1. Advanced Transmission Line Theory

Transcription

1 1 1. Avanc Tansmissin Lin T p://psna.mmu.u.m/~wlung/ads/as.m T infmain in is w as bn bain fm sucs bliv b liabl. T au s n guaan accuac cmplnss f an infmain psn in an sall n b spnsibl f an s missins amags as a sul f us f is infmain. Jun Fabian Kung Wai L 1 Pfac Tansmissin lins an wavguis a ms impan lmns in micwav RF cicuis an ssms. Tansmissin lins an wavguis a us cnnc vaius cmpnns g fm a cmpl cicui. Tis is simila lw fqunc cicui w w us wis cpp ac cnnc vaius cmpnns in an lcnic cicui. In aiin u will s la a man ps f micwav cmpnns a fabica fm s scins f ansmissin lins wavguis. F s asns a l f mpasis is plac n unsaning bavi f lcmagnic fils in ansmissin lins an wavguis. Tansmissin lin Jun Fabian Kung Wai L

2 Rfncs [1] R.. Cllin Funain f micwav ngining n iin 199 McGaw-Hill. A v avanc an in-p b n micwav ngining. Difficul a bu infmain is v cmpnsiv. A classic w. Rcmmn. [] D. M. Pa Micwav ngining n iin 1998 Jn-Wil & Sns. (3 iin 005 is als availabl fm Jn-Wil & Sns). asi a an unsan. Als a g b. Rcmmn. [3] S. Ram J.R. Winn T.D. Van Du Fil an wavs in cmmunicain lcnics 3 iin 1993 Jn-Wil & Sns. G cvag f M wi mpasis n applicains. [4] C. R. Paul Inucin lcmagnic cmpaibili Jn- Wil & Sns 199. Jun Fabian Kung Wai L 3 Rfncs Cn... [5] F. Kung Mling f ig-sp pin cicui ba. Mas g issain 1997 Univsi Malaa. p://psna.mmu.u.m/~wlung/mas/msis.m [6] F. Kung unpublis ns an ws. Jun Fabian Kung Wai L 4

3 3 1.0 Rviw f lcmagnic (M) Fils Jun Fabian Kung Wai L 5 lcic an Magnic Fils (1) In an lcnic ssm suc as n PCB assmbl a lcic cags (q). T ma u lcnic ssm ws w ssniall cnl lcic cags ( cag nsi an a f flw n vaius pin in cicui). Flw f lcic cags u pnial iffnc (V) pucs lcic cun (I). Asscia wi cag is lcic fil () an wi cun is magnic fil (H) * cllcivl call lcmagnic (M) fils. q 1 Fc F Ts cag q F I B F q Ts q q cun I Fc 1 1 4πε F Culmb s Law T c an fil w us an lcic cag. T c H fil w us a cun lp I *T magnic fil is B µh Jun Fabian Kung Wai L H is magniain. 6 H

4 4 lcic an Magnic Fils () fils Cnuc - q fils b cnvnin is ic fm cnuc wi ig pnial cnuc wi lss pnial. Dicin inicas fc pinc b a small s cag accing Culmb s Fc Law. Dnsi f fil lins cspns sng f fil. Fc n a s cag q F 1 Qq ) 4 πε Cnuc H fils H fils b cnvnin is ic accing ig-an ul. Dicin inicas fc pinc b a small s cun accing Ln s Fc Law. ( v B) F q Dnsi f fil lins cspns sng f fil. Jun Fabian Kung Wai L 7 I -I Cnuc Cnuc Mawll quains (Lina Mium) - Tim- Dmain Fm (1) Faaa s law B H J D D ρv B 0 Cnsiuiv lains Gauss s law W: H H J J ρ v ρv Mifi Amp s law N nam bu can b call Gauss s law f magnic fil F lina mium ac paam pns n 4 inpnn vaiabls ( ) ( ) ( ) ( ) H ( ) H ( ) ( ) J ( ) J ( ) ( ) Uni vc in -icin cmpnn lcic fil innsi H Auilia magnic fil D lcic flu B Magnic fil innsi J Cun nsi ρ v Vlum cag nsi ε pmiivi f f spac ( ) µ pmabili f f spac (4π 10-7 ) ε laiv pmiivi µ laiv pmabili Jun Fabian Kung Wai L 8

5 5 Mawll quains (Lina Mium) - Tim-Dmain Fm () Mawll quains as swn a acuall a cllcin f 4 paial iffnial quains (PD) a scib psical lainsip bwn lcmagnic (M) fils cun an lcic cag. T Dl pa is a san f -imnsinal (3D) iffniain: ( ) F insanc cnsi 1 s an 3 Mawll quains: Cul ~ ~ Divgnc ( B B B ) ρ ε Jun Fabian Kung Wai L 9 Gain F F ) F F ) T ul unsans is subc an als RF/Micwav cicui sign n ns av a sng gasp f lcmagnism (M). Ra fncs [1] [3] an g b n M. a: Wav Funcin an Pas (1) f ( ω ) f ( ) A gnal funcin scibing ppagaing wav in icin f ( ω ) f ( ) ( ) ( ω ) f (ω ) Wn im incass b w s a w mus incas all psiin b mainain sap. In ssnc wavfm avls in icin. Ts ms mus cancl ff i.. psiiv F a wav funcin in icin: f ω ( ) Tim Dicin f avl W s a sap mvs b wiin a pi f us pas vlci v p : ω ω ω v ( ) ( ) Jun Fabian Kung Wai L 10 p ω

6 6 a: Wav Funcin an Pas () An ampl: v( ) V cs( πf ) f 1. 0MH 1 A sinusial wav v() Pas Vlci: v p ω πf wavlng λ π Jun Fabian Kung Wai L 11 a: Wav Funcin an Pas (3) In man was fqunc f s n ca muc infmain. If a lina ssm is ci b a sinusial suc wi fqunc f w nw spns a v pin in ssm will b sinusial wi fqunc f. I is pas cnsan wic cais m infmain i mins vlci an wavlng f wav. Tus i is m cnvnin if w cnv pssins f M fils in pas Tim-Hamnic fm as swn blw: ul s fmula α cs α sin α 1 cs ( ωm ) ( f ) v( ) V cs π ( ) { ω m ω m } cs R Cnvnin: small l f im-main fm capial l f pas. Using ul s fmula m Jun Fabian Kung Wai L 1 M cmpac fm V( ) V Pas f v()

7 7 Jun Fabian Kung Wai L 13 a: Wav Funcin an Pas (4) Wav funcin an pas nain is n nl applicabl quaniis li vlag cun cag. I is als appli vc quaniis li an H fils. F insanc f sinusial fil avling in icin: T pas is givn b: Finall if w subsiu pas fm f H J an ρ in im-main Mawll s quains w wul bain Mawll s quains in im-amnic fm. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ω ω ω ω ω R cs cs cs cs ( ) ( ) ( ) ( ) Ε ω Pan funcin ( pnn) Ppgaing funcin Jun Fabian Kung Wai L 14 Mawll quains (Lina Mium) - Tim- Hamnic Fm (1) F sinusial vaiains wi im w subsiu pass f H J an ρ in Mawll s quains sul a Mawll s quains in im-amnic fm. 0 B ρ D D J H B v ω ω ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ρ ρ J J J J H H H H v v W: F lina mium Cnsiuiv lains lcic fil innsi H Auilia magnic fil D lcic flu B Magnic fil innsi J Cun nsi ρ v - Vlum cag nsi ε pmiivi f f spac ( ) µ pmabili f f spac (4π 10-7 ) ε laiv pmiivi µ laiv pmabili (1.a) (1.b) (1.c) (1.) ω ac paam pns n 3 inpnn vaiabls

8 8 lcmagnic Spcum AM FM Micwavs LF VLF LF MF (MW) HF (SW) VHF UHF SHF HF IR 100 H 10 H 100 H 1 MH 3 MH 30 MH 300 MH 1 GH 30 GH 300 GH L S C X Ku K Ka mm 1 GH GH 4 GH 8 GH 1 GH 18 GH 7 GH 40 GH 300 GH Jun Fabian Kung Wai L 15 A Lil Pspciv ~ ~ µ H ~ ~ B µ J µε ~ ρ ε ~ B 0 ~ ρ J ~ Vlag/Pnial Cun Inucanc Capacianc Rsisanc Cnucanc Kicff s Vlag Law (KVL) Kicff s Cun Law (KCL) lcnics & Miclcnics Infmain Cmpu & Tlcmmunicain Cnsvain f cag Quanum Mcanics /Psics Cmis Jun Fabian Kung Wai L 16

9 9.0 Inucin Tansmissin Lin Cncps Jun Fabian Kung Wai L 17 Dfiniin f lcical Incnnc Incnnc - mallic cnucs a is us ansp lcical ng fm n pin f a cicui an. ampl: Cnuc Incnnc Aial Dicin Tansvs Plan Tus cabls wis cnuciv acs n pin cicui ba (PCB) scs pacaging mallic ubs c. a all ampls f incnnc. 1. Usuall cnains m cnucs fm a cls cicui.. Cnucs assum b pfc lcic cnucs (PC) Ribbn cabl Caial cabl Sc Wavgui Jun Fabian Kung Wai L 18 PCB Tacs

