class nos, M. Rodwll, copyrighd 9 ECE 145A 18 C, nos s 1: Transmission in Propris and Analysis Mark Rodwll Univrsiy of California, Sana Barbara rodwll@c.ucsb.du 85-893-344, 85-893-36 fax
Transmission in Analysis class nos, M. Rodwll, copyrighd 9 Gomris Characrisic Impdancs Tim Domain Analysis aic Diagrams Frquncy Domain analysis Rflcion cofficins Movmn of Rfrnc Plan Impdanc vs Posiion Smih Char Sanding Wavs Solving wav quaions quickly
class nos, M. Rodwll, copyrighd 9 yps of ransmission lins
Transmission ins for On-Wafr Wiring class nos, M. Rodwll, copyrighd 9 microsrip lin gomry volags currns W + I H I W+S coplanar wavguid W + I I I H
Transmission ins for On-Wafr Wiring class nos, M. Rodwll, copyrighd 9 gomry volags currns coplanar srips W+S I I W + - H slolin G + - I I H
Subsra Microsrip in class nos, M. Rodwll, copyrighd 9 W Dominan Transmission mdium in III- microwav & mm-wav ICs H Ky advanag: IC inrconncs hav vry low ground-lad inducanc Ground-lad inducanc: -lads o ground-bounc -is Millr-muliplid by IC gain Ky problms: hrough-wafr grounding hols vias coupling o TM mods in subsra ia inducanc forcs progrssivly hinnr wafrs a highr frquncis.
class nos, M. Rodwll, copyrighd 9 basic hory, C, o, vlociy, Gamma
Transmission ins class nos, M. Rodwll, copyrighd 9 A pair of wirs wih rgular spacing, dilcric loading along h lngh. Ths hav inducanc pr uni lngh and capacianc pr uni lngh. Forward and rvrs wavs propaga. Rflcions will occur if lins ar no corrcly rminad
Transmission ins: Basic Thory class nos, M. Rodwll, copyrighd 9 s d gn From basic nodal analysis of lin : d d di d and di d Cd C d d from which w find, v v I, o v o v whr o C and v 1 C
class nos, M. Rodwll, copyrighd 9 Forward and Rvrs Wavs currn in rvrs wav currn in forward wav volag in rvrs wav volag in forward wav o o v v v v
lociy and Characrisic Impdanc class nos, M. Rodwll, copyrighd 9 o C and v 1 C and C ar hr quaniis pr uni lngh. v c whr c is hspd of ligh and r, ff r, ff is h ffciv dilcric consan of h lin
Rflcions class nos, M. Rodwll, copyrighd 9 s gn A nd of and T s l s lin : T o whr A bginning of lin : o s s gn l l l whr s o o 1 1 s s o o 1 1 Nd good rminaions o prvn lin rflcions and ringing
class nos, M. Rodwll, copyrighd 9 Toal inducanc & capacianc in a lngh of lin If oallin lngh is C lngh lngh o o l l lngh Thn oalcapacianc in ha lngh is and oalinducanc in ha lngh is whr l lngh v "spd of ligh dlay"on hlin
class nos, M. Rodwll, copyrighd 9 umpd modls of vry shor ransmission lins T-modl Pi-modl C C C If oallin lngh or oallin dlay l is much lss han a wavlngh v is much lss han 1 or oallin dlay is much lss han puls risim hn h lin can b approximad as a T or scion C lngh lngh o o l lngh lngh f signal
class nos, M. Rodwll, copyrighd 9 addr modls of modraly shor ransmission lins Pi-modl synhsis C C C C C C C C C C T-modl synhsis C C C C C Clarly, w can brak a lin of any lngh ino scions of lngh l lin such ha lin l lin v is much lss han a signal priod. In his fashion a ransmission - lin can b modlld by an C filr. This is a frqun subsiuion in circui simulaions
Microsrip ins class nos, M. Rodwll, copyrighd 9
Microsrip in: Approxima Propris 1 class nos, M. Rodwll, copyrighd 9 W H Wid lin fild mosly in dilcric. This givs : v c 1 r H, whr c 1 is hspd of ligh 1 r W, whr is h fr spac wav impdanc no: wid lins hav problms
Microsrip in: Approxima Propris class nos, M. Rodwll, copyrighd 9 W H ~W +H If h lin is narrowr, hand analysis only Effciv widh W H approxima v r, ff H c lis 1 r, ff 1 r W H somwhr bwn ha of dpnding upon wha proporionof only vry appoximaly air and of h fild is h dilcric, in air.
