Impedance (T) EELE 461/561 Digital System Design. Module #3 Interconnect Modeling with Distributed Elements Topics. Transmission Lines

Transcription

1 EEE 46/56 igital Systm sign Modul #3 ntrconnct Modling with istributd Elmnts Topics. mpdanc of Transmission ins Ttbook Rading Assignmnts Transmission ins mpdanc T - Transmission ins ar istributd lmnts - This mans that thr is propagation dlay from th bginning of th lin to th nd of th lin - n rality, all wirs ar istributd. Howvr, somtims thy ar so short th propagation dlay can b ignord. - W do this to simplify th circuit analysis: , , , , What you should b abl to do aftr this modul umpd Systm istributd Systm - th dpndant variabls & ar only a function of tim. - th dpndant variabls & ar a function of tim AN spac.. Know whn to us lumpd vs. distributd modling. alculat th rflctions off of an impdanc discontinuity 3. raw th timing diagram of a transmission lin with rflctions 4. raw a bounc diagram Pag - W say that any conductor that has ngth, nds to b tratd as a distributd systm. - Whn w us a Transmission in lmnt, w always spcify two paramtrs: mpdanc lngth Prop lay T T vlocity Pag umpd vs. istributd Transmission ins - Th first qustion w always ask whn looking at a systm is whthr it nds to b tratd as umpd or istributd. - W nd to trat a conductor as istributd whn: umpd vs. istributd Transmission ins - n a vacuum, th following prssion rlats frquncy and wavlngth: c f th lngth of th intrconnct is comparabl to th wavlngth prsnt in th signal if w hav a GHz sin wav in a vacuum: c f 8 3 m / s GHz.3m " lay is not larg rlativ to λ lay is larg rlativ to λ and you driv a lngth of prfct intrconnct i.., in a vacuum a full cycl of th sin wav occurs at th sourc bfor th nrgy is sn at th nd of th lin. This illustrats that th voltag on th lin is dfinitly dpndant on tim AN spac. Or said anothr way, th voltag in th lin is NOT th sam at all parts, so its distributd ffct cannot b ignord. Pag 3 Pag 4 Transmission ins Transmission in Paramtrs umpd vs. istributd - A mor prcis dscription for whn to us distributd modling is: Propagation lay - f our intrconnct rsids in a vacuum, it will travl at th spd of light: whn th tim to chang voltag on th signal is comparabl to th tim it taks to propagat down th lin c 8 m 3 s - W can us th rul of thumb that w us distributd modling whn: ngth of in sourc NOTE: som popl us λ/4, or a quartr wavlngth. W ll us λ/ for this class - This is quivalnt to: c / ns "/83ps or 83ps / in - To put in trms of digital ristims and prop dlay, th rul of thumb is: Us istributd Whn: T t ris Us umpd Whn: T t ris igital Systm EEE 46/56 sign Modul #3 Pag 5 - f it travls in a mdium that is NOT a vacuum, th vlocity is givn by: v c r and l T v Pag 6

