Topics EEE 46/56 Digital Systm Dsign. Skin Ect. Dilctic oss Modul #7 ossy ins ossy ins - Whn w divd Tlgaphs Equations, w mad an assumption that th was no loss in th quivalnt cicuit modl i.., =, = - This allowd us to simpliy th math and com up with th ollowing impotant quations o a osslss T-lin: Ttbook ading Assignmnts. 9.-9. What you should b abl to do at this modul. Dscib th physical phnomnon bhind Skin Ect and Dilctic oss. Us a modn AD tool to simulat th bhavio o lossy tansmission lins T D Pag Pag icuit Modl - th lin is lossy, w nd to includ th sis sistanc and shunt conductanc back into ou quivalnt T-lin cicuit modl icuit Modl - ts div th lationship btwn oltag & unt to tim and spac with th ull modl - W ist din th lngth o th wi using. - Sinc ou lctical componnts a dind in unit lngth, th total valus can b ound by multiplying th unit lngth valu by. = sistanc p Unit ngth = nductanc p Unit ngth = apacitanc p Unit ngth = onductanc p Unit ngth sg = sg = sg = sg = Pag 3 Pag 4 icuit Modl - W nt th sgmnt at and w it th lin at + icuit Modl - Th input voltag can b dscibd as:, - Th input cunt can b dscibd as:, - Th output voltag can b dscibd as: +, - Th output cunt can b dscibd as: +,, +, +, - + +, - Pag 5 Pag 6
icuit Modl - W can wit an pssion o th voltag dop acoss th inducto and sisto using K: d,,,, - Which is wittn as:,, d,, icuit Modl - Now w can wit an pssion o th output cunt using K: d,,,, - Which can b wittn as:,, d,, - w lt, w a lt with a dintial quation: d, d,, - w lt, w a lt with anoth dintial quation: d, d,, Pag 7 Pag 8 icuit Modl - Ths two st od dintial quations dscib th complt intaction o and on a T-lin. i.., "Tlgaphs Equations" Wav Equations - W can put th voltag pssion into th om o th Wav Equation by dintiating th ist Tlgaph quation with spct to d d d d dintiat with spct to d - ts put ths into a mo usabl om using phaso psntation wh d/ d d d d Pag 9 Pag Wav Equations - W can now plug in ou pssion o th divativ o cunt: d d d d - aanging, w gt: Wav Equation - w din g as: - w can wit th voltag pssion in Wav Equation om as: d d Pag Pag
Wav Equations - Now lts put th cunt pssion into th om o th Wav Equation by dintiating th scond Tlgaph quation with spct to dintiat with spct to d d d Wav Equations - W can now plug in ou pssion o th divativ o voltag: d d d d - aanging, w gt: d Pag 3 Pag 4 Wav Equations - Sinc, w hav alady dind g as: Wav Equations - Ths two nd od dintial quations dscib th voltag and cunt popagation on a lossy tansmission lin: - w can wit th cunt pssion in Wav Equation om as: d d d - W call g th "ompl Popagation onstant" which consists o a al and maginay pat: Pag 5 Pag 6 Attnuation onstant - W call th al pat o g th Attnuation onstant - This quantity has units o Np/m NOTE: A Napi is unit o atio. t is usd to dscib atios o voltags and cunts. Np ln - mmb that in a osslss lin, =. - n a ossy lin Phas onstant - W call th maginay pat o g th Phas onstant m m - This quantity has units o ad/m - This can b usd to calculat th wav vlocity with: vlocity p ompl Popagation onstant o Passiv T-lins - W choos th squa oot valus o g that giv positiv valus o and. - Fo passiv T-lins, Pag 7 Pag 8 3
4 Pag 9 haactistic mpdanc - Th Wav Equations hav tavling wav solutions o a lossy mdium in th om o: Fowad vs Tavling Tavling Wav Wav Pag haactistic mpdanc - w dintiat th voltag solution, w can gt th solution o cunt by plugging it back into ou oiginal Tlgaphs Equation: - aanging o, w gt: d d Pag haactistic mpdanc - mmbing th dinition o impdanc is th atio o ith th owad o vs tavling wavs: - W can now plug in ou quations o and, and only consid th owading tavling wavs: Pag haactistic mpdanc - Now w hav ou pssion o th chaactistic impdanc o a lossy tansmission lin: - Notic that this quantity is compl. This implis that th magnitud o th impdanc dpnds on th incidnc wavs quncy. - Whn a digital signal is tavling in a lossy lin, ach hamonic componnt o th squa wav will pinc a dint impdanc. Pag 3 Soucs o oss Soucs o oss - oss s to th amount o signal tansmittd that dos not ach th civ. - oss can occu du to many soucs: mpdanc Mismatchs ading to lctd Engy oupling to adacnt Tacs 3 adiation oss 4 onducto oss 5 Dilctic oss - Whn w talk about, w tnd to ocus on itms 4 & 5 - lctions & oupling can b modld using ou standad modl o a T-lin. - adiation is typically small, and is diicult to modl using cicuit modls so w tat that using EM simulatos. - onducto oss and Dilctic oss can b modld using ou complt tansmission lin cicuit. - Whn popl talk about ossy ins, thy a typically ing to onducto & Dilctic oss Pag 4 onducto oss Skin Dpth - At D, th sistanc o a conducto is popotional to: - th coss-sctional aa o th conducto - th bulk sistivity o th matial - At D, th chag is qually distibutd acoss th coss-sction o th conducto: Aa W
