Transmission Properties of Doubly Periodic LC Ladder Networks

Transcription

1 Tranmon Properte of Doubly Perod adder Networ Ana Fláva G. Greo Joaqum J. Barroo² and Joé O. o³ Aoated Plama aboratory Natonal nttute for Spae eearh NPE 7-00 São Joé do ampo SP Brazl Abtrat By developng a general formulaton derbed by a ytem of ordnary dfferental equaton n the tme doman the preent wor examne flterng and tranmon proee that our n doubly perod tranmon lne. The formulaton developed heren allow to nlude an arbtrary number of eton where even and odd numbered eton are dentfed by two et of reatve element { } and { } repetvely. t hown that lumped-element tranmon lne exhbt trong patal dperon whereby eah lne node how a dtnt frequeny petrum. Numeral reult by onderng ndutor and apator wth typal value of H and F and drvng retangular pule of 5- wh are gven and dued by hghlghtng the patal flterng feature that our on uh lne. Keyword doubly perod tranmon lne; lumpedelement rut; patal dperon. NTODUTON Wth the ontnuou nreae of operatng frequene of modern hgh-peed ntegrated rut aurate tranmon lne model are requred to quantfy the propagaton delay between deve manage tranmon lne refleton and rotal bede redung gnal loe. A mot eletron deve are tme-varyng and nonlnear the analy of thee ytem mut be performed n the tme doman. Analy n the tme doman alo mportant to predt the tranent behavor of power lne exted by external eletromagnet feld produed for ntane by lghtnng troe or by drupton uh a hort rut [][]. n fat when the ytem element are tme varyng and/or nonlnear olvng the overall networ n the frequeny doman erve no purpoe. The prmary ue of tranmon lne to tranmt F power between loaton eparated by dtane omparable to a quarter wavelength. Tranmon lne model nlude a aade of dtrbuted element ontng of ere ndutane and hunt apatane wth lo retane and ondutane and requre a large number of element and nternal node to be aurate. When the wre are muh horter than a quarter wavelength the tme lag between the endng and reevng end wll be a mall porton of yle and the ytem an be analyzed by rut theory. Expreed n term of wavelength any nteronneton of rut element that mall ompared wth the wavelength aoated wth the hghet frequeny of nteret alled lumped rut []-[5]. A long a th retrton on the ze of the rut hold the Krhhoff voltage and urrent law are vald ne th retrton reflet the fat that Krhhoff law are approxmaton of Maxwell equaton. Whle the ontnuou tranmon lne derbed by tme-nvarant partal dfferental equaton of the hyperbol type the lumped rut equaton are ordnary dfferental equaton whh n general are tme varyng and nonlnear. n lght of th the preent wor onerned wth the tmedoman analy of lumped tranmon lne. The onfguraton of the lne ondered here ont of aade ell buldng up a drete eletral networ of an arbtrary number of unt ell. A ytem of ordnary dfferental equaton n the tme doman developed n a general formulaton where eah lumped element n a gven eton agned wth arbtrary value or. The tudy foued on the propagaton haratert of doubly perod tranmon lne where odd-numbered eton ha a par of element dentfed by and and even eton have a par { }. n the agnment of the reatve element two ae are ondered. n the frt the lne ompoed wth element = o that the trant tme the ame along every unt ell n the doubly perod tranmon lne. n the eond ae / = / for whh all the eton have the ame haratert mpedane. arou tuaton are examned by both / varyng the number of eton and the wh of the nput retangular pule whh drve the lne. t hown that the doubly perod lne exhbt mportant properte manly thoe related wth t patal flterng ablty whh onttute the man ontrbuton of the wor a wll be dued n the next eton.. UT EUATONS A. Perod Tranmon ne wthout oad Fg. how a generalzed lne wthout load where the tate varable are the meh urrent flowng n the ndutor

