Chapter 4 Network Analysis

Transcription

1 Chaptr 4 Ntwork Aalysis 4. troductio Th KL ad KCL i circuit i thory ar o logr valid. Us th Maxwll s quatios to solv all microwav circuits:? Fild Problm EH, (ctors Circuit Problm, (calars (Equivalt quatitis 4. mpdac, Equivalt oltag ad Currt Equivalt voltag ad currt ar valid oly for TEM lis + - Ed H d C Exampl: For a complicats coplaar stripli (CP low-pass filtr, ca w dsig it usig circuit poit of viw? Agilt 85 with had moduls For o-tem lis, quivalt voltag ad currt ar: (anot uiqu but usful for a girig approach, (bfor crtai mod oly, * (c = powr flow of th mod, ad (dth charactristic impdac is dfid as / Th Cocpt of mpdacs A importat lik btw EM fild ad circuit thory ( = / trisic impdac of a mdium. also th wav impdac of a pla wav. ( W Wav impdac ( TE, TM,or TEM wav a charactristic for a particular mod,.g.th TE mod. (3 O Charactristic impdac, th ratio of ad for a travlig wav o a trasmissio li ( =,,, Exampl: Applicatio of Wavguid mpdac Fid th rflctio of a TE wav icidt o th itrfac from z. Lt a.86 mm, b.6 mm, ad f GHz. z, ; z, l ol: air k 58.m a 34. di k r a m k Wair 5 air 58 kdd Wdi 59.6 di 34. Wdi Wair Not: Th calculatio of Γ is valid oly wh th two wavguids hav idtical cross Wdi Wair sctio dimsios. i r r 3 4

2 O-Port Ntwork Complx powr dlivrd to th twork: * P ( EH ds P Wm W P avrag powr dissipatd by th twork W, W stord magtic ad lctric rgy m put impdac * P P ( Wm W i 4 ( W,, iductiv load m W X W m W R X, X X, Wm W, capactiv load Ev ad Odd Proprtis of i (ω ad Γ(ω it it vt ( ( d ( d * * it v ( t ( d * * ic vt ( is ral ( vt ( v( t, ( ( R ( R (, m ( m ( i( R( X( R ( R (, m ( m ( ( i( R( X( ( i( i( R( X( ( ( ( R( X( * ( ( ( i * ( ( * i ( i R ( X ( * ( R( X( * ( ( 5 6 N-Port Microwav Ntwork 4. -Matrix of a N-port Ntwork, k,,..., N at t pla k k k k k k total voltag at port k. k k total currt at port k. k ( Each ports has,,,, k,,..., N,dfid at t pla. k k k k k ( ( Thr could b may mods ar ach trasmissio li-twork uctio. (3 ( Each t pla is chos at a pla with a suitabl distac away k from th uctio surig o highr ordr mod at. t k 7 For a arbitary N -port liar twork: i i k, k = trasfr impdac btw ports i ad wh all othr ports ar op circui td. mpdac matrix 8

3 Exampl 4.3 i of a two-port -Matrix of a N-Port Ntwork Fid of th followig two-port twork: i A C, C B C BC othr words, a two-port spcifid by ca b rprsts by C, C 9... N... N N N N... NNN Admittac matrix = i i k, k = trasfr admittac btw ports i ad wh all othr ports ar short-circuitd. Exampl i of a Two-Port Ntwork Fid i of th followig two-port twork:, a b c b othr words, a two-port spcifid by ca b rprsts by b Rciprocal Ntwork... N... N N N N... N N N... N... N N N N... N N N f thr is o activ dvic or aisotropic matrials, lik BJT, FET, frrit ad plasma, isid th twork, o ca us th rciprocity thorm to prov that ad, i.. i i i i ad ar symmtric matrics.

4 Losslss Ntwork N N t t P ( * * * av m m m * * * * For N=, Pav ( / Not that th proprty of is ithr dpdt o or o Th t ral powr dlivrd to th twork is zro. Thus, R P = for all possibl combiatios.. *, m,, R m.,,, k m or, m k * * R ( mm m = R m =. Losslss twork R =, m, m av R R 3 i 4.3 Th cattrig Paramtr Matrix ad ar th amplituds of icidt ad rflctd voltag wavs, rspctivly,o port.... N... N N N N NN N i i is dtrmid by drivig port with a icidt voltag wav, ad masurig th rflctd wav, comig out from port i. k, k Wh i is valuatd, th icidt voltag wavs o all ports xpcts port ar st to zro, which mas that all othr ports should b trmiatd i matchd load to avoid rflctios. i 4 Exampl 4.4 Evaluatio of -paramtrs Covrsio Btw Ntwork Paramtrs Coditio # :, Fid th -paramtrs of th twoport. f th circuit is a 3-dB attu- RR //( R R R //( R ator, what ar th valus of R ad R //( R R( R R? R RR R R R R R ( R xr R R //( R ( i R ( R R R R Coditio#: ( R( R R R ( R R ( R R R R R ( R R R( R R x R R xrr xr x R ( x x R R R xr ( x ( x ( x R 4.4 xx ( x x R R ( x x 5 6