10 10 S Incnnc Lump Cicui F s incnnc mmn swic is cls a vlag will appa acss R L as cun flws ug i. T ffc is insananus. Vlag an cun a u lcic cag mvmn alng incnnc. Asscia wi lcic cags a saic lcmagnic (M) fil in spac suuning s incnnc. T s incnnc ssm can b ml b lump RLC cicui. V s M fil is saic quasisaic. - L lcic cag C Jun Fabian Kung Wai L 19 G R L R Saic M fil cangs unifml i.. wn fil a n pin incass fil a lcains als incass. Again w n sss a picall valus f RLCG a v small a lw fqunc i ffc can b simpl ign. Lng Incnnc (1) If incnncin is lng i as sm im f vlag an cun appa n sis R L af swic is cls. lcic cags mv fm V s sis R L. As cags mv is an asscia M fil wic avls alng wi cags. In ffc is a ppagaing M fil alng incnnc. T ppagaing M fil is call a wav an incnnc is guiing M wav is M fil is namic. Sinc an abia wavfm can b cmps in is sinusial cmpnns l us cnsi V s b a sinusial suc. Lng incnnc: Wn is an appciabl la bwn inpu an upu V s - Wiu mal cnucs M wavs will isps i.. aia u in spac. Jun Fabian Kung Wai L 0 L R L

11 Lng Incnnc () A simpl animain Psiiv cag H fil fil Aial Dicin Tansvs Plan I L V L V s - R L 3 n Jun Fabian Kung Wai L 1 Lng Incnnc (3) T cspning M fil gna wn lcic cag flws alng incnnc is als sinusial wi spc im an spac. T M fils caacisics a ica b Mawll s quains. L lcic fil () Rmmb a cun is u flw f f lcns. an H a als sinusial. Bavi f an H a ica b Mawll s quains: T Magnic fil (H) 1/T fqunc Jun Fabian Kung Wai L 11

12 1 Jun Fabian Kung Wai L 3 Lng Incnnc (4) lcic fil () Magnic fil (H) L Ppagaing M fils ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) H B analing Mawll s quains Wav quains (Appni 1) Snaps f M fils a a cain insan in im n ansvs plan Jun Fabian Kung Wai L 4 Vlag an Cun n Incnnc (1) Vlag pnial iffnc is ng n bing 1 Culmb f lcic cag fm a fnc pin (GND) an (signal). Cun is a f flw f lcic cag acss a sufac. Fm Culmb s Law an Amp s Law in lcmagnism s w quaniis a la an H fils. W a ins in ansvs vlag V an cun I as swn. Lp 1 ( ) Lp1 l H I I () V () ( ) b a l V a b

13 13 Vlag an Cun n Incnnc () V Pnial iffnc bwn w pins n ansvs plan an I Ra f flw f lcic cag acss a cnuc sufac. V an I pns n insananus an H fils n incnnc an cspn w w wul masu m psicall wi pbs V an I will b uniqu (.g. n pn n masumn sup bu nl n lcain) if an nl if M fil ppagain m in incnnc is TM quasi-tm. Masuing vlag Lin ingain pa f H fil Lin ingain pa f fil S iscussin in Appni 1: Avanc Cncps Fil T Sluins f m infmain. Tansvs plan Masuing cun Jun Fabian Kung Wai L 5 Dmnsain - lcmagnic Fil Ppagain in Incnnc (1) T fllwing ampl simula bavi f M fil in a simpl incnncin ssm. T ssm is a 3D ml f a cpp ac wi a plan n bm. A numical m nwn as Fini-Diffnc Tim-Dmain (FDTD) is appli Mawll s quains pvi appima valu f an H fils a slc pins n ml a v 1.0 picscn inval. (Sac WWW s p://psna.mmu.u.m/~wlung/p/psis.m) Fil valus a ispla a an inval f 5.0 picscns. FR4 ilcic Cpp ac 50Ω sisiv vlag suc cnnc 100Ω SMD sis GND plan Jun Fabian Kung Wai L 6

14 14 Magniu f in ilcic Innsi Scal Dmnsain - lcmagnic Fil Ppagain in Incnnc () 19.mm PCB ilcic: FR4 ε 4.4 σ 5. Ticnss 1.0mm. 0.75mm 0.5mm 0.8mm Filnam: lin1_xzplan.avi V s V 3.0V 50ps - HIGH 100ps Sw simulain using CST Micwav Jun 008 Sui 006 Fabian Kung Wai L 7 Dmnsain - lcmagnic Fil Ppagain in Incnnc (3) Pb f signal gna V 3.0V 50ps HIGH 100ps Magniu f in YZ plan Vls Filnam: lin1_yzplan.avi Jun Fabian Kung Wai L 8

15 15 Dfiniin f Tansmissin Lin A ansmissin lin is a lng incnnc wi cnucs signal cnuc an gun cnuc f uning cun. Mulicnuc ansmissin lin as m an cnucs usuall a fw signal cnucs an n gun cnuc. Tansmissin lins a a subs f a ba class f vics nwn as wavgui. Tansmissin lin as a las m cnucs wil wavguis f cllcivl an sucus a can allw M wavs ppaga alng sucu. Tis inclus sucus wi nl 1 cnuc n cnuc a all. Wil nwn wavguis inclu cangula an cicula wavguis f ig pw micwav ssm an pical fib. Wavgui is us f ssm quiing (1) ig pw () v lw lss incnnc (3) ig islain bwn incnncs. Tansmissin lin is m ppula an is wil us in PCB. Fm nw n w will b cncnaing n ansmissin lin Tlin f s. Jun Fabian Kung Wai L 9 Tpical Tansmissin Lin Cnfiguains Caial lin Micsip lin Siplin Ts cnucs a psicall cnnc smw in cicui Cnuc Dilcic Sil micsip lin Tw-wi lin Paalll pla lin C-plana lin Sl lin Jun Fabian Kung Wai L 30

16 16 Sm Muli-cnuc Tansmissin Lin Cnfiguains Cnuc Dilcic Jun Fabian Kung Wai L 31 Tpical Wavgui Cnfiguains Rcangula wavgui Cicula wavgui Opical Fib Dilcic wavgui Jun Fabian Kung Wai L 3

17 17 ampls f Micsip an C-plana Lins Micsip C-plana Jun Fabian Kung Wai L 33 Lng S Incnnc? T Wavlng Rul-f-Tumb Hw w min if incnnc is lng s i.. la bwn inpu an upu is appciabl? Rlaiv wavlng f sinusial signals. Rul-f-Tumb: If L < 0.05λ i is a s incnnc wis i is cnsi a lng incnnc. An ampl a n f is scin will illusa is pcu clal. L W call is 5% ul. Lss cnsvaiv sima will us 1/ ( 10% Rul) v p Pas vlci ppagain vlci fλ fqunc wavlng f λ f λ Incnnc (1.1) Jun Fabian Kung Wai L 34

18 18 Dmnsain Lng Incnnc 19.mm 5.8GH 3.0V Magniu f in YZ plan ac fam is ispla a 5psc inval 50mA L us assum M wav avls a sp f lig C n wavlng 5.0 mm λ C f 19. mm is ga an 5% f 5 mm Cun pfil -50mA Filnam: lin1_yzplan_5_8gh.avi Jun Fabian Kung Wai L 35 Dmnsain S Incnnc 19.mm 0.4GH 3.0V L us assum M wav avls a sp f lig C n wavlng mm Magniu f in YZ plan A an insan in im cun pfil is alms unifm alng aial icin. Incnnc Cun can b cnsi pfil lump. 50mA -50mA Filnam: lin1_yzplan_0_4gh.avi Jun Fabian Kung Wai L 36

19 Ppagain Ms Jun Fabian Kung Wai L 37 Tansvs an H Fil Pans Fil pans li in Tansvs Plan. fils Tansvs plan H fils Jun Fabian Kung Wai L 38

20 0 Nn-ansvs an H Fil Pans Fil pans s n li in Tansvs Plan. Fil cnains -cmpnn fils H fils Fil cnains -cmpnn Jun Fabian Kung Wai L 39 Ppagain Ms (1) Assuming ansmissin lin is paalll icin. T ppagain f an H fils alng lin can b classifi in 4 ms: T m - w 0. TM m - w H 0. TM m - w an H a 0. Mi m an miu f abv. ± H ( ) ( ) ± m ( ( ) ( ) ) A Tlin can supp a numb f ms a an insanc wv T TM mi m usuall ccu a v ig fqunc. T is an m nwn as quasi-tm m wic is supp b siplin sucus wi nn-unifm ilcic. S iscussin in Appni 1. Jun Fabian Kung Wai L 40 m ( )

21 1 Ppagain Ms () H H TM m H TM m H T m Mi ms Jun Fabian Kung Wai L 41 ampls f Fil Pans Ms TM quasi-tm m fil H fil Jun Fabian Kung Wai L 4

22 Appni 1 Avanc Cncps Fil T Sluins f Tansmissin Lins Jun Fabian Kung Wai L 43 Fil T Sluin T nau f an H fils in spac bwn cnucs can b sui b slving Mawll s quains Wav quains (wic can b iv fm Mawll s quains) (S [1] [] [3]). Assuming cniin f lng incnncin sluins f an H fils a ppagaing fils wavs. W assum im-amnic M fils wi ω pnnc an wav ppagain alng psiiv an ngaiv -ais. Mawll quains ωµ H H J ωε ρ ε H 0 Buna cniins Wav quains 0 H H 0 ω F insanc angnial fil cmpnn n PC mus b cninui f an H fil cmpnns acss iffn ilcic maial c. Jun Fabian Kung Wai L 44 εµ (A.1) In f spac

