Microwave Engineering

Transcription

1 Microwv Egirig hg-hsig Hsu Dprtmt of Elctricl Egirig Ntiol Uitd Uivrsity

2 Outli. Trsmissio Li Thory. Trsmissio Lis d Wvguids Grl Solutios for TEM, TE, d TM wvs ; Prlll Plt wvguid ; Rctgulr Wvguid ; oxil Li ; Stripli ; Microstrip 3. Microwv Ntwork lysis mpdc d Equivlt oltgs d urrts ; mpdc d dmittc Mtrics ; Th Scttrig Mtrix ; BD Mtrix ; Sigl Flow Grphs ; Discotiutis d Modl lysis 4. mpdc Mtchig d Tuig Mtchig with Lumpd Elmts ; Sigl-Stub Tuig ; Doubl-Stub Tuig ; Th Qurtr-Wv Trsformr ; Th Thory of Smll Rflctios 5. Microwv Rsotors Sris d Prlll Rsot ircuits ; Trsmissio Li Rsotors ; Rctgulr Wvguid vitis Dilctric Rsotors 6. Powr Dividrs d Dirctiol ouplrs Bsic Proprtis of Dividrs d ouplrs ; Th T-Juctio Powr Dividr ; Th Wilkiso Powr Dividr ; oupld Li Dirctiol ouplrs ; 80 o hybrid 7. Microwv Filtrs Priodic Structur ; Filtr Dsig by th srtio Loss Mthod ; Filtr Trsformtios ; Filtr mplmttio ; Elctroic Mtrils d Dvics pplictios Lb

3 3. Microwv Ntwork lysis mpdc d Equivlt oltgs d urrts mpdc d dmittc Mtrics Th Scttrig Mtrix BD Mtrix Sigl Flow Grphs Discotiutis d Modl lysis Elctroic Mtrils d Dvics pplictios Lb

4 ircuits oprtig t low frqucis, for which th circuit dimsios r smll rltiv to th wvlgth, c b trtd s itrcoctio of lumpd pssiv or ctiv compots. this situtio th circuit dimsios r smll ough so tht thr is gligibl phs chg from o poit i th circuit to othr. dditio, th filds c b cosidrd s TEM filds supportd by two or mor coductors. -> this lds to qusi-sttic typ of solutio to Mxwll s qutios, d to th kirchhoff voltg d currt lws d impdc cocpts of circuit thory. But ths tchiqus c ot b dirctly pplid to microwv circuits.. => Bcus th solutio to Mxwll s qutio for giv problm is complt, it givs th lctric d mgtic filds t ll poits i spc. But w r itrstd i oly th voltg or currt t st of trmils, th powr flow through dvic. othr rso for usig circuit or twork lysis is tht it is th vry sy to modify th origil problm, or combi svrl lmts togthr d fid th rspos, without hvig to ly i dtil th bhvior of ch lmt i combitio with its ighbors. fild lysis usig Mxwll s qutios for such problms would b hoplssly difficult. Elctroic Mtrils d Dvics pplictios Lb

5 mpdc d quivlt voltgs d currts Equivlt voltgs d currts t microwv frqucis th msurmt of voltg or currt is difficult (impossibl), ulss clrly dfid trmil pir (TEM typ lis such s coxil cbl, microstrip, or stripli) is vilbl. => ot xist for o-tem lis such s rctgulr, circulr, or surfc wvguid Th voltg,, of E H o (chrctrsitic impdc) chrctrsitic impdc (ssumig w kow th propgtio costt for li) y j x, y, si x, y x E d H d (mpr's lw) pply th circuit thory for trsmissio lis dvlopd to chrctri this li s circuit lmt. mor difficult for wvguids For rctgulr wvguid, TE th coductor rltiv to th - coductor j j x x mod j x, y, si h x, y j 0 x y j c b foud s Elctric fild lis for th TE0 mod of rctgulr wvguid. Elctric d mgtic fild lis for rbitrry two-coductor TEM li. Elctroic Mtrils d Dvics pplictios Lb

