Microwave Engineering

Transcription

1 Micrwav Enginring hng-hsing Hsu Dpartmnt f Elctrical Enginring Natinal Unitd Univrsity

2 Outlin. Transmissin Lin Thry. Transmissin Lins and Wavguids Gnral lutins fr TEM, TE, and TM wavs ; Paralll Plat wavguid ; Rctangular Wavguid ; axial Lin ; triplin ; Micrstrip 3. Micrwav Ntwrk Analysis Impdanc and Equivalnt Vltags and urrnts ; Impdanc and Admittanc Matrics ; Th cattring Matrix ; ABD Matrix ; ignal Flw Graphs ; Discntinutis and Mdl Analysis 4. Impdanc Matching and Tuning Matching with Lumpd Elmnts ; ingl-tub Tuning ; Dubl-tub Tuning ; Th Quartr-Wav Transfrmr ; Th Thry f mall Rflctins 5. Micrwav Rsnatrs ris and Paralll Rsnant ircuits ; Transmissin Lin Rsnatrs ; Dilctric Rsnatrs 6. Pwr Dividrs and Dirctinal uplrs Basic Prprtis f Dividrs and uplrs ; Th T-Junctin Pwr Dividr ; Th Wilkinsn Pwr Dividr ; upld Lin Dirctinal uplrs ; 8 hybrid 7. Micrwav Filtrs Pridic tructur ; Filtr Dsign by th Insrtin Lss Mthd ; Filtr Transfrmatins ; Filtr Implmntatin ; Elctrnic Matrials and Dvics Applicatins Lab

3 6. Pwr Dividrs and Dirctinal uplrs Basic Prprtis f Dividrs and uplrs ; Th T-Junctin Pwr Dividr ; Th Wilkinsn Pwr Dividr ; upld Lin Dirctinal uplrs ; 8 hybrid Elctrnic Matrials and Dvics Applicatins Lab

4 Basic Prprtis f Dividrs and uplrs Fr th scattring matrix thry, sm basic prprtis f thr and fur prt ntwrks hav bn discuss and driv. islatin, cupling, and dirctivity, which ar usful quantitis fr th charactrizatin f cuplrs and hybrids. Elctrnic Matrials and Dvics Applicatins Lab

5 Thr-prt Ntwrk (T-junctins) Th simplst typ f pwr dividr is a T-junctin, which is thr-prt ntwrk with tw inputs and n utput. -> th scattring matrix f an arbitrary thr ntwrk has nin indpndnt lmnts If th cmpnnt is passiv and cntains n anistrpic matrials, thn it must b rciprcal and its [] matrix must b symmtric ( ij = ji ). A junctin that is lsslss and matchd at all prts t avid th pwr lss. Hwvr, thr-prt ntwrk cannt b lsslss, rciprcal, and matchd at all prts. If th ntwrk is als lsslss, thn nrgy cnsrvatin rquirs that th scattring matrix b unitary => 3 Matchd, Rciprcal 3 3 frm last thr quatin at last tw f th thr paramtrs (,, 3 ) must b zr, but this cnditin will always b incnsistnt with n f first thr quatin. => if n f ths thr cnditin is rlaxd, a physical ralizabl dvic is pssibl Pwr divisin and cmbining. (a) Pwr divisin. (b) Pwr cmbining Elctrnic Matrials and Dvics Applicatins Lab

6 If th thr-prt ntwrk is nnrciprcal, thn ij ji, and th cnditins f input matching at all prts and nrgy cnsrvatin can b satisfid. => a circulatr Any matchd lsslss thr-prt ntwrk fr a circulatr must b nnrciprcal. => Th [] matrix a matchd thr-prt ntwrk has th fllwing frm: Thn if th ntwrk is lsslss, [] must b unitary => Ths quatins can b satisfid in n f tw ways, Eithr : Or : This rsult shws that ij ji fr ij,which implis that th dvic must b nnrciprcal. 3 3 => Th [] matrics fr th tw slutin ar shwn in figur, tgthr with th symbls fr th tw pssibl typs f circulatrs. (prt t, r prt t 3, r prt 3 t frm figur (a) dirctin) , 3, Th tw typs f circulatrs and thir [] matrics. (Th phas rfrncs fr th prts ar arbitrary.) (a) lckwis circulatin. (b) untrclckwis circulatin. Elctrnic Matrials and Dvics Applicatins Lab

