T RAN SMISSI ON LIN ES AND WA VE GU ID ES INT …scadec.ac.in/upload/file/EC6503-TLW LECTURE NOTES.pdfsca d engine e ring c oll eg e t ran smissi on lin es and wa ve gu id es un it

EGELLOC GNIREENIGNE DACS

NOISSIMSNART SENIL DNA SEDIUGEVAW

TINU -I

NOISSIMSNART ENIL YROEHT

NOITCUDORTNI OT TR SNA M NOISSI ENIL YROEHT

noissimsnarT seniL dna sediugevaW

A NOISSIMSNART ENIL is a ived ce ed s dengi ot ediug ele c irt c la ygrene f or m eno tniop

ot .rehtona tI is us ,de of r axe m ,elp ot snart f re eht ou tupt rf ygrene of a snart m retti ot na

nnetna a. ihT s ygrene lliw ton levart hguorht ron m la ele c irt c la eriw uohtiw t taerg ol sses.

hguohtlA eht annetna c na eb c enno c det erid c ylt to eht snart m ,retti eht annetna is us yllau

detacol s emo ecnatsid yawa morf eht snart m .retti

nO draob ,pihs eht snart m retti si detacol ni s edi a oidar oor m, dna ti s sa soc detai

annetna is m detnuo no a mas .t A snart m si s noi enil is desu ot c enno ct eht snart m retti dna eht

nnetna a. ehT snart m noissi enil ah s a s gni el oprup se f ro htob eht rettimsnart dna eht .annetna

ihT s esoprup si ot refsnart eht ygrene tuptuo fo eht snart m retti ot eht annetna htiw eht ael st

sop s elbi rewop sol s. woH llew siht si enod sdneped no eht s ep c lai physic la dna ele c irt c la

scitsiretcarahc i( ecnadepm dna )ecnatsiser fo eht noissimsnart .enil

SNART M NOISSI ENIL YROEHT

ehT lacirtcele c scitsiretcarah fo a owt - eriw snart m noissi enil dneped irp m ylira no eht

noitcurtsnoc fo eht enil . ehT owt - eriw enil acts ekil a gnol c apa c roti . ehT egnahc of sti eviticapac

ecnatcaer si elbaeciton sa eht ycneuqerf deilppa ot ti si .degnahc

ecniS eht gnol c rotcudno s evah a m itenga c dleif tuoba eht m nehw ele c irt c la ygrene si

gnieb sap s de hguorht eht m, yeht la so tibihxe eht eitreporp s fo udni c nat c .e ehT eulav s fo

dni ecnatcu dna ecnaticapac detneserp dneped no eht uoirav s isyhp c la fac rot s taht ew

sucsid s de .reilrae

roF axe m ,elp eht epyt fo enil ,desu eht eleid c cirt ni eht ,enil dna eht htgnel of eht enil

must eb c no s deredi . ehT effe cts of eht udni c evit a dn c apa c eviti r ae c nat ce of eht nil e dneped no

eht f neuqer cy deilppa . ecniS on cirtceleid si rep fec ,t ele c snort m egana ot m evo orf m eno

rotcudnoc ot eht rehto hguorht eht eleid c irt c.

hcaE epyt of owt - eriw snart m si s noi enil la so ah s a c udno c ecnat .eulav T sih

natcudnoc ce av eul erper s tne s eht eulav fo eht tnerruc f wol taht m ya eb epxe c det hguorht eht

ni s ,noitalu If eht enil si inu f ro m lla( seulav lauqe ta ae ch tinu ,)htgnel ht en eno sm lla sec noit fo


eht nil e m ya erper s tne eves r la f tee . T ih s ulli s noitart of a wt o- eriw rt sna m si s noi enil lliw eb us de

tuohguorht eht noissucsid of snart m noissi ;senil ,tub k pee ni m dni taht eht nirp c elpi s erp s detne

ylppa ot lla snart m noissi .senil We lliw nialpxe eht seiroeht gnisu DEPMUL STNATSNOC dna

DETUBIRTSID STNATSNOC ot f rehtru is m ilp fy eseht nirp c .elpi

DEPMUL STNATSNOC

A snart m noissi enil sah eht seitreporp of ,ecnatcudni ,ecnaticapac dna ecnatsiser tsuj sa

eht m ero c lanoitnevno ric c tiu s .evah Us ,yllau ,revewoh eht c stnatsno ni c lanoitnevno c ri c stiu

era ul m dep otni a si elgn ived ce ro com .tnenop roF axe m ,elp a c lio of eriw sah eht ytreporp fo

cnatcudni e. W neh a c niatre am tnuo of ecnatcudni is dedeen ni a c ri c ,tiu a c lio of eht reporp

id m snoisne si .detresni

ehT ecnatcudni fo eht c ri c tiu is ul m dep otni eht eno com .tnenop T ow m late setalp

detarapes yb a sm lla s ap c ,e c na eb desu ot s ylppu eht deriuqer c apa c nati ce rof a c .tiucri nI

hcus a ,esac most of eht c apa c nati ce of eht c ri c tiu is ul m dep otni this eno com tnenop . ,ylralimiS

a f dexi rotsiser nac eb desu ot s ylppu a c niatre eulav fo c ri cu ti er sis nat ce as a ul m dep us m.

,yllaedI a snart m si s noi enil luow d osla evah ti s stnatsnoc of udni c ,ecnat c apa c nati c ,e dna

natsiser ce epmul d rehtegot , as wohs n in ugif er 3-1. yletanutrofnU , siht is on t t eh

snarT.esac m noissi enil c stnatsno era sa debircsed ni eht f gniwollo hpargarap s.

DETUBIRTSID STNATSNOC

noissimsnarT enil ,stnatsnoc dellac detubirtsid ,stnatsnoc era daerps gnola eht eritne

htgnel fo eht noissimsnart enil dna tonnac eb dehsiugnitsid yletarapes . ehT am tnuo fo

,ecnatcudni ticapac ,ecna dna iser s ecnat sdneped no eht htgnel of eht ,enil eht ezis of eht

gnitcudnoc ,seriw eht gnicaps neewteb eht ,seriw dna eht cirtceleid ria( ro gnitalusni )muidem

neewteb eht eriw s . ehT gniwollof shpargarap lliw eb lufesu ot uoy sa uoy yduts etubirtsid d

stnatsnoc no a noissimsnart .enil


owT - eriw noissimsnart .eniI

ecnatcudnI fo a noissimsnarT eniL

W neh tnerruc f swol hguorht a ,eriw m citenga senil of rof ce era s te pu dnuora eht .eriw

As eht c tnerru ni c aer ses dna ed c aer ses ni am dutilp e, eht f ei dl dnuora eht eriw dnapxe s dna

sespalloc ylgnidrocca . ehT ygrene decudorp yb eht citengam senil fo ecrof

gnispalloc kcab otni eht eriw sdnet ot k pee eht c tnerru f gniwol ni eht same erid c .noit hT is

erper s tne s a c niatre tnuoma of udni c nat c ,e ihw ch is serpxe s de ni m yrnehorci s rep neltinu g .ht

erugiF setartsulli eht ecnatcudni dna citengam f dlei s of a snart m noissi

li .en

ecnaticapaC fo a noissimsnarT eniL

ecnaticapaC osla stsixe neewteb eht snart m noissi enil ,seriw sa detartsulli ni erugif 3-3 . itoN ce

taht eht owt lellarap seriw tca sa setalp fo a roticapac dna taht eht ria neewteb eht m stca sa a

eleid c irt c . ehT ecnaticapac neewteb eht seriw si yllausu desserpxe ni sdarafocip rep tinu .htgnel

sihT cirtcele f dlei neewteb eht seriw si sim rali ot eht f dlei taht stsixe neewteb eht owt etalp s of a

apac c .roti

ecnatsiseR fo a noissimsnarT eniL

ehT snart m noissi enil nwohs ni f erugi 3-4 sah ele c irt c la er sis nat ce gnola ti s .htgnel T sih

EGELLOC GNIREENIGNE DACS ecnatsiser si yllausu serpxe s de ni ho ms rep tinu htgnel na d is s nwoh as ixe s gnit c uounitno s yl

morf eno dne fo eht enil ot eht rehto ..

egakaeL tnerruC

ecniS yna eleid c ,cirt neve ,ria si ton a rep fect ni s ,rotalu a sm lla c tnerru k nwon as EGAKAEL

TNERRUC f wol s wteb e ne eht wt o eriw s. nI ,tceffe eht rotalusni acts as a er sis ,rot rep m gnitti

c tnerru ot sap s neewteb eht owt eriw s. erugiF 3-5 swohs iht s ael k ega htap as er sis rot s ni

lellarap detcennoc neewteb eht owt enil s. sihT ytreporp is c della OC N CUD T CNA E )G( dna si

eht etisoppo of er sis ecnat . udnoC c nat ce ni snart m noissi enil s si desserpxe sa eht er c lacorpi fo

ecnatsiser dna si yllausu nevig ni orcim sohm rep tinu .htgnel

CITENGAMORTCELE SDLEIF CITSIRETCARAHC ECNADEPMI

ehT detubirtsid noc s stnat of ,ecnatsiser udni c nat c ,e dna c apa c nati ce era ab sic

seitreporp c mo m no ot lla snart m si s noi enil s dna ixe st hw e reht or ton any cu rr tne f wol ixe sts. sA

noos sa tnerruc f wol and egatlov ixe st ni a snart m si s noi ,enil rehtona ytreporp eb comes etiuq

nedive t. sihT is eht erp s ne ce of na ele c ort m itenga c f ,dlei ro enil s o f f ,ecro tuoba ht e eriw s o f eht

noissimsnart .enil

ehT senil fo f ro ce sevlesmeht era ton iv s ;elbi ,revewoh rednu s gnidnat eht f ro ce taht na

nortcele secneirepxe elihw ni eht f dlei of eht se senil si yrev im tnatrop ot ruoy rednu s gnidnat fo

ygrene snart m noissi . T ereh era owt k dni s of f dlei s; eno si sa soc detai htiw gatlov e dna ht e rehto

htiw c tnerru . ehT f dlei sa soc detai htiw egatlov is c della eht E CEL T IR C (E) LEIF D. tI exe rts a

rof ce no yna ele c irt c egrahc alp c de ni ti . T eh f dlei sa soc detai with c ru r tne is c della a

CITENGAM )H( ,DLEIF eb c esua ti sdnet ot artxe f ro ce no yna m itenga c elop alp c de ni ti . erugiF

3-6 setartsulli eht yaw ni hcihw eht E f sdlei dna H f dlei s dnet ot tneiro eht m evles s neewteb

srotcudnoc fo a ipyt c la owt - eriw snart m noissi enil . ehT i ull s noitart s swoh a sorc s es c noit of eht

snart m noissi senil . ehT E f dlei is detneserper by s dilo enil s dna eht H f dlei by do ett d enil s. T eh

worra s idni c eta eht erid c noit of eht enil s of f .ecro B hto f dlei s ron m ylla ixe st rehtegot dna era

EGELLOC GNIREENIGNE DACS nekops fo vitcelloc yle sa eht citengamortcele .dleiF

sdIeiF teb w nee .srotcudnoc

Y uo c na ebircsed a snart m si s noi enil ni ret ms fo sti im .ecnadep ehT oitar of egatlov ot

c tnerru (E )niI/ni ta eht tupni dne is k nwon sa eht NI PUT CNADEPMI E ( niZ ). T ih s is eht

ecnadepmi detneserp ot eht rettimsnart yb eht noissimsnart enil dna sti ,daol eht .annetna

T eh oitar of egatlov ot c tnerru ta eht ptuo ut )TUOI/TUOE( dne is k nwon sa eht O TUPTU

CNADEPMI E UOZ( T) . sihT is eht im nadep ce erp s detne ot eht daol by ht e snart m si s noi enil dna

ti s s ruo ce. If na ni f yletini gnol snart m si s noi enil dluoc eb us ,de eht oitar of egatlov ot c tnerru ta

yna tniop no taht snart m noissi enil dluow be some itrap c ralu eulav of im nadep c .e hT is

im ecnadep is nwonk sa eht CITSIRETCARAHC IMPE CNAD E. ehT m ixa mum dna( most

fe f )tneici snart f re fo irtcele c la ygrene sekat alp ce nehw eht ecruos im nadep ce is m ta c deh ot eht

daol im ecnadep . sihT f tca si yrev im tnatrop ni eht yduts fo snart m si s noi enil s dna annetna s. If

eht c arah c citsiret im ecnadep of eht snart m noissi enil dna eht daol im nadep ce era ,lauqe ygrene

morf eht rettimsnart lliw levart nwod eht noissimsnart enil ot eht annetna htiw on rewop ssol

desuac yb er f .noitcel

ENIL SESSOL

ehT noissucsid fo noissimsnart senil os raf sah ton yltcerid serdda s de ENIL ;SESSOL

yllautca os me sessol rucco ni lla senil . eniL sessol yam eb yna fo eerht sepyt


1 . ,REPPOC DIE IRTCEL C,

2. AIDAR T OI N ro CUDNI T OI N SSOL E .S NO :ET noissimsnarT senil era semitemos derrefer ot sa FR senil . nI iht s txet eht ret sm era

u des .ylbaegnahcretni

C reppo sessoL

enO epyt of c reppo ssol si 2I R .SSOL nI FR senil eht er sis ecnat fo eht c udno c rot s si

reven lauqe ot ez ro. W reveneh tnerruc f swol hguorht eno fo eseht rotcudnoc s, os me ygrene si

sid s detapi ni eht f ro m of taeh . ihT s taeh ol ss is a OP W RE SSOL . W hti c reppo ,diarb ihw ch sah

a ecnatsiser rehgih naht dilos ,gnibut siht rewop ssol is .rehgih

A ehton r epyt of c reppo sol s is eud ot SKIN E CEFF T. W neh dc f wol s hguorht a

,rotcudnoc eht m evo m tne of snortcele hguorht eht udnoc c ’rot s ssorc sec noit is rofinu m, T eh

noitautis is os m tahwe tnereffid nehw ca si .deilppa T eh gnidnapxe dna palloc s gni f sdlei tuoba

ae ch nortcele ne c elcri rehto ele c nort s . sihT onehp m ,none dellac FLES CUDNI TI NO , drater s eht

tnemevom of eht delcricne .snortcele

ehT f xul ytisned ta eht retnec si so taerg taht ele c nort m evo m tne ta iht s tniop is uder c .de

As neuqerf cy is ni c ,desaer eht oppo s noiti ot the f wol of c tnerru ni ht e c retne fo eht eriw

ni c aer ses. tnerruC ni eht retnec fo eht eriw oceb m se sm rella dna most fo eht ele c nort wolf is no

eht eriw s afru c .e W neh eht neuqerf cy ilppa ed is 001 m ztrehage ro ,rehgih eht ele ctr no

m evo m tne ni eht c retne is so sm lla taht eht c retne of eht eriw c dluo eb er m devo tuohtiw any

elbaeciton effe ct no c tnerru . uoY s dluoh eb lba e ot s ee taht t eh fe fec evit c sor s-sec lanoit aera

sesaerced sa eht f ycneuqer aercni s .se

ecniS ecnatsiser is revni s yle noitroporp al to eht c sor s-sec lanoit ,aera eht resis nat ce lliw

ni c esaer as eht neuqerf cy is aercni s .de ,oslA nis ce rewop ssol aercni s se sa er sis nat ce

ercni ,sesa rewop sessol esaercni htiw na esaercni ni ycneuqerf esuaceb fo niks .tceffe

reppoC sessol c na eb m ini m dezi dna c udno c ytivit ni c aer s de ni na FR enil by p nital g eht

enil htiw s .revli S ni ce s revli is a retteb c udno c rot naht c ,reppo most fo eht c ru r tne wi ll f wol

hguorht eht revlis reyal . ehT gnibut neht sevres yliramirp sa a m lacinahce .troppus

cirtceIeiD sessoL

CIRTCELEID SESSOL tluser orf m eht gnitaeh tceffe no eht irtceleid c m laireta neewteb

eht udnoc c rot s. P rewo f or m eht s ecruo is us de ni gnitaeh eht id e el c irt c. ehT taeh udorp c de si

detapissid otni eht s gnidnuorru m uide m. W neh ereht si on laitnetop nereffid ce neewteb owt

,srotcudnoc eht ota ms ni eht eleid c cirt m laireta neewteb eht m era ron m la dna eht tibro s fo eht

snortcele era .ralucric

W neh ereht is a laitnetop fid f ecnere neewteb owt c udno c rot s, eht tibro s fo eht ele c snort

c gnah e. ehT xe c se s evi evitagen egrahc no eno udnoc c rot leper s ele c snort no eht eleid c cirt

drawot eht evitisop rotcudnoc dna uht s trotsid s eht stibro of eht .snortcele

A c egnah ni ht e pa ht of ele c nort s eriuqer s m ero ene r ,yg tni r udo c gni a rewop ol ss . T eh

ota m ci erutcurts of rebbur is m ero fid fic tlu ot trotsid naht eht s urt c erut fo some rehto eleid c cirt

m laireta s. ehT ota ms fo m ,slaireta such sa ,enelyhteylop id s trot ae s yli . ,eroferehT enelyhteylop

si of net desu sa a cirtceleid eb c ua se sel s rewop is c no sum de nehw ti s ele c rt on tibro s era

EGELLOC GNIREENIGNE DACS id s .detrot

NOITAIDAR DNA NOITCUDNI SESSOL

NOIAIDAR dna NOITCUDNI SESSOL era sim rali ni taht bo ht era c ua s de by eht f sdlei

gnidnuorrus eht c udno c rot s. dnI noitcu ol ss se rucco nehw eht ele c ort m itenga c f dlei tuoba a

rotcudnoc stuc hguorht yna ybraen m cillate bo ject dna a c tnerru si decudni ni taht ejbo ct. sA a

,tluser rewop si detapissid ni eht

tcejbO dna si ol s .t noitaidaR sessol rucco esuaceb some m itenga c l seni of ecrof tuoba a

rotcudnoc od ton nruter ot eht c udno c rot nehw eht cyc el etanretla s. ehT se enil s fo rof ce era

ejorp c det otni s ap ce as oitaidar n, dna this er s tlu ni rewop ol ss es. T tah is, rewop is s deilppu by

eht ,ecruos tub is ton elbaliava ot eht aol .d

EGATLOV EGNAHC

nI na e el c irt c c ri c ,tiu ygrene is s derot ni ele c irt c dna m itenga c f dlei s. esehT f dlei s must

eb thguorb ot eht daol ot snart m ti taht rene g .y tA eht ,daol rene gy c deniatno ni eht f dlei s si

detrevnoc ot eht derised mrof fo ygrene oissimsnarT n fo ygrenE

W neh eht daol si detcennoc yltcerid ot eht ecruos of ,ygrene ro nehw the art nsm si s noi

enil is s ,troh elborp ms noc c gninre c tnerru dna egatlov c na eb s devlo yb gniylppa Ohm’s .wal

W neh eht snart m noissi enil oceb m se gnol hguone so eht it me ereffid ecn neewteb a c egnah

gnirrucco ta eht rotareneg dna a c egnah gniraeppa ta eht daol eb com se erppa c ,elbai ylana s si

fo eht noissimsnart enil semoceb .tnatropmi

cD deiIppA ot a noissimsnarT eniL

nI erugif 3- ,7 a yrettab si detcennoc orht hgu a ylevitaler gnol owt - eriw noissimsnart enil

ot a daol ta eht raf dne fo eht enil . tA eht tnatsni eht s tiw ch

si ,desolc rehtien tnerruc ron egatlov stsixe no eht .enil

W neh eht hctiws si ,desolc tniop A semoceb a evitisop ,laitnetop dna tniop B semoceb

vitagen e . esehT stniop fo ecnereffid ni laitnetop evom nwod eht enil . ,revewoH sa eht laitini

stniop fo laitnetop evael stniop A dna ,B yeht era dewollof yb wen stniop of nereffid ce ni

,laitnetop hcihw eht yrettab sdda ta A dna B.

EGELLOC GNIREENIGNE DACS ihT s si m ylere s gniya ht ta eht ettab ry m sniatnia a c no s tnat laitnetop fid f nere ce neewteb

tniop s A dna B. A trohs it me af ret eht hctiws is c ol s ,de eht laitini iop nts fo nereffid ce ni laitnetop

evah aer c deh tniop s A’ dna B ;’ eht eriw sec noit s orf m tniop s A ot A’ dna tniop s B ot B’ are ta

eht same laitnetop as A dna B, er s ep c .ylevit T eh tniop s of c egrah era erper s detne by ulp s ( )+

dna m suni (-) s ngi s gnola eht eriw s, T eh erid c noit s of eht c tnerru s ni eht eriw s era erper s detne

yb eht sdaehworra no eht ,enil dna eht noitcerid of levart is idni c deta yb na worra leb ow eht

li .en

lanoitnevnoC senil of f ro ce erper s tne eht ele c cirt f dlei taht ixe sts neewteb eht oppo s eti

k dni s of c egrah no eht eriw sec noit s orf m A ot ’A dna B ot B’. sorC s se sliat( of )sworra idni c eta

eht m itenga c dleif c detaer yb eht ele c irt c f dlei gnivom nwod eht nil e. ehT m gnivo ele c irt c f dlei dna eht occa m gniynap m citenga f dlei etutitsnoc na ortcele m citenga evaw taht is m gnivo morf

eht rotareneg )yrettab( drawot eht .daol

ihT s evaw slevart ta ixorppa m yleta eht s deep of thgil ni f eer s ap ce. T eh ygrene aer c gnih

eht daol si lauqe ot taht depoleved ta eht ba ett ry sa( sum gni ereht era on sol ses ni eht

noissimsnart )enil . fI eht daol sbrosba lla fo eht ,ygrene eht tnerruc dna egatlov lliw eb ylneve

detubirtsiD gnola eht .enil

cA ppA deiI ot a noissimsnarT eniL

W neh eht ettab ry of f erugi 3-7 is alper c de yb na ac rotareneg (f gi . 3- ,)8 ae ch evisseccus

ni s uoenatnat s eulav of eht rotareneg egatlov is detagaporp do nw ht e il ne ta eht s deep of .thgil

ehT ac noit is sim rali ot eht evaw c detaer by eht b etta r ,y xe c tpe eht ilppa ed egatlov is s uni s ladio

ni s daet of c no s .tnat ussA me taht eht s hctiw si c ol s de ta eht mom tne eht rotareneg egatlov si

sap s gni hguorht orez dna taht eht en xt lah f cyc el m eka s tniop A op s .eviti At eht ne d of eno

elcyc fo reneg rota ,egatlov eht tnerruc dna egatlov noitubirtsid lliw eb sa s nwoh ni f ugi

nI siht noitartsulli eht c lanoitnevno enil s fo f ro ce erper s tne eht ele c irt c dleif s. roF

is m ilp c ,yti eht m citenga f sdlei era ton .nwohs stnioP fo c egrah era idni c deta yb ulp s )+( na d

m suni (-) ngis s, eht regral ngis s idni c gnita tniop s fo rehgih am edutilp of htob egatlov dna

c nerru t. trohS sworra idni c eta erid c noit of c tnerru ele( c nort f .)wol ehT evaw f ro m nward leb ow

eht noissimsnart enil stneserper eht egatlov )E( dna tnerruc )I( vaw .se

ehT enil si sa sum de ot eb ni f etini ni nel g ht so ereht is on er f el c .noit ,suhT gnilevart

unis s ladio egatlov dna tnerruc sevaw yllaunitnoc levart ni ahp se orf m eht rotareneg drawot eht

,daol ro f ra dne fo eht il ne . W seva gnilevart f or m eht rotareneg ot t eh daol era c della TNEDICNI

W EVA S. W eva s gnilevart f or m eht daol cab k ot eht rotareneg era c della LFER E DETC W SEVA

dna lliw eb denialpxe ni retal hpargarap s.

