www.Vidyarthiplus.com

T eh a tt enuation consta tn f ro a rap tic lu ar propa ag tion om de in an o tp ical f bi er, the r ae l part of the a ix al pr po agation constant.

Phase consta 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 ff icie tn is the ima ig nary compone tn of the pr po agation constant f ro a plane wave. tI rper ese tn s the chan eg ni ph esa per metre alo gn the path rt ave dell by t eh wave ta any insta tn and is equal ot the ang lu ar wavenumber of the wave. tI is rper esent de by the sy bm ol β a dn is m ae s ru ed ni tinu s of radians per metre.

From the def ini tion of a gn ralu wavenumber;

T ih s quantity is often (strict yl sp ae king incorr ce t )yl abbre iv a et d ot wavenu bm er. Pr po erly, wavenumber is vig en by,

hw ich di ff ers from angular wavenumber only by a consta tn m lu tip el of 2π, ni the same way that a gn lu ar frequency di ff ers from frequency.

For a rt ansmission nil ,e the Hea iv s di e co idn tion of the tele rg apher's equation te sll us that the wave un mber must be p or portional ot frequency for the transmission of ht e wave ot be nu dis ot rted ni the time dom nia . T ih s include ,s but is n to limited ,ot ht e di eal case of a lossle ss line. The reason for t ih s condition can be seen by consideri gn t tah a usef lu s gi nal is com op sed of ma yn differe tn wavelengths ni the frequency doma ni . For there to be no dist ro tion of the wave of rm, a ll these waves must rt avel at the same ve ol city so that they arrive at the far end of the nil e ta the same time as a rg oup. S ni ec wa ev p ah se ve icol ty is given by;

it is rp oved t tah β is req riu ed to be pro op rtional ot ω. In et rms of primary oc e ff icie tn s of the nil e, t ih s iy elds from t eh telegrapher's equation f ro a id stortio eln ss line the condition;

www.Vidyarthiplus.com www.Vidyarthiplus.com

H wo ever, rp ca tical lines can o yln be expect de ot approximate yl m ee t t ih s condition over a lim deti f er quency band.

6. tliF e sr

T eh et rm propagation consta tn or propaga it on uf nction is app il ed ot f retli s a dn other two-po tr netwo kr s used f ro si ng al p or cessing. In these ac se ,s however, the tta e un ation and phase oc e ff icie tn s are expre ss de in et rms of nepers and radians per network section rather than per me rt e. Some authors make a id st ni ction eb tween per me rt e m ae s ru es (for w ih ch "constant" is u es d) and per sec it on m ae sures (for hw ich "f cnu tio "n is use )d .

T eh propagation consta tn is a usef lu conce tp ni f tli er design which vni ariably uses a ac scaded s ce it on topolo yg . In a ac s ac ded t po ology, the p or pa ag tion consta ,tn tta e un ation consta tn and phase consta tn of indi iv dual se itc ons may eb simply added to fi dn the tot al rp po agation constant e ct .

csaC da ed ne owt rks

T rh ee networks iw th arbitrary rp opagation c no sta tn s a dn impedan ec s conn ce det in cas ac de. ehT Zi et rms represe tn image impedance and it si a ss umed t tah co nn ce tions a er betw ee n matc gnih image impedances.

T eh ratio of ou pt tu ot inp tu vo tl age f ro ea hc network is ig ven by,

www.Vidyarthiplus.com www.Vidyarthiplus.com

T eh et rms are impeda cn e s ac gnil terms[3] and the ri use is expla ni ed ni the image impedance article.

T eh overa ll vo tl age rat oi is given by,

Thus for n ac s ac d de sec it ons a ll ha iv ng matc gnih impedances fac gni ae ch other, ht e overa ll propa ag tion consta tn is ig ven by,

7. retliF fundame tn als – Pa ss and tS op bands. f retli s of a ll t py es are re uq ired in a variety of app il cations from aud oi to RF a dn ac or ss the hw o el spec rt um of frequencies. As such RF f retli s form an i pm ortant le eme tn within a variety of scenar oi s, enabli gn the req riu ed frequenci se to be pa ss ed thro gu h the circ tiu , elihw re ej ct gni those that ra e n to needed.

T eh ideal f tli er, hw ether ti is a wol pa ,ss ih gh pa ss , or band pa ss f tli er w lli ex ih b ti on loss within the pass band, i.e. ht e frequencies below the c tu o ff frequency. Then ab vo e t ih s frequency ni what is et rmed ht e stop band the f retli w lli ejer ct all signals.

In rea il ty ti is n to possib el ot ac ih eve the perf ce t pa ss f retli and there is always some ol ss within the pa ss band, and ti is not possible ot achieve ni f etini re ej ction in ht e stop band. lA so there is a transition be ewt en the pass band and the stop ban ,d

www.Vidyarthiplus.com www.Vidyarthiplus.com

hw ere eht response cur ev fa ll s away, with the level of re ej ction ris se as the frequency mo ev s from the pa ss band to the s ot p ba dn .

Basic t py es of RF f li ter

There are four t py es of f retli that can be def ni ed. Each di ff erent type re ej cts or a ecc tp s s gi nals ni a di ff ere tn way, and by us ni g t eh co err ct t py e of RF f retli ti is po ss ible ot ecca tp the required signals and re ej ct those t tah a er not wa detn . T eh fo ru basic t py es of RF f retli are:

• woL pass f retli • hgiH pass f tli er • Band pa ss f li ret • Ba dn re ej ct f etli r

As the names of these types of RF f tli er indica et , a wol ap ss fi tl er o yln a ll ows frequencies below what is et rmed the cut o ff frequency through. T ih s can also eb tho gu th of as a ih gh r eje ct filter as ti rejects h gi h frequencies. Sim li ar yl a ih gh pass f retli o yln all wo s signals t rh o gu h above the c tu o ff frequency and re ej cts those below the c tu o ff frequency. A band pass f tli er a woll s frequencies throu hg within a ig ven pass band. niF a yll the band re ej ct fil ret re ej cts s gi nals wit nih a ec tr a ni ban .d tI can be ap rtic lu arly usef lu of r re ej cting a rap tic alu r nu wan et d signal or set of signals fa gnill wit nih a ig ven ba dn w di t .h

www.Vidyarthiplus.com www.Vidyarthiplus.com

www.Vidyarthiplus.com www.Vidyarthiplus.com

f retli f er quencies

A f retli a ll ows signals through ni what is term de the ap ss ba dn . T ih s is the ba dn of frequencies be ol w the c tu o ff frequency for ht e filte .r

T eh c tu off frequency of the f retli is defined as ht e p tnio at hw ich the outp tu level from the f tli er fa ll s to 5 %0 (-3 dB) of the ni band level, assuming a constant ni put level. The cut o ff frequency is sometimes referred to as the half power or -3 dB frequency.

The stop band of the f tli er is e ss entia yll the ba dn of frequencies that is re ej c det by ht e f retli . tI is kat en as s rat ti gn at the po tni where the f tli er r ae c eh s ti s req iu red level of re ej ction.

retliF classifi ac tions

retliF s can be des gi ned ot m ee t a av riety of req iu reme tn s. lA though using t eh same bas ci c ri c tiu conf gi ru ations, t eh circ tiu val eu s id ff er when the c ri c tiu is designed to meet di ff re ent cr ti er ai . In band ripple, fas et st rt ansition to the lu tima et ro ll o ff , ih g eh st o tu of band r eje ction are some of the cr eti ria that res lu t ni di ff ere tn c ri c iu t values. These di ff ere tn f retli s ra e given names, cae h one be gni op it mised for a diffe er nt le ement fo ep r of rma ecn . T erh e c mmo on t py es of f retli are vig en below:

• B ttu er ow tr :h T ih s type of f retli pro div es the ma ix mum ni band flatness. • Be ss el: T sih f li t re provi ed s eht optimum in-ba dn phase er sponse dna ht erefore also pro div es eht best s et p response. • Chebychev: T ih s f etli r provides fast ro ll o ff af et r the cut o ff frequency is r cae hed. oH wever t ih s is at the e px ense of ni band r pi ple. The m ro e ni ba dn ripple that can be lot era det , ht e fas ret the ro ll o ff . • E ill pt ci a :l T ih s has sig in fica tn levels of ni band and o tu of band ri pp le, and as e px ce det the h gi her the degr ee of r pi ple ht at can be tolera et d, the s et eper ti r ae ches ti s lu tima et ro ll o ff .

mmuS ary

RF filters are w di e yl used ni RF design a dn in a ll manner of RF and analog eu ric c tiu s ni general. As they a ll ow thou hg no ly partic lu ar frequencies or bands of frequencie ,s they are an essential ot ol f ro t eh RF d se ign en nig ee .r

www.Vidyarthiplus.com www.Vidyarthiplus.com

8. C no sta tn k f retli

C no sta tn k f tli ers, also k-t py e f retli s, are a pyt e of el ce rt o cin f retli designed using ht e image meth do . They are the ori ig nal a dn simplest f retli s p or duced by t ih s met oh dology and consist of a ladder ne wt ork of ident ci al sec it ons of ap ss vi e compone tn s. iH s ot rica yll , they a er the first f retli s that could appr ao ch the di eal f etli r f er quency response to wit nih any prescribed limit with the addition of a su ff icie tn number of s ce tion .s oH wever, they are rarely cons di re ed of r a modern design, the principles be ih nd them ha iv ng b ee n superse ded by other meth do olo ig es hw ich are more a cc ura et ni the ri prediction of f tli er response.