7 Altrnativly, a lsslss and rciprcal thr-prt ntwrk can b physically ralizd if nly tw f its prts ar matchd. If prts and ar ths matchd prts T b lsslss, th fllwing unitarity cnditins must b satisfid: th (frnt nd) thr quatins shw = 3 => = 3 = => = 33 =. Th signal flw graph sn th ntwrk actually cnsists f tw sparat cmpnnts, n a matchd tw-prt lin and th thr a ttally mismatchd n-prt. Finally, if th thr-prt ntwrk is allwd t b lssy, it can b rciprcal and matchd at all prts => rsistiv dividr In additin, a lssy thr-prt can b mad t hav islatin btwn its utput prts A rciprcal, lsslss thr-prt ntwrk matchd at prts and. Elctrnic Matrials and Dvics Applicatins Lab

8 Fur-prt Ntwrks (Dirctinal uplrs) Th [] matrix f a rciprcal fur-prt ntwrk matchd at all prts has th fllwing frm: If 4 multiply first quatin by 3 4 multiply first quatin by 3 34 and and () () If th ntwrk is lsslss, quatins rsult frm th unitarity, r nrgy cnsrvatin, cnditin. Lt us cnsidr th multiplicatin f rw and rw, and th multiplicatin f rw 4 and rw 3 4 and scnd quatin by imilarly, th multiplicatin f rw and rw 3, and th multiplicatin f rw 4 and rw, and scnd quatin by () and () t b satisfid dirctinal cuplr , and subtract t btain, and subtract t btain Elctrnic Matrials and Dvics Applicatins Lab

9 Thn th slf - prducts f th rws f which imply that chs j β, and Dt prduct f rws and 3 givs th unitary Furthr simplificatin can b mad by chsing th phas rfrncs n thr f fur prts α,, and that 4 matrix yild th fllwing quatin whrαand βar ral, and ar phas cnstants t b dtrmind j β , θ nπ If w ignr intgr multipls f, thr ar tw particular chics that cmmnly ccur in practic: ( I ) ymmtrical uplr / ( II ) Antisymmtrical cuplr, α jβ α β α jβ α β jβ α β α jβ α β α Ntd : tw cuplrs diffr nly in th chic f rfrnc plans. Als, th amplitudαand βar nt indnpndnt α β thus, apart frm t phas rfrncs, an idal dirctinal cuplr has nly n dgr f frdm. Elctrnic Matrials and Dvics Applicatins Lab

10 lg db D lg db I lg I D db 4 4 db Th idal cuplr wuld hav infinit dirctivity and islatin ( 4 =). Bth and culd b dtrmind frm th cuplr factr,. cupling factr thrugh prt with th cfficint n pwr is dlivrd t prt 4 ( islatin β β prt) Elctrnic Matrials and Dvics Applicatins Lab

11 Hybrid cuplrs ar spcial cass f dirctinal cuplrs, whr th cupling factr is 3 db, => = = /() / Thr ar tw typs f hybrids Quadratur hybrid (9 hybrid) => 9 phas shift btwn prts and 3 (= = / ) whn fd at prt, and is an xampl f a symmtrical cuplr. Magic-T hybrid r rat-rac hybrid (8 hybrid) => 8 phas diffrnc btwn prts and 3 whn fd at prt 4, and is an xampl f an antisymmtrical cuplr. Elctrnic Matrials and Dvics Applicatins Lab

12 Dirctivity Masurmnt Elctrnic Matrials and Dvics Applicatins Lab

13 Th T - Junctin Pwr Dividr Th T-junctin pwr dividr is a simpl thr-prt ntwrk that can b usd fr pwr divisin r pwr cmbining, and can b implmntd in virtually any typ f transmissin lin mdium => cmmnly usd in wavguid, micrstrip r striplin structur. (lss r lsslss junctins) (a) E plan wavguid T. (b) H plan wavguid T. (c) Micrstrip T-junctin Elctrnic Matrials and Dvics Applicatins Lab

