IMPROVED DESIGN EQUATIONS FOR ASYMMETRIC COPLANAR STRIP FOLDED DIPOLES ON A DIELECTRIC SLAB

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1 IMPROVED DESIGN EQUATIONS FOR ASYMMETRIC COPLANAR STRIP FOLDED DIPOLES ON A DIELECTRIC SLAB H.J. Visse* * Hols Cene TNO P.O. Bo N Eindhoven, The Nehelands hui.j.visse@no.nl eywods: Coupled ansmission lines, Dielecic maeials, Dipole anennas, Modelling, Pemiiviy. Asac Design equaions fo he inpu impedance of he asymmeic sip folded dipole, developed y Lampe, [,3], depend, amongs ohes, on he chaaceisic impedance of an asymmeic coplana sip (CPS ansmission line. Lampe assumes a homogeneous suounding medium. In pacice, he anenna will e ealised on a dielecic sla. Employing he Lampe equaions fo his inhomogeneous case may lead o elaive eos in he CPS chaaceisic impedance as much as 3% and heefoe o lage eos in he inpu impedance. Impoved design equaions fo he inpu impedance ae discussed fo he inhomogeneous case. These impoved design equaions ely on an accuae calculaion of he chaaceisic impedance of an asymmeic CPS on a dielecic sla and fuhe employ coecion facos applied o he homogeneous case dipole lengh and equivalen adius. Inoducion L in ( + a d ( + a d + =. ( -C C Alhough he esonan folded dipole anenna is known fo is impoved fequency andwidh ove ha of he odinay dipole anenna, [], is main aacion a he momen lies in is ailiy o adjus he inpu impedance ove a wide ange of values. This is especially ue fo he anenna ealised as a plana folded dipole using PCB echnology, see figue, which may e employed e.g. fo RF powe scavenging o RFiD.. Lampe model Design equaions fo he inpu impedance of he asymmeic coplana sip folded dipole anenna wee developed y Lampe, [,3]. These equaions give hee means of conolling he inpu impedance: he impedance of a dipole of equivalen adius, he sep-up impedance aio ha depends on he widhs of he wo ams of he plana folded dipole anenna and he impedance of he CoPlana Sip (CPS ansmission line fomed y hese wo ams. The inpu impedance of he anenna is given y, [,3] Figue : Coplana sip folded dipole anenna. hee d is he impedance of an equivalen dipole, i.e. a cylindical dipole of equivalen adius ρ e, is he impedance of he ansmission line mode and (+a is he sep-up impedance aio. The impedance of he ansmission line mode is he impedance of a shoed CPS of lengh L/, see figue, ' = π j an β L. ( The epession eween he ackes is he chaaceisic impedance of he CPS in a homogeneous medium of elaive pemiiviy. (k is he complee ellipic funcion of he Auhoized licensed use limied o: Eindhoven Univesiy of Technology. Downloaded on Januay, a : fom IEEE Xploe. Resicions apply.

2 fis kind, (k=(k, whee k =-k. β is he wave nume in he medium. The complee ellipic funcion of he fis kind (k is appoimaed y, [] ' ln π ln and k is given y, [,3] whee e= + k + + k π + k + + k [ + e( + ] + + e( k ; k ; k k ' ' ; ; k k (3 k =, ( + ( + ( + ( + ( ( The paamee a in he sep-up impedance aio is given y [( c ( ] ( ln [( c ( ] ( ln (5 ln c+ a=, ln c+ (6 and he dipole equivalen adius ρ e is given y ( a + a a c c + + ρ e =, (7 whee c is defined in figue. As an eample, in figue he eal and imaginay pa of he inpu impedance of an asymmeic coplana sip folded dipole ae shown as funcion of fequency as calculaed wih a full wave mehod (Finie Inegaion, CST Micowave Sudio and as calculaed wih he aove equaions (Tansmission Line mehod. The dimensions of he anenna ae, wih efeence o figue : =3mm, =mm, =mm, L=6.5mm, =. The dipole impedance has een calculaed y applying he empiical doule polyfi equaions fo he ing-middleon second-ode soluion as given in [5]. Figue : Calculaed eal and imaginay pa of he inpu folded dipole anenna in fee space. The ageemen eween he wo simulaion esuls is vey good aound esonance hus demonsaing he usefulness of he Lampe model. In figue 3, we show he esuls as full-wave simulaed fo he same anenna on a dielecic sla of hickness =.6mm and having a elaive pemiiviy =.8. In he same figue he esuls of he Lampe ansmission line (TL model fo he fee space anenna ae shown fee space -fee space Figue 3: Calculaed eal and imaginay pa of he inpu folded dipole anenna on a dielecic sla and he same anenna in fee space. The figue shows ha he inpu impedance of he anenna on a dielecic sla as funcion of fequency is vey diffeen fom ha of he same anenna in fee space. Theefoe i is necessay o adap he TL model fo he dielecic sla ecs. The dielecic sla affecs oh he ansmission line mode and he anenna mode. Asymmeic coplana sip ansmission line Closed-fom equaions fo he chaaceisic impedance of asymmeic coplana sip ansmission lines on a dielecic sla of finie hickness ae no eadily availale. Fo Auhoized licensed use limied o: Eindhoven Univesiy of Technology. Downloaded on Januay, a : fom IEEE Xploe. Resicions apply.