Term ni olo yg

Some of the impedance et rms and s ce it on et rms used in t ih s article are pict ru ed in ht e diagram be ol w. Image theory def ni es quantities ni et rms of an ni f etini ac s ac ed of two-po tr s ce tions, and in t eh case of ht e f retli s being discu ss ed, na inf etini la dd er net ow rk of L-section .s Here "L" should not be confused with the inductance L – ni el ce tro in c filter ot polo yg , " "L ref re s ot ht e s ep cif ci f tli er s pah e w ih ch resemb el s i vn erted le tt er " "L .

T eh se itc ons of the hypothet ci al infin ti e f tli er are made of series eleme tn s having impedan ec 2Z and s uh tn eleme tn s with adm tti an ec 2Y. The fac rot of two is tni or duced for mathematical conve in ence, s ni ec ti is usual to work in et rms of half-

www.Vidyarthiplus.com www.Vidyarthiplus.com

s ce it ons where it disa pp ae r .s The image impeda cn e of the inp tu and outp tu po tr of a s ce it on w lli general yl not be the same. H wo ever, of r a dim -ser ei s sec it on (that is, a s ce it on rf om halfway throu hg a series eleme tn to halfway t rh ough the next es ries le eme )tn w lli have the same ima eg impedan ec on both po tr s due to sy mm etry. T sih image im ep dance is designated ZiT due ot the " "T topology of a mid-series s ce it on. kiL ewise, ht e image impedan ec of a m di -shu tn s ce it on is desi ng a det Z Πi d eu ot t eh "Π" topology. Half of such a "T" or "Π" se itc on is ca ll ed a half-s ce tion, w ih ch is la so an L-s ce tion b tu with half the element va ul es of the f llu L-s ce tion. The image impedan ec of the half-section is di ss im li ar on the inp tu and outp tu ports: on the s di e present ni g the series eleme tn ti is equal to the mid-series ZiT, b tu on the s di e rp esenting the sh tnu eleme tn ti is uqe al to eht mid-sh tnu ZiΠ . hT ere ra e thus two varia tn ways of us gni a half-s ce tion.

Der vi ation

oC nstant k low-p ssa f tli er ha fl section. Here ni ductan ec L is equal Ck2

C no stant k band-pass if lter half section. 2 2 L1 = C2k and L2 = C1k

www.Vidyarthiplus.com www.Vidyarthiplus.com

Ima eg i pm edance ZiT of a constant k p or totype low-pa ss f retli is p ttol ed vs. frequency ω. T eh impedan ec is purely resis vit e (rea )l below ωc, and pure yl r cae t evi

(ima ig nar )y above ωc.

T eh b liu di gn b ol ck of consta tn k f retli s is the half-s ce tion "L" network, co pm osed of a se ir es impedance Z, na d a s uh tn admi tt a cn e Y. The "k" ni "consta tn "k is ht e value ig ven by,[6]

Thu ,s k w lli have un ti s of im dep ance, that i ,s ohms. It is er adi yl appare tn that in order of r k to eb consta tn , Y must be ht e dual impedance of Z. A physical tni erpre at tion of k can be given by bo ser niv g that k is the limiting value of Zi as eht si ez of the section ( ni et rms of va ul es of ti s compone tn s, such as inductan sec , cap ca ti ances, e ct .) a pp roaches ez ro, w lih e ke pe ing k at ti s ini tial va ul e. Thu ,s k is ht e char ca teristic impedance, Z0, of the transmission l ni e that would be formed by ht ese ni f ini tesimally sma ll s ce tions. I t is also the image i pm edance of the s ce tion at res no anc ,e in the ac se of band-pa ss f tli er ,s or at ω = 0 in the ac se of ol w-pass f retli s.[7] F ro example, t eh pictured ol w-pass half-section ah s

.

www.Vidyarthiplus.com www.Vidyarthiplus.com

leme tn s L a dn C can be made arb rti ar yli sma ll w elih reta ini ng the same va eul of k. Z and Y however, are ob th pa p or aching ez or , a dn rf om the of rm alu e b( e ol w) for image impedan ec s,

.

Ima eg impedance

T eh image impedances of t eh se itc on are given by[8]

and

Provided that the f tli er od es n to con at ni a yn resistive eleme tn s, the ima eg impedan ec ni the pass b na d of the f tli er is pure yl real and ni the stop band ti si p ru e yl ima ig nary. F ro examp el , for ht e pict ru ed low-pa ss half-section,[9]

T eh rt ansition occurs at a uc t-o ff frequency ig ven by

Below t ih s frequency, ht e ima eg impedance is r ae ,l

Above the cut-o ff frequency the image impedan ec is ima nig ary,

T ar nsmission parame ret s

www.Vidyarthiplus.com www.Vidyarthiplus.com

T eh transf re uf nction of a consta tn k protot py e low-pass f tli er for a si lgn e half- s ce it on s oh wing a tt enuation ni nepers and hp ase change ni radians.

S ee also: Ima eg impedance#Transfer function

T eh transmission parame ret s f ro a general consta tn k half-s ce tion are vig en by 1[ 0]

and for a hc ain of n half-s ce tions

For t eh low-pa ss L-shape section, below eht cut-o ff frequency, the rt ansmission parame ret s are given by[8]

T tah i ,s the transmission is ol ssle ss ni the pass-b na d with o yln the phase of the signal chan ig ng. Abo ev t eh cut-o ff frequency, t eh transmission parameters are:[8]

Prot to ype transformations www.Vidyarthiplus.com www.Vidyarthiplus.com

T eh p er se etn d plots of image impedan ec , a tt e un ation and phase cha gn e correspo dn ot a ol w-pa ss p or totype f etli r section. The p or totype has a cut-off frequency of ωc = 1 ra /d s and a nominal im dep ance k = 1 Ω. T ih s is p or du ec d by a f retli half-section iw th inductan ec L = 1 henry and ac pac ti a cn e C = 1 far da . This p or totype ac n be impedan ec s ac led a dn frequency scaled ot the des ri ed val eu s. The low-pass protot py e can also be rt ansformed ni to ih gh-pa ,ss band-pa ss or band-stop t py es by ap ilp cation of s atiu ble f er quency trans of rm ita ons. 1[ 1]

csaC gnida s ce it ons

aG in response, H(ω) f ro a cha ni of n ol w-pa ss constant-k f tli er half-s ce tions.

Several L-sha ep half-s ce tions may eb ac s ac ded ot form a com op s eti f tli er. Like impedan ec must always f eca l ki e ni these comb ni ation .s There are t reh ef ro e t ow ric c tiu s that can be of rmed with two di entical L-shaped half-s ce tion .s Where a port of image impedan ec ZiT faces another ZiT, ht e section is ca ll ed a Π section. Where ZiΠ faces ZiΠ the se itc on so formed is a T s ce it on. Further additions of half-s ce tions ot either of t eh se section forms a ladder network w ih ch may sta tr and end with se ir es or sh tnu eleme tn s. 1[ 2] tI sho lu d be borne ni mind that the charac et ristics of the f li t re predic det by the image method are o ln y acc taru e if the sec it on is et rm ni a det with ti s ima eg impedance. T ih s is usua yll n to true of the es itc ons at eit eh r end, w ih ch are usually et rm tani ed with a f xi ed resistance. The uf rther the s ce tion is rf om the end of the f retli , the mo er ca c etaru the prediction lliw b ce ome, s ni ec t eh e ff ce ts of t eh et rm ani t gni impedances are mas dek by the tni erveni gn sections.[ 31 ]

www.Vidyarthiplus.com www.Vidyarthiplus.com

9. m- ed r vi ed f tli er m-der vi ed f tli ers or m-type filters are a type of ele rtc o cin f tli er designed using the image met oh d. They were vni e detn by Otto Zobel ni the ae r yl 1 29 0s.[1] T ih s f retli epyt w sa ro i ig na yll tni ended for use with telepho en m lu tiple ix ng and was an improveme tn on the e ix sti gn consta tn k epyt f etli r.[2] T eh ma ni p or blem being ad rd e ss ed w sa ht e n ee d ot ca ih eve a be tt er m ta ch of the f tli er i tn o the term ni ati gn impedan ec s. nI general, a ll fil ret s d se i ng ed by the ima eg method fa li ot vig e an axe ct m ta ch, b tu the m-t py e f retli is a b gi improveme tn with su ati ble choice of the parame ret m. The m-type f tli er section has a f ru ther adva tn age ni that there is a rapid transition from t eh cut-o ff frequency of t eh pa ss band ot a pole of tta enuation uj st ins edi the s ot p band. Des etip ht ese adva tn ages, ht ere is a rd awb ca k iw th m- epyt f tli ers; at frequencies past the pole of a aunett tion, the response s trat s ot rise ga a ni , and m-types have op or stop band re ej c it on. For t ih s r ae son, f retli s designed us ni g m-type s ce tions are often designed as composi et fi tl ers with a m xi t ru e of k- epyt a dn m-type s ce tions and di ff ere tn va ul es of m at di ff erent poi tn s ot get the po timum performance from both t py es. 3[ ]

Der vi ation

m-derived ser sei general fi tl er half se tc ion.

www.Vidyarthiplus.com www.Vidyarthiplus.com

m-derived shunt low-pa ss f tli er half s ce tion.