14 Lsslss Dividr Th lsslss T-junctin can all b mdld as a junctin f thr transmissin lins. => thr ar fringing filds and highr rdr mds assciatd with th discntinuity at such a junctin, lading t strd nrgy that can b accuntd fr by a lumpd suscptanc, B. => dividr t b matchd t th input lin f th charactristic impdanc f. If th transmissin lins ar assumd t b lsslss (r f lw lss), thn th charactristic impdancs ar ral. => assum B = => / +/ = / If B is nt ngligibl, sm typ f ractiv tunning lmnt can usually b addd t th dividr t cancl this suscptanc, at last vr a narrw frquncy rang. Thr will b n islatin btwn th tw utput prts, and thr will b a mismatch lking int th utput prts. Elctrnic Matrials and Dvics Applicatins Lab

15 Elctrnic Matrials and Dvics Applicatins Lab Ex: A lsslss T-junctin pwr dividr has a surc impdanc f 5. Find th utput charactristic impdancs s that th input pwr is dividd in a : rati. mput th rflctin cfficints sn lking int th utput prts. in in in P V P P V P V P 3 3

16 Rsistiv (lssy) Dividr If a thr-prt dividr cntains lssy cmpnnts it can b mad t b matchd at all prts, althugh th tw utputs may nt b islatd. => using lumpd-lmnt rsistrs t fit th circuit fr th dividr. => An qual-split (-3dB) dividr is frm, but unqual pwr divisin ratis ar als pssibl. Assuming that all prts ar trminatd in th charactristic impdanc, th impdanc, sn lking int th /3 rsistr fllwd by th utput lin, is => = /3 + = 4 /3 th input impdanc f th dividr is in = /3 + /3 = shws that th input is matchd t th fd lin => th ntwrk is symmtric frm all thr prts, th utput prts ar als matchd. => = = 33 =. If th vltag at prt is V, thn by vltag divisin th vltag V at th cntr f th junctin is V V 3 / 3 V 3 3 V V 3 V => = 3 = 3 =/, which is -6 db blw th input pwr lvl => is rciprcal, scattring matrix is symmtric / 3 3 V 4 V Half f th supplid pwr is dissipatd in th rsistrs. Elctrnic Matrials and Dvics Applicatins Lab

17 Th Wilkinsn Pwr Dividr Du t Lsslss T-junctin dividr is nt bing matchd at all prts and dsn t hav any islatin btwn utput prts, and rsistiv dividr can b matchd at all prts (but vn thugh it is nt lsslss) and still dsn t hav islatin. Hwvr, a lssy thr-prt ntwrk can b mad having all prts matchd with islatin btwn th utput prts. Th Wilkinsn pwr dividr is such a ntwrk, with th prprty f bing lsslss whn th utput prts ar matchd; that is, nly rflctd pwr is dissipatd. (can b mad with arbitrary pwr divisin, but cnsidr th qual-split 3dB cas.) Analyz this circuit by rducing it t tw simplr circuits drivn by symmtric and antisymmtric surcs at th utput prts. =>vn-dd md analysis tchniqu. Th Wilkinsn pwr dividr. (a) An qual-split Wilkinsn pwr dividr in micrstrip frm. (b) Equivalnt transmissin lin circuit. Elctrnic Matrials and Dvics Applicatins Lab

18 Evn-dd Md Analysis Nrmaliz all impdancs t th charactristic impdanc, and rdraw th circuit with vltag gnratin at th utput prts as shwn in blw figur. (A frm that is symmtric acrss th midplan; th tw surc rsistrs f nrmalizd valu cmbin in paralll t giv a rsistr f nrmalizd valu. => rprsnting th impdanc f a matchd surc.) quartr-wav lins hav a nrmalizd valu f ; shunt rsistr has a nrmalizd valu f r => th qual-split pwr dividr, ths valus shuld b = () / and r =. w dfin tw sparat mds f xcitatin fr th circuit f fllwing figur. Evn-md : V g = V g3 = V ; Odd-md : V g = -V g3 = V Using suprpsitin f ths tw-md => ffctivly hav an xcitatin f V g = 4V, V g3 =, frm which w can find th -paramtrs f ntwrk. Th Wilkinsn pwr dividr circuit in nrmalizd and symmtric frm. Elctrnic Matrials and Dvics Applicatins Lab

