Broadband Equivalent Circuit Model for a Coplanar Waveguide Line Loaded with Split Ring Resonators

Hindawi Publishing Co po a ion In e na ional Jou nal o An ennas and P opaga ion Volume 2012, A icle ID 613518, 6pages doi:10.1155/2012/613518 Resea ch A icle B oadband Equi alen Ci cui Model o a Coplana Wa eguide Line Loaded wi h Spli Ring Resona o s Vic o Sanz,1Angel Belengue ,1Alejand o L. Bo ja,1 Joaquin Cascon,1Hec o Es eban,2and Vicen e E. Bo ia2 1Depa amen o de Ingenie ´ ıa El´ ec ica, Elec ´ onica, Au om´ a ica y Comunicaciones, Uni e sidad de Cas illa-La Mancha, Escuela Poli ´ ecnica de Cuenca, Campus Uni e si a io, 16071 Cuenca, Spain 2Depa amen o de Comunicaciones, Uni e sidad Poli ´ ecnica de Valencia, 46022 Valencia, Spain Co espondence should be add essed o Vic o Sanz, ic o [email p o ec ed] Recei ed 17 Sep embe 2012; Accep ed 16 Oc obe 2012 Academic Edi o : E ic Lheu e e Copy igh © 2012 Vic o Sanz e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. A new equi alen ci cui o a coplana wa eguide loaded wi h spli ing esona o s is p esen ed. The adi ional ci cui s ha model hese de ices a e only able o cha ac e ize he le -handed p opaga ion band, and hei esponse is e y simila o he eal one wi hin a e y limi ed bandwid h. In con as , his p oposed b oadband equi alen ci cui is able o po ay no only he le - handed p opaga ion band, bu also he igh -handed one ha occu s a highe equencies. Besides, he esponse o his kind o basic cells can be adjus ed wi h he p oposed ci cui model in a bandwid h close o a decade. 1. In oduc ion Many publica ions, ega ding he design o e y compac il e s based on Coplana Wa eguide (CPW) lines loaded wi h Spli Ring Resona o s (SRRs), can be ound in scien i ic li e a u e. Diffe en examples o such il e designs a e [1– 5]. These il e s a e designed based on an equi alen ci cui whe e he alue o he diffe en componen s can be linked in a e y di ec way o he physical dimensions o he ings and he line [5–7]. The e o e, using his ci cui model— ha is ela i ely easy o calcula e—, i is possible o design complex il e s o med by he conca ena ion o a ious basic cells [8,9]. This equi alen ci cui p o ides a e y good s a ing poin , so ha an app op ia ely con igu ed op imiza ion algo i hm can con e ge o an accep able solu ion. Ini ially, he equi alen ci cui o [5] was able o cha ac e ize he uppe igh -handed bands. Howe e , in a la e publica ion he au ho s modi ied hei own equi alen ci cui [6]. The new p oposal p o ided a mo e accu a e cha ac e iza ion o he coupling and p opaga ion physical phenomena o he eal ci cui made wi h CPW and SRRs, bu un o una ely, his modi ied ci cui was no able o model he uppe igh -handed bands. In his pape a modi ica ion o he equi alen ci cui o [5–7] is p oposed. This new equi alen ci cui p ese es he igo ous desc ip ion o he physical mechanisms o he eal ci cui and, a he same ime, is able o simul aneously model he le - and igh -handed ansmission o e a bandwid h close o a decade. The pape is o ganized as ollows. Sec ion 2 discusses he equi alen ci cui de i a ion. Sec ion 3 p esen s a speci ic design and he associa ed esul s. The pe o mance is also compa ed wi h he adi ional equi alen ci cui . Finally, in Sec ion 4, he main conclusions o he s udy a e ou lined. 