23 3 a: Diving Hml Wav quains Fm Mawll quains Pfming cul pain n Faaa s Law ωµ H : ( ) ( ) ωµ ( H ) vc calculus ini ρ ω µε ωµ J ( ) A ( A) ε Ts a sucs f fil N: us wll-nwn A In f spac is n lcic cag an cun: N: Tis ivain is ω µε 0 vali f im-amnic cas Simila pcu can b us bain un lina mium nl. S m avanc f H ω µεh J gnal wav quain. F ampl: O in f spac C. A. Balanis Avanc ngining lcmagnics Jn-Wil H H 0 ω µε Jun Fabian Kung Wai L 45 Obaining pssins f an H (1) Assuming an ina iffnial quain (OD) ssm as swn: OD 0 f ( 0) C1 an ( b) C T bain a sluin abv ssm (a sluin mans a funcin a wn subsiu in OD will caus lf an ig an si b qual) man appacs can b us (f insanc s. Ksig Avanc ngining mamaics 1998 Jn Wil). On ppula appac is Tial-an-/subsiuin m w w guss a funcinal fm f () as fllws: ( ) Subsiuing is in OD: ( ) ( ) [ 0 b] 0 0 ± w 1 Sinc is is a n OD w n inuc unnwn cnsans A an B an a gnal sluin is: A B (1) ( ) T Dmain Buna cniins Ta ial-an- m ws is aibu Uniqunss Tm f lina OD. Jun Fabian Kung Wai L 46

24 4 Jun Fabian Kung Wai L 47 Obaining pssins f an H () T fin A an B w n us buna cniins. Slving (a) an (b) f A an B: S uniqu sluin is: ( ) ( ) C B A C b C B A C b b (a) (b) ( ) ( ) ( ) b C C b C C b b b b B C A B C B B C sin 1 sin ( ) ( ) ( ) b C C b C C b b sin sin 1 1 (q...) (3a) (3b) Jun Fabian Kung Wai L 48 Obaining pssins f an H (3) T sam appac can b appli Wav quains Mawll quains f Tlin. Cnsi Wav quains (A.1) in im-amnic fm. T unnwn funcins a vc pass () an H(). T iffnial quain f in Casian cina is: Tis is call a Paial Diffnial quain (PD) as ac an pns n 3 vaiabls wi iffniain subsiu b paial iffnial. T a 3 PDs if u bsv cafull. F -cmpnn is is: Bas n pvius OD ampl an als fac a w pc fil avl alng -ais fllwing fm is suggs: ( ) 0 0 ( ) ( ) ( ) 0 A funcin f an T pnn!!!

25 5 Jun Fabian Kung Wai L 49 Obaining pssins f an H (4) Caing n in is mann f an -cmpnns w aiv a fllwing fm f fil. Nic a up nw w av n slv Wav quains bu ml min funcinal fm f is sluin. W sill n fin u wa is () () () an. Using simila appac n will il simila pssin f H fil. ( ) ( ) ( ) ( ) ( ) ( ) (A.1a) ( ) 0 H ( ) ( ) ( ) ( ) ( ) ( ) H v v (A.1b) Jun Fabian Kung Wai L 50 an H fils pssins (1) ( ) ( ) ( ) ( ) ( ) ( ) Tus ppagaing M fils gui b Tlin can b win as: ( ) ( ) ( ) ( ) ( ) ( ) H Tansvs cmpnn Aial cmpnn ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) H M fils Ppagaing In icin M fils Ppagaing In - icin (A.a) (A.b) (A.3a) (A.3b) supscip inicas ppagain icin

26 6 Jun Fabian Kung Wai L 51 an H fils pssins () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) cs cs cs R ω ω ω ω W can cnv pas fm in im-main fm f insanc f fil ppagaing in icin: W ( ) ( ) ( ) ( ) Uni vc ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ω ω ω cs cs cs (A.5b) (A.4) (A.5a) Jun Fabian Kung Wai L 5 an H fils pssins (3) Usuall n nl slvs f fil cspning H fil pas can b bain fm: T pw cai b M fils is givn b Pning Tm: H H ωµ ωµ 1 ( ) s s H P S S * * R 1 R 1 S ( ) ( ) s s s H P S S S * * * R 1 R 1 R 1 Psiiv valu mans a pw is cai alng ppagain icin. (A.6) Psiiv Z icin Ngaiv Z icin

27 7 an H fils pssins (4) T a asns f csing sign cnvnins f v an -v ppagaing wavs as in (A.) an (A.3). S a 0 f b v an -v ppagaing fil (cnsisnc wi Mawll s quains). T ansvs magnic fil mus cang sign upn vsal f icin f ppagain bain a cang in icin f ng flw ( ) 0 F v icin F -v icin Jun Fabian Kung Wai L 53 Pas Vlci I is as sw a quain (A.a) an (A.b) scibs avling fil wavs (als f H). T sp w an H fils avl is call Pas Vlci v p. Pas Vlci pns n ppagain m ( b iscuss la) fqunc an psical ppis f incnnc. v p ω (A.7) Jun Fabian Kung Wai L 54

28 8 Wavlng F incnnc ci b sinusial suc if w f im a a cain insan sa an H fils pfil will va in a sinusial mann alng -ais. λ ( ) Wavlng λ f v p ω λ π ω (A.8) π Jun Fabian Kung Wai L 55 Suppsiin Tm A an insan f im a an H fils ppagaing in psiiv an ngaiv icin alng ansmissin lin. T al fils a a suppsiin f psiiv an ngaiv ic fils: H H H A pical fil isibuin a a cain insan f im f css scin f w incnncs (w-wi an c-aial cabl) is swn blw: Cnucs H Jun Fabian Kung Wai L 56

29 9 Jun Fabian Kung Wai L 57 Fil Sluin (1) T fin valu f an funcins w subsiu quains (A.a) an (A.b) in Mawll Wav quains. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) H 0 εµ ω H H 0 Buna cniins 0 H J H H ε ρ ωε ωµ Mawll quains Wav quains Jun Fabian Kung Wai L 58 Fil Sluin () H ωµ H ωε ωµ ωµ ωµ (A.10a) ωε ωε ωε (A.10b) (A.10c) (A.10) (A.10) (A.10f) T pcu ulin fllws s fm Pa []. Assum Tlin wavgui ilcic gin is suc f. Fm Mawll s quains: Subsiuing suggs sluin f () f (A.a) in (A.9) an paning iffnial quains in an cmpnns: (A.9) J B ~ ~ ~ ωµε µ 0 J

30 30 Jun Fabian Kung Wai L 59 Fil Sluin (3) Fm (A.10a)-(A.10f) w can pss in ms f an : c ωε c ωε c ωµ c ωµ λ π µε ω c (A.11a) (A.11b) (A.11c) (A.11) (A.11) Ts quains scib cmpnns f gnal M wav ppagain in a wavguiing ssm. T unnwns a () an () call Pnial in liau. S b b Cllin [1] Cap 3 f alnaiv ivain Jun Fabian Kung Wai L 60 T M Summa (1) F T m 0 (Smims is is call H m). W cul caaci Tlin in T m b M fils: Fm wav quain f H fil: ( ) H m ± ± m ± εµ ω H H 0 ( ) ( ) 0 N Onl s a n. T ansvs fil cmpnns can b iv fm 0 0 c c c Fm (A.11) ( ) m m Using fac a Tansvs Laplacian pa

31 31 Jun Fabian Kung Wai L 61 T M Summa () Sing 0 in (A.11a)-(A.11): Ts quains plus pvius wav quain f nabl us fin cmpl fil pan f T m. 0 c c buna cniins f an H fils (A.1b) c c c ωµ c ωµ Fm pvius sli (A.1a) Jun Fabian Kung Wai L 6 T M Summa (3) Fm (A.1a) an (A.1b) w can sw a: Tf w cann fin a uniqu vlag b (bu w can fin a uniqu cun): Als fm (A.1a) w can fin a wav impanc f T m. 0 ωµ 1 C C l V T Z Z (A.13) 0 ( ) c c c ωµ ωµ ωµ

32 3 Jun Fabian Kung Wai L 63 TM M Summa (1) F TM m 0 (Smims is is call m). W cul caaci Tlin in TM m b M fils: Fm wav quain f fil: H m ± ± ( ) m ± ± εµ ω 0 ( ) ( ) 0 Z 0 0 c c c Onl s a n. T ansvs fil cmpnns can b iv fm Jun Fabian Kung Wai L 64 TM M Summa () Sing 0 in (A.11a)-(A.11): Ts quains plus pvius wav quain f nabl us fin cmpl fil pan f TM m. c ωε c ωε c c (A.14a) buna cniins f an H fils (A.14b) 0 c c Fm pvius sli

33 33 Jun Fabian Kung Wai L 65 TM M Summa (3) Similal fm (A.14a) an (A.14b) w can sw a W cann fin a uniqu cun b (bu w can fin a uniqu vlag): Als fm (A.14a) w can fin a wav impanc f TM m. 0 ωε C l I Z TM ωε (A.15) 0 Jun Fabian Kung Wai L 66 TM M (1) TM m is paiculal impan caaci b 0. F v ppagaing wavs: Sing 0 in (A.11a)-(A.11) w bsv a c 0 in f a nn- sluin is. Tis implis: Appling Hml Wav quain fil: m ± H m ± ± (A.16a) (A.16b) µε ω ( ) ( ) 0 0 (A.17a) 0 0