class nos, M. Rodwll, copyrighd 9 ins in Tim Domain
aic Diagrams = Echo Diagrams class nos, M. Rodwll, copyrighd 9 Firs : Analyfor impuls rspons Thn : Us convoluion o find gnral rpons. Rcall : A nd of s whr A bginning of lin : T s gn lin : R R whr s o o 1 1 Rs o 1 o and Ts Rs o 1 o Rs
aic Diagrams = Echo Diagrams class nos, M. Rodwll, copyrighd 9
aic Diagrams = Echo Diagrams class nos, M. Rodwll, copyrighd 9 Now plas considr how h wavforms would chang if h gnraor wr a sp- funcion.
class nos, M. Rodwll, copyrighd 9 Rpad Rflcions Ringing or Exponnial Dcay If s is posiiv, pulsrsponssdcay gomrica lly xponnially If s is ngaiv, puls rsponssalso alrna in sign - -ringing. Bhavior appars vry clos o RC ringing. Why?
Tim-Domain Analysis class nos, M. Rodwll, copyrighd 9 C, Approxima modl R R s R nglc inducor RC circui charging. R s C
Tim-Domain Analysis class nos, M. Rodwll, copyrighd 9 C, Approxima modl R R s R nglc capacior R circui charging. R s C
Tim-Domain Analysis class nos, M. Rodwll, copyrighd 9 C, Approxima modl R C R C nglc nd capacior RC circui ringing S
class nos, M. Rodwll, copyrighd 9 and C ar imiing Cass of High-, low- lins C, High - lin : larg, small C. approximaly an inducor ow - lin : larg C, small. approximaly a capacior.
class nos, M. Rodwll, copyrighd 9 ins in Frquncy Domain
class nos, M. Rodwll, copyrighd 9 in Analysis in Frquncy Domain Smih Char Tim - domain analysis : inuiiiv and clar : pulss bouncing back and forh. vry difficul wih raciv,c load or gnraor impdancs Frquncy - domain analysis : lss inuiiiv. asy wihraciv,c load 1 sanding wavs Smih char or gnraor impdancs
class nos, M. Rodwll, copyrighd 9 in Analysis in Frquncy Domain: Phas Consan b Phasor noaion: s s cos o R, whr is o o is complx. Ona ransmission lin, wavs ravl as For a cosinusoidal wav ravling a vlociy v, cos v, v. v cos v cos b. b v is h phas propagaion consan.
class nos, M. Rodwll, copyrighd 9 in Analysis: Exponnial Wavs v v v v o b b - dircion : h ngaiv propagaing in wavs Exponnial h posiiv - dircion : propagaing in wavs Exponnial. wrin implicily as wavs ar sinusiodal, R cos Bcaus
class nos, M. Rodwll, copyrighd 9 olags on a Transmission in b b on hlin : Phasor volag implici. dpndnc im maks Working wih h phasor ] R[, on lin : olag
class nos, M. Rodwll, copyrighd 9 olags and Currns on a Transmission in I b b b b Phasor currn on hlin : on hlin : Phasor volag
Wav Paramrs class nos, M. Rodwll, copyrighd 9 Dfin wav ampliud a such ha if a 1, hn wav powr 1Wa. olag in forward wav : Currn in forward wav : I Powr in forward wav I * Foward wav ampliud: a Rvrs wav ampliud: b
class nos, M. Rodwll, copyrighd 9 Wav Paramrs and Powr w us R.M.S.quaniis. h nos, Throughou wav Powr in rvrs Powr in forward wav * * * * b b I a a I
Rlfcions from h oad class nos, M. Rodwll, copyrighd 9 whr and is -1 1 is h load rflcion cofficin. h* normalid * load impdanc.
Rlfcions from h Gnraor class nos, M. Rodwll, copyrighd 9 S T s s whr T S S is h sourc ransmission cofficin whr and S S S S S is -1 1 is h sourc rflcion cofficin. h* normalid * sourc impdanc. Noha h rfrnc plan has bn movd.
class nos, M. Rodwll, copyrighd 9 Movmn of Rfrnc Plan b b b b b bcaus 1 cofficin h posiion- dpndn rflcion is whr 1
Posiion-Dpndn Rflcion Cofficin Rflcion cofficin a a disanc l from load. l b class nos, M. Rodwll, copyrighd 9 Th rflcion cofficin has gon hrough a phasshif of l ngaiv radians. or ngaiv b l radians. or l ngaiv 36 dgrs....simply bcaus and undrgo 36 dgr phasshifs vry wavlngh of disanc.