SNART M NOISSI SMUIDEM

ehT yvaN sesu ynam tnereffid sepyt fo NOISSIMSNART SMUIDEM ni sti cinortcele

ilppa c noita s . hcaE muidem enil( ro )ediugevaw sah a niatrec citsiretcarahc ecnadepmi ,eulav

tnerruc - gniyrrac ,yticapac dna lacisyhp epahs dna si dengised ot teem a ralucitrap .tnemeriuqer

ehT evif sepyt fo noissimsnart smuidem taht ew lliw ssucsid ni siht .cipot

1 . RAP A LELL - NIL E,

2 . TW DETSI ,RIAP

EGELLOC GNIREENIGNE DACS 3 . S IH E DEDL IAP R,

4 . IXAOC AL IL N ,E dna

5. W EDIUGEVA S.

ehT esu fo a itrap c ralu enil ,sdneped am gno rehto gniht s, no eht deilppa ,ycneuqerf eht rewop - gnildnah ,seitilibapac dna eht epyt fo ni s .noitallat

rewoP gnidnatS - evaW oitaR IeIIaraP eniL

ehT s erauq of eht rwsv si dellac eht OP W RE enO epyt of lellarap enil is eht TWO-W ERI

GNIDNATSNEPO -W EVA OITAR )rwsp( . :eroferehT ,ENIL detartsulli ni erugif

SNART MI TT ER

TUPNI DNE NOISSIMSNART ENIL TUPTUO DNE

ANNETNA

sihT enil stsisnoc fo owt eriw s taht era yllareneg s ap c de orf m 2 ot 6 ni c eh s trapa yb

ni s gnitalu aps c re s. T ih s pyt e of enil is most of net us de f ro rewop enil s, larur enohpelet enil s,

dna hpargelet l eni s. tI si som ite mes us de as a snart m si s noi T ih s oitar is usef lu eb c ua se eht

ni s urt m tne s us de ot nil e neewteb a snart m retti dna na annetna ro neewteb de et ct s gnidnat

sevaw aer ct ot eht erauqs o f eht na nnetna a dna a er c .revie

nA egatnavda fo siht epyt of enil si sti sim elp c no s urt c noit . ehT nirp c lapi id s segatnavda

of siht epyt of enil era the hgih oitaidar n ol ss es dna ele c irt c la ion se cip k pu eb c ua se of the kcal

of .gnidleihs

noitaidaR sessol era udorp c de yb eht c gnignah f dlei s c detaer yb eht c gnignah c tnerru ni

ae ch c udno c rot . rehtonA epyt of lellarap enil si eht TWOW RI E RIB NOB T( W NI AEL D) NIL E,

detartsulli ni f erugi 3- 01 . sihT epyt fo snart m si s noi enil si moc m ylno desu ot c tcenno a noisivelet

gniviecer annetna ot a oh me ivelet s noi s .te

ihT s enil is yllaitnesse eht as me sa eht owt - eriw nepo enil xe c tpe taht rofinu m aps c gni si

derussa yb em gniddeb eht owt eriw s ni a low-loss eleid c irt c, us yllau po teyl h nely e. S ni ce eht

eriw s era em deddeb ni eht th ni nobbir of ylop eth nely e, ht e eleid c irt c s ap ce is yltrap a ri dna

yltrap .enelyhteylop


Tw detsi riaP

ehT TW DETSI RIAP snart m noissi enil si detartsulli ni f erugi 3- 11 . As eht an me im eilp s,

eht enil c stsisno fo owt ni s detalu eriw s iwt s det to rehteg ot f ro m a f elbixel enil wi tuoht eht use fo

recaps s. tI si ton desu f ro snart m gnitti hgih f neuqer cy eb c ua se of eht hgih eleid c irt c sol ses taht

rucco ni eht rebbur noitalusni . W neh eht enil si ,tew eht sessol esaercni .yltaerg

owT - eriw nobbir .eniI

dedIeihS .riap

ehT DEDLEIHS ,RIAP nwohs ni f ,erugi c no sis st of rap lella c udno c rot s detarapes f mor

EGELLOC GNIREENIGNE DACS ae ch rehto dna s dednuorru by a s dilo eleid c irt c. ehT c udno c rot s era c deniatno tiw h ni a rb a dedi

c reppo gnibut ht at acts as na ele c irt c la s dleih . ehT sa sem ylb is c derevo htiw a rebbur ro f elbixel

oc m noitisop c gnitao taht etorp c st eht enil f or m iom s erut dna mec inah c la ad m ega . ,yldrawtuO ti

skool hcum ekil eht rewop droc fo a gnihsaw enihcam ro .rotaregirfer

dedIeihS .riap

ehT lapicnirp egatnavda of eht s dedleih riap si taht eht c udno c srot era nalab c de ot

;dnuorg taht is, eht c pa ac ecnati neewteb eht eriw s is inu f ro m tuohguorht eht htgnel of eht .enil

ihT s nalab ce si eud ot eht inu f ro m s ap c gni fo eht dednuorg s dleih taht s sdnuorru eht seriw

gnola rieht eritne nel g ht . ehT dediarb reppoc s dleih is etalo s eht c udno c srot f or m s yart m nga cite

f dlei s.

IaixaoC seniL

erehT era owt sepyt of IXAOC AL ,SENIL

1. DIGIR )RIA( LAIXAOC ENIL

2. ELBIXELF LAIXAOC)DILOS( .ENIL

ehT lacisyhp noitcurtsnoc fo htob sepyt si yllacisab eht ;emas taht ,si hcae sniatnoc owt

cirtnecnoc dnoc .srotcu

DIGIR OC LAIXA ENIL

ehT digir laixaoc enil noc sists of a c ,lartne detalusni eriw renni( c udno c )rot m detnuo

ni s edi a ralubut retuo udnoc c .rot ihT s enil is s nwoh ni f erugi 3- .31 nI os me ilppa c ,snoita eht

renni c udno c rot is la so .ralubut ehT renni c udno c rot is ni s detalu f or m eht retuo c udno c rot yb

ni s gnitalu s srecap ro daeb s ta raluger lavretni s. ehT srecaps era m eda fo ,xeryP nerytsylop e, ro

os me rehto m laireta taht ah s doog ni s gnitalu c arah c iret s cit s dna wol id e el c irt c sol ses ta hgih

.seicneuqerf

A SEGATNAVD FO DIGIR OC LAIXA ENIL

ehT feihc egatnavda fo eht digir enil si sti ytiliba ot eziminim noitaidar sessol . T eh

cirtcele dna citengam sdleif ni a owt - eriw lellarap enil dnetxe otni ecaps rof ylevitaler taerg

EGELLOC GNIREENIGNE DACS natsid ces dna noitaidar sol ses rucco . oH ,revew ni a laixaoc enil on cirtcele ro m citenga f sdlei

dnetxe edistuo fo eht retuo udnoc c rot . ehT sdleif era denifnoc ot eht ecaps neewteb eht owt

,srotcudnoc gnitluser ni a yltcefrep dedleihs laixaoc enil . rehtonA egatnavda si taht retni f nere ce

morf ehto r senil si .decuder

SID SEGATNAVDA FO DIGIR OC LAIXA ENIL

ehT digir enil ah s eht f gniwollo egatnavdasid s:

)1( tI is evisnepxe ot c no s urt c ;t

)2( tI tsum eb tpek yrd ot tneverp evissecxe egakael neewteb eht owt

C udno c rot s;

)3( hguohtlA hgih -f ycneuqer ol ss es era som tahwe el ss naht ni uoiverp s yl m denoitne

,senil yeht era s llit xe c se s evi hguone ot il m ti eht arp c it c la htgnel of eht enil . aeL k ega c ua s de yb

eht c nedno s noita of m io s erut is detneverp ni some digir enil ilppa c snoita by eht use of na eni rt

ag s, s hcu as ,negortin uileh m, ro ar .nog tI is pum dep otni eht id e el c irt c s ap ce of eht enil at a

serp s eru taht c na yrav orf m 3 ot 53 dnuop s rep s auq re ni c .h T eh treni ag s is us de ot dry eht

enil nehw ti si f ri st dellatsni dna serp s eru is m deniatnia ot ne s eru taht on m io s erut sretne eht

li .en

ELBIXELF OC LAIXA ENIL

elbixelF laixaoc senil (f gi . 3- )41 era edam htiw na renni rotcudnoc taht stsisnoc fo

f elbixel eriw detalusni morf eht retuo rotcudnoc yb a ,dilos suounitnoc gnitalusni m laireta . T eh

retuo rotcudnoc si edam fo latem ,diarb hcihw sevig eht enil ytilibixelf . ylraE etta m stp ta gniniag

ytilibixelf devlovni gnisu rebbur srotalusni neewteb eht owt srotcudnoc . ,revewoH eht rebbur

srotalusni desuac evissecxe sessol ta hgih .seicneuqerf

eIbixeIF xaoc Iai .eniI

esuaceB fo eht hgih - neuqerf cy sol s se ossa c detai htiw rebbur ni s ,srotalu enelyhteylop

citsalp saw depoleved ot ecalper rebbur dna ile m etani eseht .sessol

P ylo eth enely alp s it c si a s dilo s bu s nat ce taht rem nia s f elbixel evo r a ediw r egna fo

repmet eruta s. tI si detceffanu yb s ,retawae ag s ,enilo ,lio dna most rehto diuqil s taht m ya eb

dnuof aoba rd s .pih ehT use fo ylop eth enely sa na ni s rotalu er s tlu s ni retaerg hgih -f neuqer cy

EGELLOC GNIREENIGNE DACS sol ses naht eht use of ria as na ni s otalu r. ,revewoH eht se sol ses era s llit rewol naht ht e ol s ses

detaicossa htiw m tso rehto dilos cirtceleid .slairetaM

sihT sedulcnoc ruo yduts fo noissimsnart senil . ehT tser fo siht retpahc lliw eb na noitcudortni

otni eht s ydut of ediugevaw s.

EHT NOISSIMSNART ENIL NOITAUQE – LARENEG NOITULOS

A tiucric htiw detubirtsid retemarap seriuqer a dohtem of sisylana tahwemos fid f tnere

morf taht deyolpme ni stiucric fo depmul stnatsnoc . ecniS a egatlov pord co c ru s ac sor s ae ch

seires tnemercni fo a ,enil eht egatlov deilppa ot hcae tnemercni fo tnuhs mda ecnatti si a

elbairav dna suht eht detnuhs tnerruc si a elbairav gnola eht .enil

ecneH eht enil tnerruc dnuora eht pool si ton a ,tnatsnoc sa si demussa ni ul m dep

tnatsnoc ric c tiu s, tub seirav orf m tniop ot tniop gnola eht enil . laitnereffiD tiucric snoitauqe ht at

sebircsed taht noitca lliw eb nettirw rof eht ydaets ,etats morf hcihw lareneg tiucric noitauqe lliw

eb denifed sa f .swollo

=R seires ,ecnatsiser smho rep tinu htgnel fo (enil sedulcni htob )seriw

=L seires ,ecnatcudni syrneh rep nu ti htgnel of enil

=C ecnaticapac neewteb ,srotcudnoc syadaraf rep tinu htgnel fo enil

=G tnuhs egakael ecnatcudnoc neewteb ,srotcudnoc sohm rep tinu htgnel

Of enil

Lω = seires ,ecnatcaer smho rep tinu htgnel of enil

Z = +R jωL

Lω = seires ,ecnatpecsus m soh rep tinu htgnel fo enil

Y = Cωj+G

S = ecnatsid ot eht tniop fo ,noitavresbo derusaem morf eht gniviecer dne fo eht enil

I = tnerruC ni eht enil ta any po tni

=E egatlov neewteb srotcudnoc ta yna tniop

l tgnel = h of enil

ehT woleb erugif setartsulli a enil t tah ni eht il m ti yam eb deredisnoc sa edam pu fo dedacsac

lamisetinifni T ,snoitces eno fo hcihw si .nwohs

sihT latnemercni noitces si fo htgnel fo sd dna seirrac a tnerruc I . ehT eires s enil im nadep ce

gnieb Z smho dna eht egatlov pord ni eht htgnel sd is

dE = sdZI ( )1

EGELLOC GNIREENIGNE DACS dE = ZI ( )2

sd

ehT tnuhs ecnattimda rep tinu htgnel fo enil si Y ,sohm os taht

ehT ecnattimda fo rht enil si sdY m soh . ehT tnerruc Id taht swollof ssorca eht enil ro morf eno

rotcudnoc ot eht rehto si

= EY sd

ehT noitauqe 2 dna 4 may eb taitnereffid de htiw er s tcewp ot s

d E ds 2

d I ds 2

d E ds 2

d I ds 2

esehT era eht laitnereffi snoitauqe fo eht snart m noissi ,enil latnemadnuf ot stiucric of id s detubirt

.stnatsnoc

sihT stluser setacidni owt ,snoitulos eno rof eht sulp ngis dna eht rehto rof eht sunim ngis

erofeb eht lacidar . ehT noitulos of eht fid f ;laitnere noitauqe s era

E = Ae + Be - YZ s

I = eC + De - YZ s

W ereh D,C,B,A era yrartibra stnatsnoc fo .noitargetni

ecniS eht ecnatsid si derusaem orf m eht er gniviec dne fo eht ,enil ti si elbissop ot ngissa

snoitidnoc hcus taht ta

s = 0, I = I E = ER

ehT n noitauqe 7 & 8 b ce om se

( )9

R

A dnoces tes fo yradnuob noitidnoc si ton ,elbaliava tub eht emas tes yam eb desu revo niaga fi

a wen tes fo snoitauqe a er demrof yb noitaitnereffid fo noitauqe 7 dna 8 . T suh

Id = sdYE

dI

( )3 ( )4

( )5

( )6

2 Id = Z , sd

Ed sd

2

= Y

2

= EYZ

= IYZ

2

sYA ( )7 ( )8 YZ s

R ,

ER = A + B I = C + D


( )01

- B YZ e ( )11

YZ s -

( )21

( )31

suoenatlumiS noitulos fo noitauqe 9 21, dna ,31 nola g

eht htiw f taht tca ER = IRZR dna taht Z Y sah

neeb itnedi f dei as eht Z 0 fo eht sdael,enil ot noitulos f ro eht noc s stnat o f eht evoba snoitauqe

sa

A = +

B = -

C = +

�1 - �

ehT noitulos fo eht laitnereffid snoitauqe fo eht snart m noissi enil m ya eb nettirw

Z 0 YZ s ER � +

Z R �e 2

� � � �

2 Z 0 �e 2 Z 0

�e

ehT evoba snoitauqe era yrev lufesu rof m rof eht egatlov dna c tnerru ta yna tniop no a

art noissimsn .enil

retf gniyfilpmis eht evoba snoitauqe ew teg eht lanif dna yrev lufesu mrof fo snoitauqe

rof egatlov dna tnerruc ta yna tniop no a ,enil=k dna era snoitulos ot eht evaw noitauqe

E = ER hsoc YZ s + IRZ0 hnis YZ s

I = I R hsoc YZ s + nis h YZ s

- YZ s

YZ s

YZ s

YZ s

A YZ e Ed = sd

YZ e

YZ s

ZI I =

Id sd

E =

- = A A

= C

C

- B

Y - e Z

e YZ

Y e Z

YZ Z e Y Y Z Z Y

B

YZ e YZ s e - - YZ s

Y - e Z

YZ D Y Z

Z Y

YZ s -

- B

- D

= A

= C

I

E

R

R

ER

2 ER = 2

Z Y

Z 0 � Z R

� � �

� � �1 +

I R

� 2

� � Y 2 Z R

� Z ER Z 0 =

� � � � ER I R

2 2 �1 -

� �1 +

� �

� R Y Z

Y Z

I R ZR � Z 0

�

Z R �

� �

I ER

2

ER

2

= 2

I R = 2

� � �

2

I R

2 Z � D = -

� �1 + E = � � �

� - �

YZ s

( )41

� �

Z 0 � Z R

�e � ER

2 �1 -

� � � � � �1 + �1 - I = + I R Z R YZ s I R ZR

- YZ s

� � � �

ER

Z 0


,htgneIevaW yticoIeV fo noitagaporp

W htgneIeva

ehT ecnatsid eht evaw slevart gnola eht enil elihw eht ahp se elgna si c degnah hguorht

2∏ snaidar si dellac .htgnelevaw

λ /п2= ß

ehT egnahc of п2 ni esahp elgna stneserper eno elcyc ni it me dna srucco ni a natsid ce of eno

,htgnelevaw

f/v =λ

V yticoIe

V= f λ

V=ω/ ß

sihT si eht yticolev fo noitagaporp gnola eht enil desab no eht noitavresbo of eht egnahc ni eht

esahp elgna gnola eht tI.enil si derusaem ni dnoces/selim fi ß si ni snaidar rep .retem

eW wonk taht

Z = R + j ωL

=Y +G j Cω

T neh

=γ α+j ß = YZ

GR -� CL + j� ( GL + CR)

gnirauqS no htob s sedi

α + 2 jαβ - β = GR -� CL + j� ( GL + RC)

gnitauqE laer strap dna yranigami strap ew teg

GR -� CL + ( GR -� LC )2 +� ( GL + CR) α =

2

dnA eht noitauqe f ro ß si

� CL - GR + ( GR -� LC) +� ( GL + CR)

2 nI a tcefrep enil 0=R dna G = 0 , nehT eht evoba noitauqe dluow eb

β = � LC

dnA eht yticolev of noitagaporp rof s hcu na laedi enil si nevig yb

2 =

2 2 2

2 2 2

2 2 2

β =

EGELLOC GNIREENIGNE DACS � v = β

suhT eht evoba noitauqe gniwohs taht eht enil retemarap seulav xif eht yticolev fo .noitagaporp

NOITROTSID

evaW - mrof noitrotsid

ehT eulav fo eht noitaunetta tnatsnoc α sah neeb reted m deni taht

GR -� CL + ( GR -� LC )2 +� ( GL + CR) α =

2

nI lareneg α si a nuf c noit of neuqerf cy. llA eht seicneuqerf snart m detti no a enil lliw neht ton eb

detaunetta lauqe l .y A oc m xelp deilppa atlov g ,e such as iov ce egatlov c gniniatno m yna

,seicneuqerf lliw ton evah lla seicneuqerf snart m detti htiw lauqe ,noitaunetta dna eht deviecer

rof lliw eb itnedi c la htiw eht tupni evaw f ro m ta eht s gnidne .dne sihT noitairav ic s= k nwon sa

ycneuqerf .noitrotsid

esahP noitrotsiD

ehT fo noitagaporp h sa neeb detats taht

� CL - GR + ( GR -� LC) +� ( GL + CR) 2

tI is tnerappa taht dnaω β od ton tob h evlovni f neuqer cy in same m renna dna aht t eht ve ol city

of noitagaporp lliw ni lareneg eb emos noitcnuf of .ycneuqerf

llA eht f seicneuqer deilppa ot a snart m si s noi enil iw ll n to evah eht same it me fo

snart m noissi , some neuqerf c ei s deyaled m ero naht eht o reht s. roF na deilppa iov ce egatlov

sevaw eht er c devie evaw s lliw not eb itnedi c la htiw eht tupni evaw f ro m ta eht er c gnivie ,dne

ecnis os me oc m stnenop lliw eb led a dey m ero aht n oht se fo eht rehto neuqerf c ei s. sihT

onehp m none si nwonk sa yaIed ro esahp noitrotsid .

ycneuqerF noitrotsid si uder c de ni eht snart m noissi fo hgih ytilauq ra oid daorb cast

smargorp revo eriw enil yb esu fo srezilauqe ta enil slanimret

esehT tiucric s era krowten s ohw se f neuqer cy and ahp se c arah c iret s cit s era ujda s det ot

eb revni se ot esoht fo eht ,senil gnitluser ni na revo lla inu f ro m neuqerf cy er s esnop revo eht

derised ycneuqerf .dnab

yaleD noitrotsid is ylevitaler m roni im natrop ce ot iov ce dna music art sn m si s noi eb c esua

of eht c arah c iret s cit s fo .rae tI nac eb yrev eires s ni c ri c tiu s dednetni f ro ip c erut snart m si s ,noi

dna snoitacilppa of eht oc laixa elbac evah neeb edam ot revo emoc eht fid f .ytluci

nI hcus selbac eht lanretni natcudni ce si wol ta hgih f neuqer c sei esuaceb of nniks ,tceffe

eht ecnatsiser sm lla esuaceb fo eht egral c rotcudno s, dna c apa c nati ce dna ael k na ce era sm lla

eb c esua of eht use of ria eleid c irt c tiw h a m ini mum s ap c re s. T eh olev c yti of tagaporp i no si

desiar dna edam m ero ylraen lauqe f ro lla f neuqer c .sei

ehT noitrotsid sseI eniI

fI a enil is ot evah rehtien ycneuqerf ron yaled id s neht,noitrot noitaunetta c no s tnat dna olev c yti

2 2 2

2 2 2

β =

EGELLOC GNIREENIGNE DACS fo noitagaporp tonnac eb noitcnuf fo .ycneuqerf

eW wonk taht

� v = β

nehT eht esahp tnatsnoc eb a tcerid noitcuf fo .ycneuqerf

� CL - GR + ( GR -� LC) +� ( GL + CR) 2

ehT evoba noitauqe swohs taht fi eht eht ret m rednu eht dnoces lacidar eb decuder ot lauqe

( GR +� LC )2

T neh eht deriuqer c noitidno f ro ß is deniatbo . gnidnapxE eht ret m rednu eht lanretni idar c la dna

gnicrof eht ytilauqe sevig

R 2G 2 - 2� GRCL +� L2C +� L2G 2 + 2� GRCL +� C R 2 = ( GR +� LC )2

ihT s secuder ot

- 2� GRCL +� L2G 2 +� C R 2 = 0

( GL - CR) 2 = 0

eroferehT eht noitidnoc taht lliw ekam ahp es tnatsnoc a tcerid mrof f=do ycneuqerf si

GL = RC A pyh o iteht c la l eni m thgi eb tliub ot f lu f lli iht s c oitidno n. T eh enil ow u dl th ne evah a eulav of ß

deniatbo yb esu fo eht evoba .noitauqe

ydaerlA ew wonk taht eht alumrof rof eht esahp tnatsnoc

β = � LC

nehT eht yticolev fo noitagaporp lliw eb

v = 1/ CL

sihT si eht as me rof eht lla f ,seicneuqer suht gnitanimile eht yaled .noitrotsid

We wonk taht eht noitauqe rof noitaunetta tnatsnoc

GR -� CL + ( GR -� LC )2 +� ( GL + CR) α =

2

yaM eb m eda i tnednepedn fo ycneuqerf if eht ret m rednu eht lanretni idar c la is rof c de ot ecuder

ot

( GR +� LC )2

isylanA s s woh s taht ht e c noitidno f ro eht id s noitrot sel s enil GL = RC , liw l udorp ce eht ed sir de

,tluser so taht ti is sop s elbi ot make noitaunetta noc s at tn dna olev c yti dni e tnednep of neuqerf cy

lsuoenatlumis y . gniylppA eht noitidnoc =GL CR ot eht noisserpxe rof eht noitaunetta sevig

α = RG

ihT s is eht tnednepedni of ,ycneuqerf suht ile m gnitani f neuqer cy id s noitrot no a enil . To ac eveih

2 2 2

β =

2

2 4 2 2 2 2 2 2

2 2 2 2

2 2 2

2

EGELLOC GNIREENIGNE DACS siht noitidnoc

GL = RC L R = C G

R iuqe re a ev ry al r eg eulav of ,L s ni ce G is sm .lla If G is yllanoitnetni ni cr ae s ,de

noitaunetta era ,desaercni gnitluser ni roop enil iffe c .ycnei

oT ecuder R iar s se eht ezis dna tsoc fo eht udnoc c srot evoba ec ono m ci il m ti s, so taht eht

lacitehtopyh stluser tonnac eb .deveihca

EHT ENOHPELET ELBAC

nI eht anidro ry ohpelet ne c elba eht deriw era i sn detalu htiw repap dna iwt s det ni .sriap

ihT s noc s urt c noit er s tlu s ni igilgen b el eulav s of udni c nat ce dna c udno c nat ce so taht aer s elbano

noitpmussa s ni eht oidua egnar fo seicneuqerf era taht

Z = R Y = j�C

GR -� CL + j� ( GL + CR)

γ = j� RC =

htiW ,0=L siht noitauqe semoceb

tI dluohs eb devresbo taht htob α dna eht yticolev era f snoitcnu of ,ycneuqerf hcus taht

eht rehgih seicneuqerf era detaunetta m ero dna levart retsaf naht eht rewol seicneuqerf . Very

elbaredisnoc ycneuqerf dna yaled noitrotsid si eht tluser no eht enohpelet .elbac

ECNATCUDNI GNIDAOL FO ENOHPELET ELBAC

A noitrotsid sel s enil htiw id s detubirt arap me sret s egu st a er m yde rof eht s ereve

ycneuqerf dna led ay id s noitrot neirepxe c de no gnol c elba s. tI aw s idni c deta taht ti saw

en c yrasse eht /L C itar o ot ac eveih id s noitrot less c noitidno s. edisivaeH s eggu s det taht eht

ecnatcudni eb ,desaercni

A dn P nipu leved depo eht oeht ry taht m eda elbissop this ni c esaer ni eht udni c nat ce yb

DEPMUL SROTCUDNI s ap c de ta raluger lavretni s gnola eht l eni . ihT s use of udni ctance si

c della

nI os me bus m enira elbac s, id s detubirt ro inu f ro m gnidaol is deniatbo yb niw d gni eht

elbac iw ht a hgih rep m ytilibae leets epat such sa rep m yolla . sihT m dohte is em deyolp eb c ua se

fo eht lacitcarp seitluciffid fo gningised depmul gnidaol slioc rof hcus retawrednu .stiucric

α dna

2 We wonk taht γ =

α =

β =

v =

� RC 2

� RC 2

� 2� = β RC

EGELLOC GNIREENIGNE DACS roF ,yticilpmis redisnoc f tsri a rofinu m yl dedaol c elba tiucric f ro hcihw ti m ya eb ussa m de

taht =G 0 dna rof hcihw L sah neeb desaercni os taht �L si egral tiw h er s tcep t o R . T neh

Z = R + j�L Y = j�C

niS c ,e

Z = R 2 +� L2 � - at n -1 2

T neh

Y

R 2 +� L2� - at n -1 x�C�

1 R at n -1 2 �L

nI iv ew of eht fact taht R i s sm lla tiw h er s ep ct to�L , eht mret

noitagaporp tnatsnoc oceb m se

1 R at n -1 2 �L

R �L

nat ) = is n� nat

roF a llams elgna

is nθ = at nθ = θ

oc sθ =

yllaniF eht noitagaporp tnatsnoc yam eb nettirw ,sa

γ = � CL ( oc sθ + j is nθ ) = � CL

,eroferehT rof eht inu f ylmro dedaol ,elbac

R L α = 2 C

β = � LC � v = = β

tI si ylidaer bo s devre ,taht rednu eht sa sum noitp s of G= dna0 �L egral tiw h er s ep ct ot R,

π 2 R �L

γ = =

= �

π R π 2 �L 2

2

2

π � - 2

LC 1+ R 2

� L2

4

2

R 2

� L2

m ya eb ppord e ,d dna m ya

π 2

γ = � LC� -

π 1 - at n -1

2 2 If θ =

� -1 -1 1 R 1 R 2 �L 2 �L

� �

π �

� oc sθ = soc ( - 2

R 2�L

R 2�L

� �

� � � �

+ j

1 CL

EGELLOC GNIREENIGNE DACS eht

noitaunetta dna olev c yti era htob tnednepedni of neuqerf cy dna eht dedaol c elba iw ll eb

id s noitrot sel s. ehT noisserpxe f ro noitaunetta c no s tnat s woh s taht ht e unetta a noit m ya eb

decuder yb gnisaercni ,L dedivorp taht R si ton osla desaercni oot .yltaerg

uounitnoC s ro inu f ro m gnidaol is nepxe s evi dna ac eveih s ylno a sm lla aercni se ni L rep tinu

.htgnel

uL m dep gnidaol si yliranidro erp f derre sa a snaem fo snart m si s noi evorpmi m tne f ro

elbac s. ehT im evorp m tne elbaniatbo no nepo eriw enil si yllausu ton tneiciffus ot itsuj fy eht

artxe soc t fo eht gnidaol .srotcudni

S’LLEBPMAC NOITAUQE

An ylana sis f ro eht rofrep m na ce fo a enil dedaol ta inu f ro m slavretni c na eb deniatbo by

gniredisnoc a s my m lacirte es c noit of enil orf m eht c retne of eno gnidaol lioc ot eht c retne of eht

,txen erehw eht daol gni lioc fo eht ecnatcudni si Z .c

ehT noitces enil yam eb decalper htiw na tnelaviuqe T noitces gnivah lacirtemmys seires

ra ms . gnitpodA eht noitaton fo retlif stiucric eno fo eseht seires smra si dellac Z1 2/ dna si

= Z 0 hnat 2 2

W ereh N is eht un m reb m eli s neewteb aol d gni c lio s dna γ is eht noitagaporp c no s tnat

rep m .eli nopU ni c gnidul lah f a gnidaol c ,lio the tnelaviuqe s eire s mra of eht dedaol sec noit

oceb m se

' Z1 Zc Nγ 2 2 2

ehT tnuhs z2 mra fo eht tnelaviuqe T noitces si

Z 0 Z 2 = hnis Nγ

An noitauqe gnitaler taht γ dna eht tiucric ele m tne of a T sec noit aw s aerla dy ,devired which

yam eb deilppa ot eht dedaol T es c noit sa

'

hsoc Nγ = 1+

Z1 Nγ

= + hnat

' Z1

2Z 2


Zc / 2 + Z 0 hnat (Nγ / )2 Z 0 / hnis Nγ

yB esu fo laitnenopxe ti nac eb swohs taht

Nγ hsoc Nγ -1 = 2 hnis Nγ

oS taht eht evoba uqe ati no uder ces ot

hsoc Nγ =

ihT s serpxe s noi is k nwon sa s’IIebpmaC E noitauq dna rep m ti s eht reted m noitani of a eulav f ro

' γ of a enil noitces gnitsisnoc yllaitrap fo ul m dep dnal yllaitrap of detubirtsid ele m .stne

aC m s’llebp noitauqe m eka s elbissop eht c la c noitalu of eht fe fects of gnidaol c lio s ni

gnicuder noitaunetta dna noitrotsid no .senil

roF a c elba 2Z of eht evoba f erugi is se s yllaitne c apa c eviti dna eht c elba c apa c nati ce

sulp depmul secnatcudni a raepp ralimis ot eht tiucric fo eht woI ssap retIif

tI si f dnuo taht f ro f eicneuqer s woleb wht c fotu f, nevig yb

f 0 = π

ehT noitaunetta is uder c de as epxe c ,det tub evoba c fotu f eht noitaunetta ir s se as a

res tlu of f etli r ac .noit ihT s fotuc f ycneuqerf rof ms a ed f ini te reppu il m ti ot s cu c lufsse

noissimsnart revo .selbac

tI nac eb desiar yb gnicuder L tub siht tneidepxe al eol s eht noitaunetta ot ir s .e

T eh c fotu f neuqerf cy la so eb uder c de yb s ap c gni eht c ol s re ,rehtegot thus r ude c gni C

dna m ero ylesolc gnitamixorppa eht detubirtsid tnatsnoc ,enil tub eht tsoc sesaercni .yldipar

nI ,ecitcarp a yIurt sid t noitro sseI eniI si on t o deniatb yb aoI d ,gni eb c esua R dna L

era ot emos tnetxe snoitcnuf fo rf ycneuqe . yddE tnerruc sessoI ni eht gnidaoI cudni tors

etavargga t sih idnoc t noi . How ,reve a rojam tnemevorpmi si deniatbo ni eht dedaoI eIbac

rof a eIbanosaer ycneuqerf gnar e.