onducto oss onducto oss Skin Dpth - Whn A cunt lows though th conducto, th chag is not qually distibutd within th coss-sctional aa o th conducto. At A, th cunt will attmpt to ind th path o last impdanc - This sults in two tnds: Skin Dpth - Th phnomnon o th cunt lowing though this ducd coss-sction o th conducto is dscibd using a quantity calld Skin Dpth - Th ducd coss-sction has th ct o incasing th sis sistanc o th conducto as th quncy incass. Th cunt within th conducto will spad out as a as possibl in od to minimiz th patial sl-inductanc o th conducto: Th cunt within th conducto will ty to mov as clos as possibl to th oppositly dictd tun cunt in mo to maimum th patial mutual inductanc btwn th two cunts. Pag 5 Pag 6 onducto oss onducto oss Skin Dpth - Skin Dpth is dscibd by and is pssd as: Skin Dpth - Skin Dpth dictly cts th sis sistanc o a Tansmission lin by ducing th ctiv coss-sctional aa o th conducto that th cunt can low though. S Aa wh: is th sistivity o th conducto is th conductivity o th conducto / is th magntic pmability - Skin dpth has units o mts and is th dinition o th dpth blow th suac o a conducto with dpth d at which th cunt dnsity J dcays to / ~.37 o th cunt dnsity o th suac J S: d / J J S - Notic that th skin dpth is invsly popotional to th squa oot o quncy: - Sinc skin dpth is in th dnominato o th pssion o sis sistanc o th conducto, this mans that th sistanc is popotional to th squa oot o quncy: S Pag 7 Pag 8 onducto oss onducto oss Skin Dpth - W modl th conducto loss du to skin dpth using th sis sisto in ou T-lin modl. Total onducto oss - Th complt pssion o conducto loss should also includ any D loss du to th sistivity o th bulk matial: - This sistanc valu is dpndant on quncy: S conducto D A - Whn stimulating th lossy tansmission lin with a digital signal, ach quncy componnt o th signal will pinc a dint sis sistanc. - Not that it is had to din on complt pssion o th A conducto sistanc bcaus it dpnds on th gomty o th conducto. i.., ound, squa, ctangl, tc - Fo a givn coss sctional shap, th skin dpth is thn applid to that shap in od to pdict th nw coss-sctional aa that th cunt lows though. Pag 9 Pag 3 5
Dilctic oss - Now w mov to th nd main ct in lossy lins, th dilctic. - W us th lmnt in ou modl to account o dilctic loss. Dilctic oss - An idal capacito will sto a paticula amount o chag dpnding on th voltag applid. Q - Th capacito is constuctd using two conducting suacs spaatd by a dilctic. - n th idal cas, this stuctu dos NOT allow D cunt to low btwn th two tminals o th dvic. Pag 3 Pag 3 Dilctic oss - n ality, th chag in th capacito is hld by lctic dipols within th matial. - an A voltag is applid to th capacito, th dipols will align to th diction o th applid lctic ild. - This movmnt o chag sults in A cunt low. Dilctic oss - Sinc th is no D cunt o no in-phas cun, th is no al pow dissipation in an idal capacito. - This allows us to say that th sistivity o an idal capacito is. - owv, al capacitos do hav som sistanc associatd with thm. - This mans that cunt will low though th capacito that is in phas with th voltag. - W modl this sistanc with a sisto componnt in paalll with th capacito. - Th amount o cunt that lows is popotional to th at-o-chang o voltag acoss th capacito: d i c Pag 33 Pag 34 Dilctic oss - Th dnsity o availabl dipols in a matial to hold th chag is lctd in th dilctic constant i.., th lctic pmittivity. - Th dnsity o dipols availabl in ai is dscibd with a pmittivity o - a paalll plat capacito was constuctd using an ai dilctic, th capacitanc would b givn by: A t Dilctic oss - Th dipols in th matial do not -oint instantanously upon a chang in voltag. - Fo a givn tim vaying voltag, th a going to b dipols that a pctly alignd with th applid lctic ild. Ths dipols contibut to th capacitanc o th stuctu and sult in an out o phas cunt -9 - Du to th init spd at which th dipols can chang, th will b dipols that a alignd ppndicula to th applid lctic ild which poduc an in phas cunt - This in phas cunt is th souc o th lakag cunt in a capacito - th ai was thn placd with a dint insulating matial with >, th nw capacitanc would b dscibd as: Pag 35 Pag 36 6