6 pplid to lctric fild j x si j y dy Th voltg dpds o th positio x,s wll s th lgth of th itgrtio cotour log th y dirctio, d is quit diffrt from th diffrt positio ( x=0, /). wrog to obti voltg, d th currt is lso th sm situtio. (for o-tem lis) Thr r my wys to dfi quivlt voltg, currt, d impdc for wvguid. oltg d currt r dfid oly for prticulr wvguid mod, d r dfid so tht th voltg is proportiol to th trsvrs lctric fild, d th currt is proportiol to th trsvrs mgtic fild. ordr to b usd i mr similr to voltgs d currts of circuit thory, th quivlt voltgs d currts should b dfid so tht thir product givs th powr flow of th mod. Th rtio of th voltg to th currt for sigl trvlig wv should b qul to th chrctristic impdc of th li. This impdc my b chos rbitrry, but is usully slctd s qul to th wv impdc of th li, or ls ormlid to uity. Elctroic Mtrils d Dvics pplictios Lb

7 Elctroic Mtrils d Dvics pplictios Lb d impdc. powr c b dtrmid from th rmiig two coditios for d rltioship r Th proportiolity costts for this voltg d currt wvs s Dfis quivlt, ˆ,, r rltd by th wv impdc, d Sic th trvlig wvs. r th fild mplituds of d th mod, r th trsvrs fild vritios of d whr,,,,,,,, th trsvrs filds c b writtr s rbitrry wvguid mod with both positivly d gtivly trvlig wvs, For o j j j j w w t t - j j j j t j j j j t / / / / / / with y x y x h H E, h y x h y x h y x H y x y x y x E

8 Elctroic Mtrils d Dvics pplictios Lb rch mod r th proportiolity costts for d th mod ; r wquivlt voltgs d currts for th d whr,,, ;,,, ormli th chrctristic impdc to uity th th mod, of ) th wv impdc (, is dsird to hv it f d sic Th chrctristic impdc is th wvguid th surfc itgrtio is ovr th cross sctio of : whr ˆ to powr to b qul this ˆ ˆ flow for th icidt wv is giv by Th complx powr * * * * * * * N N j j t j j t o TE TM w TE TM w o o S S S y x h y x H y x y x E i wvguid fild grl or or S ds h ds h ds h P

9 Th cocpt of mpdc Th cocpt of impdc forms s importt lik btw fild thory d trsmissio li or circuit thory Summri th vrious typs of impdc w hv usd so fr d thir ottio: =(/) / =itrisic impdc of th mdium. dpdt oly o th mtrils prmtrs of th mdium, d qul to th wv impdc for pl wvs. w =E t /H t =/Y w =wv impdc is chrctristic of th prticulr typ of wv. TEM, TM, d TE wvs ch hv diffrt wv impdc ( TEM, TM, TE ) dpd o th typ of li or guid, th mtril, d th oprtig frqucy. o =/Y o =(L/) / =chrctristic impdc is th rtio of voltg to currt for trvlig wv o trsmissio li. (th chrctristic impdc of TEM wv is uiquly, th chrctristic impdc of TM, TE wvs r i vrious wys du to hv uiquly TE d TM wvs. Elctroic Mtrils d Dvics pplictios Lb

10 Ex. osidr rctgulr wvguid with =.86 cm d b =.06 cm (X-bd guid), ir filld for < 0 d rxolit filld (dilctric costt =.54) for > 0, s show i Figur. f th oprtig frqucy is 0 GH, us quivlt trsmissio li modl to comput th rflctio cofficit of TE 0 wv icidt o th itrfc from < 0. <Sol> Th propgtio costts i th ir ( 0) d th dilctric ( 0) rgios r ko 58.0 m, b r ko 304. m whr ko 09.4 m Trt th problm s th rflctio of icidt voltg wv t th juctio of two ifiit trsmissio lis. th quivlt chrctristic impdcs for th two lis r d d 0 0 k k with this rsults, xprssios for th icidt, rflctd d trsmittd wvs c b writt i trms of filds, or i trms of quivlt voltgs d currts. ; 0 d d Elctroic Mtrils d Dvics pplictios Lb