2 Propagation lay Transmission in Paramtrs - Most microlctronic dilctrics hav rlativ prmittivity a.k.a., k or ilctric onstant btwn - what is th wav vlocity if r=4? v v.5 ft / ns 6"/ ns or c r "/67 ps m / s.5 m / s 4 Propagation lay Transmission in Paramtrs - Ristims of modrn off-chip digital drivrs ar anywhr from ps to ns if t ris=ns and r=4, whn do w nd to considr th intrconnct distributd? W considr th distributd natur whn th dlay of th intrconnct is / of th sourc ristim n/ or ps Sinc r=4, th spd of th intrconnct is 67ps/in That mans that w considr distributd at /67:.6 if t ris=ps and r=4, whn do w nd to considr th intrconnct distributd? /67 =.6 if t ris=ns and r=4, whn do w nd to considr th intrconnct distributd? /67 = 6 NOTE: 6 is prtty larg. That s why yars ago digital dsignrs didn t us transmission lins. Pag 7 Pag 8 Transmission in Paramtrs Transmission in Paramtrs Propagation lay - Today, ristims ar consistntly blow ns. That mans EERYTHNG in a systm is tratd as a transmission lin. NOTE: Whn w say systm, w ar typically talking about chip-to-chip communication using a combination of PB s, cabls, and connctors. Physical onstruction of a T-in On-hip intrconnct is still small nough to b tratd as a lumpd lmnt. ircuit Modl - A T-lin is a distributd lmnt in which voltag and currnt dpnd on both tim and spac. - A proprty of a distributd systm is that wavs can travl both in a forward and rvrs dirction. - Th voltag at any givn point is th suprposition of th forward and rvrs travling wavs. - W can crat a circuit modl of a transmission lin using RG s that will allow us to bttr undrstand th voltag and currnt on th T-lin. - A T-lin is simply a conductor. This mans that at, if its rsistanc is ngligibl, th wir is transparnt. - So w only considr th impdanc and prop dlay of th T-lin whn th wav is travling. - This allows us to quickly analyz a circuit at A and thn again at i.., stady stat to gt a fl for th bhavior at th bginning and nd of th dg transition. W can thn prform a dpr analysis to undrstand th transint bhavior. R = Rsistanc pr Unit ngth = nductanc pr Unit ngth = apacitanc pr Unit ngth G = onductanc pr Unit ngth Pag 9 Pag Transmission in Paramtrs Transmission in Paramtrs ircuit Modl - This sgmnt will modl th voltag and currnt dpndncy on tim and spac. - Using this basic RG sgmnt, w can crat a Transmission in modl that contains an infinit sris of ths lmnts. - As w add mor and mor sgmnts, th amount of "ngth" that th RG modl rprsnts will shrink to. ircuit Modl - Th R in this lmnt rprsnts th "conductor loss" mainly du to skin ffct, mor on this latr - Th G in this lmnt rprsnts th "dilctric loss" du to loss in th insulator, mor on this latr - Whn w includ th R and G in our modl, w hav a "ossy Transmission in" - f w assum thr is no loss, w can rduc our modl to an idal "osslss Transmission in" consisting of only nductanc and apacitanc. R = Rsistanc pr Unit ngth = nductanc pr Unit ngth = apacitanc pr Unit ngth G = onductanc pr Unit ngth Pag Pag

3 Transmission in Paramtrs Transmission in Paramtrs ircuit Modl - ts driv th rlationship btwn oltag & urrnt to tim and spac. ircuit Modl - W ntr th sgmnt at and w it th lin at +d - W first dfin th lngth of th wir using d. - Sinc our lctrical componnts ar dfind in unit lngth, th total valus can b found by multiplying th unit lngth valu by d. sg = d sg = d d Pag 3 d Pag 4 Transmission in Paramtrs Transmission in Paramtrs ircuit Modl - Th input voltag can b dscribd as:,t - Th input currnt can b dscribd as:,t - Th output voltag can b dscribd as: +d,t - Th output currnt can b dscribd as: +d,t, t +d, t ircuit Modl - W can writ an prssion for th voltag drop across th inductor using our prssion for th voltag on an inductor =di/dt: d, t, t d, t d dt - f w lt d, w ar lft with: +, t - + +d, t - d d d dt, t +d, t +, t - + +d, t - d Pag 5 Pag 6 d Transmission in Paramtrs Transmission in Paramtrs ircuit Modl - Now w can writ an prssion for th currnt using our prssion for th currnt in a capacitor i =dv/dt: d d, t, t d, t d dt - f w lt d, w ar lft with: ircuit Modl - Ths two st ordr diffrntial quations dscrib th intraction of and on a T-lin. - Ths ar known as th "Tlgraphrs Equations" d d d dt d d d dt d d d dt, t +d, t +, t - + +d, t - or mor formally:, t, t t, t, t t d Pag 7 Pag 8 3