34 34 TM M () T sam can b swn f : J (A.17b) quain (A.17a) is simila Laplac quains in D. Tis implis ansvs fils is simila saic lcic fils a can is bwn cnucs s w cul fin a ansvs scala pnial Φ: Φ( ) Tansvs pnial Φ( ) 0 (A.18) Als n a: S [1] Cap 3 f alnaiv ivain ( Φ) 0 (A.19) Using an impan ini in vc calculus ( F ) 0 W F is abia funcin f psiin i.. F F(). Jun Fabian Kung Wai L 67 Alnaiv Viw (TM M) Alnaivl fm (A.10c) an nwing a 0 in TM m: ωµ 0 Divgnc in D (in XY plan) Fm wll nwn Vc Calculus ini ( F ) 0 w n psula isnc f a scala funcin Φ() w Φ( ) Fm (A.10f) w can als sw a (in f spac): ωε 0 Jun Fabian Kung Wai L 68

35 35 TM M (3) Nmall w wul fin fm (A.17a) n w iv fm : In f spac 0 0 H 1 ωµ 1 Z 1 µ ε ( ) ( ) H 1 ωµ 1 ωµ 0 (A.0a) cis: s if u can iv is quain Z Ininsic impanc f f spac Z TM Wav impanc f TM m TM An impan bsvain is a un TM m ansvs fil cmpnns an fulfill simila quains as in lcsaic. µ Z Z ε (A.0b) Jun Fabian Kung Wai L 69 Vlag an Cun un TM M Du 0 an 0 in spac suuning cnucs w cul fin uniqu ansvs vlag (V ) an ansvs cun (I ) f ssm fllwing sana finiins f V an I. T V an I s fin s n pns n sap f ingain pa. I () V () S m ail vsin f is n s fncs [1] & []. V I l Lp1 b l a ( ) H ( ) b a Lp 1 Jun Fabian Kung Wai L 70

36 36 C a: Inpnnc f V an I fm Ingain Pa un TM M Lp L Aa S Using S s Tm l s C 1 0 s S L1 L1 l L l L l L l 0 L1 L l l 0 l V Lp L Sinc sap f lp L is abia as lng as i sas in ansvs plan pas L 1 L an nc ingain pa f V is abia. Simila pf can b cai u f I using lp as swn an 0 I l C 1 C Jun Fabian Kung Wai L 71 C C 1 TM M Summa (1) F TM m 0. T fin M fils f TM m: Slv Φ( ) 0 wi buna cniins f ansvs pnial. Fin fm ± m m [ Φ( ) ] Fin H fm ± 1 ± H ( ) Z W cul caaci Tlin in TM m b M fils: ± m ± m H ± ω µε Jun Fabian Kung Wai L 7

37 37 TM M Summa () O ug auilia cicui quaniis: ± m V V ± m I ± I T pw cai b M wav alng Tlin is givn b Pning Tm: P 1 R S P 1 R * ( H ) * s 1 RVI * ( ) 1 1 * Z II R( Z VV ) c c (A.1) S a n b F.Kung f pf Bcaus is alwas al cmpl (wn ilcic is lss) f all fquncis TM m alwas is fm na.c. ml ig fquncis. ω Jun Fabian Kung Wai L 73 µε Nn-TM Ms an V I (1) F nn-tm ms w cann fin b auilia quaniis V an I uniqul using sana finiin f vlag an cun (bcaus 0 ). F insanc in T m: 0 ωµ C Tus V C l will n b uniqu an will pns n lin 1 ingain pa. Tis mans if w amp masu vlag acss Tlin using an insumn aing will pn n wis an cnncin f pb! Fum f nn-tm ms: P 1 R * 1 R H s V I S ( ) * Tus w cann caaci a Tlin supping nn-tm ms using auilia quaniis suc as V an I. Jun Fabian Kung Wai L 74

38 38 Nn-TM Ms an V I () As an ampl cnsi TM m in micsip lin: c c V l C C c C Using pa C as swn in figu: ( ) C YH Y0 Un quasi-tm cniin wn 0 c als 0 n V will b a nn- valu. H V c 0 c 0 bcaus f buna cniin [ ( H ) ( 0) ] 0 In gnal is is u f abia Tlin an wavgui css scin. If w cs ingain pa an C w sill bain V 0 u 0 in TM m. Jun Fabian Kung Wai L 75 Cu-ff Fqunc f T/TM M Bcaus f T an TM ms: c T is a pssibili a bcms imagina wn c >. Wn is ccu T TM m M fils will ca pnniall fm suc. Ts ms a nwn as vanscn an a nnppagaing. Tus f T TM m is a pssibili f a cu-ff fqunc f c w f signal fqunc f < f c n ppagaing M fil will is. Jun Fabian Kung Wai L 76

39 39 Pas Vlci f TM T an TM Ms Pas vlci is ppagain vlci f M fil supp b lin. I is givn b: V ω p F TM m: V ω 1 p µε F T & TM m: V p ω 1 c (A.) ω 1 µε c (A.3) Tus w bsv a TM m is ininsicall nn-ispsiv wil T an TM m a ispsiv. Jun Fabian Kung Wai L 77 Final N n TM T an TM Ppagain Ms Finall n a fmula f TM T an TM ms appl all wavgui sucus in wic ansmissin lin is a subs. Jun Fabian Kung Wai L 78

40 40 ampl A1 - Paalll Pla Wavgui/Tlin T paalll pla wavgui is simpls p f wavgui a can supp TM T an TM ms. H w assum a W >> s a finging fil an vaiain alng can b ign. 0 Cnucing plas 0 Ppagain alng ais W Jun Fabian Kung Wai L 79 ampl A1 Cn... Div M fils f TM T an TM ms f paalll pla wavgui. Sw a TM m can is f all fquncis. Sw a T an TM ms pssss cu-ff fqunc f c w f paing fqunc f lss an f c suling M fil cann ppaga. Jun Fabian Kung Wai L 80

41 41 ampl A1 Sluin f TM M (1) TM m Sluin: Φ Φ Φ( ) ( 0) 0 ( ) V Φ Φ Φ 0 0 f 0 W 0 Buna cniins Sluin f ansvs Laplac PD: Φ Φ ( ) ( 0) Φ ( ) A B A 0 A 0 B V V B Φ 0 Gnal Sluin sinc Φ ( ) Φ( ) Tus ( ) V Φ Uniqu sluin Jun Fabian Kung Wai L 81 ampl A1 Sluin f TM M () Cmpuing an H fils: V V Φ V V V H 1 1 ε ωµ µ µ ε Cmpuing ansvs vlag an cun: V V l V 0 C1 C 1 C W I H l 0 C ε V µ ε V W µ Jun Fabian Kung Wai L 8

42 4 ampl A1 Sluin f TM M (3) Cmpuing pw flw (pw cai b M wav gui b wav -gui): 1 P R W 1 R W 1 R S H s V ε V ( ) ( ) µ V ε V ( )( ) * µ S 1 1 R R V 0 ε VW 1 * 1 [ ] [ ] V W V R V I µ µ W 0 ε µ V ε s Jun Fabian Kung Wai L 83 ampl A1 Sluin f TM M (4) Pas vlci v p f TM m: v ω ω 1 p ω µε µε T pas vlci is qual sp-f-lig in ilcic. Jun Fabian Kung Wai L 84

43 43 Jun Fabian Kung Wai L 85 ampl A1 Sluin f TM M (1) ( ) ( ) ( ) ( ) f 0 W c c ( ) ( ) Sluin f : ( ) ( ) ( ) ( ) B A c c c c cs sin 0 sinc Buna cniins Gnal sluin ( ) ( ) ( ) n c c c n n A A B B A π π 13L an 0 0 sin Tus: ( ) ( ) A n n π sin ( ) ( ) n n A π sin Appling buna cniins: Jun Fabian Kung Wai L 86 ampl A1 Sluin f TM M () ( ) ( ) ( ) ( ) A n n An n n c c c cs sin π π π ( ) ( ) n n A n cs π π Cmpuing ansvs M fils using (A.0a): ( ) ( ) H n n A n c c cs π π ε µ ωε ωε Sinc n is an abia ing TM m is usuall call TM n m.

