Chapter 8: Waveguides, Resonant Cavities, and doptical lfibers

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1 清大物理系電動力學 ( 二任課老師 : 張存續 lrodmis (II (PYS 53 Sprig Smsr, Dprm of Phsis, Niol Tsig u Uivrsi, Tiw Tl. 4978, -mil: hshg@phs.hu.du.w Offi hour: 3:3-4:3 47 助教 : 趙賢文 : , s9887@m98.hu.du.w 張家銓 : , @homil.omom.Too d Cos of h Cours: J. D. Jso, Clssil lrodmis, 3rd diio, Chprs 8-, 4. Ohr oos will rfrd i h lur os wh dd.. Codu of Clss : Lur os will projd squill o h sr durig h lss. Phsil ops will mphsid, whil lgri dils i h lur os will of sippd. usios r ourgd. I is ssumd h suds hv ls go hrough h lgr i h lur os for dig lsss (impor!. 3. Grdig Poli: Midrm (4%; Fil (4%; ui 4 (% d r pois (%. Th ovrll sor will ormlid o rfl vrg osis wih ohr ourss. 4. Lur N Nos: Srigi from si quios, h lur os follow Jso losl wih lgri dils filld i. quios umrd i h form of (8.7, ( rfr o Jso. Supplmr quios drivd i lur os, whih will lr rfrd, r umrd (, (... [rsrig from ( i h hpr.] quios i Appdis A, of h hpr r umrd (A., (A. d (., (. Pg umrs id i h (.g. p. 395 rfr o Jso. Sio umrs (.g. S. 8. rfr o Jso (p for sios i Ch.. Mi opis wihi h sio r highlighd oldfd hrrs. Som words r pd i iliid hrrs for io. Thil rms whih r irodud for h firs im r udrlid. Chpr 8: Wvguids, Rso Cviis, d dopil lfirs 8. Filds h Surf of d Wihi Good Coduor Noios:, : filds ousid h oduor; C, C: filds isid h oduor; : ui vor o oduor surf; : orml oordi io h oduor.,, i Assum: (i filds (ii good u o prf oduor, i..,,, u [S Ch. 7 of lur os, q. (4]. (iii ( is ow. Fid: (, (, d powr loss,. i rms of ( Th mi rsuls i S. 8. [ (8.9, (8., (8., (8.4, d (8.5] hv drivd wih muh simplr mhod i Ch. 7 of lur os. [S os followig q. (6]. So, w will o ovr his sio i lsss.

2 8. Filds h Surf of d Wihi Good Coduor (oiud Clulio of C (, C( : I h oduor, w hv ( i good oduor ssumpio J D ( i I good oduor, filds vr rpidl log h (3 orml o h surf, s Ch. 7 of lur os. (4 (, (, (3 i (5 C si dph i Su. (4 io (5: (8.7 i ( (.. : ( ( i i ( ( ( (6 J= (5 ( ( ( (7 i Su. (7 io (6 ( ( o ( Filds h Surf of d Wihi Good Coduor (oiud i i C i C i Su. ( ( io ( ( ( ( ( (8. ( ( ( ( ( (8... (4 dp loss d Powr Loss Pr Ui Ar: C C C im vrgd powr io oduor pr ui r R ( ( R ( ( C ( ( (8. 4 ( C 8. Filds h Surf of d Wihi Good Coduor (oiud Alriv mhod o driv (8.: ( i (8. J( ( ( i ( (8.3 im vrgd powr loss i oduor pr R J ( ( J ( ui volum (8.3 dp loss d ( ( d d J (, sm s (8. ffiv surf urr K ff f K ( i ( f f d ( i ( J d ( (8.3 (8.4 dp (8. & (8.4 loss Kff (8. 5 d : Mods i Wvguid Cosidr hollow oduor of ifi i lgh d uiform ross sio of rirr shp (s figur. W ssum h h fillig mdium is uiform, lir, d isoropi ( ; D, whr d r i grl ompl umrs. This is sruur ommol usd o guid M wvs s wll s rr s whr soluios r possil (for som simpl ross sios. Mwll quios wri (8, (9 ompl d ( ( (8 ( ( Similrl, ( 9 (3