11 osidr o - port twork d driv grl rltio btw its impdc proprtis d lctromgtic rgy stord i, d th powr dissiptd by, th twork. Th complx powr dlivrd to this twork is * P whr is E H ds P S jwm W P th vrg powr dissiptd by th twork dw dw f dfi rl trsvrs modl filds, E f -jβj x,y, x,y ; H x,y, h x,y with ormlitio : th bov qutio c b xprssd i trms of * P h d s S Th th iput impdc is th twork is losslss, th i t is purly imgiry, with rctc X lod ( W h d s * d h, ovr th trmil pl of 4 th trmil voltg d currt : W * P P j m W i R jx whr R of th iput impdc is rltd to dissiptd powr, whil X is rltd to th t rgy stord i th twork m S t P 0 d R 0. m W jββ m W ), d gtiv for cpcitiv lod ( W m W rl d rprsts rprst th stord mgtic d lctric rgy which is positiv for iductiv W ) th twork rbitrry o-port twork. Elctroic Mtrils d Dvics pplictios Lb

12 mpdc d dmittc mtrics Usig th impdc d/or dmittc mtrics of circuit thory to rlt ths trmil or port qutitis to ch othr, d thus to sstilly rriv t mtrix dscriptio of th twork. osidr rbitrry N-port microwv twork (y typ of trsmissio li or trsmissio li quivlt of sigl propgtig wvguid mod. t spcific poit o th th port, trmil pl, t, is dfid log with quivlt voltgs d currts for th icidt ( +, + ) d rflctd ( -, - ) wvs. Th impdc mtrix [] of th microwv Ntwork = [] = [][] => = wh = 0 Elctroic Mtrils d Dvics pplictios Lb

13 dmittc mtrics Similrly, w c dfi c dmittc mtrix [Y] [Y] = [] -, both th [] d [Y] mtrics rlt th totl port voltgs d currts. Y ij ij i j i j k 0 for k j 0 for k j k Drivig port j with th currt j, op-circuitig ll othr port, d msurig th op-circuit voltg t port i => ii is th iput impdc s lookig ito port i wh ll othr ports r op-circuitd Drivig port j with th voltg j, short-circuitig ll othr port, d msurig th short-circuit currt t port i => Y ii is th iput dmittc s lookig ito port i wh ll othr ports r short-circuitd grl, ch ij or Y ij lmt my b complx. Howvr, my tworks r ithr rciprocl or losslss, or both. f th twork is rciprocl, th impdc d dmittc mtrics r symmtric. => ij = ji d Y ij = Y ji ; f th twork is losslss, ll th ij or Y ij lmts r purly imgiry => Eithr of ths spcil css => c rduc th umbr of idpdt qutitis or dgrs of frdom tht N-port twork. Elctroic Mtrils d Dvics pplictios Lb

14 Ex. Fid th prmtrs of th two-port T-twork c b foud s th iput impdc of port wh port is op circuitd t port wh currt 0 0 B (trsfr impdc) c b foud msurig th op-circuit voltg 0 rciprocl B is ppld t port. Elctroic Mtrils d Dvics pplictios Lb

15 Rciprocl Ntworks Y ij = Y ji if [Y] is symmtric mtrix, its ivrs, [], is lso symmtric Losslss Ntworks Now cosidr rciprocl losslss N-port juctio; f th twork is losslss, th th t rl powr dlivrd to th twork must b ro. th lmts of th impdc d dmittc mtrics must b pur imgiry. Elctroic Mtrils d Dvics pplictios Lb

