(d) Show that the series resistance and inductance per unit length of the line are

Transcription

1 8. sissio lie osisig of Two oei iul lides of el wi oduivi d ski dep, s sow, is filled wi uifo lossless dielei,. TM ode is popged log is lie. Seio 8. pplies. () Sow e ie veged powe flow log e lie is P l wee is e pek vlue of e iul gei field e sufe of e ie oduo. () Sow e sied powe is eued log e lie s P wee P e l () Te eisi ipede Z of e lie is defied s e io of e volge ewee e lides o e il ue flowig i oe of e posiio Z. Sow fo is lie Z l (d) Sow e seies esise d idue pe ui leg of e lie e L l wee is e peeili of e oduo. Te oeio o e idue oes fo e peeio of e flu io e oduos dise of ode. Sol: () B se, d

2 l l l l. l os Fo TM wve popgig i dieio,. os B Le os os.. d d d S P S l e * () Fo kso (8.57) dl d dp O e ie sufe dl d dp i O e oue sufe dl d dp i Teefoe, powe loss

3 d dp d dp d dp ou i odig o P e P d dp P P e P d dp l () V Z d dl V l dl dl K l l V Z (d) Beuse ople odiio, d dp oside e eeg i volue, d d U vol l Teefoe, oside e eeg i wll, we kow e e // wee idied e dise io e oduo. ssue ie d e d U

4 U oue d e d U ol l L U l ol 8.3 () sissio lie osiss of wo ideil i sips of el, sow i oss seio i e ske. ssuig >>, disuss e popgio of а ТЕМ ode o is lie, epeig e deivios of Pole 8.. Sow P Z L wee e sols o e lef ve e se eigs s i Pole 8.. oose egul oodie sse wi pllel o e sip log side, pepediul o e sip d log e lie. Le, K K e ik e e sufe ue desi of e op sip. Tus, e gei field i ewee e wo sips is give ik B ik BK K e, K e Teefoe, K. Te elei field e deived fo e Mwell s equio: B kb k ik B i e i Te vege Poig veo k k k S

5 Te vege powe sied log e lie P S d es of powe P, P Te powe loss pe ui e, dp Keff d Te powe loss pe ui leg log e, dp dp P P d d P Tus, Pe wi Te poeil diffeee ewee e wo sips V V k Te wve ipede Z K V dl e k ik Te seies esise pe ui leg dp d Te idue pe ui leg L B d B d oduo wee e iegio of e seod e kig io ou e gei eeg soed iside e oduos. Noe iside e oduos, i/ i, e e / Tus, oduo B d e d Teefoe, L + () Te lowe lf of e figue sows e oss seio of iosip lie wi

6 sip of wid oued o dielei suse of ikess d dielei os, ll o goud ple. W diffeees ou ee oped o p if >>? f <<? Fo e se >>, e elei d gei fields e osl ofied i e egio ewee e sip d e goud ple d e uifo wii e egio. Teefoe, is se is ve siil o p () wi e sl d is io ige. oweve e se << is ve diffee fo (). Tis se e ppoied wie ove goudig ple. Te dielei suse sould ve lile effe o e quiies luled i () e o elei d gei fields eed osl i e egio wiou e suse. 8. Tsvese elei d gei wves e popged log ollow, ig iul lide wi ie dius d oduivi. () Fid e uoff fequeies of e vious T d TM odes. Deeie ueill e lowes uoff feque (e doi ode) i es of e ue dius d e io of uoff fequeies of e e fou ige odes o of e doi ode. Fo is p ssue e oduivi of e lide is ifiie. () lule e euio oss of e wveguide s fuio of feque fo e lowes wo disi odes d plo e s fuio of feque. Sol: () Followig e lsis o pge 369 wi e eplee e uoff fequeies e M, fo ode TM d Tee is e eo of e Bessel fuio p k, i is see d, fo ode T. d is e eo d.8. Te fudel ode is T wi, wi d. Te e fou ige odes e: M, TM wi. 36

