Jordi Selga, Student Member IEEE, Ana Rodríguez, Student Member IEEE, Vicente E. Boria, Senior Member IEEE, and Ferran Martín, Fellow IEEE

Transcription

1 Synthesis o Split Rings bsed Artiiil Trnsmission ines through New Two-Step, Fst onverging, nd Robust Aggressive Spe Mpping ASM Algorithm Jordi Selg, Student Member IEEE, An Rodríguez, Student Member IEEE, Viente E. Bori, Senior Member IEEE, nd Ferrn Mrtín, Fellow IEEE This pper is oused on the synthesis o rtiiil trnsmission lines bsed on omplementry split ring resontors SRRs. The onsidered strutures re mirostrip lines with SRRs ethed in the ground plne nd mirostrip lines loded with both SRRs nd series pitive gps. An ggressive spe mpping ASM optimiztion lgorithm, ble to utomtilly generte the lyout o these rtiiil lines, hs been developed. The tool hs been optimized in order to hieve st onvergene nd to provide urte results. The min relevnt spets o the proposed lgorithm bsed on novel two-step ASM optimiztion pproh re: i the pbility to provide the implementble iruit elements o the equivlent iruit model o the onsidered rtiiil lines, nd ii the bility to onverge in ew un-preedent itertion steps, due to new proedure to generte the initil lyouts whih re very lose to the inl ones. First, the sotwre is tested through the synthesis o severl SRR-bsed mirostrip lines, nd then some prtil pplition exmples o suh rtiiil lines re reported to illustrte the potentil o the proposed synthesis tool. Index Terms Spe mpping, omplementry split ring resontor SRR, rtiiil trnsmission lines, metmteril trnsmission lines, mirostrip. I. INTRODUTION The topi o rtiiil trnsmission lines hs ttrted muh ttention in the lst dede nd opened pth to new type o rtiiil trnsmission lines bsed on metmteril onepts []-[6]. Metmteril-bsed or inspired trnsmission lines onsist o host line loded with retive elements indutors, pitors, resontors, or ombintion o them. Suh lines exhibit ontrollble dispersion nd hrteristi impedne, this being the min relevnt dvntge s ompred to onventionl lines, or to other rtiiil trnsmission lines. In onventionl lines, the line impedne n be tilored to some extent through the geometry; however, the phse onstnt or given substrte is brely dependent on the geometry. In ertin rtiiil trnsmission lines, suh s slow wve This work hs been supported by MIIIN-Spin proets TE00-75 METATRANSFER, TE , ONSOIDER EMET SD , nd grnt AP , Generlitt de tluny proet 009SGR-, nd MITy-Spin proet TSI METASINTESIS. J. Selg nd F. Mrtín re with GEMMA/IMITE, Deprtment d Enginyeri Eletròni, Universitt Autònom de Brelon, 0893 Bellterr, Spin. E-mil: Ferrn.Mrtin@ub.es. A. Rodríguez nd V.E. Bori re with Deprtmento de omuniionesiteam, Universidd Politéni de Vleni, 60 Vleni, Spin. E- mil: vbori@dom.upv.es. trnsmission lines, host line is periodilly loded with pitors, indutors or by mens o semi-lumped retive elements, whih eetively inrese the line pitne or indutne, nd hene redue the phse veloity [7]-[9]. In metmteril trnsmission lines, the presene o retive elements inreses the design lexibility o the lines. Thus, the hrteristi impedne nd the phse onstnt s well s the phse nd group veloities n be engineered in order to ulill line speiitions or untionlities not hieved beore. Size redution [0],[], brodbnd [0],[]-[6] nd multibnd opertion [7]-[], leky wve strutures with steerble rdition bems [],[3], or novel brodbnd ilters nd diplexers with spurious suppression []-[6], inluding tunble ilters [7] re some o the pplitions o metmteril trnsmission lines. In these pplitions, devie design is bsed on both impedne nd dispersion engineering. The erly metmteril trnsmission lines were implemented by loding host line with series pitors nd shunt indutors []-[]. At low requenies, the loding elements re dominnt, nd the struture behves s bkwrd wve trnsmission line. In this regime, the phse nd group veloities re nti-prllel nd the struture supports bkwrd wves, or, in other words, it exhibits lethnded behvior. At higher requenies, the line indutne nd pitne re dominnt, nd the propgtion is orwrd or right-hnded. Thereore, these -loded lines exhibit omposite right/let hnded RH wve propgtion [8],[9]. Suh RH behvior n lso be hieved by mens o metmteril trnsmission lines bsed on eletrilly smll resontors, suh s split ring resontors SRRs [30] or omplementry split ring resontors SRRs [3]. In the ltter se, whih is the relevnt one or the purposes o the present pper, host line, typilly lthough not exlusively mirostrip line, is loded with SRRs ethed in the ground plne beneth the ondutor strip nd series pitive gps on top o the SRRs. These SRRbsed lines do lso exhibit RH behvior. However, the iruit model, depited in Fig., revels the presene o oupling pitne,, between the SRR modeled by nd nd the line, responsible or the presene o trnsmission zero. is the line indutne nd g is relted to the gp pitne. In t, ssuming tht the series gp n be modeled by -network with series pitne s nd shunt ringing pitne, it ollows [3]:

2 g s pr pr s pr s with pr = +, being the line pitne. The requeny response o -loded nd SRR-bsed RH lines exhibits pss bnd in the let-hnded nd the right-hnded regions, nd these two bnds merge i the RH line is blned, tht is, i the series nd shunt resonne requenies re identil. By removing the gp, the pss bnd behvior o SRR-bsed lines swithes to stop bnd, used by the presene o the trnsmission zero the iruit model or these strutures is lso depited in Fig. b. This stop bnd untionlity is o interest or the implementtion o noth ilters nd bnd-stop ilters [33]. Despite the numerous pplitions nd dvntges o SRR-bsed rtiiil lines, wek point towrds the urther penetrtion o these lines into the mrket onerns their synthesis. In this regrd, reent eorts hve been mde by the uthors in order to implement eiient lgorithms or lyout genertion. Speiilly, in [3] nd [35], the potentil o spe mpping optimiztion or the synthesis o SRR-bsed lines ting s stop bnd strutures i.e., without the presene o the gps ws demonstrted. In the present pper, we extend the spe mpping optimiztion to the synthesis o SRR nd gp loded lines. Moreover, s ompred to [3] nd [35], we introdue severl improvements tht provide ster onvergene nd mor robustness to the lgorithm, nd method ble to give the implementble element vlues o the equivlent iruit model. Fig.. Typil topologies nd iruit models unit ell o SRR-loded mirostrip line with nd without b series gp. The ground plne, where the SRR re ethed, is depited in grey. This pper is orgnized s ollows. In setion II, the generl ormultion o ASM is provided or ompleteness nd or better omprehension o the pper. Setion III is oused on the pplition o ASM optimiztion to the synthesis o SRR-bsed lines, where the new two-step ASM lgorithm bsed on novel method to determine the initil geometry is presented. In setion IV, the new two-step ASM-bsed tool is pplied to the synthesis o prtil mirowve iruits, inluding bnd-stop ilter nd dul-bnd power divider. Finlly, the min onlusions re highlighted in setion V. II. GENERA FORMUATION OF AGGRESSIVE SPAE MAPPING ASM Spe mpping is tehnique extensively used in the optimized design proess o mirowve devies, whih mkes proper use o two simultion spes [36]-[38]. In the optimiztion spe, X, the vribles re linked to orse model, whih is simple nd