19 Evn-md Evn-md xcitatin, V g = V g3 = V, and s V = V 3 and thr is n currnt flw thrugh th r/ rsistrs r th shrt circuit btwn th inputs f tw transmissin lins at prt. Frm th pn circuit t frm vn-md analysis, th impdanc lking int prt => in = / sinc th transmissin lin lks lik a quartr-wav transfrmr. => If = () /, prt will b matchd fr vn md xcitatin => V = V sinc in =. If w lt x = at prt and x = -/4 at prt, V(x)=V+( -jx + jx ) V = V(-/4) = jv + (-)=V = V = V() = V + (+) = jv (+)/(-) Th rflctin cfficint is sn at prt = [-() / ] / [+() / ], V = -jv () / = -j() / ymmtry f prt and 3=> V 3 =V ; 33 = ; = -j() / Bisctin f th circuit. (a) Evn-md xcitatin. (b) Odd-md xcitatin. Elctrnic Matrials and Dvics Applicatins Lab

20 Odd-md Odd-md xcitatin, V g = -V g3 = V, and s V = V 3 and thr is a vltag null alng th middl f th circuit. Thus this circuit by grunding it at tw pints n its mid-plan. Lking int prt, an impdanc f r/, sinc th paralll-cnnctd transmissin lin is /4 lng and shrtd at prt, and lks lik an pn circuit at prt. Thus, prt will b matchd fr dd md xcitatin if w slct r =. => V = V and V = xcitatin all pwr is dlivrd t th r/ rsistrs, with nn ging t prt. Finally, w must find th input impdanc at prt f th Winkinsn dividr whn prts and 3 ar trminatd in matchd lads. Th rsulting circuit is shwn in fllwing figur, it is sn that similar t an vn md f xcitatin, sinc V =V 3. => n currnt flws thrugh th rsistr f nrmalizd valu, s it can b rmvd. W nw hav th paralll cnnctin f tw quartrwav transfrmrs trminatd in lads f unity (nrmalizd) => Th input impdanc is thn in = [() / ] / = Analysis f th Wilkinsn dividr t find. (a) Th trminatd Wilkinsn dividr. (b) Bisctin f th circuit in (a). Elctrnic Matrials and Dvics Applicatins Lab

21 ummary, w can stablish th fllwing 3 ( 3 3 in V V -j at prt ) V V j ; (symmtry f (symmtry du t rciprcity) prts and 3) (du t shrt r pn at bictin) 33 paramtrs fr th Wilkinsn dividr : (prts and 3 matchd fr vn and dd mds) Whn thn dividrs is drivn at prt and th utputs ar matchd, n pwr is dissipatd in th rsistr. => th dividr is lsslss whn th utputs ar matchd; nly rflctd pwr frm th prts r 3 is dissipatd in th rsistr. sinc 3 = 3 = -> prts and 3 ar islatd. Phtgraph f a fur-way crprat pwr dividr ntwrk using thr micrstrip Wilkinsn pwr dividrs. Elctrnic Matrials and Dvics Applicatins Lab

22 Ex. Dsign an qual-split Wilkinsn pwr dividr fr a 5 systm impdanc at frquncy f, and plt th rturn lss ( ), insrtin lss ( = 3 ), and islatin ( 3 = 3 ) vrsus frquncy frm.5f t.5f. A charactristic impdanc f quartr-wav transmissin lins in dividr => = () / = 7.7, and th shunt rsistr a valu f R = = Frquncy rspns f an qual-split Wilkinsn pwr dividr. Prt is th input prt; prts and 3 ar th utput prts Elctrnic Matrials and Dvics Applicatins Lab

23 Th quadratur (9 ) Hybrid Quadratur hybrid ar 3 db dirctinal cuplrs with 9 phas diffrnc in th utputs f th thugh and cupld arm. (ftn mad in micrstrip r striplin) -> branch-lin hybrid analyz th pratin f th quadratur hybrid using an vn-dd md dcmpsitin tchniqu similar t that usd fr th Wilkinsn pwr dividr. Th branch-lin cuplr with all prt matchd, th pwr ntring prt is vnly dividd btwn prts and 3, with a 9 phas shift btwn ths utputs. Nt: => N pwr is cupld t prt 4. => th [] matrix. has a high dgr symmtry, as any prt can b usd as th input prt. utput prts always b n ppsit sid f th junctin frm th input prt 3. islatd prt rmains n th sam sid as th input prt. 4. this symmtry is rflctd in th scattring matrix as ach rw can b btaind as a transpsitin f th first rw. Gmtry f a branch-lin cuplr. Elctrnic Matrials and Dvics Applicatins Lab