44 44 ampl A1 Sluin f TM M (3) W can nw min nwing c : n c ω µε nπ ( ) Sinc TM m can nl ppaga if is al an smalls valu f Is 0 n wn 0: ω µε ( nπ ) 0 ω nπ πf µε f n µε Wn n 1 is psn cu-ff fqunc f TM m. f 1 cuff _ TM µε Jun Fabian Kung Wai L 87 ampl A1 Sluin f TM M (4) F abia n pas vlci v p f TM n m: v p ω ω ( nπ ) ω µε F f > f cuff w bsv a pas vlci v p is acuall ga an sp f lig!!! NOT: T M fils can avl a sp ga an lig wv w can sw a a f ng flw is lss an sp-f-lig. Tis a f ng flw cspns sp f pns if ppagaing M wav is a as a clus f pns. S a ns f pf. Jun Fabian Kung Wai L 88

45 45 Dminan Ppagain M F vaius ansmissin lin plg is a minan m. Tis minan m f ppagain is fis m is a lws paing fqunc. T scna ms will cm in isn a ig fquncis. T ppagain ms f Tlin pns n ilcic an css scin f ansmissin lin. F Tlin a can supp TM m TM m will b minan m as i can is a all fquncis ( is n cu-ff fqunc). Jun Fabian Kung Wai L 89 Tansmissin Lins Dminan Ppagain M Caial lin - TM. Micsip lin - quasi-tm. Siplin - TM. Paalll pla lin - TM TM (pns n mgnui f ilcic). C-plana lin - quasi-tm. N: Gnall f Tlin wi nn-mgnus ilcic Tlin cann supp TM ppagain m. Jun Fabian Kung Wai L 90

46 46 Quasi TM M (1) Lucil f plana Tlin cnfiguain ws minan m is n TM TM T minan ms can b appima b TM m a lw fqunc. F insanc micsip lin s n supp TM m. T acual m is TM. Hwv a a fw GH is muc small an an a i can b ign. W can assum m b TM wiu incuing muc. Tus i is call quasi-tm m. Lw fqunc appimain is usuall vali wn wavlng >> isanc bwn w cnucs. F pical micsip/siplin n PCB is can mans fqunc blw 0 GH lw. T an H cmpnns appac a lw fqunc an ppagain m appacs TM nc nwn as quasi-tm. Wn is appns w can again fin uniqu vlag an cun f ssm. S Cllin [1] Cap 3 f m mamaical illusain n is. Jun Fabian Kung Wai L 91 Quasi TM M () H 0 0 V an I N 0 0 V an I Ys H Nn-TM m 0 0 H Quasi-TM m V an I Ys TM m Jun Fabian Kung Wai L 9

47 47 a: W Inmgnus Sucus Ds N Supp Pu TM M (1) W will us Pf b Cnaicin. Supps TM m is supp. T ppagain fac in ai an ilcic wul b: ai ω µε i ω µεε M fils in ai will avl fas an in ilcic. F TM m ai < i v p( ai) ω > v p( i) ω ai i Nw cnsi buna cniin a ai/ilcic infac. T fil mus b cninuus acss buna fm Mawll s quain. amining cmpnn f fil: Ai ( ai) ai ( i) i Dilcic ( ai) i ( i ai ) ( i) ai Jun Fabian Kung Wai L 93 a: W Inmgnus Sucus Ds N Supp Pu TM M () Sinc lf an si is a cnsan wil ig an si is n. I pns n isanc pvius quain cann b fulfill. Wa is cnclu is a u iniial assumpin f TM ppagain m in inmgnus sucu is wng. S pu TM m cann b supp in inmgnus ilcic Tlin. Jun Fabian Kung Wai L 94

48 48 ampl A Minimum Fqunc f Quasi-TM M in Micsip Lin sima lw fqunc limi f micsip lin. H 1.6mm C λ C/f > 0H 3.0mm f < C/ GH f ciical 9.375GH H w plac >> sign wi quimn a wavlng > 0H. Yu can us lag limi as is is basicall a ul f umb. Tus bn f ciical quasi-tm appimain cann b appli. T ppagain m bn f ciical will b TM. A m cnsvaiv limi wul b us 30H 40H. Jun Fabian Kung Wai L 95 Summa f TM Quasi-TM T an TM Ms TM: H 0 ± m ± m H ± Can fin uniqu V an I. Quasi-TM: 0 H 0 ± m H ± ± m Can fin uniqu V an I. T: 0 H 0 ± m H ( ± ) ± m Cann fin uniqu I. TM: 0 H 0 ± m H ± ± m ( ± ) Cann fin uniqu V. Psical Tlin can b Ml b quivaln lcical cicui. Pas vlci. V ω 1 p µε N cu-ff fqunc. Nn-ispsiv Psical Tlin can b Ml b quivaln lcical cicui. Pas vlci. V p ω 1 µεff N cu-ff fqunc. Nn-ispsiv Psical Tlin cann b ml b quivaln lcical cicui. Pas vlci. v 1 p ω ω µε c Cu-ff fqunc. Dispsiv f c c π µε Psical Tlin cann b ml b quivaln lcical cicui. Pas vlci. v 1 p ω ω µε c Cu-ff fqunc. f c c π µε Dispsiv Jun Fabian Kung Wai L 96

49 49 W V an I is s Impan? (1) Wn w can fin vlag an cun alng Tlin ig-fqunc cicui f a ma n w can anal ssm using cicui insa f fil. Cicui is suc as KVL KCL -p nw a muc asi slv an Mawll quains wav quains. Hig-fqunc cicuis usuall cnsis f cmpnns wic a cnnc b Tlins. Tus micwav ssm can b ml b an quivaln lcical cicui wn minan m in ssm is TM quasi-tm. F is asn Tlin wic can supp TM quasi-tm is v impan. Jun Fabian Kung Wai L 97 W V an I is s Impan? () Tlin Amplifi XXX.09 Fil A cmpl psical ssm can b cas in quivaln lcical cicui. Pwful cicui simula ls can b us pfm analsis n quivaln cicui. Annna quivaln cicui Micsip annna Jun Fabian Kung Wai L 98

50 50 ampls f Cicui Analsis* Bas Micwav/RF CAD Sfwa Agiln s Avanc Dsign Ssm Appli Wav Rsac s Micwav Offic Ansf s Dsing *T sfwa swn als av numical M slv capabili fm D.5D full 3D. Jun Fabian Kung Wai L Tansmissin Lin Caacisics an lcical Cicui Ml Jun Fabian Kung Wai L 100

51 51 Disibu lcical Cicui Ml f Tansmissin Lin (1) Sinc ansmissin lin is a lng incnnc fil an cun pfil a an insan in im is n unifm alng lin. I cann b ml b lump cicui. Hwv if w ivi Tlin in man s sgmns (< 0.1λ) fil an cun pfil in ac sgmn is alms unifm. ac f s s sgmns can b ml as RLCG nw. Tis assumpin is u wn M fil ppagain m is TM quasi-tm. Fm nw n w will assum Tlin un iscussin supp minan m f TM quasi-tm. F ansmissin lin s asscia R L C an G paams a isibu i.. w us p uni lng valus. T ppagain f vlag an cun n ansmissin lin can b scib in ms f s isibu paams. Jun Fabian Kung Wai L 101 Disibu lcical Cicui Ml f Tansmissin Lin () 5.8GH 3.0V Magniu f in YZ plan Cun pfil alng cnucing ac Wiin ac sgmn cun is m lss cnsan in an u cun is simila. Als M can b cnsi saic. Jun Fabian Kung Wai L 10 I V

52 5 Disibu Paams (1) T L an C lmns in lcical cicui ml f Tlin is u magnic flu linag an lcic fil linag bwn cnucs. S Appni : Avanc Cncps Disibu RLCG Ml f Tansmissin Lin an Tlgapic quains f pfs. I L C V 1 V H Magnic fil linag lcic fil linag Jun Fabian Kung Wai L 103 Disibu Paams () Wn cnuc as small cnuciv lss a sis sisanc R can b a inucanc. Tis lss is u a pnmnn nwn as sin ffc w ig fqunc cun cnvgs n sufac f cnuc. Wn ilcic as fini cnucivi an plaiain lss a sun cnucanc G can b a in paalll capacianc. T inclusin f R an G in Tlin isibu ml is nl accua f small lsss. Tis is u ms f im as Tlin is usuall ma f v g cnuciv maial an g insula. T quains f fining L C R G un lw lss cniin a givn in fllwing sli. Cnuc lss Un lss cniin R L an G a usuall funcin f fqunc nc Tlin is ispsiv. I L R V 1 C G V Dilcic lss Jun Fabian Kung Wai L 104

53 53 Disibu Paams (3) Tus a ansmissin lin can b cnsi as a casca f man f s quivaln cicui scins. Wing wi cicui an cicui lmns a muc asi an wing wi an H fils using Mawll quains. In f is RLCG ml f Tlin b vali fm lw v ig fqunc ac sgmn lng mus appac an numb f sgmn n accual ml Tlin bcms infini. Tis lcical cicui ml f Tlin is cmmnl nwn as Disibu RLCG Cicui Ml. 0 I V Disibu RLCG cicui I V Jun Fabian Kung Wai L 105 Fining RLCG Paams µ L H v C I v V Tis inicas V V vlum nclsing (4.1a) cnucs R H G l v V σ cδs I V C1 C δs sin p (4.1b) ε ' ε " ε ε εε anδ ωσ cµ Sin p σ c cnucivi f malic bc Ts fmulas a an G pn iv fm ng n fqunc 1m cnsiain. N a cnuc lss C 1 suls in R wil ilcic lss suls Cnuc 1 S in G. S Scin 3.9 f Cllin [1]. V is vlum suuning cnucs wi lng f 1 m alng ais. C 1 an C a pas suuning sufac f cnuc 1 an. Lss angn f ilcic Cnuc Jun Fabian Kung Wai L C 106

