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AppendixA.
1MOSTransistorModelUncertaintyThenumberoftransistorprocessparametersthatcanvaryislarge.
Inpreviousresearchaimedatoptimizingtheyieldofintegratedcircuits[1,2],thenumberofparameterssimulatedwasreducedbychoosingparameterswhicharerelativelyindependentofeachother,andwhichaffectperformancethemost.
Theparametersmostfrequentlychosenare,fornandp-channeltransistors:thresholdvoltageatzeroback-biasforthereferencetransistoratthereferencetemperatureVT0R,gainfactorforaninnitesquaretransistoratthereferencetemperaturebSQ,totallengthandwidthvariationDLvarandDWvar,oxidethicknesst0x,andbottom,sidewallandgateedgejunctioncapacitanceCJBR,CJSRandCJGR,respectively.
Thevariationinabsolutevalueofalltheseparametersmustbeconsidered,aswellasthedifferencesbetweenrelatedelements,i.
e.
matching.
ThethresholdvoltagedifferencesDVTandcurrentfactordifferencesDbarethedominantsourcesunderlyingthedrain-sourcecurrentorgate-sourcevoltagemismatchforamatchedpairofMOStransistors.
TransistorThresholdVoltage:Variousfactorsaffectthegate-sourcevoltageatwhichthechannelbecomesconductivesuchasthevoltagedifferencebetweenthechannelandthesubstraterequiredforthechanneltoexist,theworkfunctiondifferencebetweenthegatematerialandthesubstratematerial,thevoltagedropacrossthethinoxiderequiredforthedepletionregion,thevoltagedropacrossthethinoxideduetoimplantedchargeatthesurfaceofthesilicon,thevoltagedropacrossthethinoxideduetounavoidablechargetrappedinthethinoxide,etc.
Inorderforthechanneltoexisttheconcentrationofelectroncarriersinthechannelshouldbeequaltotheconcentrationofholesinthesubstrate,/S=-/F.
Thesurfacepotentialchangedatotalof2/Fbetweenthestronginversionanddepletioncases.
Thresholdvoltageisaffectedbythebuilt-inFermipotentialduetothedifferentmaterialsanddopingconcentrationsusedforthegatematerialandthesubstratematerial.
Theworkfunctiondifferenceisgivenby/ms/FSub/FGatekTqlnNDNAn2iA:1A.
Zjajo,StochasticProcessVariationinDeep-SubmicronCMOS,SpringerSeriesinAdvancedMicroelectronics48,DOI:10.
1007/978-94-007-7781-1,SpringerScience+BusinessMediaDordrecht2014157Duetotheimmobilenegativechargeinthedepletionregionleftbehindafterthepmobilecarriersarerepelled.
Thiseffectgivesrisetoapotentialacrossthegate-oxidecapacitanceof–QB/Cox,whereQBqNAxdqNA2eSi2/FjjqNAs2qNAeSi2/FjjpA:2andxdisthewidthofthedepletionregion.
Theamountofimplantedchargeatthesurfaceofthesiliconisadjustedinordertorealizethedesiredthresholdvoltage.
Forthecaseinwhichthesource-to-substratevoltageisincreased,theeffectivethresholdvoltageisincreased,whichisknownasthebodyeffect.
Thebodyeffectcoccursbecause,asthesource-bulkvoltage,VSB,becomeslarger,thedepletionregionbetweenthechannelandthesubstratebecomeswider,andthereforemoreimmobilenegativechargebecomesuncovered.
Thisincreaseinchargechangesthechargeattractedunderthegate.
Specically,Q0BbecomesQ0B2qNAeSiVSB2/FjjpA:3Thevoltagedropacrossthethinoxideduetounavoidablechargetrappedinthethinoxidegivesrisetoavoltagedropacrossthethinoxide,Vox,givenbyVoxQoxCoxqNoxCoxA:4Incorporatingallfactors,thethresholdvoltage,VT,isthangivenbyVT2/F/msQ0BQoxCox/ms2/FQBQoxCoxQBQ0BCox/ms2/FQBQoxCox2qeSiNACoxr2/FjjVSBp2/FjjphiA:5Whenthesourceisshortedtothesubstrate,VSB=0,azerosubstratebiasisdenedasVT0/ms2/FQBQoxCoxA:6Thethresholdvoltage,VT,canberewrittenasVTVT0c2/FjjVSBp2/Fjjp;c2qeSiNACoxrA:7Advancedtransistormodels,suchasMOSTmodel9[3],denethethresholdvoltageasVTVT0DVT0DVT1VT0VT0TVT0GDVT0MDVT0DVT1A:8158Appendixwherethresholdvoltageatzeroback-biasVT0[V]fortheactualtransistorattheactualtemperatureisdenedasgeometricalmodel,VT0T[V]isthresholdtemperaturedependence,VT0G[V]thresholdgeometricaldependenceandDVT0(M)[V]matchingdeviationofthresholdvoltage.
Duetothevariationinthedopinginthedepletionregionunderthegate,atwo-factorbody-effectmodelisneededtoaccountfortheincreaseinthresholdvoltagewithVSBforion-implantedtransistors.
Thechangeinthresholdvoltagefornon-zerobackbiasisrepresentedinthemodelasDVT0K0uSuS0uSuSX1KK02!
K0uSXK0uS0Ku2S1KK02!
u2SXsuS!
uSX8>>>>>>>>>:A:9uSVSB/BpuS0/BpuSTVSBT/BpuSXVSBX/BpA:10wheretheparameterVSBX[V]istheback-biasvalue,atwhichtheimplementedlayerbecomesfullydepleted,K0[V1/2]islow-backbiasbodyfactorfortheactualtransistorandK[V1/2]ishigh-backbiasbodyfactorfortheactualtransistor.
Fornon-zerovaluesofthedrainbias,thedraindepletionlayerexpandstowardsthesourceandmayaffectthepotentialbarrierbetweenthesourceandchannelregionsespeciallyforshort-channeldevices.
Thismodulationofthepotentialbarrierbetweensourceandchannelcausesareductioninthethresholdvoltage.
Insubthresholdthisdramaticallyincreasesthecurrentandisreferredtoasdraininducedbarrierlowering(DIBL).
Onceaninversionlayerhasbeenformedathighervaluesofgatebias,anyincreaseofdrainbiasinducesanadditionalincreaseininversionchargeatthedrainendofthechannel.
Thedrainbiasstillhasasmalleffectinthethresholdvoltageandthiseffectismostpronouncedintheoutputconductanceinstronginversionandisreferredtoasstaticfeedback.
TheDIBLeffectismodeledbytheparameterc00inthesubthresholdregion.
ThisdrainbiasvoltagedependenceisexpressedbyrstpartofDVT1c0V2GTXV2GTXV2GT1VDSc1V2GT1V2GTXV2GT1VgDSDSA:11VGT1VGSVT1VGS!
VT10VGS!
VT1&VGTX2p=2A:12wherec1iscoefcientforthedraininducedthresholdshiftforlargegatedrivefortheactualtransistorandgDSexponentoftheVDSdependenceofc1fortheactualtransistor.