4 Tlgraphrs Equations Transmission in Paramtrs - W can sparat ths coupld quations by diffrntiating ach to gt thir nd drivativ Tlgraphrs Equations Transmission in Paramtrs - W can now substitut into ach othr using d / ddt to form on, nd ordr diffrntial quation diffrntiat with rspct to d d d dt diffrntiat with rspct to t d d d dt d d d ddt d d ddt dt d d d ddt d d ddt dt d d d dt - This is known as th "Wav Equation" - You may hav sn th Wav Equation is this form: d u d u c dt dt Pag 9 Pag Wav Propagation Transmission in Paramtrs - Th Wav Equation has a coupl of known solutions that ar hlpful to us. Th first is for th vlocity of th wav. v - Th and in this prssion can dscrib a particular sgmnt, or th total and of th lin. Both will rsult in th sam vlocity. Propagation lay Transmission in Paramtrs - W can manipulat this quation to put it in a mor usful form by rarranging and putting it in trms of "Prop lay for a givn lngth" T v T T v - This says that th prop dlay for a transmission lin lngth with a total and is: T Pag Pag haractristic mpdanc Transmission in Paramtrs - ts go back and look at Tlgraphrs Equations in th Frquncy omain. Th Frquncy omain allows us to convrt th dpndncy on tim to a dpndncy on frquncy: haractristic mpdanc Transmission in Paramtrs - W can now substitut in our st ordr diffrntial quations into our nd ordr diffrntial quations to gt a solution: d d d dt d d d dt d d d dt d d d dt - ts now diffrntiat th frquncy domain solutions with rspct to to gt two, uncoupld, nd ordr diffrntial quations: d d d d d d d d d d d d d d d d d d d d Pag 3 Pag 4 4

5 5 Pag 5 Transmission in Paramtrs haractristic mpdanc - Th gnral solution for th oltag and urrnt is: Forward Rvrs Travling Travling Wav Wav - whr w dfin th as th Wav Propagation onstant: Pag 6 Transmission in Paramtrs haractristic mpdanc - f w diffrntiat th voltag solution, w can gt th solution for currnt by plugging it back into our original Tlgraphrs Equation: - Rarranging for, w gt: d d Pag 7 Transmission in Paramtrs haractristic mpdanc - Notic that this is in a form to giv us mpdanc =/: - whr is: Pag 8 Transmission in Paramtrs haractristic mpdanc - Finally, w can plug back in our prssion for th Wav Propagation onstant to gt a valu for th impdanc in trms of and : Pag 9 Transmission in Paramtrs haractristic mpdanc - This is calld th haractristic mpdanc of th transmission lin - things to notic about a osslss T-lin: - dos not dpnd on lngth - th valu of dos not vary with frquncy - th valu of dpnds on only and, which dpnd on th T-lin gomtry and matrials - th charactristic impdanc only has maning if wavs ar travling i.., f you put an Ohm mtr across a Transmission lin, youll s ohms. Howvr, if you snd in an A signal, th signal will s Pag 3 Transmission in Paramtrs haractristic mpdanc - f w hav a ossy Transmission in, th haractristic mpdanc of th transmission bcoms: - things to notic about a ossy T-lin lin: - it is ompl! G R

6 Basic T-in Equations Transmission in Paramtrs - Th two main quations w us for T-lins ar: Basic T-in Equations Transmission in Paramtrs - Notic that ths ar rlatd using and : T T T - So for a givn lngth of Transmission lin, if w can find th total capacitanc and total inductanc, w can driv th prop dlay and charactristic impdanc: T T T T T Pag 3 Pag 3 Basic T-in Equations Transmission in Paramtrs Transmission in Paramtrs Summary of Transmission in Paramtrs - Ths prssions say that if w can know th and T of a transmission lin, w can dtrmin th total capacitanc and inductanc of th lin: haractristic mpdanc: T T Prop lay: T T-in apacitanc: T T-in nductanc: T Pag 33 Pag 34 Rflction officint Transmission in Rflctions - W drivd th haractristic mpdanc of a transmission lin as: Rflction officint Transmission in Rflctions - As th wavs travl down th T-lin, rflctions may occur that caus opposit travling wavs osslss ossy - Th ratio of th rflctd wav to th incidnt wav is dfind as th Rflction officint: R G - W also dscribd th lctrical quantitis on a transmission lin consisting of both Forward and Rvrs Travling wavs using: - This is dpndant on two things: Th impdanc that th wav is currntly in i.., Th impdanc that th wav ss dirctly in front of it i.., oad or Pag 35 Pag 36 6

