Transmission Line Theory

Transcription

1 Tranmiion in Thory Dr. M.A.Moawa nroducion: n an cronic ym h divry of powr rquir h conncion of wo wir bwn h ourc and h oad. A ow frqunci powr i conidrd o b divrd o h oad hrough h wir. n h microwav frquncy rgion powr i conidrd o b in cric and magnic fid ha ar guidd from pac o pac by om phyica rucur. Any phyica rucur ha wi guid an cromagnic wav pac i cad a Tranmiion in. Tranmiion in ar ud in powr diribuion a ow frqunci and in communicaion a high frqunci. A ranmiion in coni of wo or mor para conducor ud o connc a ourc o a oad. h ourc may b a gnraor a ranmir or an ociaor and h oad may b a facory an annna or an ociocop rpcivy. Tranmiion in incud coaxia cab a wo wir in a para pa or panar in a wir abov h conducing pan and a micro-rip in ro ciona viw of h in coni of wo conducor in figur ach of h in coni of wo conducor in para oaxia cab ar ud in crica aboraori and in conncing T. o T. annna Micro-rip in ar imporan in ingrad circui whr maic rip conncing cronic mn ar dpoid on dicric ubra. Thr ar diffrn yp of mod propagar bwn h wo conducorof ranmiion in a: - TE ranvr cric i.. E H - TM ranvr magnic i.. H E - TEM ranvr cro-magnic i.. H E - Propaga in -dircion H E

2 Tranmiion in probm ar uuay ovd uing EM fid hory and cric hory h wo maor hori on which crica nginring i bad w u circui hory bcau i i air o da wih mahmaic. Our anayi of ranmiion in wi incud h drivaion of ranmiion in quaion and characriic quanii h u of Smih char variou pracica appicaion of ranmiion in and ranin on ranmiion in. a b c d ro ciona viw of ranmiion in: a-coaxia in b-wo wir in c- panar in d- wir abov conducing pan - microrip in

3 S S E g k gnraor coaxia in oad E & H fid in h coaxia in For conducor and for dicric.

4 Tranmiion in Paramr: W mu dcrib a ranmiion in in rm of i in paramr. [Ω/m].conduciviy of conducor [H/m].f inducanc of wir [Ω -1 /m].dicric bwn wo conducor [F/m].proximiy bwn wo conducor. W hav : + i i+ + o gnraor v v+ o oad Th in paramr and ar no dicr or umpd bu diribud a hown.by hi w man ha h paramr ar uniformy diribud aong h nir ngh. - For ach in h conducor ar characrid by and h homognou dicric paraing h conducor i characrid by

5 3-1 ; i h ac rianc pr uni ngh of h conducor compriing h in and i h conducanc pr ngh du o h dicric mdium paraing h conducor. 4- Th xrna inducanc pr uni ngh; ha i x. Th ffc of inrna inducanc in ar ngigib a high frqunci a which mo communicaion ym opra. 5- For ach in and Tranmiion in Equaion: A mniond abov wo conducor ranmiion in uppor TEM wav; h cric and magnic fid on h in ar ranvr o h dircion of wav propagaion. an imporan propry of TEM wav i ha h fid E and H ar uniquy rad o voag and currn rpcivy: E. d H. d n viw of hi w wi u circui quaion and in oving h ranmiion in probm inad of oving fid quanii E and H i. oving Maxw quaion and B.. h circui mod i impr and mor convnin. u xamin an incrmna porion of ngh of a wo conducor ranmiion in. w innd o find an quivan circui for hi in and driv h in quaion. From h figur of diribud mn mod of ranmiion in w aum ha h wav propaga aong + dircion from h gnraor o h oad. By appying Kirchoff voag aw o h our oop of figur abov w obain: K: * By aking h imi of hi quaion a ad o

6 1 K: a h main nod of h circui By aking h imi of hi quaion a ad o f w aum harmonic im dpndnc o ha : ] [ ] [ Whr and ar h phaor form of and rpcivy Thn q n 1 bcom 3