Thestaticfeedbackeffectismodeledbyc1.
Thiscanbeinterpretedasanotherchangeofeffectivegatedriveandismodeledbythesecondpartof(A.
9).
Appendix159FromrstordercalculationsandexperimentalresultstheexponentgDSisfoundtohaveavalueof0.
6.
Inordertoguaranteeasmoothtransitionbetweensubthresholdandstronginversionmode,themodelconstantVGTXhasbeenintroduced.
ThresholdvoltagetemperaturedependenceisdenedasVT0TVT0RTADTATRST;VT0A:13whereVT0R[V]isthresholdvoltageatzeroback-biasforthereferencetransistoratthereferencetemperature,TA[°C]ambientorthecircuittemperature,DTA[°C]temperatureoffsetofthedevicewithrespecttoTA,TR[°C]temperatureatwhichtheparametersforthereferencetransistorhavebeendeterminedandST;VT0[VK-1]coefcientofthetemperaturedependenceVT0.
Insmalldevicesthethresholdvoltageusuallyischangedduetotwoeffects.
Inshort-channeldevicesdepletionfromthesourceanddrainjunctionscauseslessgatechargetoberequiredtoturnonthetransistors.
Ontheotherhandinnarrow-channeldevicestheextensionofthedepletionlayerundertheisolationcausesmoregatechargetoberequiredtoformachannel.
Usuallytheseeffectscanbemodeledbygeometricalpreprocessingrules:VT0G1LE1LERSL;VT01L2E1L2ERSL2;VT01WE1WERSW;VT0A:14whereLE[m]iseffectivechannellengthofthetransistor,WE[m]effectivechannelwidthofthetransistor,LER[m]effectivechannellengthofthereferencetransistor,WER[m]effectivechannelwidthofthereferencetransistor,SL;VT0[Vm]coefcientofthelengthdependenceVT0,SL2;VT0[Vm2]secondcoefcientofthelengthdependenceVT0,SW;VT0[Vm]coefcientofthewidthdependenceVT0.
Theindividualtransistorsigma'saresquarerootoftwosmallerthanthesigmaforapair.
Inthedenitionoftheindividualtransistormatchingdeviationstatedintheprocessblock,switchmechanismandcorrectionfactorisaddedaswell,DVT0MFSDVT0AIntra=2pWeLeFCpFSDVT0BIntra=2pA:15whereDVT0(AIntra)andDVT0(BIntra)arewithin-chipspreadofVT0[Vlm],FSisasortofmechanismtoswitchbetweeninterandintradiespread,forintra-diespreadFS=1,otherwiseiszero,andFCiscorrectionformultipletransistorsinparallelandunits.
TransistorCurrentGain:AsingleexpressionmodelthedraincurrentforallregionsofoperationintheMOSTmodel9isgivenbyIDSbG3VGT31d12VDS1VDS11h1VGT1h2usus0fg1h3VDS1A:16160Appendixwhered1k1usKK0KV2SBXV2SBXk2VGT1VSB2()A:17VGT32m/Tln1G1A:18G3111expVDS/TnoG1G2111G1G1expVGT22m/TG21aln1VDSVDS1VPA:19m1m0us0us1gmA:20h1,h2,h3arecoefcientsofthemobilityreductionduetothegate-inducedeld,theback-biasandthelateraleld,respectively,/Tthermalvoltageattheactualtemperature,f1weak-inversioncorrectionfactor,k1andk2aremodelconstantsandVPischaracteristicvoltageofthechannel-lengthmodulation.
Theparameterm0characterizesthesubthresholdslopeforVBS=0.
GainfactorbisdenedasbbSQTWeLeFold1SSTI1Ab=2pWeLeFCpBb=2pFSA:21wherebSQTisgainfactortemperaturedependence,SSTIisSTIstress,FSswitchingmechanismfactor,FCcorrectionfactormultipletransistorsinparallelandunitsandAbareascalingfactorandBbaconstant.
GainfactortemperaturedependenceisdenedasbSQTbSQT0TRT0TADTAgbA:22wheregb[-]isexponentofthetemperaturedependenceofthegainfactorandbSQ[AV-2]isgainfactorforaninnitesquaretransistoratthereferencetemperaturedenedasbSQ212QWeQWxWQWxW2e2q=We1bBSQ1bBSQSLeLxLLxL2e2pLe1bBSQS1bBSQ0BB@1CCAA:23Appendix161bBSQbSQTRT0TRT0TADTAgbBSQbBSQSbSQSTRT0TRT0TADTAgbBSQSA:24Fordevicesintheohmicregion(A.
24)canbeapproximatedbyIDbVGSVT12VDS1hVGSVTVDSA:25andforsaturateddevicesIDb2VGSVT21hVGSVTA:26ChangeindraincurrentcanbecalculatedbyDIDDboIDob!
DVToIDoVT!
DhoIDoh!
A:27leadingtodraincurrentmismatchDIDIDDbbixDVTnxDhA:28whereforohmicio112hVDSVGSVT12VDS1hVGSVTnoVGSVT1hVGSVTDhA:29andforsaturationis2hVGSVTVGSVT1hVGSVTnsVGSVT1hVGSVTDhA:30Thestandarddeviationofthemismatchparametersisderivedbyr2DIDIDr2Dbbi2xr2DVTn2xr2Dh2qDbb;DVTixrDVTrDbb2qDbb;DhnxrDhrDbb2qDVT;DhixnxrDhrDVA:31with[4]rDVTAVT=2pWeffLeffpBVT=2pSVTDA:32162AppendixrDbbAb=2pWeffLeffpBb=2pSbDA:33whereWeffistheeffectivegate-widthandLefftheeffectivegate-length,theproportionalityconstantsAVT,SVT,AbandSbaretechnology-dependentfactors,DisdistanceandBVTandBbareconstants.
ForwidelyspaceddevicestermsSVTDandSbDareincludedinthemodelsfortherandomvariationsintwopreviousequations,butfortypicaldeviceseparations(\1mm)andtypicaldevicesizesthiscorrectionissmall.
Mostmismatchcharacterizationhasbeenperformedondevicesinstronginversion,inthesaturationorlinearregionbutsomestudiesfordevicesoperatinginweakinversionhavealsobeenconducted.
Qualitatively,thebehaviorinallregionsisverysimilar;VTandbvariationsarethedominantsourceofmismatchandtheirmatchingscaleswithdevicearea.
Theeffectivemobilitydegradationmismatchtermcanbecombinedwiththecurrentfactormismatchterm,asbothtermsbecomesignicantinthesamebiasrange(highgatevoltage).
Thecorrelationfactorq(DVT,Db/b)canbeignoredaswell,sincecorrelationbetweenr(DVT)andtheothermismatchparametersremainslowforbothsmallandlargedevices.
ThedrainsourcecurrenterrorDID/IDisimportantforthevoltagebiasedpair.