TUPNI ECNADEPMI DNA REFSNART ECNADEPMI

ehT tupni ecnadepmi fo a noissimsnart enil sah ydaerla neeb deniatbo sa

soc h γl + Z 0 hnis γl � Z 0 soc hγl + ZR hnis γl

nI smret fo ,slaitnenopxe siht si

eγl + eγl -

fI eht egatlov ta eht gnidnes – dne ret m slani si ,nwonk ti si tneinevnoc ot evah eht refsnart

= 1+

hnat

Zc

2Z 0

' hnis Nγ + hsoc Nγ

1 CL

� � Zs = Z 0 = � � Z R

� � �

� � � ZS = Z 0 � � � Keγl

�

� Keγl

EGELLOC GNIREENIGNE DACS ecnadepmi os taht eht deviecer tnerruc nac eb detupmoc yltcerid . ehT s ov dne gnidne l E egat s

si

(eγl + eK -γl )

Es = (eγl + eK -γl )

roF hcihw eht refsnart ecnadepmi si

ZT = = (eγl + eK -γl ) R

yB gnitutitsbus rof ,K ehT evoba noitauqe semoceb

eγl + e -γl eγl - e -γl +

sihT si elbazingocer sa

ZT = Z R hsoc γl + Z 0 hnis γl

fI eht noisserpxe si derised ni ret ms of eht ilobrepyh c f noitcnu s. O nep dna trohs detiucric senil

sA il m deti sesac ti si tneinevnoc ot redisnoc senil ret m detani ni nepo tiucric ro trohs

,tiucric taht si htiw ZR = ∞ ro ZR 0= . ehT tupni im ecnadep of a enil fo htgnel l si

� ZR soc hγl + Z 0 hnis γl � � �

dnA f ro eht trohs tiucric esac ZR ,.0= s taht o

Zs = Z 0 t hna γl

erofeB eht nepo tiucric esac si ,deredisnoc eht tupni ecnadepmi dluohs eb nettirw

� soc h γl + (Z 0 / Z R ) hnis γl � Zs = Z 0 �

� �

ehT tupni im ecnadep of eht nepo c detiucri enil of htgnel ,l htiw ZR = ∞, si

Z oc = Z 0 htoc γl

yB gniylpitlum eht evoba owt snoitauqe ti nac eb nees taht

Z 0 = ZocZ cs

sihT si eht as me tluser sa saw deniatbo f ro a ul m dep rowten k. ehT evoba noitauqe seilppus a

yrev elbaulav snaem fo yllatnemirepxe reted m gnini eht eulav fo z0 fo a .enil

oslA orf eht as me owt eq snoitau

R

ER (ZR + Z 0 ) 2ZR

I (ZR + Z 0 ) 2

Es =

Es (Z + Z 0 ) I 2

R

� � � �

� � � ZT = Z R �

� � � � � �

� � 2 2

� � Z 0 soc hγl + ZR hnis γl � �

Zs = Z 0 =

� � (Z 0 / Z R ) soc h γl + hnis γl �


hnat γl =

γl = nat h -1

esU fo siht noitauqe ni latnemirepxe krow seriuqer eht noitanimreted fo eht cilobrepyh tnegnat

fo a xelpmoc elgna fI .

hnat γl = at h(α + jβ )l = U + Vj

nehT ti nac eb nwohs taht

2U 1+ U + V

2U 1-U -V

fo eulav eh β is nu c a niatre s tnardauq ot . stI reporp eulav m ya eb detceles fi eht ixorppa m eta

yticolev fo noitagaporp si .nwonk

NOITCELFER ROTCAF DNA NOITCELFER SSOL

NOITCELFER ROTCAF

2 | Z1Z 2 | | Z1 + Z 2 |

ehT ret m K setoned eht er f noitcel fac rot . sihT oitar etacidni s eht egnahc ni tnerruc ni eht daol

eud ot noitcelfer ta eht sim m dehcta noitcnuj dna si dellac eht er f el c noit f .rotca

NOITCELFER SSOL

noitcelfeR ssol si denifed sa eht rebmun fo srepen ro selbiced yb hcihw eht tnerruc ni

eht daol rednu egami dehctam tidnoc snoi dluow deecxe eht tnerruc yllautca gniwolf ni eht .daol

sihT noitcelfer ssol sevlovni eht lacorpicer fo eht noitcelfer fac rot .K

noitcelfeR ,ssol =srepen nl |

noitcelfeR ,ssol bd = 20 gol |

NOITRESNI SSOL

noitresnI ssol of eht enil ro wten kro si ed f deni as eht un m reb of srepen ro slebiced yb

hcihw eht tnerruc ni eht daol si degnahc yb eht .noitresni

Z cs

Z co

Z cs

Z co

2 hnat 2αl =

dna hnat 2βl

=

2

K =

Z1 + Z 2

2 Z1Z 2

Z1 + Z 2

2 Z1Z 2

| |


T DNA п IUQE V TNELA OT SENIL

ehT ngised fo na tnelaviuqe T noitces morf erusaem m tne no a krowten . T eh se snoitaler

w ere

Z1 = Z1 co - Z 2 co (Z1 co - Z 2 co )

Z 2 = Z 2 co - Z 2 co (Z1 co - Z 2 co )

Z 3 = Z 2 co (Z1 co - Z 2 co )

ehT tupni ecnadepmi fo nepo detiucric na d trohs detiucric senil erew ydaerla .depoleved

e + e e - e

Z = Z hnat γ l = Z

ecniS a enil si lacirtemmys ,krowten

Z1 co = Z 2 co

T eh Z3 ro tnuhs ele m tne fo a T noitces taht lliw eb ,tnelaviuqe ni os raf sa lanretxe segatlov

na d stnerruc era ,detcennoc ot eht gnol enil nac neht eb ylidaer deniatbo sa

� - Z 0 hnat γl �

hnat γl Z 0 = hnis γl

ehT seires stnemele rof eht tnelaviuqe noitces neht era

Z 1 = Z = Z 1 co = Z =

(eγl / 2 + e )

(eγl / 2 - e )(eγl / 2 + e ) Z 1 = Z = Z hnat γ l / 2

ehT T – noitces tnelaviuqe rof eht gnol ,enil m eda pu of eseht ele m ,stne si s nwoh ni eht woleb

rugif e . tI si esu f lu ni niatrec sepyt o f enil lac c noitalu s.

� � γ l

γ l

�

Z �

� � �

- γ l =

hnat γ l

� � -γ l �

0

� - γ l

� = Z Z 1 co 0

� γ l

γ l

e - e e + e - γ l

� �

� 1 cs 0 0

� Z 0 �

� � � � Z 0 Z 3 = hnat

γl

2 eγl - e

� � -γ l

- -γ l

e γl + e eγl - e

� � � � � �

2 3 -γl

� -γl 2 / 2 �

= Z � 0

� � � � �

-γl 2 / -γ l / 2

2 0


A π - noitces tnelaviuqe rof eht enil m ya esiwekil eb reted m deni orf m eht ret m lani

.stnemerusaem

esuaceB fo ,yrtemmys

Z 2oc Z1 cs Z 2 co (Z1 co - Z1 cs )

Z 0 2Z 0

eγl - e -γl

Z 0 ZA = ZC = (hnat γl / )2

ehT ZB mra is m sa deniatbo ylp

Z 2oc Z1 cs ZB = Z 2 co (Z1 co - Z1 cs )

Z 0 ZB = Z 0 / hnis γl

Z = ZC =

erehT f ero

A Z 2 co

=

2

eγl + eγl -

� � e -γl

�

� - �

� Z 0 �

�

e -γl

2

= Z 0 hnis γl


TINU – II

SENIL NOISSIMSNART YCNEUQERF HGIH

ITCUDORTNI ON

W neh a ,enil rehtie nepo eriw ro ,laixaoc si desu ta seicneuqerf fo a m ycage c el ro m ,ro ti

si f dnuo taht c niatre ixorppa m noita s m ya eb em deyolp gnidael ot si ilpm f dei ylana sis fo enil

cnamrofrep e . ehT snoitpmussa yllausu m eda :era

1. yreV elbaredisnoc niks fe fec ,t so taht c tnerru s m ya eb sa sum de as f gniwol no a

rotcudnoc ,secafrus lanretni ecnatcudni neht gnieb .orez

2. T tah � R>>L ehw com gnitup .Z sihT sa sum tp noi is uj s it f elbai eb c ua se ti is dnuof

taht eht er s ecnatsi ni c sesaer uaceb se of sk ni effe ct htiw f elihw eht enil er sis ecnat

sesaercni yltcerid htiw .f

3. ehT senil era llew hguone detcurtsnoc taht G yam eb deredisnoc orez

ehT isylana s si m eda ni rehtie of owt ways, d gnidnepe no hw e reht R is m ylere sm lla htiw

tcepser ot � L ro R si ,llams eht enil si deredisnoc yletelpmoc elbigilgen derapmoc htiw � .L

If R si sm ,lla eht enil is c no s deredi eno fo sm lla sid s ,noitapi dna iht s c no c tpe is us lufe nehw

senil era em deyolp as ric c tiu ele m tne s ro erehw er s nano ce eitreporp s era ovni .devl If sol s se

erew detcelgen neht ni f etni c tnerru ro egatlov s ow u dl raeppa ni c la c noitalu s, dna dna physic la

ytilaer ow u dl ton eb ac .deveih

nI snoitacilppa erehw sol s se m ya eb ,detgelgen sa ni snart m noissi fo rewop ta hgih iffe c nei c ,y

R m ya eb deredisnoc sa ,elbigilgen dna eht enil sa eno fo oerez sid s noitapi . esehT sdohtem lliw

eb deiduts .yletarapes

GNIDNATS SEVAW

nehW eht noissimsnart enil si ton dehctam htiw sti daol ,.e.i daol ecnadepmi si ton lauqe

ot eht citsiretcarahc nadepmi ce ( ZR = Z 0 ) , eht e ygren dereviled ot eht daol si detcelfer kcab ot

eht s ruo c .e

ehT oc m noitanib fo tnedicni dna detcelfer sevaw evig esir ot eht s gnidnat .sevaw

GNIDNATS - EVAW OITAR

ehT m erusae m tne of s gnidnat evaw s no a snart m si s noi enil dleiy s ni f ro m noita tuoba

piuqe m tne po gnitare noitidnoc s. ixaM mum rewop is debrosba yb eht lo da nehw LZ = Z0. If a

enil sah on s gnidnat ,sevaw eht ret m noitani f ro taht enil si c erro ct dna m ixa mum rewop snart f re

at k se alp c .e

| VMAX|

SV W =R

V| NIM |

oY u evah ylbaborp iton c de aht t ht e oitairav n of gnidnats evaw s s woh s woh aen r ht e fr

enil si ot gnieb ret m detani ni 0Z . A ediw noitairav ni egatlov gnola

13


eht nel g ht m nae s a ret m noitani f ra orf m .0Z A sm lla noitairav m nae s ret m noitani raen .0Z

erehT f ,ero eht oitar of eht m ixa mum ot eht m ini m mu is a m ae s eru fo eht rep fec noit of eht

ret m noitani of a .enil sihT oitar si dellac eht GNIDNATS -W EVA AR T OI S( W )R dna is wla a sy

desserpxe ni elohw un m reb s. roF axe m ,elp a oitar fo 1:1 sebircsed a enil ret m detani ni sti

citsiretcarahc ecnadepmi .)0Z(

egatIoV gnidnatS - evaW oitaR

ehT oitar of m ixa mum egatlov ot m ini mum egatlov no a enil is c della eht V EGATLO

GNIDNATS - EVAW OITAR SV( W )R . erehT f :ero ehT lacitrev senil ni eht rof m alu etacidni taht eht

ne c ol s de eititnauq s era ba s etulo dna taht eht two eulav s era at k ne with tuo ager rd ot alop rit ,y

gnidnepeD no eht erutan of eht gnidnats ,sevaw eht un m ire c la eulav fo SV WR segnar orf m a

eulav of 1 LZ( = ,0Z on s gnidnat )sevaw ot na ni f etini eulav f ro iteroeht c ylla oc m etelp er f el c .noit

S ni ce ereht is wla ays a sm lla ssol on a nil e, ht e m ini mum egatlov is reven orez dna eht

SV WR si wla a sy os me etinif eulav . ,revewoH if eht SV WR is ot eb a us lufe .ytitnauq eht rewop

sessol gnola eht enil tsum eb sm lla ni com irap s no ot eht snart m detti rewop .egatlov S ni ce

rewop is lanoitroporp ot eht s erauq of eht ,egatlov eht itar o of eht s erauq of eht m ixa mum dna

m ini mum segatlov si c della eht rewop s gnidnat - evaw itar o. nI a ,esnes eht an me is mis gnidael

esuaceb eht rewop gnola a noissimsnart enil seod ton .yrav

tnerruC gnidnatS - evaW oitaR

T eh oitar of m ixa mum ot m ini mum c tnerru gnola a snart m si s noi enil is c della TNERRUC

GNIDNATS - EVAW OITAR SI( W )R . :eroferehT ihT s oitar si eht as me sa taht f ro egatlov s. tI nac

eb desu erehw m ae s eru m tne s era m eda htiw ol ops taht sam elp eht m itenga c f dlei ola gn a nil e.

tI sevig eht emas stluser sa RWSV .stnemerusaem

GNIDNATS EVAW OITAR

ehT oitar fo eht mumixam ot muminim sedutingam fo egatlov ro tnerruc no a enil gnivah

gnidnats sevaw si dellac eht gnidnats evaw oitar ro egatlov gnidnats evaw oitar SV( W )R

S =

egatloV noitauqe si

V = [e + Ke - ]

amixaM fo egatlov srucco ta hcihw eht tnedicni dna detcelfer sevaw era ni esahp

V xam = [1 + K ] R

iniM ma of egatlov srucco ta hcihw eht ni c tnedi dna er f detcel sevaw era tuo of esahp

23

jβx

V xam I xam = V nim I nim

VR (ZR + Z 0 ) jβx

2ZR

VR (Z + Z 0 ) R

2Z


VR (ZR + Z 0 ) 2ZR

V xam = V nim 1-

V xam V nim

K = V am x V nim

V am x - V nim K =

| V nim | +V nim

|K|

SWR

sihT erugif swohs eht noitaler neewteb gnidnats evaw oitar S dna noitcelfer tneiciffeoc

ENO HTGIE ENILEVAW

roF eht art noissimsn enil eht egatlov dna tnerruc ta yna tniop x morf eht gniviecer dne fo

eht noissimsnart enil si

V = [eγx + Ke -γx ]

I = [eγx - Ke -γx ]

ehT mret htiw γx si deifitnedi sa eht tnedicni evaw gnissergorp drawrof morf eht ecruos

ot eht ,daol rehw e sa eht ret m gnivlovni e-γx si eht detcelfer evaw gnilevart morf daol kcab

sdrawot eht s ruo c .e

V nim = [1- K ]

1+ K K

-1

+1

R

VR (ZR + Z 0 ) 2ZR

I (ZR + Z 0 ) 2Z 0

EGELLOC GNIREENIGNE DACS 33

roF eht enil of orez ,noitapissid ht e noitaunetta noc s tnat si .orez

Z 0 = R0

VR (ZR + Z 0 ) 2ZR

retfA noitacifilpmis fo eht evoba noitauqe f ro ats gnidn evaw |K| = 1

V = V soc β x + Ij R nis β x .

,ylralimiS rof eht tnerruc no eht noissimsnart enil

I = I R soc βx + jVR / R0 nis βx.

ehT tupni im ecnadep fo a noitapissid enil si

Zs =

VR soc βx + Ij R0 nis βx =

I R soc βx + jVR / R0 nis βx

� ZR soc βx + jR0 nis βx � = R0 �

� �

Z + Rj R0 + Zj

roF na hthgie evaw enil

x = λ /8,βx =

ZR + jR0 nat (π / )4 R0 + jZR nat (π 4 / )�

ZR + jR0 R0 + jZ R

�

fI hcus a enil si ret m detani htiw erup ecnatsiser

ZR = RR

� RR + jR0 � ZS =

� �

,ecniS htob eht rotaremun dna rotanimoned evah lacitnedi ,sedutingam neht

γ = jβ dna

jβx [e + Ke - jβx ] V =

R R 0

V I

R

� R0 soc βx + jZR nis βx � � �

� �

nat β x � Or

R 0 Z = R nat β

x

2π λ . λ 8

Zs = R0 � Zs = R0 �

� �

� � � �

� �

� R0 + jRR �


| Zs =| R0

suhT na hthgie – evaw enil m ya eb us de ot refsnart any er sis nat ce ot im nadep ce htiw a

m edutinga lauqe ot R0 of eht ,enil ro tbo a ni a m edutinga m ta ch neewteb a er sis nat ce of yna

dna eulav R0 , eht lanretni resis nat ce of eht ecruos

RETRAUQ ENILEVAW DNA ECNADEPMI GNIHCTAM

ehT tupni ecnadepmi fo a noitapissid snart m noissi enil si

ZR + jR0 nat βx R0 + jZR nat βx

ZR / nat βx + jR0 R0 / nat βx + jZ R

�

roF a retrauq evaw enil

x = λ / 4, βx = 2π / λ * λ / 4 = λ / 2

gnitutitsbuS eht arap m rete eulav ni eht evoba noitauqe eht s gnidne dne im ecnadep of eht

retrauq evaw remrofsnart si

R0 Zs = ZR

A retrauq evaw sec noit of enil m ya eb c no s deredi sa a rofsnart m re ot m ta ch a daol fo ' ZR ot a s ruo ce fo Zs . uS ch a m ta ch c na eb deniatbo if eht c arah c iret s it c im nadep ce R0 of eht

m ta c gnih retrauq evaw noitces of eht enil si ylreporp c oh s .ne

R0 =| Zs ZR |

� Zs = R0 � �

� �

�

Zs = R0 � � �

� �

2

'


III TINU

YCNEUQERF HGIH NI GNIHCTAM ECNADEPMI

LPA SNOITACIL FO RETRAUQ EVAW REAMROFSNART

A retrauq evaw remrofsnart yam osla eb desu fi eht daol si ton a erup ecnatsiser

.

ehT retrauq evaw remrofsnart si a elgnis ycneuqerf ro worran dnab

ived ce . ehT htdiwdnab m ya eb desaercni yb gnisu owt ro m ero

retrauQ evaw noitces ni .seires

A retrauq evaw rofsnart m re yam eb deredisnoc sa na im ecnadep

retrevnI ni taht ti nac mrofsnart a wol ecnadepmi otni a hgih

Im ecnadep dna eciv rev s .a

53

sZ ZR

λ 4

FLAH EVAW ENIL

ydaerlA ew wonk taht ehT tupni ecnadepmi fo a oitapissid n ssel enil si

Z R + jR0 nat βx R0 + jZR nat βx

roF a retrauq evaw enil

x = λ / 2, βx = 2π / λ * λ / = 2 π

+ jR0 at nπ � R0 + jZR at nπ

ZR R0

� Zs = R0 � � �

� �

Zs = R0 � � � �

� ZR

R0

EGELLOC GNIREENIGNE DACS Zs = ZR

sihT enil yam eb deredisnoc sa eno ot eno .remrofsnart

:NOITACILPPA

tI is desu ni gnitcennoc a daol ot a ruos ce ni cas se nehw eht aol d dna s ruo ce c tonna eb

edam ajda c .tne

BUTS GNIHCTAM

deeN rof buts gnihctam

oT hctam eht daol im ecnadep ot eb lauqe ot eht tupni im .ecnadep

SEPYT

1. ELGNIS BUTS GNIHCTAM

2. ELBUOD BUTS IHCTAM NG

63

ELGNIS BUTS TAM GNIHC NO A ENIL

roF tsetaerg fe f neici cy dna dereviled po ,rew a h hgi neuqerf cy snart m si s noi enil s dluoh

eb detarepo as a sm htoo nil e ro htiw an R0 et rm .noitani H ,revewo eht us lau daol s, s hcu sa

annetna s, od ton ni lareneg evah er sis nat ce fo eulav lauqe ot R ,0 so taht m yna cas se ti si

en c yrasse ot ecudortni os me f ro m fo im ecnadep – rofsnart m gni ac noit neewteb enil na d daol ot

ekam eht daol raeppa ot eht enil sa a ecnatsiser eulav .0R

ehT retrauq evaw enil ro remrofsnart dna eht derepat enil era s hcu im nadep ce m gnihcta

ived ces . rehtonA snaem fo occa m gnihsilp eht derised tluser si eht esu of na nepo ro desolc

buts enil fo elbatius htgnel sa a ecnatcaer s detnuh sorca s eht snart m noissi enil ta eht daol ot

ecnanoser htiw na tnanoseritna ecnatsiser lauqe ot .0R

oV egatl muminim erofeb noitresni fo eht buts

S1

d S1

A

ZR

Ys Yd

A

L


niS ce eht tupni c udno c nat ce fo eht of a enil is /S R0 ta a egatlov m ixa mum dna S/R0 ta

a egatlov m ini mum, neht ta os me retni m etaide tniop[ A eht laer trap fo ht e tupni da m ecnatti

yam eb na retni m etaide eulav fo 0R/1 ro eht tupni ecnattimda ta A sah a eulav

Ys =

ehT sus c natpe ce B si eht s tnuh eulav ta eht tniop ni euq s .noit Af et r eht tniop gnivah a

ecnatcudnoc lauqe ot 0R/1 si ol c ,deta a s troh s but enil nivah g tupni s su c natpe ce fo � β m ya

eb detcennoc ssorca eht noissimsnart enil . ehT tupni da m ecnatti ta siht tniop neht si

± jβ � jβ =

rO eht tupni ecnadepmi fo eht enil ta tniop A gnikool sdrawot eht daol si

Zs = R0

73

S ni ce htob eht ol c noita dna gnel ht of eht s but must eb reted m ,deni owt tnednepedni

stnemerusaem tsum eb edam no eht lanigiro enil dna daol ot eruces tneiciffus .atad

ehT m tso ae s yli tbo a ni ed m ae s eru m tne s era eht s gnidnat evaw oitar S dna eht op s noiti

of a egatlov m ini mum, yllausu eht m ni imum eraen st ot eht .daol A egatlov m ini mum is c nesoh

rehtar naht a ,mumixam ecnis sti noitisop yllausu nac eb denimreted erom .yletarucca

If eht noitacol fo eht buts is f dexi htiw er s ep ct ot na lanigiro egatlov m ini mum, on

egdelwonk fo eht daol ecnadepmi si .dedeen

esuaceB fo eht gnilellarap of ele m tne s, ti is most tneinevnoc ot row k htiw da m natti ces.