7 7 Pag 37 Transmission in Rflctions Rflction officint - W can driv using our T-lin sgmnt modl by placing a load impdanc at th nd of th circuit. - W also dfin th spatial location on th T-lin ust as bfor. Ecpt this tim, it maks th math asir if w dfin = as th location of, and -d as th location of d = =-d Pag 38 Transmission in Rflctions Rflction officint - Across th load impdanc, w hav: - And w dscrib th voltag and currnt as: - Substituting into th prssion for, w hav: d = =-d Pag 39 Transmission in Rflctions Rflction officint - Now w can rarrang to gt in th form - / + : Pag 4 Transmission in Rflctions Rflction officint - Th rsult is th dfinition of th Rflction officint in trms of and : Pag 4 Transmission in Rflctions Rflction officint - W can us this to dscrib th prcntag of th incidnt voltag inc that is rflctd rfl and th prcntag that is transmittd trans - As th wav travls down th T-lin, th incidnt wav is continuously valuating considring th impdanc it is currntly in, and th impdanc dirctly in front of it. - f th impdanc of th load matchs th impdanc that th wav is travling in, thr will b no rflctd nrgy. Or said anothr way, % of th incidnt wav will continu down th lin. rfl inc inc trans Pag 4 Transmission in Rflctions Rflctions on a T-lin ampl #, stp Th voltag arrivs at th nd of th T-lin B ns aftr th drivr A snds it. Th voltag at B is th instantanous suprposition of th incidnt wav v and any rflctd voltag. Sinc =infinity, thn is found using: This rsults in v of rflctd voltag. 3 Th total voltag obsrvd at ns is v+v=v. 5 5 ns

8 Transmission in Rflctions Rflctions on a T-lin ampl #, stp Transmission in Rflctions Rflctions on a T-lin ampl #,, stp 3 Th rflctd voltag v travls backwards down th lin and arrivs ns latr at A t=ns. Th nd rflction -v arrivs at th nd of th T-lin B ns latr t=3ns This rflctd wav valuats at A. Sinc = now is th sourc, thn is found using: 5 ns Again is calculatd: 3 5 ns 3 Th rflctd voltag obsrvd at ns is v+-v=v, so w don t s any nt chang at B. NOTE: A rflction still occurrd at A which travls back down th lin with a magnitud of -v. 3 This mans that % of th -v is rflctd, so th nt voltag at B = -v + -v=-v. This -v is suprimposd on th isting voltag, which was +. Th total rsult is = v. Pag 43 Pag 44 Transmission in Rflctions Rflctions on a T-lin ampl #, stp Transmission in Rflctions Rflctions on a T-lin ampl #, stp nitially, th voltag at A will b dictatd by th rsistiv dividr ntwork formd by R s and : 5 A. R 5 inc 833 This is th magnitud of th wav that will b initially launchd down th T-lin S v Th voltag.833v arrivs at th nd of th T-lin B ns aftr th drivr A snds it. Th voltag at B is th instantanous suprposition of th incidnt wav.833v and any rflctd voltag. Sinc =infinity, thn is found using: 5 ns 3 % of th.833v incidnt wav is rflctd so th obsrvd at ns is.833v+.833v=.667v. Pag 45 Pag 46 Transmission in Rflctions Rflctions on a T-lin ampl #, stp 3 Transmission in Rflctions Rflctions on a T-lin ampl #, stp 4 Th rflctd voltag.833v travls backwards down th lin and arrivs ns latr at A t=ns. 5 This wav valuats at th sourc using: ns Th rflctd voltag -.667*.833v=-.556v is suprimposd on top of th incidnc wav.883v which yilds a nt addition of.833v+-.556v=+.78v on top of th initial.833v that th drivr put on th lin. This yilds a total voltag at A of:.833v +.78v =.v. Th nd rflction -.556v arrivs at th nd of th T-lin B ns latr t=3ns Again is calculatd: 5 3 ns 3 This mans that % of th -.556v is rflctd, so -.556v v=-.v is suprimposd on th isting voltag at B.667v which yilds a total voltag of v= +.555v. NOTE: Th rflction of -.556v occurrd at A and now travls back down th lin toward B Pag 47 Pag 48 8

