14 i. Experiment 14 LUMPED-PARAMETER DELAY LINE. Dispersion Relation 1. Characteristic Impedance 3. Cutoff Frequency 5

Transcription

1 4 i Experimet 4 LUMPED-PARAMETER DELAY LINE Itrodutio Diperio Relatio Charateriti Impedae 3 Cutoff Frequey 5 Propagatio o the Lie ad Refletio at Termiatio 6 Steady-State Repoe ad Reoae 9 Prelab Problem 0 Proedure Data Aalyi 3 Appedix A: Effet of Idutor Self-Reoae 4

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3 4 9/7/0 INTRODUCTION I thi experimet you will explore the propertie of a avity reoator otruted from a diperive tramiio lie. Thi lumped parameter delay lie i made up of 0 oateated opie of the uit ell how i figure. To proeed with the aalyi of the reoator we mut firt derive the propagatio harateriti of the tramiio lie; to udertad the followig diuio you mut tudy Geeral Appedix A: Tramiio Lie Reoae due to Refletio, whih may be foud at the ed of the lab maual (followig Experimet 7). L C C Figure : A uit ell of the lumped parameter delay lie. Several of thee ell are oeted together to form a tramiio lie, whih ould be termed a ladder. Adjaet apaitor i two oeted ell are evidetly i parallel, whih i equivalet to a igle apaitor with value C. Thi i how the atual lie i otruted, exept for apaitor of value C/ at the two ed of the lie. The lab uit have L = 5 mh ad C = F. DISPERSION RELATION The tramiio lie reoator will be of uit legth (a i Geeral Appedix A) ad will be ompried of N opie of our uit ell how i figure. For thi aalyi we will oider the lie to be lole, o the idutor ad apaitor makig up the lie are aumed to be ideal. To derive the diperio relatio ( k) for wave propagatig o the lie, we ue Ohm law to relate the voltage at adjaet ode (oetio betwee uit ell; refer to figure ). V I L + V + V ( I L ) ( ) I C wave a( x) x = N Figure : Defiitio of the variou voltage ad urret ued to derive the diperio relatio. A wave a( x ) propagate to the right. The voltage V i at the ode oetig the th ad ( + )th uit ell. The urret I C i the um of the two idetial urret through the two parallel uit ell apaitor. The lie ha uit legth, ad the total umber of uit ell i N.

4 4 9/7/0 With voltage ad urret a defied i figure, we ee that ( ) ( ) ( ) ( ) = = jl I = V V ; j L I = V V ; L L + + I I I jcv L L + C () Sie the lie i aumed to be lole, the right-goig wave ax ( ) ha a propagator ( x) = exp( jkx) (ee Geeral Appedix A), o the voltage are alo related by V = ( xv ) = Ve ; V = ( + xv ) = Ve ; x= N () jk ( x ) jk ( + x ) + Combiig () ad () to fid the required relatiohip betwee ad k give ( ) ( ) jl I j L I = V V + V L L + + V V + V = jli = jl jc V = V jk N jk N ( ) ( )( ) + C e + e V = V k k o 4i N N = = k k ( k ) = i = i N N ; = (3) where the total umber of uit ell i N, ad we idetify the utoff frequey (reall that our uit of legth i the total legth of the lie, o the wave umber k i the total radia of phae alog the etire lie legth). Note that doe t deped o N, the umber of uit ell, but oly o the harateriti of the uit ell. I fat, it i evidet that i jut the reoat frequey of the L ad C aroud the loop i the uit ell (figure ) ie the two apaitor are i erie (for urret flowig aroud the loop), the equivalet apaitae i C /4, givig a reoat frequey of /4 =. Equatio (3) i the diperio relatio for thi tramiio lie. It deribe how the wavelegth ad frequey are related for a wave propagatig o the lie. Sie the phae veloity vφ = / k varie with frequey, the lie i diperive, ad, aordig to (3), lower frequey wave have a greater phae veloity tha high frequey wave. Thi futioal relatio ( k) i plotted i figure 3.