ehT tupni im ecnadep noitauqe si gnikool awot rds eht daol f or m yna p tnio no eht nil e, m ya

sa nettirw

| K�φ - 2βs � | K�φ - 2βs

gnitirW 0R/1=0G dna gnignahc ot ralugnatcer setanidrooc sevig

1- | K | soc (φ - 2βs) - j | K | nis (φ - 2βs) | K | soc (φ - 2βs) + j | K | nis (φ - 2βs) �

1- | K |2 -2 j | K | nis (φ - 2βs)

� �

gnisserpxE eht s tnuh c ecnatcudno sa a id m ne s sselnoi oitar Gs ,oG/ ro no a rep u tin ab s ,si

�1 - 2 | K | (soc φ - 2βs)+ | K |2

dnA eht tnuhs ecnatpecsus no a rep tinu sisab si

1 R0

± jβ

1 R0

1 R0

Ys =

� 1- Ys = 1

R0 � � � � 1+ � �

� Ys = G0 �1-

�

dnA nopu ,gnizilanoitar

�

�

Ys = G0 �1- � �

| K |2 2+ j | K | soc (φ - 2βs) �

�

�

1- | K |2 Gs = G0

� �

�

Bs = G0 �1- � �

EGELLOC GNIREENIGNE DACS 1- | K |2 -2 j | K | nis (φ - 2βs)

� �

retfA gniyfilpmis eht evoba snoitauqe ew teg eht noitacol dna ecnatsid of eht buts

ehT ecnatsid d morf eht ov egatl muminim ot eht tniop fo buts noitcennoc si

d = s2 - s1

oc s -1 | K | 2β s -1 λ s +1� 4 π

eB f ero noitcennoc of eht ehtbuts noitauqe si

λ S nat 2π S -1

83

ehT buts htgnel s dluoh eb

L' =

ehT eIcriC margaid rof noitapissid sseI eniI

A tahwemos ralimis elcric margaid yam eb ,deniatbo ,revewoh taht sevlos eht

ecnadepmi noitauqe dna seifilpmis eht ngised fo sselnoitapissid senil ylbaredisnoc . T eh tupni

ecnadepmi noitauqe rof a noitapissid ssel enil yam eb nettirw sa

Zs 1+ | K | �φ - 2βs = R0 1- | K | �φ - 2βs Zs R0

nA lautca elcric lliw evah eht suidar

S -1 r = =

2S dnA eht retnec of eht c elcri no eht evitisop si

S +1 c = =

2S A fam yli of selcric yam eb nward rof evisseccus seulav of S a s ni f gi . nI gniward ralucitrap

selcric ti si gnitseretni ot eton taht f ro yna elcric eht retni c tpe raen nigiro is ta 1/S, dna taht f ra

devomer morf eht nigiro si ta S stinu no eht ar sixa

| K |2 2+ j | K | soc (φ - 2βs) �

d =

oc s -1 � � � �

� =

-1 L =

λ 2

- L

= ra + jxa

1 S 1 S

S -

2

2

2 S +

2


1.5 S=3 5.2=s

1 222

0.5 222 2 2=s

1111 1 =s 5.1

- 5.0

-1

- 5.1

ehT uminim m eulav rof S si ehT.ytinu evoba erugif swohs taht lla S selcric m tsu

dnuorrus eht 0,1 tniop . nI ,tcaf eht elcric rof S = 1 si detneserper yb eht 0,1 .tniop

93

ehT umixam m eulav of S si ni f ,ytini rof eht esac fo nepo tiucric ro trohs

tiucric enil ret m ani oit n. sA S ,esaercni eht suidar fo eht S ,sesaercni dna eht retnec sevom ot eht ;thgir f ro

eht il m gniti esac fo S = ,ytinifni eht c elcri eb com se eht xa .sixa

ehT enil ecnadepmi si mumixam . A dn

Zs 1+ | K | R0 1- | K |

ret m setani ta eht elcric tpecretni ,S/1 eht enil im ecnadep sah a m ini m mu

1 1- | K | = S 1+ | K |

retfA emos noitacifilpmis ew teg eht lanif ,noitauqe taht

ra

2 + � xa +

1 = nis 2βs

eniL s of lauqe βs era eht nees ot eb c ri c sel o f suidar

1 = is n2 2βs

= S =

Zs

R0 W neh

,eulav dna

Zs = R0

1 � 2

� = 1+ 1 at n 2

2βs � � � �

� nat

2βs 2

EGELLOC GNIREENIGNE DACS htiW eht tfihs fo retnec drawnwod no eht ax sixa )etanidro(

1 = -

at n 2 2βs

S- ELCRIC 2

1 S +1

� � �

noitaciIppA fo eht eIcric margaid

desU ot if dn eht tupni ecnadepmi of a enil of yna nesohc htgnel

∞ , eht gnidnopserroc S c ri c el gniraeppa as eht

itrev c la ixa s. ehT tupni im ecnadep is neht erup aer c nat c ,e htiw eht eulav f ro

suoirav irtcele c la shtgnel reted m deni yb eht snoitcesretni fo eht nopserroc gnid

βs htiw eht lacitrev .sixa

ehT tupni da m ecnatti fo eht enil m ya eb dnuof yb siht .dohtem

A trohs detiucric nil yam eb devlos yb gninimreted sti ecnattimda . ehT S ric c el si

niaga eht lacitrev ,sixa dna ecnatpecsus seulav yam eb daer ffo ta airporppa et

snoitcesretni of eht βs selcric htiw eht lacitrev Sixa

HTIMS TRAHC

� � � � � �

2

�� 2 �

�

�

�

2

+ ba = 2S � g -

�

� S + 2S

2 � a �

� �

An nepo c ri c detiu enil ah s =S


nA ecnadepmi htimS trahc htiw( on atad )dettolp

. ehT Sm hti trahC c na eb us de ot erper s tne many arap m rete s ni c gnidul secnadepmi ,

secnattimda , er noitceIf stneiciffeoc , tacs ter gni ap rameters, esion f ugi re c ri c ,sel c no s tnat

niag ruotnoc s dna noiger s rof nu c lanoitidno s ytilibat . T eh Sm hti trahC is om st yltneuqerf us de ta

ro tiw h ni eht ytinu suidar noiger . ,revewoH eht er m rednia si llits m ehta m ita c ylla naveler t, gnieb

,desu rof ,elpmaxe ni rotallicso ngised dna ytilibats sisylana ehT Sm hti trahC is dettolp no eht com xelp er f el c noit c iffeo c tnei enalp ni owt id m ne s snoi

dna si delacs ni ron m dezila im ecnadep eht( most c mo m ,)no ron m dezila da m natti ce ro ,htob

gnisu tnereffid c srolo ot id s iugnit sh neewteb ht em. esehT era netfo k nwon as eht ,Z Y dna ZY

Sm hti trahC s er s .ylevitcep ]7[ roN m dezila sc gnila lla ows eht Sm hti trahC ot eb us de rof elborp ms

gnivlovni yna itsiretcarahc c im ecnadep ro ys s et m im ,ecnadep ohtla u hg by f ra eht most

moc m ylno us de is 05 mho s. W hti ylevitaler is m elp lacihparg noc s noitcurt ti is s thgiart f drawro ot

trevnoc neewteb dezilamron im nadep ce ro( ron m dezila da m natti c )e dna eht gnidnopserroc

xelpmoc egatlov er f noitcel .tneiciffeoc

ehT Sm hti hC tra sah c laitnerefmucri gnilacs ni nelevaw g sht dna eerged s. ehT

htgnelevaw s elacs si us de ni id s detubirt com tnenop elborp ms dna erper s stne eht id s nat ce

m ae s deru gnola eht snart m si s noi enil c enno c det neewteb eht rotareneg or ruos ce dna eht daol

ot eht tniop rednu c no s .noitaredi ehT seerged elacs erper s tne s eht elgna fo eht egatlov

er f noitcel feoc fic tnei ta taht .tniop ehT Sm hti trahC m ya la so eb us de f ro ul m dep ele m tne

gnihctam dna isylana s elborp ms.

Use of eht Sm hti C trah dna eht in noitaterpret of eht er s tlu s deniatbo us gni ti eriuqer s a

doog rednu s gnidnat fo CA c tiucri oeht ry dna snart m si s noi enil oeht r ,y htob of ihw ch era erp -

setisiuqer rof FR .sreenigne

As im nadep c se dna da m natti c se egnahc htiw neuqerf c ,y elborp ms us gni eht Sm hti

trahC nac ylno eb s lo dev m yllauna us gni eno f neuqer cy at a it m ,e eht er s tlu gnieb r pe res detne

yb a tniop . sihT si netfo etauqeda f ro worran dnab ilppa c snoita yt( pic ylla up ot tuoba %5 ot %01

htdiwdnab ) tub f ro ediw r shtdiwdnab ti is us yllau yrassecen ot ylppa Sm hti trahC nhcet seuqi ta

m ero naht eno ycneuqerf ac sor s eht gnitarepo neuqerf cy nab d. P dedivor eht neuqerf c ei s are

yltneiciffus olc s ,e eht gnitluser Sm hti trahC stniop m ya eb denioj by s thgiart enil s ot etaerc a

ol cus.

A sucol fo stniop no a htimS trahC gnirevoc a egnar fo seicneuqerf nac eb desu ot yllausiv

erper s :tne

• woH eviticapac ro woh evitcudni a daol si ssorca eht ycneuqerf egnar

EGELLOC GNIREENIGNE DACS • woH tluciffid gnihctam si ylekil ot eb ta suoirav seicneuqerf

• woH llew m dehcta a ralucitrap tnenopmoc .si

ehT ycarucca fo eht Smi ht trahC is uder c de rof elborp ms gnivlovni a egral s daerp fo

im secnadep ro da m natti ces, hguohtla eht sc gnila c na eb m inga f dei rof laudividni aera s ot

etadommocca .eseht

snoigeR fo eht Z S htim trahC

If a ralop argaid m is m deppa no ot a c etra s nai c etanidroo sys met ti is lanoitnevnoc ot

m erusae selgna evitaler ot eht evitisop x- ixa s us gni a c retnuo -c col k iw se erid c noit f ro op s eviti

elgna s. T eh m edutinga of a com xelp un m reb si eht htgnel of a s thgiart nil e ward n f or m eht

nigiro to eht tniop gnitneserper .ti

hT e Sm hti trahC esu s eht as me c ,noitnevno gniton ,taht ni eht ron m dezila im nadep ce

,enalp eht op sit evi x- sixa dnetxe s orf m eht c retne of eht Sm hti C ah rt ta ot t eh tniop . ehT r noige

evoba eht x- ixa s tneserper s udni c evit im secnadep dna eht noiger leb ow eht x- ixa s erper s stne

eviticapac im secnadep . evitcudnI im nadep ces evah evitisop im yraniga strap dna eviticapac

secnadepmi evah evitagen yranigami .strap

fI eht ret m noitani si yltcefrep m ,dehcta eht noitcelfer tneiciffeoc lliw eb ,orez detneserper

ylevitceffe yb a elcric fo orez suidar ro ni fact a tniop ta eht ertnec o f eht Sm hti trahC I . f eht

ret m noitani saw a tcefrep nepo tiucric ro trohs tiucric eht m edutinga fo eht noitcelfer c tneiciffeo

dluow eb ,ytinu lla rewop dluow eb er f detcel dna eht tniop dluow il e ta emos tniop no eht ytinu

ecnerefmucric .elcric

seIcriC fo tnatsnoC deziIamroN ecnatsiseR dna tnatsnoC deziIamroN ecnatcaeR

ehT ron m dezila im ecnadep Sm hti trahC si com desop fo owt fam seili of ric c el s: c ri c el s fo

tnatsnoc ron m dezila er sis nat ce dna c ri c el s of c no s tnat ron m dezila aer c nat c .e nI eht com xelp

er f el c noit c feo fic tnei enalp eht Sm hti trahC eipucco s a c ri c el of ytinu r uida s c deretne ta eht

nigiro . nI C etra s nai c etanidroo s ereht f ero eht c ri c el dluow ap ss hguorht eht tniop s )0,1( dna (-

)0,1 no eht x-a ix s dna eht stniop )1,0( dna ,0( - )1 no eht y- .sixa

W gnikro w hti htob eht Z htimS trahC dna eht Y htimS strahC

nI FR tiucric dna m gnihcta elborp ms som emite s ti is m ero c tneinevno to row k htiw

da m natti ces gnitneserper( c udno c s’ecnat dna s secnatpecsu ) dna som ite m se ti si m ero

tneinevnoc ot krow htiw im secnadep gnitneserper( er s secnatsi dna aer c nat c ’e s .) S gnivlo a

ipyt c la m gnihcta elborp m lliw of net eriuqer s lareve c egnah s neewteb bo ht epyt s of Sm hti C ,trah

gnisu ron m dezila im ecnadep f ro s seire ele m stne dna n ro m dezila adm natti ces f ro lellarap

ele m tne s. roF eht se a laud ron( m )dezila im nadep ce dna da m natti ce Sm hti trahC m ya be us .de

,ylevitanretlA eno epyt m ya eb desu dna eht s gnilac detrevnoc ot eht rehto nehw .deriuqer

EGELLOC GNIREENIGNE DACS nI redro to c egnah f or m ron m dezila im dep ecna ot ron m dezila da m natti ce ro iv ce rev s ,a

eht tniop erper s gnitne eht eulav of er f el c noit c iffeo c tnei rednu c no s noitaredi is m devo hguorht

34

yltcaxe 081 seerged ta eht same uidar s. roF axe m elp eht tniop 1P ni eht axe m elp erper s gnitne

a er f noitcel oc iffe c tnei of ah s a ron m dezila im nadep ce of. To ihparg c ylla c egnah iht s ot eht

tnelaviuqe ron m dezila da m ecnatti ,tniop yas ,1Q a enil si nward htiw a relur f or m 1P hguorht eht

Sm hti C trah c ertne ot ,1Q na uqe al uidar s ni ht e o opp s eti id rec noit . sihT si iuqe tnelav ot m gnivo

eht tniop hguorht a c ralucri htap of axe c ylt 81 0 eerged s. R gnidae eht eulav orf m eht Sm hti

trahC rof ,1Q gnirebmemer taht eht gnilacs si won ni ron m dezila da m ,ecnatti sevig .

ecnO a noitamrofsnart morf ecnadepmi ot ecnattimda sah neeb rep demrof eht gnilacs segnahc

ot dezilamron ecnattimda litnu hcus emit taht a retal noitamrofsnart kcab ot dezilamron

im ecnadep si .demrofrep

SMELBORP NO ELGNIS BUTS GNIHCTAM

enimreteD.1 eht htgneI dna eht ecnatsid fo eht buts morf eht daoI . Gi nev t ah t a

moc xeIp daoI =LZ 05 - 001j si ot eb dehctam ot a 57 mho noissimsnart eniI gnisu a trohs

detiucric st .bu

neviG

citsiretcarahC ecnadepmi fo eht noissimsnart enil Z0= mho57

daoL im ecnadep ot eb dehctam ot eht snart m si s noi enil ZL= 05 - 001j

oT fi dn

ecnatsiD fo eht buts morf eht daol

htgneL fo eht buts morf eht daol

ehT ron m dezila im ecnadep si reted m deni yb gnidivid eht daol im nadep ce yb eht

citsiretcarahc ecnadepmi fo eht noissimsnart .enil

= 766.0 - j 3.1 3

T eh ron m dezila im nadep c ,e ZL is dettolp no eht smi ht c trah yb reted m gnini eht iop nt

of noitcesretni neewteb eht noc s tnat R c ri c el htiw R = 766.0 dna noc s tnat X c ri c el

htiw 33.1 =X

ehT ecnadepmi elcric si .nward

esuaceB eht sbuts era detcennoc ni lellarap htiw eht ,daol da m natti ces c na eb m hcu

ylisae desu rehtar naht im secnadep ot ilpmis fy eht .snoitaluclac

ehT ron m dezila da m ecnatti si reted m deni orf m eht sm hti c trah by sim ylp nitator g eht

im ecnadep ,tolp by 81 0 .eerged sihT si is m ylp d eno by rd a gniw a enil f or m tniop A

.1

.2

noituIoS

.1

.2

.3

ZL

Z 0 ZL = = 05 - j 001

57

EGELLOC GNIREENIGNE DACS hguorht eht retnec of eht trahc ot eht etisoppo edis fo eht ,elcric tniop .B

4. eht da m ecnatti tniop si detator esiwkcolc ot a tniop no eht im ecnadep elcric erehw ti

tcesretni s eht arahc c iret s cit im nadep ce Z0 . At eht tniop C. ehT laer oc m tnenop fo

eht tupni im nadep ce ta siht tniop is lauqe ot eht c arah c iret s it c im ecnadep Z0 . tA siht

tniop ,C eht ecnattimda si .7.1j+1=y

5. ehT ecnatsid orf m tniop B ot tniop ,C ni ret ms fo eht nelevaw g ht si woh f ra f or m daol

eht buts must eb alp c ,de

ehT s but m tsu evah a orez iser s evit com tnenop im nadep ce dna su natpecs ce taht

sah eht etisoppo .ytiralop

6. oT reted m eni eht htgnel of eht detrohs buts taht sah na etisoppo aer c evit oc m tnenop

ot eht tupni da m ,ecnatti eht edistuo fo eht Sm hti c trah )0=R( is m devo dnuora htiw

eht s gnitrat tniop ta D {s ni ce ta tniop D =t 0 dna neh ce γ = ∞ ,} litnu na da m natti ce y

= si7.1 f dnuo

7. T eh id s nat ce neewteb p tnio D dna E is eht gnel th of eht s .but roF iht s itnauq ty eht

morf eht htims ,trahc

ELBUOD BUTS GNIHCTAM

2. gnisU D eIbuo buts ,gnihctam hctam a xeIpmoc daoI of LZ = 52.65j+57.81 to a

iI en htiw citsiretcarahc ecnadepmi 0Z = .mho57

enimreteD eht buts ,shtgneI gnimussa a rauq ter w gneIeva th gnicaps are

deniatniam neewteb eht owt trohs detiucric sbuts .

A s ap c gni fo λ / 4 is m deniatnia neewteb eht s but s, s 2but dna s .1but roF sm htoo enil

repo noita of eht snart m si s noi enil eht tupni im ecnadep gnikool otni eht ret m lani s 2,2 fo

eht enil dluohs ,eb

Y2,2 = 1 / Z 0

ehT buts ta 1,1 tsum eb elbapac ot mrofsnart eht ecnattimda ta eht gnitanimret

ecnadepmi dne ot eht elcric B hcihw si decalpsid morf t eh elcric ;A 1=R yb ‘ λ / 4 .’

ehT retrauq htgnelevaw enil lliw rehtruf mrofsnart eht da m ecnatti otni a eulav ta 2,2

hcihw tolp no eht ric c el A . suhT eht enil ot daol natsid ce neewteb noitisop 2,2 si ton

deriuqer ot eb reted m .deni

54


TZ

gnitanimreT im cnadep e

Ls2

ehT dezilamron daol im ecnadep

Zi ZL = = Z 0

ZL 52.0 = + j 57.0

gnittolP eht dezilamron ecnadepmi no eht htimS ,trahc eht ecnadepmi elcric si nward

htiw ecnatsid neewteb eht tniop )0,1( dna eht tniop fo eht ezilamron d ecnadepmi sa eht

suidar ,ecnatsid{ }AO

1. gnivoM yb 081 eerged 52.0( λ ) no eht im ecnadep elcric , taht si ta a aid m yllacirte

etisoppo tniop ot eht tniop ,A ,.e.i tniop B lliw evig eht dezilamron da m .ecnatti

morF eht htims trahc =LY 4.0 - 2.1j

2. elcriC A si eht tnatsnoc R elcric rof R = 1 . riC c el B si eht ol c su of lla ht e op stni no eht

elcric A si id s decalp yb λ ,4/ retrauq htgnelevaw . ehT buts 1 sdda a ecnatapecsus fo lla

eht tniop s no eht elcric .B

ecniS buts 1 c tonna retla eht c ecnatcudno , ot a tniop no eht ric c el B, tniop ,C

Ls1

57.81 + j 52.65 57


ta(Y tniop =)C 4.0 - 5.0j

3. rrefsnarT gni eht tniop C ot eht tniop D no eht elcric ,A ecnis eht enil neewteb 1,1 dna

2,2 si a retrauq evaw enil taht smrofsnart eht ecnattimda ta 1,1 ot 2,2 hcus taht eht

ecnatcudnoc slauqe eht citsiretcarahc ,ecnatcudnoc /1 .0Z

tA(Y tniop D ) = 2.1j+0.1

4. ehT buts htgnel ta 2,2 dluohs lecnac eht im yraniga trap fo eht evoba da m ecnatti fo eht

buts ta 2,2 m tsu eb - .2.1

5. oT f dni eht htgnel of eht buts htiw na da m ecnatti ,

)a( +j 7.0 dna )b( – 2.1j

ehT edistuo elcric fo eht htims trahc eht( ,elcric ,)0=R si devom ra dnuo gnivah a

ecnerefer ta a tniop ,P litnu

nA ecnattimda =y - 2.1 si dnuof ta tniop E dna

nA ecnattimda =y 7.0+ si dnuof ta tniop .F

6. morF eht sm hti ,trahc

htgneL fo eht buts =1 ecnatsid neewteb P dna F 843.0=1sL λ

htgneL of eht s but =2 id s ecnat wteb nee P dna F Ls 11.0=2 λ 3 . enimreteD eht IIof w :gni

)a( S gnidnat w eva )RWSV(oitar

)b( daoL ecnattimdA

)c( ecnadepmI fo eht noissimsnart eniI ta eht am x mumi dna muminim of eht

yranoitats sevaw gnoIa eht eniI

)d( ecnatsiD teb w nee daoI dna tsrif egatIov mixam mu . roF a noissimsnart eniI w hti

citsiretcarahc ecnadepmi fo 05 mho htiw a gniviecer dne fo 121j+001 . T eh

w htgneIeva fo eht IacirtceIe Iangis gnoIa eht eniI si .m5.2

:neviG

citsiretcarahC ecnadepmi 05=0Z mho

daoL ecnadepmi =LZ mho121j+001

htgnelevaW fo eht lacirtcele langis λ 5.2=

noituloS 1. dezilamroN im ecnadep = 24.2j+2=

001 + j 121 05


gnitolP eht tniop p no eht htims trahc . ehT im ecnadep elcric si nward htiw )0j+1(O ertnec dna

suidar sa ,)PO( eht ecnatsid neewteb ertnec dna eht dezilamron egatlov gnidnats evaW oitaR =

5

2 . ehT op tni Q yllacirtemaid etisoppo ot eht dezilamron im ecnadep tniop no eht im ecnadep

elcric si eht dezilamron da m ecnatti fo eht .daol

= 22.0 - j 52.0

YZ 0 22.0 = - j 52.0

22.0( - j )52.0

= 0 4400. - j0 00. 5 ohm 3. ecnadepmI ta eht tsrif egatlov mumixam morf daol =

5× Z 0

= 052 mho Im ecnadep ta eht tsrif egatlov m =mumini

2.0 × Z 0

= 10 mho

ecnatsiD neewteb daol dna tsrif egatlov =mumixam

0 40. 2λ = 240,0 × 2.5m = 0. 01 5m

Y G0

1 05

daoL im ecnadep

Y =


UN TI - VI SRETLIF EVISSAP

1. Neper A pen er (Sy bm o :l Np) is a logari ht cim tinu of ratio. It is ton an SI tinu b tu si a ecc detp of r use alo gn s edi the SI. It is u es d ot e rpx e ss rat oi s, such as ga ni a dn ol ss,

dna relat vi e va eul s. T eh name is der vi ed rf om Jo nh Napier, the vni e rotn of logari ht ms.

ekiL the dec bi el, ti is a tinu ni a logarit mh ic s ac le, the d ffi erence be gni that whe er ht e dec bi el uses base- 01 logarit mh s to comp etu rat oi s, ht e neper uses base e ≈

2. 2817 8. The va ul e of a rat oi ni nepers, pN , is vig en by

hw ere x1 a dn x2 are eht va ul es of tni erest, a dn nl is the nat ru al logarit .mh T eh neper is o tf en used to e rpx e ss rat oi s of vo tl age a dn current amp il t du es in le e tc r ci al c ri c tiu s (or rp ess ru e ni ca oust ci s), hw er ae s ht e dec bi el is used ot express

wop er ar t oi .s O en k dni of rat oi may be co vn e detr otni the other. Cons di e gnir that wave wop er is orp p ro tional ot the s uq are of ht e amp il t du e, we have and


T eh de bic el a dn the neper have a dexif ratio ot cae h hto er. The (vo tl age) level si

ekiL the d ce bi el, ht e ne rep is a mid ensio ln e ss tinu . The UTI r ce o zing se ob th tinu s.