9 Transmission in Trminations Transmission in Trminations Trminations - W know that rflctions occur on a transmission lin any tim thr is an impdanc discontinuity. - W dscrib th prcntag of th incidnt wav that is rflctd using th rflction cofficint: Trminations - W can plac rsistors in our circuit in ordr to crat an impdanc match and rduc rflctions. - Rmmbr a Rsistor has an impdanc that is constant ovr all frquncis so it is an idal componnt to us for impdanc matching. - Whn w us rsistors to rduc rflctions, this is calld "trminating th transmission lin". - W call th componnt th trmination rsistor or trmination impdanc. - Thr ar a varity of trminations tchniqus, ach with advantags and disadvantags. whr th amplitud of th rflctd and transmittd voltag is rlatd to by: trans inc rfl inc Pag 49 Pag 5 Tchniqu #: oad Trmination Transmission in Trminations - ts s what happns if w plac a trmination rsistor at th nd of th transmission lin. - W will choos a rsistanc that is qual to th charactristic impdanc of th transmission lin. - W assum that th sourc impdanc of th drivr is and th charactristic impdanc of th transmission lin is 5ohms. Tchniqu #: oad Trmination Transmission in Trminations nitially, th full voltag stp dvlops at th bginning of th transmission lin sinc th sourc impdanc is. Th wav travls down th transmission lin in a constant impdanc and arrivs at th load on prop dlay T latr. 3 Th wav ss th trmination rsistanc and valuats : Sinc thr ar no rflctions, thr ar no mor transints on th transmission lin and w ar don. Sinc w ar now at i.., no transints, th drivr only ss th rsistanc of th trmination rsistor as its load. Pag 5 Pag 5 Tchniqu #: oad Trmination Transmission in Trminations Tchniqu #: oad Trmination Transmission in Trminations Advantags: Simpl f th rcivr is capacitiv which it is, th trmination rsistor will rduc th ffctiv tim constant of th load. mor on this latr 3 Th full drivr voltag is dlivrd to th rcivr isadvantags: Whn th transints hav ndd, th drivr now has a load that it is driving. This incrass powr consumption. W assumd an idal sourc impdanc Rs=, but in rality th sourc has output impdanc so aftr th transints hav ndd, thr will b a rsistiv dividr btwn R S and R. This mans that th full voltag of th drivr will not b sn at th rcivr. Pag 53 Pag 54 9