5 4 3 9/7/0 Figure 3: A plot of the diperio relatio (3) for a lie with 0 ell ad legth. The dahed lie i the aymptoti, odiperive relatio at low frequeie: k / N. Note that the group veloity d/dk vaihe at the utoff frequey =, where we alo have k = Nπ. The phae delay (phae hift) aro a igle uit ell i φ = k x= k / N, o the diperio relatio give phae delay aro a igle uit ell: φ = i ( ) propagatio time delay aro a igle uit ell: t = φ (4) At low frequeie ( ) the ie futio i (3) i approximately equal to i argumet, ad the lie i approximately odiperive with phae veloity: ( ) Low frequey phae veloity : v φ uit ell e (5) CHARACTERISTIC IMPEDANCE The other importat defiig harateriti of the tramiio lie i it harateriti impedae Z 0, defied i Geeral Appedix A. To determie Z 0 we mut derive the relatio betwee the voltage at a ode joiig two uit ell ad the urret flowig from oe ell to the ext. A before, oider a lie with a right-goig wave paig a ode i the lie (figure 4).

6 4 4 9/7/0 V I L + V + V ( I L ) ( ) I I C / I C / wave a( x) x = N Figure 4: A light modifiatio of the hemati i figure defie the urret I flowig through the wire joiig the th ad ( + )th uit ell. Itead of the igle urret I C there are ow two idetial urret I C / flowig through the two parallel uit ell apaitor, eah with value C /. The voltage V i at the ode oetig the two uit ell. A i figure, a wave a( x ) propagate to the right. The harateriti impedae of the tramiio lie i Z0 = V / I. The derivatio of Z 0 i thu: ( ) ; ( ) ( ) ( ) L C/ L + C/ I = I + I I + ( ) ( k N) jk / N jk / N ( ) jli = V V = e e V LI thu, I = I I I = I + I Z L L V V V V = + jl jl = = i L i k N V To proeed we ote that i ( k N) i ( k N) o( k N) =, o that Z 0 L L = = i o i i ( k N) ( k N ) ( k N) ( k N) Z 0 = ( ) ( ) L Z 0 = ( ) (6) where we ve ued the diperio relatio (3) to implify the fial reult. At low frequeie ( ) Z0 L/ C, but Z0 a.

7 4 5 9/7/0 CUTOFF FREQUENCY Now to diu the igifiae of the utoff frequey. Coider the phae delay expreio (4). A from below, the phae delay φ π, ad φ ( ) = π, o at the voltage at eah ode i 80 out of phae with the voltage at the two adjaet ode. What happe at frequeie >? I thi ae the argumet of the arie i (4) i greater tha, o the reult mut be a omplex umber. We a exted the diperio relatio for frequeie above utoff by demadig that: () i real, ad () the diperio relatio i otiuou at =. Here i the derivatio, tartig with (3): k = i = i N ( ) real part ( A jb) i A jb = i Aoh B jo Aih B imagiary part o Aih B 0 A = π / lim ( A jb) = π / B 0 k π = oh B; = jb N k π for > : = j oh (7) N Thi reult lead to the followig expreio for the voltage a a futio of poitio o the lie for a right-goig wave x N jkx j k N j ( ) = / ; = ( / ) = π oh / V > = e = ( ) jk/ N for : exp oh / V V for : / / ( ) = ( ) V (8) ad we ee that beyod utoff frequey there i o wave propagatio, but rather the voltage phaor hage ig from ode to ode ad deay geometrially with ditae dow the lie. We hoe ( A jb) i the derivatio of (7) to eure the geometri deay of V with ditae, rather tha it growth.

8 4 6 9/7/0 The harateriti impedae beyod utoff frequey a be derived almot immediately from equatio (6) ad i learly imagiary. We mut be areful to hooe the proper ig of j i the reult; thi i doe by oiderig the value of i ( k N ) i the derivatio of (6), give our reult (7). The value of Z 0 beyod utoff i thu for > : Z = j 0 ( ) ( ) = ( ) for : Z0 j j C (9) Agai, o wave propagatio i poible beyod utoff frequey beaue the harateriti impedae i imagiary. The impedae beyod utoff i apaitive; thi hould alo be apparet if you oider the uit ell i figure learly at very high frequeie the urret eterig the uit ell will flow through the earet parallel apaitor (with value C/). PROPAGATION ON THE LINE AND REFLECTIONS AT TERMINATIONS Armed with the diperio relatio (3) ad (8) ad the harateriti impedae (6) ad (9), we are ready to oider the propagatio of igal o the tramiio lie ad their refletio from termiatio. We kow from our readig of Geeral Appedix A that the voltage refletio oeffiiet Γ from a termiator with impedae Z i give by Z Z Γ= Z + Z 0 0 (0) Z Z Z = 0 Γ= (horted) = Γ=+ (ope) = Z Γ= 0 0 (properly termiated) The apparatu for thi experimet allow you to adjut the termiatio impedae at both ed of the lie. From the diuio i Geeral Appedix A we kow that the termiated tramiio lie will be a effiiet avity reoator if the termiatio have Γ. Coider the time domai behavior of the termiated lie i repoe to a tep i the iput voltage (a diagram of the ofiguratio for thi aalyi i how i figure 5). By applyig a lowfrequey quare-wave iput uig the igal geerator, a erie of idepedet voltage tep are ijeted oto the tramiio lie o that it propagatio propertie ad the termiator refetio propertie may be tudied.