5 Dtrmi [] from [] ad [] Matrics : charactristic impdac of port. All th ports ar idtical ad for covic. which ca b writt as U U whr U = idtity matrix U U 7 Ntwork Paramtrs for Rciprocal & Losslss Ntworks (, ar symmtric matrics for rciprocal tworks ad a purly imagiary for losslss tworks. ( is a symmtric matrix for rciprocal tworks ad a uitary matrix for los slss two rks. T * What is a uitary matrix? f = U, whr, i i.. * * * * * * * T * * i idicats that tth rgy is cosrvd. mas th diffrt colums of ar orthogoal. * * 8 Exampl 4.5 Proprtis of -Paramtrs Dtrmi if th twork is rciprocal or losslss. What is th rflctio cofficit at port if port is shor-circuitd? T symmtri T * U ol :( c matrix rciprocal twork, ( (3 f port is short-circuitd, t is ot a losslss twork. / ( /( hift i Rfrc Plas Rfrc plas should b dfid bfor a twork is charactrizd. Th plas ca b shiftd. ad,, Q Q ad Q Q Q Q Q Q Q

6 Gralizd cattrig Paramtrs a b ( a b ( a b 4.4 Th Trasmissio (T or ABCD Matrix A B (Dfiitio of th ABCD matrix is basd o th total port voltag ad currt. (Th dirctio of is diffrt from thos of ad matrics. For a cascad of two two-port tworks: b a i b i i oi a o ak, k k, k f is xprssd i trms of ad, th rsult dpds o th charactristic impdac of port,. A B A B A B 3 3 Th rsult ca b applid to a cascad of N two-port tworks. Rlatio btw [] ad ABCD Matrics Th ABCD Paramtrs of om Usful Two-Ports A B A B ( / ( / A B f th twork is rciprocal,, AD BC 3 A B cos si si cos A B A B N A B C D N Try to driv thm by yourslf. 4

7 om Proprtis of ymmtric Ntworks om Proprtis of ymmtric Ntworks 5 6 om Proprtis of ymmtric Ntworks -Paramtr of ymmtric Ntworks v v, v v, v v v v v v, v v v v v v 7 v v v v v v v / v / v v v 8

8 -Matrix of ymmtric Ntworks -Matrix of ymmtric Ntworks v v v v o dd o dd v v v v v v v v v / / 9 v v v v v v v v v / / Try to driv thm by yourslf. 3 Exampl Th charactristic aact stcadmittac cof th twoλ/ λ/4-sctio o is. Fid th -paramtrs of th two-port with rfrc impdac. Basdoth -matrix, show that th circuitislosslss is losslss. Fid th iput rflctio cofficits i ad io wh th symmtric pla is placd with op ad short circuits, rspctivly. Usig th impdac trasformatio proprty, w hav ad /, ad i ad io ( / (. i ( i io ( i io O ca validat that th circuit is losslss sic io T* = idtity matrix. 3 Exampl Lt. Fid its matrix ad at port A (rfrc impdac = if port is short- circuitd. Th -martix ca b obtaid via th ABCD matrics i p. 4: Applyig th v- aalysis, v v i i v i i v v i i v v i i v v i i i i v v i i i i ta45 4 ta45 4 v A o i A ta 45 i o o v i i 4 v i i 4 4 3

9 Equivalt Circuit for Two-Port Ntworks Thory of TRL Ntwork Aalyzr Calibratio Why calibratio is cssary i a microwav masurmt? A coaxial-to-microstrip trasitio Trasitio rprstd by a black box. A possibl quivalt circuit for a trasitio ( f th twork is rciprocal, thr ar six dgrs of frdom, so th quivalt circuit has six idpdt paramtrs. ( A orciprocal twork caot b rprstd by a passiv quivalt circuit usig rciprocal lmts. (a Th -paramtrs iclud magitud ad phas masurmt. (b Diffrt circuits may hav diffrt rfrc plas, trasitios, coctors, housig structurs, ad/or frqucy bads Th TRL Calibratio Thru---Rmov th DUT ad Lt th -paramtrs b [T] Error Box : Losss ad phas dlays causd by th ffcts of coctors, cabls, ad/or trasitios. A calibratio procdur is usd to charactriz th -paramtrs of th rror box bfor masurmt of DUT. 35 a a ( (Rciprocal. Th rror box ar symmtric ad idtical. u a v a v a a a a a v u a a b a v a ( a a Ta T T T T, Ta 36