54 54 Fining RLCG Paams Fm ng Cnsiain i() T insananus pw absb b an inuc L is: ( ) v( ) i( ) P in Assuming i() incass fm 0 a 0 I a al ng s v() L b inuc is: i() ( ) ( ) ( ) i in P ( ) ( ) 0 in τ τ v τ i τ τ 0 L τ i τ τ i 0 τ I 1 in Li i LI 0 0 Tis ng s b inuc is cnain wiin magnic fil ca b cun (f insanc s D.J. Giffis Inuc lcnamics Pnic Hall 1999). Fm M s ng in magnic fil is givn b: µ H H I V in H B ng a sam nc: 1 µ LI H H V µ L H Jun Fabian Kung Wai L I 107 V Muli-Cnuc Tansmissin Lin Smbl an Cicui Lng incnnc: l > 0.1λ Paams: P uni lng R L C G maics. Usuall as a funcin Fqunc. L11 L1 R11 R1 L1 L C11 C1 R1 G11 R G1 TM quasi-tm m C1 C G1 G R 11 L 11 C 1G C1 G C11 C1 C G C C1 C 1 L 1 C G Jun Fabian Kung Wai L 108 R L Disibu RLCG cicui lcical Smbl

55 55 Tlgapic quains f V an I Muc li M fil in psical ml f Tlin is gvn b Mawll s quains w can sw a insananus ansvs vlag V an cun I n isibu RLCG ml a gvn b a s f paial iffnial quains (PD) call Tlgapic quains (S ivain in Appni ). F simplici w will p subscip fm nw. Fui In im-main Tansfms In im-amnic fm V I V RI L ( R ωl) I ZI Invs Fui (4.b) I V Tansfms I GV C ( G ωc) V YV I (4.a) V Disibu RLCG cicui Jun Fabian Kung Wai L 109 Sluins f Tlgapic quains (1) T pssins f V() an I() a saisf im-amnic fm f Tlgapic quains (4.b) a givn as: Wav avlling in - icin Wav avlling in icin I γ γ ( ) I I (4.3a) Pas fac γ α V ( ω) ( ω) ( R ωl)( G ωc) Ppagain cfficin Anuain fac γ γ ( ) V V (4.3c) (4.3b) Jun Fabian Kung Wai L 110

56 56 Sluins f Tlgapic quains () V V - I I - a unnwn cnsans. Wn w su ansmissin lin cicui w will s w V V - I I - can b min fm buna f Tlin. F s f is iscussins ac valus f s cnsans a n n. T buna f lin cicui Z s I V s V Z L Jun Fabian Kung Wai L 111 Signal Ppagain n Tansmissin Lin Cnsiing sinusial sucs pssin f V() an I() can b win in im main as: φ θ V V I I v ( ) ( ) α ( ) α V cs ω φ V cs ω φ ( ) I cs( ω θ ) α I cs( ω θ ) α i (4.4a) (4.4b) Fm sluin f Tlgapic quains w can uc a fw ppis f quivaln vlag v() an cun i() n a Tlin sucu. v() an i() ppaga a signal will a fini im avl fm n lcain an. On can fin an impanc call caacisic impanc f lin i is ai f vlag wav v cun wav. Ta avling V an I pinc ispsin an anuain. O ffcs suc as flcin b iscuss in la pa. Jun Fabian Kung Wai L 11

57 57 Caacisic Impanc (Z c ) An impan paam in Tlin is ai f vlag v cun call Caacisic Impanc Z c. Sinc vlag an cun a wavs is ai can b nl b cmpu f vlag an cun avling in simila icin. V ZI Fm Tlgapic quains O V γ V Z c I γ I V γ Zc I γ γv γ ZI γ V Z I γ Z γ R ωl G ωc R ωl G ωc (4.5) A funcin f fqunc Jun Fabian Kung Wai L 113 Ppagain Vlci (v p ) Cmpa pssin f v() f (4.4a) wi a gnal pssin f a avling wav in psiiv an ngaiv icin: V α α ( ) V cs( ω φ ) V cs( ω φ ) f ( ω ) Cmpa An cgniing a b v() an i() a ppagaing wavs pas vlci is givn b: v p A gnal funcin scibing ppagaing wav in icin ω (4.6) Jun Fabian Kung Wai L 114

58 58 Anuain (α) T anuain fac α cass ampliu f vlag an cun wav alng Tlin. F v avling wav: V cs α ( ω φ ) F -v avling wav: Z0 V cs α ( ω φ ) Z0 Jun Fabian Kung Wai L 115 Dispsin (1) Sinc γ γ ( ω ) α( ω ) ( ω) v p ω ( ω ) Caus f ispsin W bsv a ppagain vlci is a funcin f wav s fqunc. Cmpnns Diffn cmpnn f signal ppagas a iffn vlci (an als anua a iffn a) suling in nvlp f signal bing is a upu. Lw ispsin Tansmissin Lin v in Vi v u v in v u Jun Fabian Kung Wai L 116

59 59 Dispsin () Dispsin causs isin f signal ppagaing ug a ansmissin lin. Tis is paiculal vin in a lng lin. Vi Hig ispsin Tansmissin Lin v in vu v in v u Jun Fabian Kung Wai L 117 Dispsin (3) A upu sinusial cmpnns vlap a wng iming causing isin f puls Vin( i 1) Vin( i 3) 0 Vin( i 5) Vu( i 1) Vu( i 3) 0 Vu( i 5) 0.5 Vin( i ) i v in In is ampl ig amnic fqunc lag is pas vlci i.. ig fqunc signal as lss im avl lng f Tlin. v u Vu( i ) i i Jun 008 i 006 Fabian Kung Wai L 118

60 60 T Implicains Tlin supping TM quasi-tm m Jun Fabian Kung Wai L 119 T Lsslss Tansmissin Lin Wn lin is lsslss R 0 an G 0. W av: γ ω LC Z c L C v p 1 LC 1 µε S lsslss ansmissin lin as n anuain n ispsin an caacisic impanc is al. Sinc lsslss Tlin is an ial in pacical siuain w uc lss as small as pssibl b using gl-pla cnuc an using g quali ilcic (lw lss angn). Jun Fabian Kung Wai L 10

61 61 Appni Avanc Cncps Disibu RLCG Ml f Tansmissin Lin an Tlgapic quains Jun Fabian Kung Wai L 11 Disibu Paams Ml (1) F TM quasi-tm m ppagain alng icin: l s S s C S Tm l l l l s1 s s3 s4 s l l S s s 4 S Cnuc V 1 V C ( µωh ) s V1 V L is inucanc p m V V H s 1 ω µ S V V1 ω( L I ) L V 1 V Jun Fabian Kung Wai L 1 s Lp C s 3 s 1 s 4 Tis mans lain bwn V 1 an V is as if an inuc is bwn m Flu linag (Dfiniin f Inucanc) Wn is small as cmpa wavlng H ( ) ( ) Can b psn in cicui as:

62 6 Disibu Paams Ml () C 1 I 1 C Tis mans lain bwn I 1 an I is as if a capaci is bwn m A 1 A Vlum W I 1 I C V C I Sufac S S fllwing sli f m pf. V ρ W W ε W W Using Divgnc Tm ρ S W ε S W 1 ρ S W ε S W S 1 JW ε S W ε S J S J S J S S S A1 A ε S I1 I Wn S is small (( C ) V ) ( C ) V as cmpa I I 1 wavlng Can b psn in cicui as C is p uni lng capacianc bwn cnucs f Tlin. Jun Fabian Kung Wai L 13 C 1 I 1 Disibu Paams Ml (3) C Vlum W Hnc ai bcms: 1 B A () I Cnsi ai: ε s S Q B B l A A l Sufac S F saic quasi-saic cniin fil is givn b: σ ( ) ( ) s1 1 σ s s s 4πε 1 4πε sufac n C 1 sufac n C f ( ) f ( ) Q s 1 1 s s s 4πε 1 4πε sufac n C 1 sufac n C Q is al cag n cnucs C 1 C σ s is sufac cag nsi wil f s is nmali sufac cag nsi wi spc Q. ε s S Q B C s n pn n l B ( ) ( ) A Q fs fs 1 A 1 s s l al cag n 4πε sufac n 1 4πε C1 sufac n C cnucs w call 1 C is cnsan Capacianc. ε B fs ( ) fs ( ) s 1 A 1 s s l 4πε 4 sufac n 1 πε C1 sufac n C S Pnial iffnc bwn A an B Jun Fabian Kung Wai L 14 CV

63 63 Disibu Paams (4) Cmbining lainsip bwn L C an ansvs vlags an cuns quivaln cicui f a s scin f ansmissin lin supping TM quasi-tm ppagaing M fil can b psn b quivaln cicui: I L C V 1 V Tus a lng Tlin can b cnsi as a casca f man f s quivaln cicui scins. Wing wi cicui an cicui lmns a muc asi an wing wi an H fils using Mawll quains. Jun Fabian Kung Wai L 15 Disibu Paams Ml (5) Wn cnuc as small cnuciv lss a sis sisanc R can b a inucanc. Wn ilcic as fini cnucivi a sun cnucanc G can b a in paalll capacianc. T inclusin f cnsan R an G in Tlin s isibu cicui ml is nl accua f v small lsss*. Tis is u ms f im as Tlin is usuall ma f v g cnuciv maial. T quains f fining L C R G un lw lss cniin a givn in fllwing sli. *Un lss cniin R L an G a usuall funcin f fqunc nc Tlin is ispsiv. R I L V 1 C G V Jun Fabian Kung Wai L 16