16 Th scttrig mtrix rprsttio mor i ccord with dirct msurmts, d with th ids of icidt, rflctd, d trsmittd wvs, is giv by th scttrig mtrix. Lik th impdc d dmittc mtrics for N-port twork, th scttrig mtrix provids complt dscriptio of th twork s s t its N ports. => Whil th impdc d dmittc mtrics rlt th totl voltgs d currts t th ports, th scttrig mtrix rlts th voltg wvs icidt o th ports to thos rflctd from th ports. => scttrig prmtrs c b msurd dirctly with vctor twork lyr. + is th mplitud of th voltg wv icidt o port, d - is th mplitud of th voltg wv rflctd from port. => scttrig mtrix [S] mtrix S ij is foud by drivig port j with icidt wv of voltg j+, d msurig th rflctd wv mplitud, i-, comig out of port i.=> th icidt wvs o ll ports xcpt th jth port r st to ro, which ms tht ll ports should b trmitd i mtchd lods to void rflctios. (Notd S ii ) Elctroic Mtrils d Dvics pplictios Lb

17 Ex. Fid th S prmtr of th 3dB ttutor circuit show i Figur. From S From S f S S c b foud s th rflctio cofficit s t port wh port is trmitd i mtchd lod ( S 8.56 [ ]/ i ij S i S j k 0 d 0 th iput powr is 0 for k j 0 wh port is trmitd i / th th output powr is which is o - hlf( 3dB) of th iput powr -, 0 o, i i o o o o port 50, so S o 50, d tht o 0 S S 0 (symmtry) 0 o S o o 50 ) 4 o mtchd 3B ttutor with 50 Ω hrctristic impdc Elctroic Mtrils d Dvics pplictios Lb

18 Th [s] mtrix c b dtrmid from th [] or [Y] mtrix. = = = ( 0 =) =>[][] = [][ + ] [][ - ] = [] = [ + ] + [ - ] which c b rwritt s ([]+[U])[ - ] = ([]-[U])[ + ] whr [U] is uit or idtity mtrix [U]= => [S]=([]+[U]) - ([]-[U]) givig th scttrig mtrix i trms of th impdc mtrix. => [] = ([U]+[S])([U]-[S]) - Rciprocl Ntworks [U] is digol, so [U] t =[U] => [] t =[] => [S] mtrix for rciprocl twork is symmtric. => [S] = [S] t Losslss Ntworks [S] mtrix for losslss twork is uitry. => [S] t [S] * = [U] => [S] * = {[S] t } - N * i j SkiSkj 0 i j k Elctroic Mtrils d Dvics pplictios Lb

19 shift i Rfrc Pls Du to th S prmtrs rlt mplituds (mgitud d phs) of trvlig wvs icidt o d rflctd from microwv tworks, phs rfrc pls must b spcifid for ch port of th twork. Now cosidr th S prmtrs r trsformd wh th rfrc pls r mov from thir origil loctios. lt th w [S ] is dotd s w scttrig mtrix ( =l ), d [S] is dotd s scttrig mtrix for th twork with this st of trmil pls ( =0). Th icidt d rflctd port voltgs : [-]=[S][+] [ -]=[S ][ +] For th thory of trvlig wvs o losslss trsmissio lis + = + j ; - = - -j = l lctricl lgth of th outwrd shift of th rfrc pl of port Shiftig rfrc pls for N-port twork. Elctroic Mtrils d Dvics pplictios Lb

20 ombitio bov two qutios S = -j S, mig tht th phs of S is shiftd by twic th lctricl lgth of th shift i trmil pl, bcus th wv trvls twic ovr this lgth upo icidt d rflctio. Elctroic Mtrils d Dvics pplictios Lb

21 Grlid Scttrig Prmtrs grl cs, scttrig prmtrs of tworks with th sm chrctristic impdc for ll ports (50). Howvr, th chrctristic impdcs of multi-port twork my b diffrt, which rquirs grlitio of th scttrig prmtrs s dfid up to this poit. 0 is th (rl) chrctristic impdc of th th port ; + d - rprst th icidt d rflctd voltg wvs t port. icidt (powr) wv : P S ij b ; b * * * R b b b * * th qutity b b is bi j 0 0 for k j k f xprssd i trms of S ij 0 Now th vrg powr dlivrd to th th R d - i j 0 th vrg powr dlivrd through port is qul to th powr i th icidt wv Grlid scttrig mtrix : [b] [S][] ; rflctd (powr) wv : b 0 j 0i port is k 0 for k j 0 b purly imgiry mius th powr i th rflctd wv (physiclly stisfyig) dpdt o th chrctristic impdc of th th port., th corrspodig rsult would b - 0 N-port twork with diffrt chrctristic impdcs. Elctroic Mtrils d Dvics pplictios Lb