10 Rflct---Trmiat th Error Box to a Kow Load Li---Rmov th DUT ad srt a ctio Li b a u v v u b a Th two masurmt ports ar dcoupld, R = R = éb ù é ùéa ù a =, G = úê L êëb úû êë úê ûëa úû ( Error box b b = a+ a = a+ G Lb = a - G L æ G ö L b= a+ a = a+ G Lb = + aº Ra ç è - GL ø GL R = + = R - G L 37 v u a ( a v a a a ( v a v a b a v a a a L L L L L L, 38 TRL Calibratio (cot d T L T olv,,, L, ad 5 ukows L R L L Aftr som simpl algbraic maipulatios, o ca obtai TRL Calibratio (cot d ( Th -paramtrs for th rror boxs ar kow. ( Th -paramtrs for DUT ca th b obtaid as Am Bm C m D m Masurd A B A B A B DUT Error Box Error box ( ( 4 L T T L L T T L L T L T T L, T T, T T L L R ( R 39 4

11 Optioal Exampl (Midtrm 4, part Fid th -paramtrs of th twork ad th iput admittacs s at port wh port is trmiatd i.5,. Th rfrc impdac is. Optioal Exampl (Midtrm 4 part Fid th -paramtrs of th twork ad th iput admittacs s at port wh port is trmiatd i.5,. Th rfrc impdac is. f ca b kow that,. Wh, ( ta cot ( si csc caot s th op stub? 4 v ta ta ta cot v v v v v v v v v ( ( v v v v v v v v v 4 ta cot ta (ta cot (ta cot v 4 csc 4 Part Excitatio of a TE m Wavguid Mod (Pozar 4.7&4.8 This study prsts a mthodology of xcitig a pur circular TE m mod usig cascadd -typ powr dividrs. Th dividrs partitio th iput sigal ito svral parts which ar th coupld to a circular wavguid through aprturs. Th couplig aprturs iduc magtic dipols. With propr arragmt of th magtic dipols o th circumfrc of th circular wavguid, thy th oitly xcit th dsird mod. Th couplig strgth ad th mod purity ar calculatd usig th rciprocity thorm ad th magtic currt sourcs. Thr mod covrtrs, TE,TE,adTE 4, wr dsigd, built, ad tstdt at W-bad. b d Back-to-back k trasmissio i masurmts xhibit xcllt agrmt to th rsults of computr simulatios wh th coductor loss is tak ito cosidratio. Th masurd trasmissios ar high ad th badwidths ar broad. Ths -typ covrtrs ar structurally simpl but th machiig rrors ar critical. Th factors affctig th prformac will b discussd i dtail. T. H. Chag*, C.. Li, C. N. Wu, ad C. F. u, Excitig circular TEm mods at low trahrtz rgio, Appl. Phys. Ltt. 93, 53 (8. 43 Part Applicatios Excitig a spcific wavguid mod plays a ky rol i may applicatios, such as th gyrotro travlig-wav tub (gyro- TWT ad th gyrotro backward-wav oscillator (gyro-bwo. th gyro-twt, th mod covrtr lauchs a wav of a spcific mod ito th itractio structur to itract with th lctro bam; whil, i th gyro-bwo, th mod covrtr xtracts th wav powr at th upstram d. additio to th gyrotro applicatios, th mod covrtrs ca b usd i microwav/plasma systms, radar/atas systms, ad rotary oits. 44

12 Part Tchiqus s to Excit a pcific Wavguid Mod Part Currt hts ad Arbitrary Currt ourc By couplig mthod, thy ca b classifid ito two typs: srpti/corrugatd structur ad sidwall couplig structur. Takig gyrotro as a xampl, th gral rquirmts for th mod covrtr/lauchr iclud high covrtig fficicy, high mod purity, broad badwidth, high-powr capability, ad short covrtig lgth. -typ covrtrs ar suprior ovr othr covrtrs i th gyrotro applicatios Part Proprtis ad Charactristics of th Wavguid Mods Tabl summarizs th dsird mods ad thir corrspodig couplig structurs, radii, ad parasitic mods. 47 Part Difficultis to Excit a Highr-Ordr Mod Cutoff frqucy vs wavguid radius. For th W-bad opratio, th dsird wavguid radii ar.74,.5, ad 3. mm, for TE, TE, ad TE 4, rspctivly. Cu utoff frq. (G GHz T TE&TM TE TE4&TE TM TE3 TM TM TE Radius (mm 48