3 Mods i Wvguids (oiud i (, ( i : oordis rsvrs o, L,.g. (, or ( r, i i (, ( hr i Jso whr, i grl, d r ompl oss. To spifi, w ssum h h rl prs of d r oh posiiv. Th, i i i d i hv forwrd d wrd phs vloiis, i rspivl. As will s i (3, i i d i lso hv forwrd d wrd group vloiis, rspivl., w ll ii i i forwrd wv d wrd wv. Wih h ssumd d dpds, w hv +, Crsi r, r r r r, lidril Mods i Wvguids (oiud Thus, ( (8.9 ( ( (4 ( I is i grl o possil o oi from (8.9. So our srg hr is o solv (4 for ( d (, d h prss h ohr ompos of h filds [ ( d ( ] i rms of ( d ( hrough qs. (7 d (8. ris : Wriig ( r r d usig h lidril oordi ssm, driv h quios for r d from (8.9. (hi: =, = r r Griffihs 9.5 Guidd Wvs 95W 9.5. Wv Guids C h lromgi wvs propg i hollow ml pip? Ys, wv guid. Wvguids grll md of good oduor, so h = d = isid h mril. Th oudrodiios hir wll r: // d Th gri form of h moohromi wvs: (,,, ˆ ˆ ˆ (, ( (,,, ˆ ˆ ˆ (, ( i ( i ( i( i( Griffihs Grl Propris of Wv Guids I h irior of h wv guid, h wvs sisf Mwll s quios: Wh f d J f? whr v v W oi i (i i (iv i (ii i (v i (iii i (vi

4 Griffihs T, TM, d Wvs Drmiig h logiudil ompos d, w ould quil lul ll hohrs ohrs. i ( ( / W oi v v i ( ( / i ( ( / i ( ( / Tr o driv hs rlios ourslf. If T (rsvrs lri wvs; If TM (rsvrs mgi wvs; If d wvs. 3 L i Grl Approh r r r, Crsi i i (5, lidril i i ( Usig h rlios: ( ompos of (5 d (6: i i (6, w oi from h rsvrs (7 i i (8 uppr forwrd lowr wrd I (5-(8, h sig pplis o h wv. Rwri (7 d ( Mods i Wvguids (oiud i i (7 i i (8 Si d hv lrd solvd from (4, (7 d (8 r lgri (rhr h diffril quios. W ow mipul (7 d (8 o limi i d hus prss i rms of d. (7 ( i i If, r oh idpd of, h i i (9 Su. (9 io ( Mods i Wvguids (oiud i i i i i ( Mulipl ( i : i i i (8.6 Simi lrl, i (8.6 Thus, o d hv solvd from (4, h soluios for d r giv (8.6 d (8.6.

5 Mods i Wvguids (oiud Disussio: i (i,,, i (8.6 d (8.6 r fuios of ol. (ii d ompl. Im( or Im( implis dissipio. (iii lig =, w m oi s of soluios for,, d from (4, 8.6, d (8.6, rspivl. I show h if h oudr odiio o is sisfid, h oudr odiios o d r lso sisfid., his givs s of vlid soluios lld h TM (rsvrs mgi mods. Similrl, lig =, w m oi s of vlid soluios lld h T (rsvrs lri mods. (iv is h grig fuio for h TM mod d is h grig fuio for h Tmod. Th grig fuio is dod Ψ i Jso Mods i Wvguids (oiud TM Mod of Wvguid ( =: (s pp s ( wih oudr odiio ( i ( Assum prfl oduig wll. ( Z Z wv impd, of TM mods ( T Mod of Wvguid ( =: (s pp s i Z h, wv s impd of T mods.. s Zh s ( s s ( wih oudr odiio ( ( ( ( Mods i Wvguids (oiud Disussio: (i ihr ( or ( osius igvlu prolm (s lur os, Ch. 3, Appdi A. Th igvlu will ifii s of disr vlus fid h oudr odiio, h rprsig igmod of h wvguid (A mpl will providd low. (ii ( d ( show h is prpdiulr o (lso ru i vi. (iii ( d ( show h d r i phs if,,, r ll rl (o ru i vi. (iv ( [or (] is h disprsio rlio, whih rls d for giv mod. (v Th wv impd, Z or Z h, givs h rio of o i h wvguid. Fild Prs of Cirulr Wvguid Mods -fild ( Isrio o Loss (d d ( Is srio Loss (d surf urr d (mm L (mm T T T 3 T TM TM T T T 3 T TM TM