2. Dec bi el T eh dec bi el (dB) is a lo ag rit imh c inu t of m ae s ru eme tn that e rpx esses t eh ma ing t du e of a p yh sical quantity (usua yll p wo er or tni ensity) relat vi e to a spec ifi ed

ro mi p il ed referen ec level. S ni ec ti expresses a ra oit of owt quantities with t eh sa em tinu , ti is a d mi ensio ln ess inu t. A ed c bi el is one tenth of a bel, a sel od m-used

inu t . T eh dec bi el is w di e yl known as a m ae s ru e of so dnu pr se s ru e level, b tu is also used

rof a w di e variety of other m ae s ru eme tn s ni science a dn e enign er ni g (partic lu arly ac uo stics, el ce tro in cs, a dn c rtno ol theory) a dn other discip nil es. tI c fno ers a nu bm er of adva tn a eg s, such as the ab ili ty ot co vn e in ent yl rper ese tn very al rge ro sma ll bmun er ,s a logarith cim sca gnil that ro ylhgu cor er spo dn s ot the human pe cr eption of so dnu a dn thgil , a dn the ab ili ty ot ac rr y o tu lum tip cil ation of rat soi

yb s mi ple addition a dn s rtbu ca tion. T eh d ce bi el s my bol is o tf en qua deifil iw th a s xiffu , w ih ch ni di ac et s ihw ch re ref en ec quantity or rf equency we hgi t ni g nuf ction h sa eb en us de . For example,

EGELLOC GNIREENIGNE DACS "dBm" ni di ac et s t tah the referen ec qua itn ty is o en willim a tt , w elih "dB "u si re ref en ec d ot 0. 577 vo tl s RMS. 1[ ] T eh de inif tions of the dec bi el a dn bel use base- 01 logari ht ms. For a s limi ar tinu us gni an t ru al lo smhtirag ot esab ,e ees .repen

snoitinifeD A ed c bi el is one-tenth of a bel, i.e. 1 B=10 Bd . The bel ( )B is the logarithm of t eh rat oi of owt wop er quantities of 10 1: , a dn for two if e dl quantities ni the ar t oi

. A if e dl quantity is a uq antity such as tlov age, curre tn , s dnuo press ru e, le e tc r ci if e dl s rt e gn th, ve ol city a dn charge ned sity, the square of w ih ch ni nil ear

systems is rp o rop tional ot p wo er. A power quantity is a wop er or a quantity erid ct yl p opor rtional ot p wo er, e.g. ener yg density, ca oustic tni ensity a dn mul ino su

tni ensity. T eh calc alu tion of the rat oi ni dec bi els varies ped end ni g on w eh ther the quantity be gni m ae s ru ed is a wop er quantity or a if e dl quantit .y

woP er quantities When referr gni to em as ru eme tn s of wop er ro tni ensity, a rat oi can be e rpx e ss ed in d ce bi els by eva ul at gni ten t mi es the ba es -10 logarit mh of the ratio of the m ae s ru ed

uq antity ot the referen ec level. T uh ,s fi L re rp ese tn s the rat oi of a wop er va ul e P1 ot ona ther power va ul e P0, then L Bd rep er sents ht at ratio e rpx e ss ed ni dec bi els a dn si

calc lu ated us gni the for lum a:

P1 and P0 um st have the same dimension, i e. . ht ey um st m ae s ru e t eh same t epy of

uq antity, and the same tinu s be rof e calc gnitalu the ratio: however, the choi ec of s ac le rof t ih s co mm on tinu is rri eleva tn , as ti cha egn s ob th quantities by the same f ca t ro , a dn t uh s cancels ni the ratio—the r oita of owt quantities is sc la e- vni aria tn .

toN e that fi P1 = P0 ni the abo ev equation, then L Bd = 0. If P1 is rg ae ter than P0 then L Bd is op sit vi e; fi P1 is le ss ht an P0 then LdB is negat evi .

aeR rr a gnign t eh above equation vig es the fo oll w gni of rmu al rof P1 ni terms of P0

[ ]8

EGELLOC GNIREENIGNE DACS dna LdB:

.

S ni ec a bel is equal to ten dec bi els, the co rr es nop d gni formul ea rof m ae s ru eme tn in bels (LB) ra e

.

iF e dl uq a itn t sei When refe rr gni ot em as ru eme tn s of if e dl amp il t du e ti is usual ot cons di er the rat oi of t eh s uq ares of A1 (meas ru ed amp il t du e) a dn A0 r( eference amp il t du e). T ih s si b ace use ni om st pa p il ac tions wop er is prop ro tional to the square of amp il t du e, a dn it is des ri able of r the t ow dec bi el of rmula it ons to vig e the sa em resu tl ni such yt pical ac se .s uhT s the fo oll w gni de inif tion is u es d:

T ih s of r alum is somet mi se ca dell the 02 log r lu e, a dn si lim ar yl the of r alum for rat oi s of wop ers is the 01 log r lu e, a dn s limi ar yl rof other f ca tors. c[ itati no edeen d] ehT

viuqe alence of and si of the sta dn a dr porp ert ei s of logarithms. T eh of r alum may eb re rra a gn ed ot evig

S limi ar yl , ni el ce trical c ri c tiu s, di ss pi a et d wop er is typ ci a yll p por ortional ot t eh s rauq e of vo tl age ro c rru e tn when the mi pedance is he dl constant. Tak gni vo tl a eg as an example, t ih s l ae ds ot ht e equation:


hw ere V1 is t eh vo tl age be gni meas ru ed, V0 is a spec ifi ed referen ec vo tl age, a dn G Bd is ht e wop er ga ni exp er ss de ni dec bi els. A s mi ilar of r alum ho dl s for c rru e tn .

nA examp el s ac le s oh w gni x a dn 10 log x. tI is ae sier ot arg sp a dn compare 2 ro 3 tigid bmun ers ht an ot co pm are pu ot 01 id g ti s.

toN e that a ll of these examples iy eld d mi ensio eln ss answers ni dB b ce ause they are

relat evi rat oi s e erpx ss de in deci eb ls.

• oT calc lu a et the ar t oi of 1 kW (one k li owa tt , ro 01 00 wa tt s) ot 1 W ni d ce bi els, use the of r lum a


• oT calc lu a et the ar t oi of ot ni dec bi el ,s use eht of r lum a

oN ti ec that , ulli s rt at gni the consequen ec rf om t eh de inif tions above that G Bd h sa t eh same va ul e, , regardle ss of w eh t reh ti si

tbo a ni ed with the 10-log ro 20-log r lu es; pr dedivo t tah ni t eh sp ce cifi system being noc s di ered wop er rat oi s are equal ot amp il t du e rat oi s s uq a .der

• oT calc lu a et the ar t oi of 1 mW (one im lliwa )tt ot 01 W ni ed c bi el ,s use t eh

of r alum

• oT nif d the p wo er ratio c rro espond gni to a 3 dB cha egn ni level, use t eh of r lum a

A cha gn e ni wop er rat oi by a f ca tor of 10 is a 01 dB cha gn e. A cha gn e ni wop er rat oi by a f ca t ro of owt is a rpp oximate yl a 3 Bd cha gn e. M ero rp ce ise yl , t eh f ca tor is 103/ 01 , ro 1. 3599 , a tuob 0. %42 d ffi ere tn rf om exact yl 2. Si lim ar yl , an ni c er ase of 3 Bd imp il es an ni c er ase ni vo tl age by a fac rot of ap rp oximate yl , ro ba o tu 1. ,14 na rcni ae se of 6 dB c erro spo dn s ot a pp roximate yl fo ru times the wop er a dn twice ht e vo tl age, a dn so on. In ex ca t et rms the wop er rat oi is 106/10, ro a tuob 3. 1189 , a

relat evi e rr or of abo tu 0. %5 .

Me stir T eh use of the dec bi el has a bmun er of mer ti s:

• The dec bi el's logari ht im c nat ru e means that a very large ra egn of rat oi s can be re rp ese detn by a co vn e in e tn nu rebm , ni a s limi ar ma nn er to scient cifi no at tion. T ih s a woll s one ot c el ar yl iv sua il ze eguh cha gn es of some quantit .y S( ee doB e P tol a dn ha fl logarit mh rg aph.)


• The mathemat ci al porp erties of logari ht ms mean that the overa ll dec bi el gain

of a lum ti-component system (such as cons ce ut evi amp reifil s) can eb ac l uc l deta s mi p yl by s nimmu g t eh dec bi el ga ni s of t eh ividni dual

co pm one tn s, rather than need ni g ot lum tiply amp il f ci ation af ct ro s. E ss entia yll t ih s is b ce ause log(A × B × C × ...) = log(A) + log(B) + lo (g C) + .. .

• The muh an rep ec ption of, rof examp el , s dnuo or thgil , i ,s ro ylhgu sp ae k ,gni such t tah a doub gnil of ca tual tni ensity cau es s perce vi ed tni ensity ot a wl a sy ni cr ae se yb the same amo tnu , rri esp ce t vi e of the or nigi al level. The dec bi e s'l

logari ht cim scale, ni w ih ch a doub gnil of wop er ro tni ensity a wl ays ac uses an ni c er ase of a pp roximate yl 3 dB, c rro es dnop s ot t ih s rep ec ptio .n

bA so tul e a dn relat evi dec bi el em as ru ements

lA tho hgu dec bi el em as ru eme tn s are wla ays relat evi ot a referen ec level, fi t eh

numerical va ul e of t tah referen ec is exp il cit yl a dn ex ca t yl s tat ed, then the ed c bi el m ae s ru eme tn is ac dell an "abso tul e" m ae s ru eme tn , ni the sense ht at the ex ca t va eul of the m ae s ru ed quantity can be r ce overed us gni the of r alum vig en ear il er. roF xe amp el , s ni ec dBm ni dic eta s wop er meas ru eme tn relat vi e ot 1 im l wil a tt ,

• 0 dBm means no cha gn e rf om 1 mW. T uh ,s 0 dBm is the wop er level

c rro se pond gni to a wop er of e ax ct yl 1 mW. • 3 dBm means 3 dB rg ae ret than 0 dBm. T uh ,s 3 dBm is the power level

c rro se pond gni to 103/10 × 1 Wm , ro a pp ro mix ate yl 2 mW. • -6 mBd em a sn 6 dB ssel ht an 0 dBm. ,suhT -6 dBm si t eh p rewo leve l

c rro se pond ni g to 10-6/ 01 × 1 mW, ro a rpp oximate yl 052 Wμ 0( .25 mW). If the mun erical va ul e of the referen ec is ton exp il cit yl s tat ed, as ni the dB ga ni of na amp reifil , then the d ce bi el em as ru e tnem is p ru e yl relat evi . The rp ca tice of tta ca gnih a s xiffu to the basic dB tinu , of r gnim co pm o dnu tinu s such as dBm, dB ,u

dBA, cte , is ton per im t det by IS . ]01[ oH wever, tuo s di e of doc mu e tn s adher gni ot SI tinu ,s ht e rp act ci e is very co mm on as ulli stra det yb the fo oll w gni examples.

bA so tul e m ae s ru eme stn

El ce tric wop er dBm or dB Wm

Bd (1 mW) — wop er m ae s ru eme tn re al t evi ot 1 willim a tt . X mBd = X WBd + .03


WBd

Bd (1 )W — s limi ar ot dBm, ex ec tp the referen ec level is 1 wa tt . 0 WBd = 03+ ;mBd - 03 d WB = 0 ;mBd XdBW X = d mB - .03

tloV a eg

S ni ec the dec bi el is de denif with resp ce t to wop er, on t amp il t du e, co vn ersions of vo tl age rat oi s ot d ce bi els um st sq rau e the amp il t du e, as id scu ss ed a vob e. A schematic s oh w gni the relations pih betwe ne dBu (the vo tl age so ru ce) a dn dBm (t eh wop er diss pi a det as eh ta by the 006 Ω resisto )r

VBd

Bd (1 VRMS) — vo tl age relat evi ot 1 vo tl , re ag rd el ss fo pmi edan ec . ]1[

dBu ro dBv

Bd (0. 577 VRMS) — vo tl age relat evi to 0. 577 vo tl s. 1[ ] Or nigi a yll dBv, it was cha gn ed ot dBu to avoid co ufn sion with dBV. ]11[ The "v" co em s rf om "volt ,"

elihw " "u comes rf om " olnu aded" . dBu can be used re rag dless of mi peda cn e, tub is der devi rf om a 06 0 Ω daol dissipa gnit 0 dBm (1 mW). Reference

vo tl a eg dB Vm

Bd (1 mVRMS) — vo tl age relat vi e ot 1 mi ll ivo tl ac or ss 57 Ω[ 21 ]. W di e yl used ni ac b el tele iv sion ne wt o kr ,s rehw e the no nim al s rt e gn th of a s lgni e TV

s ngi al ta the r ce e vi er et r nim als is a tuob 0 dB Vm . aC ble VT uses 57 Ω ixaoc al ac b el , so 0 dB Vm erroc spo dn s ot - 87 . 57 WBd (- 84 . 57 dB )m ro ~13

nW.

EGELLOC GNIREENIGNE DACS μBd V or dB Vu

Bd (1 μV SMR ) — vo tl age relat evi ot 1 im crovo tl . W di e yl used ni tele iv sion dna irea al amp ifil er acificeps tio sn . 06 Bd μV = 0 dB .Vm

3. eporP tr ies of Sy mm etr ci al Ne wt o kr s a dn Cha ar c et ristic pmi edan ec of S emmy trical Ne wt o skr

A two-po tr ne wt ork a( k dni of four- et r nim al netw ro k or quadr pi ole) is an el ce tr ci al

ric c iu t or de iv ec with wt o pa ri s of et r im nals co nn ec det together tni erna yll by an le e tc r ci al ne wt or .k T ow et r nim als constit etu a op rt fi they satis yf the e ss ential

req riu eme tn k on wn as the po tr condition: the same curre tn um st e tn er and el ave a rop t. Examples ni c dul e sma ll -s ngi al dom els of r transist ro s (such as the yh brid-pi

mode )l , retlif s a dn matc gnih ne krowt s. The na a yl sis of pa ss vi e wt o-p tro ne skrowt is an uo t rg owth of recip or city theorems rif st de devir by roL e tn z ]3[ . A two-po tr wten o kr makes possible the iso ital on of either a comp etel c ri c tiu or part of it a dn er p al c gni ti by ti s c rah ca et ristic parame ret .s On ec t ih s is done, the isola det pa tr of the c ri c tiu b ce omes a "bl ca k box" with a set of dist ni ct vi e porp erties,

ane b gnil us ot abs rt ca t away ti s sp ce ifi c p yh s ci al b liu d pu , t uh s simp iyfil ng sylana is. ynA enil ar circ tiu with fo ru et rminals can be rt ans of r dem otni a wt o-p tro

netw kro pro dediv that ti does ton co tn a ni na pedni e dn ent so cru e a dn satis eif s t eh trop conditions.

T reh e are a nu bm er of a tl ernat vi e sets of parame ret s that can be us de ot descr bi e a

nil ear wt o-p tro ne wt o kr , ht e usual sets ra e res ep c vit e yl ca dell ,z y, h, g, a dn BA CD parame ret ,s cae h descr bi ed ni d ivi dua yll below. These a er lla li tim ed ot nil ae r netw kro s s ni ec an dnu er gniyl ass pmu tion of the ri der vi ation is t ah t a yn vig en

ric c tiu condition is a nil ear s pu e opr sition of various s roh t-c ri c tiu a dn po en c ri c tiu noc ditions. They are usua yll e rpx e ss ed ni matr xi aton tion, a dn they es at b il sh

relations be ewt en t eh var ai bles

Inp tu vo tl a eg uO tp tu vo tl a eg

Inp tu curre tn uO pt tu c rru e tn

hT ese c rru ent and vo tl age var ai bles a er om st use luf ta ol w-to- dom era et

rf equencies. tA hgih rf equencies (e.g ,. mic wor ave erf quenc ei s), t eh use of wop er dna ener yg var ai bles si erom ap orp pria ,et a dn t eh wt o- op rt current–vo tl a eg

EGELLOC GNIREENIGNE DACS orppa ach is re lp aced by an a aorpp ch bas de pu on s ac tt er gni para tem ers.

T eh et rms four- et r nim al ne wt ork a dn uq adr pi ole ( ton ot eb c ufno sed with

uq adr pu ole) a er also used, eht la tt er partic lu ar yl ni rom e mathematical tr ae tme stn la tho hgu the et rm is b ce o gnim archaic. oH wever, a pa ri of et r nim als can be ca ll ed

a trop ylno fi the curre tn e etn r gni one et r nim al is e uq al ot the curre tn lea gniv t eh ot ;reh t ih s de inif tion is ca ll ed the p ro t condition. A four- et r nim al ne owt rk can only be porp er yl ca ll ed a two-po tr when the ter nim als a er co nn ce et d ot t eh e etx rnal

ric c tiu ry ni two riap s ob th m ee t gni the op rt c no diti no .

4. vo tl age a dn c rru e tn rat soi In dro er ot s mi p il fy calc alu tion ,s s uni s dio al vo tl age a dn curre tn wa ev s era co mm o yln re rp ese detn as complex-va ul ed nuf ctions of t mi e de deton as dna . ]8[]7[ I pm edan ec is de denif as eht rat oi of t eh se quantities.

S bu stitut gni t eh se otni O 'mh s law we ha ev

oN t gni that t ih s um st ho dl for a ll t, we yam equa et the ma ing t du es a dn phases ot tbo a ni

T eh ma ing tude equation is the fa ilim ar Oh 'm s law pa p deil ot the vo tl age a dn uc rre tn amp il t du e ,s lihw e the seco dn equat noi de enif s ht e phase re al tions ih p.

Va il dity of complex re rp ese atn tion T ih s erper se atn tion us gni complex e px one itn als may eb just deifi by on t gni that (by

EGELLOC GNIREENIGNE DACS E elu r's of r lum )a : i.e. a real-va deul s uni so di al nuf ction (w ih ch may re rp se e tn o ru vo tl age ro c rru e tn wave of rm) may be rb oken otni owt comp el x-va ul ed nuf ction .s By the pr ni ciple of s repu position, we may ana yl se ht e beha iv o ru of the s uni so di on the left-ha dn s edi

yb ana yl s gni the beha iv o ru of the owt co lpm ex et rms no the r hgi t-ha dn s di e. G vi en ht e s emmy rt y, we ylno n ee d ot per of rm the ana yl sis for one r hgi t-ha dn et r ;m t eh

res tlu s lliw eb di entical rof t eh other. tA the e dn of a yn calc lu ation, we may er turn ot r lae -va ul ed s uni so di s by uf rther on t gni that

In ot reh drow ,s we s mi p yl take ht e real rap t fo the res .tlu Phas sro A p ah s ro is a consta tn complex nu bm er, usua yll e rpx e ss ed ni e px onential for ,m re rp esent gni the complex amp il t du e (ma duting e a dn phase) of a s uni so di al nuf ction of it me. Phas ro s a er used by e el ctr ci al e enign ers to s mi p yfil comp atu tio sn

vni o gnivl s uni so di ,s where they can o tf en re ud ec a d ffi erential equation orp blem ot na a rbegl a eno ci .

T eh i pm eda cn e of a ric c tiu eleme tn ac n be nifed ed as ht e ar t oi of the p ah sor vo tl age ac or ss eht eleme tn ot the phasor curre tn t rh o hgu the eleme tn , as de et r denim

yb the relat vi e amp il t du es dna phases of the vo tl age a dn curre tn . T ih s is di entical ot ht e de inif tion rf om O mh 's law vig en above, r ce o ing s gni that the af ctors of

cancel

5. porP agation consta tn T eh p por agation constant of an elec rt oma ng et ci wave is a meas ru e of t eh cha egn

dnu ergone by t eh amp il t du e of the wave as ti p por aga et s ni a vig en d ri ce tion. T eh


uq antity be gni m ae s ru ed ac n be the vo tl age ro curre tn ni a ric c tiu or a if e dl ve tc or

such as electric if e dl s rt e gn th or xulf density. The p por a ag tion constant ti self m ae s eru s cha gn e rep me ert b tu is otherwise mid ensio eln ss. T eh porp agation consta tn is expre ss ed logarit imh ca yll , a oml st vinu ersa yll ot t eh base e, ra ht er than t eh rom e usual base 01 used ni te el co umm ni ac tions ni to her situations. The uq antity em as ru ed, such as vo tl age, is e rpx e ss ed as a s uni s doi al

hp asor. The phase of t eh s uni so di varies iw th distance w ih ch er s tlu s ni t eh porp agation consta tn be gni a complex nu ebm r, ht e mi a nig ary pa tr be gni caused by

ht e phase cha gn e.

etlA rnat vi e names T eh et rm pr po agation consta tn is so em w tah of a im snomer as ti usua yll varies s rt o ylgn with ω. tI is rp obab yl the om st diw e yl used et rm b tu there are a al r eg var ei ty of a tl ernat vi e names used by various aut roh s for t ih s quantity. These ni c edul , rt ans im ssion parame et ,r rt ans im ssion cnuf tion, pr po agation parame et r,

porp agation oc e iff cient a dn trans im ssion c no stan .t In p rul al, ti is usua yll imp eil d ht at α a dn β are be gni re ref enc de sepa ar te yl b tu co ll ce t vi e yl as ni rt ans im ssion

parame ret s, porp agation parame ret s, porp a oitag n coe iff cie tn s, rt ans im ssion noc sta tn s a dn s ce o adn ry coe iff cie tn s. T ih s last occ ru s ni rt ans im ssion nil e theory,

ht e et rm s ce o dn ary be gni used to contrast ot the primary nil e coe iff cie tn s. The pr mi ary oc e iff cie tn s be gni the p yh sical pr repo ties of the nil e; ,R C,L a dn G, rf om

ihw ch the sec dno ary coe iff cie tn s may be der devi us gni the tele rg apher's equation. toN e tha ,t at l ae st ni t eh if e dl of trans im ssion nil es, eht et rm trans im ssion

oc e iff cient has a d reffi e tn mea gnin ed spite the s limi arity of name. reH e ti is t eh roc o ll ary of re lf ce tion coe iff cien .t

inifeD tion

T eh porp agation consta tn , s my ob l γ, for a vig en system is de nif ed by the rat oi of ht e amp il t du e at the so ru ec of t eh wa ev to ht e amp il t du e at some distance x, such ht at,

S ni ec t eh porp a ag tion consta tn is a complex quantity we ac n wr ti e;


hw e er

α, ht e er al trap , is ca ll ed the a tt e un ation cons tnat

,β ht e mi a anig ry trap , is ac ll ed the phase c no sta tn T tah β od se dni ee d erper se tn phase can be s ee n from E lu er's for lum a;

ihw ch is a s uni soid w ih ch varies ni p ah se as θ av ries ub t od es ton av ry ni amp il t du e b ace use;

T eh r ae son of r the use of base e is also on w made clear. The ima nig ary phase

noc sta tn , iβ, ac n be a dd ed rid ect yl ot the a tt e un ation consta tn , α, ot form a s lgni e complex nu bm er that ac n be handled ni one mathematical po eration rp o dediv they ra e ot the same ab se. lgnA es m ae s deru ni ra id ans req riu e ab se e, so the a tt enuation

is kil ewise ni ab se e.

roF a c ppo er rt ans im ssion nil e, ht e pr po aga it on c no sta tn ac n eb calc lu a det rf om t eh pr mi ary enil coe iff cie tn s yb means of the relations pih ;

hw ere;

, ht e se ir es pmi edan ec of the nil e rep metre ,dna

, ht e s tnuh ad ttim ance of the enil per me rt e. Atte un ation consta tn In teleco cinumm ations, the et rm a tt enua it on constant, also ca ll ed tta e un ation parame ret ro coe iff cient, is the a tt e un ation of an electroma ng et ci wave p por agating

rht o hgu a medium per tinu distance rf om the so ru ec . tI is the er al pa tr of th e porp a itag on consta tn a dn is m ae s ru ed ni n srepe per metre. A ne rep si rppa oximate yl 8.7 Bd . ttA e un ation consta tn can be de nif ed by t eh amp il t du e rat ;oi


T eh porp agation consta tn per tinu le gn th is denifed as the nat ru al lo ag rithm ci of rat oi of the send gni e dn c rru e tn or vo tl a eg to ht e r ce e gnivi e dn curre tn ro vo tl age. C ppo er nil es T eh a tt e un ation consta tn rof reppoc (or a yn other conduct )ro nil es can eb calc lu a det rf om t eh primary nil e coe iff cie tn s as s oh wn ba ove. roF a nil e meeting ht e distortio ln e ss condition, iw th a conduc nat ec G ni the ni s lu at ro , eht a tt e un ation

noc sta tn is vig en b ;y

woh ever, a real nil e is kilnu e yl to m ee t iht s condition witho tu the addition of ol ad gni co sli a dn , ruf thermore, ht e er a er so em dec di ed yl non- nil ae r e ff e stc po erat gni on the pr mi ary "consta tn s" w ih ch ac use a rf equency depe dn en ec of t eh ol ss . There a er owt ma ni co pm one tn s to ht ese ol ss e ,s ht e metal ol ss a dn t eh id el ce tr ci ol ss.

T eh ol ss of om st rt ans im ssion enil s ra e do nim a det by the metal ol ss , w ih ch ac uses a rf equency depe dn ency due ot inif te co udn ct ivi ty of tem als, a dn the sk ni e eff ct ni s di e a conduc rot . The sk ni e ff ce t causes R alo gn the conductor ot eb

rppa oximate yl depe dn e tn on rf equency ac roc d gni ot ;

oL ss es ni the diele tc r ci ped end on the ol ss gnat ent t( anδ) of the material, w ih ch depe dn s revni se yl on the wavele gn th of the s ngi al a dn is d ri ce t yl orp p ro tional to

eht ycneuqerf .

pO tical erbif


T eh a tt e un ation consta tn rof a rap tic lu ar pr po a ag tion dom e ni an o tp ical bif er, eht r ae l part of the a ix al p por agation cons nat t. Phase c no sta tn In el ce rt oma ng etic theory, the phase constan ,t also ca ll ed phase cha gn e constant, parame ret or coe iff cie tn is the ima nig ary co pm one tn of the p por agation consta tn

rof a plane wave. tI erper se tn s the cha egn ni ph esa per metre alo gn the path rt ave dell by t eh wave ta a yn ni sta tn a dn is e uq al ot the a lugn ar wavenu bm er of t eh

wave. tI is erper se detn by the s bmy ol β a dn is m ae s ru ed ni tinu s of radians per metre.

rF om the de inif tion of a ralugn wavenu bm er; T ih s quantity is o tf en (strict yl sp ae k gni rrocni ce t )yl abbre iv a et d ot wave bmun e .r

porP er yl , wavenu bm er is vig en b ,y

ihw ch d ffi ers rf om a lugn ar wavenu bm er o yln by a consta tn lum tip el of 2π, ni t eh sa em way that a ralugn rf equency d ffi ers rf om rf equenc .y

roF a rt ans im ssion nil ,e the Hea iv s di e co idn tion of the tele rg apher's equation te sll us that the wave bmun er um st be p por ortional ot rf equency for the rt ans im ssion of ht e wave ot be nu dis ot r det ni the t mi e do niam . T ih s ni c edul ,s ub t is ton li tim ed ,ot ht e di eal case of a ol ss el ss nil e. The reas no of r t ih s condition can be seen by

noc s di er gni t tah a usef lu s ngi al is co opm sed of ma yn differe tn wavele gn ths ni t eh rf equency doma ni . For there to be no dist ro tion of the wave of rm, a ll these waves um st rt avel at the sa em ve ol city so that they arr vi e at the far e dn of the nil e ta t eh

sa em t mi e as a rg oup S . ni ec wa ev p ah se ve icol ty is vig en b ;y it is rp oved t tah β is req deriu ot be pr ropo tional ot ω. In et rms of primary oc e iff cie tn s of the nil e, t ih s iy e sdl from t eh tele rg apher s' equation rof a id stortio eln ss nil e the conditio ;n


woH ever, rp ca tical nil es can o yln be e px ce t de ot rppa oximate yl m ee t t ih s condition vo er a detimil erf quency ba dn .

