte Green s functions used for solving the cpcitive equivlent problem re 4of11

5 Tble 1. Green s Functions Components Needed for Solving the Cpcitive Equivlent Problem xx G A yy G A zz G A summrized. The generl expression of these Green s functions in their spectrl form is given by: G ppw ðz z ; y; y x f n g n m 0 sin(k y y) sin(k y y ) m 0 cos(k y y) cos(k y y ) m 0 sin(k y y) sin(k y y ) G V 1/" 0 sin(k y y) sin(k y y ) Þ ¼ " n b X n¼0 e f n k y y gn k y y jkz ð z z Þ jk z ð11þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k z ¼ k0 2 k2 x k2 y ; k y ¼ n ; k x ¼ ð11bþ b 1 n ¼ 0 where " n =, f 2 n 6¼ 0 n nd g n re trigonometric functions, nd x is constnt depending on the constitutive prmeters of the medium filling the rectngulr wveguide. It is interesting to observe the definition of the longitudinl wvenumber (k z ), which is modified with respect to the inductive cse with the fixed oblique incident ngle (k x = p/) used in this formultion. All the relevnt definitions for the Green s functions needed in order to solve the cpcitive equivlent problem re given in Tble 1. [22] It is worth stressing the low convergent behvior of the series represented in equtions (11) nd (11b) when the observtion point long the propgtion z xis is very close to the source t z. For these situtions, one hs to employ summtion ccelertion techniques like the Kummer s method used by Quesd Pereir et l. [2007] nd Levitn et l. [1983] for n efficient computtion of the Green s functions in the implementtion of the integrl eqution technique Electromgnetic Fields nd Scttering Prmeters Evlution [23] The scttered electric field inside the rectngulr wveguide is computed by using the following mixed potentils expression: ~E s ð~ Þ ¼ j!~a ð~ Þ rf e ð~ Þ ð12þ [24] One cn write the previous eqution in the spectrl domin s follows: ~~E s ð~ Þ ¼ j! G A ð~;~ Þ ~J x ðc Þ^x þ ~J c ðc Þ^c dc c jk x^x þ d " dc ^c G V ð~;~ Þ jk x J ~ x ð c Þþ d ~J # c ð c Þ dc dc ð13þ c j! [25] Due to the dydic nture of eqution (13), we hve different expression for ech electric field component. The min component of the electric field will be long the y xis when only the fundmentl mode is propgting inside the wveguide. This is becuse the x component is null, nd the z component is only significnt in the ner field region surrounding the conducting obstcles. This lst component is generted by the cpcitive discontinuities, nd it will be significnt only close to them. [26] It is importnt to note tht one hs to compute the y nd z derivtives of the electric sclr potentil Green s function G V (~,~ ) for computing the y nd z components of the electric field. Although these Green s function derivtives cn be computed directly from its spectrl form, presented in section 2.1, the performnce of the convergence of the series in equtions (11) nd (11b) is worse thn the originl Green s function, even fter the ppliction of the Kummer s ccelertion technique. For better convergence performnce, one cn switch to direct computtion of the z derivtives in the sptil domin s series of sptil imges [see Quesd Pereir et l., 2007], or to the ppliction of the Ewld ccelertion technique s proposed by Cpolino et l. [2005] nd Quesd Pereir et l. [2006]. [27] The computtion of the scttered mgnetic field cn be crried out by pplying similr procedure to the eqution ~H ð~ Þ ¼ 1 r~a ð~ Þ 0 ð14þ [28] For the cpcitive problem the previous expression cn be written in the spectrl domin s ~~H ð~ Þ ¼ 1 jk ^y G A ð~;~ Þð; k x1 Þ ~J x ðc Þ^x þ ~J c ðc Þ^c dc ð15þ c [29] Tking into ccount the dydic nture of eqution (15), different expression is gin obtined for ech component of the mgnetic field. [30] Finlly, to compute the fields in the sptil domin, the bove expressions re first evluted in the spectrl domin with the electric current corresponding to ech sptil frequency k x =±. After tht, the individul results re trnsformed into the sptil domin nd they re summed. Since the electric current component weights ~J c (c ) nd ~J x (c ) re relted in our problem to the hrmonics k 1 x = p/ nd k 2 x = p/, s shown in equtions (9) nd (9b), similr reltionships to (10) cn be estblished for the different electromgnetic fields components in the sptil domin. 3. Results [31] The first exmple is low pss filter with 12.8 GHz cutoff frequency presented by Levy [1973]. The geometry nd dimensions of the filter cn be seen in Figure 4. The design is composed of two different sections. The inner section corresponds to tpered corrugted wveguide lowpss filter derived from olotrev hlf stub prototype. The terminl prts re impednce trnsformers for mtching the wveguide impednce of the tpered corrugted filter to the input nd output ports of stndrd WR 90 wveguide. [32] The results obtined with the new integrl eqution technique hve been compred gin to those provided by the commercil softwre FEST3D from Auror Softwre 5of11

