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首页 SI经典名著之三+蓝宝书——数字信号完整性和建模

SI经典名著之三+蓝宝书——数字信号完整性和建模.pdf

SI经典名著之三+蓝宝书——数字信号完整性和建模

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2010-01-15 0人阅读 举报 0 0 0 暂无简介

简介:本文档为《SI经典名著之三+蓝宝书——数字信号完整性和建模pdf》,可适用于IT/计算机领域

ChapterSIGNALINTEGRITYDigitalsystemsrelyonsignalingfromdriverstoreceiverstopassinformationbetweentheircomponentsReliablesignalingisachievedwhenthesignalingspecificationsaremetunderfulladversenoiseconditionsaswellasdevicebehaviorvariationsduetobothprocessdeviationsindevicemanufacturingandnormalchangesovertheoperatingtemperatureAbreakdownincommunicationleadstoglitcheswhereunintendedorincorrectdataistransferred,asituationcalledfalsesignalingCriticaldatapathswithinsystemsoftencontainsafeguardsagainstfalsesignaling,andtheprimarysystemforthisisdataredundancythroughparityParitycanenablethesystemtodetectsmallscalesignalingfailures,whileanerrorcorrectionschemecanbeoptionallyincludedtocorrectdatafaultsdetectedusingparityinformationHowever,parityanderrorcorrectioncannotbereliedupontomakeanoisysystemstableandreliableInaddition,itisoftennotpracticalintermsofcostorperformancetoincludeparityanderrorcorrectiononeverycircuitSuperimposedonthedesiredsignalsareunwantedwaveforms(ie,noise)generatedfrommanysourcesTheprincipalsourcesarecrosstalk,impedancemismatch,simultaneousswitchingnoise,andmultiplereflectionsEachcanbeindependentlycharacterizedtofacilitateanunderstandingofthemechanismsthatdegradesignalqualityandtohelpguidedesigndecisionsInrealsystems,allactsimultaneouslyandrequiredetailedcircuitsimulationtoobtaingoodestimatesofthetotalwaveformoneachsignallineSignalIntegrityChapterTransmissionLinesAtransmissionlineisatwoconductorinterconnect(sothatitcancarrysignalfrequencycomponentsdowntoDC)thatislongcomparedtotheconductorcrosssectionanduniformalongitslengthBecausemanyinterconnectsaredominatedbylongrunsoverunbrokengroundplanes(tominimizeradiationandEMIsusceptibility),theycanbeaccuratelymodeledastransmissionlines,andmuchofsignalintegrityanalysisisbasedonthemIfashortlengthofatransmissionlineisconsidered,thenthelumpedapproximationappliesandthetransmissionlinecanbemodeled,asshowninFigure,withseriesresistanceandinductanceandwithshuntcapacitanceandconductanceApplyingKirchhoff’svoltagelawaroundtheloop,thenv(z∆z,t)−v(z,t)=−Ri(z,t)−Ldi(z,t)dt()Similarly,Kirchhoff’scurrentlawappliedatz∆zyieldsi(z∆z,t)−i(z,t)=−Gv(z∆z,t)−Cdv(z∆z,t)dt()Dividethroughby∆zandlet∆z→,then()and()transformfromdifferenceequationstothedifferentialequationsi(zz,t)∆v(zz,t)∆−−zz∆zLRGv(z,t)i(z,t)CFigureLumpedmodelofashortlengthofatransmissionlineSectionTransmissionLines∂v(z,t)∂z=−ri(z,t)−)∂i(z,t)∂t()and∂i(z,t)∂z=−gv(z,t)−c∂v(z,t)∂t,()wherethelumpedcomponentvaluestransitiontotheperunitlengthquantitiesr,),c,andgduetonormalizationby∆zSimultaneoussolutionofthetransmissionlineequations()and()yieldsthevoltageandcurrentatanypointonthetransmissionlineTimeDomainSolutionTransmissionlinesfordigitalsignalingoftenhavelowlossesForexample,theeffectofhalfanOhmoflossonaΩtransmissionlinedrivenwithaΩdriverisnegligibleformostapplicationsTofacilitateinvestigationsoftheeffectsofvarioussystemimperfectionsonsignalintegrity,losseswillbeneglectedThelosslesstransmissionlineequationsarerecoveredfrom()and()bysettingr=g=yielding∂v(z,t)∂z=−)∂i(z,t)∂t()and∂i(z,t)∂z=−c∂v(z,t)∂t()AnimportantpropertyofthelosslesstransmissionlineisthatpulsespropagateundistortedalongthelengthofthelineConsideranarbitrarywaveformsuchastheoneinFigure,wherethewaveshapeisdescribedbythefunctionf(τ)withτanindependentvariableSincetherearenolossesandnofrequencydependenceto)orc,thewaveformmustmovedownthelineunchangedinshape(seesectionforaproof)andcanbedescribedmathematicallybyf(z,t)=f(τ)=f(z−νt),whereτ=z−νtAmaximumorminimumofthewaveformoccurswhen∂τ∂t=,so∂τ∂t==dzdt−ν,SignalIntegrityChapterτf()τFigureArbitrarywaveformforpropagationdownatransmissionlineorν=dzdt,indicatingthatthemaximumorminimumpointismovinginthezdirectionwithvelocityνThepartialderivativesin()and()canberewrittenintermsofτbynotingthat∂∂z=∂∂τ∂τ∂z=∂∂τ∂(z−νt)∂z=∂∂τand∂∂t=∂∂τ∂τ∂t=−ν∂∂τ,then∂v(τ)∂τ=)ν∂i(τ)∂τ()and∂i(τ)∂τ=cν∂v(τ)∂τ()Eliminating∂i(τ)∂τbetween()and()andcancelling∂v(τ)∂τyieldsν=√)c,()SectionTransmissionLinessothevelocityofanarbitrarypulsecanbedirectlycomputedfromtheperunitlengthinductanceandcapacitanceofthetransmissionlineIntegrating()withrespecttoτwhileassumingthatnostaticchargeisonthetransmissionline(sothattheintegrationconstantvanishes),thenv(τ)=√)ci(τ),where()isusedTherefore,thevoltagesandcurrentsofanarbitrarywaveformonalosslesstransmissionlineareinphaseandrelatedbytheimpedanceZo=√)c,calledthecharacteristicimpedanceofthetransmissionlineForatransmissionlineoflengthd,thetimeforawavetotravelthelengthofthelineiscalledthedelayorthetimeofflight(TOF)andcanbecomputedasTOF=d√)c()ThelosslesstransmissionlineiscompletelyspecifiedbyitscharacteristicimpedanceanddelayNotethat)=ZoTOFd()andc=ZoTOFd()Theanalysescanberepeatedwithτ=zνtwithidenticalresults,exceptthatthewaveformtravelsinthe−zdirectionwithvelocityν=√)candthevoltageandcurrentarerelatedbyv(τ)=−Zoi(τ)EffectiveDielectricConstantSincetheTOFcanbefoundgiventhelengthofatransmissionlineandthevelocityofawaveonit,thevelocityisoftentheunknownparameterthatmustbefoundSignalIntegrityChapterInatransmissionlinewheretheelectricandmagneticfieldsarecompletelyencasedinadielectricwithdielectricconstantr,thenthevelocityisν=co√r,()whereco=×msisthevelocityofawaveinavacuum(r=)whichisalsocalledthefreespacespeedoflightFortransmissionlinessuchasstripline,dualstripline,embeddedmicrostrip,andcoax,thevelocityiseasilycomputedoncethedielectricconstantofthefillermaterialisknownWhentheelectricandmagneticfieldsrunthroughtwodielectrics,thewavestillpropagateswithsomevelocityGeneralizing()yieldsν=co√eff,whereeffisaneffectivedielectricconstantForatransmissionlinelikemicrostrip,thetwodielectricsareairwithr=andthesubstrateTheeffectivedielectricconstantmustliebetweenthesetwo,andsincemostofthefieldisbelowthestripinthesubstrate,theeffectivedielectricconstantmustbeclosertothedielectricconstantofthesubstratethantothatofairEffectivedielectricconstantsareconvenientbecausetheyofferahandyshortcutinmanysituationsandcanbeeasilyestimatedforapproximatecalculationsForexample,ifr=forthesubstrateinmicrostrip,theneff≈Forthelosslesscase,theformulasin()and()areeasilymodifiedfortheeffectivedielectricconstanttobe)=Zo√effco()andc=√effZoco()DirectionalIndependenceTheanalysisoflosslesstransmissionlinescanbecarriedfurthertoshowthattwowaveformstravelinginoppositedirectionsdonotinteractLetvdenoteavoltageSectionTransmissionLineswaveformlaunchedinthezdirection,whilev−indicatesonetravelinginthe−zdirectionDuetothelinearityofMaxwell’sequations,andassuminglinearmaterials,thetotalvoltagemustbethesuperpositionofthesetwo,sov(z,t)=v(z−νt)v−(zνt)()Thetotalcurrentistheni(z,t)=Zov(z−νt)−Zov−(zνt)()Substitutingtheseexpressionsinto()and()yields∂v∂z∂v−∂z=−)Zo∂v∂t)Zo∂v−∂tand∂v∂z−∂v−∂z=−cZo∂v∂t−cZo∂v−∂tAddingthesetworesultsin∂v∂z=−()ZocZo)∂v∂t()Zo−cZo)∂v−∂t,whilesubtractingprovides∂v−∂z=−()Zo−cZo)∂v∂t()ZocZo)∂v−∂tThesecanbefurthersimplifiedbynotingthat(�ZocZo)=νwhile(�Zo−cZo)=,then∂v∂z=−ν∂v∂tand∂v−∂z=ν∂v−∂tNotethatthesetwoequationsaredecoupled,witheachequationafunctiononlyofvorv−therefore,thetwowavescannotinteractandtravelalongthetransmissionlinewithoutinfluencingeachotherThewavesinteractonlyatboundariesbetweentransmissionlineswithdifferentimpedancesorbetweenatransmissionlineandacomponentThispropertyisexploitedtofacilitateanalysesandtoconstructbouncediagramsSignalIntegrityChapterFrequencyDomainSolutionWhenlossesaresignificantontransmissionlines,thefrequencydomainisconvenientforanalyticsolutionsAfterFouriertransformationwith(),thelossytransmissionlineequationsin()and()become∂V(z,ω)∂z=−(rω))I(z,ω)()and∂I(z,ω)∂z=−(gωc)V(z,ω),()whereVandIaretheFouriertransformsofvandiEliminatingIfrom()and()providesthesecondorderdifferentialequation∂V∂z−(rω))(gωc)V=()Defineγ=(rω))(gωc),thenγ(ω)=√(rω))(gωc),()andthesolutionto()isV(z,ω)=A(ω)e−γ(ω)zB(ω)eγ(ω)z()NoteinparticularthatA,B,andγarefunctionsofωγiscalledthecomplexpropagationconstantThecomplexpropagationconstantcanalwaysbewrittenintermsofitsrealandimaginarypartsasγ(ω)=α(ω)β(ω)then()becomesV(z,ω)=A(ω)e−α(ω)ze−β(ω)zB(ω)eα(ω)zeβ(ω)zSectionTransmissionLinesThetimedomainvoltagecanthenbefoundusingtheinverseFouriertransformgiveninchapterby()toobtainf(t)=π∫∞−∞(A(ω)e−α(ω)ze−β(ω)zB(ω)eα(ω)zeβ(ω)z)eωtdω=π∫∞−∞(A(ω)e−α(ω)ze−(β(ω)z−ωt)B(ω)eα(ω)ze(β(ω)zωt))dω,()whereitcanbeseeninthesecondformthatthetotalvoltageconsistsofthesuperpositionofmanycomplexexponentialsEachA(ω)e−α(ω)ze−(β(ω)z−ωt)representsaweightedsinusoidalwavetravelinginthezdirectionthatisattenuatedexponentiallywithdistanceAfixedpointontheexponentialcanbeidentifiedwhenβ(ω)z−ωt=constantSolvingforzandtakingthederivativewithrespecttotimeyieldsthephasevelocityνp=dzdt=ωβ(ω)()Sinceβisafunctionoffrequency,thenthephasevelocityisafunctionoffrequencyInasimilarfashion,eachB(ω)eα(ω)ze(β(ω)zωt)representsaweightedandattenuatedwavetravelinginthe−zdirectionDuetotheirrolesinwavepropagation,αiscalledtheattenuationconstantandβiscalledthepropagationconstantThephasevelocityandattenuationofthesinusoidalwavesaredifferentateveryfrequency,sotheshapeofthetimedomainwaveformmustchangeasitmovesdownthelineAttenuationisstrongerforhigherfrequencycomponents,sothewaveformtendstospread,ordisperse,withdistanceaslowfrequencycomponentstakeoverForthisreason,theeffectsoflossesonwaveshapeiscalleddispersionSometimesdispersionisattributedtothesourceofthelosses,suchasconductorlossdispersionordielectriclossdispersionBecauseeachfrequencycomponenthasadifferentphasevelocity,thewavevelocityisnotequaltothephasevelocityofanygivencomponentForlossylines,SignalIntegrityChapterthewavevelocityisfoundbyfindingtheTOFofapulsebetweentwopointsanddividingintothedistancetraveledThecharacteristicimpedancecanbefoundbysolvingforIin()withthegeneralvoltagesolutionin()toobtainI(z,ω)=√gωcrω)(A(ω)e−γ(ω)z−B(ω)eγ(ω)z)()ThecharacteristicimpedanceforalossytransmissionlineisthenZo(ω)=√rω)gωc,()anditis,ingeneral,frequencydependentThedirectionalcomponentsofvoltageandcurrentareclearlyapparentwithachangeofnotationin()toV(z,ω)=V(ω)e−γ(ω)zV−(ω)eγ(ω)z,()thenthecurrentfrom()isI(z,ω)=Zo(ω)V(ω)e−γ(ω)z−V−(ω)eγ(ω)z()ThesetwoexpressionsareoftenagoodstartingpointinsolvingcircuitproblemsinvolvingtransmissionlinesLowLossTransmissionLinesWhenlossesonatransmissionlinearesmall,additionalmathematicalmanipulationcanyieldgoodinsightintowaveformpropagationontransmissionlinesThecomplexpropagationconstantcanberearrangedasγ=√(rω))(gωc)=√ω)(rω))ωc(gωc)=ω√)c√rω)√gωc()Fornarrowbandsignals,thegroupvelocitycanbecomputedfromthefrequencydependenceofthephasevelocitytofindthevelocityofawavepacketDigitalsignalsarebaseband,sothegroupvelocityisnotapplicablesincethenarrowbandassumptiondoesnotapplySectionTransmissionLinesForlowlosses,theapproximations√rω�≈rω�,rω��√gωc≈gωc,gωc�()canbeappliedto()toobtainγ≈ω√)c(rω)gωc−rgω)c)≈ω√)c(√c)r√)cg),()wherethetermrgω�cisdroppedasanegligiblesecondordersmalltermTheattenuationandpropagationconstantsforlowlossesarethengivenbyα=(√c)r√)cg)()andβ=ω√)c,()andthephasevelocityisνp=√)cNotethattheattenuationconstantandthephasevelocityarefrequencyindependent,soallfrequencycomponentsofthewaveformtraveltogetherwithuniformattenuationTherefore,thewaveformpropagatesdownthetransmissionlinewithnochangeinwaveshapeexceptforreductioninamplitudeInthiscase,thewavevelocitydoesequalthephasevelocityThispropagationisdispersionlessSubstitutingtheattenuationandpropagationconstantsin()and()forthelowlosslineinto()yieldsf(t)=πe−αz∫∞−∞A(ω)e−ω√�czeωtdωπeαz∫∞−∞B(ω)eω√�czeωtdω=e−αza(t−√)cz)eαzb(t√)cz),()SignalIntegrityChapterwherea(t)=F−A(ω)andb(t)=F−B(ω)ThetimeshiftingtheoremforFouriertransforms,Ff(t−ξ)=Ff(t)e−ωξ,isalsoutilizedTheresultin()generalizestolowlosslinesthedirectionalindependenceestablishedinsectionforlosslesslinesInaddition,theresultshowsthatwaveformstravelunchangedexceptforamplitudeonlowlosslinesConductorLossDominatedTransmissionLinesLossesareoftendominatedbyconductors,sowhilerisoftensignificant,gisoftennotSettingg=inthelossyexpressionsforZoandγyieldsZo=√rω)ωc()andγ=√(rω))ωc()Atlowfrequencies,r�ω),soZo≈√rωce−πandγ≈√ωrceπForbothZoandγ,therealandimaginarypartsareequal(ignoringthesign)withasquarerootdependencyonfrequencyTheseresultsareverydifferentfromthoseobtainedfromthelosslesstransmissionlinemodel,yetthelosslessmodelisoftenusedinsignalintegrityworkwherebasebanddigitalsignalsincludesignificantlowfrequencycontentAnapproximatemetricisavailablefordeterminingiflosslesstransmissionlinemodelingisappropriateAtlowfrequencies,thelumpedapproximationapplies,SectionTransmissionLinesandthelossytransmissionlineissimplyaloopofwirethatcanbemodeledasaseriesRLCcircuitForlossestobenegligible,theRLCcircuitmustbestronglyunderdamped,aconditionthatoccurswhenR�√LCUsingperunitlengthquantitiesforatransmissionlinewithlengthd,thend�r√)c()Ifthelinelengthissufficientlyshort,thenlosslesstransmissionlinemodelingcanbeappropriateTodemonstratethelinelengthdependence,consideraΩtransmissionlinewitheff=andΩmoflossTheperunitlengthquantitiesfrom()and()arenHmandpFm,respectivelyThelinelengthlimitfrom()isd�mSimulationresultsforlossyandlosslessversionsofthistransmissionlineareshowninFigureTheresultsshowthatthelosslessmodelisaccurateforlinelengthslessthanm,sothepredictionfrom()worksverywellforthiscaseBoardlevelsignalintegrityworkdealswithlineslessthanameterlong,solosslesstransmissionlinemodelingcanbeappropriateiftheedgeratesaresufficientlyslowWaveformswithfastedgeratesexperiencesignificantfrequencydependentlossesduetotheskineffect,sofrequencydependentlossysimulationscanberequired(seechapter,section)Athighfrequencies,r�ω),thenZo≈√)c(−rω))andγ≈ω√)c(−rω))Inthiscase,therealpartofZoisconstantanddominatestheimaginarypart,whileforγtheimaginarypartdominateswithasimplelineardependenceonfrequencyWhenhighfrequencylossesarenegligible,thecharacteristicimpedanceandpropagationconstantsimplifytothatofthelosslesstransmissionlineSignalIntegrityChapterLosslessLossyVoltageatpFLoad,VTime,nsLineLengthmmmmmmnsnsnsVpFΩmΩmornHmpFmΩFigureTheaccuracyofthelosslesstransmissionlinemodeldependsonthelinelengthGeneralFormulasUsingRealMathWithoutassumptionsonrandω),complexmathisrequiredforcalculatingZoandγ,socalculatorbasedcalculationsusing()and()aresomewhatdifficultConvenientformulasbasedonlyonrealmatharepossibleaftersomemanipulationFirst,Zo=√)c√−rω),SectionTransmissionLinesthentherealpartisReZo=√)c(√−rω)√rω))Theterminparenthesiscanberearrangedas√−rω)√rω)=±√(√−rω)√rω))=±√√√√(√(rω)))TherealpartofthecharacteristicimpedanceisthenReZo=√)c√√√√(√(rω))),wherethepositiverootistakentorepresentapassivestructureFortheimaginarypartofthecharacteristicimpedance,ImZo=√)c(√−rω)−√rω))Theterminparenthesiscanberearrangedas√−rω)−√rω)=±√(√−rω)−√rω))=±√√√√(−√(rω))),thenImZo=−√)c√√√√(−√(rω))),wherethenegativesignistakentomatchuptothelowlossapproximationaboveThecomplexpropagationconstantcanbesimilarlyaddressedtofindthatReγ=α=ω√)c√√√√(−√(rω))),SignalIntegrityChapterandImγ=β=ω√)c√√√√(√(rω)))ImpedanceBounda

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