10 Tchniqu #: oad Trmination Trmination oltag Transmission in Trminations - Th voltag that w trminat to dosnt hav an ffct on th impdanc matching sinc in A analysis w ignor sourcs. - W can choos th voltag w trminat to. - A common approach is to trminat to Ground sinc w hav mor accss to grounds in our systm. Transmission in Trminations Tchniqu #: Thvnin Equivalnt - A tchniqu to provid a trmination impdanc to an arbitrary voltag is to us two rsistors to form a Thvnin quivalnt circuit. - W ti on rsistor btwn th signal lin and R and th othr btwn th signal and ground R. - W slct th valus of th rsistors to giv us our dsird trmination impdanc and trmination voltag. - W can trminat to a voltag in th middl of our voltag swing in ordr to rduc powr consumption i.., /. This prvnts th full voltag swing from bing dvlopd across th trmination rsistor at. - On drawback to trminating to a voltag is that you nd to produc th trmination voltag. ommonly, w only hav ground and powr in our systm so w would nd to add mor circuitry to gnrat th trmination voltag. Pag 55 Pag 56 Transmission in Trminations Tchniqu #: Thvnin Equivalnt Transmission in Trminations Tchniqu #: Thvnin Equivalnt W hav a 5 transmission lin that nds to b load trminatd. W hav a systm with a 3.3v powr supply. W want to us a Thvnin quivalnt ntwork to form a trmination impdanc of 5 to v. What ar th valus of R and R? th R R R// R 5 R R R th v R R W solv using th two quations and two unknowns to gt R= ~8 and R = ~35. A sourc of rror in this tchniqu is that physical rsistors only com in crtain valus. So w hav to choos rsistor valus that ar availabl. Pag 57 Pag 58 Transmission in Trminations Tchniqu #: Thvnin Equivalnt Advantags: A trmination voltag can b cratd without adding an additional voltag gnration circuit in your systm. Tchniqu #3: Sris Trmination Transmission in Trminations - What would happn if w addd a rsistor at th sourc and lft th nd of th transmission lin opn? - W will choos a rsistanc that is qual to th charactristic impdanc of th transmission lin. - W will assum th impdanc of th transmission lin is 5ohms. isadvantags: Rquirs an additional rsistor compard to a load trmination to Ground approach. Sinc rsistors only com in pr-dfind valus, th quivalnt trmination impdanc w gt might not b actly matchd to and rflctions may occur. Pag 59 Pag 6

11 Tchniqu #3: Sris Trmination Transmission in Trminations Tchniqu #3: Sris Trmination Transmission in Trminations nitially, half of th sourc voltag dvlops at th bginning of th transmission lin du to th rsistiv dividr formd btwn th sourc rsistor and th impdanc of th transmission lin. Sinc th impdancs ar qual, th voltag that dvlops is actly half of th sourc voltag: init Th half wav travls down th transmission lin in a constant impdanc and arrivs at th load on prop dlay T latr. 3 Th wav ss th opn nd of th T-lin and valuats : Stp RS _ tlin _ tlin 5 5 Stp or % 4 Th % positiv rflction du to th opn nd of th t-lin is suprimposd on th incidnt wav. Sinc th incidnt wav is / of th intndd voltag, th voltag stp that is sn at th rcivr is actually th intndd voltag swing i.., / + /. 5 Th rflctd nrgy travls backwards down th transmission lin toward th sourc. Whn it arrivs at th sourc it valuats. t now ss ar sris rsistor as which w choos to match th charactristic impdanc of th T-lin: Sinc thr ar no r-rflctions, thr ar no mor transints on th transmission lin and w ar don! or % Pag 6 Pag 6 Tchniqu #3: Sris Trmination Transmission in Trminations Tchniqu #3: Sris Trmination Transmission in Trminations ts look at what happnd - W wantd to transmit a voltag stp with an arbitrary magnitud to th rcivr. - By placing a rsistor at th sourc with th sam impdanc as th transmission lin, ONY HAF of th voltag travld down th T-lin. - HOWEER, sinc th nd of th lin was opn, it princd a % positiv rflction which whn suprimposd on th incidnt wav, producd a voltag at th rcivr that was actly what w intndd! - Whn th rflction travld back to th sourc, it was trminatd with th sourc rsistor, thus nding any furthr transints. Pag 63 Pag 64 Tchniqu #3: Sris Trmination Advantags: Transmission in Trminations Th transmission lin is trminatd but w didnt add any path to th circuit. This rsults in no additional powr consumption. isadvantags: t dosn t dcras th tim constant of th rcivr s load. Tchniqu #4: oubl Trmination Transmission in Trminations - W v sn that a load trmination will rduc rflctions at th nd of th lin. - W v sn that a sris trmination will rduc rflctions of wavs travling backwards toward th sourc. - f w put both a Sris and a oad trmination, thn w would b abl to liminat th most rflctd nrgy. NOTE: n ths simpl systms, our idal trmination rsistors ar liminating all rflctions. n a ral systm, thr will b othr impdancs that caus rflctions mor latr Pag 65 Pag 66