22 Elctroic Mtrils d Dvics pplictios Lb

23 Elctroic Mtrils d Dvics pplictios Lb Th trsmissio (BD) mtrix Th, Y, d S prmtr rprsttios c b usd to chrctri microwv twork with rbitrry umbr of ports, but i prctic my microwv cosist of cscd coctio of two or mor two-port tworks. Th BD mtrix is dfid for two-port twork i trms of th totl voltgs d currts to x trsmissio. For two-port twork For cscd coctio of two two-port tworks Rltio to impdc Mtrix Kowig th prmtr of twork -> dtrmi th BD mtrix D B D B ; D B D B D B D B () two-port twork; (b) cscd coctio of two-port tworks. D-B th f th twork is rciprocl, D B / ; ; ;

24 Elctroic Mtrils d Dvics pplictios Lb

25 Equivlt ircuits for Two-Port Ntworks Th spcil cs of two-port microwv twork occurs so frqutly i prctic tht is dsrvs furthr tttio. => Discuss th us of th quivlt circuits to rprst rbitrry two-port twork. Elctroic Mtrils d Dvics pplictios Lb

26 Figur shows trsitio btw coxil li d microstrip li, d srvs s xmpl of two-port twork. Trmil pls c b dfid t rbitrry poits o th two trsmissio lis. =>bcus of th physicl discotiuity i th trsitio from coxil li to microstrip li, lctric d/or mgtic rgy c b stord i th viciity of th juctio, ldig to rctiv ffcts which c b obtid by msurmts or by thorticl lysis (two-port blck box :, Y, S, or BD), suitd trsitio from o typ of trsmissio to othr or trsmissio li discotiuity. Th rbitrry two-port twork c b dscribd i th trms of impdc prmtrs s = + ; = + or = Y + Y ; = Y + Y f th twork is rciprocl, th = d Y = Y cox-to-microstrip trsitio d quivlt circuit rprsttios. () Gomtry of th trsitio. (b) Rprsttio of th trsitio by blck box. (c) possibl quivlt circuit for th trsitio [6]. Elctroic Mtrils d Dvics pplictios Lb

27 Othr quivlt circuits c lso b usd to rprst two-port twork such s T d quivlt circuits. f th twork is rciprocl, thr r six dgrs of frdom (th rl d imgiry prts of thr mtrix lmts), so th quivlt circuit should hv six idpdt prmtrs f th twork is losslss, which is good pproximtio for my prcticl two-port juctios, som simplifictio c b md i th quivlt circuit. => This rducs th dgrs of frdom for such twork to thr, d implis tht th T d quivlt circuits c b costructd from purly rctiv lmts. Equivlt circuits for rciprocl two-port twork. () T quivlt. (b) π quivlt. Elctroic Mtrils d Dvics pplictios Lb

28 Sigl Flow Grphs this sctio w discuss th sigl flow grph, which is dditiol tchiqu tht is vry usful for th lysis of microwv tworks i trms of trsmittd d rflctd wvs. first discuss th fturs d th costructio of th flow grph itslf ; scod prst tchiqu for th rductio, or solutio, of th flow grph. Th primry compots of sigl flow grph r ods d brchs: Nods: Ech port, i, of microwv twork hs two ods, i d b i. => i is idtifid with wv trig port i, whil od b i is idtifid with wv rflctd from port i. => th voltg t od is qul to th sum of ll sigls trig tht od. Brchs: brch is dirctd pth btw two ods, rprstig sigl flow from o od to othr. Evry brch hs ssocitd S prmtr or rflctio cofficit. Elctroic Mtrils d Dvics pplictios Lb