Forthecurrentbiasedpair,thegate-sourceorinputreferredmismatchshouldbeconsidered,whoseexpressioncouldbederivedsimilarlyasfordrainsourcecurrenterror.
Changeingate-sourcevoltagecanbecalculatedbyDVGSDVToVGSoVT!
DboVGSob!
A:34leadingtothestandarddeviationofthemismatchparametersisderivedbyr2DVGSVGSr2DVT#2r2Dbbwhere#VGSVT2A:35MOStransistorcurrentmatchingorgate-sourcematchingisbiaspointdependent,andfortypicalbiaspoints,VTmismatchisthedominanterrorsourcefordrain-sourcecurrentorgate-sourcevoltagematching.
TransistorwidthWandlengthL:Theelectricaltransistorlengthisdeterminedbythecombinationofphysicalpolysilicontrackwidth,spacerprocessing,mask-,projection-andetch-variationsLeLDLvarLDLPS2DLoverlapA:36whereLeiseffectiveelectricaltransistorchannellength,determinedbylinearregionMOStransistormeasurementsonseveraltransistorswithvaryinglength,Ldrawnwidthofthepolysilicongate,DLvartotallengthvariation,DLPSlengthvariationduetomask,projection,lithographic,etch,etc.
variationsandDLoverlapeffectivesource/gateordrain/gateoverlappersideduetolateraldiffusion.
Theelectricaltransistorwidthisdeterminedbythecombinationofphysicalactiveregionwidth,mask,projectionandetchvariationsAppendix163WeWDWvarWDWOD2DWnarrowA:37whereWeiseffectiveelectricaltransistorchannelwidth,determinedbylinearregionMOStransistormeasurementsonseveraltransistorswithvaryingwidth,Wdrawnwidthoftheactiveregion,DWvartotalwidthvariation,DWODwidthvariationsduetomask,projection,lithographic,etch,etc.
variationsandDWnarrowdiffusionwidthoffset:effectivediffusionwidthincreaseduetolateraldiffusionofthen+orp+implementation.
Oxidethickness:Themodelingofoxidethicknesstoxhasimpacton:totalcapacitancefromthegatetotheground:Cox=eox(WeLe)/tox,gainfactor:b—gainfactor,SL;h1R—coefcientofthelengthdependenceofh1,h1R—coefcientofthemobilityreductionduetothegate-inducedeld,subtresholdbehaviour:m0R—factorofthesubthresholdslopeforthereferencetransistoratthereferencetemperature,overlapcapacitances:CGD0=WE9Col=WE9(eoxLD)/tox,andCGS0=CGD0,andbulkfactors:K0R—low-backbiasbodyfactorandKR—high-backbiasbodyfactor.
Junctioncapacitances:Thedepletion-regioncapacitanceisnonlinearandisformedby:n+–p-:n-channelsource/draintop-substratejunction,p+–n-:p-channelsource/drainton-welljunctionandn-–p-:n-welltop-substratejunction.
Depletioncapacitanceofapnornpjunctionconsistsofbottom,sidewallandgateedgecomponent.
CapacitanceofbottomareaABisgivenasCJBCJBRABVDBRVRVDBPBA:38whereAB[m2]isdiffusionarea,VR[V]voltageatwhichparametershavebeendetermined,VDB[V]diffusionvoltageofbottomareaAB,VDBR[V]diffusionvoltageofthebottomjunctionatT=TRandPB[-]bottom-junctiongradingcoefcient.
Similarformulationsholdforthelocos-edgeandthegate-edgecomponents;onehastoreplacetheindexBbySandG,andtheareaABbyLSandLG.
CapacitanceofthebottomcomponentisderivedasCJBVCJBR1VVDBPBV\VLBCLBCLBPBVVLBVDB1FCBV!
VLB8>:A:39whereCLBCJB1FCBPBFCB11PB31PBVLBFCBVDBA:40164AppendixandVisdiodebiasvoltage.
SimilarexpressionscanbederivedforsidewallCJSVandgateedgecomponentCJGV.
Thetotaldiodedepletioncapacitancecanbedescribedby:CCJBVCJSVCJGVA:41A.
2ResistorandCapacitorModelUncertaintyTypicalCMOSandBiCMOStechnologiesofferseveraldifferentresistors,suchasdiffusionn=presistors,n=ppolyresistors,andnwellresistor.
Manyfactorsinthefabricationofaresistorsuchastheuctuationsofthelmthickness,dopingconcentration,dopingprole,andthedimensionvariationcausedbythephotolithographicinaccuraciesandnon-uniformetchratescandisplaysignicantvariationinthesheetresistance.
However,thisisbearableaslongasthedevicematchingpropertiesarewithintherangethedesignsrequire.
Theuctuationsoftheresistanceoftheresistorcanbecategorizedintotwogroups,oneforwhichtheuctuationsoccurringinthewholedevicearescaledwiththedevicearea,calledareauctuations,anotheroninwhichuctuationstakesplaceonlyalongtheedgesofthedeviceandthereforescaledwiththeperiphery,calledperipheraluctuations.
ForamatchedresistorpairwithwidthWandresistanceR,thestandarddeviationoftherandommismatchbetweentheresistorsisrfafpWr,WRpA:42wherefaandfpareconstantsdescribingthecontributionsofareaandperipheryuctuations,respectively.
Incircuitapplications,toachieverequiredmatching,resistorswithwidth(atleast2–3times)widerthanminimumwidthshouldbeused.
Also,resistorswithhigherresistance(longerlength)atxedwidthexhibitlargermismatching.
Toachievethedesiredmatching,ithasbeenacommonpracticethataresistorwithlonglength(forhighresistance)isbrokenintoshorterresistorsinseries.
Tomodela(poly-silicon)resistorfollowingequationisusedRRshLWDWReWDWA:43whereRshisthesheetresistanceofthepolyresistor,Reistheendresistancecoefcient,WandLareresistorwidthandlength,DWistheresistorwidthoffset.
Thecorrelationsbetweenstandarddeviations(r)ofthemodelparametersandthestandarddeviationoftheresistancearegiveninthefollowingr2Rr2RshdRdRsh!
2r2RedRdRe!
2r2DWdRdDW!
2A:44Appendix165r2Rr2RshL2WDW2r2Re1WDW2r2DWLRshWDW2ReWDW2"#2A:45Todenetheresistormatching,r2DRRr2RshLLRshRe!
2r2Re1LRshRe!
2r2DW1WDW2"#2A:46rRshARshWLprReARerDWADWW12pA:47CurrentCMOStechnologyprovidesvariouscapacitanceoptions,suchaspoly-to-polycapacitors,metal-to-metalcapacitors,MOScapacitors,andjunctioncapacitors.
Theintegratedcapacitorsshowsignicantvariabilityduetotheprocessvariation.
ForaMOScapacitor,thecapacitancevaluesarestronglydependentonthechangeinoxidethicknessanddopingproleinthechannelbesidesthevariationingeometries.
SimilartotheresistorsthematchingbehaviorofcapacitorsdependsontherandommismatchduetoperipheryandareauctuationswithastandarddeviationrfafpCr,CpA:48wherefaandfparefactorsdescribingtheinuenceoftheareaandperipheryuctuations,respectively.
Thecontributionoftheperipherycomponentsdecreasesasthearea(capacitance)increases.
Forverylargecapacitors,theareacomponentsdominateandtherandommismatchbecomesinverselyproportionaltoCp.
Asimplecapacitormismatchmodelisgivenbyr2DCCr2pr2ar2drpfpC34rafaC12rdfddA:49wherefp,faandfdareconstantsdescribingtheinuenceofperiphery,area,anddistanceuctuations.
Theperipherycomponentmodelstheeffectofedgeroughness,anditismostsignicantforsmallcapacitors,whichhaverelativelylargeamountofedgecapacitance.
Theareacomponentmodelstheeffectofshort-rangedielectricthicknessvariations,anditismostsignicantformoderatesizecapacitors.
Thedistancecomponentmodelstheeffectofglobaldielectricthicknessvariationsacrossthewafer,anditbecomessignicantforlargecapacitorsorwidelyspacedcapacitors.
166AppendixA.
3Time-DomainAnalysisThemodernanalogcircuitsimulatorsuseamodiedformofnodalanalysis[5,6]andNewton-Raphsoniterationtosolvethesystemofnnon-linearequationsfiinnvariablespi.
Ingeneral,thetime-dependentbehaviorofacircuitcontaininglinearornonlinearelementsmaybedescribedas[7]q0Ev0q0q0fq;v;w;p;t0A:50Thisnotationassumesthattheterminalequationsforcapacitorsandinductorsaredenedintermsofchargesanduxes,collectedinq.
TheelementsofmatrixEareeither1or0,andvrepresentsthecircuitvariables(nodalvoltagesorbranchcurrents).
Allnonlinearity'sareincorporatedinthealgebraicsystemf(q,v,w,p,t)=0,sothedifferentialequationsq0-Ev=0arelinear.
Theinitialconditionsarerepresentedbyq0.
Furthermore,wisavectorofexcitations,andpcontainsthecircuitparameterslikeparametersoflinearornonlinearcomponents.
Anelementofpmayalsobea(nonlinear)functionofthecircuitparameters.
Itisassumedthatforeachpthereisonlyonesolutionofv.
ThedcsolutioniscomputedbysolvingthesystemEv00fq0;v0;w0;pi;00A:51whichisderivedbysettingq0=0.
Thesolution(q0,v0)isfondbyNewton-Raphsoniteration.
Ingeneral,thistechniquendsthesolutionofanonlinearsystemF(v)=0byiterativelysolvingtheNewton-RaphsonequationJkDvkfvkA:52whereJkistheJacobianoff,with(Jk)ij=qfi/qvjk.
Iterationstartswithestimatev0.
AfterDvkiscomputedinthekthiteration,vk+1isfoundasvk+1=vk+Dvkandthenextiterationstars.
TheiterationterminateswhenDvkissufcientlysmall.
Forthe(A.
51),theNewton-Raphsonequationis0Eofoq0ofov0!
Dq0Dv0!
Evf!
A:53whichissolvedbyiteration(forsimplicityitisassumedthattheexcitationswdonotdependonpj).
Thisschemeisusedinthedcoperatingpoint[5–7],dctransfercurve,andeventime-domainanalysis;inthelastcase,thedependenceupontimeiseliminatedbyapproximatingthedifferentialequationsbydifferenceequations[7].
Onlyfrequency-domain(smallsignal)analysesaresignicantlydifferentbecausetheyrequire(foreachfrequency)asolutionofasystemofsimultaneouslinearequationsinthecomplexdomain;thisisoftendonebyseparatingtherealandimaginarypartsofcoefcientsandvariables,andsolvingatwiceaslargesystemoflinearequationsintherealdomain.
Appendix167Themaincomputationaleffortofnumericalcircuitsimulationintypicalapplicationsisthusdevotedto:(i)evaluatingtheJacobianJandthefunctionf,andthen(ii)solvingthesystemoflinearequations.
Afterthedcsolution(q0,v0)isobtained,thedcderivativesarecomputed.
Differentiationof(A.
51)withrespecttopjresultsinlinearsystem0Eofoq0ofov0!
oq0opjov0opj"#0ofopj!
A:54The(A.
51)canbesolvedefcientlybyusingtheLUfactorization[8]oftheJacobianthatwascomputedatthelastiterationof(A.
53).
Nowthederivativesof(A.
50)topjiscomputed.
Differentiationof(A.
50)topjresultsinlinear,time-varyingsystemoq0opjEovopj0oq0opjoq0opjofoqoqopjofovovopjofopj0A:55Ateachtimepointthecircuitderivativesareobtainedbysolvingprevioussystemofequationaftertheoriginalsystemissolved.
Suppose,forexample,thatakthorderBackwardDifferentiationFormula(BDF)isused[9,10],withthecorrectorq0nk1DtXk1i0aiqnkiA:56wherethecoefcientsaidependupontheorderkoftheBDFformula.
Aftersubstituting(A.
56)into(A.
50),theNewton-Raphsonequationisderivedasa0DtEofoqofov"#DqnkDvnk!
1DtPk1t0aiqnkiEvnkfqnk;vnk;wnk;pj;tnk2435A:57Iterationonthissystemprovidesthesolution(qn+k,vn+k).
SubstituitingakthorderBDFformulain(A.
55)givesthelinearsystema0DtEofoqofot"#oqopjnkotopjnk2643751DtPk1t0aioqopjnkiofopj264375A:58Thus(A.
57)and(A.
58)havethesamesystemmatrix.
TheLUfactorizationofthismatrixisavailableafter(A.
57)isiterativelysolved.
Thenaforwardandbackwardsubstitutionsolves(A.
58).
Foreachparametertheright-handsideof(A.
58)isdifferentandtheforwardandbackwardsubstitutionmustberepeated.
168AppendixIfrandomtermN(p,t)g,whichmodelsthetoleranceeffectsisnon-zeroandaddedtotheequation(A.
50)[11–15]fq;v;w;p;tNp;tg0A:59Solvingthissystemmeanstodeterminetheprobabilitydensityfunctionoftherandomvectorp(t)ateachtimeinstantt.
Fortwoinstantsintime,t1andt2,withDt=t1t0andDt2=t2-t0wheret0isatimethatcoincideswithdcsolutionofcircuitperformancefunctionv,Dtisassumedtosatisfythecriteriathatcircuitperformancefunctionvcanbedesignatedasthequasi-static.
Tomaketheproblemmanageable,thefunctioncanbelinearizedbyrst-orderTaylorapproximationassumingthatthemagnitudeoftherandomtermpissufcientlysmalltoconsidertheequationaslinearintherangeofvariabilityofporthenonlinearitesaresosmooththattheymightbeconsideredaslinearevenforawiderangeofpasexplainedinSect.
2.
2.
A.