2 and the orrepondng harge tored n the apator n eton [6]-[8]. t mportant from the numeral tandpont that the tate varable { } are taen o that the dfferental equaton are all frt order. oe are alo ondered by nludng the ndutor retane a ere retane for the apator and the retane of the voltage oure. Fg.. Unloaded ladder networ wth N eton. We begn by referrng to an unloaded lne wth only three eton a hown n Fg.. Fg.. rut wth three eton. For eah eton the Krhhoff voltage and urrent law are appled to the mehe and at the node to gve 0 d d d where t t and t are the voltage at the node and t t and t the meh urrent. Sne 0 ombnng and yeld the followng ytem of ordnary dfferental equaton wth and a output varable: d d d. d To generalze a ytem of equaton for an arbtrary number of unt ell eparate oupled equaton are wrtten for three nd of eton: a Startng eton; d d 4 where = and the nput voltage whh an be a nuodal gnal or a pule of arbtrary hape ether retangular trangular or a Gauan modulated pule. b ntermedate eton; d... ; N d 5 Endng eton; d d 6 B. Perod Tranmon ne wth oad Fg.. oaded lne wth N eton. A loaded lne wth an arbtrary number of eton hown n Fg. and analogouly to the proedure prevouly developed we arrve at the followng equaton: a Startng eton; d d 7 b ntermedate eton;

3 d d ;... N Penultmate eton; d d d Endng eton; ; N d load load d ; N e] the frt paband goe up to f /. 0 Hz where f / wth the eond one tartng at ~ f /. Fg. 4 alo demontrate that the wh of the paband ndependent of the haratert mpedane / whh ha however a trong effet on the heght of the reonane pea a t learly apparent n the mpedanemmathed ae n Fg. 4e.. ESUTS n the nvetgaton of frequeny repone and flterng haratert of the perod lne we frt loo at low-lo = = 0.00 Ω; = 0. Ω and 0-eton lne drven by a - ampltude nuodal gnal. Sngly perod onventonal lne agned wth = H and = F whle the doubly perod lne are pefed for two ae: wth { = H = /F} or { = H = F} n the even-numbered eton and wth { = H = F} n the odd eton. The frequeny repone of the onventonal lne llutrated n Fg. 4a and a expeted the lne behave a a low-frequeny pa flter n whh the rtal frequeny determned by f / 0. 0 Hz. The repone urve learly how the drete nature of the 0-eton lne by notng ten reonant pea where the eparaton between adjaent pea dereae progrevely toward the rtal frequeny thu demontratng that dperve effet beome tronger a f approahed. On the far left de of the frequeny band f < 50 Hz the ollaton turned out mooth ndatng that the perod networ behave a a homogenou lne. Analyzng Fg. 4a-4e t een that the frequeny repone of the doubly perod lne greatly dffer from that of the onventonal lne. We verfy the preene of a band gap eparatng fve pea to the left and fve to the rght. n the ae = [Fg. 4b and 4d] there appear a new rtal frequeny f / 6. 0 Hz at the end of the frt paband wth the eond paband extendng up to 40.0 Hz. n the eond ae where / = / [Fg. 4 and Fg. 4. Sgnal petrum at node #0 of lne: a Sngly onventonal perod lne wth = H = µf and load = Ω; b Doubly perod lne wth = µh = 0.5 µf and load = Ω; Doubly perod wth = µh = µf and load = Ω; d Doubly perod wth = µh = 0.5 µf and load = Ω; e Doubly perod wth = µh = µf and load = Ω.