T eh b liu ding b ol ck of m-der vi ed f tli er ,s as iw th a ll image impedan ec f tli er ,s is t eh "L" network, ca ll ed a half-section and composed of a series impedance Z, and a s uh tn adm tti ance Y. T eh m-der vi ed f tli er is a derivat evi of the consta tn k f etli r. T eh s rat ting poi tn of the ed sign is t eh val eu s of Z and Y der vi ed rf om the consta tn k protot py e and are ig ven by

hw ere k is t eh nominal im dep ance of the f tli er, or R0. hT e des gi ren now multip il es Z a dn Y by an arb ti rary consta tn m 0( < m < 1). There a er two di ff ere tn kinds of m-der vi ed section; series and shu tn . To o niatb the m-derived series half s ce tion, ht e designer determ eni s the impedan ec that must be added ot 1/mY to make the image impedan ec ZiT ht e same as ht e image impedan ec of the ori ig nal consta tn k sec it on. rF om the general form alu of r ima eg impedance, the additional impedance req riu ed can be s oh wn ot be[9]

oT tbo ain the m-derived s uh tn half s ce tion, an adm tti an ec si a dd ed ot 1/mZ ot make the image imp de an ec ZiΠ the same as ht e image impedance of the ori ig nal half s ce tion. The additional admittance req riu ed ac n be shown ot be[ 01 ]

www.Vidyarthiplus.com www.Vidyarthiplus.com

T eh general arra gn eme tn s of these c ri c tiu s are shown ni the diagrams to the ri hg t la ong with a sp ce ific example of a low pass s ce it on.

A consequen ec of t ih s desi ng is that the m-der vi ed half section w lli match a k-type s ce it on on o en s di e o yln . lA so, an m-t py e sec it on of one value of m w lli n to match another m-type s ce tion of another value of m ex ec tp on the s di es which o ff er the Zi of the k-type.[ 11 ]

pO erating frequency

For t eh low-pa ss ha fl s ce tion shown, ht e uc t-o ff frequency of the m-t py e is the same as ht e k-ty ep and is given by

T eh pole of a tt enuation occurs a ;t

From t ih s ti is cl ae r that sma ll er values of m w lli rp oduce c ol ser to the cut-off frequency and hen ec w lli have a sha pr er cut-o ff . Desp eti t ih s cut-o ff , ti also brings the nu wa tn de s pot band response of the m-type closer ot the cut-o ff frequency, making ti m ro e di ff ic tlu f ro t ih s ot be f deretli with subseque tn s ce tions. T eh value of m chosen is usually a co rpm omise be ewt en th ese conf il cti gn requirements. There is also a pr ca t ci al limit ot oh w sma ll m can be ma ed due ot the ni here tn resistan ec of the induc ot r .s T ih s has ht e e ff ce t of causi gn t eh po el of tta e un ation ot be le ss d ee p (that is, ti is no ol gn er a gen niu ely i fn in ti e pole) and the slope 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 gu ht olc ser ot , and ht ere ceases ot be

Ima eg impedance

www.Vidyarthiplus.com www.Vidyarthiplus.com

m-der vi ed prototype shu tn ol w-pa ss f retli ZiTm image impedan ec f ro various values of m. Va eul s below cut-o ff frequency on yl s oh wn for clarity.

T eh fo ll owing expressions of r image impeda cn es are a ll referenced ot t eh low- ap ss protot py e s ce it on. They a er scaled ot the nominal im ep dance R0 = 1, and ht e frequencies ni those expressions are a ll s ac l de ot ht e cut-o ff frequency ωc = 1.

Ser ei s s ce it ons

T eh image im ep dances of t eh se ir es s ce it on are vig en by[ 41 ]

and is the same as taht of the consta tn k sec it on

Sh tnu s ce tions

T eh image impedances of t eh s uh tn s ce it on are ig ven by[ 11 ]

and is the same as taht of the consta tn k s ce it on

www.Vidyarthiplus.com www.Vidyarthiplus.com

As with the k-type s ce tion, the image impedance of the m-type ol w-pa ss se itc on is p ru e yl er al below the cut-o ff f er quency and pure yl ima nig ary a ob ve ti . rF om t eh ahc rt ti can be seen that ni the passba dn the c ol sest impeda cn e match ot a constant pure resistan ec et rmination occurs at app or ximate yl m = 0.6. 1[ 4]

T ar nsmission parame ret s

m-Derived low-pass f retli transfer function f ro a s lgni e half-section

For an m-der vi ed section ni general the transmission parameters for a half-section ra e ig ven by 1[ 4]

and for n half-sections

www.Vidyarthiplus.com www.Vidyarthiplus.com

For the partic lu ar example of the low-pass L section, the rt ansmission parame et sr solve di ff erently ni thr ee frequency bands. 1[ 4]

oF r ht e rt ansmission is lo ss less:

oF r ht e rt ansmission parame ret s are

oF r eht rt ansmission parame ret s ra e

Pr oto type transformations

T eh p tol s s oh wn of image impedan ec , attenuation a dn p ah se change ra e the pl to s of a ol w-pa ss prototy ep f retli s ce tion. The p or totype has a uc t-off frequency of ωc = 1 ra /d s and a nom ni al impeda cn e R0 = 1 Ω. T ih s is p or du ec d by a f etli r half-s ce tion hw ere L = 1 henry a dn C = 1 farad. T ih s pr oto type can be impedan ec sca el d and frequency scaled ot the desired val eu s. hT e low-pass rp oto type ac n also be rt ansform de ni to high-pa ss , band-pa ss ro band-stop types by app il ac tion of s tiu able frequency rt ans of rmations. 1[ 5]

csaC gnida s ce tions

Several L half-s ce tio sn may eb ac s ac ded ot of rm a compos eti fil et r. iL ek impedan ec must always f eca l ki e ni these comb ni ation .s There are t reh ef ro e two ric c tiu s t tah can be formed with two di entical 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 tion formed is a T sec it on. uF rther additions of half-s ce tions to either of these forms a la dd er network hw ich may s at rt and e dn with series ro shunt eleme tn s.[ 61 ]

www.Vidyarthiplus.com www.Vidyarthiplus.com

tI sho dlu be born in mi dn that the charac et ristics of the f tli er predic det by the ima eg method are o yln ca curate if t eh s ce it on si et rm ni at de with sti image impedance. T ih s is usua yll not true of the s ce it ons at either end w ih ch are usually et rm tani ed with a f xi ed resistance. The uf rther the s ce tion is rf om the end of the f retli , the m ro e ca c ru a et t eh rp ediction lliw become s ni ec t eh e ff ce ts of t eh et rm ani t gni impedan ec s a er masked by ht e tni erve in ng s ce tions. tI is usual to provide half half-s ce tions at the ends of the fi tl er with m = 0.6 as t ih s va ul e vig se ht e f ettal st Zi ni the p sa sband and hen ec the best match ni to a resistive et rm ani tion.[17]

01 . C yr stal fil ret

A crystal filter is a sp ce ial form of quartz crystal used ni el ce tro in cs systems, in partic lu ar co umm in cations de iv ec .s tI pr vo i ed s a very precise yl defined ec ntre frequency a dn very s et ep ba dn pass hc ar ca teristic ,s ht at is a very ih gh Q f ca rot —far ih gher than can be bo niat ed iw th co vn entional lumped c ri c tiu s.

A crystal f tli er is very often fou dn ni the etni rmedia et f er quency I( F) stages of ih hg -qua il ty radio rece vi ers. Cheaper sets may use ec ramic retlif s (w ih ch also xe plo ti the pi ze oelectric e ff ec )t , or tuned LC circ tiu .s The use of a f dexi IF s at ge frequency a woll s a crystal f retli to eb used b ace use it has a very rp eci es f exi d frequency.

T eh most co mm on use of crystal f retli s, is at frequencies of 9 M zH or 10.7 MHz to provide sel ce t ivi ty ni co umm cin ations re ec revi s, or at higher frequencies as a roofing f tli er in r ece vi ers using up-convers noi .

eC ram ci f retli s te dn to be used at 10.7 MHz ot prov di e s le ect ivi ty ni b or ad ac st FM r ece vi er ,s or at a wol er frequency 54( 5 kHz) as ht e s ce ond tni ermedia et frequency f retli s ni a co mm u in cation er ceiver. eC ramic f retli s at 455 kHz can achieve sim li ar ba dn widths to crystal fi retl s at 01 .7 MHz.

www.Vidyarthiplus.com www.Vidyarthiplus.com

UN TI V

WAV GE U DI ES aW veg diu es are basica yll a de iv ec ("a diug e") of r transp ro ti gn el ce rt oma ng etic ene rgy rf om one region ot another. Typica yll , waveg ediu s a er ho ll ow metal tubes (often r ce tan lug ar ro c ri c lu ar in cro ss es itc on). hT ey are capab el of direct ni g p wo re rp ecise yl ot where ti is n ee ded, can handle large amou tn s of power and uf cn tion as a gih h-pa ss f retli .