omputtionlly eiient, lthough not urte. On the other hnd, the vribles orresponding to the vlidtion spe, X, re linked to ine model, typilly more omplex nd PU intensive, but signiintly more preise. In eh spe, we n deine vetor ontining the dierent model prmeters. et us denote suh vetors s x nd x or the ine model nd orse model prmeters, respetively. Using the sme nomenlture, R x nd R x will denote the responses o the ine nd orse models, respetively. For mirowve pplitions, the model response is relted to the evlution o the devie behviour, e.g., sttering prmeter, suh s S or S, omputed in ertin requeny rnge. The key ide behind the spe mpping lgorithm is to generte n pproprite prmeter trnsormtion x P x 3 mpping the ine model prmeter spe to the orse model prmeter spe suh tht R x R x in some predeined region, being ertin suitble norm nd smll positive number lose to zero. I P is invertible, then the inverse trnsormtion: x P x 5 is used to ind the ine model solution, whih is the imge o the orse model solution, x, tht gives the trget response, R x. The determintion o P ording to the proedure reported in [36] ollows n itertive proess tht is rther ineiient. However, the eiieny o the method n be improved by introduing qusi-newton type itertion [37]. This method ggressively exploits eh ine model EM nlysis with the result o ster onvergene. Hene, the new pproh ws lled ggressive spe mpping ASM [37], nd it is the optimiztion proedure onsidered in this work. Essentilly, the gol in ASM is to solve the ollowing set o non-liner equtions: x P x x 0 6 For better understnding o the itertive optimiztion proess, supersript is dded to the nottion tht tully indites the itertion number. Hene, let us ssume tht x is the -th pproximtion to the solution o 6 nd the error untion orresponding to x. The next vetor o the itertive proess x + is obtined by qusi-newton itertion ording to x x h 7 where h is given by: h B 8

3 nd B is n pproh to the Broyden mtrix [37]: T h B B 9 T h h whih is lso updted t eh itertive step. In 9, + is obtined by evluting 6 using ertin prmeter extrtion method providing the orse model prmeters rom the ine model prmeters, nd the super-index T stnds or trnspose. The implementtion o the ASM lgorithm is well reported in [37]. I the ine nd orse models involve the sme spe prmeters, then the irst vetor in the ine spe is typilly set equl to the trget vetor in the orse spe, nd the Broyden mtrix is initilized by oring it to be the identity. However, sine this is not the se we re onsidering in this pper, we hve ollowed dierent pproh setion III.. III. SYNTHESIS OF SRR-BASED ARTIFIIA TRANSMISSION INES BY MEANS OF THE NOVE TWO-STEP ASM APPROAH From now on, nd or ler distintion, mirostrip lines loded only with SRRs will be lled SRR-loded metlines, nd those lines loded with both SRRs nd series gps will be designted s SRR-gp-loded metlines. et us irst desribe the two spes involved in the proposed ASM lgorithm or both line types. The optimiztion spe, X, is the iruit model, nd the spe vribles re the iruit elements ppering in the equivlent iruits o Figs. nd b or the SRR-gp- nd SRR-loded lines, respetively. The vlidtion spe, X, is the eletromgneti EM model o the physil struture tht provides the requeny response rom the geometry nd substrte prmeters. The substrte prmeters will be set to ertin vlues nd thereore these prmeters re not onsidered s vribles o the ine model. The hosen substrte is the Rogers RO300 with dieletri onstnt r =0. nd thikness either h=.7mm or h=0.635mm losses re not onsidered in the EM model. Some onstrints will be pplied to the geometry prmeters in order to redue the degrees o reedom nd thus work with the sme number o vribles in both spes, s it ws done in [3] the mtrix B is thus squre nd invertible, nd the omputtionl eort is minimized. The most relevnt dierene between the ASM lgorithm reported in this pper, with regrd to the one reported in [3], is the determintion o the initil lyout. With the new proposed strtegy, the initil estimted lyout i.e. x 0 is lredy very lose to the optimum inl solution x, leding to st nd very sot onvergene rte o the whole synthesis proedure. Moreover, our previous implementtions o ASM lgorithms [3],[35],[39] re not ble to gurntee i there is mpping between the optiml orse model solution i.e., set o iruit prmeters orresponding to the trget response nd ine model point providing physilly implementble lyout. In this pper, we overome this prtil limittion through the determintion o onvergene region in the orse model spe. Hene, we will irst present this novel strtegy or the determintion o the onvergene regions, whih involves pre-optimiztion ASM sheme. For this reson, our proposed synthesis tool is bsed on two-step ASM pproh see Fig.. Begin Determintion o the onvergene Region R Vlues o, Implementble ASM pre-optimiztion [, ] inside R No End Yes Determintion o the initil lyout x 0 Using est-squre pproh ASM Full-optimiztion ORE SRR-loded metline x = [,,, ] x = [W, l,, d] SRR-gp- loded metline x =[,,,, g ] x =[W, l, s,, d] Fig.. Shemti o the new proposed two-step ASM lgorithm see the text or better omprehension. A. Determintion o the onvergene Region: Preoptimiztion ASM Algorithm The implementble lyouts o SRR bsed lines re limited to ertin ombintions o the element vlues o the iruit models. Despite some eorts hving mde by mens o prmetri nlysis [3], there is not yet systemti proedure to determine whether given ombintion o iruit model elements n be trnslted to n implementble geometry or not. This is undmentl problem to solve in order to implement prtil synthesis tool, nd it is the subet o the present subsetion. et us irst onsider the se o SRR-loded metlines. The geometril vribles re the mirostrip line length, l, nd width, W, nd the width,, nd seprtion, d, o the slot rings see Fig. b. In order to del with the sme number o vribles in the orse nd ine model spes, we onsider s split = nd l = r ext, where r ext is the externl rdius o the SRR. This hoie is ustiied to void distributed eets tht my pper i l > r ext nd re not ounted or by the iruit model nd to ensure tht the ports o the struture re essible. Given set o iruit trget vribles,,, nd, proedure to determine i this set o vribles hs physilly implementble lyout is needed. Aording to the iruit model, nd re physilly relizble i these vlues re not too extreme nd mirostrip line with resonble width nd length results. The importnt spet is thus, given pir o implementble vlues o nd, to determine the onvergene region or the iruit vlues modeling the SRR nd. So the strtegy onsists on lulting the line geometry tht provides the element vlues nd under our dierent senrios orresponding to the extreme vlues o nd d, nmely, min -d min, mx -d mx, min -d mx, nd mx -d min. The prmeters min nd d min re the minimum hievble slot nd strip widths, respetively, with the tehnology in use we hve set min = d min = 0.5 mm in this work, unless otherwise speiied. On the other hnd, mx nd d mx re set to resonble mximum vlue tht gurntees the vlidity o the model in the requeny region o interest. It hs been ound tht or vlues exeeding mm, the oupling between the slot rings is very limited nd hene we hve set mx = d mx = mm lrger vlues expnd the onvergene region, but t the expense o less ury in the inl solutions.

4 For eh se, the single geometril vribles re l nd W nd d re ixed nd r ext = l/. These vribles must be optimized with the gol o reovering the nd vlues orresponding to the trget nd. The extrtion o the elements o the iruit model rom the EM simultion o the SRR-loded line lyout whih is strightorwrd tehnique hs been reported beore [35], but it is reprodued in Appendix A or ompleteness prmeter extrtion is undmentl building blok o the proposed ASM lgorithm. For the determintion o l nd W, we do lso onsider n ASM optimiztion sheme, whih is very simple sine only two vribles in eh spe re involved. The initil vlues o l nd W re obtined rom the hrteristi impedne nd phse onstnt resulting rom nd, with the help o trnsmission line lultor. With the resulting line geometry, we extrt the prmeters o the iruit model nd lso initilize the Broyden mtrix the initition o the Broyden mtrix ollows similr pproh to tht orresponding to the ore ASM lgorithm, whih is reported in Setion III.. Then the proess itertes until onvergene is hieved. In this pre-optimiztion ASM proess, the stopping riterion is usully tighter thn the one onsidered or the ull-asm optimiztion, i.e. smller vlues o the error untions re ored. This more restritive riterion is hosen not only to urtely determine the verties o the onvergene region, but lso to obtin better estimte o the initil lyout o the ore ASM lgorithm however this does not require muh omputtionl eort due to the very smll number o terms orse model prmeters involved in the error lultion. One the l nd W vlues orresponding to the trget nd or given nd d ombintion or instne min -d min re ound, the whole geometry is known, nd the element vlues nd n be obtined by mens o the prmeter extrtor. These element vlues nd orrespond to the onsidered SRR geometry min -d min, nd tully deine the irst vertex, P 50d50, o the polygon whih deines the onvergene region, see Fig. 3. Notie tht the nomenlture used or identiying the verties indites subsript the vlues o nd d in mirons. Then, the sme proess is repeted or obtining the next points, i.e, the nd pirs orresponding to mx -d mx, min - d mx, nd mx -d min with the trget nd or these ses the initil vlues o l nd W re set to the solutions o the previous vertex, sine it provides muh better pprohes to the optiml inl solution. As result, we obtin set o points in the orse model sub-spe o nd tht deines polygon see Fig. 3. This polygon is rough estimte or the region o onvergene in the - subspe, or the trget vlues nd. The riterion to deide i the trget element vlues n be physilly implemented is the pertinene or not o the point i.e.,, to the region enlosed by the polygon. It is possible to reine the onvergene region by lulting more points, mening urther vlues/ombintions o nd d or instne those orresponding to ombintion o extreme nd n intermedite vlue. As representtive exmple o SRR-loded line model, we hve hosen =.86 nh nd =.88 pf. The onvergene region or the subspe - lulted ollowing the proedure explined bove is depited in Fig. 3. Obviously, deresing the mximum llowble vlue o mx nd d mx hs the eet o reduing the re o the polygon, s Fig. 3b illustrtes. However, mx = d mx = 0. mm is resonble vlue tht represents trdeo between ury relted to the requeny responses o the ine nd orse models nd size o the onvergene region. b Fig.3. onvergene region or SRR-loded line model with =.86 nh nd =.88 pf, deined by our points in the - subspe, nd onsidering min= d min= 0.5 mm nd mx= d mx= 0. mm. b Vrious regions deined with dierent onstrints or the sme vlues o nd. The thikness o the substrte is h=.7mm. The other se under study, i.e., SRR-gp-loded metlines, presents orse model whih depends on 5 vribles, those o the line without gp plus g Fig.. The dditionl geometril vrible is simply the gp spe, s. The onvergene region n be obtined ollowing similr proedure, but now the trget response is given by the eletril prmeters tht hrterize the mirostrip line with gp, nd g. The pre-optimiztion ASM lgorithm is thus more omplex sine it involves three vribles, being the line length, l, width, W, nd gp spe, s, the unknown dimension prmeters to be lulted. As in the previous se, the initil lyout or the irst vertex is ound nlytilly, s it is detiled in Appendix B. One the geometry o the line with gp is estimted, the eletril prmeters re extrted ording to the proedure given in Appendix A, the Broyden mtrix is initited, nd the proess is iterted until onvergene is hieved. The result is set o iruit model element vlues orresponding to the trget elements o, nd g, nd nd vlues tht depend on the onsidered SRR geometry. One dierentil spet o SRR-gp-loded lines, s ompred to SRR-loded lines, is the number o neessry points in the - subspe to urtely determine the onvergene region. It hs been ound tht our points do not suie in this se. Hene, the pre-optimiztion ASM lgorithm hs been pplied eight times: our o them by onsidering the SRR geometries with extreme vlues o nd d; the other our ses by onsidering the ollowing ombintions: min -d men, mx -d men, men -d mx, nd men -d min, where d men =d min + d mx / nd men = min + mx /. In some osions, where lrge gp pitne is required, it is onvenient to use gp with T-shped geometry Fig.. In this se, in order to preserve the number o vribles in the vlidtion spe, the gp distne, s, is set to smll vlue this helps to inrese the series pitne o the gp, nd the gp width, W g, is onsidered s the geometry vrible ssoited to the presene o the gp. Thus, the gp distne is set to s = 0. mm notie tht l g see Fig. is lso

5 set to l g = 0. mm to void n exess o vribles. However, the pre-optimizer nd ore ASM lgorithms do not experiene ny hnge by onsidering T-shped gps. Fig.. SRR-bsed mirostrip line with T-shped gp. Figure 5 depits the onvergene region in the - subspe tht results by onsidering = 9.5 nh nd = 7.93 pf nd g =.0 pf or SRR-gp-loded line with T-shped gp. In view o this igure, it is ler tht our points re not enough to determine the onvergene region notie tht the - points orresponding to the SRR geometries given by min -d men, mx -d men, men -d mx, nd men -d min re signiintly misligned with the lines orming the polygon resulting by onsidering the our ses with extreme vlues o nd d nh pf P 50d50 P 50d35 P 50d500 P 35d500 P 500d500 P 500d50 P 35d50 P 500d35 T V U Fig. 5. onvergene region or SRR-gp-loded line model with = 9.5 nh, =7.93 pf, nd g =.0 pf, deined by eight points in the - subspe, nd onsidering min= d min= 0.5 mm nd mx= d mx= 0.5 mm. The thikness o the substrte is h= 0.635mm. B. Determintion o the Initil yout et us onsider the synthesis o given set o iruit prmeters o the iruit model trget orse model solution, or SRR-loded metline, nd let us ssume tht the previous nlysis revels tht suh trget prmeters in the orse model spe hve n implementble ine model solution. The next step is the determintion o the initil lyout, unless nd oinide with ny o the verties o the onverging polygon in the - subspe in this se the lyout is lredy known nd hene no urther optimiztion is neessry. From the previous nlysis, it is expeted tht the dimensions o the SRR ter ASM optimiztion depend on the position o the - point in the onvergene region. Nmely, i the - point is lose to vertex, it is expeted tht nd d re similr to the vlues orresponding to tht vertex. In this pper, new proedure to determine the initil lyout, neessry to strt-up the ore ASM lgorithm, is proposed. First, the determintion o the initil lyout or SRR-loded lines is onsidered. The im is to express ny o the geometril vribles, d, l or W s untion o nd. To obtin the initil vlue o eh geometril dimension involved in the optimiztion proess, we will ssume it hs liner dependene with nd. For instne, the initil vlue o or the other vribles we will use identil expressions will be estimted ording to:, D B A 0 The previous expression n be lterntively written s: 3 0, where the onstnts i determine the untionl dependene o the initil vlue o with nd. To determine the onstnts i, our onditions re needed. et us onsider the ollowing error untion:, v N error where the subsript is used to dierentite between the dierent verties, nd hene is the vlue o in the vertex, nd, the orresponding vlues o nd or tht vertex. We hve onsidered number o verties equl to orresponding to SRR-loded lines, but the ormultion n be generlized to higher number o verties. Expression n then be written s: 3 0 error To ind the vlues o the onstnts i, we obtin the prtil derivtives o the previous error untion with regrd to i, nd ore them to be equl to zero [0]:, i i error 0,, i or i =,, 3,. Following this lest-squres pproh, our independent equtions or the onstnts i re obtined. Suh equtions n be written in mtrix orm s ollows: 3 o W d l g l W g s