6 Fild Prs of Cirulr Wvguid Mods -fild ( Isrio o Loss (d d ( Is srio Loss (d surf urr d (mm L (mm T T T 3 T TM TM T T T 3 T TM TM Chrriio of Cirulrl Smmri T Mod ( simulio S S (d - -4 = o = 6 o = 9 o = 45 o = 8 o Frq (G ( msurm (d S G = o = 45 o = 6 o 8.5 G = 9 o = 8 o = 8 o Th Frq (G T.. Chg d. R. Yu, Yu igh-powr Millimr-Wv Ror Joi, Rv. Si. Isrum. 8, 347 ( Mods i Wvguids (oiud Mod of Coil d Prlll-Wir Trsmissio Lis ( = = : (s Jso p. 34 i Rwri i (8.6 (8.6 Ths quios fil for diffr lss of mods, lld h (rsvrs lromgi i mod, for whih hih. owvr, h giv h odiio for h is of his mod: quios i rgulr os r. (8. si quios for h mod. 7 (8.7 is lso h disprsio rlio i ifii sp. This ms h mod vr usful us i propg frqu. To lul d, w d o go o Mwll quios Mods i Wvguids (oiud ( ( L d, i i h, us, h -ompo of givs (, A ( d, us, givs A ( ( (, wh r is h grig fuio for h mods. us o h surf of prf oduor, is suj o h oudr odiio os. o h oduo r. This giv s os. or vrwhr, if hr is ol o oduor. So, mods is ol i -oduor ofigurios, suh s oil d prlll-wir rsmissio lis. Fill, is giv h rsvrs ompos of :.

7 Wh sigl oduor o suppor wvs? (I L s osidr h propr of D Lpl quio. Suppos Φ dpds o wo vrils. pril diffril quio (PD; o ordir diffril quio (OD. rmoi fuios i wo dimsios hv h sm propris s w od i o dimsio: Φ Φ hs o lol mim or miim. All rm our h oudris. (Th surf m o qulpoil. If Φ =os., = & = (No flwlss rgum Wh sigl oduor o suppor wvs? (II Dvid Chg s plio. Chp., p.55.. Th mgi flu lis lws los upo hmslvs. For wv, h mgi fild li would form losd loops i rsvrs pl.. Th grlid Ampr s lw rquirs h h li igrl of h mgi fild roud losd loop i rsvrs pl mus qul h sum of h logiudil oduio d displ oduio urr isid h wvguid. 3. Thr is o logiudil oduio urr isid h wvguid d o logiudil displ urr ( =. 4. Thr o losd loops of mgi fild lis i rsvrs pl. (w olusio Th wv o is i sigl-oduor hollow wvguid of shp. (Agi, o prf rgum Mods i Wvguids (oiud I summr, h mods r govrd h followig s of quios: ( (3 ( (3 (3 (or Y (3 whr Y ( / is h (irisi dmi of h fillig mdium dfid i Ch. 7 of lur os (h ls pg of S. II. Disussio: : Mods i Wvguids (oiud (i For h mods, w solv -D quio ( for (. u his is o -D prolm us is o h (, full soluio Th full soluio is ( i i (, ( full soluio. Th full soluio is ( wih ( d. For ul -D lrosi prolm [ ( ( ], w hv (, whih h givs h full soluio ( (. (i i No h diffr w h slr poils disussd hr d i Ch. d Ch. 6. ( rgrd s mhmil ool. ( ( rgrd s phsil qui. (, (, A (, rgrd d As mhmil ools.

8 Mods i Wvguids (oiud mpl : T mod of rgulr wvguid Rwri h si quios for h T mod: s ( wih oudr odiio ( i ( Zh ( ( Rgulr gomr Crsi ssm, h wv quio i ( oms: ( Mods i Wvguids (oiud Rwri (4: (4 i i Assumig dpd for I ordr for, w mus hv,, w oi i i whih his sisfid dfor,,. Si (,, i i i i (,, d (, r ll lirl idpd pirs, h ompl soluio for is i i i i i A A i i C C ( Mods i Wvguids (oiud Applig oudr odiios [s (] o (5: i i i i i i i i A A C C i AiA A A ii o i i ii os os C C si m, m,,, si,,,, m i os os i ii C C (6 forwrd wv wrd wv Su. m, io, w oi m, m,,,, (7 Griffihs 9.5. T Wvs i Rgulr Wv Guid, d (, X( Y( sprio of vrils X Y X Y v ( X Y d X Y wih Griffihs drivio v uss diffr oudr odiio --- =. X ( A si os Y( Csi Dos

9 Griffihs T Wvs i Rgulr Wv Guid (II Cos Dsi i (@ C (@ si, (,,,... Aos si (@ A m (@ si,, ( m,,,...,,, (, os( m / os( / ht mod m v m ( / [( / ( / ] Mods i Wvguids (oiud Rwri (7 s, (8 m / whr m (, m,,,, (9 m h pir of ( m, givs orml mod (T m mod of h wvguid. m d o oh, us h will r siuio o (8.6 or (, mig d idrmil. m is h uoff frqu (h frqu whih of h wvduid for h T m mod. Wvs wih m o propg s T mod us oms purl imgir. m (8 is h T m mod disprsio s rlio of wvgud id filld wih dilri mdium wih os (i grl ompl d. Forhusulsofufilldwvguid of ufilld wvguid, w hv d (, d (8 (9 wri / for ufilld wih ( m m m (3 wvguid m m, m Mods i Wvguids (oiud m g guid wvlgh uoff wvlgh m dpd mod & wv o gomr frq. f fr sp wvlgh g s s m rl propgig wvs s o m g f o s m imgir vs filds T ( ( =.9 T, T ( ( = T ( ( = usl dwidh ( f usio : A pil wvguid hs =. Wh? usio : C w us wvguid o rspor wvs 6? f g Mods i Wvguids (oiud Ohr quiis of irs: ( Diffriig m wih rsp o d m d v g d group vloi i ufilld wvguid d v g vg s m ( Th rmiig i fild ompos (,,, d oid from hrough i ( m s ( d (3. ( uppr forwrd whr h sig pplis o h wv. lowr wrd

10 Mods i Wvguids (oiud T mod fild prs of rgulr wvguid Mods i Wvguids (oiud TM mod fild prs of rgulr wvguid from. L. Gio, "Mirowv msurms". : uoff frqu solid urv: -fild lis; dshd urvs: -fild lis from. L. Gio, "Mirowv msurms". : uoff frqu solid urv: -fild lis; dshd urvs: -fild lis Mods i Wvguids (oiud Disussio : Wvguid d mirowvs A pil wvguid hs o mimi h usl dwidh ( f ovr whih ol h T mod propg p d h mod puri is miid. Wvs r ormll rspord h T mod ovr his frqu rg. Wvguids om i diffr sis. Usl dwidhs of wvguids of pril dimsios (. m m ovr h ir mirowv d (3 M o 3 G. Comprd wih oil rsmissio lis, h wvguid is pl of hdlig muh highr powr., i is ommol usd i high- powr mirowv ssms. I rdr ssm, for mpl, i is usd o rspor mirowvs from T ( ( = h gror o h. T, T ( ( = T ( ( = usl dwidh ( f Mods i Wvguids (oiud mpl : : mods of oil rsmissio li mods r govrd h followig s of quios: ( (3 ( (3 Y (3 (3 (3 givs r ( r. r r r Ngl h mods r r ( r r CI( r C. ( r V C V / I( / I( r/ Appl.. V. / I( ( r C CI( V (, / i i I r r (3, h giv ( YV ( / i (3 (, i I r

M j P m iig Spifi Mod j ˆ A ol 4 4 P 4 p 4 ( p J ( p ol 4 4 P4 p4 ( p4 4 J4 ( p4 8 d ( T ( T sio (d Trsmiss ( T 4 (d Trsmissio d smissio (d Tr - - -3-4 -5 Th. u 8.3 G Th. u Th. u p.

Phs. L. 93, 53 (8. Diffiulis of iig ighr-ordr Mod: T T s mpl Dsird mod T Couplig sruur o-fd Wvguid rdius.

11 M j P m iig Spifi Mod j ˆ A ol 4 4 P 4 p 4 ( p J ( p ol 4 4 P4 p4 ( p4 4 J4 ( p4 8 d ( T ( T sio (d Trsmiss ( T 4 (d Trsmissio d smissio (d Tr Th. u 8.3 G Th. u Th. u p Frqu (G Th. u 4. G Th. u Th. u p Frqu (G Th. u Th. u. G Th. u p Frqu (G T.. Chg, C. S. L, C. N. Wu, d C. F. Yu, iig irulr Tm mods low rhr rgio, Appl. Phs. L. 93, 53 (8. Diffiulis of iig ighr-ordr Mod: T T s mpl Dsird mod T Couplig sruur o-fd Wvguid rdius.86 mm Prsii mods T,A, T, T,A, T, T T 3,A, T 3, T,A, T, TM TM,A, TM, T M,A, TM, T 4,A, T 4, T,A, T, TM TM 3,A, TM 3, T 5,A, T 5, T,A, T, TM,A, TM, Appliios of Wvguid Mods (I Appliios of Wvguid Mods (II Plsm hmrs Mril prossig Mod ovrrs Ror jois Appl. Phs. L. 94, 4 (9 T wvguid, irulor, isolor, powr dividr,

12 8.7 Mods i Cviis W osidr h mpl of rgulr vi (i.. rgulr wvguid wih wo ds losd oduors, for whih w hv wo ddiiol oudr odiios hds. Rwri (7: m i os os i i i C C ross -siol.. (i: ( C C viw i m os os si o.. (ii: d sid viw ( si d l d, l,, o (3 d i m l m,,,, os os si, (33 d l,, Su. (3 io, whr ( m m m ml ( m l ml d / :rsofrq u (34 of h T ml mod 8.5 rg Flow d Auio i Wvguids Powr i Losslss Wvguid : Cosid r TM mod (, i mdium wih rl, (h rl,. TM S [ ] ompl Poig vor [ ( ( ] for TM mods (, [ i ] 4 rl d ( i [ ] 4 P TM im vrgd powr i h -dirio A [R STM] d [ A: rossiol r] ( (35 4 A d 8.7 Mods i Cviis (oiud C+ m i From (6 : os os i ii C C, w s h vi mod is formd of forwrd wv d wrd wv of qul mpliud. Th forwrd wv is rfld h righ d o om wrd wv, d urs io forwrd wv gi h h lf d. Th forwrd d wrd wvs suprpos io sdig wv [s (33]. Thus, w m oi sid viw h ohr ompos of h vi fild suprposig h ohr ompos of forwrd wv wrd wv h wo rvlig wvs, s i (6. d Compriso wih viriol mods of srig: dpd vril(s idpd vrils mod id srig (osillio i mp., l vi,,,, (or,,, m,, l C 8.5 rg Flow d Auio i Wvguids (oiud 3 ( d d d Gr's firs idi: v s (.34 L d idpd of d ppl (.34 o sl of d surf r A (o h - pl d ifiissiml hiss i, surf igrls o A( d dl wo ds of h C sl, whih vish. A ( d dl C L d, h d A ( d [ dl ] C A d = oudr odiio i dl (4 A d. (36 Su. (36 io (35: PTM (, w oi 4 A d PTM, [whr ] (37 A d

13 8.5 rg Flow d Auio i Wvguids (oiud (i.. is h (38 uoff frq. of h mod. ( ( (39 Su. (38 d (39 io (37 TM ( ( P d [f. (8.5] (4 A Similrl, for h T mod d rl,,, d, w oi from (, (, d (, i T 4 [ ] T [ R T] A S A ( ( A d S (4 P d d [f. (8.5] (4 No: P d P r prssd i rms of h grig fuio. TM T 8.5 rg Flow d Auio i Wvguids (oiud : m rg i Losslss Wvguid : s S d v Jd i v w w d (6.34 w D if, r rl, w d wm r 4 4 lso rl d rprs im (6.33 w m 4 4 vrgd fild rg dsiis. Appl (6.34 o sio of losslss S, wvguid [i.., r rl d h wll = oduivi ]. S ( 3 (isid volum J o oh ds v J d o h sid wll S o h sid wll, (h, r rl d r i phs [ (&(] is rl S is rl o oh ds s Sd is rl R[(6.34] s S d (o powr io or ou of volum 3 3 Im[(6.34] v wd v wmd (-fild rg -fild rg ( 8.5 rg Flow d Auio i Wvguids (oiud For h TM mod ( : UTM fild rg pr ui lgh ( A( w wm d A wmd A d A d 4 ( (43 A d A d A d ( A d (36 Similrl, for h T mod ( : UT A Wd A d ( (44 A d From (4, (4, (43, d (44 Us (, d P TM PT Gr s s idi ( vg (8.53 UTM U T v (39 d (. p vpvg d ( rg Flow d Auio i Wvguids (oiud Auio i Wvguids Du o Ohmi Loss o h Wll: W prss for losslss ( d loss ( wvguid ( (, s (8.55 i, ( whr is h soluio of h disprsio rlio for, i.. [drivd i (8] (45 Th prssio for i (8.55 ssums h h wll loss ( modifis smll rl pr d smll imgir pr, whr d r o drrmid. d Phsil rso for α : ffiv wvguid rdius irss mou si dph. A lrgr wvguid hs smllr.,. Phsil rso for β : Powr dissipio o h wll.

14 ( 8.5 rg Flow d Auio i Wvguids (oiud I i, is o of primr irs us i modifis h guid wvlgh slighl. owvr, rsuls i uio, whih vr sigifi ovr log dis. W ouli low how vlud. powr dissipio/ui lgh i i P powr flow R[ ] i dp P d dp d P (8.56 fild uio os (8.57 (8.5 ff dl K (46 (8.4 K ff dl (47 (46(47 dp (8.58 d dl Si h wll loss rgrdd s smll prurio, w m us h ro-ordr drivd for i S.8. o lul dp. d 8.5 rg Flow d Auio i Wvguids (oiud Spifill, w lul h ro-ordrordr d, d us h ro-ordr d o lul P from (8.5 d dp/d from (8.58. is h foud from (8.57. Formul for for rgulr d lidril wvguids r ld uld i m mirowv oos,.g. R.. Colli, Foudio of Mirowv girig ( d d. p. 89 & p.97 (whr h uio os is dod isd of. No: (i hs luld prurio mhod. Th mhod is ivlid r h uoff frqu, whih hr is lrg prurio. S 8.6 givs mhod whih luls oh d (du o wll loss vlid for ll frquis. (ii Ohr ps of losss (.g. loss fillig mdium or ompl lso oriu o d. 8.5 rg Flow d Auio i Wvguids (oiud (iii No hr r wo dfiiios of h uio os. I Ch. 8 of Jso, h uio os for h wvguid is dod d i is dfid s P dp, (8.57 d This is h fild uio os, i..,. I Ch. 7 of Jso, h uio os for uiform mdium is dod [s (7.53] d i is dfid s P dp d This is h powr uio os, i.. P Oviousl, h powr uio os is wi h vlu of h fild uio os. Trhr Wvguid (I K. Wg d D. M. Milm, Ml wirs for rhr wv guidig, Nur, vol.43, No. 8, p.376, 4.

15 (d d/m.... Trhr Wvguid (II: Usig Th Lows Loss T Mod T 5 T 4 T 3 T T Frq(G Rfrs. Por, p.6.. Colli, p.97. : ow o i h T mod d fri i h rhr rgio? A possil soluio: X-r miro-friio (LIGA. 8.8 Cvi Powr Loss d Dfi Dfiiio ii of f : W hv so fr ssumd rl for M wvs i ifii sp or wvguid. Si filds r sord i vi, i dmps i im if hr r losss, rprsd ompl. Thus, filds poi i h vi hv h im dpd giv i, ( ( i i i( (8.88, whr is h rso frqu [.g. (34] wihou h wll loss. (8.88 ssums h h wll loss modifis smll rl pr d smll imgir pr, whr d d r o drrmid. Phsil rso for Δ ω : ffiv vi si irss mou si dph. A lrgr vi hs lowr frqu.,. Ph sil rso for : powr dissipio o h wll 8.8 Cvi Powr Loss d (oiud i i sord rg i h vi [ i ] U i ( ( du U (powr loss (8.87 d i U sord rg (im-sp dfiiio of (8.86 powr loss (8.88 rprss dmpd osillio whih dos o hv sigl frqu. To m h frqu of (. w wri whr i ( ( d, i Us (8.88, ssum ( = for < i ( ( ( i d d i( 8.8 Cvi Powr Loss d (oiud Th frqu sprum is s s form h fild rg disriuio i -sp m, ( (8.9 m, ( full widh hlf-mimum pois (frqu-sp dfiiio of (8.9 No: Fig. 8.8 is h rso frqu of h vi i h s of loss. is h rso frqu i h prs of losss. I mos ss, h diffr is isigifi.

16 8.8 Cvi Powr Loss d (oiud Phsil Irprio of : sord rg (i Us h im-sp dfiiio: powr loss f wv priod sord rg d im of d powr loss sord rg sord rg d (48 powr loss (48 shows h, whih h rsuls from h powr loss, is pproiml ims h umr of osillios durig h d im. A lrgr vlu implis h h fild rg sord i h vi for logr im., is of rfrrd o s h quli for. 8.8 Cvi Powr Loss d (oiud (ii Us h frqu-sp dfiiio: (s Fig. 8.8 For loss vi, rso mod id o jus o frqu (s is h s wih losslss vi u rg of frquis (. Th rso frqu ( + s Fig of loss vi is h frqu whih h vi id wih h lrgs isid-fild mpliud, giv h sm sour powr. Th rso widh of mod is qul o h rso frqu dividd Fig. 8.8 h vlu of h mod (s Fig No h h mod hs diffr vlu. Figur 8.8 sil grd i prim o msur h vlu. 8.8 Cvi Powr Loss d (oiud sord rg powr loss Usig h rsuls of S. 8., w lul (u o du o h ohmi loss. W firs lul h ro ordr d of spifi vi ssumig, h us h ro ordr d o lul U d powr loss, 3 3 d v 3 3 wd 3 v sord rg v ( w wm d v wmd (8.5 v powr loss ff s K d (6.33 s d (8.4 d 8.8 Cvi Powr Loss d (oiud Formul for (du o ohmi loss for rgulr d lidril viis foud i, for mpl, R.. Colli, "Foudio of Mirowv girig", i p. 53 d p. 56. du o oh r ps of losss : If hr r svrl ps of powr losss i vi (.g. du o Im d ouplig losss, prssd s follows: sord rg (49 (powr loss -h p of powr loss (5 whr ( du o h -h p of powr loss is giv sord rg (powr loss

17 8.8 Cvi Powr Loss d (oiud A Compriso w Wvguids d Cviis Wvguid Cvi. Fuio rspor M rg sor M rg igh- Mirowv/Mril Applior Chrri- disprsio rlio d rso frqu io uio os d mpls of rspor of high ( pril lrio ppliios powr mirowvs ( frqu msurm (mosl for (suh s muli-w mirowvs, wvs for log-rg G rdrs d ommuiios sord rg (powr loss Coduor loss, dilri loss, rdiio loss, diffrio loss omwor of Chp. 8 Prolms:, 3, 4, 5, 6, 8, 9,

LINEAR 2 nd ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

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