2. Equi alen Ci cui The basic cell used o he il e s o [1,2,4,8]canbeseenin Figu e 1 and i s equi alen ci cui in Figu e 2. In o de o simpli y he ci cui , he symme y along he CPW longi udinal axis has been applied, so in his equi alen ci cui , only hal cell needs o be po ayed. Since he elec ical size o he cell is qui e small, he CPW line can be modeled using a se ies induc o and a pa allel capaci o . I symme y had no been applied, hese induc ance and capaci ance alues could ha e been com- pu ed simply by mul iplying he induc ance and capaci ance 2 In e na ional Jou nal o An ennas and P opaga ion wl g w w wc gcgc Figu e 1: Basic cell geome y. Black is me al on bo om laye ; g ay is me al on op laye . LlLl 2Lp Ca/2 La/2 La/2 2Z02Z0 kk Cl 4 Cl 4 Figu e 2: Equi alen ci cui . pe uni leng h o he CPW, Lline and Cline, by he line leng h, ll. Howe e , since a symme y simpli ica ion has been conside ed, he induc ance mus be mul iplied by 2. The e o e, he line induc o will become 2Ll=2Llinell. Co espondingly, he capaci ance mus be di ided by he same ac o . So he line capaci o will be now Cl/2=Clinell/2. Fu he mo e, as can be clea ly seen in Figu e 1, he CPW is longi udinally di ided by a di ec connec ion o he cen al line and he g ound planes. This connec ion can be modeled wi h an induc o , Lp. Since only hal a ci cui is conside ed because o he symme y applied, his induc o is, indeed, equal o 2Lp. The cen al induc o equi es ha he line be di ided in o 2 hal es. Each hal can be modeled wi h a se ial induc o , whose alue is Ll=Llinell, and a pa allel capaci o equal o Cl/4=Clinell/4. Finally, i should no be o go en ha he equi alen ci cui mus be exci ed wi h an adap ed po . The cha - ac e is ic po impedance mus be equal o 2Z0,because he comple e coplana line has been designed wi h a cha ac e is ic impedance equal o Z0. As a as he ing is conce ned, i is simply a esona o . Ini ially [5], i was modeled as an isola ed pa allel LC ank. Bu in his model, he induc o is magne ically coupled wi h he CPW and ha coupling exci es he ing, and, since he line is b oken in o wo pa s by Lp, he ing mus be di ided as well. The e o e, he ing induc ance is di ided in o wo se ies-connec ed induc o s, and e e y hal o he ing has o be coupled o he app op ia e hal o he CPW. This ing model is alid o an isola ed ing [10]. E en o he mo e complica ed ci cui models like, o example, he model p esen ed in [11], p esume an isola ed ing as well. Howe e , in he basic cell o Figu e 1, one can see ha hese ings canno be conside ed in isola ion. The line ac s as a g ound plane ha , in his case, is e y close o he ings. This g ound plane p oduces an addi ional capaci i e coupling be ween he ings. This coupling is clea ly diffe en om he di ec capaci i e coupling which is modeled by Ca/2 (see Figu e 2). In o de o ake his new capaci i e coupling in o conside a ion, i is necessa y o sligh ly modi y he ci cui ha models he ing. This modi ica ion can be explained om he physical p ocess esponsible o he ing esonance. Wi hin he SRR, i s , he cu en only lows along one o he ings. As his cu en lows alongside his i s ing i p og essi ely c osses o he second one due o hei mu ual capaci i e coupling. When he cu en eaches he slo in he i s ing, i en i ely lows along he second ing so ha i is no in e up ed. F om his momen onwa ds he p ocess in e s i sel . The cu en lows along he second ing and hen, due o he capaci i e coupling, i is p og essi ely ans e ed o he i s ing. Finally, when he cu en eaches he second ing slo , i lows, again, en i ely along he i s ing, jus as i did a he beginning o he p ocess. A ci cui ha li e ally models his phenomenon can be seen in Figu e 3. (i) The i s ing can simply be eplaced by an induc o . (ii) Then, a se ies capaci o would eplace he capaci i e coupling ha ans e s he cu en o he second ing. (iii) O cou se he second ing is also modeled by an induc o . Since he second ing is he cu en des ina ion, i mus be connec ed o he p e ious capaci o . (i ) Nex , ano he capaci o can model he capaci i e coupling ha b ings he cu en back o he i s ing. ( ) Finally, he ing induc ances a e sepa a ed in o wo se ies induc o s in o de o pe o m an app op ia e coupling o he line di ided by Lp, jus as explained be o e. Ob iously, his ing model is exac ly he same as Figu e 2. Howe e , when he ings a e explici ly sepa a ed, i is possible o inco po a e o he model he capaci i e coupling h ough he line. In o de o do ha , an addi ional capaci o , Cg, connec ing he ings, is added o he ci cui , as can be seen in Figu e 4. 