9 4 7 9/7/0 Sigal Geerator R = 50Ω S Tramiio Lie Z L Z 0 Z R Figure 5: The tramiio lie i termiated at eah ed with a adjutable impedae, Z L o the left (oure) ed ad Z R o the right ed. The igal geerator at a a ideal voltage oure i erie with a 50Ω reitor, R S (whih i muh maller tha the harateriti impedae Z 0 ). Applyig teady iuoidal igal of variou frequeie a exite the may reoae of the lie; applyig a low-frequey quare wave itrodue a erie of early idepedet voltage tep iput whoe propagatio dow the lie ad refletio from the termiator may be tudied. Aume that the igal geerator i figure 5 i ued to ijet a igle voltage tep at time t = 0 ; the voltage tep i from a iitial value of V to a fial value of + V at the termial of the igal geerator. Aume that Z L ha bee et to the value L/ C, whih i very early equal to Z 0 for frequeie below the utoff frequey,. Sie Z L ad Z 0 form a voltage divider (with ZL Z0 ), the voltage tep applied to the tramiio lie i omially from /V to + /V, ad it i thi maller tep whih itrodue wave at the left ed of the lie whih the propagate toward the termiatio Z R. Figure 6: A plot of the tep repoe of the tramiio lie. The dark lie i the repoe at the left ed of the lie immediately followig a tep iput by the igal geerator (figure 5); the gray lie i the repoe 5 uit ell dow the lie a the tep propagate by that poitio. A metioed i the text, the ripple i the repoe waveform are aued by the varyig harateriti impedae with frequey, Z 0 ( ). The repoe i le harp at poitio further dow the lie beaue the phae veloity i lower for the higher-frequey ompoet of the tep. The time axi i i uit of the low-frequey propagatio time delay/ell, equatio (4) ad (5). Figure 6 how the reult for the voltage waveform at the iput to the tramiio lie ad at a poitio 5 uit ell dow the lie. The Fourier traform of a voltage tep oit of a

10 4 8 9/7/0 otiuouly ifiite et of ie waveform, all paig through 0 at time t = 0. The amplitude of the ie are iverely proportioal to their frequeie, o all of thee waveform have the ame lope ( dv / dt ) at t = 0 (thee lope all add to reate the voltage diotiuity repreeted by the tep). If our experimetal ytem i liear, the the evolutio of eah of thee idividual ie i idepedet of all the other; at ay poit o the lie at ay later time we a jut itegrate the itataeou voltage of all the evolved wave (eah at a differet frequey) to determie the voltage at that poit due to the propagatio of the origial tep iput. Thi proe eem pretty ompliated (ad it a be!), but lukily we a ue a tool like Mathematia to do the tediou algebra ad umerial itegratio. Beaue Z 0 varie with frequey (beomig apaitive above ), the voltage divider formed by Z L ad Z 0 hage the relative amplitude ad phae of the variou ie makig up the origial voltage tep. A a reult, the waveform at the iput to the tramiio lie i ot a perfetly-harp tep, but ha ripple, a how i figure 6. Eah of the variou wave the propagate dow the lie at it ow phae veloity. The diperio i phae veloitie delay the higher-frequey ompoet more tha the low frequeie, o the tep beome ever more pread-out a it propagate dow the lie (figure 6). Note that util the igal ha had time to propagate dow the lie to the right-had termiatio ad bak agai, the tramiio lie behave a though it exted to ifiity, beaue there a be o refleted wave to modify the repoe. Now oider the effet of a ope or of a horted termiatio at the right-had ed. The ope termiatio ha Γ=+, o the refleted tep ha the ame ig a the iomig tep. The horted termiatio ha Γ=, o it ivert the iomig tep. The reultig waveform are how i figure 7. Figure 7: arrival of a refletio from the right-had ed of the lie. Both plot how the waveform at the iput ed of the tramiio lie: the dark lie i the repoe with the right-had ed horted ( Γ= ), o the refletio i iverted; the gray lie i the repoe with the right-had ed ope ( Γ=+ ), o the refletio reifore the origial tep. The legth of the lie i 0 uit ell, o the refletio arrive 0 time uit after the iitial timulu. The oure ed of the lie i properly termiated, o o further refletio take plae at that ed. The fial, equilibrium voltage o the lie ( t ) i 0 for the horted ae ad for the ope ae (igal geerator output = to +).