4ParameterExtractionOncethenominalparametervectorp0isfoundforthenominaldevice,theparameterextractionofalldeviceparameterspkofthetransistorsconnectedtoparticularnodencanbeperformedusingalinearapproximationtothemodel.
Letp=[p1,p2,…,pn]T[Rndenotetheparametervector,f=[f1,f2,…,fm]T[Rmperformancevector,zk=[z1k,z2k,…,zmk]T[Rmthemeasuredperformancevectorofthekthdeviceandwavectorofexcitationsw=[w1,w2,…,wl]T[Rl.
Consideringequation(A.
50)q0Ev0q0q0fq;v;w;p;t0A:60generalmodelcanbewritten.
Themeasurementscanonlybemadeundercertainselectedvaluesofw,andiftheinitialconditionsq0aremet,sothemodelcanbesimplydenotedasfp0A:61Toextractaparametervectorpkcorrespondingtothekthdevicepkargminpk2Rnfpkzk&'A:62isfound.
Theweightedsumoferrorsquaresforthekthdeviceisformedas[7]epk12Xmi1wifipkzki212fpkzkTWfpkzkA:63Appendix169ifcircuitperformancefunctionvisapproximatedasalinearfunctionofparoundthemeanvalue"pvfp"pJp"p,fp0Dp%fp0Jp0DpA:64whereJ(p0)istheJacobianevaluatedatp0,alinearleast-squaresproblemisformedforthekthdevice[10]asminDpk2RneDpk12Jp0Dpkf0zkTWJp0Dpkf0zk&'A:65So,forthemeasuredperformancevectorzkforthekthdevice,anapproximateestimateofthemodelparametervectorforthekthdeviceisobtainedfrompk0p0Dpk0A:66whereDpk0Jp0TWJp0T1Jp0TWf0zkA:67A.
5PerformanceFunctionCorrectionTomodeltheinuenceofmeasurementerrorsontheestimatedparametervariationconsideracircuitwitharesponsethatisnonlinearinnparameters.
ChangesinthenparametersarelinearlyrelatedtotheresultingcircuitperformancefunctionDv(nodevoltages,branchcurrents,…),iftheparameterchangesaresmallDvovopDpA:68withDv=v(p)-v0andvpv0ovopTDp12DpTHDpDv0DvA:69whereHistheHessianmatrix[16],whoseelementsarethesecond-orderderivativeshijo2vp=opiopjA:70NowdeneDvrCrrDprewhereCrrDprlDv1.
.
.
lDvkTA:71170Appendixwhichistherelationbetweenmeasurementerrorse,parameterdeviationsandobservedcircuitperformancefunctionv.
AssumethatDvrisobtainedbykmeasurements.
NowanestimatefortheparameterdeviationsDprmustbeobtained.
Accordingtoleastsquareapproximationtheorem[11],theleastsquaresestimateD^profDprminimizestheresidualDvrCrrD^prkk22A:72TheleastsquaresapproximationofDprcanbeemployedtondinuenceofmeasurementerrorsontheestimatedparameterdeviationsbyD^prCrrCrr1CrrDvrA:73whichmaybeobtainedusingthepseudo-inverseofCrr.
Asstatedin[16],thecovariancematrixC^prmaybedeterminedasC^prCrrCrr1A:74Thisexpressionmodelstheinuenceofmeasurementerrorsontheestimatedparametervariation.
ThemagnitudeoftheithdiagonalelementofC^prindicatestheprecisionwithwhichthevalueoftheithparametercanbeestimated:alargevariancesignieslowparametertestability.
Likethisaparameterisconsideredtestableifthevarianceofitsestimateddeviationisbelowacertainlimit.
Theoff-diagonalelementsofC^prcontaintheparametercovariances.
Ifanaccuracycheckshowsthattheperformancefunctionextractionisnotaccurateenough,theperformancefunctioncorrectionisperformedtorenetheextraction.
Thebasicideaunderlyingperformancefunctioncorrectionistocorrecttheerrorsofperformancefunctionextractionbasedonthegivenmodelandtheknowledgeobtainedfromthepreviousstagesbyiterationprocess.
Denotingvkipv0DvkiA:75theextractedperformancefunctionvectorforthekthdeviceattheithiteration,performancefunctioncorrectioncanbefoundbyndingthesolutionforthetransformationvki1Fivkisuchthatmoreaccurateperformancefunctionvectorscanbeextracted,subjecttovki1vk\vkivkA:76wherevkargminvk2Rnevk&'A:77istheidealsolutionoftheperformancefunction.
TheerrorcorrectionmappingFiisselectedintheformofAppendix171vki1pvkipdiDvkiA:78wherediiscallederrorcorrectionfunctionandneedstobeconstructed.
Thedatasetdki;Dvki;k1;2;KnoA:79givestheinformationrelatingtheerrorsduetoinaccurateparameterextractiontotheextractedparametervalues.
AquadraticfunctionispostulatedtoapproximatetheerrorcorrectionfunctiondtPnj1cDpjPnj1Pnl1cDpjDpl;t1;2;nA:80whered=[d1,d2,…,dn]T,Dp=[Dp1,Dp2,…,Dpn]T,cjandcjlarethecoefcientsoftheerrorcorrectionfunctionattheithiteration.
Thecoefcientscanbedeterminedbyttingequationtothedatasetunderleastsquarecriterion.
Oncetheerrorcorrectionfunctionisestablished,performancefunctioncorrectionisperformedasvki1pvkipDvki1A:81Dvki1vkipdiDvkiA:82A.
6SampleSizeEstimationTheproblemofstatisticalanalysisconsistsindeterminingthestatisticalpropertiesofrandomtermN(p,t)g,whichmodelsthetoleranceeffects#Np;tgfq0;t0;w0;pi;0A:83asshowninA.
3.
InMonte-Carloanalysisanensembleoftransfercurvesiscalculatedfromwhichthestatisticalcharacteristicsareestimated.
Fromestimationtheoryitisknown,thattheestimateforthemean^l1nXni11iA:84withcondencelevelc=1-alieswithintheintervalprobability[17]1z1d2rnp^l1z1d2rnpA:85172AppendixofaN(0,1)distributedrandomvariablef.
FromthiswithgivenintervalwidthDl2z1d2rnpA:86thenecessarysamplesizenisobtainedasn2z1d2rDl2A:87If,forexampleameanvaluehastobeestimatedwitharelativeerrorDl/r=0.
1andacondencelevelofc=0.
99(z1-d/2&2.
5)thesamplesizeisn=2500.
Similartothatwehavefortheestimateofthevariance^r21n1Xni11il2A:88anecessarysamplesizeofn22pz1d2r2Dr222z1d22rDr2A:89inordertoprovidethattheestimate^r2fallswithprobabilitycintotheintervalr2Dr22^r2r2Dr22A:90Forexample,therequirednumberofsamplesforanaccuracyofDr/^r0:1andacondencelevelof0.
99isn=1250.
A.
7FrequencyDomainAnalysisThebehaviorofasystem(A.