29 osidr th flow grph of rbitrry two-port twork. Th flow grph givs ituitiv grphicl illustrtio of th twork bhvior. icidt t port, prt goig through S d out port s rflctd wv d prt trsmittd through S to od b. f lod with oro rflctio cofficit is coctd t port, this wv (b wv) will b prtly rflctd d rtr th two-port twork t od. => prt of th wv c b rflctd bck out port vi S, d prt c b trsmittd out port through S. Two othr spcil tworks, o-port twork d voltg sourc. it is sy mttr to solv for th rtio of y combitio of wv mplituds. => usig four bsic dcompositio ruls, but th sm rsults c lso b obtid usig Mso s rul from cotrol systm thory. Th sigl flow grph rprsttio of two-port twork. () Dfiitio of icidt d rflctd wvs. (b) Sigl flow grph. Th sigl flow grph rprsttios of o-port twork d sourc. () o-port twork d its flow grph. (b) sourc d its flow grph. Elctroic Mtrils d Dvics pplictios Lb

30 Dcompositio of Sigl Flow Grphs sigl flow grph c b rducd to sigl brch btw two ods usig th four bsic dcompositio ruls blow, to obti y dsird wv mplitud rtio. Rul (Sris Rul): two brchs, commo od hs oly o icomig d o outig wv. => to form sigl brch whos cofficit is th product of th cofficits of th origil brchs. 3 = S 3 = S 3 S Rul (Prlll Rul): two brchs, from commo od to othr commo od my b combid ito sigl brch => whos cofficit is th sum of th cofficits of th origil brchs. = S + S b = (S + S b ) Rul 3 (Slf-Loop Rul): Wh od hs slf loop ( brch tht bgis d ds o th sm od) of cofficit S, th slf loop c b limitd by multiplyig cofficits of th brchd flig tht od by / (-S) => = S + S, 3 = S 3 =>limitig : 3 = [S 3 S / ( S )] Rul 4 (Splittig Rul): od my b split ito two sprt ods s log s th rsultig flow grph cotis, oc d oly oc, ch combitio of sprt (ot slf loops) iput d output brchs tht coct to th origil od. => 4 = S 4 = S S 4 Dcompositio ruls. () Sris rul. (b) Prlll rul. (c) Slf-loop rul. (d) Splittig rul. Elctroic Mtrils d Dvics pplictios Lb

31 Ex: Us sigl grphs to driv xprssios for i d out for th two-port twork. b i out b S S SS S SS S s s Dcompositios of th flow grph to fid Γ i = b/ d Γ out = b/. () Usig Rul 4 o od. (b) Usig Rul 3 for th slf-loop t od b. (c) Usig Rul 4 o od b. (d) Usig Rul 3 for th slf-loop t od. Elctroic Mtrils d Dvics pplictios Lb

32 pplictio to TRL Ntwork lyr librtio For furthr pplictio of sigl flow grphs, clibrtio of twork lyr usig th Thru-Rflct Li (TRL) tchiqu. Th grl problm is itdd to msur th S-prmtrs of two-port dvic t th idictd rfrc pls. =>Th primry rfrc pl for such msurmts is grlly t som poit withi th lyr itslf. th msurmt will iclud losss d phs dlys cusd by th ffcts of th coctors, cbls, d trsitios tht must b usd to coct th dvic udr tst (DUT) to th lyr. ths ffcts r lumpd togthr i two-port rror box ( clibrtio procdur is usd to chrctri th rror boxs bfor msurmt of th DUT) plcd t ch port btw th ctul msurmt rfrc pl d th dsird rfrc pl for th two-port DUT. => th ctul rror-corrctd S-prmtr of th DUT c b clcultd. (Msurmt of o-port twork c b cosidrd s rducd cs of th two-port) Th simplst wy to clibrt twork lyr is to us thr or mor kow lods, such s shorts, ops, d mtchd lods. => uss thr simpl coctios to llow th rror boxs to b chrctrid compltly. => [] Thru coctio is md by dirctly coctig port to port, s th rfrc pls [] Rflct coctio uss s lod hvig lrg rflctio cofficit, L, such s omil op or short. [] Li coctio ivolvs coctig ports d 3 togthr through lgth of mtchd trsmissio li Elctroic Mtrils d Dvics pplictios Lb