11 4 9 9/7/0 STEADY-STATE RESPONSE AND RESONANCES Geeral Appedix A provide a brief itrodutio to the topi of avity reoae ad derive the oditio o the wave umber (k) for reoae to our, depedig o the harater of the avity termiatio. Thee oditio are ummarized here: Same termiatio at both ed (both horted, both ope): k = m π m () m Oppoite termiatio at the ed (oe horted, oe ope): m ( ) k = m π m () where, a i Geeral Appedix A, k m i the total phae alog the lie for the m-th reoae (or, equivaletly, the lie i take to have uit legth, ad k m i the wave umber). So for the ame termiatio at both ed, at a reoae the tramiio lie i a eve umber of ¼ wavelegth log; if the termiatio are differet, the there are a odd umber of ¼ wavelegth at reoae. Thi oditio eure that the refleted wave, oe it ha bee refleted by both ed of the lie, i agai i phae with the origial wave. There i aother iteretig iterpretatio of thee reoae oditio: if the far ed of the tramiio lie i horted, ay, the wheever the lie legth i a eve umber of ¼ wavelegth the lie preet a hort iruit to the oure drivig it; it preet a ope iruit wheever the legth i a odd umber of ¼ wavelegth. Thee reult would be the other way aroud if the far ed of the lie i ope-iruited. For a give ofiguratio of ed termiatio, the umber of ditit reoae i the ame a the total umber N of uit ell makig up the lie. Thi reult i relatively eay to derive ad i left to the problem. Thi reult alo follow by oiderig the umber of dyamial degree of freedom o the tramiio lie. There are N degree of freedom, oe for eah uit ell (the itataeou dyamial tate of a ell ould be peified, for example, by peifyig the urret flowig aroud the loop formed by the idutor ad the two apaitor of the ell ad it time derivative [ee figure ]). A you will lear i your advaed laial mehai oure, the umber of ormal mode of a dyamial ytem i the ame a it umber of degree of freedom; a i other imilar ytem, it ormal mode are jut the reoat mode.

12 4 0 9/7/0 PRELAB PROBLEMS. The phae veloity i v ( ) / φ = k. Ue the diperio relatio (3) to how that: v φ ( ) = v ( ) π φ. Show that if the total umber of uit ell i N, the thi i alo the umber of direte reoat frequeie of a avity otruted from the tramiio lie, a tated i the diuio followig equatio (). Coider both ed termiatio ae, equatio () ad (). Hit: eah reoat frequey mut have i k / N (equatio (3))., o that ( ) 3. If L= 5.0 mh ad C =.0 F, the what are Z 0,, ad f = π? What would be the low-frequey ( ) phae veloity i uit ell/e? 4. Give the reult of the previou problem, ad if the tramiio lie ha 0 uit ell ( N = 0 ), what hould be the lowet reoat frequey for eah of the followig termiatio oditio: a. Both ed horted b. Soure ed horted, far ed ope Sketh the amplitude v. poitio, ( ) V x, at that reoat frequey for eah of the above ae.

13 4 9/7/0 PROCEDURE O Off R L L L L L Termiated Ope Short C C C C C C R R Figure 8: The lumped-elemet tramiio lie apparatu oit of N = 0 uit ell joied a how. Two variable reitor, R L o the left (oure) ed ad R R o the right ed, permit adjutmet of the refletio oeffiiet of the lie termiatio. A with o the right ed allow a additioal eletio of horted or ope termiatio at that ed. The igal geerator i oeted to the BNC oetor o the left ed of the lie; it may be iolated from the iruit uig the other with o that the reitor value may be meaured without damagig a ohmmeter. L = 5.0 mh, C =.0 F, R L ad R R are adjutable 0 0 kω. Oe the igal geerator i attahed, the bottom odutor will be oeted to groud, a how. Figure 9: The etup howig oetio of the igal geerator ad omputer data aquiitio (DAQ) hardware. Not how i the hortig jumper wire ued to hort the reitae of R L (figure 8) durig teady-tate avity reoae meauremet. Computer data aquiitio ad otrol of the experimet i imilar to that of Experimet. The AI igal lead (+) may be oeted to the variou ode joiig uit ell; the ( ) iput hould be oeted to the ommo groud odutor.