6. sretliF

T eh et rm rp opagation consta tn ro p por aga it on nuf ction is pa p il ed ot retlif s a dn ot reh two-po tr netw kro s used rof s ngi al p or ce ss gni . In these ac se ,s woh ever, t eh

tta e un ation a dn phase oc e iff cie tn s are e px re ss de ni et rms of nepers a dn radians per netw kro section rather than per me rt e. Some authors make a id st ni ction wteb een per me rt e m ae s ru es (for w ih ch "constant" is u es d) a dn per sec it on m ae s ru es ( rof

ihw ch " cnuf tio "n is use .)d T eh p por agation consta tn is a use luf conc tpe ni tlif er des ngi w ih ch avni riab yl uses a ac scaded s ce it on pot olo yg . In a ac s ac ded pot olo yg , the porp a ag tion consta ,tn

tta e un ation consta tn a dn phase consta tn of ni d ivi dual se itc ons may eb simp yl dda ed ot if dn the tot al porp agation constant e .ct

csaC da ed ne skrowt

T rh ee ne wt o kr s iw th arbitrary rp opagation c no sta tn s a dn pmi edan ec s co nn ce det ni cas ac de. ehT Zi et rms re rp ese tn mi a eg pmi edance a dn ti si a ss mu ed t tah

nnoc ce tions a er be wt ee n matc gnih image pmi edances. T eh rat oi of ou pt tu ot ni p tu vo tl age rof ea hc ne wt o kr is vig en b ,y

EGELLOC GNIREENIGNE DACS T eh et rms ra e i pm eda cn e s ac gnil terms ]3[ dna the ri use is expla deni ni t eh image i pm edance article. T eh overa ll vo tl age rat oi is vig en b ,y

T uh s for n ac s ac ded sec it ons a ll ha gniv matc gnih i pm edances fac gni ae ch othe ,r ht e overa ll porp a ag tion consta tn is vig en b ,y

7. retliF dnuf ame tn als – Pa ss a dn tS op ba dn s.

retlif s of a ll t py es are re riuq ed ni a variety of app il cations from aud oi ot RF a dn ac or ss ht e hw o el sp ce rt um of rf equenc ei .s As such RF retlif s form an i opm rta tn le eme tn wit nih a variety of scenar oi ,s enab gnil the req riu ed rf equenci se to eb

pa ss ed rht o hgu the c ri c tiu , elihw re ej ct gni t oh se that ra e ton n ee ded. T eh ideal tlif er, hw ether ti is a wol pa ,ss hgih pa ,ss ro dnab pa ss tlif er w lli e ihx b ti

on ol ss wit nih the pa ss ba dn , i.e. ht e rf equencies be wol the c tu o ff rf equency. Then voba e t ih s rf equency ni what is et rmed ht e s pot ba dn the retlif w lli ejer ct all

s ngi als. In rea il ty ti is ton op ssib el ot ac ih eve the perf ce t pa ss retlif a dn there is a wl a sy some ol ss wit nih the pa ss ba dn , dna ti is not op ss ible ot ac ih eve etinifni re ej ction in ht e s pot ba dn . lA so there is a rt ansition be ewt en the pa ss dnab a dn the s pot ba ,dn


hw ere eht response cur ev fa ll s wa ay, with the level of re ej ction ris se as eht

rf equency mo ev s from the pa ss ba dn ot the s pot ba .dn Basic t py es fo RF retlif There are four t py es of retlif that can be de nif ed. aE ch d ffi erent t py e re ej cts ro a ecc tp s s ngi als ni a d ffi ere tn way, a dn by us gni t eh co err ct t py e of RF retlif ti si

op ss ible ot ecca tp ht e req riu ed s ngi als a dn re ej ct oht se t tah a er not wa detn . T eh fo ru basic t py es fo RF retlif are:

• woL pa ss retlif • hgiH pa ss tlif er • Ba dn ap ss lif ret • Ba dn re ej ct retlif

As the names of these t py es of RF tlif er ni dica et , a wol ap ss tlif er o yln a swoll rf equencies below what is et rmed the c tu o ff rf equency t rh o hgu . T ih s can also eb ht o thgu of as a hgih ejer ct tlif er as ti re ej cts hgih erf quencies. S limi ar yl a hgih pass

retlif o yln a woll s s ngi als t rh o hgu above ht e c tu o ff rf equency a dn re ej cts oht se below ht e c tu o ff rf equency. A ba dn pa ss tlif er a woll s rf equencies t rh o hgu wit nih a

vig en pa ss ba dn . niF a yll the ba dn re ej ct retlif re ej cts s ngi als wit nih a ec tr a ni ba .dn tI can be ap rtic lu ar yl use luf rof re ej ct gni a rap tic alu r wnu a etn d s ngi al ro set of

s ngi als fa gnill wit nih a vig en ba dn w di t .h



retlif erf quencies

A retlif a woll s s ngi als thro hgu ni what is ter dem the ap ss ba dn . T ih s is the ba dn of rf equencies be wol the c tu o ff rf equency of r ht e tlif e .r

T eh c tu off rf equency of the retlif is de nif ed as ht e tniop at ihw ch the o tu p tu level rf om the tlif er fa ll s ot 5 %0 (-3 d )B of the ni band level, ass imu ng a constant ni put

level. The cut o ff rf equency is so em t mi es referred to as the ha fl wop er or -3 dB rf equency.

The s pot ba dn of the tlif er is e ss entia yll t eh dnab of rf equenci se that is re ej c det by ht e retlif . tI is kat en as s rat t gni at the po tni where the tlif er r ae c eh s ti s req riu ed

level of re ej ctio .n

retliF class ifi ac tio sn

retliF s can be des dengi ot m ee t a av riety of re riuq eme tn s. lA tho hgu us gni t eh sa em bas ci c ri c tiu co rugifn ations, t eh c ri c tiu va eul s ffid er when the c ri c tiu is des ngi ed to m ee t d reffi ent cr ti er ai . In ba dn r pi ple, fas et st rt ansition ot the lu tima et ro ll o ,ff

ehgih st o tu of ba dn ejer ction ra e some of the cr eti ria that res tlu ni d ffi ere tn c ri c tiu va ul es. These d ffi ere tn retlif s ra e vig en na em ,s cae h one be gni po it im sed for a

tnereffid le eme tn fo ofrep rma ecn . T erh e c mmo on t epy s of retlif are vig en be ol w:

• B ttu e owr tr :h T ih s t py e of retlif pro div es the ma mix um ni ba dn lf atness. • Be ss e :l T sih retlif pro ediv s eht opt mi um in-ba dn phase er sponse dna

ht erefore also rp o div es eht best s pet res nop se. • Chebychev: T ih s etlif r pro div es fast ro ll o ff a etf r the cut o ff rf equency si

r cae hed. woH ever t ih s is at the e px ense of ni ba dn r pi ple. The rom e ni ba dn r pi ple that can be lot era det , ht e af s ret the ro ll o ff .

• E ill pt ci a :l T ih s has s ingi fica tn levels of ni ba dn a dn o tu of ba dn ri pp le, dna as e px ce det the rehgih the de rg ee of r pi ple ht at can be tolera det , the s et eper ti r ae ches ti s lu t mi a et ro ll o ff .

mmuS ary

RF tlif ers are w di e yl used ni RF des ngi a dn ni a ll ma nn er of RF a dn analo eug

ric c tiu s ni general. As they a ll ow tho hgu ylno rap tic lu ar rf equencies ro ba dn s of rf equencie ,s ht ey are an e ss ential ot ol rof t eh RF d se ngi e enign e .r


8. C no sta tn k retlif C no sta tn k tlif ers, also k-t py e retlif s, are a pyt e of el ce rt o cin retlif ed s ngi ed using ht e mi age em t doh . They are the or nigi al a dn s mi plest retlif s p or duced by t sih

met doh olo yg a dn consist of a la dd er ne rowt k of di ent ci al sec it ons of ap ss evi co pm one tn s. iH s ot rica yll , they a er t eh if rst retlif s that co dlu app aor ch the di eal

etlif r erf quency response to wit nih a yn prescr bi ed li tim with the addition of a s iffu cie tn nu bm er of s ce tion .s woH e rev , they are rare yl cons redi ed rof a dom ern des ngi , the pr ni ciples be dnih them ha gniv b ee n s pu erse ded by other met doh olo ig es

ihw ch are more a cc ru a et ni the ri rp ediction of tlif er response. Ter nim olo yg Some of the pmi eda cn e et rms a dn s ce it on et rms used ni t ih s article are pict ru ed in ht e dia rg am be wol . Image theory de nif es quantities ni et rms of an etinifni ac s ac ed

of owt - trop s ce tions, a dn ni t eh case of ht e retlif s be gni discu ss ed, na etinifni la dd er ne rowt k of L-section .s Here " "L sho dlu not be c ufno s de with the ni ductance L – ni el ce tro in c tlif er pot olo yg , " "L re ref s ot ht e s ep c cifi tlif er s pah e w ih ch resemb el s vni er det le tt er " "L .

T eh se itc ons of the opyh thet ci al ifni n ti e tlif er are made of ser ei s eleme tn s ha iv ng i pm edan ec 2Z a dn s tnuh eleme tn s with ad ttim an ec 2Y. The f ca rot of two si

ortni duced for mathematical co vn e in ence, s ni ec ti is usual ot ow rk ni et rms of half-


s ce it ons where it disa pp ae r .s The mi age i depm a cn e of the ni p tu a dn o tu p tu po tr of a s ce it on w lli genera yll not be the same. woH ever, rof a dim -ser ei s sec it no (that is, a s ce it on rf om ha wfl ay t rh o hgu a series eleme tn to ha wfl ay t rh o hgu the next es ries le eme )tn w lli have the same ima eg i pm edan ec on ob th po tr s due to s mmy etry. T sih

image depmi ance is des ngi a det ZiT due ot the " "T pot olo yg of a im d-series s ce it o .n ekiL wise, ht e mi age i pm edan ec of a dim -s tnuh s ce it on is des ngi a det Z Πi d eu ot t eh

"Π" topolo yg . Ha fl of such a "T" or "Π" se itc on is ca ll ed a ha fl -s ce tion, w ih ch si la so an L-s ce tion b tu with ha fl the element va ul es of the lluf L-s ce tion. The ima eg

i pm edan ec of the half-section is di ss i lim ar on the ni p tu a dn o tu p tu po tr s: on t eh s di e p er sent gni the series eleme tn ti is equal to the im d-series ZiT, b tu on the s edi

rp esent gni the s tnuh eleme tn ti is uqe al to eht im d-s tnuh Z Πi . hT ere ra e uht s two varia tn ways of us gni a half-s ce tio .n

Der vi ation

tsnoC a tn k ol w- ssap retlif ha ces fl tion. Here ni ductan ec L is equal Ck2

C no stant k band-pass tlif er ha fl sectio .n L1 = C2k2 a dn L2 = C1k2


Ima eg pmi edance ZiT of a constant k p otor t py e ol w-pa ss retlif is p dettol vs. frequency ω. T eh i pm edan ec is p ru e yl resis vit e r( ea )l below ωc, a dn p ru e yl r cae t evi ( mi a nig ar )y above ωc . T eh b liu d gni b kcol of consta tn k retlif s is the half-s ce tion "L" ne wt o kr , co opm sed of a se ir es pmi edance Z, dna a s tnuh admi tt a cn e Y. The "k" ni "consta tn "k is eht va ul e vig en by,[6] T uh ,s k w lli have tinu s of i depm ance, that i ,s o mh s. It is er ad yli a rapp e tn that in

dro er rof k to eb consta tn , Y um st be ht e dual pmi edance of Z. A p yh sical tni e rpr e at tion of k can be vig en by bo ser gniv that k is the li im t gni va ul e of Zi as eht

si ez of the section ( ni et rms of va eul s of ti s compone tn s, such as ni ductan sec , cap ca ti ances, e ct .) a orpp aches ez or , lihw e ke gnipe k at ti s ini tial va ul e. T uh ,s k is ht e char ca teristic pmi edance, Z0, fo the trans im ssion l ni e that wo dlu be of rmed by ht ese inifni tesima yll sma ll s ce tions I . t is also the image pmi eda cn e of the s ce tion at

res no anc ,e ni the ac se of band-pa ss tlif er ,s or at ω = 0 ni the ac se of ol w-pa ss retlif s. ]7[ roF example, t eh ip ct ru ed ol w-pa ss ha fl -section ah s

.

EGELLOC GNIREENIGNE DACS leme tn s L a dn C can be made arb rti ar yli sma ll w elih reta gnini the same va eul of k. Z a dn Y however, are ob th pa p or ac gnih ez or , a dn rf om the of r alum e b( e ol w) for image i pm edan ec s,

.

Ima eg depmi ance T eh image depmi ances of t eh se itc on ra e vig en by 8[ ] and

rP o dediv that the tlif er od es ton con niat a yn resist evi eleme tn s, the mi a eg i pm edan ec ni the pa ss dnab of the tlif er is rup e yl real a dn ni the stop ba dn ti si

rup e yl mi a nig ary. roF examp el , for ht e pict ru ed ol w-pa ss ha fl -section, 9[ ] T eh rt ansition occ ru s at a uc t-o ff rf equency vig en by

Be wol t ih s erf quency, ht e mi a eg i pm edance is r ae ,l

bA ove the cut-o ff rf equency the mi age mi pedan ec is ima anig r ,y

narT s sim s noi rap ame sret


T eh rt ans ref nuf ction of a consta tn k p or tot py e ol w-pa ss tlif er for a s lgni e half- s ce it on s oh w gni a tt e un ation ni nepers a dn hp ase cha gn e ni radians.

S ee also: Ima eg pmi edanc #e Transfer nuf ct noi T eh rt ans im ssion parame ret s rof a general consta tn k half-s ce tion are vig en by ]01[

dna for a hc a ni of n ha fl -s ce tio sn

roF t eh ol w-pa ss L-shape section, be wol eht cut-o ff rf equency, the rt ans im ssion parame ret s are vig en by 8[ ] T tah i ,s the trans im ssion is ol ssle ss ni the pass- nab d with o yln the phase of t eh s ngi al cha gnign . bA o ev t eh cut-o ff rf equency, t eh rt ans im ssion parameters are: 8[ ]

torP to py e rt ansformatio sn


T eh p er se etn d plots of mi age i pm edan ec , a tt e un ation a dn phase cha gn e oc rrespo dn ot a ol w-pa ss torp ot py e etlif r section. The p otor t py e ah s a cut-o ff rf equency of ωc =

1 ra /d s a dn a nominal depmi ance k = 1 Ω. T ih s is orp du ec d by a retlif half-section iw th ni ductan ec L = 1 henry a dn ac pac ti a cn e C = 1 far da . T ih s torp to py e ac n eb

i pm edan ec s ac led a dn rf equency scaled ot the des ri ed va eul s. The ol w- ap ss otorp t py e can also be rt ansformed tni o hgih -pa ,ss band- ap ss or band-s pot t py es by ilppa cation of s atiu ble erf quency rt ans of r itam ons. 11[ ]

csaC gnida s ce it o sn

niaG pser onse, H(ω) rof a cha ni of n ol w-pa ss noc s nat t-k tlif er ha fl -s ce tions. Several L-sha ep ha fl -s ce tions may eb ac s ac ded ot of rm a co opm s eti tlif er. Li ek i pm edan ec um st a wl ays f eca kil e ni these comb ni ation .s There are t reh e rof e owt

ric c tiu s that can be of rmed with wt o di entical L-shaped ha fl -s ce tion .s rehW e a po tr of mi age pmi edan ec ZiT f eca s a on ther ZiT, ht e section is ca ll ed a Π section. Where Z Πi faces Z Πi the se itc on so of rmed is a T s ce it on. uF rther a dd itions of half-s ce tio sn ot either of t eh se s ce tion of rms a ladder netw kro w ih ch may sta tr a dn end with

se ir es ro s tnuh eleme tn s. 21[ ] tI sho dlu eb ob r en ni dnim that the charac et ristics of the retlif rp edic det by t eh

image met doh are o ln y acc taru e fi the sec it on is et r nim a det with ti s ima eg i pm edan ec . T ih s is usua yll ton true of the es itc ons at eit eh r e dn , w ih ch ra e usually et r tanim ed with a xif ed resistance. The ruf ther the s ce tion is rf om the e dn of t eh

retlif , the mo er ca c etaru the prediction lliw b ce o em , s ni ec t eh e ff ce ts of t eh et r anim t gni i pm edan ec s are mas dek by the tni erve gnin sections. 31[ ]

EGELLOC GNIREENIGNE DACS 9. m- ed r devi tlif er

m-der devi tlif ers ro m-t py e tlif ers are a yt pe fo ele rtc o cin tlif er des dengi us gni t eh image met doh . They were vni e detn by Otto boZ el ni the ae r yl 0291 s. 1[ ] T ih s retlif

epyt w sa nigiro a yll tni e dn ed for use with telepho en lum tiple gnix a dn w sa an i rpm oveme tn on the e ix st gni consta tn k epyt etlif r.[ ]2 T eh ma ni orp blem being

rdda e ss ed w sa ht e n ee d ot ca ih eve a be tt er tam ch of the tlif er otni the et r nim at gni i pm edan ec s. nI general, a ll retlif s d se ngi ed yb the mi a eg met doh fa li ot evig an

axe ct tam ch, b tu the m-t py e retlif is a b gi mi proveme tn with s atiu ble choice of t eh parame ret m. The m-t py e tlif er section has a ruf ther adva tn age ni that there is a rap di rt ansition from t eh cut-o ff rf equency of t eh pa ss band ot a pole of tta e un ation uj st ni s edi the s pot band. D se etip ht ese a vd a tn age ,s ht ere is a rd a bw ca k iw th m-

epyt tlif ers; at rf equencies past the pole of a aunett tion, the response s trat s ot rise ga a ni , a dn m-t py es have op or s pot ba dn re ej c it on. roF t ih s r ae son, retlif s des ngi ed

us gni m-t py e s ce tions are o tf en des ngi ed as co opm si et tlif ers with a xim t ru e of k- epyt a dn m-t py e s ce tions a dn d ffi ere tn va ul es of m at d ffi ere tn po tni s ot get the

po t mi um performance rf om both t py es. 3[ ] Der vi ation m- devired seires ge ren al retlif ha fl itces o .n


m-der evi d s nuh t low- ap ss etlif r ha fl s ce tio .n

T eh b liu d gni b ol ck of m-der vi ed tlif er ,s as iw th a ll image i pm edan ec tlif er ,s is t eh "L" ne wt o kr , ca ll ed a half-section and composed of a series pmi eda cn e Z, a dn a s tnuh ad ttim ance Y. T eh m-der vi ed tlif er is a de vir at evi of the consta tn k etlif r . T eh s rat t gni po tni of the ed s ngi is t eh va eul s fo Z a dn Y der vi ed rf om the noc sta tn k

otorp t py e a dn are vig en by

hw ere k is t eh nominal i depm ance of the tlif er, ro R0. hT e des rengi won multip il es Z a dn Y by an ra b rti ary consta tn m 0( < m < 1). T reh e a er t ow d ffi ere tn k dni s of m-der devi section; series and s tnuh . To o niatb the m-der vi ed series ha fl s ce tio ,n ht e des ngi er de et r enim s the pmi edan ec that um st be added ot 1/ Ym ot make t eh

image i pm edan ec ZiT ht e same as ht e image pmi edan ec of the or nigi al consta tn k sec it on. rF om the general for alum rof ima eg i pm edance, the additional i pm edance req deriu can be s oh wn ot be[9]

oT tbo a ni the m- ed r vi ed s tnuh ha fl s ce t noi , an ad ttim an ec si a dd ed ot 1/mZ ot make the image mi p de an ec ZiΠ the sa em as ht e image i pm edance of the or nigi al ha fl s ce tion. The additional ad im ttance req riu ed ac n be s oh wn ot be 01[ ]


T eh general arra gn eme tn s of these c ri c tiu s are s oh wn ni the dia rg ams ot the r thgi la o gn with a sp ce ifi c example of a l wo pa ss s ce it on.

A consequen ec of t ih s ed s ngi is that the m-der vi ed ha fl section w lli match a k-t epy s ce it on on o en s di e o yln . lA so, an m-t py e sec it on of one va ul e of m w lli ton match

ona ther m-t py e s ce tion of a on ther va ul e of m ex ec tp on the s di es which o ff er the Zi of the k-t py e. 11[ ]

pO erat gni erf quency

roF t eh ol w-pa ss ha fl s ce tion shown, ht e uc t-o ff rf equency of the m-t py e is t eh sa em as ht e k-t epy a dn is vig en by T eh pole of a tt e un ation occ ru s a ;t

rF om t ih s ti is cl ae r that sma ll er va ul es of m w lli orp duce c ol ser ot the cut-off rf equency a dn hen ec w lli have a sha pr er cut-o ff . Desp eti t ih s cut-o ff , ti also

br gni s the wnu a detn s pot ba dn response fo the m-t py e c ol s re ot the cut-o ff rf equency, mak gni ti rom e d iffi c tlu rof t ih s ot be deretlif with s bu seque tn s ce tions.

T eh va ul e of m chosen is usua yll a co rpm o im se be ewt en th ese co ilfn ct gni riuqer e tnem s. There is also a pr ca t ci al li im t ot oh w sma ll m can be ma ed due ot t eh

hni ere tn resistan ec of the ni duc rot .s T ih s has ht e e ff ce t of caus gni t eh po el of tta e un ation ot be le ss d ee p (that is, ti is on gnol er a ge niun e yl ifni n ti e pole) a dn t eh

s pol e of cut-o ff ot be el ss s et ep. T ih s e ff ce t eb comes more ma kr ed as is bro thgu olc s re ot , a dn ht ere aec ses ot eb

Ima eg depmi ance


m-der devi orp tot py e s tnuh ol w-pa ss retlif Zi mT mi age i pm edan ec rof various va ul es of m. Va eul s be wol cut-o ff rf equency o yln s oh wn for clarit .y T eh fo oll w gni expre ss ions rof image pmi eda cn es are a ll refere cn ed ot t eh ol w- ap ss

otorp t py e s ce it on. They a er scaled ot the no nim al depmi an ec R0 = 1, a dn eht rf equencies ni t oh se expre ss ions are a ll s ac del ot ht e cut-o ff rf equency ωc = 1.

Ser ei s s ce it o sn T eh image depmi ances of t eh se ir es s ce it on are vig en by ]41[

dna is the same as taht of the consta tn k se itc on

S tnuh s ce tio sn T eh image depmi ances of t eh s tnuh s ce it on are vig en by ]11[

dna is the same as taht of the consta tn k s ce it on


As with the k-t py e s ce tion, the mi age pmi edance of the m-t py e ol w-pa ss se itc on si

rup e yl er al be wol t eh cut-o ff erf quency and p ru e yl mi a anig ry a ob ve ti . rF om t eh ahc rt ti can be seen that ni the pa ss ba dn the c ol sest i pm eda cn e match ot a consta tn rup e resistan ec et r anim tion occ ru s at a xorpp imate yl m = 0.6. ]41[

narT s sim s noi rap ame sret

m-Der vi ed ol w-pa ss retlif transfer nuf ction rof a s lgni e half-section

roF an m-der devi section ni general the rt ans im ssion parameters for a ha fl -section ra e vig en by 41[ ]

dna for n ha fl -sectio sn


roF the partic lu ar example of the ol w-pa ss L section, the rt ans im ssion parame sret

s vlo e d ffi erent yl ni t rh ee rf equency ba dn s. ]41[

oF r ht e rt ans im ssion is ol ss less:

oF r eht rt ans im ssion parame ret s era

oF r eht rt ans im ssion parame ret s era

otorP t py e trans of rmatio sn T eh p tol s s oh wn of image pmi edan ec , a tt e un ation a dn p ah se hc a gn e ra e the p stol of a ol w-pa ss otorp t epy retlif s ce tion. The p otor t py e has a uc t-o ff rf equency of ωc = 1 ra /d s a dn a no nim al pmi eda cn e R0 = 1 Ω. T ih s is p or du ec d by a etlif r half-s ce tion

hw ere L = 1 henry a dn C = 1 farad. T ih s otorp t py e can be mi pedan ec sca el d and rf equency scaled ot the des ri ed va eul .s hT e low-pa ss otorp t py e ac n also eb rt ans of rm de otni hgih - ap ss , band-pa ss ro band-s ot p t py es by app il ac tion of s tiu ab el rf equency rt ans of rmations. 51[ ]

csaC gnida s ce tio sn

Several L ha fl -s ce tio sn may eb ac s ac ded ot of rm a compos eti etlif r. ekiL i pm edan ec um st a wl ays f eca kil e ni these comb ni ation .s There are t reh e rof e wt o

ric c tiu s t tah can be of rmed with owt di ent ci al L half-s ce tion .s Whe er ZiT faces ZiT, ht e s ce tion is c lla ed a Π s ce tion. Where ZiΠ f eca s ZiΠ ht e s ce it on of rmed is a T

sec it on. ruF ther additions of half-s ce tions to either of these of rms a la dd er ne owt rk ihw ch may s at rt a dn e dn with series ro shu tn eleme tn s. ]61[


tI sho dlu be ob rn ni mi dn that the charac et ristics of the retlif predic det by t eh

ima eg met doh are o yln ca c ru ate fi t eh s ce it on si et r nim a det with sti ima eg i pm edan ec . T ih s is usua yll on t true of t eh s ce it ons at either e dn w ih ch are usually et r tanim ed with a xif ed resistance. The ruf ther the s ce tion is rf om the e dn of t eh

retlif , the rom e cca ru a et t eh rp ediction lliw become s ni ec t eh e ff ce ts of t eh et r anim t gni pmi edan ec s a er masked by ht e tni erve gnin s ce tions. tI is usual to rp o div e ha fl ha fl -s ce tions at the e dn s of the tlif er with m = 0.6 as t ih s va ul e sevig ht e ettalf st Zi ni the p sa s dnab a dn hen ec the best match ni to a resist evi et r anim tion. 71[ ]

01 . C yr stal retlif

A crystal tlif er is a sp ce ial of rm of quartz crystal used ni el ce tro in cs systems, in partic lu ar co cinumm ations de iv ec .s tI pr edivo s a very rp ecise yl de nif ed ec nt er rf equency a dn very s et ep ba pdn ass hc ar ca teristic ,s ht at is a very hgih Q f ca rot —far

rehgih than can be niatbo ed iw th co vn entional lu pm ed c ri c tiu s. A crystal tlif er is very o tf en fo dnu ni the etni rmedia et erf quency I( )F stages of

hgih -qua il ty radio er ce vi ers. Che pa er sets may use ec ra im c retlif s (w ih ch also xe plo ti the pi ze oelectric e eff c )t , ro t nu ed LC c ri c tiu .s T eh use of a dexif IF s gat e rf equency a woll s a crystal retlif to eb used b ace use it has a very rp eci es exif d rf equency.