T eh wavegui ed a tc s as a ih gh pa ss f etli r in that most of the e en rgy above a ec rta ni frequency (the cuto ff frequency) w lli pa ss through t eh waveg iu de, wher ae s most of ht e e en rgy that is be ol w the cuto ff frequency w lli be a tt e un a et d by the waveg ediu . aW veg diu es are often u es d at microwa ev frequencies (gr ae ter than 03 0 MHz, with 8 GHz a dn above be ni g more co mm on .)

aW veg diu es are iw deband de iv ces, dna c na ac rry o( r transmit) either op wer or co umm cin ation si ng al .s An example of a ho ll ow metal r ce ta lugn ar wavegui ed si s oh wn ni the fo ll owing f gi ure.

www.Vidyarthiplus.com www.Vidyarthiplus.com

aW veg diu es ac n bend if t eh des ri ed ap ilp cation req riu es ,ti as s oh wn ni the fo oll wing Figure.

T eh ba ove waveg iu des c na be used with wavegui ed ot coaxial ac ble ada tp er ,s as s oh wn ni the ne tx F gi ure:

eW on w k on w what a wave ug ide is. Lets exam ni e metal ca iv ties with a r ce tang lu ar rc o ss s ce it on, as s oh wn in iF ug re 1. Ass mu e ht e wave ug i ed is f delli with v ca cuum, ria or some diel ce tric with the permeab ili ty vig en by a dn the perm ti t ivi ty vig en yb .

T eh waveguide has a width a ni the x-direc it on, a dn a heig th b ni the y-d eri ction, iw th >a b. hT e z-a ix s is eht d ri ce tion ni w hcih the waveg diu e is ot ac rry op wer. www.Vidyarthiplus.com www.Vidyarthiplus.com

iF g eru 1. Cro ss se itc on of a wavegui ed with long id mension a and sho tr dimension b.

On t ih s page, I'm going to give t eh general "r lu es" of r wavegu di se . T tah i ,s I' ll give ht e equations for key parame ret s and let you know wh ta the parame ret s mean. On ht e ne tx page, we' ll go ni to the mathema it ac l der avi tion (w ih ch you would od in en nig eering rg adua et cs hoo )l , b tu you can get away with ton know ni g a ll that math if you don't wa tn to know ti .

riF st and possibly om st importantly, t ih s wave ug i ed has a c tu o ff frequency, fc. The uc to ff frequency is the frequency at which a ll wol er frequencies are a tt e un ated by ht e wavegui ed , and above the cuto ff frequency a ll ih g reh rf equencies orp pagate

within t eh waveg iu de. The c tu o ff frequency de if nes t eh ih gh-pass fi tl er rahc ca teristic of the wave ug di e: above siht frequency, the waveg ediu pa ss es power, bel wo t ih s frequency the wavegu di e ta et nua et s ro blocks power.

T eh cuto ff frequency depends on t eh shape and si ez of the cro ss s ce it on of the wave ug ide. The larger the wave ug i ed is, ht e lower the cuto ff frequency f ro that wavegui ed i .s The form alu f ro the cuto ff frequency of a r ce ta gn ralu cross s ce tioned aw veg ediu is given b :y

In the above, c is the sp ee d of il g th within the waveg ediu , um is the ep rmeab ili ty of ht e ma et rial that f lli s the wavegui ed , and epsilon is the perm ti t ivi ty of the ma et rial ht at f lli s t eh wavegui ed . etoN ht at the c tu o ff f er quency is i dn epe dn e tn of the short length b of the waveg ediu . www.Vidyarthiplus.com www.Vidyarthiplus.com

T eh c tu o ff f er quency for a waveg ediu with a c ri cular cross sec it on of radius a is ig ven b :y

D eu to Maxwell's Equations, t eh fie dl s iw t nih the waveguide always have a sp ce ific "form" or "wavesh pa e" to them - these a er ca ll ed m do e .s Assume the wavegui ed is orient de such that the ener yg is ot be transm tti ed alo gn the wavegui ed a ix ,s the z-axis. The modes ra e classified as either TE ('transverse le e tc ric' - w ih ch ind ci a et s t tah t eh E-fie dl is orthogonal ot the a ix s of the wavegui ed , so that E =z )0 ro TM ('transverse magnetic' - which indi ac et s t ah t the H-field is ro thogonal ot the a ix s of the wavegui ed , so Hz = )0 . The om des are f ru ther classif dei as TE ji , where the i a dn j indica et the number of wave osc lli ations f ro a partic lu ar fie dl d eri ction ni the long d ri ection (dimension a ni iF g eru )1 and s oh tr dir ce tion (dimension b ni iF g ru e )1 , resp ce tive yl .

Me lat waveg ediu s ac nn to s ppu o tr the TEM ('transverse el ce tric a dn magnetic' - hw en Ez and zH are ez ro) mode. The ri e ix sts on solution ot Maxwe 'll s equations ht at also satisfy the req riu ed bou dn ary conditions f ro t ih s om ed to occ .ru

Maxwe ll 's Eq au tions are ton ae sy to solve. Hence, eve ry math tr kci someo en can iht nk of w lli be used ni order ot make ht e analysis tr ca at ble. We' ll s trat with id scussing the el ce tric v ce t ro p to ential, F. In a s uo rce-f er e region (i.e., an area ht ro gu h w ih ch wa ev s p or paga et ht at is away rf om so ru ec s), we know t ah t:

In the above, D is the El ce tric xulF Density. If a v ce t ro quantity is d vi ergenceless (as ni the above), then it can be express de as the curl of another quantity. T ih s means that we can wr eti t eh solution for D and the correspond gni ele tc r ci fie dl E as:

In the above, epsilon is t eh permitt ivi ty fo the medium thro gu h w ih ch the wave www.Vidyarthiplus.com www.Vidyarthiplus.com

propagates. We are purely in the world of mathematics now. The quantity F is not physical, and is of little practical value. It is simply an aid in performing our mathematical manipulations. tI tur sn o tu that waves ( ro le ectromagnet ci ener yg ) ac n not p por aga et ni a wavegui ed when both zH and Ez are qe ual ot ez .or Hence, ahw t field conf gi uratio sn t tah a er a woll ed w lli be classified as either MT (Transverse Ma ng etic, ni which H =z 0) a dn TE (Transverse El ce tric, ni which Ez= )0 . The r ae son that waves ac nnot be TEM (Trans ev rse Ele rtc omagnetic, H =z E =z 0) w lli be s oh wn towards the end of t ih s derivation.

oT perform our ana yl si ,s we' ll assume that E =z 0 (i.e ,. we are looking at a TE mode ro field config aru tion). In t ih s esac , ow rk ni g through Maxwe 'll s equations, ti can be s oh wn t tah the E- a dn H- fie dl s ac n be ted erm deni rf om the fo ll owing equations:

T eh re of re, if we can f dni Fz (the z-compone tn of t eh v ce tor )F , then we can fi dn ht e E- and H- fie dl s. In the a ob ve equation, k is the wavenumber.

Work ni g throu hg the math of Ma wx e ll 's Equation ,s it can be shown that ni a so ru ce-fr ee region, ht e v ce tor top ential F must satisfy the v ce tor wave equatio :n

[1]

oT br ae k t ih s equation down, we w lli look o yln at the z-co pm one tn of the ba ove equation (that i ,s Fz). eW lliw also a ss ume that we are look ni g at a sin elg frequency, so ht at the t mi e ped enden ec is assumed ot be of the form ig ven by (we ra e now usi gn phasors ot analy ez the uqe at oi n):

www.Vidyarthiplus.com www.Vidyarthiplus.com

Then the e uq ation [1] ac n be simp il f dei as f ollo ws:

[2]

oT solve t ih s eq au tion, we w lli use the et ch in q eu of separation of variable .s He er we a ss ume that the f cnu tion Fz(x, y, z) can be wr tti en as the rp oduct of three f nu ctions, each of a si lgn e variable. T tah is, we assume that:

[3]

(You mig th ask, how do we know that t eh separation of variab el s assumption ab vo e is valid? We don't - we just a ss ume ti s corre tc , dna if it so vl es the id ff erential equation when we are done doing t eh analysis then the assumption is va dil ). Now we p ul g ni our assumption for Fz (equation [3]) ni to equation [2], and we end up wit :h

[4]

In the above equation, the prime re rp ese tn s the der vi at vi e with respect to the var ai b el ni the equation (for instan ec , Z' re rp ese tn s the deri av tive of the Z-function iw th resp ce t to z). We w lli rb ae k up the variable k^2 ni to co pm one tn s (aga ni , just ot make our math ae sie )r :

[5]

Using equation [5] to b er akdown equation [4], we ac n wr ti e:

www.Vidyarthiplus.com www.Vidyarthiplus.com

[6]

T eh r ae son that the equations in [6] are va il d is b ce ause they are o ln y uf nctions of dni epende tn variables - hen ec , ea hc eq au tion must hold f ro [5] ot be tr eu eve rywhere in the waveg diu e. Sol iv ng t eh above equations us ni g ordinary id ff erential equations theory, we eg t:

[7]

T eh form of the solution ni the abo ev equa it on is di ff erent for )z(Z . The r ae son is ht at both forms (that f ro X a dn Y, and that for )Z , are both equa yll va dil solutions f ro the di ff erential equations ni equation [6]. However, ht e complex exponential typica yll represe tn s rt ave ill ng wave ,s and eht [rea ]l s uni so di s re rp ese tn standing waves. Hence, we choose the forms given ni [7] f ro the so ul tion .s No math r elu s ra e violated here; aga ni , we are just choos ni g forms t tah w lli ma ek our ana yl sis easie .r

For now, we can set c5=0, because we wa tn to anal zy e waves p or pa ag ting ni the z+ -dire tc ion. The ana yl sis is identical of r waves p por agat ni g in the -z-d ri ce tion, so iht s is fair yl arbitrary. The s ulo tion of r Fz can be wr tti en as:

[8]

If you remem reb anyt ih ng abo tu di ff erential equations, you k on w there en eds ot be some bo nu dary conditions app il ed ni o dr er to de et rm ni e the consta tn s. aceR ll ing our p yh sic ,s we k on w that the tangential El ce tr ci fie dl s at a yn perfe tc conductor must be ez or (w yh ? because , so if the conducti iv ty app or aches inf ini ty www.Vidyarthiplus.com www.Vidyarthiplus.com

(perfect conductor), then if the tangential E-field is not zero then the induced current would be infinite). T eh tangential fie dl s must be ez ro, so Ex must be ez or when y=0 and when y=b (see Figure 1 abov )e , no ma tt er what the va ul e for y and z are. I n addition, Ey must be z ore hw en x=0 and when x=a (independe tn of x a dn z). eW lliw calc lu a et E :x

www.Vidyarthiplus.com www.Vidyarthiplus.com

Ex is g vi en by the above equation. The bou dn ary condition given by

Ex( x, y=0, z)=0 [9] imp eil s t tah c4 must eb equal to ez ro. T ih s is the o yln way t tah bo nu dary condition ig ven ni [9] will be tr eu f ro a ll x and z position .s If you don't be il eve t ih s, try to s oh w t tah ti is ni co rr ce t. You w lli quickly de et rm eni that c4 must be zero for the b nuo ad ry condition in [9] ot be sa it sfied everywhe er ti is req riu ed.

Ne tx , the sec no d boundary condition,

Ex(x, y=b, z)=0 [ 01 ] imp eil s somethi gn very unique. The o yln way for the c no dition ni [10] ot eb true f ro a ll va ul es of x a dn z whenever y=b, we must have:

If t ih s is ot be true everywhe er , c3 could eb ez ro. However, if 3c is ez ro (and we have a rl ae dy eted rm ni ed that c4 is ez ro), ht en a ll of the fie dl s would end pu being ze or , b ace use the function Y(y) in [7] would eb ez ro everywhere. Hen ec , 3c ca nn ot be ez or if we ra e l oo king for a non ez ro solution. He cn e, ht e o ln y a etl r an t evi is if ht e above equation imp il es that:

T ih s last equation is fu dn ame tn al ot dnu erst na ding wave ug ide .s tI s tat es t ah t the no ly solutions for Y( )y uf nction must end up bei gn sinusoids, that an ni teger number of m lu tiples of a half-wavelength. These are the o yln type of f nu ctions that

www.Vidyarthiplus.com www.Vidyarthiplus.com

sa it sfy the di ff erential equation ni [6] and the required bo nu dary condition .s T ih s is na e rtx eme yl i pm o tr a tn conce .tp

If we invo ek our other two bou dn ary conditions:

Ey(x=0, y, z)=0

Ey(x a= , y, z)=0

Then (us ni g di entical er asoni gn to that ab vo e), we can ted erm ni e that 2c =0 and aht t:

T ih s s at teme tn imp il es that t eh o yln func it ons of x t tah satisfy the di ff re ential equation and the requ ri ed bo nu dary conditi no s must be an i tn eger m lu tiple of half- sinusoids wit nih the wavegui ed .

Combi in ng these resu tl s, we ac n wr eti the s ulo tion for zF as:

In the above, we ha ev comb ni ed the rema ni i gn non ez ro consta tn s 1c , c3, and c6 otni a s ni elg consta ,tn ,A f ro simp il cit .y We have fo dnu that only ce tr ain id stributions (or field confi ug rations) w lli sa it sfy the required d ffi erential eq au tio sn and the bo nu dary condition .s Each of these fie dl conf gi urations w lli be known as a mode. B ce ause we derived the res tlu s abo ev for the ET ac se, the modes w lli be known as TEmn, where m ni di ac et s the nu bm er of half-cyc el av riations within the wavegui ed f ro X(x), a dn n indi ac et s the number of half-cycle var ai tions within the wavegui ed of r Y( .)y

Using the field relationships:

www.Vidyarthiplus.com www.Vidyarthiplus.com

eW can wr ti e the a woll able fie dl configurations for the TE ( rt ansverse el ce tric) modes iw t nih a wave ug ide:

www.Vidyarthiplus.com www.Vidyarthiplus.com

In the ba ove, the consta tn s are written sa Amn - t ih s imp eil s that t eh amp il tude for e ca h m do e can be indepe dn e tn of the others; however, the field com op ne tn s for a sin lg e mode must lla be re detal t( h ta is, xE and yH do n to have independent oc efficients).

C tu o ff rF equency (fc)

tA this po tni ni the ana yl sis, we a er able to say somethi gn i tn e gill e tn . aceR ll that ht e compone tn s of the wavenumber must sa it sfy the relations ih p:

[3]

S ni ec kx a dn ky are res rt a ni ed ot o yln take no ec tr a ni value ,s we ac n plug t ih s fact i :n

[4]

nA i tn eresting question arises at t ih s po ni t: What is the lowest frequency in w ih ch ht e waveg ediu lliw pr po a etag the TE dom e?

For pr po agation to o cc ,ru . If t ih s is true, then kz is a real number, so that ht e field compone tn s (equations [1] and [2]) w lli conta ni complex exponentials, hw ich repres tne pr po agating waves. If on the other hand, , ht en zk will be na ima ig nary number, ni w ih ch ac se the complex e px onential above ni equations [1-2] b ce omes a deca iy ng real e px onen it al. In t ih s ac se, ht e fie dl s lliw not p or paga et b tu ni s et ad q iu ck yl die o tu within the waveguide. Ele tc r mo agnet ci fields ht at die o ff ni s et ad of propaga et ra e referred to as evanescent waves.

oT f dni the lowest rf equency ni w ih ch propagation can occur, we set k =z 0. T ih s is ht e rt ansition be wt een t eh cuto ff region (evanesce tn ) and the p or pagation re ig on. Sett gni k =z 0 in equation [4], we atbo i :n

www.Vidyarthiplus.com www.Vidyarthiplus.com

[5]

If m a dn n are both ez or , then a ll of the fie dl compone tn s ni [1-2] become ez or , so we cannot have this condition. The lowest value the left hand s di e of equation [5] can take occurs when m=1 and n=0. The s ulo tion to equation [5] when m=1 a dn n=0, vig es the cuto ff f er quency of r t ih s waveg diu e:

Any frequency be ol w the cuto ff frequency (fc) w lli only res tlu ni evanesce tn or d ace iy ng modes. The wave ug ide w lli not rt anspo tr energy at these frequencie .s In ad id tion, if the wave ug ide is operating at a frequency just above fc, then t eh only mode that is a p or pagat ni g om de w lli eb the TE10 om de. llA other om des w lli be d ace ying. Hen ec , the TE10 om ed , s ni ce ti has the ol west cuto ff frequency, si referred to as eht dom ni a tn mode.

E ev ry mode ht at can e ix st iw t nih the wave ug i ed has ti s own cuto ff frequency. That i ,s for a ig ven mo ed to propaga et , the po er gnita frequency must eb above the cutoff frequency for that m do e. By sol iv ng [5] ni a more general form, the cuto ff frequency for the TEmn m do e is given by:

lA though we haven't discu ss ed the TM (transverse magnetic) om de, ti w lli turn out ht at the domina tn TM m edo has a higher c tu o ff frequency than the domina tn TE mode.

De et rm ni i gn the fields for the TMz (Transverse Magnet ci to the z d ri ce tion) om des fo woll s a sim li ar pro ec d ru e ot that f ro the ET z ac se. oT begin, we' ll s trat by id scussing t eh ma ng etic v ce rot p to ential, A. T ih s is a non-physical quantity that si often in used tna enna theory ot simp il fy the mathematics of Maxwe 'll s Equations.

www.Vidyarthiplus.com www.Vidyarthiplus.com

oT nu derstand t eh magnet ci v ce t ro p to e itn al, n eto that s ni ce the ma ng etic flux density B must always be diverge cn ele ss :

If a v ce tor quantity is divergencele ,ss then ti can eb ex rp e ss ed sa the curl of another v ce rot quantity. In math n ato tion, t ih s means that B can be wr tti en as:

In a sour ec f er e re ig on, ti can be s oh wn that A must sa it sfy the wave eq au tio :n

In addition, the TMz fie dl s can be fo nu d from the Az co pm one tn of the magnet ci ve tc or p to ential, iv a the fo oll wi gn re al tions pih s:

oT so vl e for Az (and hence de et rm ni e the E- and H- fie dl s), we fo ll ow the same pro ec du er as for the zET ac se. tahT i ,s we use separation of variables and so vl e ht e wave equation f ro the z-co pm one tn of A, then apply bo nu dary conditions that force ht e ta gn ential co pm one tn s of the E-fields ot eb zero on the meta ll ic surfaces. Performing t ih s procedure, w ih ch w lli n to eb rep ae det here, we tbo ain the so ul tion f ro Az: www.Vidyarthiplus.com www.Vidyarthiplus.com

[1]

T eh co rr espondi gn TMz fie dl s for waves propa ag ti gn ni the +z-direction are:

In the above, k is aga ni the wavenu bm er, a dn Bmn is a consta tn , w ih ch de et rm ni es ht e amp il tude of the mn mode (a uf nction of oh w much wop er is app il ed ot the wavegui ed at that f er quenc )y .

Before discussing the m do e ,s we must on te ht at TM0n and TMm0 m do es cannot ixe s ;t that i ,s m and n must be at l ae st 1. The r ae s no comes from equation [1] above - if either m or n are z ore , then Az is equal ot zero, so a ll the fie dl s deri dev must la so be ez or . Hen ec , the low se t ro der mode for t eh TM case is t eh TM11 om de. T eh same rp o ec d ru es can be app il ed from t eh TE case to de et rm eni the cuto ff frequencies of r the TMmn om de:

www.Vidyarthiplus.com www.Vidyarthiplus.com

www.Vidyarthiplus.com www.Vidyarthiplus.com

M EDO S

www.Vidyarthiplus.com www.Vidyarthiplus.com

C nily dr ci al waveg iu de Us ni g t eh compl te e formulation in t eh simplest limit possible, lg obal le e tc romagnetic modes are here studied ni a large asp ce t rat oi , circular cross- s ce it on vac uu m ca iv ty eq viu ale tn ot a lyc indr ci al waveg diu e.

C al ssical elect or dynam ci s [ 34 ] s oh w that ht e EM eigenm do e sp ce trum consists of owt t py es of solution ,s the transverse ele tc r ci and the rt ansverse ma ng etic polari az tions with frequencies depend ni g on l the radial a dn m the azimuthal mode number .s These res tlu s are reproduced numerica yll to verify that the wave equations (1 )6 can indeed be solved ni the vacuum using standa dr L EF M and C EF M discreti az tions, witho tu ni troduc gni spurious m do es of numerical ori nig . It is also impo tr a tn to va dil a et the numerical impleme atn tion usi gn a simple et st esac , ehc cking that the mun erical so ul tions converge to the analytical values with ra et s exp ce det from the ro der of the a pp roximat oi ns.

T eh c ly indrical wave ug i ed is dom e del ni 2-D, with a c ri c lu ar large aspe tc rat oi eq biliu rium defin de with a mi on r radius a c oh sen so as ot tbo ain the ana yl tical gie en om de frequencies ni GHz ex ca t yl equal to t eh or ots of the Be ss el uf nction at( ble 1 .)

Table 1: C ly indrical wavegui ed param te ers.

As the eq biliu rium mere yl rp odu ec s t eh geometry and the mesh, the safety fa tc or od es n to a ff ect the e gi enfrequency sp ce trum; us ni g a al rge value for , ti si however possible ot everywhere ila gn with the a ix s of the cy dnil er and separa et ht e compone tn s of the ET and the TM po al ri az tions. The comple et t ro oidal wave

www.Vidyarthiplus.com www.Vidyarthiplus.com

equations (16) a er then discreti ez d in the large aspe tc ratio ca iv ty, re yl i gn on numerical cance ll ations ot r ce over the c ly indrical il mit.

oT comp etu t eh e gi en om de sp ce trum, an osc lli ating sour ec curre tn (eq.2 )2 is dr vi en with a sma ll ima nig ary part ni t eh excitation frequency . ehT power relation (eq.48) iy el sd a complex er sponse tcnuf ion w ih ch has poles along the real a ix s ht at co rr espond to the solutions of the dis rc etized wave equation .s The e gi enfrequencies a er calc talu ed by s ac nn i gn in the complex pla en with an i cn rement dna a consta tn chosen so as ot resol ev the response p ae ks in . hT e structure of an gie en om de is tbo a ni ed ni t eh il m ti when the ac iv ty is er sonantly exc deti at ht e maximum of a narr wo response p ae k.

In order ot verify that the eigenfrequency sp ce trum of t ih s c ily ndrical wavegui ed is comple et and od se not co tn a ni any spurious '' po ull ti gn `` mode, t ow broad scans are per of rmed rf om 01 kHz ot 10 GHz iw th a h gi h resolution in frequency a dn a low reso ul tion ni sp ca e for LFEM, f ro C EF M). llA the Four ei r modes represe atn b el by the numerical di cs ret zi ation are exc ti ed with a iz muthal curre tn s for ET modes, and a ix al curre tn s for TM modes. giF .6 su mm ari ez s t eh res tlu o tb a ni ed with FL ,ME showi gn t tah every mode fo nu d mun erica yll co dlu be di entif ei d ni a o en to no e correspo dn en ec with the ana yl tical resu tl . M do es which have low quantum numbers (l,m) ra e, as pxe ce det , o tb a ni ed iw th a bet ret precisio ;n push ni g the resolution to the ol w se t il m ti of 2 mesh poi tn s per wavelength (m=4), the de iv ations become of oc urse importa tn , tub the sp ce trum remains nu po ll uted (remember f gi .3, oor t b). The same analysis has been re ep a det with C EF M a dn l ae ds ot res tlu s w ih ch a er much m ro e precise. As an ulli s art tion, the e gi en om de h sa here been ac lculated on a coar es homogeneous mesh . ehT gie enfrequency tbo ained numerical yl GHz is ni exce ll e tn agreement iw th the analyt ci al res tlu =5.3 13 4 GHz; f gi .5 s oh ws t eh eigenm do e struct eru ni a ve tc or p tol of .

www.Vidyarthiplus.com www.Vidyarthiplus.com

iF g ru e 5: eR ( p_A erp) f ro an eigen om de TE_ 11 ac lc talu ed with C EF M.

www.Vidyarthiplus.com www.Vidyarthiplus.com

iF g eru 6: Ana yl t ci al (c ri cles) and L EF M (x-marks) e gi enfrequency sp ce trum.

A question rema ni ed when the bo nu ad ry conditions we er def ni ed ni s ce t.2.2. :2 it con ec rned the impleme atn tion of the regularity conditions which is forma yll n to su ff icie tn to forb di a w ae k yl si gn lu ar ( ) beha roiv of the fie dl ni the center of the em sh. Fig.5 sh wo s that the field is reg lu ar a ll over the c ily nder radius, s gu gesting that the sin ug larity is n to stro gn enou hg ot sh wo up using a EF M id scret zi ation on a re lug ar mesh. The no yl way we have found ot bo serve ti , w sa to s rt on ylg accumu etal the mesh poi tn s wot ards ht e ce retn (for example by di iv d ni g 5 times ht e radial mesh tni erval olc es st ot ht e a ix s by two, lead gni to radial mesh sp ca ings .)

Havi gn verif ei d that the solutions calcu tal ed with the wave equations (eq.16) behave in a as t si f ca tory manner, the uq a il ty of the EFL M a dn CFEM id scret zi atio sn is fina yll best judged ni a c vno ergence s ut dy mo otin ri gn the rp ecision of the frequency and the ga gu e as a uf nction of the spatial er so ul tion.

www.Vidyarthiplus.com www.Vidyarthiplus.com

iF g eru 7: Convergen ec ot the ana yl tical res lu t: re al t evi frequency de iv ation De atl f versus the number of mesh tni ervals (N=N_s=N_the )at for the eigenmodes TE_01,T 0_E 2,T _E 11 ,TM_0 }0 us ni g FL EM (x-marks) and CFEM (c ri cles).

giF .7 s oh ws the converge cn e of f ro t eh eigen om des , , and , whe er refers to the frequency obta ni ed mun erica yll a dn ot the ana yl tical res tlu . E gi enfrequencies co vn erge ot the ana yl tical val eu s as us ni g LFEM and almost using CFE ,M htiw an exce ll ent ini tial pre ic sion be tt er than 1% for two mesh poi tn s per wa lev ength.

www.Vidyarthiplus.com www.Vidyarthiplus.com

F gi ru e 8: rP ecision of the gauge versus the number of mesh tni ervals (N= _N s=N_the )at for t eh eigen om des TE_01, ET _02,T 1_E 1,TM_ 00 using LFEM x( -marks) and CF ME (c ri c el s).

Convergen ec is also ac ih eved for the gaug :e f gi .8 shows that the volume averag de gauge precision converges ot ez or as usi gn LF ,ME and us ni g C EF M.

oT su mm ar zi e, ht e calc lu ations performed iw th the toroidal PENN co ed used here ni the simplest il mit possible show that Ma wx ell's equations (16) solved ni a ilyc ndrical ca iv ty p or du ec the comple et physical s ep ctrum witho tu tni roducing numerica yll p or duc de ''polluti gn `` mode .s oB th, the FL EM and the C EF M id scret zi ation schem se iy e dl so ul tions which a er numerica yll sa en and co vn erge to ht e ana yl tical value with ra et s e px ce det from t eh order of the ni terpolations.