33 Us sigl flow grphs to driv th st of qutios cssry to fid th S-prmtr for th rror boxs i th TRL clibrtio procdur. Th Thru, Rflct, d Li coctios t th rfrc pl for th DUT, d msur th S-prmtr for ths thr css t th msurmt pls. =>ssum th sm chrctristic impdc for ports d, d th rror boxs (chrctrid by th S-mtrix [S], d ltrtivly by th BD mtrix) r rciprocl d idticl for both ports. => S =S d ivrs rltio btw th BD mtrics of th rror boxs for ports d. [T] ; [R] ; [L] mtrics Thru coctio Th thru coctio d th corrspodig sigl flow grph. Obsrv tht w hv md us of th fct tht S =S d tht th rror boxs r idticl d symmtriclly rrgd. c b sily rducd usig th dcompositio ruls to giv th msurd S-prmtrs t th msurmt pls i trms of th S-prmtrs of th rror boxs T T b b 0 0 S S S SS S () () By symmtry to T =T, d by rciprocity to T =T Elctroic Mtrils d Dvics pplictios Lb

34 Rflct coctio Not tht this rrgmt ffctivly dcoupls th two msurmt ports, so R =R =0 b SL => c b sily rducd to show tht R 0 S (3) S L By symmtry to R =R Li coctio rductio similr to tht usd for th thru cs L L b b 0 0 S SS S S S (4) (5) By symmtry d rciprocity to L =L d L =L Elctroic Mtrils d Dvics pplictios Lb

35 Now hv fiv qutios for fiv ukows S, S, S, L, -l => Sic qutio 3 is th oly qutio tht cotis L, w c first solv th othr four qutios for th othr four ukows. => qutio c b limit S from qutios, 4 d 5, d th S c b limitd from qutios d 4. => L -l L S = T -l T S -l -l (T -S T )-T S = L ( -l S ) S T L T ( T L ) [ L T ( T L ) ] l rl d imgiry prts of b positiv, or by kowig th phs of L to withi 80º W multiply qutio by S d subtrct from qutio to gt T = S + S T, d similrly multiply qutio 5 by S d subtrct from qutio 4 to gt L = S + S L -l Elimitig S from ths two qutios givs S i trms of -l s S = (T -L ) / (T -L -l ) S = T -S T d solvig qutio for S givs S = T (-S ) L Filly qutio 3 c b solvd fro L to giv L = (R -S ) / [S + S (R -S )] T From th rsults giv th S-prmtrs for th rror boxs, s wll s th ukow rflctio cofficit, L, d th propgtio fctor, -l.=> this complts th clibrtio procdur for th TRL mthod. 4L T Elctroic Mtrils d Dvics pplictios Lb

36 Elctroic Mtrils d Dvics pplictios Lb Sic w r ow workig with cscd of thr two-port tworks, it id covit to us BD prmtrs. W covrt th rror box S-prmtrs to th corrspodig BD prmtrs, d covrt th msurd S-prmtrs of th cscd to th corrspodig m B m m D m prmtrs. f w us B D to dot th prmtrs for th DUT, th D B D B D B D B BD D B D B D B D B m m m m m m m m ' ' ' ' prmtrs for th DUT s from which w c dtrmi th ' ' ' '

37 Discotiuitis d Modl lysis By ithr cssity or dsig, microwv tworks oft cosist of trsmissio lis with vrious typs of trsmissio li discotiuitis. Discotiuitis r uvoidbl rsult of mchicl or lctricl trsitios from o mdium to othr (.g.: juctio btw two wvguids, or cox-to-microstrip trsitio). Dpdig o th typ of discotiuity, th quivlt circuit my b simpl shut or sris lmt cross th li or, i th mor grl cs, T- or -quivlt circuit my b rquird. dditio, th compot vlu of quivlt circuit dpd o th prmtrs of th li d th discotiuity, s wll s th frqucy of oprtio. Elctroic Mtrils d Dvics pplictios Lb