6 Figure 4. Geometry nd dimensions of the low pss filter presented by Levy [1973]. The filter structure is composed of impednce converters in the input nd output ports nd olotrev prototype for the inner section. The input nd output wveguides re WR 90 ( = mm, b = mm). nd Testing, S.L. [2009], which is bsed on the integrl eqution technique presented by Álvrez Melcón et l. [1996], showing very good greement, s cn be observed in Figure 5. The results re lso in good greement with respect to the mesured results reported by Levy [1973], thus vlidting the softwre tool developed. In this cse, only 15 modes hve been enough for the Green s functions computtion, wheres the number of unknowns for expnding the surfce current density hs been set to The evlution time hs been 0.51 s per frequency point in 64 bits computer with 2.0 GHz clock. [33] Although we hve employed generl purpose bsis functions for expnding the unknown electric current density on the discontinuities (tringulr bsis functions), we hve lso tken cre of the fst electromgnetic field vritions close to conducting shrp corners. In this cse, the electric current is better modeled with finer mesh density round these corners. We hve employed cosine like mesh pttern for stright edges forming corner. This pttern provides fster convergence of the solution thn uniform mesh, since it concentrtes more mesh cells in the proximity of the corners. [34] The new technique hs been employed for the nlysis nd design of different low pss filter implementtion. In this cse, the filter is composed of different cpcitive irises cting s impednce inverters, inside constnt height rectngulr wveguide section with dimensions = mm nd 6.0 mm (see Figure 6). In this design, ll wveguide sections re selected of the sme chrcteristic impednce, nd the impednce inverters re vried to recover the desired trnsfer function s described by Cmeron et l. [2007]. The results obtined by the new numericl technique hve been compred to those provided by the commercil softwre tool FEST3D from Auror Softwre nd Testing, S.L. [2009], showing very good greement s cn be observed in Figure 7. The mximum number of modes used for computing the Green s functions presented in section 2.1 is 10, wheres 380 bsis functions hve been enough for chrcterizing the unknown electric currents, yielding to n ccurte response in the frequency rnge of the nlysis. This structure is prticulrly convenient for the softwre tool, since ll impednce inverters Figure 5. Scttering prmeters of the low pss filter proposed by Levy [1973]. The geometry nd dimensions of the filter re shown in Figure 4. The results computed by the new integrl eqution technique hve been compred to those provided by the commercil softwre tool FEST3D [Auror Softwre nd Testing, S.L., 2009]. 6of11

7 Figure 6. Cpcitive low pss filter composed of rectngulr irises inside rectngulr wveguide ( = mm nd b = 6.0 mm). The dimensions of the filter re: b 1 = 3.4 mm, b 2 = 1.7 mm, b 3 = 1.26 mm, l 1 = 6.55 mm, l 2 = 5.0 mm nd w = 4.0 mm. shre common wveguide section. As consequence, only the cpcitive steps need to be meshed, leding to reduced number of unknowns. With these considertions, the simultion time is less thn 0.2 s per frequency point in the sme computer s used with the previous exmple. [35] To show the usefulness of the procedure presented in section 2.2 for the evlution of the electromgnetic fields inside the cpcitive structure, the totl electric field inside the rectngulr wveguide hs been computed t 15 GHz on the (y, z) plne for x = /2. This frequency is within the pssbnd of the cpcitive filter. As cn be seen in Figure 8, the highest electric field mgnitude is locted inside the shortest gp t the center of the structure. This is, therefore, the criticl gp of most concern in power hndling cpbility pplictions. In Figure 8 we cn lso clerly observe the fringing fields t the corners of the cpcitive windows, where they show fst vritions. Due to these fst vritions we hve verified for this exmple tht improved ccurcy is obtined when cosine pttern mesh is used to define the cpcitive windows. The field plots shown in Figure 8 re computed with this cosine pttern discretiztion. Similr considertions cn lso be pplied to the totl mgnetic field shown in Figure 9. [36] The next exmple is sixth order cpcitive lowpss filter composed of six circulr conducting posts inside constnt height rectngulr wveguide, with the sme dimensions s in the previous design ( = mm nd b = 6.0 mm) (see Figure 10). To the uthors knowledge, this is the first cpcitive low pss filter design using circulr conducting post insted of corrugted rectngulr wveguide sections. Results obtined with the new integrl eqution technique hve been compred gin to those estimted by the commercil full wve simultion tool FEST3D (Figure 11). The greement exhibited by the two results is lso good, despite of following very different numericl pproches. [37] The mximum number of modes used for computing the Green s functions presented in section 2.1 is 10, wheres 360 bsis functions hve been enough for chrcterizing the unknown electric currents, yielding to n ccurte response in the studied frequency rnge. The simultion time is less thn 0.18 s per frequency point in the sme computer. [38] The electromgnetic fields hve lso been computed for this filter t the sme cut plne s done before, nd t 15 GHz, well inside the pssbnd of the filter. As cn be seen in Figure 12 the totl electric field is concentrted on the top nd bottom res of the circulr conducting posts. The nrrower gps support the highest electric field intensities. However, the contour lines tend to bend due to the curvture of the circulr posts. This might be beneficil effect for power hndling cpbilities issues. Agin, similr considertions cn be pplied to the mgnetic field presented in Figure 13 for completeness. 4. Conclusions [39] A new integrl eqution technique hs been presented for the nlysis of rbitrrily shped cpcitive microwve wveguide circuits. For the first time, the scttering prmeters of these kind of devices hve been computed by formulting 2 D scttering problem with oblique incident ngle. The boundry conditions of the originl host wveguide hve been tken into ccount through the use of the Figure 7. Scttering prmeters of the low pss filter shown in Figure 6. Results computed by the new proposed numericl method hve been compred to those dt provided by the commercil softwre pckge FEST3D. 7of11