T eh om st co mm on use of crystal retlif s, is at rf equencies of 9 M zH or 01 .7 MHz to

rp o div e sel ce t ivi ty ni co cinumm ations re ec revi ,s or at ehgih r rf equencies as a or o gnif tlif er ni r ece vi ers us gni up-co vn ers .noi

eC ra cim retlif s te dn to be used at 10.7 MHz ot pro div e s le ect ivi ty ni orb ad ac st MF

r ece vi er ,s ro at a wol er rf equency 54( 5 kHz) as ht e s ce o dn tni ermedia et erf quency retlif s ni a co mm u cin ation er ce vi er. eC ra im c retlif s at 54 5 kHz can ac ih eve si lim ar

ba dn widths ot crystal retlif s at 01 .7 MHz.


TINU V

WAV GE U DI ES aW ve ediug s are basica yll a de iv ec ("a ediug ") rof trans rop t gni el ce rt oma ng et ci ene r yg rf om one re ig on ot another. Typica yll , wave ediug s a er ho woll metal t bu es

(o tf en r ce ta ralugn ro c ri c lu ar ni c or ss es itc o )n . hT ey era apac b el of d ceri t gni p rewo rp ecise yl ot where ti is n ee ded, can handle large amo tnu s of wop er a dn cnuf tion as

a gih h-pa ss .retlif T eh wave ediug a tc s as a hgih ap ss etlif r ni that om st of the e en r yg ba ove a ec rta ni rf equency (the c tu o ff rf equenc )y w lli pa ss t rh o hgu t eh wave ediug , wher ae s om st of ht e e en r yg that is be wol the c tu o ff rf equency w lli be a tt e un a et d by the wave .ediug

aW ve ediug s are o tf en u es d at im c wor a ev rf equencies ( rg ae ter than 003 MHz, with 8 HG z a dn above be gni om er co mm o .)n

aW ve ediug s are diw eba dn de iv ces, dna c na ac rry ro( trans im t) either wop er ro co cinumm ation s ngi al .s nA examp el of a ho woll metal r ce ta ralugn wave ediug si s oh wn ni the fo oll w gni rugif e.


aW ve ediug s ac n be dn fi t eh des ri ed ap ilp cation req riu es ,ti as s oh wn ni t eh fo oll w gni ugiF re .

T eh ba ove wave ediug s c na be us de with wave ediug ot coa ix al ac ble ada tp er ,s as s oh wn ni the ne tx rugiF e:

eW on w k won what a wave diug e is. Lets exa nim e metal ca iv ties with a r ce ta lugn ar orc ss s ce it on, as s oh wn ni rugiF e 1. A ss emu ht e wave ediug is dellif with v ca cuu ,m ria or some diel ce tric with the per em ab ili ty vig en by a dn the per tim t ivi ty vig en yb .

T eh wave diug e has a width a ni the x-d eri c it on, a dn a he thgi b ni the y-d eri ctio ,n

iw th >a b. hT e z-a ix s is eht d ri ce tion ni w hcih the wave diug e is ot ac rry wop er.


erugiF 1. Cro ss se itc on of a wave ediug with lo gn id mension a a dn sho tr d mi ension .b

nO t ih s page, 'I m go gni to vig e t eh general "r lu es" of r wave diug se . T tah i ,s I ll' evig

ht e equations for key parame ret s a dn let uoy k won w tah the parame ret s mean. On ht e ne tx page, we ll' og otni the mathema it ac l der avi tion (w ih ch you wo dlu od ni

nigne eer gni rg adua et cs oh o )l , b tu you can get away with ton k on w gni a ll that math fi you do t'n wa tn ot know ti .

riF st a dn op ssib yl om st i opm rtant yl , t ih s wave ediug has a c tu o ff rf equency, fc. T eh tuc o ff rf equency is the rf equency at w ih ch a ll wol er rf equencies are a tt e un ated by

ht e wave ediug , a dn above the cu ot ff rf equency a ll rehgih rf equencies porp aga et

iw t ih n t eh wave diug e. The c tu o ff rf e uq ency de nif es t eh gih h-pass tlif er rahc ca teristic of the wave diug e: above siht erf quency, the wave ediug pa ss es

wop er, be wol t ih s rf equency the wave diug e ta unet a et s ro blocks wop e .r T eh c tu o ff rf equency depe dn s on t eh s pah e dna si ez of the c or ss s ce it on of t eh wave diug e. The larger the wave ediug i ,s ht e wol er the cuto ff rf equency rof that wave ediug i .s The for alum rof the c tu o ff fre uq ency of a r ce ta ralugn cro ss s ce tioned

aw ve ediug is vig en b :y In the above, c is the sp ee d fo thgil wit nih the wave ediug , um is the ep rmeab ili ty of ht e ma et rial that llif s the wave ediug , and eps li on is the per tim t ivi ty of the ma et rial ht at llif s t eh wave ediug . etoN ht at the c tu o ff erf quency is dni epe dn e tn of the short

le gn th b of the wave .ediug

EGELLOC GNIREENIGNE DACS T eh c tu o ff erf quency for a wave ediug with a c ri c lu ar c or ss sec it on of ar d ui s a si

vig en b :y

D eu ot Ma wx e 'll s Equations, eht if e dl s iw t nih the wave diug e a wl ays have a sp ce fi ic "form" or w" aves pah e" to them - these a er ca ll ed dom e .s Ass mu e t eh wave ediug is orie detn such that the ene ygr is ot be rt ans ttim ed alo gn t eh wave ediug a ix ,s t eh z-a ix s. The modes ra e class ifi ed as either TE ('transverse le e tc ric' - w ih ch ni d ci a et s t tah t eh E-fie dl is orthogonal ot the a ix s of t eh

wave ediug , so that E =z )0 ro TM ('transverse ma ng etic' - w ih ch ni di ac et s t ah t the H- if eld is ro thogonal ot the a ix s of the wave ediug , so Hz = )0 . The dom es are

ruf ther cl ssa deifi as TE ji , w reh e the i a dn j ni dica et the nu bm er of wave osc lli ations rof a partic lu ar if e dl d eri ction ni the lo gn d ri ection (d mi ension a ni

erugiF )1 a dn s troh d ri ce tion (d mi ension b ni rugiF e )1 , resp ce t vi e yl . Me lat wave ediug s ac tonn s ppu o tr the T ME ( rt' ansverse el ce tric a dn ma ng etic' -

hw en Ez a dn zH a er ez ro) mo ed . The ri e ix sts on so ul tion ot Ma wx e 'll s equatio sn ht at also satis yf the er q riu ed bo dnu ary conditions rof t ih s edom ot occ .ru

Ma wx e 'll s Eq au tions are ton ae sy ot so vl e. Hence, eve ry math tr kci someo en can

kniht of w lli be used ni o dr er ot make ht e ana yl sis rt ca at ble. We ll' s trat with id scuss gni the el ce tric v ce t ro top ential, F. nI a s ruo ce- erf e re ig on (i.e., an area rht o hgu w ih ch wa ev s porp aga et ht at is away rf om so ru ec s), we wonk t ah t:

In t eh above, D is the El ce tric xulF Density. If a v ce t ro quantity is d vi ergenceless (as ni the ba ove), then it can be exp er ss de as the curl of a on ther quantity. T ih s means that we can wr eti t eh so ul tion of r D a dn the correspond gni ele tc r ci if e dl E as: In ht e above, eps li on is t eh per im tt ivi ty fo the medium t rh o hgu w ih ch the wa ev

EGELLOC GNIREENIGNE DACS porp a etag s. eW are p ru e yl ni the wor dl of mathe am tics now. The quantity F is not

yhp s ci al, a dn is of til tle rp ca tical va ul e. tI is s mi p yl an a di ni per of r gnim o ru mathe am tical ma in p lu ations. tI tur sn o tu that waves ro( le ectroma ng et ci e en r )yg ac n not porp aga et ni a

wave ediug when ob th zH and Ez a er uqe al ot ez .or Hence, ahw t if e dl rugifnoc atio sn t tah a er a woll ed w lli eb class ifi ed as either MT ( rT ansverse

Ma ng etic, ni w ih ch H =z )0 a dn TE (Transverse El ce tric, ni w ih ch E =z )0 . T eh r ae s no that waves ac nn ot be TEM (Trans rev se Ele rtc oma ng etic, H =z E =z )0 w lli eb s oh wn wot a dr s ht e e dn of t ih s der vi ation .

oT perform our ana yl si ,s we ll' a ss ume that E =z 0 (i.e ,. we are ol ok gni at a TE mo ed ro if e dl co arugifn tio )n . In t ih s esac , gnikrow t rh o hgu Ma wx e 'll s equation ,s ti can eb

s oh wn t tah the E- a dn H- if e dl s ac n be ted er denim rf om the fo oll w gni equations:

T reh e rof e, fi we can dnif Fz (the z-co pm one tn of t eh v ce tor )F , then we can dnif ht e E- a dn H- if e dl s I . n the a vob e equation, k is the wavenu bm e .r

W ro k gni t rh o hgu the math of Ma wx e 'll s Equation ,s it can be shown that ni a s ruo ce- rf ee re ig on, ht e v ce tor top ential F um st satis yf the v ce t ro wave e uq atio :n

]1[

oT br ae k t ih s equation down, we w lli ol ok o yln at the z-co pm one tn of the ba o ev uqe ation (that i ,s Fz). eW lliw also a ss ume that we are ool k gni at a sin elg

rf equency, so ht at the t mi e ped e dn en ec is a ss mu ed ot be of the of rm vig en by ( ew ra e won us gni phas ro s ot ana yl ez the uqe at noi ):

EGELLOC GNIREENIGNE DACS Then the e uq ation [1] ac n be simp deifil as wollof s:

]2[

oT so vl e t ih s eq au tion, we w lli use the et c inh q eu of separation of variable .s He er

we a ss mu e that the cnuf tion Fz(x, y, z) can eb wr tti en as the orp duct of t rh ee nuf ction ,s each of a s lgni e variable. T tah i ,s we ass mu e that:

]3[ Y( ou im g th sa k, woh od we know that t eh separation of variab el s a ss pmu tion

voba e is va il d? eW do t'n - we just a ss ume ti s corre tc , dna fi ti so vl es t eh ffid erential equation when we are done do gni t eh ana yl sis then the ass mu ption si

va dil ). oN w we gulp ni o ru a ss mu ption of r Fz (equation 3[ ]) otni equation ]2[ , a dn we e dn pu wit :h

]4[ In the above equation, the pr mi e re rp ese tn s the der vi at vi e with respect ot the var ai b el ni the equation (for ni stan ec , Z' re rp ese tn s the der avi t vi e of the Z- nuf ction

iw th resp ce t to z). We w lli rb ae k pu the var ai ble k^2 ni to co pm one tn s (aga ni , just ot make o ru math ae sie )r :

]5[ Us gni equation ]5[ ot erb ak od wn equation [ ]4 , we ac n wr ti e:


]6[ T eh r ae son that the equations ni ]6[ are va il d is b ce ause they are o yln nuf ctions of

pedni e dn e tn variables - hen ec , ea hc eq au tion um st ho dl rof [ ]5 ot be tr eu eve rywhere ni the wave diug e. So nivl g t eh above equations us gni ord ni ary ffid erential equations theory, we eg t:

]7[ T eh of rm of the so ul tion ni the abo ev equa it on is d ffi erent for )z(Z . The r ae son si ht at b to h of rms (that rof X a dn Y, a dn that for )Z , are ob th equa yll va dil so ul tio sn

rof the d ffi erential equations ni equation [ ]6 . woH ever, ht e complex exponential yt pica yll represe tn s rt ave gnill wave ,s and eht [rea ]l s uni so di s re rp ese tn standing

waves. Hence, we c ooh se the of rms vig en ni ]7[ rof the so ul tion .s oN math r selu ra e iv ola det here; aga ni , we are uj st c oh os gni forms t tah w lli ma ek o ru ana yl s si

easie .r

roF won , we can set c5=0, because we wa tn to ana zyl e waves porp a ag t gni ni the z+ - oitcerid n. The ana yl sis is di entical rof waves porp agat gni in the -z-d ri ce tion, so iht s is fair yl arbitrary. The s ulo tion rof Fz can be wr tti en as:

[ ]8 If you reme rebm a yn t gnih abo tu d ffi erential equation ,s you k won there en eds ot eb some bo dnu ary conditions app il ed ni dro er ot de et r nim e the consta tn s. aceR ill ng

ruo p yh sic ,s we k won that the ta gn ential El ce tr ci if e dl s at a yn perfe tc conductor

um st be ez or (w ?yh because , so fi the conduct ivi ty pa p or aches inifni ty

EGELLOC GNIREENIGNE DACS p( er ef ct conduc ot r), ht en fi the ta gn ential E- if e dl is not ze or then the ni d cu ed uc rre tn wo dlu be tinifni e .)

T eh ta gn ential if e dl s um st be ez ro, so Ex um st be ez or when y=0 a dn when y=b (see rugiF e 1 abo )ev , no ma tt er what the va ul e of r y a dn z are I . n addition, Ey um st be z ore hw en x=0 a dn when x=a ( dni epende tn of x a dn z). eW lliw calc lu a et E :x


Ex is vig en by the ba ove equation. The bo dnu ary condition vig en by Ex( x, y=0, z)=0 ]9[ imp eil s t tah c4 um st eb equal ot ez ro. T ih s is the o yln way t tah bo dnu ary condition

vig en ni ]9[ will be tr eu rof a ll x a dn z position .s If you do t'n be il eve t ih s, try to s woh t tah ti is ni co rr ce t. You w lli q iu ck yl de et r enim that c4 um st be z ore for t eh

adnuob ry condition ni [9] ot be sa it s if ed eve yr whe er ti is er q riu e .d Ne tx , the sec dno bo adnu ry conditio ,n Ex(x, y=b, z)=0 ]01[ imp eil s somet gnih ev ry inu que. The ylno way for the noc dition ni 1[ 0] ot eb tr eu

rof a ll va ul es of x a dn z hw enever y=b, we um st have: If t ih s is ot be true everywhe er , c3 co lu d eb ez or . oH wever, fi 3c is ez or (a dn ew have a rl ae dy eted r denim that c4 is ez )or , ht en a ll of the if e dl s dluow dne pu being ze or , b ace use the nuf ction Y( )y ni [7] would eb ez ro everywhere. Hen ec , 3c ca tonn be ez or fi we ra e ool k gni for a non ez or so ul tion. He cn e, ht e o yln a etl r an t evi is fi ht e above equation imp il es that:

T ih s last equation is dnuf ame tn al ot dnu ers nat d gni wave ediug .s tI s tat es t ah t t eh

ylno so ul tions of r Y( )y nuf ction um st end up be gni s uni so di s, that an tni eger nu bm er of lum tiples of a half-wavele gn th. These are the o yln t py e of nuf ctions that


sa it s yf the d ffi erential equation ni [6] and the req riu ed bo dnu ary condition .s T ih s si na e rtx eme yl i pm o tr a tn conc .tpe

If we ekovni o ru other owt bo dnu ary conditions: Ey(x=0, y, z)=0 Ey(x a= , y, z)=0 Then (us gni di entical er aso gnin ot that ab vo e), we can ted er nim e that 2c =0 a dn

aht t : T ih s s tat eme tn imp il es that t eh o yln uf nc it ons of x t tah satis yf the d reffi ential

uqe ation and the req riu ed bo dnu ary condit noi s um st be an tni eger lum tiple of ha fl - s uni so di s wit nih the wave ediug . Comb gnini these res tlu s, we ac n wr eti the s ulo tion for zF as:

In the above, we ha ev comb ni ed the rema gnini non ez or consta tn s 1c , 3c , a dn c6

otni a s elgni consta ,tn ,A rof simp il cit .y We ha ev fo dnu that o yln ce tr ain id stributions (or if e dl co rugifn ations) w lli sa it s yf the req riu ed d ffi erential eq au tio sn dna the bo dnu ary condition .s Each of these if e dl con rugif ations w lli be k on wn as a

mode. B ce ause we der vi ed the res tlu s abo ev rof the ET ac se, the modes w lli eb nk own as TE nm , where m ni di ac et s the bmun er of half-cyc el av riations wit nih t eh

wave ediug rof X( )x , a dn n ni di ac et s the nu bm er of half-cycle var ai tions wit nih t eh wave ediug rof Y( .)y Us gni the if e dl relations pih s:


eW can wr ti e the a woll able if e dl co gifn urations of r the TE ( rt ansverse el ce tric) modes iw t nih a wave diug e:


In the ba ove, ht e consta tn s are written sa nmA - t ih s mi p eil s that t eh amp il t du e for e ca h dom e can be dni epe dn e tn of the others; however, the if e dl com op ne tn s for a s lgni e mode um st lla be re detal t( tah is, xE a dn yH od ton have dni epe dn e tn oc e iff cie tn s .)

C tu o ff erF quency (fc)

tA this po tni ni the ana yl sis, we a er able to s ya so em t gnih tni e gill e tn . aceR ll that ht e compone tn s of t eh wavenu rebm um st sa it s yf the relations ih p:

]3[ S ni ec kx a dn ky are res rt a ni ed ot o yln kat e no ec tr a ni va ul e ,s we ac n p gul t ih s af ct

:ni

]4[

nA i retn est gni question arises at t ih s po ni t: What is the lowest rf equency ni w ih ch ht e wave ediug lliw porp a etag the TE dom e?

roF pr po agation to o cc ,ru . If t ih s is true, ht en zk is a real bmun er, so ht at

ht e if e dl com op ne tn s (equations [1] and [ )]2 w lli conta ni complex e px onentials,

ihw ch repres tne porp agat gni waves. If on the other ha ,dn , ht en zk w lli be na ima nig ary nu bm er, ni w ih ch ac se the complex e px onential above ni equatio sn 1[ -2] b ce omes a deca gniy real e px onen it al. In t ih s ac se, ht e if e dl s lliw on t

porp aga et b tu ni s et ad q iu ck yl die o tu wit nih the wave diug e. mortcelE a eng t ci if e sdl ht at die o ff ni s et ad of porp aga et ra e refe rr ed ot as evanescent waves.

oT dnif the wol est rf equency ni w ih ch propagation can occ ru , we set k =z 0. ihT s si

ht e rt ansition be wt een t eh c tu o ff re ig on (evanesce tn ) a dn the porp a ag tion re ig o .n Sett gni k =z 0 ni equation [4], we :niatbo


]5[ If m a dn n are ob th ez or , then a ll of the if e dl co pm one tn s ni [1- ]2 become ez or , so we ca nn ot have t ih s condition. The wol est va ul e the le tf ha dn s di e of equation [ ]5 can take occ ru s when m=1 a dn n=0. The s ulo tion ot equation 5[ ] when m=1 a dn n=0, vig es ht e c tu o ff erf quency rof t ih s wave diug e:

ynA rf equency be wol the c tu o ff rf equency (fc) w lli o yln res tlu ni evanesce tn or d ace gniy modes. ehT wave diug e w lli not rt anspo tr ener yg at these rf equencie .s In

idda tion, fi the wave diug e is po erat gni at a rf equency just above fc, then t eh only mode that is a porp agat gni dom e w lli eb the T 01E dom e. llA other dom es w lli be d ace niy g. Hen ec , the ET 10 edom , s ni ec ti has the wol est c tu o ff rf equency, si re rref ed ot as eht do nim a tn mode . E ev ry mode ht at can e ix st iw t nih t eh wave ediug ah s ti s own cuto ff rf equency. That i ,s for a vig en mo ed ot p por aga et , the po e gnitar rf equency um st eb above the c tu fo f rf equency for that dom e. By so gnivl [5] ni a more general of rm, the c tu o ff rf equency for the TE nm dom e is vig en b :y

lA tho hgu we have t'n discu ss ed the TM (transverse ma ng etic) dom e, ti w lli turn o tu ht at the domina tn TM edom has a hgih er c tu o ff rf equency than the do nim a tn TE

mode . De et r nim i gn the if e dl s for the TMz (Transverse Ma ng et ci to the z d ri ce tion) dom es fo woll s a s limi ar orp ec d ru e ot that rof the ET z ac se. oT be nig , we ll' s trat by

id scuss gni t eh ma ng etic v ce rot top ential, A. ihT s is a non-p yh sical quantity that si o tf en ni used tna e nn a theory ot simp yfil the am thematics of Ma wx e 'll s Equations.


oT dnu ersta dn t eh ma ng et ci v ce t ro top e itn al, eton that s ni ce the ma ng etic lf ux

density B um st a wl ays be vid erge cn e el ss : If a v ce tor quantity is d vi ergencele ,ss neht ti can eb e erpx ss de sa the curl of

ona ther v ce rot quantity. In math aton tion, t ih s em ans that B can be wr tti en as:

In a s ruo ec erf e re ig on, ti can be s oh wn that A um st sa it s yf the wave eq au tio :n

In addition, the TMz if e dl s can be fo dnu from the Az co pm one tn of the ma ng et ci ve tc or top ential, iv a the fo oll w gni re al tions pih s:

oT so vl e for Az (and hence de et r nim e the E- a dn H- if e dl s), we fo woll the sa em orp ec d eru as for the zET ac se. tahT i ,s we use separation of variables a dn so vl e eht

wave equation rof the z-co pm one tn of A, t neh app yl bo dnu ary conditions that force ht e ta gn ential co pm one tn s of the E- if elds ot eb z ore on t eh meta ill c surfaces.

Perfor im ng t ih s pro ec d ru e, w ih ch w lli ton eb r pe ae det here, we tbo a ni the so ul tion rof Az:


]1[ T eh c rro espond gni TMz if e dl s for waves pr po a ag t gni ni the +z-d eri ction ra e:

In the above, k is aga ni the wave bmun er, a dn B nm is a consta tn , w ih ch de et r nim es ht e amp il t du e of the nm mode (a nuf ction of woh um ch wop er is app il ed ot t eh

vaw e ediug ta taht erf quenc )y . Before discuss gni the dom e ,s we um st on te ht at TM0n a dn TMm0 dom es ca nn ot

ixe s ;t that i ,s m a dn n um st be at l ae st 1. hT e r ae s no comes from equation [1] abo ev - fi either m ro n are z ore , then Az is equal ot ze or , so a ll the if e dl s der devi um st la so be ez or . Hen ec , the wol se t dro er mode of r t eh TM case is t eh TM 11 dom e.

T eh same orp ec d ru es can be app il ed from t eh TE case to de et r enim the cuto ff rf equencies rof the TM nm dom e:



M SEDO

EGELLOC GNIREENIGNE DACS C dnily r ci al wave ediug

Us ni g t eh comp tel e alumrof tion ni t eh s mi plest li im t op ssible, lg obal le e tc roma ng etic modes are here studied ni a large asp ce t rat oi , c ri cu ral c or ss-

s ce it on av c muu ca iv ty eq viu ale tn ot a ilyc ndr ci al wave diug e. C al ssical el ce t or d ny a cim s [ ]34 s woh that ht e EM e gi en dom e sp ce tr mu consists of

owt t py es of so ul tion ,s the transverse ele tc r ci a dn the rt ansverse ma ng et ci polari az tions with rf equencies depend gni on l the radial a dn m the az umi thal

mode nu bm er .s These res tlu s are re rp oduced mun erica yll to ev r yfi that the wave uqe ations ( )61 can edni ed be so vl ed ni the vac muu us gni sta dn a dr EFL M and

C EF M discreti az tions, witho tu ni t or duc gni s up rious dom es of mun erical or nigi . It is also mi p tro a tn to va dil a et the mun erical impleme atn tion us gni a s mi ple et st esac ,

ehc ck gni that the emun rical so ul tions co vn erge ot the ana yl tical va ul es with ra set pxe ce det rf om the dro er of the a rpp oximat noi s.

T eh c ly dni rical wave ediug is dom e del ni 2-D, with a c ri c lu ar large aspe tc rat oi

biliuqe rium de denif with a onim r rad ui s a c oh sen so as ot tbo a ni the ana yl tical gie en dom e rf equencies ni GHz ex ca t yl equal to t eh or ots of the Be ss el nuf ction at( ble 1 .)

