Voltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!

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1 Considr a pair of wirs idal wirs ngh << ngh >>, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds!

2 Thr is capacianc bwn any wo pics of conducors. A pair of plas, wirs, c., or h cor and shilds of a co-ax cabl. A pic of wir is acually an inducor A pair of wirs, coupld, bu similar.

3 To mak hings simpl, w firs considr a pair of idal wirs. No rsisanc, no shun (lakag. Pay clos anion. W ak a diffrn approach han dos h book. Now, oom in on on sgmn:

4 nducor Capacior Tak drivaivs wih rgard o,

5 Parial diffrnial quaions Do hs 2 quaions look familiar o you? Wha ar hy?, w hav v = f (v p is h gnral soluion o his quaion. Do i on your own: vrify his.

6 , w hav v = f (v p is h gnral soluion o his quaion. This is h wav quaion! Mor sricly, h losslss, disprsionlss, linar wav quaion. Assum: no rsisanc, no lakag; v p indpndn of frquncy; v p indpndn of volag v s his amaing? W arrivd a h wav quaion from circui hory, rgardlss of frquncy. Why dos his approach work? On mor agrmn, for co-ax cabls: S Tabl 2-1, pp. 45 in 7/E, pp.53 in 6/E Consisn wih EM hory! Chck his offlin for ohr ransmission lins. (Thr will b homwork problms for you o larn abou ohr yps of ransmission lins, as wll as non-idal co-ax cabls

7 (h quaion for i is in h sam form formally h sam. Th wav quaion. Mor sricly, h losslss, disprsionlss, linar wav quaion. Assum: no rsisanc, no lakag; v p indpndn of frquncy; v p indpndn of v or i v = f (v p is h gnral soluion o his quaion. Wha ar h singl frquncy, harmonic soluions?

8 v = f (v p is h gnral soluion o his quaion. Singl frquncy, harmonic soluions: v(, cos( i(, cos( (h quaion for i is in h sam form formally h sam. Hr, + and + ar complx ampliuds ha w will alk abou lar. For h wavs, hy ar no h phasors of h wavs. W will alk abou h disincion. + and + ar h ral ampliuds, or simply ampliuds. W hav no shown h volag and currn wavs ar in phas. Bu hy ar. You can ak his as a conclusion for now. Or, if you ar inrsd, rad h proof nx pag.

9 Hr w show ha h volag and currn wavs ar in phas wih ach ohr: cos(, ( v v cos(, ( i i call ha sin( v v sin( i i For his o hold for any arbirary, w mus hav i v So, in phas! W also g a by-produc: v p Uni: Ω H/s m H s m Ths conclusions ar imporan! Anywhr, any im v(,/i(, = consan Do no us rad hrough. Us scrach papr!

10 Dfin Considr: v p ( is ral, i.., purly rsisiv i(, i(, + + vs. v(,... i(, v(, nfinily long Thr is no way o ll h diffrnc us by masuring v and i. Enrgy propagaing away vs. nrgy dissipad Analogy: lasr bam going o infiniy or hiing a oally black wall You may also us Doing h drivaivs in a similar way as in las pag, you will also s h volag and currn wavs ar in phas. You will hav a similar by-produc abou h v/i raio. may look diffrn, bu you should b abl o show hy ar qual. Do i on your own. Hin: us Us scrach papr!

11 Now you may work on HW2: P1, P2 Som Clarificaions Class nos and ohr informaion on cours wbsi All class nos, homwork, homwork answrs, qui answrs (afr h qui, of cours. ad h nos, in conuncion wih h xbook scions indicad in h schdul (also on wbsi, bfor you do homwork Th nos conains mah drivaions ha do no go ovr in dail and ll you o go ovr offlin Th nos also conain conns ha call h xra mil, for you o pondr upon Homwork Do i on your own bfor chcking h answr shs Homwork problms ar for your xrcis. Effor lvl ndd o achiv sam larning oucom diffrs by individual. Thrfor homwork is no gradd. Quis You will do wll on quis if you do h abov. Exampl: Qui 1. Gradd (gnrously. No a gocha hing. Don wih purposs. Will giv answrs righ afr in-class quis. Primary purpos is o mak sur you ar prpard for h conn o b covrd in class. Proc On going. por o b submid whn don (afr Ts 1. Tak nos, sav rsuls. You will br undrsand h rsuls lar; you can go back o any sp and do/obsrv mor whn your undrsanding dpns. TA offic hour for CAD ool suppor?

12 Proc A circui simulaion proc o ransiion you from lumpd componn-basd circui hory n Par 1 and Par 2, you buil an C nwork: And, you did ransin simulaions of h following circuis wih h gnraor signal bing volag sps wih diffrn ris ims (.1 ns and 1 ps: nh.12 pf 5 5 Par 3: Now, cra a nw nwork ha is a cascad of 1 insancs of h abov C nwork. You may cra a symbol for his nw nwork for convninc. Using h sam inducanc and capacianc valus o do h sam simulaions you hav don for h abov singl C nwork. (Sam gnraors wih sam inrnal impdanc. Simula for boh opn circui and 5-ohm loads, h wo ris ims for ach cas, as don for h singl C. Ongoing proc. Say und for nx sps.

13 wri: cos(, ( v cos(, ( i n wha dircion do hs wavs propaga?