7 Ao d d 4 By aking d/d of 3: d d d d Subiu in 4: d d d d d d Wav q n for voag 5 Ao d d Wav q n for currn 6 Th ar wav quaion for voag and currn imiar in form o h wav quaion obaind for pan wav in prviou chapr. 7 i h propagaion conan. i anuaion facor [Np/m db/m]. i pha conan [rad/c]. i wavngh [m]. f u i wav vociy [m/]. Souion of wav quaion:

8 Th ouion of h inar homognou diffrnia quaion 5 and 6 ar: 8 and Whr : ar wav ampiud; + and ign rpcivy dno wav raving aong + and dircion a i ao indicad by h arrow. Thu w obain h inananou xprion for voag a : co co ] [ 1 W dfin h / impdanc of h in a h raio of poiivy raving voag wav o currn wav a any poin on h in. i anaogou o η inrinic impdanc of h mdium of wav propagaion. By ubiuing q 8 and 9 ino q. 5 6 and quaing cofficin of rm and a: W hav: By quaing cofficin of h xponnia hn: / impdanc of h in

9 W hav ao: d d By quaing cof of h xponnia hn: X Whr [Ω] i ra par of X [Ω] i imaginary par of. Th propagaion conan and h / impdanc ar imporan propri of h in bcau hy boh dpnd on h in paramr and and h frquncy of opraion. h rciproca of i h /c admianc Y ha i Y = 1/. W may now conidr wo pcia ca of o ranmiion in and diorion- in. A- o in: A ranmiion in i aid o b o if h conducor of h in ar prfc and h dicric mdium paraing hm i o. For uch in : ncary condiion for a in o b o.

10 Thn: Ao 1 u f X X i dirab for powr ranmiion B- Diorion in : A igna normay coni of a band of frqunci; wav ampiud of diffrn frquncy componn wi b anuad diffrny in a oy in a α i frquncy dpndn. Thi ru in diorion. A diorion- in i on in which h anuaion conan α i frquncy indpndn whi h pha conan β i inary dpndn on frquncy. From h gnra xprion for α and β. i i vidn ha a diorion- in ru if h in paramr ar uch ha : u 1 f Showing ha α do no dpnd on frquncy whra β i a inar funcion of frquncy.

11 X 1 / 1 / X No ha: 1. Th pha vociy i indpndn of frquncy bcau h pha conan inary dpnd on frquncy. W hav hap diorion of igna un u and ar indpndn on frquncy.. u and rmain h am a for o in. 3. A o in i ao diorion- inbu a diorion- in i no ncariy o. Ahough o in ar dirab in powr ranmiion Tphon in ar rquird o b diorion-.

12 Microwav Enginring Sh #3-a Tranmiion in Q1: Dfin T..? Q: Sa yp of mod which propaga in T..? Q3: Dduc h formua of propagaion conan? Q4: Dduc h formua of / impdanc? Q5: Skch E/H fid in coaxia cab? Q6: An air in ha =7 Ω and pha conan =3 rad/m a f= 1MH acua: - h inducanc/m - h capacianc/m Q7: A ranmiion in opraing a f=5 MH ha =8 Ω.4N p / m 1.5rad / m Find h in paramr and. Q8: A diorion- in ha = 6 Ω mn p / m u=.6 cvociy of igh in vacuum. Find: and a f=1 MH? Q9: A phon in ha = 3 Ω/km = 1 mh/km = and = µf/km a f= 1 kh obain: a- / impdanc of h in b- propagaion conan c- h pha vociy Dr. M.A.Moawa

13 - npu mpdanc Sanding Wav aio and Powr: onidr a ranmiion in of ngh characrid by and conncd o a oad a hown in figur: g ' = g _ in in _ = = a- npu impdanc du o a in rminad by a oad g g + in _ b- Equivan circui for finding and in rm of in a h inpu ooking ino h in h gnraor h in wih h oad a an inpu impdanc. i our innion in hi cion o drmin h inpu impdanc inp h anding wav raio SW and powr fow on h in. h ranmiion in xnd from = a h gnraor o = a h oad. Fir of a w nd and a mnion abov