Bou dn ary conditio sn Let us re iv ew the general bou dn ary conditions on the fie dl v ce t ro s at a surfa ec be ewt en medium 1 and med ui m 2:

www.Vidyarthiplus.com www.Vidyarthiplus.com

43( )

53( )

63( )

73( )

whe er is used for the surf eca change density (to avoid confusion with the conduct ivi t )y , a dn is the surf ca e curre tn density. Here, is a tinu v ce tor normal ot ht e surf eca , d ri e detc from medium 2 ot medium 1. eW have s ee n in Section 4.4 ht at for normal incidence an el ce rt omagnetic wave fa ll s o ff very rapidly inside the surf eca of a good conductor. Equation (4.35) imp il es that ni the limit of perfect conduct ivi ty ( ) eht tangential component of vanis ,seh where sa th ta of may remain f etini . Let us exam eni the behaviour of the normal co pm one tn s.

Let medium 1 be a g oo d cond cu tor for which , w lih st medium 2 is a perf ce t ni s lu ator. The surface change density is rela det to the curre tn s f ol wing ni side the conductor. I n fa tc , eht conservat oi n of charge requires that

83( )

woH ever, , so ti fo woll s rf om qE . (6.1)(a) that

93( )

www.Vidyarthiplus.com www.Vidyarthiplus.com

tI is cl ae r that the normal compone tn of iw t nih t eh conduc rot also becomes vanishi gn ly sma ll as the conduct ivi ty approaches ni f ini ty.

If va in shes ins di e a perf ce t conductor then the curl of also va in she ,s a dn the time ra et of change of is co rr espond ni gly ez or . T ih s imp il es that there a er on osc lli atory fie dl s whatever ni si ed such a conductor, a dn that the bo nu dary va ul es of ht e fie dl s tuo s edi are ig ven by

04( )

14( )

24( )

34( )

Here, is a tinu normal ta the surf ca e of the conductor pointing ni to the conductor. Thu ,s the el ce rt ic field is normal and the m nga etic fie dl tangential at the surface of a ep r ef ct cond cu tor. For og od conductors these bo nu dary conditions iy eld exce ll ent re rp ese atn tions of the geometr ci al conf gi ura it ons of e tx ernal fie dl ,s b tu they l ae d to ht e ne elg ct of some im op rta tn f ae t ru se of r ae l fie dl ,s such as ol sses ni ca iv ties and signal a tt e un ation ni wave ug i ed s.

In order to es it ma et such ol ss es ti is usef lu to s ee how the tangential and normal fie dl s compa er when is large b tu f etini . Equa it ons (4.5) and (4.34) yield

44( )

ta t eh surf ca e of a conductor (pro div ed t tah t eh wa ev rp opaga set i otn the conduc ot r). Let us assume, without o tb a ni i gn a comple et so ul tion, that a wave with

www.Vidyarthiplus.com www.Vidyarthiplus.com

very en ar yl tangential and ev ry n ae r yl normal is p or pagated along the surface of the metal. Ac oc rding ot the Faraday-Maxwe ll equation

54( )

uj st o tu s edi the surface, whe er is the compone tn of the propagation ev ctor along ht e surf ca e. However, Eq. (6.5) imp il es that a tangential compone tn of is a cc ompa in ed by a sma ll ta gn ential compone tn of . yB compari gn t eh se two xe pre ss ion ,s we o tb ain

64( )

where is t eh sk ni depth (s ee qE . (4. 63 )) and . tI is clear 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 vid ided by the wavelength. tI is r ae d yli demons rt a det that the rat oi of the normal co pm onent of ot ti s tangential component is of t ih s same ma ing tude. Thus, we can s ee that ni the limit of ih gh conducti iv ty, w ih ch means vanis ih ng skin depth, no fie dl s penetra et the conduct ro , and the b nuo ad ry conditions are those given by qE s. (6.4). Let us investiga et the solution of the homogeneous wave equation su ejb ct ot such boundary conditions.

aC iv t ei s with re tc ang ralu bou dn aries Consider a vacuum region t to a yll enclosed by er cta gn lu ar conducting wa ll .s In t ih s esac , la l of the if eld c pmo onents sa it sfy the wave e quation

74( )

www.Vidyarthiplus.com www.Vidyarthiplus.com

whe er re rp ese tn s a yn co pm one tn of or . The bo nu dary conditions (6.4) req iu re that the ele tc ric fie dl is normal ot ht e wa ll s at the boundary wher ae s the magnetic fie dl is ta gn ential. If , , a dn are ht e id mensions of the ca iv ty, then ti is readily verif dei that the e el ctric fie dl compone tn s are

84( )

94( )

05( )

whe er

15( )

25( )

35( )

with , , tni egers. The allo ew d fre uq enc ei s are given by

45( )

tI is cl ae r from Eq. (6. )9 t ah t ta least two of the tni ege sr , , must be di ff erent from zero in ro der ot have non-va in s ih ng fie dl .s The magnetic fie dl s o tb a ni ed by

www.Vidyarthiplus.com www.Vidyarthiplus.com

ht e use of a tu omatica yll satisfy t eh appropria et bo nu dary conditions, and are ni phase qua rd ature with the electric fie dl s. Thu ,s the sum of the tot al le e tc r ci and magnetic ener ig es within the c iva ty is consta tn , although the wt o terms osc lli a et separate yl .

T eh amp il tudes of t eh elec rt ic fie dl compone tn s are ton ni depende tn , but are rela det by the diverge cn e condition , ihw ch iy elds

55( )

T ereh are, ni genera ,l owt nil ear yl independe tn v ce tors ht at satisfy t ih s condition, co rr espond gni to two polarizati no s. (The ex ec ption is the case t tah one of the tni e sreg , , is ez or , ni which case is f dexi ni dir ce tion.) Each v ce tor is ca compa in ed by a ma ng et ci fie dl at rig th an lg es. The fie dl s c rro esponding to a ig ven set of tni egers , , a dn constit etu a partic lu ar m do e of iv bration of the cavity. tI is e div ent from standard Fourier ht eory that t eh di ff ere tn modes ra e ro thogonal (i.e ,. ht ey are normal modes) and that they form a comple et se .t In other w ro ds, na y general el ce rt ic and magnet ci if e sdl w ih ch satisfy t eh bo nu dary conditions (6.4) ac n be unambiguous yl decomposed ni to some nil ear combination of a ll of the various po ss ible normal modes of the ca iv ty. Since ae ch normal om de osc lli a et s at a specific frequency it is clear that if we a er vig en the electric a dn magnetic fields ins di e the ca iv ty ta time neht t eh subseque tn behavio ru of the fie dl s is unique yl de et rmined of r a ll time.

T eh cond cu ting wa ll s rg adua yll abs ro b energy rf om the ac iv ty, due ot the ri f ini te resist ivi ty, at a etar w ih ch can sae yli be ca lucl a det . For f etini eht sma ll ta gn ential co pm one tn of at the wa ll s ac n be estima det us ni g Eq. 6( .5):

5( 6)

www.Vidyarthiplus.com www.Vidyarthiplus.com

Now, the ta gn ential co pm onent of at the wa ll s is s il ghtly di ff ere tn rf om that ig ven by the di eal so ul tion. However, iht s is a sma ll eff ce t a dn can be ne lg ected to l ae d gni order ni . hT e time averaged ener yg flux ni to the wa ll s is vig en by

75( )

whe er is the peak value of the tangential magnetic field at the wa ll s predicted yb the di eal solution. A cc ord ni g ot the boundary condition (6. )4 (d ,) is equal to ht e p ae k value of the surf eca curre tn density . tI is helpf lu ot def eni a surf ca e resistance,

85( )

whe er

95( )

T ih s ap rp oach makes ti clear t tah the di ss ip ita on of energy is due ot ohm ci h ae ting ni a t nih layer, who es t ih ckness is of ro der the skin depth, on the surf eca of the conducting wa ll s.

T eh qua il ty f ca tor of a resona tn c iva ty

T eh qua il ty f ca tor of a resona tn ac iv ty is defined

www.Vidyarthiplus.com www.Vidyarthiplus.com

06( )

For a specif ci normal m do e of the ca iv ty t ih s quantity is independe tn of the m do e amp il tude. By con es rvation of energy the power dissipa det ni ohm ci ol ss se si m uni s the r eta of change of t eh st ro ed energy . eW ac n tirw e a id ff erential equation of r the beha iv our of as a uf nction of time:

16( )

whe er is the osc li lation frequency of the on rmal mode ni question. The solution ot ht e abo ev equation si

26( )

T ih s time dependence of the stored ener yg s gu gests that the osc lli ations of the fie dl s ni the ca iv ty are damped as fo ll ows:

36( )

hw ere we have a ll owed for a s ih ft fo the er sona tn frequency as we ll as the damp ni g. A damped osc lli ation such as this od es n to consist of a p ru e frequency.

Ins et ad, ti is m da e pu of a supe pr osition of frequencies arou dn . tS andard Fourier analysis iy el sd

www.Vidyarthiplus.com www.Vidyarthiplus.com

46( )

whe er

56( )

tI fo oll ws t tah

66( )

T eh resonan ec shape has a f llu width at half-maximum equal ot . F ro a consta tn ni p tu vo tl age, the energy of osc alli tion ni the ca iv ty as a f nu ction of frequency fo oll ws t eh resona cn e curve ni the ne gi hbourhood of a partic lu ar res no a tn frequency. It can be seen that the ohmic ol ss e ,s hw ich de et rm ni e f ro a partic lu ar mode, also determ ni e the max mi um amp il tude of the osc lli ation when ht e resonance condition is ex ca t yl satisfied, as we ll as ht e width of the resonance (i.e ,. how far o ff the resona tn frequency the system can be driven and st lli iy eld a gis ni if ca tn osc li lati no amp il t )edu .