3 symmeic CPS ansmission lines, analyic fomulas may e found in [6] and [7]. In a fis aemp o deive he equied equaions, we could y o modify he equaion fo he chaaceisic impedance of an asymmeic CPS ha is used in equaion (. Unifom medium ' π = (8 Fis, in a vey cude appoimaion, we could susiue fo in equaion (8 he elaive pemiiviy of he dielecic sla. This means ha we assume he coplana sips o e pesen in a unifom medium wih elaive pemiiviy equal o ha of he dielecic sla. The chaaceisic impedance fo diffeen values of dielecic sla pemiiviies, heighs, sip sepaaions and widhs have hus een calculaed. Resuls fo symmeic CPS ansmission lines have een compaed wih full wave simulaion esuls as epoed in [7], see ale. In his ale, is he heigh of he dielecic and is he sepaaion of he idenical sips of widh full wave (Ohm analyic (Ohm Relaive eo (% Tale : Calculaed chaaceisic impedances fo diffeen symmeic CPS ansmission lines and elaive diffeences. The ale eveals ha he elaive diffeence may e as high as 3%. The impac of his CPS chaaceisic impedance appoimaion used on he inpu impedance of an asymmeic coplana sip folded dipole anenna on a dielecic sla is shown in figue fo =3mm, =mm, =mm, L=6.5mm, =.6mm and =.8. I should e noed ha he dielecic has only e aken ino accoun fo he CPS ansmission line mode of he anenna. Howeve, he ec on he anenna mode eveals iself meely as a shif in fequency of he impedance cuve and a change of impedance levels, no a change in shape. Alhough he impedance cuves show a disinc impovemen wih espec o hose shown in figue 3, hee is sill oom fo impovemen, even when aking ino accoun ha he impedance has no ye een coeced fo he anenna mode. Since he anenna will, mos likely, e conneced o a ansceive y a lengh of CPS ansmission line, a need eiss fo calculaing he CPS chaaceisic impedance wih an accuacy highe han 68% Figue : Calculaed eal and imaginay pa of he inpu folded dipole anenna on a dielecic sla and he same anenna in a unifom medium wih elaive pemiiviy equal o ha of he dielecic sla.. Half spaces A moe ealisic appoimaion han assuming he whole space eing filled wih sla dielecic is o assume he dielecic sla o fill up a half space. Then we may eplace in equaion (8 wih an aihmeic aveage of he elaive pemiiviies of wo dielecic half spaces on oh sides of he anenna, [8] + = sla. (9 Chaaceisic impedance calculaions fo symmeic CPS ansmission lines, having adaped his ecive elaive pemiiviy in equaion (8, have een compaed wih full wave simulaion esuls as epoed in [7], see ale full wave (Ohm analyic (Ohm Relaive eo (% Tale : Calculaed chaaceisic impedances fo diffeen symmeic CPS ansmission lines and elaive diffeences. The ale eveals ha he elaive diffeence is now less han 8.5%. The impac of his CPS chaaceisic impedance appoimaion used on he inpu impedance of an asymmeic Auhoized licensed use limied o: Eindhoven Univesiy of Technology. Downloaded on Januay, a : fom IEEE Xploe. Resicions apply.

4 coplana sip folded dipole anenna wih he same dimensions as analysed in secion. is shown in figue 5. Again i should e noed ha only he ansmission line mode of he anenna has een modified fo he dielecic sla Figue 5: Calculaed eal and imaginay pa of he inpu folded dipole anenna on a dielecic sla and he same anenna on a a half space wih elaive pemiiviy equal o ha of he dielecic sla. k =, ( + k sinh = sinh π ( π ( [ + ]. (3 The chaaceisic impedance of an asymmeic CoPlana aveguide (CP, see figue 6, on a half space dielecic sla ( of elaive pemiiviy is given y, [9] Upon a close inspecion of figues and 5 we see ha he impedance cuves in figue 5 fo he ansmission line model ae apa fom a shif in fequency in close ageemen wih he full wave esuls as hose shown in figue. e aim a developing closed fom equaions fo he inpu impedance of asymmeic coplana sip folded dipole anennas on dielecic slas wih a easonale accuacy. Theefoe, having aived a his poin, we should coninue wih he calculaion of he CPS chaaceisic impedance using equaions (8 and (9. Howeve, o e ale o impove upon he accuacy of he model if his will appea o e necessay, we ake he calculaion of he CPS chaaceisic impedance one sep fuhe and again impove upon accuacy..3 Analogy wih asymmeic coplana waveguide The chaaceisic impedance of a symmeic CPS on a dielecic sla of heigh and elaive pemiiviy is given y, [6] whee and ' π =, ( ( k' ( k ' = +, ( Figue 6: Asymmeic coplana waveguide. ' 3π =, ( whee k is given y equaion ( and is given y equaion (9. Fo a CP on a dielecic sla of heigh, he chaaceisic impedance is sill given y equaion (, u is now given y, [9] whee ( k' ( k ' = +, (5 k ( + α A B =, (6 B + αa ( A sinh π =, (7 ( [ ] π B = sinh +, (8 Auhoized licensed use limied o: Eindhoven Univesiy of Technology. Downloaded on Januay, a : fom IEEE Xploe. Resicions apply.