14 1 Whr q n of ha bn incorporad 3 To find and.g. a h inpu a = : h rmina condiion mu b givn. 4 5 Subiu 4 5 ino 1 ru in : f h inpu impdanc a h inpu rmina i inpu currn ar aiy obaind from figur a: inp h inpu voag and h in g in g in g 8 g a = : On h ohr hand if w ar givn h condiion a h oad ay:

15 9 Subiu hi ino q n 1 giv: a any poin on h in: Nx w drmin inpu impdanc / in a any poin on h in. a h gnraor for xamp in 1 Subiuing q n 1 11 ino 1 yid: in anh anh anh anh coh inh inh coh coh inh inh coh ] coh inh inh coh [ coh inh inh coh 13 in anh anh * for oy in 14

16 in anh anh * for o in 15 Thi indica ha h inpu impdanc vari priodicay wih dianc from h oad. Th quaniy β i uuay rfrrd o a h crica ngh of h in and can b xprd in dgr or radian. oag rfcion cofficin : W now dfind a h voag rfcion cofficin a h oad. i h raio of h voag rfcion wav o h incidn wav a h oad: by q n 1 11 giv: 16 n gnra h voag rfcion cofficin a any poin on h in can b dfind a h raio of h magniud of h rfcd voag wav o ha of h incidn wav ha i a. Bu ' ubiuing and combining wih qn w g: ' ' 17

17 currn rfcion cofficin Th currn rfcion cofficin a any poin on h in i ngaiv of h voag rfcion cofficin a ha poin. Thu h currn rfcion cofficin a h oad i : / a w did in pan wav w dfin h anding wav raio : max min max min i ay o how ha max max / and min min /. Th inpu impdanc inp in q n 14 ha maxima and minima ha occur rpcivy a h maxima and minima of h voag and currn anding wav. i aiy hown ha: in max max 19 min And min in min max A a way of dmonraing h concp conidr a o in wih characriic of 5 rminad in a pur riiv oad. For h ak of impiciy w aum ha h in i 1 and h voag a h oad i 1 rm. Th condiion on h in rpa hmv vry haf wavngh. Th avrag inpu powr a a dianc from h oad i givn by an quaion imiar o q n of P av P 1 [ * ] av 1

18 voag &currn wav parn on a o in rminad by a riiv oad. W now conidr pcia ca whn h in i conncd o oad and. Th pcia ca can aiy b drivd from h gnra ca. A. Shord in =

19 For hi ca q. 15 bcom: c in an Ao 1 Thi impdanc i pur racanc which coud b capaciiv or induciv dpnding on h vau of. Th variaion of in wih i hown in figur a. npu impdanc of a o in: a whn hord b whn opn B. Opn ircuid in n hi ca q 15 bcom

20 oc im in an co 3 1 Th variaion of in wih i hown in figur b. noic from q 3 ha:. Machd in in occ 4 Thi i h mo dird ca from h pracica poin of viw. For hi ca q15 rduc o: in 5 And 1 Tha i h who wav i ranmid and hr i no rfcion. h incidn powr i fuy aborbd by h oad. Thu maximum powr ranfr i poib whn h ranmiion in i machd o h oad. Examp: A 3-m ong o ranmiion in wih =5 Ω opraing a MH i rminad wih a oad 6 4. f u. 6c on h in. Find: a- Th rfcion cofficin b- Th anding wav raio. c- Th inpu impdanc.

21 ang max max min min in anh anh 3-Th Smih har i a oo o ov probm of ranmiion in i i commony ud of h graphica chniqu. i baicay a graphica indicaion of a ranmiion in a on mov aong h in. W cacua and in and o on. Examp : a 1+15 Ω oad i conncd o a 75 Ω o in. find : a- b- S c- Th oad admianc Y d- a.4 from h oad in - Th ocaion of max and min w.r.. h oad if h in i.6 λ ong f- in a h gnraor.

22 Smih har

23 Sh #3-b Tranmiion in npu impdanc rfcion cofficin & anding wav raio Q1: a ranmiion in ha c/c impdanc =5 Ω i conncd wih a gnraor ha frquncy f=5 MH and rminad by a oad impdanc a =5 m from h oad in =5-1. quird: