UNIEITY OF TOONTO Dparmn of Elcrical and ompur Enginring Filds and Wavs aboraory ourss EE 3F and EE 357 III Yar EONANT AITY upplmnary Insrucions
EONANE Pag of 3
Pag 3 of 3. Inroducion, gnral rsonanc A linar, singl inpu rsonan sysm is an assmbly of obcs xhibiing all of h following propris: (i) Powr inpu by xrnal sourc a a frquncy producs a sady sa rspons a h sam frquncy. (ii) om of h nrgy supplid by h sourc is sord in h sysm. (iii) Thr xiss a las on frquncy, such ha no porion of powr absorbd by h sysm a his frquncy is rurnd o h sourc. Th dfiniion givn abov will b illusrad for hr sysms considrd blow. In all hr cass h sourc of nrgy is a gnraor of EMFo cos and inrnal rsisanc s. as (i): Th gnraor driving a load rsisanc as shown in Fig.. i A O - B Figur. Gnraor driving a load rsisanc. (a) ysm rspons: currn i, i cos. () (b) Powr supplid by h sourc p s, p v i cos. () s AB ( )
(c) Powr absorbd by h load p, Pag 4 of 3 p v i cos. (3) ( ) (d) Enrgy sord by h load, T T ( p p ). s d d o Th sysm is linar bu no rsonan. o (4) as (ii) Th gnraor driving an, circui shown in Fig. i v A O - v B Figur. Gnraor driving, circui. (a) ysm rsponss: currn i, capacior volag v, rsisor volag v and load volag v AB. i ( ) ( ) { [ ] }. (5a) wih an ( ). i cos(. { [( ) ] } ) (5b)
Pag 5 of 3 [ ] [ ] { } 9 cos v o (5c) (b) Powr supplid by h sourc p s, [ ] [ ] { }. wih an, cos cos p i i v p s i AB s (6a) [ ] [ ] ( [ ] ) [ ] [ ] { }. cos cos cos cos ps o o (6b) Th rm [ ] cos cos varis in h cours of a cycl bwn posiiv valus, indicaing nrgy flow ino h sysm, and ngaiv valus indicaing flow of nrgy ino h sourc. (d) Enrgy sord in h load, [ ]. 9 cos o v c (7) I is apparn from h abov ha nrgy varying wih im is sord in h sysm du o h prsnc of h capacior, a circui lmn capabl of soring lcric nrgy. Th sysm hus is linar, capabl of soring nrgy bu is no a rsonan sysm bcaus i xchangs nrgy wih h sourc. as (iii) Th gnraor driving an,, circui shown in Fig. 3.
Pag 6 of 3 O - v v A B l Figur 3. Gnraor driving,, circui. (a) ysm rsponss: currn i, capacior volag v c, inducor volag v and load volag v AB. Th currn i, i (8a) wih. an If on inroducs h symbol h xprssion for i bcoms [ ] { } [ ] { } cos. i (8b) Th capacianc volag v is
Pag 7 of 3 [ ] { } [ ] { }. 9 cos v o (8c) Th inducanc volag is, by analogous procdur, [ ] { }. 9 cos v o (8d) Th load volag v AB is,. an cos AB v wih (8) (b) Powr supplid by h sourc, p s is [ ] [ ] { } ( i v p AB s cos cos ) (9a) On obsrvs ha for h cas of boh phas angls and ar zro so ha h xprssion for powr p s dlivrd by h sourc is proporional o cos and is always posiiv, implying ha no powr is rurnd o h sourc. Th rquirmn ha can b rformulad in h form. Bcaus of h prsnc of capacianc and inducanc h sysm is capabl of soring nrgy. I is hus apparn ha i saisfis all hr rquirmns (i), (ii), and (iii) characrizing a rsonan sysm, and is hrfor an xampl of such, rsonaing a frquncy f, π π f. ()
Pag 8 of 3 om addiional faurs of h... circui considrd will b prsnd in paragraphs (c) and (d) blow. (c) Powr p dissipad by h load a rsonanc, wih is p i cos, () [ ( )] which is qual o h powr supplid by h sourc a rsonanc as vidn from Eq. 9a. (d) Enrgy sord by h load a rsonanc, i.. a, is i v c, wih i and v c dsignaing currn and capacianc volag ampliuds rspcivly a. (a) ubsiuion from Eq. 8b and 8c rducs h abov rlaion o cos sin (b) ( ) ( )( ) Bu, so ha ( ) ( ) ( ) cos sin (c) I is apparn from h abov rsul ha in h,, rsonan sysm considrd h oal nrgy sord a rsonanc dos no vary wih im. This faur as obsrvd in h spcial sysm discussd is an illusraion of gnral propry of all linar rsonan sysms, sad hr wihou proof, of soring im indpndn oal nrgy a rsonanc.. ris rsonan circui
Pag 9 of 3 Th cas (iii) sysm dscribd abov and shown in Fig. 3 is a sris rsonan circui, a sysm commonly mployd as such, or as a convnin approximaion of ohr rsonan sysms. For hs rasons som addiional propris hrof will b dscribd blow. A faur of inrs is h bhaviour of h circui a frquncis clos o rsonanc. In h discussion o follow w shall mploy sandard phasor quaniis I and insad of insananous circui quaniis i and v usd arlir. Th common loop currn I is, from h diagram of Fig. 3, givn by h rlaion I (3a) [( )] ( ) Whn h xprssion for h rsonan (angular) frquncy quaion h xprssion for h currn I bcoms I I ( ) ( ) ( ) ( ) ( ). is inroducd ino h abov (3b) Th final form for I obaind abov will now b formulad in convnin dimnsionlss paramrs. W shall also limi h frquncis considrd o valus clos o rsonan frquncy. I is usful o obsrv ha h rm Furhrmor h rm is h magniud of h racanc of h capacianc a frquncy. and is h raio of lossy porion of h circui X impdanc and a rprsnaiv of raciv porions. I should b born in mind ha for h magniud of h capaciiv racanc X. Th rciprocal of h rm is qual o h magniud of h induciv racanc X ( X ( ) X X ( ) ), is calld h qualiy facor of h circui and will b shown blow o b a masur of h rlaionship bwn powr dissipad in h circui and nrgy sord hrin, and will also govrn h frquncy bhaviour of h circui in h viciniy of h rsonanc.
Pag of 3 I should also b nod ha for frquncis clos o rsonanc i is convnin o approxima h valu by, xcp in h rm -. Whn h rm and h approximaion of o is inroducd ino Eq. 3b, h xprssion for h currn I rducs o I X [ ( ) ]. (4a) Th abov rlaion may also b rformulad by saing ha h loop impdanc Z in h viciniy of rsonanc is [ ( ) ] Z X. (4b) Th quaion for currn I drivd abov xhibis svral faurs characrisic of all rsonan phnomna: (i) A rsonanc h driving volag and h rspons I ar in phas du o h disapparanc of h imaginary rm ( ) and, (ii) h drivn currn ampliud is maximum, I max X. Th wo obsrvaions can b considrd o b h consqunc of h fac ha a rsonanc h loop impdanc is pur rsisanc. I should b mniond ha alhough in h circui considrd h rspons currn ampliud is maximum a rsonanc, hr ar circuis whr h currn ampliud is minimum. I is, howvr, in all cass in phas wih h driving volag. W shall nx considr h off rsonanc bhaviour of h circui. As h frquncy movs away from h rsonanc h imaginary rm & incrass h magniud of h dnominaor in Eq. 4a for h currn I, rducing h ampliud hrof and inroducing a phas shif bwn h phasors of h driving volag and h currn I. Inrsing condiions ar obaind whn h magniud of h imaginary rm in h loop impdanc rachs h valu of h ral valu hrin, i.. whn
Pag of 3 ±. (5a) A his frquncy h magniud of h impdanc incrass from is rsonan valu of X o X, rducing h valu of loop currn ampliud o of is rsonan valu. Th powr dissipad in h circui is ( ) I and h rducion of h currn ampliud by h facor is accompanid by h rducion of h powr dissipad o on-half of is valu a rsonanc. Exprssd in dcibl unis h lvl of powr rducion is 3dB. Th frquncy shif - from rsonanc o 3dB powr rducion is, from Eq. 5a, ±. Th rang of frquncis for which powr absorbd (and dissipad) lis bwn maximum a rsonanc and 3dB powr rducion is hus, (5b) and is calld h 3dB bandwidh of h circui, and is ofn mployd as a masur of frquncy rang of h rsonan circui ffciv loop impdanc. I follows from Eq. 4b ha h loop impdanc Z, approximad in h nighbourhood of rsonanc by xprssion X incrass rapidly as h frquncy movs away from h rsonanc wih h raciv par dominaing h magniud. Th xac valu of h raciv par, ( ) X is sn o approach larg [ ] capaciiv racancs for low frquncis, and larg induciv for high frquncis. As a consqunc h abiliy of h circui o absorb powr a frquncis significanly rmovd from h rsonanc is srongly rducd. An imporan aspc of h circui prformanc is h phas bwn h driving volag and h drivn currn I(angl of Eq. 8a). As was mniond arlir, a rsonanc h wo quaniis ar in phas. As h frquncy movs away from h rsonanc h loop impdanc acquirs raciv componns affcing h phas rlaionships bwn inpu volag and currn. A 3dB poins, i.. a ± h phas is ±45, h currn lading h volag a h lowr dg of h band, whr h raciv componn of h inpu is capaciiv, and h currn lagging h volag whr h raciv componn is induciv. Th faurs of h rsonan circui mniond abov ar diagrammaically rprsnd in Fig. 4.
I I max,phas Pag of 3.5 45. - -.5.5 o o - 45 Figur 4. Ampliud and phas rlaionships nar rsonanc. Bcaus h abiliy of h rsonan circui o absorb powr is frquncy-snsiiv, i is commonly usd as a dvic o slc a dsirabl frquncy from a manifold hrof. In communicaions applicaions i is ofn mployd as a frquncy slciv lmn of a unr. I may b usful o considr in som dail h significanc of h qualiy facor. I govrns wo imporan propris of a rsonan circui: h frquncy slciviy as xprssd in 3dB bandwidh and h valu of impdanc a rsonanc, X. I is imporan o no ha h facor dpnds no only on h rsisiv lmn of h load, bu also on h rsisiv lmn of h sourc. Thus frquncy slciv propris of a rsonan circui do no dpnd solly on h loss mchanism of h load, bu also on h loss mchanism of h sourc. Anohr imporan propry of h facor is is ffc on h capacianc and inducanc volags a rsonanc. A rsonanc h
Pag 3 of 3 impdanc of h capacianc and inducanc ar and. Th circui currn I a rsonanc is ( ) ( ), and, X. Thus h racanc volags ar X ( ). (6) Th magniuds of racanc volags ar hus ims largr han h volags ha would hav appard across hm, had hy bn drivn by h sourc indpndnly. Bcaus h raciv volags ar of opposi polariy h oal volag across h sris combinaion of h inducanc and capacianc is zro and h volag apparing across h load is and dpnds only on h loss lmns of h loop. Th qualiy facor also bars on h nrgy balanc in h circui a rsonanc. Th insananous nrgy sord in h inducanc and capacianc ar i and υc rspcivly. A rsonanc h phasor of h currn i is [ ( )] ( ) whil h phasor of h capacianc volag v c is. Th currn i and volag v c ar hus 9 ou of phas so ha i cos and, c sin [ ( )] v. (7) Th oal nrgy sord W is W cos ( ) sin [ ( )]. (8a) Bu, bcaus and ( ) ( ),
h quaion for W rducs Pag 4 of 3 W ( ) (8b) Th following conclusions can b drawn from h las quaion: (i) A rsonanc h oal nrgy sord in a sris rsonan circui is indpndn of im. (ii) is h M valu of h ampliud of h driving EMF. Th rm ( ) M ( ) circui. Eq. 8b can hus b xprssd in h form is h powr P dissipad in h oal rsisiv porion of h W. (9) P Th rsul obaind abov can b formulad in rms of h samn ha h qualiy facor of a sris rsonan circui is h raio a rsonanc of h oal nrgy sord muliplid by angular rsonan frquncy, and h oal powr dissipad in h circui. Th rsuls (i) and (ii) lisd abov hav bn drivd analyically for a sris rsonan circui. Thy apply, howvr, o all rsonan circuis. Also, xprssion for h 3dB bandwidh is xacly valid for sris and paralll rsonan circuis. For ohr rsonan configuraion h rlaion is of h form γ whr γ is a numrical facor commonly lying bwn on and wo. Th rlaionships involving wr drivd for sady sa condiions. Thy also occur in analysis of ransin bhaviour. Equaion of moion for a harmonically xcid sris rsonan circui is d i di i. () d d c Th gnral soluion incorporaing ffcs of iniial condiion is β i I I, () whr β I is h soluion of h homognous quaion
Pag 5 of 3 c i d di d i d, (a) which, on subsiuion for i I β bcoms β β (b) or, β β. (c) Th soluions of his quaion ar β ±. (3a) For h common cas of low loss sysms h rm / is smallr han and i is convnin o xprss β in h form β ±. (3b) ubsiuion of for h rm / rducs h quaion o h form β ±. (3c) Th ransin soluion i is hus [ ] τ I I i (4) whr and τ.
Pag 6 of 3 I is apparn from Eq. 4 ha h im consan τ for h linar ransin is, whr is h 3dB bandwidh of h sady sa rspons. In as much as h quadraic quaniis such as nrgy and powr ar proporional o h squars of linar quaniis, h dcays hrof ar drmin by h rm (5) Th rciprocal of h 3dB bandwidh is hus h im consan of h ransin bhaviour of sord nrgy and powr in a sris rsonan circui. imilar rlaions also obain for mor complicad rsonan sysms. 3. Paralll rsonan circui A circui dual o a sris rsonan circui is a paralll rsonan circui shown in Fig. 5. I O G G v - Figur 5. Paralll rsonan circui suls obaind for sris rsonan circui can b adapd o dscrib propris of a paralll rsonan circui by carrying ou h dualiy ransformaion swiching h words sris paralll, impdanc admianc and volag currn. Thus whras in sris rsonan circui h driving quaniy was volag and h rsponding quaniy was currn as common o all componns of h circui, in paralll rsonan circui h drivr is currn and rsponding quaniy is h common volag. Prsnd in Tabl ar dual rlaionships praining o h wo circuis discussd.
Pag 7 of 3 Tabl uaniy ris rsonan circui Paralll rsonan circui impdanc - admianc Z Y G G impdanc - admianc nar rsonanc Z ( ) Y ( ) impdanc - admianc a rsonanc Z Zmin Y Ymax Y G G Ymin Z Zmax qualiy facor G G rsonan frquncy f f π f π 3dB bandwidh f f f f f
EETOMAGNETI AITY Pag 8 of 3
. Elcromagnic caviy Pag 9 of 3 Elcromagnic caviy is a volum of spac nclosd by lcromagnically impnrabl, usually mallic walls. If h caviy is o inrac wih ousid spac, h caviy walls ar brachd by a small opning commonly calld an iris hrough which nrgy can pass ino or ou of h caviy. Exampls of caviis ar scions of ransmission sysms such as coaxial lins or wavguids rminad a boh nds by lmns impnrabl, or narly so for lcromagnic mods of h corrsponding ransmission sysms. A prooyp of a ransmission sysm caviy is a scion of ransmission lin rminad a boh nds by dvics inhibiing oally or parially passag of lcromagnic nrgy hrough hm. Th prooyp may ofn srv as an quivaln circui of wid rang of caviis and will b analysd blow.. Transmission lin caviy wih singl iris Th sysm considrd is a scion of ransmission lin of characrisic impdanc Z and phas vlociy u, of lngh l. On nd of h scion is rminad in a shor circui whil h ohr nd is conncd o a sourc of frquncy hrough idnical ransmission lin. An iris allowing passag of som lcromagnic nrgy ino h caviy is insrd bwn h fd lin and h caviy scion. Elcromagnic propris of h iris ar quivaln o a suscpanc B. Th circui rprsnaion of h sysm is shown in Fig.. l B Figur. Transmission lin caviy. Th paramr dscribing mos of h circui propris of h caviy is h inpu impdanc Z in which, howvr, mus b associa wih a spcific pair of rminals. A usful rlaionship in h discussion of h problm is h rlaion givn blow in Fig. and provd in h Appndix.
B Y co Z Z an Z Pag of 3 B Figur. Invrr circui. Th quivaln circui of h caviy as shown in Fig. can b modifid o h form shown in Fig. 3. l-d l-d l d M in M ' B N N ' B Y co Figur 3. Equivaln circui of ransmission lin caviy. Th amndd circui idnifis convnin rminals of driving poin impdanc Z in and allows on o mploy sandard ransmission lin circui analysis, as will b carrid ou blow. Th driving poin impdanc Z in a rminals MM is rlad o h impdanc Z a rminals NN by h rlaionship givn in Equaion in Fig., wih n co and B Y co. Z Z in (a) n Z
Bu Z is h impdanc of a shor circuid scion of lngh d and is propagaion consan u. Thus h driving poin impdanc Z in is Pag of 3 Z an βd, whr β is h Z in Z n an βd (b) W shall invsiga h rsonanc bhaviour of h circui, i.. whn Z in or. Th choic of h wo xrm possibiliis is suggsd by h bhaviour of convnional losslss sris or paralll rsonan circuis. W shall sar h analysis of h circui bhaviour in h frquncy rang clos o whr Z in, i.. whr mulipl of half wavlngh λ i.. whr mod m, i.. whn d λ. co βd or, whr an β d. This occurs whnvr d is an ingral d m λ. W shall invsiga h lows longiudinal W obsrv ha in h viciniy of co βd an βd is β d π, i.. d π h approxima powr sris xpansion of u co βd co βd cos βd & sin βd βd π d π u u u & d d π u d (3) wih π u. d Th xprssion for Z in bcoms ( u d ) Z Z Z in & (4) n πn Whn on obsrvs ha for a losslss paralll rsonan circui h inpu impdanc nar rsonanc is quivaln circui shown in Fig. 4. h xprssion for Z in drivd in Eq. 4 can b considrd o b h inpu impdanc of
in : πn M M ' Pag of 3 γ o o Z o o Figur 4. umpd quivaln circui of ransmission lin caviy nar rsonanc. W no ha inpu impdanc of a lossy paralll rsonan circui is bhaviour of circui shown in Fig. 5., dscribing h ( ) in Figur 5. ossy paralll rsonan circui. Thus a good approximaion of h impdanc of a lossy ransmission lin caviy is, by analogy o h paralll rsonan circui h modificaion of h xprssion for Z in in Equaion 4 Zin π Z n ( ) (5) lading o h quivaln circui shown in Fig. 6.
: πn Pag 3 of 3 Z Figur 6. Equivaln circui of a lossy ransmission lin in h viciniy of rsonanc. In discussion o follow i will b ncssary o considr frquncis subsanially rmovd from rsonanc. Undr hs circumsancs on is rmindd ha h xac xprssion for caviy inpu impdanc is givn in Equaion, Z Z Z β in co d co π. n n Th inpu impdanc of a rsonan caviy a a frquncy far rmovd from rsonanc is vry narly zro as viwd a rminals MM, i.. h caviy in hs frquncy rangs bhavs as a shor circui. I is convnin o dscrib h frquncy rspons of h caviy in rms of h W producd on h inpu lin. Th coordinas mployd ar shown in Fig. 3 and posiion of W minimum will b dsignad i. i λ λ 4 Figur 7. ocaion of W minimum in h viciniy of rsonanc for undrcoupld and ovrcoupld caviis.
Pag 4 of 3 Th bhaviour of h volag minimum posiion as h frquncy is swp hrough rsonanc and is convninly visualizd by comparing i wih h posiion of volag minimum producd by a shor circui locad a obsrvaion rminals MM of Fig. 3. Plod in Fig. 7 as a dod lin is h posiion of volag minimum wih rfrnc o a poin λ away from h posiion of h shor a MM. As h frquncy incrass wavlngh bcoms shorr and h minimum movs closr o h posiion of h shor, h procss indicad by h slop of h dod lin. Whn h shor is rplacd by h caviy, a frquncis sufficinly rmovd from rsonanc on h low sid of h caviy impdanc approximas zro as vidn from Eq. and h locaion of volag minimum follows h dod lin of Fig. 7. As h frquncy approachs rsonanc h condiions chang. aviy impdanc blow rsonanc is induciv as is apparn from quivaln circui of Fig. 6 and h disanc i of volag minimum from rminals MM bgins o drop fasr han would b h cas of shor circui rminaion. Th condiions chang whn h frquncy approachs rsonanc bcaus of h ffc of rsisiv rm Z as vidn from Fig. 7 and Fig. 6. A rsonanc h impdanc of h caviy is purly rsisiv and is valu is Z Z () in. πn Exprssion π n will occur frqunly is subsqun discussions and i will b convnin o inroduc a symbol for i, i.. π n calld h xrnal. Thus Zin Z. Dpnding on whhr h rsonan rsisiv impdanc is smallr han, largr han, or qual o Z hr will obain hr diffrn condiions as lisd blow. (i) Zin Z < Z : in his cas volag minimum will occur a h sam locaion as volag null producd by shor circui rminaion. (ii) Zin Z > Z : volag maximum will occur a h null locaion producd by shor circui rminaion. (iii) Zin Z Z : h caviy is machd o h lin, no sanding wav parn is prsn. Th hr cass considrd abov ar dsignad undrcoupld for Z < Z, ovrcoupld for Z > Z, and criically coupld for Z Z. As h frquncy is incrasd byond rsonanc h inpu impdanc acquirs capaciiv characr. For h undrcoupld cas h disanc of h obsrvd minimum, movs iniially away from h rfrnc rminals. In high frquncis h inpu impdanc bgins o approxima shor circui and minimum approachs h locaion of slcd minimum of h shor-circuid rminaion as shown in Fig. 7.
Pag 5 of 3 Th parn of bhaviour for ovrcoupld cas is diffrn in ha a rsonanc, whn inpu impdanc Z in is ral, and is largr han Z a rfrnc rminals h volag is maximum and minimum occurs λ 4 away. Whn h bhaviour of volag minima is racd in his cas as h frquncy is incrasd from is iniial off rsonanc valu h minimum movs owards h rfrnc poin bu as h frquncy approachs rsonanc i dos no rvrs is moion as was h cas for undrcoupld caviy, bu sops a λ 4 disanc from rfrnc rminals, which bcom h locaion of volag maximum as mniond arlir. As h frquncy is incrasd byond rsonanc h minimum coninus o mov oward h locaion of shor circui minimum, bu no h on from which i sard bu on λ closr o rfrnc rminals, as shown in Fig. 7. Exrnal is hus sn as a paramr which quanifis h inracion of h insid of h caviy wih xrnal nvironmn. 3. oadd Transmission in aviy In many insancs h caviy has wo inpu-oupu porals. Th inpu poral conncs h drivr o h caviy whil h oupu poral, usually an iris a h original shor circui wall of h caviy conncs h insid of h caviy o h load which absorbs a porion of powr supplid by h sourc, modifid by inrposiion of h caviy. A common applicaion of his naur is h us of h caviy as a bandpass filr. Whn h shor circui wall of h caviy is rplacd by an iris h quivaln circui of h caviy as shown in Fig. is modifid o h configuraion shown in Fig. 8. l B B ' Figur 8. oadd ransmission lin caviy. Th ffc of h suscpanc B can b convninly valuad by mploying h impdanc ransformaion of Equaion as shown in Fig. and shown in Fig. 9.
M M ' d N B Yγ co N ' Pag 6 of 3 Figur 9. Applicaion of invrr circuis o a ransmission lin caviy. Th rsulan quivaln circui of h loadd caviy is givn in Fig.. M : πn N n Z : Z M ' N ' n co n co Figur. umpd quivaln circui of loadd ransmission lin caviy. Analysis of h circui of Fig. 9 is simplifid if on xprsss h rminal load Z Z n in h form Z Z an β δ. (8) Th caviy inpu impdanc Z in is hn givn by Z Z in n an β ( d δ ). (9a) Z n an βd an βδ an βd an βδ Inasmuch as anβd in h viciniy of rsonanc is a small numbr and Z is usually a small prurbaion of h shor circui rminaion h xprssion for Z in can b approximaly rducd o
Pag 7 of 3 Z Z in ( ) (9b) n an β d an βδ Powr sris xpansion of angn funcion u in h viciniy of rsonanc rducs h xprssion for Z in, in a mannr analogous o ha mployd o dvlop Equaion 4, o h form Zin Z. (a) π n ( ) n π Inrnal losss in h caviy can b accound for by h addiion of h rm i o h dnominaor. Th rms of h form n π hav bn dsignad xrnal,. Th final xprssion for Z in hn bcoms Zin Z i ( ) (b) An quivaln circui appropria for h xprssion for Z in as dvlopd abov is givn in Fig.. M N i Z Z M ' N ' Figur. ducd lumpd quivaln circui of a loadd ransmission lin caviy. Th rciprocal of h rm in Equaion b is commonly dsignad h loadd of h caviy i and incorporas h ffc of xrnal loading on h prformanc of h caviy. 4. Frquncy rspons of a rsonan caviy. I is ofn imporan o know h frquncy rspons of a caviy. I may b dfind as h raio of powr absorbd by h caviy a frquncy, usually lying clos o h rsonan frquncy, h powr absorbd a rsonanc, h maximum powr. A common masur of h ffc is h sprad of frquncis δ in which h raio is abov, h 3 db bandwidh.
Pag 8 of 3 Powr absorbd a frquncy, P() is qual o h incidn powr minus rflcd powr, so ha wih ρ() h rflcion cofficin a frquncy, ρ i P P, () wih P i h incidn powr. Th 3 db bandwidh is hrfor givn by h rlaion. ± ρ δ ρ () Th rflcion cofficin ρ() is Z Z Z Z in in ρ (3a) ubsiuion from Equaion b yilds ρ (3b) whr is h loadd, h rciprocal of i. Whn h abov xprssions ar inroducd ino Equaion, i rducs o δ δ. (4) Th valu of δ obaind from h abov is, i δ (5) I is apparn from h abov quaion for δ ha h powr dlivrd o xrnal sourc and load impdancs as govrnd by xrnal s has h sam ffc on h frquncy rspons as h powr dlivrd o inrnal loss mchanism.
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Appndix Pag 3 of 3 Transmission lin invrr circui. onsidr a ransmission lin nwork shown in h figur blow: I I Z B Z - - Th rlaionship bwn h column vcors and is givn by h produc of hr I I componn nwork marics (), (B) and () so ha, I I ( )( B)( ) (6) Th marics ar: cos, Y sin, Z sin cos ( ) and ( B). B cos, Z sin, cos, Z sin Thus. (7) I Y sin, cos B, Y sin, cos I Whn h marix muliplicaion is xcud h rlaionship bwn h wo ss of circui variabls bcoms, I cos Y ZB ( ) sin, Z sin BZ sin BZ sin BZ cos, cos sin. (8) I For h cas of B Y co, h abov quaion rducs o I, Y co, Z an, (9) I or,
Z an, I Pag 3 of 3 I Y co. (a) Th rsulan rlaion bwn h impdancs Z I and Z I follows, Z Z Z an. (b).e.d.