24 Evn-dd Md Analysis Th schmatic circuit f th branch-lin cuplr in nrmalizd frm as th fllwing figur. => assum that a wav f unit amplitud A = is incidnt at prt. Bcaus f th symmtry r antisymmtry f th xcitatin, th fur prt ntwrk can b dcmpsd int a st f tw dcupld tw-prt ntwrk. => sinc th amplituds f th incidnt wavs fr ths tw-prts ar ±/, th amplituds f th mrging wav at ach prt f th branch lin hybrid can b xprssd as : B = / + / ; B = T / + T / ; B 3 = T / T / ; B 4 = / - / ircuit f th branch-lin hybrid cuplr in a nrmalizd frm. Elctrnic Matrials and Dvics Applicatins Lab

25 Firstly, cnsidr th calculatin f and T, fr th vn-md tw-prt circuit. dn by multiplying th ABD matrics f ach cascad cmpnnt in th circuit. => shunt /4TL shunt Y = j Y = j Frm ABD paramtrs can b cnvrt t -paramtrs ( =) imilarly, with th dd-md. => B = (prt is matchd) B = -j / () / (haft-pwr, -9 phas shift frm t ) B 3 = - / () / (haft-pwr, -8 phas shift frm t 3) B 4 = (n pwr t pwr 4) Using multi-sctin BL hybrid => incras BW. Dcmpsitin f th branch-lin cuplr int vn- and dd-md xcitatins. (a) Evn md (). (b) Odd md (). Elctrnic Matrials and Dvics Applicatins Lab

26 upld Lin Dirctinal uplrs Whn tw unshildd transmissin lins ar cls tgthr, pwr can b cupld btwn th lins du t th intrsctin f th lctrmagntic filds f ach thr. nsist f thr cnductrs in cls prximity In additin, cupld transmissin lins ar usually assumd t prat in TEM md. rigrusly valid fr striplin structurs and apprximatly valid fr micrstrip structurs. ( a thr-wir lin can supprt tw distinct prpagating mds -> can b usd t implmnt dirctinal cuplrs, hybrids, and filtrs.) Varius cupld transmissin lin gmtris. (a) upld striplin (planar, r dg-cupld). (b) upld striplin (stackd, r bradsid-cupld). (c) upld micrstrip. Elctrnic Matrials and Dvics Applicatins Lab

27 upld Lin Thry If w assum TEM prpagatin, thn th lctrical charactristics f th cupld lins can b cmpltly dtrmind frm th ffctiv capacitancs btwn th lins and th vlcity f prpagatin n th lin. If th strip cnductrs ar idntical in siz and lcatin rlativ t th grund cnductr, thn =. A thr-wir cupld transmissin lin and its quivalnt capacitanc ntwrk. nsidr tw spcial typs f xcitatins fr th cupld lin: vn-md, whr th currnts in th strip cnductrs ar qual in amplitud and in th sam dirctin. dd-md, whr th currnts in th strip cnductrs ar qual in amplitud but in ppsit dirctin. Evn- and dd-md xcitatins fr a cupld lin, and th rsulting quivalnt capacitanc ntwrks.(a) Evn-md xcitatin. (b) Odd-md xcitatin. Elctrnic Matrials and Dvics Applicatins Lab

28 Evn - md : lctric fild has vn symmtry abut th cntr lin, Lad t th quivalnt circuit shwn, whr Odd md : and n currnt flws btwn th tw strip cnductrs. th rsulting capacitanc f ithr lin t grund fr th vn md is assuming that th tw strip cnductrs ar idntical in siz and lcatin charactristic impdanc fr vn - md is lctric fild lins hav an dd symmtry abut th cntr lin, imagin this as a grund plan thrugh th middl f ffctiv capacitanc btwn ithr strip cnductr and grund is th charactristic impdanc fr th dd - md is is ffctivly pn - circuitd. L v p L and a vltag null xists btwn th tw strip cnductrs. v p Elctrnic Matrials and Dvics Applicatins Lab