7 , T wi. 659, T wi. 8, TM wi. 8 M () T : We equie o lule e powe P fo q. 8.5 i kso, d ll gei fields o lule dp dl d i e ik ik e i i wi, d, k d. Ug q. 8.5 P,, d d k dp d k Te euio os fo ollow ss guide, fo wi d is e foud o e, P dp d,,,, TM : Te equied fields e, ug k d Z

8 ik ik ik ik Z Z ug q. 8.5 lulig dp d d evluig ), i is foud dp fo ollow ss guide ( d P d, 3 M, 8.5 wveguide is osued so e oss seio of e guide fos ig igle wi sides of leg,,, s sow. Te ediu iside s. () ssuig ifiie oduivi fo e wlls, deeie e possile odes of popgio d ei uoff fequeies. geel, o solve pole like is, we eed o oside e Diile o Neu pole fo oud wiou sdd (i.e. egul o iul) se. piul, is es ee is o ul oodie sse o use fo e wo-diesiol elol equio o llows fo sepio of viles d espes e se of e oud sufe (wi would llow siple speifiio of e oud d). geel pole of is fo (wi o siple oud se) is quie uples o solve. is se, we ik of e igle s lf of sque.

9 piul, e ke sep o is pole is o oe e igle e oied fo e sque ipog efleio se log e = digol. Tis se is efleio o e oodies of e fo :, igefuios, e e lssified s eie -eve o -odd :,, Te odd fuios vis log e digol, so e uoill sisf Diile o e digol. Siill, e eve fuios ve odiios vig ol deivive o e digol d ee uoill sisf Neu odiios. We will use is f o osu TM d T odes fo e igle. We egi wi e TM odes. Ug egul oodies, i is ul o wie soluios of e elol equio s e ik k wee k k. Tis es we epd e eigefuios i es of es d oes. Fo TM odes sisfig e Diile odiio S, we s wi eigefuios o e sque wi uoill sisf e oud odiios o e fou wlls of e sque. Tis gives so e uoff fequeies e () ode o sisf e Diile odiio o e digol, we ke e -odd oiio (TM) is siple o veif,,,, so ll oud

10 odiios o e igle e ideed sisfied. Te uoff fequeies e give (). Noe ee e pojeio eoves e = odes d lso iseies wi. s esul, e iege lels d e ke o sisf e odiio >>. Te lsis fo T odes is siil. oweve, fo Neu odiios, we ke oe oiios s well s -eve eigefuio. Tis gives (T) os os os os wi ideil uoff fequeies s i (). Tis ie, oweve, e lels d e ke o sisf (eep == is o llowed). () Fo e lowes odes of e pe lule e euio os, ssuig e wlls ve lge, u fiie, oduivi. ope e esul wi fo sque guide of side de fo e se eil. Te euio oeffiies e deeied powe d powe loss. We egi wi TM odes. Fo e powe, we eed o opue d k k k k d () is peps esies o opue is iegig ove e sque d e dividig wo fo e igle. Tis is euse e iegio sepes io d iegls, d we use oogoli k i kjd i, j Tis gives d Te fo of / is fo e igle, wile e fo of is euse wo o-vig es ise we squig e iegd i (). (ell fo TM odes.) Tis gives epessio fo e powe / / d P wee / is e e of e igle. lulig e powe loss ivolves

11 iegig ol deivive dl We ek is io ee ps: log =, log = d log e digol =. log e = wll, we ve d s esul k k d (3) siil lulio, o use of se, will esul i ideil epessio fo e iegl log e = wll. Fo e digol, we use o opue os k kos k k Tis gives k k dl d oiig is digol wi (3) fo e sides, we oi dl wee is e iufeee of e igle. Tis gives TM ode powe loss of dp dl d Te euio oeffiie is us

12 / dp Pd so e geoeil fo is ivil. Noe e eeg loss lulio log e digol of e igle gives e se esul s log e sque edges. s esul, e geoeil fo is idepede of wee e wveguide is sque o ig igul. Tis is w e igul TM esul is ideil o e sque TM esul, les up o e ios igle d / / fo e sque. / / 6.83/ fo e Te powe loss fo e T odes is soew de o del wi euse of e possiili of speil ses. oside os k os k os k os k () wee. f =, we ed up wi is se os k os k (>) d d dos k os k wile e peiee iegls e 3 d d osk d dosk wi gives d dl 3 d d d d k

