Analysis of Microstrip Coupling Gap to Estimate Polymer Permittivity

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1 Analysis of Microsrip Couplin Gap o Esimae Polymer Permiiviy Chanchal Yadav Deparmen of Physics & Elecronics Rajdhani Collee, Universiy of Delhi Delhi, India Absrac A ap in he microsrip line can be modeled as a π- nework of capaciances. The series ap capaciance depends on he permiiviy of he polymer ha fills he ap. We propose modificaion o he model so ha permiiviy of he polymer es incorporaed in i. Usin his model, we can esimae he permiiviy of he polymer from he ransmission coefficien of wo-por nework. Keywords Couplin ap; π-nework; ransmission coefficien; oranic polymer I. INTRODUCTION Gaps in he conducin srips of microsrip ransmission lines are used in microwave circuis such as capaciors, DC blocks, radiain elemens, and in measuremen sysems. They are very suiableelemens for monolihic and hybrid microwave ineraed circuis. The sudy of ap in microsrip line is useful in he desin of DC blocks, coupled filers and couplin elemen o resonaors ec. In a microsrip circui, he dielecric media above he circui is usually air and he dielecric below he circui is he subsrae maerial. Wih air as he dielecric above he circui, he dielecric consan ε r of he subsrae and he effecive dielecric consan ε eff of he microsrip are relaed by a fillin facor ha weihs he amoun of he field in air and he amoun of field in he dielecric subsrae. As in a microsrip ransmission line, he elecric field is concenraed beween he srip and he round plane and a weak frinin field exiss beyond he dielecric subsrae. One of he echniques o increase he ineracion of he polymer wih he fields would be o inroduce he polymer in he ap of he ransmission line as here are sron frinin fields a he open ends of he ap. In his paper, we analyze sraih ap disconinuiy in a microsrip ransmission line, and he effec of chanin he permiiviy of he maerial which fills he ap, on he scaerin parameers of he wo-por srucure. The polymer chanes he ap capaciance, which is a funcion of he permiiviy of he polymer. So by measurin he erminal scaerin parameers, i is possible o esimae he ap capaciance and hence o esimae he permiiviy of he polymer. One of he rea advanaes of his mehod compared o he oher echniques is ha very small quaniy (as much required o fill he ap) of sample is required o make he measuremen. II. STRAIGHT GAP IN MICROSTRIP TRANSMISSION LINE A. Gap Disconinuiy- Equivalen Circui The ap disconinuiies in microsrip lines have he abrup chane in he dimension of he srip conducor, which ives rise o a chane in he elecric field and maneic field disribuions. The sraih ap disconinuiy in microsrip line is shown in Fi.1, enerally represened as a π-equivalen circui wih hree capaciive elemens [1]. The sandard equivalen circui represenaion of ap disconinuiy in microsrip ransmission line is as shown Fi.2, where denoes he ABCD marix for he ransmission line secion and A is he ABCD marix of π-equivalen circui of he ap disconinuiy. h w ε r μ r L Fi.1. Gap disconinuiy in microsrip ransmission line. TABLE1. PARAMETERS OF SUBSTRATE USED TO ANALYZE GAP DISCONTINUITY Parameer Symbol Value Frequency f 2-3GHz Widh of line w 0.373mm Heih of subsrae h 0.254mm Thickness of copper srip 0.035mm Couplin ap 0.2mm Toal lenh L 35mm Dielecric consan of he subsrae ε r