12 Tchniqu #4: oubl Trmination Transmission in Trminations Tchniqu #4: oubl Trmination Transmission in Trminations nitially, half of th sourc voltag dvlops at th bginning of th transmission lin du to th rsistiv dividr formd btwn th sourc rsistor and th impdanc of th transmission lin. 4 Sinc thr ar no r-rflctions, thr ar no mor transints on th transmission lin and w ar don! Sinc th impdancs ar qual, th voltag that dvlops is actly half of th sourc voltag: init Stp RS _ tlin _ tlin Stp Th half wav travls down th transmission lin in a constant impdanc and arrivs at th load on prop dlay T latr. 3 Th wav ss th load trmination and valuats : or % Pag 67 Pag 68 Transmission in Trminations Tchniqu #4: oubl Trmination Advantags: Both forward and rvrs travling wavs ar trminatd isadvantags: Th voltag lvl that is dlivrd to th load is ½ of what th drivr outputs. Rquirs two trmination rsistors. Transmission in Bounc iagrams Bounc iagrams - A graphical way to kp track of rflctions on a T-lin. - W plot Tim vs. ocation. - W can calculat th at ach impdanc discontinuity - for ach boundary and wav dirction will b th sam throughout th ntir analysis. - Th bounc diagram givs us a procss to tabulat all of th information for ach rflction. - W continu th diagram until th transints ar small nough for us to ignor < -%. Pag 69 Pag 7 Bounc iagrams Transmission in Bounc iagrams Rflctions from s Transmission in Bounc iagrams - Until now, w hav considrd rflctions from idal T-lin lmnts. - An idal T-lin has an impdanc that is constant with frquncy. - Whn w hav ractiv lmnts in our systm i.., s and s, th rflctions hav a mor dynamic rspons bcaus th impdanc of ths lmnts chang with frquncy: f f - f w stimulatd our systm with sin wavs, w could calculat th impdanc of s and s dirctly in ordr to find. - f w ar stimulating our systm with a prfct squar wav, ach spctral componnt will s a diffrnt impdanc and will rsult in a diffrnt. Pag 7 Pag 7

13 Transmission in Rflctions from apacitors Rflctions from apacitors Transmission in Rflctions from apacitors Rflctions from apacitors - f w us a prfct stp i.., t ris=, thn a capacitor will instantanously look lik a short circuit to th incidnt wav. 5ps ristim w/ pf oad apacitor in a doubl trminatd systm. - As th capacitor chargs, its impdanc will gt largr until it ultimatly looks lik an opn - f w hav a finit ristim which w always do, w can calculat th avrag impdanc of a capacitor for a givn chang in voltag i.., t ris d dt t 9 tris - Th impdanc of th capacitor must b combind with any othr impdancs that th incidnt wav ss: - Th rsultant rflction cofficint rprsnts an stimat of th maimum magnitud of th rflctd nrgy. 9 ris tris 5 p 5 p oad 5// % % rprsnts th maimum magnitud of th rflction that will b sn. Pag 73 Pag 74 Transmission in Rflctions from apacitors Rflctions from apacitors - th rsultant systm ristim can b stimatd using th Sum-of-Squars mthod. - th load ristim is found using:. t t ris ris t t T T oad t. - n this ampl, th load tim constant is th capacitanc tims th paralll combination of th 5ohm load rsistor and th 5 ohm transmission lin: t ==5//5 p=5p ris t 743ps ris 5 p. 5 p Transmission in Rflctions from nductors Rflctions from nductors - f w us a prfct stp i.., t ris=, thn an inductor will instantanously look lik an opn circuit to th incidnt wav. - As th inductor chargs, its impdanc will gt smallr until it ultimatly looks lik a short - f w hav a finit ristim which w always do, w can calculat th avrag impdanc of an inductor for a givn chang in voltag i.., t ris d dt t 9 - Th impdanc of th inductor must b combind with any othr impdancs that th incidnt wav ss: - Th rsultant rflction cofficint rprsnts an stimat of th maimum magnitud of th rflctd nrgy. ris 9 t ris Pag 75 Pag 76 3