8 Figure 8. Totl electric field mgnitude inside the rectngulr wveguide t 15 GHz (x = /2). The nodes of the mesh employed for the nlysis hve been lso plotted s white circles. Figure 9. Mgnitude of the mgnetic field x component inside the rectngulr wveguide t 15 GHz (x = /2). The nodes of the mesh employed for the nlysis hve been lso plotted s white circles. 8of11

9 Figure 10. Cpcitive low pss filter composed of six circulr conducting posts inside rectngulr wveguide ( = mm nd b = 6.0 mm). The dimensions of the structure re: d 1 = 3.0 mm, d 2 = 4.2 mm, d 3 = 4.68 mm, l 1 = 10.2 mm, l 2 = 9.5 mm nd l 3 = 9.2 mm. Figure 11. Scttering prmeters of the low pss filter shown in Figure 10. Results computed by the new proposed numericl method hve been compred to those dt provided by the commercil softwre pckge FEST3D. 9of11

10 Figure 12. Totl electric field mgnitude (V/m) inside the rectngulr wveguide t 15 GHz (x = /2). The nodes of the mesh employed for the nlysis hs been lso plotted s white circles (A 10 = " 0 /(2jp) in eqution (1)). Figure 13. Mgnitude of the totl mgnetic field inside the rectngulr wveguide t 15 GHz (x = /2). The nodes of the mesh employed for the nlysis hve been lso plotted s white circles (A 10 = " 0 /(2jp) in eqution (1)). 10 of 11