13 d d d k wi gives Ug dl P / d d dp dl d wi e ove iegls gives euio oeffiie / dp 3 Pd wee d / /. ee e geoeil fos e, (>=) Fo e egul wveguide, oe s ised, we Tis is diffee euse e powe loss lulio is o loge uivesl, givig diffee oeffiies log e digol s log e sque edges. Te eiig T ses o oside e odes () wee => d >>. ee we sipl se e esuls. Fo => we ve os k os k

14 (we ve eoved uipo fo of wo) so Tis gives d 8 3 dl 8 dl 8, (=>) O e oe d, fo e geel se >> we fid d dl dl wi ields, (>>) ll ses,, wi is e se fo e igle o e sque wveguide. Fo, e fo is esseill geoei oiio of oiuios log e peiee of eie o / depedig o e piul ode d is degeeies. 8.6 eso vi of oppe osiss of ollow, ig iul lide of ie dius d leg L, wi fl ed fes. () Deeie e eso fequeies of e vi fo ll pes of wves. Wi s ui of feque, plo e lowes fou eso fequeies of e pe s fuio of L of L. Does e se ode ve e lowes feque fo ll L? () f =, L = 3, d e vi is de of pue oppe, w is e ueil vlue of Q fo e lowes eso ode? Sol:

15 () Fo kso (8.8) d (8.83), TM p : L p p p =,,, =,,, =,, 3 T p : L p p p =,, 3, =,,, =,, 3 Te feque of TM is idepede of L. Te foudel ode is eie T o TM depede o L. () 3 L, e fudel ode is TM, L L L L P U Q L d L d P L d U i i L 8 os,

16 fo TM. L Q L wee 8.8 () Fo e use of Gee's eoe i wo diesios sow e TM d ТЕ odes i wveguide defied e oud-vlue poles (8.3) d (8.36) e oogol i e sese,, fo d fo TM odes, d oespodig elio fo fo ТЕ odes. Oogoli is geel pope of e eigefuios of e wve equio. Te geel wo-diesiol equio is give wee eie o TM odes S S T odes To pove oogoli, oe d sisf e equios, Muliplig e fis d e seod d suig gives egig is ove e oss-seiol e, d ug Gee s eoe ields d d dl

17 wee we ve used iwd poiig ol dieio. We ow oe e ig d side vises fo eie TM o T oud odiios. Tus, povided, we ed up wi d ( ) Fo o-degeee eigevlues, olude d fo so log s d e o TM odes (o e o T odes). Noe, fo TM odes, wile, fo T odes. Fo degeee eigevlues, liei of e wve equio guees we fid oogol sis ug, e.g., G-Sid oogoliio poess. () Pove e elios (8.3)-(8.3) fo osise se of oliio odiios fo e fields, iludig e iuses we is TM ode d is а ТЕ ode. We s wi elio (8.3), wi ses d wee,,,, e eie TM o T ode. To dle is epessio, we oe e svese fields fo TM d T odes e give ik TM:,, Z k Z i T:,, Z Z k ee fo wo TM odes, we ed up wi () k k d d dl d,,,,, S,,, Te sufe e vises euse of Diile oud odiios, wile e e e e siplified ug. ee we ive,,

18 k d d,,,, fo () We popel olied fo, is gives (8.3) fo wo TM odes. Te se of wo T odes is siil. We ve d d,,,,,,,, d,, d (3) We ve oed ideill (e e svese gdie is oogol o ẑ ). Te poof of oogoli of wo T odes e follows ug e se iegio eod ws used ove fo e TM odes (u wi epled, d wi / vig o e oud). Fill, fo oe T ode d oe TM ode, we ve k d d,,,, k,, d k,, d k,, dl S Tis iegl vises euse, vises o e oud. s esul, ll T odes e oogol o ll TM odes. Pope oliio e esuls i (8.3). We ow u o elio (8.3), wi ses d,,, Z Te es w o pove is is o oe fo (),, Z fo eie TM o T odes, povided Z is ose odigl. is se