6 5 One the onstnts i re obtined solving the previous equtions, the initil vlue o is inerred rom. The proess is repeted or d, W nd l, nd the initil geometry neessry or the initition o the ASM lgorithm is thus obtined. For the SRR-gp-loded lines, similr pproh is ollowed, lthough the number o verties is eight, rther thn our. As ompred to the lgorithm reported in [3],[35], the initil geometry derived rom the proedure explined in this sub-setion is very lose to the inl geometry, nd this is undmentl to redue the number o steps towrds ASM onvergene, s it will be shown lter. As ompred to previous works by the uthors, the min relevnt ontributions o the present pper re the determintion o the initil lyout rom the lest squres pproh, nd the preoptimizer ASM lgorithm to determine the onvergene region in the - spe. These re undmentl spets to mke the synthesis pproh prtil nd ompetitive.. The ore ASM lgorithm One the initil geometry is lulted, the response o the ine model is obtined through EM nlysis, nd the iruit prmeters re extrted, rom whih the error untion 6 n be obtined. To initite the Broyden mtrix, we slightly perturb eh geometril vrible rom the vlue orresponding to the initil lyout, nd we obtin the iruit prmeters resulting rom eh geometry vrition. The reltive hnges n be expressed in mtrix orm s ollows: W W W W l l l l d d d d 6 whih orresponds to the initil Broyden mtrix vlues. One the Broyden mtrix is known, the geometry o the ollowing itertion n be derived rom 7, nd the proess is iterted until onvergene is obtined. To void tht the vribles in the ine model spe exeed the limits o the implementble rnge o vlues with the vilble tehnology, geometril onstrints nd shrinking tor re introdued. As it n be seen in Fig. 6, the vetor x is tested, nd i some o the geometril vribles exeed the geometril onstrints, the lgorithm uses the shrinking tor in order to obtin new vetor x inside the limits. The itertive lgorithm is ontrolled by the MATAB R ommeril sotwre [], whih is the ore tool o the implemented ASM lgorithm. The three min building bloks o the MATAB progrm re the initil geometry lultor tht previously determines the onvergene regions in the - subspe s explined beore, the EM solver, nd the prmeter extrtor. For the EM simultion, we hve onsidered two lterntive options: the Agilent Momentum simultor inorported in the Agilent ADS ommeril sotwre [], nd Ansot Designer [3]. The EM solver retes the geometry o the struture, whih is then exported to either Agilent Momentum or Ansot Designer to rry out the EM simultion. The results o this simultion the S- prmeters re then imported bk to MATAB nd proessed by the prmeter extrtion module, whih llows us to extrt the iruit prmeters. These vlues re ompred with the trget prmeters, nd the error untion is obtined ording to: norm From this, the next geometry o the itertive proess is inerred ollowing the updte proedure desribed beore. The low digrm o the omplete ASM lgorithm inluding onstrints to void unwnted solutions is depited in Fig. 6. Begin Initition Trget: x =0; Estimtion x Geometril onstrins x min ; x mx Shrinking tor δ Typ. vlue 0.3, 0<δ< x = Px PE norm Finite Di. Sheme B h = δ h norm < ŋ Yes No =0 h = -B - No Yes x test = x + h No End =+ Updte B x test < x mx x test > x min x = x test Fig.6. Flow digrm o the ASM lgorithm. The stopping riterion pplied in this work is norm < η <<. D. Eets o the new ASM lgorithm on onvergene speed The min novelty o the proposed ASM lgorithm is the lultion o very good initil geometry. First, we hve pplied the new ASM lgorithm to the SRR-loded metlines whose optiml orse prmeters re detiled in Tble I, using Ansot Designer s the EM-solver. In Fig. 3 the lotion o these three dierent ses with respet to the shred onvergene region they hve the sme nd vlues is shown. Notie tht in the se tht the point is out o the onvergene region but lose to it, we let the lgorithm to ontinue nd hnge the limiting onstrins o nd d. The evolution o the error untion with the itertion number is plotted in Fig. 7, nd ompred to tht o our previous ASM lgorithm [3], [35]. onvergene is muh ster with the new ASM pproh sine the initil lyout is loser to the inl solution. It is notieble tht there re points, suh s S, tht do not onverge to inl solution with Yes

7 our previous ASM lgorithm, wheres now onvergene is obtined with the new pproh. The number o itertions tht needs the ASM lgorithm to onverge to the inl lyout dimensions nd its orresponding evluted error re summrized in Tble II, or the three dierent exmples onsidered η hs been set to 0.0. TABE I. OPTIMA OARSE SOUTION x [nh] [pf] [nh] [pf] S G M sine the trget S ws pled out o the onvergene region see Fig. 3. However, we hve let the lgorithm to ontinue towrds onvergene by relxing the limiting vlues o nd d. Thus, the t tht the trget is out o the onvergene region does not neessrily men tht onvergene is not possible, but tht the resulting geometri vlues might be beyond or below the onsidering limits o nd d. For the S point, the lose proximity to the line o the polygon orresponding to =0.mm Fig. 3 explins tht the inl vlue o is 0.mm. The greement between the requeny response obtined by EM simultion o the inl lyout obtined ter single itertion nd the iruit simultion o the trget prmeters is exellent, s it n be seen in Fig. 8. This is resonble sine the vlue o is not r rom the onsidering limiting vlue tht gurntees tht the SRRs re urtely desribed by the models depited in Fig.. Figure 9 depits the omprison o the requeny responses orresponding to the point G, where, gin, very good greement between iruit simultion trget prmeters nd EM simultion o the inl lyout results. TABE II. FINA AYOUT x em Iter. Error se l W d no. norm Anlyti S Fig. 7. Evolution o the error untion o the ASM lgorithm or the points o Tble I. Dshed lines orrespond to the evolution o the error untion by onsidering the initil lyout resulting rom nlytil expresssions [3],[35]; solid lines give the error untion or the initil lyout lulted rom the lest-squres pproh proposed in this pper. M blue, G green nd S pink points. est-squres S Anlyti G est-squres G Anlyti M est-squres M All dimensions in mm b b Fig. 8. Mgnitude nd phse b o the sttering prmeters S nd S t initil solution x em, x em, nd iruitl simultion o the x or point S. The inl lyout obtined or the irst exmple S is slightly out o the lower boundry = 0. mm, s it ws expeted Fig. 9. Mgnitude nd phse b o the sttering prmeters S nd S t initil solution x em, x em, nd iruitl simultion o x or point G.