64 64 a: Lss Dilcic Assuming ilcic is nn-magnic n ilcic lss is u laag (nn- cnucivi) an plaiain lss*. Plaiain lss is u vibain f plai mlculs in ilcic wn an a.c. lcic fil is imps. B mcanisms can b ml b cnsiing an ffciv cnucivi σ f ilcic a paing fqunc. Tis is usuall vali f small lcic fil. *W sul als inclu sisis lss in fmagnic maial. Lss cun nsi H J ωε ε H σ ωε ε H σ ωε 1 ε ωε ε ω H ( ) ( ε ' ε " H ) ωε ε 1 anδ σ ε ' ε ε ε " ε ε anδ anδ ωε ε N a σ is a funcin f fqunc Tis is call Lss Tangn Jun Fabian Kung Wai L 17 Fining Disibu Paams f Lw Lss Pacical Tansmissin Lin Wn lss is psn ppagain m will n b TM anm (Can u plain w is is s?). Hwv if lss is v small w can assum ppagaing M fil b simila M fils un lsslss cniin. Fm an H fils w cul iv RLCG paams fm quains (4.1a) an (4.1b). Alug RLCG paams un is cniin is nl an appimain is usuall small. Tis appac is nwn as pubain m. Jun Fabian Kung Wai L 18

65 65 Divain f Tlgapic quains (1) F Tlin supping TM an quasi-tm ms V an I n lin is sluin f a pblic paial iffnial quain (PD) nwn as lgapic quains. Cnsi fis lsslss lin: I 1 L I I 1 I Us Kicff s Vlag Law (KVL): ωl I V C 1 V Obsving a: V V lim 1 V 0 V 1 V V V1 ωl I V V 1 ωli V ωli Jun Fabian Kung Wai L 19 Divain f Tlgapic quains () Nw cnsiing cun n b ns an using Kicff s Cun Law (KCL): Using KCL : I 1 L I ωl I 1 I 1 I V C 1 V (ωc ) -1 V V Again bsving a: I I1 V C I ( ωc ) I1 V I I I 1 ωcv ωcv Lsslss Tlgapic quains Jun Fabian Kung Wai L 130

66 66 Rlainsip Bwn Fil Sluins an Tlgapic quains Incnnc S incnnc Lump RLCG cicui Lng incnnc Wavgui Mawll s quains Wav quains Tansmissin lin KVL & KCL & H wav TM T TM Hbi Onl f siplin sucus Quasi-TM V I an Disibu RLCG cicui Tlgapic quains Z c v p anuain fac ispsin pw anling. Jun Fabian Kung Wai L 131 ampl A3 Fin RLCG paams f lw lss paalll pla wavgui in ampl A1. Assuming cnucivi f cnuc is σ an ilcic bwn plas is cmpl (is mans ilcic is lss ): ε ε ' ε" Us pssins f an H as iv in ampl A1. µ ' W L H/m C ε F/m W ωε σ R Ω/m G C W Ω σ c δ s W ε ' " 1 /m Jun Fabian Kung Wai L 13

67 Tansmissin Lin Snsis On Pin Cicui Ba (PCB) An Rla Sucus Jun Fabian Kung Wai L 133 Siplin Tcnlg (1) Siplin is a plana-p Tlin a lns islf wll micwav inga cicui (MIC) an pligapic fabicain. Siplin can b asil fabica n a pin cicui ba (PCB) smicnuc using vaius ilcic maial suc as p sin glass fib suc as FR4 plafluln (PTF) cmmnl nwn as Tfln Plimi aluminium i ianium i an camic maials pcsss f insanc lw-mpau c-fi camic (LTCC). T ms cmmn Tlin cnfiguains using siplin cnlg a micsip lin siplin an c-plana siplin. Jun Fabian Kung Wai L 134

68 68 Siplin Tcnlg () A vai f subsas Tin an Tic-Film cnlgis can b mpl. F m infmain n micsip lin cicui sign u can f T.C. was Funain f micsip cicui sign n iin 199 Jn Wil & Sns. (3 iin 000 is als availabl). F m infmain n siplin cicui sign u can cnsul H. Hw Siplin cicui sign 1974 Ac Hus. F m infmain n micwav maials an fabicain cniqus u can f T.S. Lavga Micwav maials an fabicain cniqus 3 iin 000 Ac Hus. Jun Fabian Kung Wai L 135 T Subsa Lamina f Siplin Tcnlg Cpp icnss Dilcic icnss Dilcic Sana maial cnsis f p wi glass fib infcmn Cpp (Usuall gl pla pc agains iain) Tpical ilcic icnss a 3mils (0.80mm) 6mils (1.57mm) f ubl si ba. F muli-la ba icnss can b cusmi fm 6 mils in 1 mils sp. Cpp icnss is usuall pss in ms f mass f cpp spa v 1 squa f. Sana cpp icnss a an.0 /f. 0.5 /f 0.7mils ic. 1.0 /f 1.4mils ic..0 /f.8mils ic. Jun Fabian Kung Wai L 136

69 69 Facs Affcing Cics f Subsas Opaing fqunc. lcical caacisics -.g. nminal ilcic cnsan anisp lss angn ispsin f ilcic cnsan. Cpp wig (affc lw fqunc sisanc). T g glass ansiin mpau. Cs. Tlanc. Manufacuing Tcnlg - Tin ic film cnlg. Tmal quimns -.g. mal cnucivi cfficin f mal pansin (CT) alng an ais. Mcanical quimns - flanss cfficin f mal pansin mal-film asin (pl sng) flam aain cmical an wa sisanc c. Jun Fabian Kung Wai L 137 Cmpaisn bwn Vaius Tansmissin Lins Micsip lin Siplin C-plana lin Suffs fm ispsin Pu TM m Suffs fm ispsin an nn-tm ms an nn-tm ms as fabica Difficul fabica Fail ifficul fabica Hig nsi ac Mi nsi ac Lw nsi ac Fai f cupl lin sucus N ug ls cnnc gun G f cupl lin sucus N ug ls cnnc gun N suiabl f cupl lin sucus N ug l qui cnnc gun Jun Fabian Kung Wai L 138

70 70 Fil Sluin f M Wavs n Siplin Sucu (1) In micsip an c-plana Tlin ilcic maial s n cmpll suun cnuc cnsqunl funamnal m f ppagain is n a pu TM m. Hwv a a fqunc blw a fw GH (<10GH a las) M fil ppagain m is quasi- TM. T micsip Tlin can b caaci in ms f is appima isibu RLCG paams. F siplin minan m is TM nc i can b caaci b is isibu RLCG paams v ig fquncis. Unfunal is n simpl cls-fm analic pssins a f M fils RLCG paams f a plana Tlin. A m nwn as Cnfmal Mapping is usuall us fin appima cls-fm sluin f Laplac paial iffnial quain f TM/quasi-TM m fils. T pssin can b v cmpl. S Ram [3] Cap 7 f m infmain n Cnfmal Mapping m. Cllin [1] Cap 3 pvis mamaical ivain f fil sluins f paalll pla wavgui wi inmgnus ilcic an micsip lin. Jun Fabian Kung Wai L 139 Fil Sluin f M Wavs n Siplin Sucu () An appac is us numical ms slv f saic an H fil alng css scin f Tlin. Fm an H fils RLCG paams can b bain fm quains (4.1a) an (4.1b). T a numus cmmcial an nn-cmmcial sfwa f pfming is analsis. Onc RLCG paams a bain ppagain cnsan γ an caacisic impanc Z c f Tlin can b bain. Z c can n b pl as a funcin f Tlin imnsins ilcic cnsan an paing fqunc. Man aus av slv saic fil pblm f siplin sucus using cnfmal mapping an appacs slv scala pnial φ an vc pnial A. T fllwing slis sw sm usful suls as bain b sacs in pas f signing plana Tlin. Sm f quains a bain b cuv-fiing numicall gna suls. Jun Fabian Kung Wai L 140

71 71 Tpical Iaiv Flw f Tansmissin Lin Dsign Sa Daw Tlin psical css scin Slv f TM m an H fils a fqunc f ins N Dmin R L C G fm (4.1) Cmpu Z c α an a fqunc f ins Ciia m? n Ys Dsign quains f Tlin Jun Fabian Kung Wai L 141 Dsign quains B vaing psical imnsins an using flw f pvius sli n can bain a cllcin f suls (Z c α ). Ts suls can b pl as pins n a gap. Cuv-fiing cniqus can n b us iv quains a mac suls wi psical paams f Tlin W µ ε µ ε Zc( 1) Zc( ) 00 Zc( 3) Z c ε 1 Zc( 4) ε Zc( 5) 100 ε 3 Zc( 6) ε W/ 8 Jun Fabian Kung Wai L 14

72 7 Dsign quains f Micsip Lin Micsip Lin (s fnc [3] Cap 8) µ ε ffciv ilcic cnsan (S Appni 3) ε 1 1 ε 1 ff w w w Zc 1.98 εff (5.1a) (5.1b) W µ ε v p µε 1 ff ε (5.1c) Onl vali wn quasi-tm appimain an v lw lss cniin applis. Jun Fabian Kung Wai L 143 Dsign quains f Siplin Siplin (s fnc [3] Cap 8): v p 1 µε ( ) Z K Z c 4 K 1 π 1 φ πw K( ) cs sin φ Cmpl llipic ingal f n in (5.c) (5.a) Z µ ε (5.b) µ ε Onl vali wn TM quasi-tm appimain an v lw lss cniin applis. w Jun Fabian Kung Wai L 144