Dilctic oss - w apply an abitay sinusoidal voltag to a capacito: th cunt that sults is dscibd as: t Th ompl Dilctic onstant - W now can us th lativ pmittivity o th matial to dscib both th in phas cunt and th out o phas cunt by making it a compl quantity: wh: d - W can us ou dinition o a non-ai dilctic capacito to includ th dilctic constant in this quation: d = th compl dilctic constant = th al pat o th compl dilctic constant, psnting th out o phas cunt low in th capacito. This is what w hav always calld simply th dilctic constant. mmb that this cunt is always dind as bing -9 o out o phas with th voltag = th imaginay pat o th compl dilctic constant, psnting th in phas cunt low in th capacito. This is th nw quantity that wv addd to psnt th cunt du to th loss in th dilctic. Sinc this cunt is in phas, w nd to apply a "-" sign in th compl quantity to -align it back to th voltag mmb by dinition th cunt in th capacito stats at -9 o Pag 37 Pag 38 Th loss angl - w dscib th angl btwn th al and imaginay pats o th compl dilctic constant as th loss angl - Not: this is NOT th sam as th skin dpth, it is ust an unotunat coincidnc that dlta is usd o both. Somtims popl will us o th loss angl Th oss Tangnt, tan - By convntion, w us th atio o th imaginay to th al pats o th dilctic constant to dscib th loss in a matial: - This is calld th oss Tangnt o Dissipation Facto m tan tan m Pag 39 Pag 4 Th oss Tangnt Dissipation Facto - At this point, things can gt a littl conusing. mmb that: - th dilctic constant is dscibd as a compl quantity in od to dscib both: - th out o phas cunt du to an applid voltag - th in phas cunt du to an applid voltag - Th out o phas cunt that sults om an applid voltag is actually th pctd spons o an idal capacito. That is why this quantity is dscibd using th al pat o th compl dilctic constant. - Th in phas cunt that sults om an applid voltag is du to th loss in a al capacito and is dscibd using th imaginay pat o th compl dilctic constant. Th oss Tangnt Dissipation Facto - Now lts dscib th loss in th dilctic using th oss Tangnt. - Th complt pssion o th cunt in ou capacito is givn by: m - Th sistanc o th matial is dscibd as: lakag tan tan Pag 4 Pag 4 7
Th oss Tangnt Dissipation Facto - W can now put his pssion o lakag into tms o bulk conductivity o th matial: lakag tan - Th dinition o sistanc & capacitanc o a tst stuctu is looking om th top to bottom: ngth Aa Aa Aa Aa - Ths two pssions can b combind to om a lationship btwn th loss tangnt & th bulk conductivity o th matial: tan Aa tan Th oss Tangnt Dissipation Facto - Now w hav th bulk conductivity and sistivity in tms o th oss Tangnt and quncy: tan tan - This can w usd to dscib th conductanc o sistanc o th shunt sisto in ou modl: - W typically us to modl th dilctic loss bcaus it scals dictly with lngth. That way all 4 o ou T-lin paamts will scal with lngth Pag 43 Pag 44 Dilctic oss - W modl th conducto loss du to skin dpth using th shunt conductanc in ou T-lin modl. - This conductanc valu is dpndant on quncy: shunt Total oss - w now s that th a two quncy dpndant soucs o loss in ou lossy T-lin. - This sults in high quncis bing attnuatd mo than low quncis. - Th two quncy dpndant soucs a spcd in tms o unit lngth i..,, dilctic tan S D A - Whn stimulating th lossy tansmission lin with a digital signal, ach quncy componnt o th signal will pinc a dint shunt conductanc. S shunt Pag 45 Pag 46 Signal locity & Dispsion - w hav sn that th vlocity o a sin wav in a lossy mdium dpnds on th quncy. u p - Fo a digital signal, this mans that dint quncy componnts will tavl though th lossy lin at dint ats. - Th nd sult will b a distotd signal at th civ. Appoimations - Th a a coupl appoimations that can b mad to simpliy th analysis o lossy lins: ow oss gim - whn S<<X and <<X - w can us th losslss quations o and vlocity ood onducto - whn S<<X i.., skin ct is ngligibl - This phnomnon is calld Dispsion ow oss Dilctic - whn <<X i., th dilctic lakag is ngligibl Pag 47 Pag 48 8