2 The shun capaciance C 1 is he resul of he disorder in elecric field disribuion a he ede of he srip. The series capaciance C 2 arises from he couplin beween he srip conducors consiuin he ap. C 2 reduces wih he increase in ap spacin and for infinie spacin C 2 approaches zero and C 1 equals he end-capaciance for an open-ended line. The parameers of he subsrae used o analyze he microsrip ap disconinuiy are lised in TABLE1, which saisfy he condiions 2.5 ε r 15 and 0.5 w/h 2 for π- nework sandard equivalen circui formulaion, as iven by Eq. (1) o Eq. (12). C 1 C 2 C 1 The sandard equivalen circui capaciances C 1 and C 2 are expressed in erms of C even and C odd as iven by Eq. (1) o Eq. (4) [2]. A C even = 2 C 1 (1) C odd = 2 C 2 + C 1 (2) C 1 = 0.5 C even (3) C 2 = 0.5 C odd 0.25 C even (4) C sa C sp C sd wherec even and C odd are he equivalen circui parameers for he ap when i is excied symmerically and anisymmerically. The closed form expression for C even and C odd when 2.5 ε r 15 and 0.5 w/h 2 are saisfied, is iven by Eq. (5) and Eq. (6). C odd ε r = w ε r 9.6 C even ε r = w ε r 9.6 Here w w m o exp ko m e exp ke m o = w h lo w 10 h (for 0.1 /w 1.0) k o = lo w 10 h (for 0.1 /w 1.0) m e = (for 0.1 /w 0.3) 0.12 k e = w h (for 0.1 /w 0.3) m e = w h (for 0.3 /w 1) k e = w h (for 0.3 /w 1) (5) (6) (7) (8) (9) (10) (11) (12) C p The sandard equivalen circui capaciances C 1 and C 2 arecalculaed usin bahl_formula.m m-file for subsrae parameers as lised in TABLE1. This formulaion does no really represen he ap because no allowance is made for he disconinuiy filled via polymer in he equivalen lumped parameers. Hence a muli-elemen equivalen nework is proposed as shown in Fi.2 [3], where denoes he ABCD marix for he ransmission line secion and A P is he ABCD marix of π-equivalen circui of he ap disconinuiy filled wih polymer. The improved equivalen circui has ap capaciance correspondin o he disconinuiy and akes he effec of chanin he permiiviy of he maerial ha fills he ap ino accoun. B. Gap Filled Wih Polymer Improved Equivalen Circui The improved equivalen circui of he microsrip line havin ap filled wih polymer is shown in Fi.2. Fi.3 shows he ap disconinuiy in microsrip ransmission line filled wih oranic polymer and Fi.4 shows he correspondin cross secional view. A P Fi.2. Equivalen circui represenaion of ap disconinuiy in microsrip ransmission line sandard equivalen circui and improved equivalen circui. C p 795

3 Gap filled via polymer w C 21 C 11 C 11 L h ε r μ r A 1 Fi.3. Gap disconinuiy in microsrip ransmission line filled wih oranic polymer. C sa1 Polymer C sp1 C sd1 hε r, μ r Fi.4. Cross secional view of ap disconinuiy in microsrip ransmission line filled wih oranic polymer. To calculae he ap capaciances for he disconinuiy filled via polymer, semi-empirical relaions are developed. Sarin wih an approximaion ha he sandard formulaion is valid for air (ε r =1), he ap capaciances (C sa1, C sp1, C sd1, C p1 ) are calculaed for he disconinuiy in microsrip ransmission line havin air as dielecric subsrae, hen he ap capaciances (C sa2, C sp2, C sd2, C p2 ) for he disconinuiy inmicrosrip ransmission line havin dielecric subsrae wihε r =6.15 are formulaed, usin hese he ap capaciances for he disconinuiy filled wih polymer (C sa3, C sp3, C sd3, C p3 ), in microsrip ransmission line havin dielecric subsrae for εr=6.15 are calculaed. Theapcapaciances (C sa3, C sp3, C sd3, C p3 ) are hen used o calculae wo-por scaerin parameers of microsrip line ap filled wih polymer by usin new_equi.m m-file. Fi.5 shows he equivalen circui represenaion of he ap disconinuiy in microsrip ransmission line havin air as dielecricsubsrae. As C sa1, C sp1, C sd1, C p1 are he ap capaciances for he disconinuiy, he sandard equivalen circui series ap capaciance C 21 can be spli ino hree pars as follows. C 21 = C sa1 + C sp1 + C sd1 (13) where C sa1 is capaciance of he frinin fields in air, C sp1 is parallel plae capaciance of he ap as filled wih air, C sd1 is capaciance of he frinin fields inside he dielecric subsrae maerial which is air in his case. C p1 Fi.5. Represenaion of ap disconinuiy in microsrip ransmission line havin air as dielecric subsrae sandard equivalen circui and improved equivalen circui. As C sa1 depends on he physical parameers and C sd1 is a funcion of he dielecric properies of he maerial beween he srip and he round plane of he microsrip line (air as dielecric subsrae in his case). C sd1 = C sa1 (14) C sp1 = ε o w (15) C sa1 = 0.5 C 21 C sp1 (16) C p1 = C 11 (17) where C p1 is capaciance of he frinin fields beween he srip ede and he round plane hrouh air. Frinin capaciance is he capaciance of a line s edes i.e. he increased capaciance beyond he ideal parallel plae capaciance due o ede fields ha do no reach from one ede o he oher. Fi.6 shows he equivalen circui represenaion of he ap disconinuiy in microsrip ransmission line havin dielecric subsrae correspondin o ε r =6.15. The sandard equivalen A P1 C p