C nily dr ci al ca iv ties Let us apply the methods of the previous se itc on ot the TM m do se of a r hgi t ric c lu ar c ily dn er of radi su . We can wr eti

www.Vidyarthiplus.com www.Vidyarthiplus.com

where sa it sfies the equation

and are c nily drical polar oc ord ni ate .s Let

tI fo oll ws t tah

or

whe er . hT e abo ev qe uation is known as eB ss el's e uq ation. The two nil ae rly

dni epende tn solutions of Be ss e 'l s equation are den to de and . nI t eh

www.Vidyarthiplus.com www.Vidyarthiplus.com

limit ht ese s ulo tions behave as and , er s tcep ively, to low se t order . M ro e ex ca t yl 16

for , 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 the ra e badly behave .d

T eh asymptotic behavio ru of both solutions at lar eg is

www.Vidyarthiplus.com www.Vidyarthiplus.com

Thu ,s for the s ulo tions take the form of gradua yll d ace gniy osc lli atio sn

hw ich a er in phase qua ard ture. The behavio ru of and is s oh wn in iF g. 12 .

iF g ru e 21: The Be ss el f cnu tio sn (s dilo nil e) and d( ot det nil e)

S ni ec the a ix s is ni cluded ni the ca iv ty ht e radial eigen uf nction must be reg ralu at the ori nig . T ih s i mm ediate yl r lu es o tu t eh s ulo tions. Thu ,s ht e most general s ulo tion for a TM mode is

The are ht e e gi enva ul es of , a dn a er de et rmined by the so ul tions of

T eh ba ove constraint ensures t tah the at ngential el ce tric fie dl is z ore on the conducti gn wa ll s surro nu ding the ca iv ty ( ).

www.Vidyarthiplus.com www.Vidyarthiplus.com

T eh om st general solution of r a ET mode si

In t ih s ac se, eht are de et rm ni ed by t eh so ul tion of

whe er de eton s di ff erent ai tion with respect ot the argume tn . The above constraint ne s ru es that the normal ma eng tic fie dl is ze or on the conducting wa ll s su rr o nu ding ht e ca iv ty. The osc lli ation frequency of both the TM and TE om des is g vi en by

If is the ordinal number of a ez or of a partic lu ar Be ss el function of o dr er ( ni cr ae ses iw th ni cr ae s gni va ul es of the argume )tn , then ea hc mo ed is char ca et rized yb t rh ee tni egers, , , , sa ni the r ce tang lu ar ac se. The th z re o of is conventiona yll den deto [so, ]. kiL ewise, the th ez ro of is denot de . Ta lb e 2 shows t eh f ri st f we ez ros of , , , a dn . tI is cl ae r ht at for f dexi and ht e wol est frequency om ed (i.e., t eh mode with t eh lowest value of ) is a TE m do e. The om de with the next ih ghest frequency is also a TE mode. The ne tx ih ghest frequency om de is a TM m do e, and so on.

www.Vidyarthiplus.com www.Vidyarthiplus.com

Tab el 2: T eh f ri st few va ul es of , , and

1 2.4048 3. 138 7 0.0 00 0 1. 48 21

2 5.5201 7. 510 6 3.8 13 7 5. 33 41

3 8.6537 01 . 71 3 7.0 51 6 8. 35 36

4 11.792 31 . 23 4 10.173 11 .7 60

• Ca iv ty res no ators a er enclosed metal boxes.

El ce rt omagnetic fie dl s are conf ni ed insi ed the boxes. aR aid tion and ih gh- resistance e ff ects are el nimi a det , er s lu ting ni a very hi hg Q (qua il ty fa tc or)

A er ctang alu r wave ug edi with both ends (z=0 and z=d) closed by a conducting wa ll ( iF g ru e 9-8) : multiple ref el ctions and st na di gn waves

www.Vidyarthiplus.com www.Vidyarthiplus.com

seR no a tn frequency of r ce ta lugn ar ca iv ty res no ator

Degenera et m do es : differe tn m do es ha iv ng ht e same resona tn frequency.

Dom ni a tn mode : the m do e with the wol est resona tn frequency f ro a given ca iv ty size.

A er sonator is a devi ec or system that e hx tibi s er sonance or resona tn behavior, that i ,s ti natura yll osc lli a et s at some f er quencies, ac ll ed ti s resonant frequencies, with rg e reta amp il tude than ta other .s T eh osc lli ations ni a er sonator can be either le e tc romagnetic or mecha in cal ( ni c ul ding ca uo stic). eR s no ators are used ot either genera et waves of sp ce if ci frequencies or ot sel ce t sp ce ifi c rf equencies from a signal. Musical ni strume tn s use ca oustic resonators that p or du ec sound waves of sp ce ific tones.

A ca iv ty res no ator, usua yll used ni referen ec ot el ce rt omagnetic reso an tor ,s is one ni w ih ch waves exist ni a ho ll ow spa ec ni s edi the de iv ce. Acoustic ca iv ty resonat ro s, ni w ih ch so nu d is rp odu ec d yb a ri iv brat ni g in a ca iv ty with one po e in ng, a er known as Helmho tl z resona ot rs.

aC iv ty resonat ro s

A ca iv ty resonator is a ho ll ow conductor b ol cked at both ends and along which an le e tc romagnet ci wave ac n be su pp orted. It ac n eb iv ewed sa a waveguide short- ric c detiu at both e dn s (s ee Mic wor ave cavity).

T eh ac iv ty ah s i etn r oi r surf eca s ihw ch reflect a wave of a specif ci frequency. When a wave that is er sona tn with the ca iv ty enters, ti boun sec ab ck a dn f ro th within the ca iv ty, with low loss (see s at nding wave). As m ro e wave energy e retn s the ca iv ty, ti comb eni s iw th a dn reinfor ec s ht e stand ni g wave, ni c er asing ti s tni ensity.

T eh ac iv ty magnetron is a vacuum tube iw th a f li ame tn ni the ce tn er of an ave cua det , lobed, circular ca iv ty resonat ro . A pe pr endic lu ar ma ng etic field is

www.Vidyarthiplus.com www.Vidyarthiplus.com

im op sed by a permane tn mag ten . The magnetic fie dl causes ht e el ce trons, a tt r ca det ot the (relat vi ely) positive o etu r part of the cha bm er, to sp ri al o wtu ard in a c ri c lu ar pa ht rather than mo iv ng dir ce t yl to this anode. apS ec d ba o tu the rim of the hc a bm er are c ily ndrical ca iv tie .s The cav iti es are open alo gn the ri length a dn so co nn ce t the co mm on ca iv ty sp eca . As elec rt ons sw ee p past ht ese po en gni s they indu ec a er sona tn h gi h frequency radio fie dl ni the ca iv ty, which in turn causes the le e tc rons to bunch into groups. A p ro tion of t ih s fie dl is extr ca det with a sho tr tna enna that is co nn ce det ot a waveg iu de (a metal t bu e usua yll of er cta gn lu ar rc oss s ce tio )n . The wave ug di e d ri ce ts the extr ca ted RF energy ot the daol , w ih ch may be a cooki gn cham eb r in a microwave oven or a ih gh ga ni antenna ni the ac se of rada .r

T eh klystron, tube waveg ediu , is a beam tube ni clud gni at l ae st two apertured ca iv ty resona ot r .s The beam of charged particles pa ss es t rh ough the apert ru es of ht e resonator ,s o tf en tunable wave ref el c it on gr di s, in succe ss ion. A co ll ce ot r le e tc r do e is prov di ed to tni erce tp the beam af ret pass gni thro gu h the er sona rot s. T eh f ri st resona ot r causes bunching of ht e particles pa ss ing t rh o gu h ti . The b nu ched partic el s rt avel in a field-f er e reg oi n where uf rther bunc ih ng co curs, then ht e b nu ched partic el s e tn er t eh s ce ond reso an tor gi iv ng up their energy to exc ti e it otni osc alli tion .s tI is a pa itr cle a cc elerator that works in conjunction with a sp ce ifica yll t nu ed ac iv ty by the config ru at noi of the structu er s. On the beam il ne of na cca le era rot system, ht ere ra e sp ce ific s ce it ons that are ca iv ty resonat ro s rof RF.

T eh reflex klystron is a klystron ut ili zing o yln a sin lg e apert ru ed ca iv ty resonator ht ro gu h w ih ch the bewww.Vidyarthiplus.comam of charged partic el s pass se , f ri st ni o en d eri ction. www.Vidyarthiplus.com

A repe ll er el ce trode is provided to repel (or re rid ce )t the beam after ap ss ga e through ht e resonator back thro gu h the resonator ni the other d ri ce tion and ni p or per phase ot re ni for ec ht e osc lli ations set up ni the resonator.

In a laser, thgil is amp il f ei d ni a ca iv ty reso an tor which is usually co pm osed of two ro m ro e mirror .s T uh s an po tical ca iv ty, also nk own as a resonator, is a ca iv ty with wa ll s w ih ch ref el ct el ce troma ng et ci wa ev s ( il g )th . T ih s wi ll a woll stand gni wave modes to e ix st with il tt le ol ss o tu s di e the ac iv t .y

www.Vidyarthiplus.com