38 Figur to c, thi mtllic diphrgms (or iriss ) c b plcd i th cross sctio of wvguid to yild quivlt shut iductc, cpcitc, or rsot combitio. Similr discotiuitis c b lso b md i circulr wvguid Rctgulr wvguid discotiuitis. Elctroic Mtrils d Dvics pplictios Lb

39 Som typicl microstrip discotiuitis d trsitios r show i figur. Similr gomtris xist for stripli d othr pritd trsmissio lis such s slotli, covrd microstrip, coplr wvguid, tc. Som commo microstrip discotiuitis. () Op-dd microstrip. (b) Gp i microstrip. (c) hg i width. (d) T-juctio. () ox-to-microstrip juctio. Elctroic Mtrils d Dvics pplictios Lb

40 Modl lysis of H-Pl Stp i Rctgulr Wvguid Th fild lysis of most discotiuity problms is vry difficult, d th tchiqu of modl lysis is rltivly strightforwrd d similr i pricipl to th rflctio/trsmissio problms. dditio, modl lysis is rigorous d vrstil tchiqu tht c b pplid to my cox, wvguid, d plr trsmissio li discotiuity problms, d lds itslf wll to computr implmttio. => w will prst th tchiqu of modl lysis by pplyig it to th problm of fidig th quivlt circuit of H-pl stp (chg i width) i rctgulr wvguid. ssum tht oly th domit TE 0 mod is propgtig i guid ( < 0), d tht such mod is icidt o th juctio from < 0. it is lso ssumd tht o mods r propgtig i guid, lthough th lysis to follow is still vlid if propgtio c occur i guid. Th trsvrs compots of th icidt TE 0 mod c b writt, for < 0, i x j i x j Ey si ; H x si whr k0 is th propgtio costt of th TE d k mod i guid(of width ) is th wv impdc of th TE 0 mod i guid. Gomtry of H-pl stp (chg i width) i rctgulr wvguid. Elctroic Mtrils d Dvics pplictios Lb

41 Du to th discotiuity t = 0 thr will b rflctd d trsmittd wvs i both guids, cosistig of ifiit sts of TE 0 mods i guids d. => oly th TE 0 mod will propgt i guid, but th highr-ordr mods r lso importt bcus thy ccout for stord rgy, loclid r = 0. => dditio, thr is o y vritio itroducd by this discotiuity, TE m mods for m 0 r ot xcitd, or r y TM mods. (Howvr, mor grl discotiuity my xcit such mods. Th rflctd mods i guid my th b writt, for 0, s E E r y t y whr B x si c j c j H Similrly, th trsmittd mods ito guid c b writt, for 0,s r x H t x Th rflctd cofficit of th icidt TE x si 0 B whr th propgtio costt i guid is ; ; d th wv impdc i guid is x si c c c j is ukow mplitud cofficit of th rflctd TE mod is th. x si c j k 0 k 0 0 c c 0 c mod i guid. Elctroic Mtrils d Dvics pplictios Lb

42 Th trsvrs fild ( E dditio, E Eforcig ths boudry codistios lds to th followig qutios : E H y x x si x si Th fct tht th quivlt circuit of th H - pl stp looks iductiv is rsult of th ctul vlu of th rflctio cofficit,, but w c vrify this rsult by computig th complx powr ito th vsct mods o ithr sid of P c x0 bc b y y0 must b ro for c E H B c* jbc k ) must b cotiuous for 0 Equtio 3 shows tht th complx powr y bcus of x x si 0, si B for x c () c 0 for c x, x B x si si for 0 x c. () c c * 0, H 0 x 0 th discotiuity. (3) drivd for th vsct mods i guid. (this is lft dxdy b B x c c x0 E y H * x dx b x th stp. c B x0 c t 0. flow ito guid is purly iductiv. similr s problm) x si c m B * m c * m mx si dx c rsult c b Elctroic Mtrils d Dvics pplictios Lb

43 Microstrip Discotiuity ompstio Elctroic Mtrils d Dvics pplictios Lb