4 C 22 iven by Eq. (23) and plae capaciance C sp3 is iven by Eq. (24). C 23 C 12 C 12 C 13 C 13 A 2 A 3 C sa2 C sp2 C sa3 C sd2 C sp3 C sd3 C p2 A P2 Fi.6. Represenaion of ap disconinuiy in microsrip ransmission line havin dielecric subsrae (εr=6.15) sandard equivalen circui and improved equivalen circui. C p2 C p3 A P3 C p3 circui series ap capaciance C 22 can be spli ino hree ap capaciances as C sa2, C sp2 and C sd2 as iven by Eq. (18). Fi.7. Represenaion of ap disconinuiy filled wih polymer in microsrip ransmission line havin dielecric subsrae (εr=6.15) sandard equivalen circui and improved equivalen circui. C 22 = C sa2 + C sp2 + C sd2 (18) C sa2 = C sa1 (19) C sp2 = C sp1 = ε o w (20) C sd2 = C 22 C sp2 C sa2 (21) C p2 = C 12 (22) where C p2 is he frinin field capaciance for a dielecric subsrae. Fi.7 shows he equivalen circui represenaion of he ap disconinuiy in microsrip ransmission line havin dielecric subsrae correspondin o ε r =6.15, filled wih oranic polymer. C sa3, C sp3, C sd3 and C p3 are he improved equivalen circui ap capaciances when disconinuiy in microsrip ransmission line havin dielecric subsrae, is filled wih oranic polymer. Now he sandard equivalen circui series ap capaciance C 23 is C 23 = C sa3 + C sp3 + C sd3 (23) C sp3 = κ 0.85 ε o w (24) where κ is he permiiviy of he polymer fillin he microsrip line ap, ε 0 is he free space dielecric consan, is hickness of he conducin srip a he ap, is he widh of he ap and w is he widh of he conducin srip. C sa3 = C sa2 (25) C sd3 = C sd2 (26) C p3 = C p2 (27) The series capaciance in he improved equivalen circui depends on he permiiviy of he polymer ha fills he ap, so has an effec on he scaerin parameers of he wo-por 797

5 srucure. Now we can compue he ABCD marices correspondin o he ransmission line ( ) and π-equivalen circui of he ap disconinuiy filled wih polymer (A P3 ). The marix produc ( A P3 ) ives he overall ABCD marix of he microsrip ransmission line srucure havin ap disconinuiy. From his, we can compue he wo-por scaerin parameers of he srucure. Usin EM simulaion ool (IE3D) we can compued he scaerin parameers. III. RESULT AND DISCUSSION A comparison of S 12 compued usin he equivalen circui and EM simulaion ool is shown in Fi.8. Fi.8 shows he variaion of S 12 wih κ for microsrip ap disconinuiy when ap is filled wih polymer, where κ symbolizes he permiiviy of he polymer fillin he ap. The plos wih markers on lines correspond o simulaed S 12 for differen values of κ, while plos wihou markers correspond o S 12 response, obained usin new_equi.m m-file. The difference beween he S 12 obained usin EM simulaion and S 12 compued usin proposed equivalen circui can be accouned due o he approximaion made in he sandard formulaion for air (ε r =1), o find he ap capaciances (C sa1, C sp1, C sd1, C p1 ) for he disconinuiy in microsrip ransmission line havin air as dielecric subsrae. The wo-por scaerin parameer response is sensiive o he chane in he permiiviy of he polymer (κ) fillin he disconinuiy in microsrip ransmission line and can be used o esimae he permiiviy of he polymer. For example, if he measured S 12 versus frequency is provided, we can esimae permiiviy by minimizin he difference beween he measured and calculaed scaerin parameers ( in press Error! Reference source no found.). In his opimizaion process, he only unknown is he permiiviy of he polymer. The main advanae of usin he equivalen circui raher han an EM simulaion ool o compue he scaerin marix is he speed. IV. CONCLUSION In his paper we presen a modified equivalen circui for a ap disconinuiy in a microsrip line. The modificaion akes ino accoun he effec of dielecric fillin he ap. We demonsrae he accuracy of he equivalen circui by comparin he ransmission coefficien compued usin EM simulaion and S 12 compued usin proposed equivalen circui. Usin he equivalen circui o compue he scaerin parameers raher han EM simulaion will considerably reduce he ime required o esimae he permiiviy usin an opimizaion procedure. The proposed equivalen circui is able o predic he ransmission coefficien reasonably well, houh here is scope for improvemen.i is raher difficul o fill he ap compleely wih polymer due o surface ension, here could be voids in he fillin and hence affecs he accuracy of esimaion of permiiviy. Fi.8. Maniude of ransmission coefficien of ap disconinuiy filled wih polymer showin he effec of κ. REFERENCES [1] R. Gar and I. J. Bahl, Microsrip Disconinuiies, Inernaional Journal of Elecronics, vol. 45, no. 1, pp , [2] B. C. Wadell, Transmission Line Desin Handbook, Arech House, Inc, [3] K. Ozmehme, New frequency dependen equivalen circui for ap disconinuiies of microsriplines, IEEE Proceedins, vol. 134, p. H., no. 3, pp , June [4] C. Yadav, Transmission-reflecion mehod o esimae permiiviy of polymer, Inernaional Journal of Emerin Technoloy and Advanced Enineerin, in press