59)inthefrequencydomainfqjx;vjx;wjx;pjx;jxNpjx;jx10A:91isdescribedbyasetoflinearcomplexequations[7]Tp;jxXp;jxWp;jxA:92whereT(p,jx)isthesystemmatrix,X(p,jx)andW(p,jx)arenetworkandsourcevectors,respectivelyandxisthefrequencyinradianspersecond.
ToevaluatenetworkvectorX(p,jx)totheparameterp,thepreviousequationisdifferentiatedwithrespecttoptoobtainoXp;jxopT1p;jxoTp;jxopXp;jxoWp;jxop!
A:93Appendix173Thecircuitperformancefunctionv=f(p,jx)isobtainedfromv=f(p,jx)=dTX(p,jx)usingtheadjointortransposemethod[18]wherethevectordisaconstantvectorthatspeciesthecircuitperformancefunction.
ThederivativesofthecircuitperformancefunctionwithrespecttoVTandbarethencomputedfromovVTi;jxoVTidTT1VTi;jxoTVTi;jxoVTiXVTi;jxoWVTi;jxoVTi!
A:94ovbi;jxobidTT1bi;jxoTbi;jxobiXbi;jxoWbi;jxobi!
A:95TherstorderderivativesofthemagnitudeofthecircuitperformancefunctionarecomputedfromovjxjjoVTivVTi;jxjjRe1vVTi;jxovVTi;jxoVTi!
A:96ovbi;jxjjobiovbi;jxjjRe1vbi;jxovbi;jxobi!
A:97where'Re'denotestherealpartofthecomplexvariablefunction.
Thesecondorderderivativesarecalculatedfromo2vVTi;jxjjoV2TivVTi;jxjjRe1vVTi;jxovVTi;jxoVTi!
2vVTi;jxjjRe1vVTi;jxo2vVTi;jxoV2Ti1vVTi;jx2ovVTi;jxoVTi2"#2A:98o2vbi;jxjjob2ivbi;jxjjRe1vbi;jxovbi;jxobi!
2vbi;jxjjRe1vbi;jxo2vbi;jxob2i1vbi;jx2ovbi;jxobi2"#2A:99Thecircuitperformancefunctionv(jx)canbeapproximatedwiththetruncatedTaylorexpansionsasvjxlvjxJ^tjxl^tjxhiA:100whereJistheR9MNJacobainmatrixofthetransformationwhosegenericijelementisdenedas174AppendixJijovi^t;jxo^tjxj^tl^ti1;R;j1;MNA:101ThemultivariatenormalprobabilityfunctioncanbefoundasPv12pRCvvjxqexp12vjxlvjxhiTCjx1vvvjxlvjxhi!
A:102wherethecovariancematrixofthecircuitperformancefunctionCvv(jx)isdenedasCvvjxJjxC^t^tjxJjxTA:103andcovariancematrixisC^t^tC^p1^p1C^p1^p2.
.
.
C^p2^p1C^p2^p2.
.
.
2435A:104whereC^p1^p1ij1WiLiWjLjZxiLixiZxjLjxjZyiWiyiZyjWjyjRp1p1xA;yA;xB;yBlp1xA;yAlp1xB;yBdxAdxBdyAdyBA:105C^p1^p2ij1WiLiWjLjZxiLixiZxjLjxjZyiWiyiZyjWjyjRp1p2xA;yA;xB;yBlp1xA;yAlp2xB;yBdxAdxBdyAdyBA:106andRp1p1(xA,yA,xB,yB),theautocorrelationfunctionofthestochasticprocessp1,isdenedasthejointmomentoftherandomvariablep1(xA,yA)andp1(xB,yB)i.
e.
,Rp1p1(xA,yA,xB,yB)=E{p1(xA,yA)p1(xB,yB)},whichisafunctionofxA,yAandxB,yBandRp1p2(xA,yA,xB,yB)=E{p1(xA,yA)p2(xB,yB)}thecross-correlationfunctionofthestochasticprocessp1andp2.
TheexperimentaldatashowsthatthresholdvoltagedifferencesDVTandcurrentfactordifferencesDbarethedominantsourcesunderlyingthedrain-sourcecurrentorgate-sourcevoltagemismatchforamatchedpairofMOStransistors.
Appendix175Thecovariancerpipj=0,fori=j,ifpiandpjareuncorrelated.
ThusthecovariancematrixCPofp1,…,pkwithmeanlpiandavariancerpi2isC^p1;.
.
.
^pkdiag1;1A:107In[4]theserandomdifferencesforthesingletransistorhavinganormaldistributionwithzeromeanandavariancedependentonthedeviceareaWLarederivedasforijC^p1^p1ijrDVTAVT=2pWeffLeffpBVT=2pSVTD;fori6jC^p1^p1ij0A:108forijC^p2^p2ijrDb=bAb=2pWeffLeffpBb=2pSbD;fori6jC^p2^p2ij0A:109whereWeffistheeffectivegate-widthandLefftheeffectivegate-length,theproportionalityconstantsAVT,SVT,AbandSbaretechnology-dependentfactors,DisdistanceandBVTandBbareconstants.
Assumingtheaccomponentsassmallvariationsaroundthedccomponent,thefrequencyanalysistolerancewindow,consideringonlytherstandsecond-ordertermsoftheTaylorexpansionofthecircuitperformancefunctionv=f(VT(jx),b(jx)),aroundtheirmean(=0),themeanlvandrvofthecircuitperformancefunctionforq=0,canbeestimatedaslvv012Xni1o2vVTi;jxjjoV2Tir2VTio2vVbi;jxob2ir2bi()A:110r2vXni1o2vVTi;jxjjoV2Tir2VTio2vVbi;jxob2ir2bi()A:111wherenistotalnumberoftransistorsinthecircuitandlvisthemeanofv=f(VT(jx),b(jx))overthelocalorglobalparametricvariations.
A.
8DicriminationAnalysisDerivationofanacceptabletolerancewindowisaggravatedduetotheoverlappedregionsinthemeasuredvaluesoftheerror-freeandfaultycircuits,resultinginambiguityregionsforfaultdetection.
Lettheone-dimensionalmeasurementspacesCGandCFdenotefault-freeandfaultydecisionregionsandf(wn|G)andf(wn|F)indicatesthedistributionsofthewnunderfault-freeandfaultyconditions.
Then,176AppendixaPwn2CFjGZCFfwnwnjGdwnPw!
cjw$NlG;r2=nPZ!
clGr=npA:112bPwn2CGjFZCGfwnwnjFdwnPw\cjw$NlF;r2=nPZ\clFr=npA:113whereZ*N(0,1)isthestandardnormaldistribution,thenotationaindicatestheprobabilitythatthefault-freecircuitisrejectedwhenitisfault-free,andbdenotestheprobabilitythatfaultycircuitisacceptedwhenitisfaultyandccriticalconstantofthecriticalregionoftheformCw1;wn:w!
cA:114andPGPwn2CGjGZCGfwnwnjGdwn1ZCGfwnwnjFdwn1bA:115PFPwn2CFjFZCFfwnwnjFdwn1ZCFfwnwnjGdwn1aA:116Recallthatifw*N(l,r2),thenZ=(w-l/r)*N(0,1).