3. Resul s In o de o es he alidi y o he p oposed new equi alen ci cui a speci ic p o o ype cell has been designed. This cell has been implemen ed o e a Roge s 4003C subs a e o 1.524 mm hick and designed o show a le - handed ansmission band o a ound 3 GHz and an inpu In e na ional Jou nal o An ennas and P opaga ion 3 CaCa La/4 La/4 La/4 La/4 Figu e 3: Ring equi alen ci cui . k/√2 k/√2k/√2 k/√2 Ca CaCg La/4 La/4 La/4 La/4 LlLl 2Z02Z0 2LpCl 4 Cl 4 Figu e 4: Final equi alen ci cui . impedance o Z0=50 Ω. The physical dimensions o he ci cui ha e been calcula ed by applying he exp essions ound in [10]. Then, his basic cell has been simula ed by using he comme cial so wa e Anso HFSS and was sligh ly modi ied in o de o une he esponse exac ly a 3 GHz. These modi ied dimensions a e shown in Table 1. Simila ly, ollowing he ecommenda ions o [10], he componen s and he coupling coefficien o he o iginal equi alen ci cui (Figu e 2) ha e been compu ed. These alues can be seen in he i s ow o Table 2. Finally, he alues ha e been modi ied by op imiza ion in o de o make he HFSS esponse and he equi alen ci cui ma ch a ound he passband. The op imized alues a e e y simila o he o iginal ones. This ac clea ly alida es his equi alen ci cui . In Figu e 5(a), he HFSS and he equi alen ci cui esponses a e compa ed and, as men ioned in he in oduc- ion, bo h esponses a e i ually iden ical a ound 3 GHz. Un o una ely, he igh -handed band ha clea ly appea s in he HFSS esponse a ound 7 GHz is comple ely missing in he equi alen ci cui esponse. In Figu e 5(b) he new equi alen ci cui esponse (Figu e 4) and he HFSS-simula ed esponse a e compa ed. The esul s, a his s age, a e clea ly be e han he esul s 2468 F equency (GHz) −30 −20 −10 0 Sca e ing pa ame e s (dB) HFSS—s11 HFSS—s21 Ci cui —s11 Ci cui —s21 (a) 2468 F equency (GHz) −30 −20 −10 0 Sca e ing pa ame e s (dB) HFSS—s11 HFSS—s21 Ci cui —s11 Ci cui —s21 (b) Figu e 5: (a) O iginal equi alen ci cui (Figu e 2) e sus HFSS and (b) equi alen ci cui wi hou conside ing equency dependence e sus HFSS. o he o iginal equi alen ci cui (compa e Figu e 5(b) wi h Figu e 5(a)). In ac , he esponse o his new model and he eal esponse a e almos iden ical unde 6 GHz. Un o una ely, he equi alen ci cui esponse p og essi ely deg ades, as he equency is inc eased. E en so, he igh -handed band, o iginally missing, now clea ly mani es s i sel . The a o e- men ioned p og essi e disc epancy is due o he ac ha he dispe si e beha io o he CPW line has been comple ely igno ed. The e o e, i is necessa y o conside he equency dependence o he CPW pe -uni -leng h induc ance and capaci ance i a mo e accu a e esponse is desi ed. On 4 In e na ional Jou nal o An ennas and P opaga ion Table 1: Basic cell dimensions. wc(mm) gc(mm) (mm) w (mm) g (mm) wl(mm) 10 0.64 3.83 0.4 0.4 1 Table 2: Equi alen ci cui elemen s. Ll(nH) Cl(pF) llLp(nH) La(nH) Ca(pF) k 1.793 0.65 2 0.2 18.45 0.299 0.271 1.793 0.65 2 0.2 18.96 0.299 0.246 −30 −20 −10 0 Sca e ing pa ame e s (dB) 2468 F equency (GHz) HFSS—s11 HFSS—s21 Ci cui —s11 Ci cui —s21 (a) −30 −20 −10 0 Sca e ing pa ame e s (dB) 2468 F equency (GHz) HFSS—s11 HFSS—s21 Measu ed—s11 Measu ed—s21 (b) Figu e 6: (a) Op imized equi alen ci cui (Figu e 4) e sus HFSS, (b) Resul s e sus HFSS. he o he hand, since he coupling ac o depends on he pe -uni -leng h induc ance o he line [10], his makes i equency dependen oo. Figu e 7: Fab ica ed p o o ype. 2 4 6 8 HFSS Measu ed F equency (GHz) 0123 Phase ( ad) Ci cui Figu e 8: Dispe sion diag ams. To achie e a be e cha ac e iza ion o he ac ual beha io o his kind o cells, i has been decided o ake in o accoun he equency dependency o hese pa ame e s. To simpli y he cha ac e iza ion o he abo e-men ioned equency dependency, i has been assumed ha i is basically linea . The e o e, o ully cha ac e ize his equency dependencies In e na ional Jou nal o An ennas and P opaga ion 5 Table 3: F equency dependency o some elemen s o ci cui o Figu e 4. 