14 4 9/7/0 Eure that the apparatu ha bee aembled a i figure 8 ad 9. Familiarize yourelf with the delay lie aembly ad it withe ad variable reitor. The ivetigatio to be ompleted durig the experimet are:. Ivetigate the propagatio ad refletio of a tep iput.. Determie the harateriti impedae (i the low-frequey limit). 3. Set-up ad meaure the lowet reoat frequey for the horted ope lie. 4. Examie the hape of the wave o the lie at thi reoat frequey. 5. Meaure the detailed frequey repoe of thi avity reoator over a wide rage of frequeie. Determie the utoff frequey. 6. Ivetigate the behavior of the tramiio lie at frequeie beyod the utoff frequey. 7. Fid the elf-reoae frequey of the idutor ued i the tramiio lie. 8. (optioal) Meaure the frequey repoe at ode 9 of the horted horted ofiguratio. What follow are additioal ote ad guidae for ompletig thee ivetigatio. Ue the Traiet Repoe program ad iput a low-frequey (~00 Hz?) quare-wave to ivetigate the propagatio propertie of the tramiio lie ad the refletio produed by variou ettig of the termiatio impedae. Examie the igal at the left ed of the tramiio lie (ode 0). With the right termiatio et to Short ad the Ope, ompare the igal with that predited by the theory (figure 7). Proper adjutmet of the oure termiatio (R L ) hould reult i oly a igle refetio; if it i ot adjuted to L/ C, you hould ee everal refletio of dereaig amplitude, eah arrivig after aother roud-trip delay of the tramiio lie. Whe R L i et properly, the iitial tep amplitude at ode 0 hould alo have approximately ½ the amplitude of the quare-wave iput tep (AI 0). How i the roud-trip delay time for the refletio related to the lowet reoat frequey (both ed horted)? What i the low-frequey delay time per uit ell? How doe thi ompare to the low-frequey phae veloity you alulated (problem 3)? Next et the right-ed with to Termiated ad adjut R R util the refletio vaihe, properly termiatig both ed of the lie. Note the hape of the igal at ode 0 ad it relatio to the iput quare-wave from the igal geerator. Doe it amplitude math the theory? Look at the igal at ode 5 ad ompare it to figure 6. Tur the left with to Off, dioetig the igal geerator, before you meaure R L ad R R uig a ohmmeter. How doe the 50Ω output impedae of the igal geerator affet the value of R L whe properly et to elimiate refletio? For the avity frequey repoe ivetigatio, hort R L uig a jumper wire o that the igal geerator ad AI 0 are oeted diretly to the iput of the tramiio lie. Coet AI at

15 4 3 9/7/0 the other ed of the lie with the termiatio with to Ope. Ue a mall-amplitude ie-wave iput from the igal geerator to fid the firt reoat frequey for thi ofiguratio (horted ope). Make ure your iput igal amplitude i ot et too high! The phae at the lowet reoae hould be 90, ad the gai i (4 / π )Q (Geeral Appedix A equatio () ad (9), with k = π ad x = ). Ue the Frequey Repoe program to aurately fid the lowet reoat frequey. Note the gai ad phae at the reoae (thi would be V ( ) V (0) uig the otatio of Geeral Appedix A). Reord the gai ad phae at eah of the ode to 0 for later aalyi, o that you a ompare to Geeral Appedix A equatio () for V ( x) V (0) at a reoae (remember, x goe from 0 at left ed to at the right ed of the lie, o x= N for ode ). Sie the right ed of the lie i a ati-ode for all reoae i the horted ope ofiguratio, you hould oet AI at that loatio for your frequey weep (up to ad iludig the utoff frequey). At ome frequey lightly above utoff, you hould reord the gai ad phae for eah of the firt few ode to ompare with the theory (equatio (8)). Frequey weep above utoff hould be doe with AI oeted at ode. Fid the idutor elf-reoae frequey by lookig for a harp ull i the output at ode at a frequey a few time higher tha the utoff frequey (refer to figure i Appedix A of thee ote, pg 4 6). The phae i 80 below thi frequey ad 0 above it. DATA ANALYSIS Do the tep-iput obervatio math the theory preeted o page 4 6 to 4 8? Compare the low-frequey phae veloity to. Do the gai ad phae v. poitio data (proedure tep 4) math the theory (Geeral Appedix A equatio ())? Aume the wave umber for eah of the horted ope reoae i that give i equatio (). Calulate xm i ( km N) for eah of the reoae ad fit thi modified diperio relatio m v. xm uig your oberved reoat frequeie. Aordig to the theory (equatio (3)), hould the fit be a trit proportio ( m = xm)? What doe thi fit give for the utoff frequey,? I there a patter to the reidual? Traform your data to ( / m) v. ( / xm) ad try a liear fit, that i: ( / m) = a + b( / xm). / I the fit better? Compare to b ad your oberved idutor elf-reoae frequey,, to a /. For a exteio to the theory whih ilude ee Appedix A of thi experimet.