11 prllel plte Green s functions of infinite line sources. Results re vlidted with the nlysis of severl prcticl cpcitive wveguide devices, such s low pss filters with different geometries. The technique hs lso been employed for the efficient evlution of the electromgnetic fields inside the proposed structures. A new cpcitive low pss filter composed of circulr conducting posts hs been designed using the softwre tool. In ll cses, simultion results compred to dt from commercil softwre tools nd the technicl literture hve shown the vlidity nd ccurcy of the new method. [40] Acknowledgments. This work hs been developed with finncil support from SENECA project reference 08833/PI/08, nd CICYT project reference TEC C03. References Álvrez Melcón, A., G. Connor, nd M. Guglielmi (1996), New simple procedure for the computtion of the multimode dmittnce or impednce mtrix of plnr wveguide junctions, IEEE Trns. Microwve Theory Tech., 44(3), Arcioni, P., M. Bressn, G. Conciuro, nd L. Perregrini (1996), Widebnd modelling of rbitrrily shped E plne componentes by the boundry integrl resonnt mode expnsion method, IEEE Trns. Microwve Theory Tech., 44(11), Aud, H., nd R. F. Hrrington (1984), Inductive posts nd diphrgms of rbitrry shpe nd number in rectngulr wveguide, IEEE Trns. Microwve Theory Tech., 32(6), Auror Softwre nd Testing, S.L. (2009), Full wve electromgnetic simultion tool 3D v6.0, Vlenci, Spin. [Avilble t com.] Blnis, C. A. (1989), Advnced Engineering Electromgnetics, John Wiley, Chichester, U. K. Bunger, R., nd F. Arndt (2000), Moment method nlysis of rbitrrily 3D metllic N port wveguide structures, IEEE Trns. Microwve Theory Tech., 48(4), Cmeron, R. J., R. Mnsour, nd C. M. Kudsi (2007), Microwve Filters for Communiction Systems: Fundmentls, Design nd Applictions, Wiley Interscience, Chichester, U. K. Cpolino, F., D. R. Wilton, nd W. A. Johnson (2005), Efficient computtion of the 2 D Green s function for 1 D periodic structures using the Ewld method, IEEE Trns. Antenns Propg., 53(9), Conciuro, G., P. Arcioni, M. Bressn, nd L. Perregrini (1996), Widebnd modeling of rbitrrily shped H plne wveguide components by the boundry integrl resonnt mode expnsion method, IEEE Trns. Microwve Theory Tech., 44(7), Estebn, H., S. Cogollos, V. Bori, A. A. Sn Bls, nd M. Ferrndo (2002), A new hybrid mode mtching/numericl method for the nlysis of rbitrrily shped inductive obstcles nd discontinuities in rectngulr wveguides, IEEE Trns. Microwve Theory Tech., 50(4), Guglielmi, M., nd G. Gheri (1994), Rigurous multimode equivlent network representtion of cpcitive steps, IEEE Trns. Microwve Theory Tech., 42(4), Hrrington, R. F. (1968), Field Computtion by Moment Methods, McMilln, New York. Kirilenko, A. A., S. L. Senkevich, V. I. Tkchenko, nd B. G. Tysik (1994), Wveguide diplexer nd multiplexer design, IEEE Trns. 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Álvrez Melcón (2007), Efficient integrl eqution formultion for inductive wveguide components with posts touching the wveguide wlls, Rdio Sci., 42, RS6002, doi: /2006rs Peterson, A. F., S. L. Ry, nd R. Mittr (1998), Computtionl Methods for Electromgnetics, chp. 8, pp , IEEE Press, Pisctwy, N. J. Quesd Pereir, F. D., F. J. Pérez Soler, B. Gimeno Mrtínez, V. E. Bori Esbert, J. Pscul Grcí, J. L. Gómez Tornero, nd A. Álvrez Melcón (2006), Efficient nlysis tool of inductive pssive wveguide components nd circuits using novel sptil domin integrl eqution formultion, pper presented t 36th Europen Microwve Conference, Inst. of Electr. nd Electron. Eng., Mnchester, U. K. Quesd Pereir, F. D., V. E. Bori Esbert, J. Pscul Grcí, A. Vidl Pntleoni, A. Álvrez Melcón, J. Gómez Tornero, nd B. Gimeno Mrtínez (2007), Efficient nlysis of rbitrrily shped inductive obstcles in rectngulr wveguides using surfce integrl eqution formultion, IEEE Trns. Microwve Theory Tech., 55(4), Ro, S. M., D. R. Wilton, nd A. W. Glisson (1982), Electromgnetic scttering by surfces of rbitrrily shpe, IEEE Trns. Antenns Propg., 30(5), Rojs, R. G. (1988), Scttering by n inhomogeneous dielectric/ferrite cylinder of rbitrry cross section shpe, oblique incidence cse, IEEE Trns. Antenns Propg., 36(2), Slzr Plm, M., T. K. Srkr, L. E. G. T. Roy, nd A. Djordjevic (1998), Itertive nd Self Adptive Finite Elements in Electromgnetic Modeling, Artech House, Norwood, Mss. Twersky, V. (1962), On scttering of wves by the infinite grting of circulr cylinders, IRE Trns. Antenns Propg., 10(6), Young, L. (1962), Stepped impednce trnsformers nd filter prototypes, IEEE Trns. Microwve Theory Tech., 10(5), A. Álvrez Melcón, F. D. Quesd Pereir, nd P. Ver Cstejón, Informtion nd Signl Theory Deprtment, Technicl University of Crtgen, Cmpus de l Murll del Mr, Curtel de Antigones, E Crtgen, Spin. (fernndo.quesd@upct.es) V. E. Bori Esbert, Communictions Deprtment, Technicl University of Vlenci, Cmino de Ver s/n, E Vlenci, Spin. B. Gimeno, Applied Physics Deprtment, University of Vlenci, C. Dr. Moliner 50, Burjssot, E Vlenci, Spin. 11 of 11