73 73 Dsign quains f C-plana Lin C-plana Lin ([3] assum is lag cmpa s): ffciv ilcic cnsan (S Appni 3) ε 1 ε ff Z Z c π εff πz Z c 4 εff Z ln 1 ln 1 µ ε a w f 0 < < w a w a w a v p 1 w f < < 1 a µε (5.3a) 1 ff ε (5.3b) (5.3c) s a w Onl vali wn quasi-tm appimain an v lw lss cniin applis. Jun Fabian Kung Wai L 145 Dispsiv Pp f Micsip an C-plana Lins (1) T acual ppagain m f micsip an c-plana lins a a cmbinain TM an T ms. B ms a ispsiv. T pas vlci f M wav is pnn n fqunc (s fncs [1] an [] iscussin in Appni 1). Tis cang in pas vlci is flc b ffciv ilcic cnsan a cangs wi fqunc. A lw fqunc (f < f ciical ) wn ppagain ms f micsip an c-plana lins appacs quasi-tm pas vlci is alms cnsan. Appni 1 sws a simpl m sima f ciical f micsip an c-plana lins. Tus f ciical is usuall an as upp fqunc limi f micsip an c-plana lins. Tpical valu is GH pning n ilcic icnss. Siplin s n pinc is ffc as icall i can supp pu TM m. Jun Fabian Kung Wai L 146

74 74 Dispsiv Pp f Micsip an C-plana Lin () ε ff Micsip lin Rgin w (5.1a) an (5.1b) applis. ε ff Siplin ε ε 1 1 f f f ciical Limi f quasi-tm appimain s ampl A1 an n w sima f ciical v p µε 1 ff ε F micsip lin M fil is pal in ai an ilcic. Hnc ffciv ilcic ε ff cnsan is bwn s f ai an ilcic. N: Bn f ciical cncp f caacisic impanc bcms maninglss. Jun Fabian Kung Wai L 147 Tl f Siplin Dsign Yu can asil f bs unals magains c. us appimain quain vlp b s an wi u wn sfwa ls fm a simpl spas Visual Basic JAVA C bas pgams. A fw f sfwa a availabl nlin m nabl is AppCAD b Agiln Tcnlgis. Yu can als cc u f m public main ls Jun Fabian Kung Wai L 148

75 75 ampl Micsip Lin Dsign (a) Dsign a 50Ω micsip lin givn a 1.57 mm an ilcic cnsan 4.6 (H i mans fin w). Sps... Pl u Z c vsus (w/). Fm cuv w s a w/ 1.8 f 50Ω. Tus w mm.8 mm 4 Z S i ε S i Ω W/S i S W/ i 1.8 Jun Fabian Kung Wai L Micsip Lin Dsign ampl Cn... (b) If lng f Tlin is 6.5 cm fin ppagain la. (c) Using l < 0.05λ ul fin fqunc ang w micsip lin can b psn b lump RLCG cicui. Fm ε ff vsus w/ w s a ε ff 3.55 a w/ 1.8. Tf: v ms 1 la εµ 3.51 v p p 406 ps T b psn as lump wavlng λ mus b > 0 Lng: λ > 0 l 1.30 m v p f f > 1.30 v p < 13. MH 1.30 f lump 13. MH Jun Fabian Kung Wai L 150

76 76 Micsip Lin Dsign ampl Cn... () Wn lw lss micsip lin is cnsi s iv is quivaln LC nw. L 1 1 Zcv p C LC C 1 1 C 14.6pF/m Z cv p L Z c C 313.7nH/m F s incnnc w cul ml Tlin as: 1 L nH 10.18nH L 0. 36nH C pf L nH C pf 4.048pF 1 C pf Jun Fabian Kung Wai L 151 Micsip Lin Dsign ampl Cn... () Finall sima f ciical limi w quasi-tm appimain bgins ba wn. Again using ciia a wavlng > 0 f quasi-tm m ppaga: λ v p fciical > fciical v p < 5.10GH W s a incas f ciical small icnss sul b us. Jun Fabian Kung Wai L 15

77 77 Micsip Lin Dsign ampl Cn... µ ε 65.0 mm.8 mm 1.57 mm µ µ ε 4.6ε Z c 50Ω v p m/s la 406psc Maimum usabl fqunc f ciical 5.10 GH. S incnnc limi 13. MH Jun Fabian Kung Wai L 153 ampl 5. simaing ffc f Tac Wi an Dilcic Ticnss n Z C Cnsi fllwing micsip lin css scins assuming lsslss Tlin ma a cmpaisn f caacisic impanc f ac lin. TL 1 TL TL 3 TL 4 Jun Fabian Kung Wai L 154

78 78 ampl Siplin Dsign ampl Using D M Fil Slv Pgam H w mnsa us f a pgam call Mawll D b Ansf Inc. sign a siplin. T vsin us is call Mawll SV V 9.0 a f vsin wic can b wnla fm cmpan s wbsi. T sfwa uss fini lmn m (FM) cmpu wimnsinal (D) saic an H fil f an aa f mallic bcs. I is assum a siplin is lsslss. Tw pcs a ca n is lcsaic pblm f calculain f saic lcic fil an isibu capacianc is Magnsaic pblm f calculain f saic magnic fil an isibu inucanc. Caacisic impanc f siplin can n b cmpu fm isibu capacianc an inucanc. Jun Fabian Kung Wai L 155 ampl Scn S Jun Fabian Kung Wai L 156

79 79 ampl Siplin Css Scin Daw css scin f ml an assign maial. GND GND1 0.6mm FR4 Subsa subsa Signal cnuc (PC) Tac1 GND plans (PC) 8.0mm S awing unis micn. S awing si 10000um f an 4000um f. S gi U V 100um. Daw ml us ic n m f cnucing sucus li GND plans an signal. Nam signal cnuc Tac1 an GND plans GND1 an GND. Nam FR4 as subsa n gup b GND1 an GND as n bc GND. 0.3mm 0.3mm 0.036mm 0.4mm 0.036mm Jun Fabian Kung Wai L 157 ampl lcsaics: Sup Buna Cniins S buna cniins. All buna a Diicl p i.. vlags a spcifi. L gs f ml main main as Balln Buna. 0V 1V 0V 0V 0V Balln Bunais Jun Fabian Kung Wai L 158

80 80 ampl lcsaics: Sup cuiv Paams Un Sup cuiv Paams ab slc Mai an pc pfm capacianc mai sup as swn. Jun Fabian Kung Wai L 159 ampl lcsaics: Sup Slv an Slv f Scala Pnial φ Sup slv an slv f appima pnial sluin. Us Suggs Valus if u a n su. Jun Fabian Kung Wai L 160

81 81 ampl Fini lmn M (1) In fini-lmn m (FM) an bc is ug cnsis f man small lmns usuall iangl f D bc an an f 3D bc. FM is us slv f appima scala pnial V ( φ) f lcsaic pblm an vc pnial A f magnsaic pblm a v f ac iangl. T paial iffnial quains (PD) slv a Pissn s quains. ρ V ε A J µ Pnial valu insi iangl can b sima via inplain. F D pblm PD can b win as: ρ V ) ) ε A µ J V V ( ) ρ ρ ( ) A A ( ) J J ( ) Jun Fabian Kung Wai L 161 ampl Fini lmn M () D quasi-saic fil can n b bain b: ( ) V ( ) Similal magnic flu innsi H can b bain fm: ) H ( ) 1 [ A ( ) µ ] F m infmain f T. I (i) Numical cniqus f micwav an milimwav passiv sucus Jn-Wil & Sns P. P. Silvs R. L. Fai Fini lmns f lcical ngins Cambig Univsi Pss O nw bs. Jun Fabian Kung Wai L 16

82 8 ampl lcsaics: T Tiangula Ms Jun Fabian Kung Wai L 163 ampl lcsaics: Pl f fil Magniu Jun Fabian Kung Wai L 164

83 83 ampl lcsaics: Pl f Vlag Cnu Jun Fabian Kung Wai L 165 ampl lcsaics: Capacianc T sima isibu capacianc is n cmpu using: C ε ' v V V Appimain ingain using summain is pfm b sfwa. T sul is swn blw: C F/m pf/m. Jun Fabian Kung Wai L 166

84 84 ampl Magnsaics: Sup Buna Cniins Sli suc 1A Sli suc 1A (al f plans) Balln Bunais Jun Fabian Kung Wai L 167 ampl Magnsaics: Sup cuiv Paams Un Sup cuiv Paams ab slc Mai/Flu an pc pfm inucanc mai sup as swn. Jun Fabian Kung Wai L 168

85 85 ampl Magnsaics: Pl f B fil Magniu Jun Fabian Kung Wai L 169 ampl Magnsaics: Inucanc T sima isibu capacianc is n cmpu using: L µ I V H v L H/m nh/m. Jun Fabian Kung Wai L 170

86 86 ampl Divain f Paams f Siplin Z c L C Ω v ms 1 p LC Jun Fabian Kung Wai L 171 ampl 5.4 Tlin Dsign Using Agiln s AppCAD V3.0 Jun Fabian Kung Wai L 17