13 Part Mod ythsizig Excitatio of a giv mod ca b achivd usig proprly arragd lctric or magtic currt sourcs. Ths currt sourcs com from th prob fds, th loop fds, or th couplig aprturs. Th sidwall couplig aprturs, capabl of high powr opratio, o thus ar th bst choic c for prst study. Mod xcitatio from a arbitrary lctric ad magtic currt sourc ca b foud i Pozar s Chap.4. Hr w prst a complt approach to xcitig a cylidrical TE m mod usig multipl magtic currt sourcs. Part Elctric/Magtic Dipols ad Currts duc currt sourcs ythsiz th dsird mod Aalyz mod purity P Eˆ ( xx P m m H t ( x x 49 5 Part duc Currt ourcs Th ida of th sidwall couplig structur is to quip th wavguid with som dipols. Figur shows th gomtry of th sidwall couplig structur udr study. ( (a (b TE/TM E E H Pm Pm Pm Pm cutoff sctio Th couplig aprtur iduc quivalt lctric ad magtic dipol momts. Th lctric dipol momt is proportioal to th ormal lctric fild whil th magtic dipol momt is proportioal to th tagtial magtic fild P Eˆ ( xx P H ( x x m m t whr x ad x ar positios of th obsrvr ad th aprtur; ad m ar costats that dpd o th siz ad shap of th aprtur, ad ε is th prmittivity. Part Equivalt Polarizatio Currts Th lctric ad magtic dipol momts ar P E ˆ ( xx P H ( xx m m t Th quivalt lctric ad magtic polarizatio currts ar J P M P H zˆ ( x x m m z i i t whr th wav is assumd to b siusoidal tim-dpdt ( x i idicats th positio of th ith couplig hol ad H z is th amplitud of th tagtial magtic fild. 5 5

14 Part ythsiz th dsird mod: th rciprocity thorm (, Pozar 9.9 Th currt sourcs ust iducd might xcit dsird mod as wll as parasitic mods. this sctio w provid a mthod to show how th dsird mod is sythsizd ad how th parasitic mods ar supprssd. A usful thorm i lctromagtism is to b mtiod---th rciprocity thorm. Two sts of currt sourcs: ( J, M ad ( J, M Thir corrspodig filds: ( E, H ad ( E, H Assum M to b th oly ozro currt sourc. Th, th rciprocity thorm rads: ( E H E H ds ( M H dv 53 Part ythsiz th dsird mod: th rciprocity thorm ( ic J M, th corrspodig E ad Har assumd to b crtai circular TE m wavguid mod. E ad H ar th lctric ad magtic filds du to M. Th rsultig lctric ad magtic filds i th cylidrical wavguid ca. b xprssd as: ˆ z z z ˆ z z z E A H A h h zˆz h zˆhz whr ad ar th ormalizd filds of th th mod ad A rprsts th amplitud of th th mod. 54 Part (a TE (b TE (TE (c TE4 ythsiz th dsird mod: How may dipols ar dd? id viw Top viw. id viw of th magtic fild ad top viw of th lctric fild for (a TE mod, (b TE mod, ad (TE (c 4 mod. Th fild profils ar calculatd usig HF. Two coupligs ca xcit TE mods, whil four coupligs with propr phas cotrol ca xcit ithr TE or TE 4 mod. 55 Part ythsiz th dsird mod: Dmostratio How th magtic dipols ar formulatd? Th cas for th TE mod is dmostratd. 3 Pm mhzzˆ ( ( zz ( ( ( ( whr H z is th tagtial H fild (alog z dirctio ad α m is a costat dpdig. o th gomtry of th hol. Four magtic dipols vly distributd i th circumfrc ar assumd. With th lgthy calculatio, w would fid th solutio for th amplitud of th th mod: 3 A ( zh P d H h ( h ( h ( h ( ˆ z z m m z z z z z P P whr P (( h zˆ ds is a ormalizatio costat proportioal to th powr flow of th th mod ad h ( ( Asi mbcos m J ( p. z m m 56