Inthepresentcase,thesamplemeanofw,w*N(l,r2/n),sincethevariablewisassumedtohaveanormaldistribution.
Sinceaandbrepresentprobabilitiesofeventsfromthesamedecisionproblem,theyarenotindependentofeachotherorofthesamplesize.
Evidently,itwouldbedesirabletohaveadecisionprocesssuchthatbothaandbaresmall.
However,ingeneral,adecreaseinonetypeoferrorleadstoanincreaseintheothertypeforaxedsamplesize.
Theonlywaytosimultaneouslyreducebothtypesoferrorsistoincreasethesamplesize.
However,thisprovestobetime-consumingprocess.
TheNeyman-PearsontestisaspecialcaseoftheBayestest,whichprovidesaworkablesolutionwhentheaprioriprobabilitiesmaybeunknownortheBayesaveragecostsofmakingadecisionmaybedifculttoevaluateorsetobjectively.
TheNeyman-PearsontestisbasedonthecriticalregionC*(X,whereXissamplespaceoftheteststatisticsCw1;wn:lw1;wnjG;FkfgA:117Appendix177whichhasthelargestpower(smallestb—probabilitythatfaultycircuitisacceptedwhenitisfaulty)ofalltestswithsignicancelevela.
IntroducingtheLagrangemultiplierktoaccountfortheconstraintgivesthefollowingcostfunction,J,whichmustbemaximizedwithrespecttothetestandkJ1bka0aka0ZCGfwnwnjFkfwnwnjGdwnA:118TomaximizeJbyselectingthecriticalregionCG,weselectwn[CGsuchthattheintegrandispositive.
ThusCGisgivenbyCGwn:fwnjFkfwnwnjG[0A:119TheNeyman-Pearsontestdecisionrule/(wn)canbewrittenasalikelihoodratiotest/wn1passiflw1;.
.
.
;wnjG;F!
k0failiflw1;.
.
.
;wnjG;F\k&A:120Supposew1,…,wnareindependentandidenticallydistributedN(l,r2)randomvaluesofthepowersupplycurrent.
ThelikelihoodfunctionofindependentandidenticallydistributedN(l,r2)randomvaluesofthepowersupplycurrentwherelF[lGisgivenbylw1;wnexp12r2Xni1wilG2(),exp12r2Xni1wilF2()exp12r2Xni1wilF2Xni1wilG2!
()A:121Now,Xni1wilF2Xni1wilG2nl2Fl2G2nwlFlGA:122UsingtheNeyman-PearsonLemma,thecriticalregionofthemostpowerfultestofsignicancelevelaisCw1;.
.
.
;wn:exp12r2nl2Fl2G2nwlFlG&'k&'w1;.
.
.
;wn:w!
r2nlFlGlogklFlG2&'w1;.
.
.
;wn:w!
kA:123178AppendixForthetesttobeofsignicancelevelaPw!
kjw$Nl;r2=nPZ!
klGr=npa)klGz1arnpA:124whereP(Z\z(1-a))=1-a,whichcanbealsowrittenasU-1(1-a).
z(1-a)isthe(1-a)—quantileofZ,thestandardnormaldistribution.
Thisboundaryforthecriticalregionguarantees,bytheNeyman-Pearsonlemma,thesmallestvalueofbobtainableforthegivenvaluesofaandn.
Fromtwopreviousequations,wecanseethatthetestTrejectsforTwlGr=np!
z1aA:125Similarly,toconstructatestforthetwo-sidedalternative,oneapproachistocombinethecriticalregionsfortestingthetwoone-sidedalternatives.
Thetwoone-sidedtestsformacriticalregionofCw1;wn:wk2;w!
k1A:126k1lGz1a2rnpk2lGz1a2rnpA:127Thus,thetestTrejectsforTwlGr=npz1a2orTwlGr=np!
z1a2A:128Ifthevariancer2isunknown,acriticalregioncanbefoundCw1;wn:twlGS=np!
k1&'A:129wheretisthet-distributionwithn-1degreesoffreedomandSisunbiasedestimatorofther2condenceinterval.
k1*ischosensuchthataPwlGS=np!
k1wlGS=np$tn1A:130togiveatestofsignicancea.
ThetestTrejectsforTwlGS=np!
tn1;aA:131Acriticalregionforthetwo-sidedalternativeifthevariancer2isunknownoftheformCw1;wn:twlGS=npk2;t!
k1&'A:132Appendix179wherek1*andk2*arechosensothataPwlGS=npk2wlGS=np$tn1PwlGS=np!
k1wlGS=np$tn1A:133togiveatestofsignicancea.
ThetestTrejectsforTwlGS=nptn1;a2orTwlGS=np!
tn1;a2A:134A.
9HistogramMeasurementofADCNonlinearitiesUsingSineWavesThehistogramoroutputcodedensityisthenumberoftimeseveryindividualcodehasoccurred.
ForanidealA/Dconverterwithafullscalerampinputandrandomsampling,anequalnumberofcodesisexpectedineachbin.
ThenumberofcountsintheithbinH(i)dividedbythetotalnumberofsamplesNt,isthewidthofthebinasafractionoffullscale.
Bycompilingacumulativehistogram,thecumulativebinwidthsarethetransitionlevels.
Theuseofsinewavehistogramtestsforthedeterminationofthenonlinearitiesofanalog-to-digitalconverters(ADC's)hasbecomequitecommonandisdescribedin[19]and[20].
Whenaramportrianglewaveisusedforhistogramtests(asin[21]),additivenoisehasnoeffectontheresults;however,duetothedistortionornonlinearityintheramp,itisdifculttoguaranteetheaccuracy.
Foradifferentialnonlinearitytest,aonepercentchangeintheslopeoftherampwouldchangetheexpectednumberof,codesbyonepercent.
Sincetheseerrorswouldquicklyaccumulate,theintegralnonlinearitytestwouldbecomeunfeasible.
Frombriefconsiderationitisclearthattheinputsourceshouldhavebetterprecisionthantheconverterbeingtested.
Whenasinewaveisused,anerrorisproduced,whichbecomeslargernearthepeaks.
However,thiserrorcanbemadeassmallanddesiredbysufcientlyoverdrivingtheA/Dconverter.
Theprobabilitydensityp(V)forafunctionoftheformAsinxtispV1pA2V2pA:135IntegratingthisdensitywithrespecttovoltagegivesthedistributionfunctionP(Va,Vb)PVa;Vb1psin1VbA!
sin1VaA!
&'A:136180Appendixwhichisinessence,theprobabilityofasamplebeingintherangeVatoVb.
Iftheinputhasadcoffset,ithastheformVo+AsinxtwithdensitypV1pA2VVo2qA:137ThenewdistributionisshiftedbyVoasexpectedPVa;Vb1psin1VbVoA!
sin1VaVoA!
&'A:138Thestatisticallycorrectmethodtomeasurethenonlinearitiesistoestimatethetransitionsfromthedata.