14 Wavs propagaing h ohr way ar also soluions o h sam quaions: cos(, ( v cos(, ( i Of cours, any linar combinaions of wavs in opposi dircions ar also soluions: cos( cos(, ( v cos( cos(, ( i Thy may rprsn combinaions of incidn and rflcd wavs. call ha w hav a mahmaical ool o 1. Avoid h pain of daling rigonomric funcions, and 2. Turn parial diffrnial quaions o ordinary diffrnial quaions by puing asid h known im variaion

15 Exprss wavs wih phasors phasor cos( cos(, ( v Whr do h phass go? call h dails how w convr a im-domain funcion o a phasor: By adding an imaginary parnr, ( v ] ( [( ( ( This is no h phasor y. Throw away h known im variaion and dfin h complx ampliuds and W g h phasor: Posiiv going Ngaiv going Noic sign and dircion

16 Exprss wavs wih phasors Th currn wav is similar i(, cos( cos( phasor This ool maks our lif much asir whn w dal wih a mor complicad siuaion. Hr s h mor complicad siuaion: No wirs ar idal. Any wir has som rsisanc. sisanc pr lngh Thr is always som shun conducanc bwn wo wirs Noic ha and G dsribs wo diffrn hings. 1/G Shun conducanc pr lngh

17 Analy h circui in h phasor way d ( d d d lim lim ( G C Tak drivaivs on on quaion and insr i ino h ohr, you g and a formally sam quaion for h currn. This is an ordinary diffrnial quaion. Bcaus w usd phasors. W hav arrivd a his quaion us by circui analysis using phasors. You could also firs o h circui analysis in h im domain, arriving a parial diffrnial quaions, and hn convr quaniis o phasors and arriv a h sam ordinary diffrnial quaions, as don in h book (Scions 2-3 & 2-4.

18 Th parial diffrnial quaions for h gnral, mor complicad siuaion ar, wll, oo complicad. W don vn bohr o ackl hm. s look a h simplr ordinary diffrnial quaion: Bfor solving his quaion, l s firs hav a digrssion back o h idal cas call ha w hav Noic ha hs ar acually wo soluions. Wha ar h diffrnc bwn h wo soluions?

19 Wavs propagaing in wo opposi dircions: No surpris. Th soluions w go arlir for h idal cas. Now, back o h mor complicad, gnral cas w can wri Compar his o h idal cas 2 Ths wo quaions ar formally h sam, xcp 2 is complx. 2 2 hus and So, h soluions ar Wha kind of wavs ar hy?

20 Wha kind of wavs ar hy? = + (w ar doing nohing. Any complx numbr can b wrin as his, w hav h firs soluion ( Wha is his? Similarly, ( Why do h wavs anua whn hr is rsisanc or shun lakag? (Why is hr no anuaion whn h wir is mad of a prfc conducor and h mdium bwn hm is a prfc insulaor? call ha, in circui hory, raciv vrsus rsisiv...,... How o xprss h dcaying wav propagaing in h dircion?

21 Again, hr can b wavs going h ohr way. Now, w discuss h mos imporan concp of h firs half of h smsr: d d ( ( ( lim d d Tak drivaivs From circui analysis For h idniy o hold for all, w mus hav h following: For h rm: ( ( ( C G C G

22 For h rm: ( ( ( C G C G Noic ngaiv signs. Jus bcaus of sign convnion (s circui diagram Dfin C G h characrisic impdanc Complx and xplicily dpndn on frquncy in h gnral (lossy cas.

23 For h wav ravling owards +, A any, ( ( For h wav ravling owards, A any, ( ( Again, noic his ngaiv sign. n gnral, C G is complx and xplicily dpndn on frquncy. For h losslss ransmission lin, = and G =, C al. No xplici frquncy dpndnc. Ths rlaions ar sparaly hld by h wo wavs in opposi dircions.

24 Tak-hom mssags olag v and currn i follow h sam diffrnial quaion Thrfor sam soluion Thrfor hr is a consan raio bwn hir ampliuds and hr is a consan shif bwn hir phass for harmonic wavs going in on dircion n h phasor form, h complx ampliud raio is Bing a volag/currn raio, has h dimnsion of impdanc n gnral (lossy cas, is complx and xplicily dpnds on n h losslss cas, is ral w/o xplici frquncy dpndnc bing ral mans h volag and h currn ar in phas. also mans h quivaln circui for a (smi-infinily long ransmission lin is simply a rsisor wih a rsisanc valu s nx pag. A his poin, rviw xbook up o Scion 2-4. Finish P1 hrough P6 of HW1.

25 ... vs. nfinily long Thr is no way o ll h diffrnc us by masuring v and i. Enrgy propagaing away vs. nrgy dissipad Analogy: lasr bam going o infiniy or hiing a oally black wall mpdanc mach By h way, ransmission lin (hick lin vrsus wir wirs (hin lins Th sam as h infinily long lin! W wan impdanc mach! (asons? A his poin, rviw xbook up o Scion 2-4.

26 Now, l s look a a ransmission lin wih a sourc and a load. f =, impdanc machd. All nrgy dlivrd o load. Good! (w can viw his from h vanag poin of quivaln circuis Wha if? Th load says ( ( ( f hr is only h incidn wav, ( Somhing has o happn o rsolv his conflic. Tha somhing is rflcion. Sign du o convnion Now, w us focusd on h load. Will alk abou h lin and gnraor lar. By dfiniion, Solv i and w hav

27 f, hr has o b a rflcd wav. Th load dos no g all nrgy carrid by h incidn wav. Whr dos h rs of h nrgy go? Considr analogy: lasr bam hiing wall no oally black/dark. Dfin h volag rflcion cofficin 1 1 On-o-on mapping bwn and / Th raio / mor imporan han islf 1 1

28 Thrfor w dfin h normalid load impdanc Thus, Noic h on-o-on mapping. This is vry imporan! For h currn Whr dos h ngaiv sign com from? is ral for a losslss lin, bu C is complx in gnral. G C is complx in gnral. Thus, is complx in gnral. ad h xbook: Scion 2-6 ovrviw, 2-6.1