4 The tranmon and refleton haratert of the networ were nvetgated by endng through the lne a 40- -wh pule; the wh of 40 wa eleted o that the harmon ontent of the nput pule ould not be trongly attenuated durng t travel along low-lo = = 0.00 Ω; = 0. Ω and 00-eton lne wth rtal frequene ~0 Hz. We frt refer to a onventonal unloaded lne for whh the trant tme aro eah eton. t een n Fg. 5a that the refleted pule return ba to node # after a tme delay of 400 orrepondng to a round-trp tme of x00. The dtorton on the pule are from lne mmathng whh alo tranlated n the frequeny petrum n Fg. 5b. n the followng loong upon the tranmon of the 40- pule along the doubly perod lne wth = and loaded wth a -Ω retane we verfy n Fg. 8 the ourrene of refleted pule. Now loadng the lne wth a - Ω retane the haratert mpedane of a ngly perod lne wth = H = F refleton at both end tll pert a llutrated n Fg. 9 where the refleted pule have polarty oppote of the orrepondng pule n Fg. 8. But mplementng a lne wth / = / termnated wth the mathng retane of / we ee n Fg. 0 that refleton are eentally uppreed n th example; note alo that the trant tme for the pule to reah node #00 nreae to 8.0 [Fg. 0b]. Fg. 5. Sgnal at node # of a ngly perod lne wth = µh = µf and load = Ω a Tranent repone and b Fourer petrum. Fg. 6 and 7 how tranmtted and refleted pule alulated at node #00 and #00; the pule refleted from the load preerve t hape whle the pule refleted off the endng end emerge wth negatve ampltude lghtly attenuated thrd pule from the left n Fg. 6 and the nverted pule n Fg. 7. n th example wthout retve load and the lne termnaton beng reatve the refleton oeffent ha a unty magntude but wth a phae hft aoated wth the reatve end. n th way the pule ompletely refleted at the termnaton of the lne wth no energy dpaton. Fg. 8. etangular 40-µ pule tranmtted through a doubly perod lne wth = µh = 0.5 µf and load = Ω: a Tranent repone at node #00 and b tranent repone at node #00. Fg. 9. The ame a n Fg. 8 but wth load = Ω. Fg. 0. The ame a n Fg. 8 but wth / = / = µh; = µf. Fg. 6. Tranent repone at node #00 of a ngly perod lne wth = H and = µf. Fg. 7. Tranent repone at node #00 of a ngly perod lne wth = H and = µf. To nvetgate the flterng properte of the doubly perod lne we now onder a 5- retangular pule uh that t Fourer petrum ha multple delobe wth wh of 00 Hz to enompa the rtal frequene of 0 Hz and 60 Hz a dued n Fg. 4. egardng the lne wth = then we ee n Fg. that the petrum of the tranmtted pule exhbt dfferent feature at eah node. For example n Fg. a and b orrepondng to node # and #0 there appear n the frt delobe an olated paband whh progrevely narrowed a the order of the node nreae. Th phenomenon nterpreted a a loal effet at eah node whh an be dentfed a a proe that ontrol the level of flterng. Suh a mehanm alo manfet telf n the lne wth / = / Fg. but n th ae the man lobe harply utoff at 60 Hz.

5 ntal ondton for the dtrbuton of harge of the apator and the urrent through the ndutor. n nvetgatng the flterng properte of the doubly perod lne drven by a retangular pule t wa found that an olated frequeny paband n the Fourer petrum beome progrevely narrowed toward the end of the lne. n omparon wth a ngly perod lne th phenomenon are from the fat that the number of degree of freedom of the doubly perod lne doubled and hene there appear twe a many branhe eparated by a band gap n the frequeny petrum. Fg.. Spatal flterng n a doubly perod lne wth = µh = 0.5 µf and load = Ω: a at node # b at node #0 at node #00 and d at node #00. AKNOWEDGMENT Ana Fláva G. Greo than the oordnaton for the mprovement of Hgher evel Peronnel APES/Brazl for ther fnanal upport. EFEENES Fg.. The ame a n Fg. but wth / = / = µh; = µf.. ONUSON A tranent and frequeny-repone analy of drete tranmon lne wa performed by formulatng a ytem of generalzed rut equaton n the tme doman for the lumped element of an ladder networ. The ytem of dfferental equaton wa numerally olved wth gven [] Tom Dhaene and Danel De Zutter Seleton of umped Element Model for oupled oy Tranmon ne EEE Tran. omput. Ad. D. vol. no. 7 pp July 99. [] Jyh-Mng Jong Bozdar Jano and ja Trpath Equvalent rut Modelng of nteronnet from Tme-Doman Meaurement EEE Tran. ompon. Hybr. vol. 6 no. pp. 9-6 February 99. [] M. emoenet Wave alled Solton: onept and Experment. Berln: Sprnger 999. [4] P.. Magnuon G.. Alexander. K. Trpath and A. Wehaar Tranmon ne and Wave Propagaton. New Yor: Pre 00. [5]. aloz and T. toh Eletromagnet Metamateral: Tranmon ne Theory and Mrowave Applaton. New Yor: Wley 004. [6] G.. Eleftherade A.K. yer and P.. Kremer Planar negatve refratve ndex meda ung perodally - loaded tranmon lne EEE Tran. Mrow. Theory vol. 50 pp De. 00. [7]. u B. Zhao X.. n and T. J.u Expermental obervaton of evaneent-wave amplfaton and propagaton n mrowave regme Appl. Phy. ett. vol 89 Art. D [8] A. F. G. Greo J. J. Barroo and J. O. o Modelng and Analy of adder-networ Tranmon ne wth apatve and ndutve umped Element J. Eletromagnet. Anal. Appl vol. 5 pp.-8 May 0.