3 GHz Slope Op imized slope Ll1.793 nH 0.0531 nH/GHz 0.115 nH/GHz Cl0.65 pF −0.003 pF/GHz −0.003 pF/GHz k0.246 0.0039 GHz−10.033 GHz−1 i is sufficien o es ablish he alue o hese pa ame e s o a speci ic equency, in his case 3 GHz, and de e mine he slope. These pa ame e s ha e been es ima ed by using he HFSS po sol e . The esul ing alues a e e lec ed in Table 3 (columns “3 GHz” and “slope”). The capaci o modeling he capaci i e coupling o he ings h ough he CPW, Cg, has been conside ed a design pa ame e . By pe o ming a pa ame ic analysis, an ini ial alue o his capaci o has been compu ed. Speci ically Cg= 0.125 pF has been selec ed as he s a ing alue. Once he ini ial alues ha e been calcula ed, he esponse o he ci cui has been op imized. In his case only he slope o he equency dependences o o he diffe en pa ame e s and Cghas been op imized. The es o alues a e iden ical o hose ob ained a e op imizing he o iginal equi alen ci cui (Table 2). The op imized slopes can be seen in he las column o Table 3 and, inally, Cg=0.14 pF. The esul s ob ained wi h his new equi alen ci cui a e qui e simila o he HFSS esul s (see Figu e 6(a)). This ac alida es he p oposed b oadband equi alen ci cui . Finally, a p o o ype o his cell has been buil . In o de o be able o measu e he cell, a ape and a connec o ha e been added o he inpu and ou pu po s. A pho og aph o his p o o ype ( op and bo om) can be seen in Figu e 7. Nex , in Figu e 6(a) a compa ison be ween he sca e ing pa ame e s gi en by HFSS and he measu ed ones is shown. In he igu e, i can be seen how he expe imen al esul s a e sligh ly displaced upwa ds in equency. This is basically a consequence o he ab ica ion p ocess. The p o o ype has been made using a milling machine. I is impossible o adjus he milling dep h wi h o al accu acy. The e o e, p o o ypes a e always sligh ly o e milled. The main consequence o his o e milling is his upwa d equency displacemen ha can be no iced in he measu emen s. Finally, in Figu e 8 one can see a compa ison o he dispe sion diag ams o he equi alen ci cui , HFSS esul s, and expe imen al da a. The dispe sion diag ams ha e been calcula ed om (1) as explained in [12]: βll=cos−11−s11s22 +s12s21 2s21 .(1) I he equency displacemen o HFSS esul s is no aken in o accoun , one can say ha he dispe sion diag ams eally esemble each o he . I can be no iced ha a le - handed ansmission band appea s a ound 3 GHz and a igh -handed ansmission band a ound 7 GHz. 4. Conclusions The adi ional equi alen ci cui ha models an SRR- loaded CPW has been acco dingly modi ied in o de o ake in o accoun he addi ional capaci i e coupling ha happens h ough he CPW. When his addi ional coupling is conside ed, he ci cui is able o cha ac e ize no only he le -handed p opaga ion h oughou he cell, bu also he igh -handed p opaga ion. This igh -handed band could no be modeled wi h he p e ious equi alen ci cui . I he igh -handed p opaga ion can be modeled by he equi alen ci cui , he design o balanced cells wi h his s uc u e will be no ably easie . Wide bandwid hs will hus be achie ed, and he applica ion ange o hese de ices will be imp o ed. Acknowledgmen s This wo k was suppo ed by he Minis e io de Ciencia e Inno aci´ on, Spanish Go e men , unde Resea ch P ojec s TEC2010-21520-C04-03 and -01, and by he Au onomous Go e nmen o Cas illa-La Mancha unde Resea ch P ojec s PPII10-0047-0220 and PPII10-0027-1277. 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