16 4 4 9/7/0 APPENDIX A: EFFECTS OF INDUCTOR SELF-RESONANCE A idutor i the lumped-parameter tramiio lie i made by widig a log, thi wire ito a tightly-wrapped oil ad iertig it ito a dout-haped ferrite form. Beaue the may widig of the oil are very loe to oe aother ad are eparated by oly a thi layer of iulatio, eletri field form betwee adjaet widig ad attrat harge i the wire. I other word, adjaet widig form little apaitor whih provide a alterate path for the varyig urret flow through the wire. Coequetly, the oil of wire ha a apaitae whih i ituated i parallel with the idutae reultig from urret flow whih follow the tur i the oil. The equivalet iruit of the oil whih ilude thi apaitae i how i figure 0. C S L Figure 0: The equivalet iruit of a idutor formed from a loely-paked oil of wire ilude a parallel apaitae, how here with value C S. The impedae of the oil i the the parallel ombiatio of the two ompoet impedae, jlad j CS. The parallel L ad C form a tak iruit. S The parallel ombiatio of the idutor ad apaitor ha a equivalet impedae whih beome ifiite at the elf-reoae frequey of the oil, = S ; thi impedae i: oil ( ) ( ) Z = j L j C jl = = S ( ) jcs ( ) So for frequeie < the oil behave a a idutor, but for frequeie > it ha a apaitive impedae. A ee i (3), Zoil at. To ilude thi effet i the theory of the tramiio lie diperio relatio, we ubtitute Z oil for j L i the derivatio leadig to equatio (3) o page 4 : V V + V + = Zoil IC = Zoil ( jcv ) = V e jk N jk N + e = ( ) k 4 i = = N ( ) ( ) ( ) ( ) (3) = + (4) 4 i ( k N)

17 4 5 9/7/0 Equatio (4) i the ew diperio relatio whih ilude the effet of idutor elfreoae. Cutoff till our whe the argumet of the ie i π /; the group veloity vaihe here, ad k mut beome omplex at higher frequeie. The utoff frequey, however, i learly differet from that defied by the origial diperio relatio (3). Whe the ie i (4) i, the modified utoff frequey i lower; ee (5) ad figure, below. = = + 4 C S ( 4 + ) (5) Figure : A ompario of the diperio relatio (3) (dahed) with (4) (olid) for C = 0 CS. The utoff frequey i redued by ~0% by the preee of C S. The wave umber at utoff i k = Nπ for eah form of the diperio relatio. The lat expreio i (5) how that, agai, i jut the reoat frequey of the L ad total equivalet C i the iruit loop of a uit ell the oil tray apaitae C S i i parallel with the other apaitor, o it apaitae add to their equivalet erie ombiatio, C 4. Whe >, the wave umber k mut be omplex, a we foud previouly, o wave do ot propagate o the lie at frequeie above utoff. For < < the behavior i imilar to that of equatio (8), o there i a 80 phae hift from ode to ode, ad the amplitude fall geometrially with ode umber: V for < < : = exp oh V ( ) (6) From the above expreio it i lear that V = 0 at =, a metioed before. Thi fat provide the mot traightforward method to experimetally idetify. For frequeie above, the above expreio mut be modified. Without proof, we tate the behavior i (7).

18 4 6 9/7/0 V for > : = exp ih V V for : V ( ) CS C+ C S ; C C S (7) Above the phae hift i 0 (all ode are i phae), ad a otiue to ireae, the atteuatio per ell approahe a otat value, whih i jut the atteuatio due to a apaitive voltage divider ladder oitig of apaitor C ad C (figure ). S Figure : The atteuatio of the iput igal above utoff frequey, iludig the effet of the idutor tray apaitae, C S. The vertial gridlie i at ; the horizotal gridlie at the apaitive voltage divider ratio give by the fial expreio i (7).