5 and ( [ ] π E = sinh +, (9 B E ( ( BE α = A A B + E A ( Given he analogy eween a coplana waveguide and a coplana sip ansmission line, we may easily ansfom he chaaceisic impedance equaions fo an asymmeic CP on a finie hickness dielecic sla o hose of a CPS. The chaaceisic impedance of an asymmeic coplana sip ansmission line on a finie hickness dielecic is given y equaion (, whee k is calculaed wih equaions ( and (5 and is calculaed wih equaions (5 ill (. The complee ellipic funcion of he fis kind (k is calculaed wih equaion (3. The elaive diffeences wih he full wave esuls fo he impedances hus calculaed fo he symmeic CPS sucues as saed in ales and ae less han %. The impac of he CPS chaaceisic impedance hus calculaed on he inpu impedance of an asymmeic coplana sip folded dipole anenna wih he same dimensions as in secion. is shown in figue 7. Also fo his figue i should e noed ha only he ansmission line mode of he anenna has een modified fo he dielecic sla Figue 7: Calculaed eal and imaginay pa of he inpu folded dipole anenna on a dielecic sla. The esuls ae vey simila o he peviously shown esuls. The enefi of eing ale o calculae he CPS chaaceisic impedance wih a high accuacy lies in he oppouniy o calculae he inpu impedance of he folded dipole also wih a high accuacy. Fo he momen howeve, we will develop an appoimae mehod o calculae his inpu impedance wih a easonale accuacy. To ha pupose, also he appoimae chaaceisic impedance calculaion fo a CPS ased on wo half spaces may e used. e will use he laes discussed equaions howeve. 3 Dipole lengh and adius coecions Alhough i is possile o accuaely accoun fo he sip dipole on he dielecic sla, we will develop an appoimae mehod ased on coecion ems applied o he fee space analysis of he folded dipole anenna. If a highe accuacy in he end-esuls is equied, he accuae analysis of he ansmission line mode of he anenna needs o e used ogehe wih an accuae analysis of he dipole mode. Such a dipole analysis mehod will e oulined in he ne secion. 3. Sip dipole analysis An accuae way o accoun fo he sip dipole anenna eing siuaed on a dielecic sla is o sa wih a hee-em model fo a cylindical dipole anenna ha models a non-pefec conduco y means of a disiued impedance, [,]. By viue of his disiued impedance, i will e possile o model a dielecic o magneic coaing of a cylindical dipole anenna hough susiuing a disiued inducance fo he disiued impedance, []. A sip dipole on a dielecic sla will now me modeled as an equivalen, magneically coaed, cylindical dipole anenna, []. In his analysis, [], he saic capaciance of a coupled sip ansmission line is needed, whee he sip widhs ae equal o he dipole sip widh. This capaciance value is calculaed y he mehod descied in [3]. This analysis mehod, howeve, will no e applied o he polem a hand now. Insead we will aemp o coec he impedance cuves esuling fom accouning fo he dielecic in he ansmission line mode of he anenna y inoducing coecion ems applied o he fee space dipole lengh and equivalen adius. 3. Sip dipole appoimaion e have seen ha accouning fo he dielecic sla in he ansmission line mode of he asymmeic coplana sip folded dipole anenna has lead o impovemen of he impedance vs. fequency cuves. The cuves esemle he ones oained fom full wave analyses apa fom a fequency shif and a change in impedance level. e know ha one of he main ecs of a dielecic on a dipole anenna will e a loweing of he esonance fequency. Theefoe, we could y y lenghening he dipole, in he dipole mode analysis pa of he anenna, o make he esonance fequency coincide wih ha oained y full wave analysis. Fuhe, y inceasing he equivalen adius we could y o make he impedance levels coincide. Thus L' = αl ρ' = χρ e e. ( Fo a lage nume of asymmeic coplana sip folded dipole anennas, having diffeen dimensions, eing posiioned on dielecic slas of diffeen heighs having diffeen elaive pemiiviies, he coecion facos α and χ have een deemined. Auhoized licensed use limied o: Eindhoven Univesiy of Technology. Downloaded on Januay, a : fom IEEE Xploe. Resicions apply.