15 Part Aalyz mod purity Usig th abov mtiod approachs, w could sythsiz th dsird mod. Howvr, som uwatd mods may ivitably b gratd ad rsult i a srious mod-comptitio problm i a gyrotro xprimt. Thrfor, mod purity is a importat issu i th dsig of a mod covrtr. TE mod P H ( ( zz [ ( ( ] m m z z ˆ A H BJ p TE,B m z m ( m P TE,A TE,A TE,B TM A, A A, A Th arragmt of dual iputs oly xcits th liar polarizatio of th TE mod,.g., ad luckily, it limiats o two-fold symmtric mods. o th maor parasitic mod TE mod caot b xcitd. 57 Part Aalyz mod purity Th fild pattr of TE mod is azimuthally symmtric. W ca choos dual fds or quad fds. Quad-fd structur is sufficit to limiat ths two uwatd mods. Th magtic dipols ca b xprssd as: TE mod 3 Pm mhz( ( zz zˆ [ ( ( ( ( ] 4 TE ( ˆ z B A zhz Pm dv mhz J( p P P ATE,A ATE,B, ATE, A ATE, B A, A A TM TM,A TM,B Th quad-fd arragmt avoids xcitig th parasitic mods. o th mod purity would b vry 58 Part Aalyz mod purity Th fild pattr of TE 4 mod suggsts a quad-fd structur. Th magtic dipols ca b xprssd as: TE 4 mod 3 Pm mhz( ( zz zˆ [ ( ( ( ( ] 4B ATE4,B mhz J4( p4 P 4 4B ATE mhz J( p P total k 4 P4 ( p J4 ( p4 6 ( mhz J4 ( p4 kc,4 4 ttl total k P ( p 4 J ( p 6 ( mhz J ( p k c, total total P p ( p J ( p Th powr ratio is: P p ( p 4 J ( p 59 Part (a TE purity (db Mod imulatio Rsults Usig HF (b TE - (c TE 4 ty (db Mod purit Mod purity (db B trasmissio rflctio Frqucy (GHz -4-6 trasmissio trasmissio rflctio Frqucy (GHz TM rflctio TE Frqucy (GHz Calculatd fild pattr ad th mod purity for (a TE mod, (b TE mod, ad (c TE 4 mod. Th trasmissios of th dsird mods ar show i blu lis ad dth rflctios ar show i dashd lis. Th mior ad maor parasitic mods ar show i gray ad rd, rspctivly. 6

16 Part Dsig ad Fabricatio Part Exprimtal tup (a TE (b TE (c TE4 Two idtical covrtrs oid back-to-back for th thr mods of fitrst. t Each st cosists of thr pics mad of oxyg- fr high-coductivity coppr. (a had modul (b hp 85c DUT had modul (a Photo of th xprimtal stup for dirctly masurig th back- to-back trasmissio. Th tst st is coctd to th had moduls which ar calibratd. Th tst st is largd. (b ( lottd plat of TE covrtr as a xampl. 6 6 Part (a TE (b TE Trasm missio (db (c TE 4 imulatd ad Masurd Rsults GHz -4 Th. cu Th. cu -5 Th. cu Exp o (db Trasmissi Tra smissio (d db Frqucy (GHz Th. cu 4. GHz Th. cu Th. cu Exp Frqucy (GHz Th. cu Th. cu. GHz Th. cu Exp Frqucy (GHz Fild pattr ad trasmissio for (a TE, (b TE, ad (c TE 4, rspctivly. Th fild pattrs ar HF s simulatio rsults. Th solid dt dots rprst tth masurd rsults ad th lis ar th simulatios. Thr diffrt rsistivity of th coppr ar displayd. 63 Part Coclusio Usig -typ powr dividr to xcit pur TE m mods was rportd. Thr mod covrtrs wr dsigd, fabricatd, ad tstd. Ths covrtrs fatur a high back-to-back covrtig fficicy, high mod purity, broad badwidth, ad compact covrtig sctio. uch a covrtr is suitabl for a varity of applicatios, spcially th gyrotros to grat low-trahrtz radiatio [34]. At highr frqucy, lik trahrtz rgio, th micro-fabricatio tchiqu is d, which is currtly udr ivstigatio. Th authors would lik to thak th tchical support of Mr. C. L of Asoft, Taiwa Brach. T. H. Chag, C. H. Li, C. N. Wu, ad C. F. u, " Gratig pur circular TE m mods usig -typ powr dividrs", EEE Tras. Microwav Thory Tch. 58, 543 (. 64

17 g{x XÇw Éy {tña G 65