29 Fr symmtry cupld striplin (supprt purly TEM md), th dsign graph can b usd t dtrmin th ncssary strip widths and spacing fr a givn st f charactristic impdanc,,, and dilctric cnstant. Fr micrstrip, th rsults d nt scal with dilctric cnstant, s dsign graphs must b mad fr spcific valus f dilctric cnstant. Anthr difficulty with micrstrip cupld lins is th fact that th phas vlcity is usually diffrnt fr th tw mds f prpagatin => sinc th tw mds prat with diffrnt fild cnfiguratins in th vicinity f th air-dilctric intrfac=> dgrading ffct n cuplr dirctivity. Nrmalizd vn- and dd-md charactristic impdanc dsign data fr dg-cupld striplins. Evn- and dd-md charactristic impdanc dsign data fr cupld micrstrip lins n a substrat with r =. Elctrnic Matrials and Dvics Applicatins Lab

30 Dsign f upld Lin uplrs Apply an vn-dd md analysis t a lngth f cupld lin t arriv a th dsign quatins. This fur-prt ntwrk is trminatd in th impdanc at thr f its prts, and drivn with a vltag gnratr f V and intrnal impdanc at prt. => cuplr can b dsignd with arbitrary cupling such that th input (prt )is matchd, whil prt 4 is islatd. Prt is th thrugh prt, and prt 3 is th cupld prt. => a grund cnductr is undrstd t b cmmn t bth strip cnductrs. A singl-sctin cupld lin cuplr. (a) Gmtry and prt dsignatins. (b) Th schmatic circuit. Elctrnic Matrials and Dvics Applicatins Lab

31 Th input impdanc at prt f th cuplr can thus b xprssd as => in = V /I = (V +V )/(I +I ) Prt is matchd. Nw V =V => th vltag at prt 3 is V 3 = V 3 + V 3 = V V = V [( in )/( in + ) ( in )/( in + )] in j tanθ whr j tanθ in in in j j tanθ tan θ => in = Dcmpsitin f th cupld lin cuplr circuit int vnand dd-md xcitatin. (a) Evn md. (b) Odd md. Elctrnic Matrials and Dvics Applicatins Lab

32 Elctrnic Matrials and Dvics Applicatins Lab If th charactristic impdanc,, and th vltag cupling cfficint,, ar spcific, thn th fllwing dsign quatins fr th rquird vn- and dd-md charactristic impdancs can b asily drivd. =>

33 Lang cuplr Gnrally th cupling in a cupld lin cuplr is t ls t achiv cupling factrs f 3 db r 6 db. Incrasing th cupling btwn dg-cupld lins is t us svral lins paralll t ach thr, s that th fringing filds at bth dgs f a lin cntribut t th cupling. => Lang cuplr, fur cupld lins ar usd with intrcnnctins t prvid tight cupling, and can asily achiv 3 db cupling ratis, with an ctav r mr bandwidth. => Thr is a 9 phas diffrnc btwn th utput lins (prts and 3) => Lang cuplr is a typ f quadratur hybrid. Th main disadvantag f th Lang cuplr is difficult t fabricat th ncssary bnding wirs acrss th lins, and th lins ar vry narrw This typ f cupld lin is als rfrrd t as intrdigitatd; can b usd fr filtr dsign. Th Lang cuplr. (a) Layut in micrstrip frm. (b) Th unfldd Lang cuplr. Elctrnic Matrials and Dvics Applicatins Lab

34 Th unfldd Lang cuplr prats ssntially th sam as th riginal Lang cuplr, but is asir t mdl with th quivalnt circuit. => cnsist f a fur-wir cupld lin structur => all th lins hav th sam width and spacing. If th rasnabl assumptin that ach lin cupls nly t its narst nighbr, and ignr mr distant cuplings => ffctivly hav a tw-wir cupld lin circuit. If w can driv th vn- and dd- md charactristic impdancs, 4, 4, f th fur-wir circuit in trms f, and, th vn- and dd-md charactristic impdancs f any adjacnt pair f lins. Th capacitanc f th fur lins t grund ar diffrnt dpnding n whthr th lin is n th utsid ( and 4), r n th insid ( and 3) => in = x ( x m ) / ( x + m ) Equivalnt circuits fr th unfldd Lang cuplr. (a) Fur-wir cupld lin mdl. (b) Apprximat tw-wir cupld lin mdl. Effctiv capacitanc ntwrks fr th unfldd Lang cuplr quivalnt circuits. (a) Effctiv capacitanc fr th fur-wir mdl. (b) Effctiv capacitanc fr th tw-wir mdl. Elctrnic Matrials and Dvics Applicatins Lab