6 Figue 8 shows a ypical eample of he hus calculaed inpu impedance of a folded dipole anenna ogehe wih full wave analysis esuls Figue 8: Calculaed eal and imaginay pa of he inpu folded dipole anenna on a dielecic sla. =mm, =mm, =.5mm, L=6.5mm, =.6mm, =.8, α=.3, χ=.9. Figue 9 shows a good eample of he hus calculaed inpu impedance Figue 9: Calculaed eal and imaginay pa of he inpu folded dipole anenna on a dielecic sla. =.75mm, =.5mm, =.5mm, L=3mm, =5.6mm, =.8, α=.5, χ=.9. The ageemens eween he hus calculaed inpu impedances and he full wave simulaion esuls end o ge ee fo smalle sip widhs and sepaaions. Fo he esed fequency ange (GHz-6GHz he coecion facos appea o e fequency independen and may e appoimaed y α = 3 { + log( } χ =.9 Conclusions.5 ( Impoved design equaions ae deived fo an asymmeic coplana sip folded dipole anenna on a dielecic sla ased on he analysis of he same anenna in a unifom medium. A fai o good ageemen wih full wave simulaion esuls fo he inpu impedance may e oained y aking he dielecic sla ino accoun in he ansmission line mode of he anenna and y coecing fo lengh and equivalen adius of he dipole anenna mode as calculaed fo fee space. Refeences [] C. A. Balanis, Anenna Theoy, Analysis and Design, nd ediion, John iley & Sons, New Yok, (996. [] R.. Lampe, Design Fomulas fo an Asymmeic Coplana Sip Folded Dipole, IEEE Tans. An. Popagaion, Vol. AP-33, No. 9, pp. 8-3, (985. [3] R.. Lampe, coecions o [], IEEE Tans. An. Popagaion, Vol. AP-3, No., p. 6, (986. []. Hileg, Fom Appoimaions o Eac Relaions fo Chaaceisic Impedances, IEEE Tansacions on Micowave Theoy and echniques, Vol. MTT-7, No. 5, pp , (969. [5] R. S. Ellio, Anenna Theoy and Design, Revised Ediion, John iley & Sons, New Yok, (3. [6] M. Y. Fankel, R. H. Voelke and J. N. Hilfike, Coplana Tansmission Lines on Thin Susaes fo High-Speed Low-Loss Popagaion, IEEE Tansacions on Micowave Theoy and Techniques, Vol., No. 3, pp. 396-, (99. [7] T. Q. Deng, M. S. Leong, P. S. ooi and T. S. Yeo, Synhesis Fomulas fo Coplana Lines in Hyid and Monolihic MICs, Eleconics Lees, Vol. 3, No., pp. 53-5, (996. [8] S. B. Cohn, Slo Line on a Dielecic Susae, IEEE Tansacions on Micowave Theoy and Techniques, Vol. MTT-7, No., pp , (969. [9] V. F. Hanna and D. Theaul, Theoeical and Epeimenal Invesigaion of Asymmeic Coplana aveguides, IEEE Tansacions on Micowave Theoy and Techniques, Vol. MTT-3, No., pp , (98. [] R.. P. ing and T. T. u, The Impefecly Conducing Cylindical Tansmiing Anenna, IEEE Tans. An. Popagaion, Vol. AP-, No. 5, pp. 5-53, (966. [] R.. P. ing, C.. Haison and E. A. Aonson, The Impefecly Conducing Cylindical Tansmiing Anenna, Numeical Resuls, IEEE Tans. An. Popagaion, Vol. AP-, No. 5, pp , (966. [] B. D. Popovic and A. Nesic, Genealisaion of he Concep of Equivalen Radius of Thin Cylindical Anennas, IEE Poceedings, Vol. 3, P. H, No. 3, pp , (98. [3] E. Ymashia and S. Yamazaki, Paallel-Sip Line Emedded in o Pined on a Dielecic Shee, IEEE Tansacions on Micowave Theoy and Techniques, pp , (968. [] J. Mooe and M. A. es, Simplified Analysis of Coaed ie Anennas and Scaees, IEE Poceedings Micowaves, Anennas and Popagaion, Vol., No., pp. -8, (995. Auhoized licensed use limied o: Eindhoven Univesiy of Technology. Downloaded on Januay, a : fom IEEE Xploe. Resicions apply.

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t Lecue 6: Fiis Tansmission Equaion and Rada Range Equaion (Fiis equaion. Maximum ange of a wieless link. Rada coss secion. Rada equaion. Maximum ange of a ada. 1. Fiis ansmission equaion Fiis ansmission