19 d d,,,, ZZ,,,, d Z Z d Z Z Z Z Z,,,, ee we ve de use of e f vises euse is svese o e ẑ dieio. Te ls lie follows fo pplig (8.3), wi we poved ove. Te powe flow elio (8.33) d Z follows siill. Speifill, we ve,,,,, d,, d Z = Z,,,, d d Z Z Z,,,, Te elio (8.3) esseill olies e odes fo e TM d T se. iio of () fo TM odes d (3) fo T odes idies e pope oliio is TM: T:,,, d k d kz,,, 8.9 Te figue pole 8.9 sows oss seiol view of ifiiel log egul wveguide wi e ee oduo of oil e lie eedig veill dise io is ieio. Te ue log e poe osilles usoidll i ie wi feque, d is viio i spe e

20 ppoied s. Te ikess of e poe e egleed. Te feque is su ol e T ode popge i e guide. () lule e pliudes fo eiio of o T d TM odes fo ll, d sow ow e pliudes deped o d fo, fo fied feque. () Fo e popgig ode sow e powe died i e posiive dieio is X P k wi equl ou i e opposie dieio. ee k is e wve ue fo e T ode. () Disuss e odifiios ou if e guide, ised of uig off o ifii i o dieios, is eied wi pefel oduig sufe L. W vlues of L will iie e powe flow fo fied ue? W is e diio esise of e poe (defied s e io of powe flow o oe lf e sque of e ue e se of e poe) iu? Sol: () ( ) M T k ik M e 3 d k os os wee, o,,eie,, o,

21 ik ik ik ik d e d M e M e e M 3 os os os os d d d os os os os os os os os os os os os os os os os ik T e os os wee e ik e N d N k i ik ik TM 3 ) (

22 ik TM ik e i e ik N os os os ) ( () 8 TM T k P ol T,,. k k k 8 8 os os os () Te oigil field is o odified. Bu ddiiol efleed field wi pse diffeee is supeiposed. Teefoe iefeee will ou. Te iu powe will ou osuive iefeee.

23 P d i (), so P P i (). P os 3 8 wee 3 8 os 8 8. ifiiel log egul wveguide s oil lie eiig i e so side of e guide wi e i el oduo foig seiiul loop of dius wose ee is eig ove e floo of e guide, s sow i e opig oss-seiol view. Te lf-loop is i e ple = d is dius is suffiiel sll e ue e ke s vig os vlue evewee o e loop. () Pove o e ee e ue is os oud e lf-loop, e TM

24 odes e o eied. Give psil eplio of is lk of eiio. Te field i e wveguide e wie s wee e oeffiies e give q. (8.6): Z 3 Z d dl V oose e oo-lef oe of e guide s e oodie oigi wi e -is log e edge d e -is log e edge. os ee is e pol gle wi espe o e ee-of-e loop. Tus Z / / os ( d ) Z os d / / wee d e - d - opoes of e eige-field log e loop. Fo TM wves, e elei field opoes e give q. (8.35): ee Teefoe, os os os os / Z os os os os os / Z / d os os d d / d d d Z / d os d / d / Z os / ee, o TM odes e eied. Tis is euse iul ue i e

25 svese ple will lws esul i o-vig logiudil opoe of, i.e.,. () Deeie e pliude fo e lowes ТЕ ode i e guide d sow is vlue is idepede of e eig. Fo T wves, os os os os wi e oliio edued fo of if = o =. Tus / Z os os os os os / d Te lowes odes ( =, = ): Z os Z d, /,, 3 os / 3,, wee, /. ee we ve used e iegl epeseio of Bessel fuios: / d d os os / Te pliude is idepede of e eig. Fo, Z 3,, 5, 8 () Sow e powe died i eie dieio i e lowes ТЕ ode is P Z 6 wee Z is e wve ipede of e T ode. ee ssue,. Te vege powe died i eie dieio

26 P d d d Z is se, P, Z, Z, 6