8 PhseS, S deg S, S db TABE III. OPTIMA OARSE SOUTION x [nh] [pf] g[pf] [nh] [pf] T U V We hve lso obtined the error untion with the itertion number or SRR-gp T-shped loded lines with the trgets indited in Fig. 5 nd given in tble III the trget T orresponds to dul-bnd impedne inverter, whih is the rtiiil line used in one o the exmples disussed in setion IV. The resulting geometril vlues nd number o required itertions re given in Tble IV in this se η hs been set to 0.0, still providing very urte results. Figure 0 depits the requeny response or the initil nd inl lyout, s well s the iruit simultion o the trget prmeters or the T se the onsidered eletromgneti solver or these SRRgp-loded lines is the Agilent Momentum S S iruit Sim. x T Initil lyout x em T Finl lyout x em T Frequeny GHz b S S iruit Sim. x T Initil lyout x em T Finl lyout x em T Frequeny GHz Fig. 0. Mgnitude nd phse b o the sttering prmeters S nd S t initil solution x em, x em, nd iruitl simultion o x or point T. Devitions between the iruit simultion nd the eletromgneti response o the inl lyout re due to the et tht the iruit model is not vlid t high requenies. It should be noted tht the initil lyout inerred rom the pre-optimiztion lgorithm is lredy very lose to the inl solution. However, it n be observed progressive mismth between the eletromgneti response o the inl lyout nd the iruit simultion s requeny inreses due to the inury o the iruit model t high requenies. We do not provide the onvergene urves tht led to the number o itertions nd error untions indited in Tble IV, sine we nnot ompre suh urves with those urves tht result by using the previous version o the ASM lgorthm i.e., tht reported in [39]. The reson is tht our old ASM lgorithm ws not dpted or T-shped gps. Nevertheless, onvergene speed is good ew itertions suie to obtin smll error untion, lthough not so st, s ompred to the strutures without gp Fig. 7. se TABE IV. x em FINA AYOUT l W d W g Iter. no. Error norm T U V A. Stop-bnd ilter IV. APPIATION EXAMPES. All dimensions in mm A prtil pplition o SRR-loded lines without gps is the implementtion o stop-bnd ilters with ontrollble response bndwidth nd reetion level. ompt plnr stop-bnd ilters n be designed by sding severl SRRloded line unit ells. et us illustrte this possibility through the design proedure nd experimentl veriition o three unit ell stop-bnd ilter in mirostrip tehnology. The irst step onsists on the synthesis o the entrl unit ell, whose trnsmission zero requeny, z, is hosen to be the entrl requeny o the stop-bnd, i.e.,.5ghz. Sine the other ells involved in the design will hve similr trget responses, nd very lose trnsmission zero requenies i.e..36 GHz nd.53 GHz, the optiml orse solutions or ll the involved ells were ored to be pled in the sme onvergene region i.e. with the sme nd vlues, s it is seen in Tble V in order to speed up the design proess. By sding the three ells with the iruit elements indited in Tble V, stistory stop-bnd response t the iruit simultion level results. By using the ASM lgorithm desribed in Setion III, nd onsidering the Rogers RO300 substrte with thikness h=.7 mm nd dieletri onstnt r =0., the three lyouts summrized in Tble VI hve been obtined. TABE V. OPTIMA OARSE SOUTIONS OF UNIT ES x [nh] [pf] [nh] [pf]

9 TABE VI. SYNTHESISED AYOUTS FOR THE DIFFERENT UNIT ES ell z x em Iter. Error l W d no. norm.36 GHz GHz TOP.53 GHz All dimensions in mm However, it is expeted tht by diretly sding the three designed ells, oupling eets between the SRRs pper this hs been veriied through EM simultion, not shown. Thereore, it is neessry to insert trnsmission line setions between the synthesized unit ells, s depited in Figs. nd. The length o the trnsmission line setions between dent SRRs hs been set to 3/ the SRR rdius the width being identil to tht o the mirostrip on top o the SRR, indited in Tble VI. With these trnsmission line lengths, oupling eets between dent SRRs re not present, while the iruit response does not substntilly hnge s ompred to the se with diret onnetion between unit ells t the iruit level. BOTTOM Fig.. Photogrph o the brited SRR-bsed stop-bnd ilter. Fig. 3. Frequeny response o the stop-bnd ilter: the trget response i.e., tht orresponding to the equivlent irtuit simultion is depited in greendotted line, the ull-wve EM simultion in blue-dshed line, nd the mesurements in blk-solid lines. Fig.. Equivlent iruit nd lyout o the implementer stop-bnd ilter. The smll blue retngles orrespond to lines o physil length equl to.mm nd width ording to the vlue or eh unit ell see Tble VI. The ompleted mnutured stop-bnd ilter n be seen in Figure, where tpers between the input/output 50 Ω trnsmission lines needed to perorm the mesurements nd the designed struture re inluded. It ws brited by using the iruit bord plotter PKF ProMt S03. The mesured ilter response, displyed in Fig. 3, is ompred with the equivlent iruit nd ull-wve eletromgneti simulted responses. The mesured reetion is better thn 0dB within 35MHz requeny bnd. The greement between iruit nd eletromgneti simultion is very good, s expeted on ount o the smll errors tht pper in Tble VI. The smll disrepnies in mesurement nd the dditionl relexion zero re ttributed to brition relted tolernes. B. Dul bnd SRR-bsed power divider et us now onsider the pplition o the proposed two-step ASM lgorithm to the synthesis o dul-bnd power splitter bsed on dul-bnd impedne inverter implemented by mens o the SRR-gp-loded line. To hieve the dulbnd untionlity, the inverter must provide phse shit o 90 o nd +90 o t the operting requenies, nd, respetively. We re thus exploiting the omposite right/let hnded behvior o the struture, with nd loted in the let hnded nd right hnded regions, respetively see [9]. The inverter impedne is set to t both requenies, in order to gurntee good mthing when the inverter output port is loded with pir o mthed lods in prtie two 50 ess lines to implement the divider. Notie tht these our onditions do not univolly determine the ive iruit elements o the model o the SRR-gp loded line Fig.. However, we n set the trnsmission zero see eqution A. in ppendix A to ertin vlue, nd the ive prmeters thus hve unique solution. Speiilly, the design requenies o the inverter hve been set to = 0.9 GHz nd =.8 GHz nd the trnsmission zero to z = 0.5 GHz.

10 With the previous speiitions, the element vlues n be esily ound rom A., A.3 nd the hrteristi impedne o the symmetri T-network, given by: Z Z [ Z Z ] 8 o s s where Z s nd Z p re the series nd shunt impedne, respetively, o the T-iruit model o Fig.. The resulting element vlues re =.5 nh, =.9 pf, g =.5 pf, = 3.38 nh nd = 5.0 pf. With the element vlues o, nd g, we hve obtined the onvergene region in the - plne ording to the method reported beore, nd the trget vlues o nd do not belong to suh region. This mens tht it is not possible to implement the dul-bnd impedne inverter by merely onsidering the SRR-gp-loded line some element vlues re too extreme. However, it is expeted tht by sding trnsmission line setions t both sides o the SRR-gp loded line, the element vlues o the ell re relxed, nd solution within the onvergene region rises. Thereore, we hve sded two identil trnsmission line setions t both sides o our SRR-gp loded line. The width o these line setions is.7 mm, orresponding to the indited hrteristi impedne in the onsidered substrte the Rogers RO300, with thikness h= mm nd dieletri onstnt r =0.. Notie tht by sding suh lines, the eletril length t the operting requenies is the sum o the eletril lengths o the lines nd the SRR-bsed ell. Thus, the phse ondition tht must stisy the SRR-gp loded line n be expressed s = 90 o nd = +90 o, where nd re the eletril lengths o the SRR-bsed ell t the design requenies nd, nd is the phse introdued by the line t the indited requeny. We hve mde prmetri nlysis onsisting on obtining the element vlues o the SRR-bsed ell or dierent vlues o the length o the sded trnsmission line setions nd hene nd. The results re depited in Fig.. It n be observed tht or smll vlues o, nd re too lrge or being implemented. rge mens smll vlue o W, but this is not omptible with lrge vlue. On the other hnd, the vlues o nd without sded line setions, i.e., =0 o, give extreme vlues o d nd, tht is, lrge vlue o d nd smll vlue o. However, by inresing or the length o the sded lines, the vrition o the elements o the entrl SRR-gp bsed ell goes in the orret diretion or their implementtion. Speiilly, we hve onsidered pir o trnsmission line setions with =5 o, whih mens tht the required eletril lengths or the SRR-bsed ell t the operting requenies re = 0 o nd = +30 o. The element vlues orresponding to these phses re = 9.5 nh nd = 7.9 pf, g =.0 pf, =.85 nh nd =.95 pf, nd these vlues led us to n implementble lyout. The reson is tht this phse shit gives the minimum vlue o see Fig. nd resonble smll vlue o, with not so smll nd no so lrge. Notie tht g does not experiene signiint vritions with. We hve pplied the proposed two-step ASM lgorithm to the previous element vlues, nd we hve synthesized the lyout o the SRR-gp-loded line, onsidering T-shped p geometry or the gp. The geometril prmeters o the synthesized struture re in reerene to Fig. l=. mm, W= 0.87 mm, = 0.3 mm, d= 0.0 mm, W g = 7.3 mm, nd onvergene with n error o 0.0 hs been obtined ter 6 itertions. Fig.. Dependene o the element vlues o the SRR-gp loded line with the phse o eh sded trnsmission line setion t. The omprison o the eletril length nd hrteristi impedne inerred rom EM simultion o the synthesized impedne inverter the SRR-bsed ell plus the sded trnsmission line setions nd the ones inerred rom iruit simultion re shown in Fig. 5. The greement is exellent in the let hnded region, where the model desribes the struture to very good pproximtion, nd progressively degrdes s requeny inreses, s it is well known nd expeted. Nevertheless, the phse shit nd the hrteristi impedne t re resonbly lose to the nominl vlues, nd hene we do expet tht the untionlity o the power divider t is preserved. Fig. 5. omprison between the eletromgneti nd iruit simultion orresponding to the hrteristi impedne nd eletril length o the designed dul-bnd impedne inverter. We hve sded two output 50 ess lines or onnetor soldering, nd the struture hs been brited by mens o photo/msk ething proess Fig. 6. Fig. 7 shows the requeny response o the divider, where it n be ppreited tht optimum mthing ours t nd slightly below, or the resons explined. Nevertheless, the untionlity o the power divider overs both design requenies. The disrepny between the mesured response nd the trget is not due to ilure o the ASM lgorithm,

11 S, S, S 3 db but to the t tht the iruit model o the SRR-gp loded line does not urtely desribe the struture t high requenies, inluding prt o the right hnded bnd. A more urte model inreses the omplexity o the ASM lgorithm. / g g / Z0 =50Ω Port Z =35.35Ω Z =35.35Ω Z 0 =50Ω Port Port 3 Fig. 6. Shemti nd photogrph o the brited dul-bnd power divider TOP S, S 3 S S, S, S 3 iruit Sim. S, S, S 3 EM Sim. S, S, S 3 Mes Frequeny GHz Fig. 7. Frequeny response iruit nd eletromgneti simultion nd mesurement o the designed nd brited dul-bnd power divider. V. ONUSIONS BOTTOM In onlusion, new tool or the utomted synthesis o SRR-bsed trnsmission lines, bsed on ggressive spe mpping ASM, hs been proposed nd developed. As ompred to previous reported tools, the proposed lgorithm is ble to provide the onvergene region in the orse spe; nmely, it is ble to predit whether given set o element vlues o the equivlent iruit model o the unit ell is physilly implementble or not. Moreover, the proposed tool is two-step ASM lgorithm, where the irst nd st ASM optimizer is used not only to obtin the onvergene region, but lso to provide the neessry inputs geometril vribles or the determintion o the initil lyout rom novel developed lest-squres pproh. Thus, the initil lyout is very lose to the trget, nd the onvergene time o the ore seond step ASM lgorithm is drmtilly deresed s ompred to previous proposls in some ses, two itertions hve been enough to hieve onvergene. To illustrte the potentil o the new two-step ASM lgorithm to the synthesis o SRR bsed devies, the tool hs been used or the utomted synthesis o stop bnd ilter nd dul-bnd power divider. In the ormer se, the ilter is hieved by sding three unit ells. In order to void oupling eets, tht the equivlent iruit does not predit, smll trnsmission line setions between dent ells hve been dded to the struture. Good greement between the iruit simultion nd the eletromgneti simultion with the lyout o the SRR-bsed ells obtined with the proposed lgorithm hs been hieved. The mesured results re lso in good greement with the iruit nd eletromgneti simultions, hene vlidting the designing proess in the region o interest. For the dul-bnd power divider, the preoptimizer ASM hs reveled tht the trget element vlues inerred rom the onsidered speiitions re not physilly implementble. However, it hs been ound tht by sding two trnsmission line setions to the input nd output ports, the element vlues o the unit ell re not so extreme, resulting in n implementble SRR-gp loded unit ell tht hs been synthesized by mens o the proposed tool. The power divider hs been brited nd hs been demonstrted to exhibit its untionlity t the design requenies. The onsidered eletromgneti solvers o the new proposed ASM tool hve been the Agilent Momentum nd the Ansot Designer ommeril sotwre pkges i.e., two versions o the two-step ASM lgorithm hve been developed. The tool n be extended to other vilble eletromgneti solvers. The results o this work open the pth to the integrtion o this new ASM lgorithm in suh ommeril tools or the st nd eiient optimiztion o SRR-bsed iruits. APPENDIX A PARAMETER EXTRATION The prmeters o the iruit o Fig. n be extrted rom the EM simultion o the SRR-loded line ording to the ollowing proedure. First, the reletion oeiient, S, is represented in the Smith hrt. At the interept o S with the unit resistne irle, the shunt brnh opens nd hene we n determine the resonne requeny o the SRR o A s well s the vlue o the line indutne rom the simulted series retne t o. To univolly determine the three iruit elements o the shunt brnh we need two dditionl onditions, prt rom expression A. One o them is the trnsmission zero requeny, z, whih n be esily determined rom the representtion o the mgnitude o S with requeny. The trnsmission zero obeys the expression: z A The third required ondition n be derived rom the ollowing expression providing the eletril length o the struture:

12 Zs os A3 Z where Z s nd Z p re the series nd shunt impedne o the T-iruit model. Foring = /, it ollows Z Z A s / p / where / is the ngulr requeny where = /, whih mens tht the phse o the trnsmission oeiient is lso = / nd n be esily omputed. Thus, rom A, A nd A we n determine the three retive element vlues tht ontribute to the shunt impedne. The iruit o Fig. b hs n dditionl prmeter g, thereore, we need n dditionl ondition to ully determine the iruit prmeters. This ondition n be the resonne requeny o the series brnh s A5 whih is given by the interept o S with the unit ondutne irle, where the impedne o the series brnh nulls. APPENDIX B INITIA INE AND GAP GEOMETRY Inversion o expressions nd gives: g s B p g g g pr B g I we ssume tht the ringing pitne is smll s ompred to the line pitne, it ollows tht = pr, nd rom the vlue o nd, we obtin the line impedne ording to: Z 0 B3 We use this impedne vlue to determine the line width, W, ording to well known expressions the line width is inerred by mens o trnsmission line lultor. From the substrte prmeters nd the line width, the gp spe, s, neessry to obtin the vlue o s given by expression B n be inerred ording to well known expressions []. Finlly, the line length, l, is inerred by inverting the phse veloity given by: l vp B e where is the speed o light in vuum nd e is the eetive dieletri onstnt tht n be esily omputed rom trnsmission line lultor. REFERENES [] D.R. Smith, W.J. Pdill, D.. Vier, S.. Nemt-Nsser nd S. Shultz, omposite medium with simultneously negtive permebility nd permittivity, Phys. Rev. ett., vol. 8, pp. 8-87, 000. [] A. K. Iyer nd G. V. Eletherides. 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13 with tunble ngle nd bemwidth, IEEE Trns. Mirowve Theory Tehniques, vol. 5, No., pp , Deember 00. [3] F.P. sres-mirnd,. mho-penlos,. loz, High-gin tive omposite right/let-hnded leky-wve ntenn, IEEE Trnstions on Antenns nd Propgtion, vol. 5, no.8, pp.9-300, August 006. [] J. Bonhe, I. Gil, J. Grí-Grí, F. Mrtín, Novel mirostrip bnd pss ilters bsed on omplementry split ring resontors, IEEE Trnstions on Mirowve Theory nd Tehniques, vol. 5, pp. 65-7, Jnury 006. [5] M. Gil, J. Bonhe, J. Grí-Grí, J. Mrtel nd F. Mrtín, omposite right/let hnded RH metmteril trnsmission lines bsed on omplementry split rings resontors SRRs nd their pplitions to very wide bnd nd ompt ilter design, IEEE Trnstions on Mirowve Theory nd Tehniques, vol. 55, pp , June 007. [6] M. Durán-Sindreu, A. Vélez, G. Sisó, J. Selg, P. Vélez, J. Bonhe, nd F. Mrtín, Reent dvnes in metmteril trnsmission lines bsed on split rings, Proeedings o the IEEE, vol. 99, pp , Otober 0. [7] D Bouyge, D. Mrdivirin, J. Bonhe, A. runtenu, A. Pothier, M. Durán-Sindreu, P. Blondy, F. Mrtín, Split ring resontors SRRs bsed on miro-eletro-mehnil deletble ntilever-type rings: pplition to tunble stopbnd ilters, IEEE Mirowve nd Wireless omponents etters, vol., pp. 3-5, My 0. [8]. loz, nd T. Itoh, Eletromgneti Metmterils: Trnsmission ine Theory nd Mirowve Applitions, John Wiley & Sons, In., USA, 006. [9] R. Mrques, F. Mrtín, nd M. Soroll, Metmterils with Negtive Prmeters: Theory, Design nd Mirowve Applitions, John