35 Elctrnic Matrials and Dvics Applicatins Lab Fr vn md xcitatin, all fur cnductrs ar at th sam ptntial, s m has n ffct and th ttal capacitanc f any lin t grund is 4 = x + in. Fr dd-md xcitatin, lctric walls ffctivly xist thrugh th middl f ach m, s th capacitanc f any lin t grund is 4 = x + in + 6 m. vn- and dd-md charactristic impdancs ar 4 = / (v p 4 ) ; 4 = / (v p 4 ) nsidr any islatd pair f adjacnt cnductrs in th fur-lin md; th ffctivly capacitancs ar = x ; = x + m Th vn-dd md capacitancs f th fur-wir lin in trms f tw-wir cupld lin: / ; / is whil th vltag cupling cfficint 3 3 pair. th tw - cnductr and vn - md charactristic impdancs f ar th dd -, whr 3 ; 3 inc 3 ; 3 v p

36 Th 8 hybrid Th 8 hybrid junctin is a fur-prt ntwrk with a 8 phas shift btwn th tw utput prts. A signal applid t prt will b vnly split int tw in-phas cmpnnts at prt and 3, and prt 4 will b islatd. => If th input is applid t prt 4, it will b qually split int tw cmpnnts with a 8 phas diffrnc at prts and 3, and prt will b islatd. Whn pratd as a cmbinr, with input signals applid at prts and 3, th sum f th inputs will b frmd at prt, whil th diffrnc will b frmd at prt 4. => prts and 4 ar rfrrd t as th sum and diffrnc prts, rspctivly. Th scattring matrix fr th idal 3 db 8 hybrid => unitary and symmtry Hybrid junctins. (a) A ring hybrid, r rat-rac, in micrstrip r striplin frm. (b) A taprd cupld lin hybrid. (c) A wavguid hybrid junctin, r magic-t. ymbl fr a 8 hybrid junctin. Elctrnic Matrials and Dvics Applicatins Lab

37 Evn-dd Md Analysis nsidr a unit amplitud wav incidnt at prt (th sum prt). => th ring junctin this wav will divid int tw cmpnnts, which bth arriv in phas at prts and 3, and 8 ut f phas at prt 4. Using th vn-dd md analysis tchniqu, dcmps this cas int a suprpsitin f th tw simplr circuits and xcitatins. Amplituds f th scattrd wavs : B = / + / ; B = T / + T / ; B 3 = / - / ; B 4 = T / T / Evn- and dd-md dcmpsitin f th ring hybrid whn prt is xcitd with a unit amplitud incidnt wav. (a) Evn md. (b) Odd md. Elctrnic Matrials and Dvics Applicatins Lab

38 Using th ABD matrix fr vn- and dd-md tw-prt circuits t valuat th rquird rflctin and transmissin cfficints. B = ; B = -j / () / ; B 3 = -j / () / ; B 4 = => th input prt is matchd, prt 4 is islatd, and th input pwr is vnly dividd and in phas btwn prts and 3. Elctrnic Matrials and Dvics Applicatins Lab

39 nsidr a unit amplitud wav incidnt at prt 4 (th diffrnc prt). => th tw wav cmpnnts n th ring will arriv in phas in phas at prts and 3, with a nt phas diffrnc f 8 btwn ths prts. Amplituds f th scattrd wavs : B = T / T / ; B = / - / ; B 3 = T / + T / ; B 4 = / + / Evn- and dd-md dcmpsitin f th ring hybrid whn prt 4 is xcitd with a unit amplitud incidnt wav. (a) Evn md. (b) Odd md. Elctrnic Matrials and Dvics Applicatins Lab

40 Elctrnic Matrials and Dvics Applicatins Lab Using th ABD matrix fr vn- and dd-md tw-prt circuits t valuat th rquird rflctin and transmissin cfficints.,,,,,, ; 4 3 B j B j B B j T j j T j j j D B A j j D B A Input prt is matchd, prt is islatd, and th input pwr is vnly dividd int prts and 3 with a 8 phas diffrnc. Th bandwidth f th ring hybrid is limitd by th frquncy dpndnc f th ring lngths, bur is gnrally n th rdr f ~3% => incrasing th BW additinal sctins r a symmtric ring circuit.

41 Elctrnic Matrials and Dvics Applicatins Lab