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. So, his 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 v s his amaing? W arrivd a h wav quaion from circui hory, rgardlss of frquncy. Why dos his approach work? On mor agrmn: 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.

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 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

10 Dfin v p ( is ral, i.., purly rsisiv Considr:... 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 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

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

12 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 h 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

13 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

14 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

15 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.

16 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?

17 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 Ths wo quaions ar formally h sam, xcp 2 is complx. So, h soluions ar Wha kind of wavs ar hy?

18 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...,...

19 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

20 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.

21 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.

22 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 raion, 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.

23 ... nfinily long vs. 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?

24 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 h rflcion. Sign du o convnion By dfiniion, Solv i and w hav

25 f =, hr has o b a rflcion 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

26 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.

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!

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! Considr a pair of wirs idal wirs 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! Thr

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