Wiley & Sons, In., USA, 008. [30] J.B. Pendry, A.J. Holden, D.J. Robbins nd W.J. Stewrt, Mgnetism rom ondutors nd enhned nonliner phenomen, IEEE Trnstions Mirowve Theory Tehn., vol. 7 pp , Nov [3] F. Flone, T. opetegi, J.D. Ben, R. Mrqués, F. Mrtín nd M. Soroll, Eetive negtive- stop-bnd mirostrip lines bsed on omplementry split ring resontors, IEEE Mirowve nd Wireless omponents etters, vol., pp. 80-8, June 00. [3] J. Bonhe, M. Gil, O. Grí-Abd nd F. Mrtín, Prmetri nlysis o mirostrip lines loded with omplementry split ring resontors, Mirow. Opt. Tehn. ett., vol. 50, pp , August 008. [33] J. Grí-Grí, F. Mrtín, F. Flone, J. Bonhe, I. Gil, T. opetegi, M.A.G. so, M. Soroll, R. Mrqués, Spurious pssbnd suppression in mirostrip oupled line bnd pss ilters by mens o split ring resontors, IEEE Mirowve nd Wireless omponents etters, vol., pp. 6-8, September 00. [3] J. Selg, A. Rodríguez, M. Gil, J. rbonell, V.E. Bori, nd F. Mrtín, Synthesis o Plnr Mirowve iruits through Aggressive Spe Mpping using ommerilly vilble sotwre pkges, Int. J. RF Mirow. omputer-aided Engineering, vol. 0 5, pp , September 00. [35] J. Selg, A. Rodríguez, M. Gil, J. rbonell, J. Bonhe V.E. Bori, nd F. Mrtín, Towrds the Automti yout Synthesis in Resonnt-type Metmteril Trnsmission ines, IET Mirowves Antenns nd Propgtion, vol. 8, pp , Aug. 00. [36] J.W. Bndler, R.M. Biernki, S.H. hen, P.A. Grobelny nd R.H. Hemmers, Spe mpping tehnique or eletromgneti optimiztion, IEEE Trnstions on Mirowve Theory nd Tehniques, vol., pp , De. 99. [37] J.W. Bndler, R.M. Biernki, S.H. hen, R.H. Hemmers, nd K. Mdsen, Eletromgneti optimiztion exploiting ggressive spe mpping, IEEE Trnstions on Mirowve Theory nd Tehniques, vol. 3, pp , De [38] J. W. Bndler, Q. S. heng, S. A. Dkroury, A. S. Mohmed, M. H. Bkr, K. Mdsen, nd J. Søndergrd, Spe mpping: the stte o the rt, IEEE Trns. Mirowve Theory Teh., vol. 5, pp , 00. [39] J. Selg, A. Rodriguez, V. E. Bori nd F. Mrtín, Applition o Aggressive Spe Mpping to the Synthesis o omposite Right/et Hnded RH Trnsmission ines Bsed on omplementry Split Ring Resontors, Europen Mirowve on. Dig., pp , Ot 0. [0] D.M. Btes, D.G. Wtts, Nonliner regression nlysis nd its pplitions, John Wiley, New York, 998. [] MATAB version 00, The MthWorks In., 00. [] Agilent Momentum version , Agilent Tehnologies, 008. [3] Ansot Designer version 6..0, Ansys In., 00 [] T.. Edwrds, M.B. Steer, Foundtions o interonnet nd mirostrip design, 3 rd Ed., John Wiley, New York, 000. Jordi Selg S ws born in Brelon, Spin, in 98. He reeived the Teleommunitions Engineering Diplom in 006 nd the Eletronis Engineering degree in 008 rom the Universitt Autònom de Brelon UAB, Brelon, Spin. In 008 he oined IMITE-UAB, reserh entre on Metmterils supported by TENIO tln Government. He obtined the Mster degree in Miro nd Nno Eletronis Engineering in 009 rom UAB, nd he is urrently working towrds his PhD degree in subets relted to metmterils, AD design o mirowve devies nd EM optimiztion methods t the UAB. He ws holder o ntionl reserh ellowship rom the Formión de Proesordo Universitrio Progrm o the Edution nd Siene Ministry Reerene AP An Rodriguez S 0 ws born in ugo, Spin. She reeived the Teleommunitions Engineering degree rom the Universidde de Vigo UV, Spin, in 008. As student, she prtiipted in the Ersmus exhnge progrm, developing the Mster Thesis t the University o Oulu, Finlnd. Sine the end o 008, she hs oined the Institute o Teleommunitions nd Multimedi Applitions iteam, whih is prt o the sientii prk t the Universitt Politèni de Vlèni UPV, Spin. She obtined Mster en Tenologí, Sistems y Redes de omuniiones" in 00 rom UPV, nd urrently is on the wy o pursuing Ph.D. degree. Her min reserh interests inlude AD design o mirowve devies, EM optimiztion methods nd metmterils. Viente E. Bori S 9-A 99-SM 0 ws born in Vleni, Spin, on My 8, 970. He reeived his Ingeniero de Teleomuniión degree with irstlss honors nd the Dotor Ingeniero de Teleomuniión degree rom the Universidd Politéni de Vleni, Vleni, Spin, in 993 nd 997, respetively. In 993 he oined the Deprtmento de omuniiones, Universidd Politéni de Vleni, where he hs been Full Proessor sine 003. In 995 nd 996, he ws holding Spnish Trinee position with the Europen Spe Reserh nd Tehnology entre, Europen Spe Ageny ESTE-ESA, Noordwik, The Netherlnds, where he ws involved in the re o EM nlysis nd design o pssive wveguide devies. He hs uthored or o-uthored 7 hpters in tehnil textbooks, 75 ppers in reereed interntionl tehnil ournls, nd over 50 ppers in interntionl onerene proeedings. His urrent reserh interests re oused on the nlysis nd utomted design o pssive omponents, let-hnded nd periodi strutures, s well s on the simultion nd mesurement o power eets in pssive wveguide systems. Dr. Bori hs been member o the IEEE Mirowve Theory nd Tehniques Soiety IEEE MTT-S nd the IEEE Antenns nd Propgtion Soiety IEEE AP-S sine 99. He is member o the Editoril Bords o the IEEE Trnstions on Mirowve Theory nd Tehniques, IEEE Mirowve nd Wireless omponents etters, Proeeding o the IET Mirowves, Antenns nd Propgtion, IET Eletronis etters nd Rdio Siene. He is lso member o the Tehnil ommittees o the IEEE-MTT Interntionl Mirowve Symposium nd o the Europen Mirowve onerene. Ferrn Mrtín M 0-SM 08-F ws born in Brkldo Vizy, Spin in 965. He reeived the B.S. Degree in Physis rom the Universitt Autònom de Brelon UAB in 988 nd the PhD degree in 99. From 99 up to 006 he ws Assoite Proessor in Eletronis t the Deprtment d Enginyeri Eletròni Universitt Autònom de Brelon, nd rom 007 he is Full Proessor o Eletronis. In reent yers, he hs been involved in dierent reserh tivities inluding modelling nd simultion o eletron devies or high requeny pplitions, millimeter