Table 1: C nily drical wave ediug para tem ers. As the eq biliu rium mere yl rp odu ec s t eh geometry a dn the em sh, the safety fa otc r

od es ton a eff ct the e gi e rfn equency sp ce tr ;mu us gni a al rge va ul e for , ti si

woh ever op ssible ot everywhere ngila iw th the a ix s of the c dnily er a dn separa et ht e compone tn s of the ET a dn the TM po al ri az tions. The comple et t ro o di al wa ev


uqe ations (16) a er then discreti ez d ni the lar eg aspe tc rat oi ca iv ty, re gniyl on

numerical cance ll ations ot r ce over the c ily ndrical imil .t

oT comp etu t eh e gi en dom e sp ce trum, an osc lli at gni so ru ec c rru e tn (eq. )22 si dr vi en with a sma ll ima nig ary part ni t eh exc ati tion rf equency

. ehT power relation (eq.48) iy e sdl a complex er sponse

itcnuf on w ih ch has poles alo gn the real a ix s ht at co err spo dn to t eh s ulo tions of the dis rc etized wave equation .s The e gi e rfn equencies a er calc talu ed by s ac gninn ni the complex pla en with an cni reme tn dna a consta tn

ohc sen so as ot res evlo t eh response p ae ks ni . hT e struct eru of an gie en dom e is tbo a ni ed ni t eh timil when the ac iv ty is er sonant yl exc deti at

ht e maximum of a ran r wo response p ae .k In dro er ot ev r yfi that the e gi e rfn equency sp ce tr mu of t ih s c dnily rical wave ediug si comple et and od se on t co tn a ni a yn spurious p'' o ull t gni `` mode, owt b or ad scans are per of rmed rf om 01 kHz ot 01 G zH iw th a hgih reso ul tion ni rf equency

a dn a wol reso ul tion ni sp ca e of r FL EM, rof C EF M). llA the Four rei modes re rp ese atn b el by t eh

numerical di cs ret zi ation are exc deti with a miz uthal curre stn rof ET modes, dna a ix al c rru e stn for TM modes. giF .6 s mmu ari ez s t eh res tlu tbo a deni with

FL ,ME show gni t tah every mode fo dnu mun erica yll co dlu be di ent eifi d ni a o en to no e co rr espo dn en ec with the ana yl tical resu tl . M do es w ih ch have wol quantum

nu bm ers (l,m) ra e, as pxe ce det , tbo a eni d iw th a bet ret rp ecisio ;n pus gnih t eh res ulo tion to the wol se t timil of 2 mesh po tni s per wavele gn th (m=4), eht de iv ations become of ruoc se i opm rta tn , tub the sp ce tr mu rema ni s nu po ll uted r( eme bm er gif .3, toor b). The same ana yl sis has been re ep a det with C EF M a dn

l ae ds ot er s tlu s w ih ch a er um ch rom e precise. As an ulli s art tion, the e gi en edom h sa here eb en ac lc lu ated no a coar es homogeneous mesh . ehT

gie e rfn equency tbo a ni ed mun erica yll HG z is ni exce ll e tn a rg eeme tn

iw th the ana yl t ci al res tlu =5.3 413 GHz; gif .5 s woh s t eh e gi en dom e struct eru ni a ve tc or p tol of .


erugiF 5: eR ( p_A e )pr rof an e gi en dom e TE 11_ ac lc detalu with C EF M.


erugiF 6: nA a yl t ci al (c ri cles) a dn L EF M (x-ma kr s) e gi e rfn equency sp ce tru .m A question rema ni ed when the bo adnu ry c no ditions we er de denif ni s ce t.2.2. :2 ti

noc ec rned the impleme atn tion of the regularity conditions w ih ch is of rma yll ton s iffu cie tn ot of rb di a w ae k yl s lugni ar ( ) beha roiv of the if e dl ni t eh

etnec o r f the em sh. giF .5 s woh s that the if eld is re lug ar a ll over the c dnily er rad ui s, s ggu est gni that the s lugni arity is ton str gno eno hgu ot s woh pu us gni a MEF

id scret zi ation on a re ralug mesh. The ylno way we have fo dnu ot bo serve ti , w sa ot s rt o ylgn acc mu u etal the mesh po tni s wot ards ht e ce retn (for example by d ivi d gni 5 times ht e radial mesh tni erval olc es st ot ht e a ix s yb owt , lead gni ot radial em sh sp ca gni s .) Ha gniv ver eifi d that the so ul tions calc talu ed with the wave equations (eq.16) behave in a rotcafsitas y ma nn er, the uq a il ty of the EFL M a dn C EF M

id scret zi atio sn is nif a yll best judged ni a c vno ergence s ut dy mo otin r gni t eh rp ecision of the rf equency a dn the ga gu e as a nuf ction of the spatial er so ul tio .n


erugiF 7: Co vn ergen ec ot the ana yl tical res lu t: re al t evi rf equency de iv ation De atl f versus the nu bm er of mesh etni rvals (N= _N s= _N the )at of r the e gi enmodes

ET _ 10 ,T 20_E ,T 11_E ,TM }00_ us gni FL EM x( -ma kr s) a dn C EF M (c ri cles) .

giF .7 s woh s the co vn erge cn e of rof t eh e gi en dom es ,

, and , whe er re ref s to the rf equency obta ni ed mun erica yll a dn

ot the ana yl tical er s tlu . E gi e rfn equencies c vno erge ot the ana yl tical va eul s as us gni FL EM a dn al om st us gni C EF ,M htiw an exce ll e tn ini tial pre ic sion

be tt er than %1 for owt mesh po tni s per wa lev e gn th.


rugiF e 8: rP ecision of the ga gu e versus the nu bm er of mesh tni erva sl N( = _N s= _N the )at rof t eh e gi e domn es TE 10_ , ET _ 20 ,T 11_E ,TM_ 00 us gni FL EM x( -ma )skr a dn C MEF (c ri c )sel .

C vno ergen ec is also ac ih eved of r the gaug :e gif .8 shows that the volume avera deg

ga gu e p er cision c vno erges ot ez or as us gni FL ,ME a dn us gni C EF M.

oT su amm r zi e, ht e calc lu ations performed iw th the rot o di al P NE N c edo used he er ni the s mi plest timil po ss ible s oh w that Ma wx e ll 's equations ( 61 ) so vl ed ni a

dnilyc rical ca iv ty p or du ec the comple et yhp sical s ep ctrum witho tu tni roducing numerica yll p or duc de p'' o ull t ``gni mode .s htoB , the EFL M and the C MEF

id scret zi ation schem se iy e dl so ul tions w ih ch a er mun erica yll sa en a dn co vn erge ot ht e ana yl tical va ul e with ra et s e px ce det rf om t eh order of the retni op lations.

Bo dnu ary conditio sn Let us re iv ew the general bo dnu ary conditions on the if e dl v ce t ro s at a surfa ec be ewt en medium 1 dna med mui 2:


)43(

)53(

)63(

)73(

hw e er is used of r the surf eca cha gn e density t( o avoid co ufn sion with t eh noc duct ivi t )y , a dn is the surf eca curre tn density. Here, is a tinu v ce tor on rmal

ot ht e surf eca , d ri e detc from medium 2 ot em dium 1. eW have s ee n ni Section 4.4 ht at for normal ni c di ence an el ce rt oma ng etic wave fa ll s o ff very rapid yl ni s di e t eh

surf eca of a oog d conduct ro . Equation (4. )53 imp il es that ni the li im t of perfect noc duct ivi ty ( ) eht ta gn ential compone tn of va in s ,seh w eh saer t tah of

may rema ni etinif . Let us exa enim the beha iv o ru of the on rmal co pm one tn s.

Let medium 1 be a doog cond otcu r for w ih ch , lihw st em dium 2 is a perf ce t ni s lu ator. The surface cha gn e density is rela det ot the curre tn s olf wing ni s di e the conductor I . n fa tc , eht conservat noi of charge req riu es that

)83(

woH ever, , so ti fo woll s rf om qE . 6( . ()1 a) that

)93(

EGELLOC GNIREENIGNE DACS tI is cl ae r that the on rmal co pm one tn of iw t nih t eh c no duc rot also b ce omes

va in s ylgnih sma ll as ht e conduct ivi ty a pp roaches fni ini t .y If va in shes ni s di e a perf ce t conduc rot then the curl of also va in she ,s a dn t eh time ra et of cha gn e of is co rr espond lgni y ez or . T ih s imp eil s that there a er on osc lli atory if e dl s whatever ni s edi such a c no duc rot , a dn that the bo dnu ary va ul es of ht e if e dl s tuo s edi are vig en by

)04(

)14(

)24(

)34(

Here, is a tinu normal ta the surf ca e of the c no duc rot po ni t gni otni the conducto .r T uh ,s the el ce rt ic if e dl is on rmal a dn the ngam etic if e dl ta gn ential at the sur af ce of a tcefrep cond cu t ro . For doog conductors these bo dnu ary conditions iy e dl exce ll e tn re rp ese atn tions of the geometr ci al co ugifn ra it ons of e tx ernal if e dl ,s b tu they l ae d to ht e ne elg ct of so em i opm rta tn f ae t ru se of r ae l if e dl ,s such as ol sses ni ca iv ties a dn

s ngi al a tt e un ation ni wave ediug s. In o dr er ot es it ma et such ol ss es ti is use luf to s ee woh the ta gn ential a dn on rmal if e dl s co pm a er when is large b tu etinif . Equa it ons 4( .5) a dn (4. )43 iy e dl

)44(

ta t eh surf ca e of a conductor ( rp o div ed t tah t eh wa ev rp opaga set otni t eh

noc duc ot r). Let us ass mu e, without tbo a ni i gn a comple et so ul tion, that a wave with


very en ar yl ta gn ential a dn ev ry n ae r yl on rmal is porp aga det alo gn the sur af ce

of the metal. Ac roc d gni ot the Faraday-Ma wx e ll equation

)54(

uj st o tu s edi the surface, hw e er is the comp no e tn of the rp opagation ev ctor al no g ht e surf ca e. woH ever, qE . 6( . )5 mi p il es that a ta gn ential compone tn of is

a cc o pm a in ed by a sma ll ta gn ential comp no e tn of . yB compar gni t eh se owt xe pre ss ion ,s we tbo a ni

64( )

hw ere is t eh sk ni depth (s ee qE . 4( . ))63 a dn . tI is c el ar that the ar t oi of ht e ta gn ential co pm one tn of ot ti s normal co pm one tn is of order the sk ni depth

divid ed by t eh wavele gn th. tI is r ae d yli demons rt a det that the rat oi of the normal co pm onent of ot ti s ta gn ential component is of t ih s same ma ing t du e. T uh s, ew can s ee that ni the li im t of hgih conduct ivi ty, w ih ch em ans va in s gnih sk ni dept ,h

on if e dl s penetra et the conduc rot , a dn the adnuob ry conditions are t oh se vig en by qE s. 6( .4). Let us ni vest gi a et the s ulo tion of t eh homogeneous wave equation

s ejbu ct ot such bo dnu ary conditions.

aC iv t ei s with re tc a ralugn bo dnu aries

C no s di er a vac muu re ig on t to a yll enc ol sed yb er cta lugn ar conduct gni wa ll .s In t sih ,esac lla fo eht if e dl pmoc one stn itas s yf eht wa e ev q au tio n

)74(


hw e er re rp ese tn s a yn co pm one tn of or . The bo dnu ary conditions 6( .4) req riu e that the ele tc ric if e dl is normal ot ht e wa ll s at the bo dnu ary wher ae s t eh ma ng etic if e dl is ta gn ential. If , , a dn are ht e id mensions of the ca iv ty, then ti is read yli ver deifi that the e el ctric if e dl com nop e tn s a er

)84(

)94(

)05(

hw e er

)15(

)25(

)35(

iw th , , tni egers. ehT a dewoll uqerf enc sei era vig en yb

)45(

tI is cl ae r from qE . 6( . )9 t ah t ta least two of the tni ege sr , , um st be d ffi ere tn rf om ze or ni dro er ot have non-va in s gnih if e dl .s The ma ng etic if e dl s tbo a ni ed by


ht e use of a tu omatica yll satis yf t eh app or pria et bo dnu ary conditions,

dna are ni phase qua rd at ru e with the electric if e dl s. T uh ,s the s mu of the tot al le e tc r ci a dn ma ng etic ener ig es wit nih the c iva ty is consta tn , altho hgu the wt o et r sm

osc lli a et separate .yl T eh amp il t du es of t eh elec rt ic if e dl compone tn s are ton pedni e dn e tn , but a er rela det by the vid erge cn e condition , ihw ch iy e sdl

)55( T ereh are, ni genera ,l owt nil ear yl in ped e dn e tn v ce tors ht at satis yf t sih

noc dition, co rr espond gni ot two polarizat noi s. T( he ex ec ption is the case t tah o en of the tni e sreg , , is ez or , ni w ih ch ac se is dexif ni d ri ce tion.) Each v ce tor is ca co pm a in ed by a ma ng et ci if e dl at rig th a lgn es. The if e dl s c rro espond gni ot a

vig en set of tni ege sr , , a dn noc stit etu a partic lu ar dom e of rbiv ation of t eh vac ity. tI is e div e tn rf om sta dradn Fourier ht eory that t eh d ffi ere tn mo ed s era

ro thogonal (i.e ,. ht ey are on rmal modes) a dn that they of rm a comple et se .t In to her drow s, yna general el ce rt ic a dn ma ng et ci if e sdl w ih ch satis yf t eh bo dnu ary

noc ditions 6( .4) ac n be nu amb ugi ous yl deco opm sed otni some nil ear comb ni ation of a ll of the various op ss ible normal modes fo the ca iv ty. S cni e ae ch on rmal edom osc lli a et s at a spec cifi rf equency it is clear that fi we a er vig en the elec rt ic a dn ma ng etic if e dl s ni s di e the ca iv ty ta t emi neht t eh s bu seque tn beha iv o ru of t eh if e dl s is inu que yl de et rmined rof a ll t mi e.

T eh cond cu t gni wa ll s rg adua yll abs bro ene ygr rf om the ac iv ty, due ot the ri inif te resist ivi ty, at a etar w ih ch can sae yli be ca lucl a det . For etinif eht sma ll ta gn ential co pm one tn of at ht e wa ll s ac n be estima det us gni qE . 6( .5):

5( )6


oN w, the ta gn ential co pm onent of at the wa ll s is s hgil t yl d ffi ere tn rf om that vig en by the di eal so ul tion. However, iht s is a sma ll e ff ce t a dn can be ne lg ec det ot

l ae d gni o dr er ni . hT e t mi e avera deg ener yg xulf otni the wa ll s is vig en by

75( )

hw e er is the p ae k va ul e of the ta gn ential ma ng etic if eld at the wa ll s predicted

yb the di eal so ul tion. A cc ro d gni ot the bo dnu ary condition 6( . )4 ( ,)d is equal ot

ht e p ae k va ul e of t eh surf eca curre tn density . tI is help luf ot de enif a surf ca e resistan ec ,

)85(

hw e er

)95(

T ih s a orpp ach makes ti clear t tah the di ss ip ita on of ener yg is due ot o cimh h ae ting ni a t nih layer, w oh es t ih ckne ss is of ro der the s nik depth, on the surf eca of t eh

noc duct gni wa ll s.

T eh qua il ty f ca tor of a res no a tn c iva ty T eh qua il ty f ca tor of a res no a tn ac iv ty is de denif


)06(

roF a spec cifi normal dom e of the ca iv ty t ih s quantity is pedni e dn e tn of the edom am edutilp . By con es rvation of ener yg the wop er diss pi a det ni o cimh ol ss se si

unim s the r eta of cha gn e of t eh s rot ed ener yg . eW ac n tirw e a ffid erential uqe ation rof the beha iv o ru of as a nuf ction of t mi e:

)16(

hw e er is the osc lli ation rf equency of the on rmal mode ni question. The so ul tion ot ht e abo ev equation si

)26(

T ih s time depe dn ence of the stored ener yg s ggu ests that t eh osc lli ations of t eh if e dl s ni the ca iv ty are da pm ed as fo woll s:

)36(

hw ere we have a woll ed of r a s tfih fo the er sona tn rf equency as we ll as t eh damp ni g. A da pm ed osc lli ation such as t ih s od es ton consist of a p ru e rf equenc .y

Ins et ad, ti is dam e pu of a s pu e opr sition of rf equencies aro dnu . tS a dn a dr Fourier ana yl sis iy e sdl


)46(

hw e er

)56(

tI fo woll s t tah

)66(

T eh resonan ec shape has a lluf w di th at ha fl -maximum equal ot . roF a

noc sta tn ni p tu vo tl age, the ener yg of osc alli tion ni the ca iv ty as a nuf ction of rf equency fo woll s t eh resona cn e cur ev ni the ne hgi bourho do of a partic lu ar

res no a tn rf equency. It ac n be seen that the imho c ol ss e ,s ihw ch de et r nim e rof a partic lu ar mode, also deter nim e the ma mix um amp il t du e of the osc lli ation when ht e resonance condition is ex ca t yl satis if ed, as we ll as ht e w di th of the resonance

(i.e ,. oh w far o ff the resona tn rf equency the system can be dr vi en and st lli iy eld a cifingis a tn so c lli at noi amp il t .)edu

C dnily r ci al ca iv ties Let us app yl the methods of the pre iv ous se itc on ot the TM dom se of a r thgi

ric c lu ar c dnily er of rad sui . We can wr eti


hw e er sa it s if es the equation and are c nily drical polar oc ord ni a et .s Let

tI fo woll s t tah

ro

hw e er . hT e abo ev uqe ation is nk own as eB ss 'le s e uq ation. The owt nil ae rly

pedni e dn e tn so ul tions of Be ss e 'l s equation a er de deton and . nI t eh


li im t ht ese s ulo tions behave as and , er s vitcep e yl , ot wol se t dro er . M ero ex ca t yl 61

of r , hw e er

and is E lu er's consta tn . Cl ae r yl , t eh are we ll behaved in ht e li tim , hw e er as eht ra e ba yld beha .dev

T eh as my p ot tic beha iv o ru of ob th so ul tions at lar eg is


T uh ,s for the s ulo tions kat e the of rm of rg adua yll d ace gniy osc lli atio sn

ihw ch a er ni phase q ardau t ru e. T eh heb a iv o ru of and is s oh wn in iF g. .12

rugiF e 21: The Be ss el cnuf tio sn (s dilo enil ) a dn od( t det nil e) S ni ec the a six is ni c dedul ni the ca iv ty ht e radial e gi e nufn ction um st eb

re ralug at the or nigi . T ih s i mm ediate yl r lu es o tu t eh s ulo tions. T uh ,s eht om st general s ulo tion for a TM mode is

The are ht e e gi e vn a ul es of , a dn a er de et rmined by the so ul tions of

T eh ba ove c no stra tni ens ru es t tah the gnat ential el ce tric if e dl is z ore on t eh

noc duct gni wa ll s s rru o nu d gni the ca iv ty ( .)


T eh om st general so ul tion rof a ET mode si In t ih s ac se, eht are de et r denim by t eh so ul tion of

hw e er de eton s d ffi erent ai tion with respe tc ot the ar mug e tn . The ba ove constra tni ne s ru es that the normal ma eng tic if e dl is ze or on the conduct gni wa ll s s rru o nu ding ht e ca iv ty. The osc lli ation rf equency of both the TM a dn TE dom es is vig en by

If is the ord ni al nu bm er of a ez or of a partic lu ar Be ss el nuf ction of dro er ( ni cr ae ses iw th rcni ae s gni va ul es of the argume )tn , then ea hc mo ed is char ca et rized

yb t erh e tni egers, , , , sa ni the r ce tang lu ar ac se. The th z re o of is

vnoc entiona yll de deton [so, ]. kiL ewise, the ht ez ro of is denot de . Ta lb e 2 s woh s eht rif st wef ez ros fo , , , a dn . tI is cl ae r ht at for dexif and ht e wol est erf quency edom (i.e., t eh dom e with t eh wol est

va ul e of ) is a TE dom e. The dom e with the next gih hest rf equency is also a TE mode. The ne tx ehgih st rf equency dom e is a TM dom e, a dn so o .n


Tab el 2: T eh rif st few va ul es of , , a dn

1 2. 404 8 3. 7138 0. 0000 1. 2148

2 5. 025 1 7. 6510 3. 7138 5. 4133

3 8. 356 7 01 . 371 7. 6510 8. 3635

4 11 . 97 2 31 . 423 01 . 371 11 . 607 • aC iv ty res no a rot s a er enc ol sed metal boxes.

El ce rt oma ng etic if e dl s are co denifn ni s edi the boxes . aR aid tion a dn gih h-resistance

effe cts are e nimil a det , er s lu t gni ni a very hgih Q (qua il ty fa tc or)

A er cta ralugn wave ediug with both e dn s (z=0 a dn z=d) c ol sed by a conduct gni wa ll

( rugiF e 9- )8 : multiple re elf ctions a dn s nat d gni waves


seR no a tn rf equency of r ce ta lugn ar ca iv ty res no ator

Degenera et dom es : d ffi ere tn dom es ha niv g ht e sa em resona tn rf equenc .y

Do nim a tn mode : the dom e with the wol est res no a tn rf equency rof a vig en ca iv ty size . A er sonator is a de iv ec or system that e tibihx s er s no ance or resona tn beha iv or, that i ,s ti nat ru a yll osc lli a et s at some erf quenc ei s, ac ll ed ti s resona tn rf equencies, with

rg e reta amp il t du e than ta other .s T eh osc lli ations ni a er sona ot r can be either le e tc roma ng etic or mecha in cal ( ni c ul d gni ca uo stic). eR s no ators are used ot eit reh

genera et waves of sp ce cifi rf equencies or ot sel ce t sp ce ifi c rf equencies rf om a s ngi al. Musical ni strume tn s use ca uo stic res no ators ht at orp du ec so dnu waves of sp ce ifi c tones. A ca iv ty res no a rot , usua yll used ni referen ec ot el ce rt oma ng etic reso tan ro ,s is o en ni w ih ch waves e ix st ni a ho woll sp eca ni s edi the de iv ec . Acoustic ca iv ty

res no at ro ,s ni w ih ch so nu d is rp odu ec d yb a ri rbiv at gni ni a ca iv ty with o en po e gnin , a er k on wn as Hel hm o tl z res no a ot rs.

aC iv ty res no a srot

A ca iv ty resonator is a ho ll wo conduc rot b ol cked at both e dn s a dn alo gn w ih ch an le e tc roma ng et ci wave ac n be s oppu r det . tI ac n eb iv ewed sa a wave diug e short- ric c detiu at ob th e dn s (s ee Mic wor ave ca iv ty).

T eh ac iv ty ah s etni r roi surf eca s ihw ch re lf ect a wave of a spec cifi rf equency. When a wave that is er sona tn with the ca iv ty e tn ers, ti bo nu sec ab ck a dn rof th wit nih t eh ca iv ty, with wol ol ss (see s nat d gni wave). As rom e wave ener yg e retn s the ca iv t ,y ti comb eni s iw th a dn re fni or ec s ht e stand gni wave, ni c er as gni ti s tni ensit .y

T eh ac iv ty ma ng etron is a av cuum t bu e iw th a lif ame tn ni the ce tn er of an

ave cua det , debol , c ri c lu ar ca iv ty resonat ro . A pe pr endic lu ar ma ng etic if eld si


i opm sed by a permane tn ma teng . The ma ng etic if e dl causes ht e el ce trons, a tt r ca det ot the r( elat vi ely) op sit vi e o etu r part of the cha bm er, ot sp ri al o wtu ard ni a

c ri c lu ar pa ht rather than mo gniv d ri ce t yl to t ih s a don e. apS ec d ba o tu the r mi of t eh hc a bm er are c dnily r ci al ca iv tie .s The ca itiv es are open alo gn the ri le gn th a dn

so nnoc ce t the co mm on ca iv ty sp eca . As elec rt ons sw ee p past ht ese po e gnin s

they ni du ec a er sona tn hgih rf equency ar d oi if e dl ni the ca iv ty, w ih ch ni turn causes

t eh le e tc rons ot b nu ch ni to rg o pu s. A rop tion of t ih s if e dl is e rtx ca det with a

sho tr tna e nn a that is co nn ce det ot a wave diug e (a metal t bu e usua yll of er cta lugn ar orc ss

s ce it o )n . The wave diug e d ri ce ts the extr ca ted RF ener yg ot the daol , w ih ch may eb

a c oo k gni cha ebm r ni a im c wor ave oven ro a hgih ga ni a tn e ann ni the ac se of ra .rad T eh k yl s rt on, t bu e wave ediug , is a beam t bu e ni c ul d gni at l ae st wt o apert ru ed ca iv ty resona rot .s The beam of charged particles pa ss es rht o gu h the pa ert ru es of ht e resona rot ,s o tf en t nu able wave re elf c it on gr di s, ni succe ss ion. A

co ll ce rot le e tc r do e is pro div ed to tni erc tpe the beam a retf pa ss gni t rh o hgu the er sona rot s.

T eh rif st resona ot r causes b nu c gnih of ht e partic el s pa ss gni t rh o hgu ti . T eh

nub ched partic el s rt avel ni a if eld- erf e re noig where ruf ther b nu c gnih co c ru s, then ht e b nu ched partic el s e etn r t eh s ce ond reso tan or gnivig up their ener yg to exc eti ti

otni osc alli tion .s tI is a pa itr cle a cc elera ot r that krow s ni conj nu ction with a sp ce ifi ca yll t nu ed ac iv ty by the co arugifn t noi of the struct eru .s nO the bea nilm e of na cca le era rot system, ht ere ra e sp ce ifi c s ce it ons ht at are ca iv ty resonat ro s rof

R .F T eh re lf ex k yl s rt on is a k yl stron ut ili z gni ylno a s elgni apert ru ed ca iv ty resonator

rht o hgu w ih ch the beam of charged partic el s pa ss se , rif st ni o en d eri ction.

EGELLOC GNIREENIGNE DACS A repe ll er el ce t dor e is pro div ed ot repel o( r re rid ce )t the beam a tf er ap ss ga e t rh ough ht e resona ot r b ca k t rh o hgu the resonat ro ni the other d ri ce tion a dn ni p por er

phase ot re fni or ec ht e osc lli ations set pu ni the er s no at ro .

In a laser, thgil is amp eifil d ni a ca iv ty reso tan or w ih ch is usua yll co opm sed of two

ro rom e im r ror .s T uh s an po tical ca iv ty, also nk own as a res no ator, is a ca iv ty with wa ll s w ih ch re elf ct el ce troma ng et ci wa ev s ( )thgil . T ih s w lli a woll stand gni wa ev modes ot e ix st with il tt le ol ss tuo s di e the ac iv t .y

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T RAN SMISSI ON LIN ES AND WA VE GU ID ES INT …scadec.ac.in/upload/file/EC6503-TLW LECTURE NOTES.pdfsca d engine e ring c oll eg e t ran smissi on lin es and wa ve gu id es un it