class nos, M. Rodwll, copyrighd 9 Impdanc vs. Posiion 1 1 a any poin Normalid impdanc 1 1 Impdanc a any poin I I I
class nos, M. Rodwll, copyrighd 9 in Inpu Impdanc unnormali d. 1 1 normalid. 1 1 a impdanc Inpu l l l I l l l l l I l l l
class nos, M. Rodwll, copyrighd 9 Impdanc and Rflcion Cofficin vs Posion ool. Work wih a graphical bu dious mah. simpl, Concpually impdanc normalid 1 1 1 cofficin s rflcion wavs I I b b b b b
Dvloping h Smih Char class nos, M. Rodwll, copyrighd 9 Th rlaionship 1 1 1 1 is ky. Th rlaionship is a 1-1 mapping bwn h complx #s and ; a conformal ransformaion. This rlaionship can b graphd. In h - dimnsional plan of - h a rflcion hr, a rd do. cofficin is plan - rprsnd by a poin.
Moving Rfrnc Plans---on h Smih Char class nos, M. Rodwll, copyrighd 9 As w mov a disanc l away from h vcor roas by an angl h load, bl o l - 36 on whol roaion in h plan for ach half - wavlngh movmn on h ransmission lin.
Finding Impdancs class nos, M. Rodwll, copyrighd 9 1 1 I 1 This is a 1: 1 rlaionship bwn rflcion cofficin magniud and phas and normalid impdanc ral and imaginary pars. Plo hunis of on h plan!
class nos, M. Rodwll, copyrighd 9 Finding Impdancs x 1 1 1 x 1 3 x 3 R X r x x Ral and imaginary r 1 3 r 1 r 3 parsof impdanc x 13 can b rad from h curvd impdanc x 3 axs on h char. x 1
Finding Rflcion Cofficin Th magniud and angl of ar simply rad from h char radius and angl class nos, M. Rodwll, copyrighd 9 - plan his masurmn can b don using a rulr and a proracor*. *hough oday hcadsofawardos hmasurmn from a cursor.
Using h Smih Char class nos, M. Rodwll, copyrighd 9 Saring wih h load impdanc wcompu.,, W hn find his poin on hsmih char. This drmins cofficin. h load rflcion
Using h Smih Char class nos, M. Rodwll, copyrighd 9 W hn roa h vcor hrough an angl 36 o l. This locas cofficin. hinpu rflcion W can now rad off h inpu impdanc. in in l, l
Impdanc-Admianc Char class nos, M. Rodwll, copyrighd 9 Impdanc R X Normalid impdanc Admianc Y 1 G B Normalid impdanc Y Y r Y Y x g Smih chars can hav axs for, Y, or boh. o o o b
class nos, M. Rodwll, copyrighd 9 Solving Wav Equaions Quickly
class nos, M. Rodwll, copyrighd 9 Wavs and Fourir Transforms 1 y x x x y k y k x k x x y x y x E y x E k k E E E y x E E E E E J J E E b,,, wriing 's ar complx : h Somims,, and hsam for,,, Tosolv his asily, assum. nuraliy assum charg, Assum nonro conduciviy, ar uniform and if wav quaion : us a giv M axwll's quaions 1
class nos, M. Rodwll, copyrighd 9 Wavs and Fourir Transforms sa sady h wav quaion in h sinusiodal This is whr simply This bcoms,, and hsam for,,, Givn, and hsam for w hav : So k k k k k E E E y x E E E E E E y E x E y x y k y k x k x x y x x x x x y x
class nos, M. Rodwll, copyrighd 9 Wavs and Fourir Transforms 3 Y I I I I Y I I paralll sris paralll sris and Thn, and, Tosolv his asily, assum and sysmransmission - lin Now considr a 1- dimnsional
Wavs and Fourir Transforms 4 class nos, M. Rodwll, copyrighd 9 sris I and I Y paralll Muliply hs : I sris Y paralll I sris Y paralll Divid hs: I sris I Y paralll I sris Y paralll Th bfor h roo indicas ha hforward currn has hsam sign as h forward volag, whil h rvrs currn has sign opposiha of h rvrs volag.
Wavs and Fourir Transforms 5 class nos, M. Rodwll, copyrighd 9 in has sris inducanc and sris R rsisranc pr uni lngh. in has paralll capacianc C and paralll conducanc G pr uni lngh. Thn sris R Y paralll G C So : R G C R G C
class nos, M. Rodwll, copyrighd 9 Wavs and Fourir Transforms 6 in S- paramr calibraion somims Imporan slighly complx. bcoms No ha 1 1 1 1 1 Us1. and Suppos C G R C C G R C O N C G R C G R C G R N
class nos, M. Rodwll, copyrighd 9 Wavs and Fourir Transforms 7 b C G R C C G C R C G R C C G R C O N C G R C G R C G R N 1 1 1 1 1 Us 1. and Suppos