TheratioofbinwidthtotheidealbinwidthP(i)isthedifferentiallinearityandshouldbeunity.
SubtractingonLSBgivesthedifferentialnonlinearityinLSB'sDNLiHi=NtPi1A:139ReplacingthefunctionP(Va,Vb)bythemeasuredfrequencyofoccurrenceH/Nt,takingthecosineofbothsidesof(A.
138)andsolvingfor^Vb,whichisanestimateofVb,andusingthefollowingidentitiescosabcosacosbsinasinbA:140cossin1VAA2V2pAA:141yieldsto^V2b2VacospHNt^VbA21cos2pHNtV2a0A:142Inthisconsideration,theoffsetVoiseliminated,sinceitdoesnoteffecttheintegralordifferentialnonlinearity.
Solvingfor^Vbandusingthepositivesquareroottermasasolutionsothat^VbisgreaterthanVa^VbVacospHNtsinpHNtA2V2aqA:143Thisgives^VbintermsofVa.
^VkcanbecomputeddirectlybyusingtheboundaryconditionVo=–AandusingCHkXki0HiA:144Appendix181theestimateofthetransitionlevel^VbdenotedasaTkcanbeexpressedasTkAcospCHk1Nt;k1;N1A:145Aisnotknown,butbeingalinearfactor,alltransitionscanbenormalizedtoAsothatthefullrangeoftransitionsis±1.
A.
10MeanSquareErrorAstheprobabilitydensityfunctionassociatedwiththeinputstimulusisknown,theestimatorsoftheactualtransitionlevelTkandofthecorrespondingINLkvalueexpressedinleastsignicantbits(LSBs)arerepresentedasrandomvariablesdened,respectively,foracoherentlysampledsinewavesmdAsin2pDMmh0m0;1;M1A:146TkdAcospCHkM;k1;N1INLkTkTik=Dk1;N1A:147whereA,d,h0arethesignalamplitude,offsetandinitialphase,respectively,Misthenumberofcollecteddata,D/Mrepresentstheratioofthesinewaveoverthesamplingfrequencies.
Tkiistheidealkthtransitionvoltage,andD=FSR/2Bistheidealcode-binwidthoftheADCundertest,whichhasafull-scalerangeequaltoFSR.
Acommonmodelemployedfortheanalysisofananalog-todigitalconverteraffectedbyintegralnonlinearitiesdescribesthequantizationerroreasthesumofthequantizationerrorofauniformquantizereqandthenonlinearbehavioroftheconsideredconverteren.
Forsimplicityassumingthat|INLk|\D/2,wehaveenXN1k1DsgnINLkis2IkA:148wheresgn(.
)andi(.
)representthesignandtheindicatorfunctions,respectively,sdenotesconverterstimulussignalandthenon-overlappingintervalsIkaredenedasIk^TikINLk;Tik;INLk[0Tik;TikINLk;INLk\0&A:149Thenonlinearquantizermean-square-error,evaluatedundertheassumptionofuniformstimulationofallconverteroutputcodes,isgivenbymseZ11eqsens2fssdsA:150182AppendixwherefsrepresentPDFofconverterstimulus.
Stimulatingalldeviceoutputcodeswithequalprobabilityrequiresthatfss1VMVmiVms\VMA:151Thus,msebecomesmse1VMVmZVMVme2qs2eqsense2ns!
dsA:152AssumingD=(VM-Vm)/N,andexploitingthefactthemseassociatedwiththeuniformquantizationerrorsequenceisD2/12mseD2121NDXN1k1ZIk2DsgnINLkeqsD2!
dsA:153Since,foraroundingquantizer,eq(s)=D/2-D(s/D-1/2),itcanbeveriedthatsgn(INLk)eq(s)\0,sothatmseD2121NXN1k1INL2kA:154WhencharacterizingA/DconverterstheSINADismorefrequentlyusedthanthemse.
TheSINADisdenedasSINAD20log10rmssignalrmsnoisedBA:155LettheamplitudeoftheinputsignalbeAdBFS,expressedindBrelativefullscale.
Hence,thermsvalueisthenrmssignalD10AdBFS202b12pA:156Therms(noise)amplitudeisobtainedfromthemseexpressionabovesothatrmsnoisemsepSINADINL20blog10210log1032AdBFS10log10mseD2=12dBA:157TocalculatetheeffectivenumberofbitsENOB,rstlyexpresstheSINADforanidealuniformADCandthansolveforbSINADideal20log106pA2bFSRA:158Appendix183ENOBlog21020SINADlog2FSR6pAA:159LettingtheamplitudeA=10A(dBFS)/20FSR/2,andincorporatingaboveequation,theENOBcanbeexpressedasENOBINLb12log2mseD2=12dBA:160A.
11MeasurementUncertaintyToestimatetheuncertaintyontheDNLandINLitisnecessarytoknowtheprobabilitydistributionofthecumulativeprobabilityQitorealizeameasurementV\UBi,withUBitheuperboundoftheithlevelQiPV\UBiZUBiVoVpVdVA:161andusinglineartransformationUBicospQiA:162Thevarianceandcross-correlationofUBiisderivedusinglinearapproximations.
TorealizethevalueQi,itisnecessarytohaveNimeasurementswithavalue\UBi,and(N-Ni)measurementswithavalue[UBi.
ThedistributionofQiisabinomialdistribution,whichcanbeverywellapproximatedbyanormaldistribution[20]PQ0iCNiNPV\UBiNi1PV[UBiNNiCNiNQNii1QiNNiA:163withQi0theestimatedvalueofQi.
ThemeanandthestandarddeviationisgivenbylQ0iQirQ0iQi1Qi=NpA:164whichstatesthatQi0isanunbiasedestimateofQi.
TocalculatethecovariancebetweenQiandQj,rstly,let'sdeneQ0PV[UBjQijPUBi\V\UBj1QiQjA:165andtherelation184AppendixNjNiNijNiNijN0Nr2NiNjr2Nir2NiNijr2N0r2Nir2Nij2r2NiNijA:166whichleadstor2NiNjr2Nir2N0r2Nij=2A:167withr2NiNQi1Qir2N0NQ01Q0r2NijNQij1QijA:168orr2NiNjNQiQ0NQi1Qjr2QiQjQi1Qj=NA:169TocalculatethevariancerUB2r2UBiEdUBidUBip2sin2pQir2Qip2sin2pQiQi1Qi=NA:170Similarly,r2UBiUBjEdUBidUBjp2sinpQisinpQjQi1Qj=NA:171SincethedifferentialnonlinearityoftheithlevelisdenedastheratioDNLiUBiUBi1LR1A:172whereLRisthelengthoftherecord,theuncertaintyinDNLiandINLimeasurementscanbeexpressedasr2DNLir2UBir2UBi12r2UBiUBjq=LRr2INLirUBi=LRA:173ThemaximaluncertaintyoccursforQi=0.
5,thusthepreviousequationcanbeapproximatedwithr2DNLi%p=LRp1=Npr2INLip=2LR1=NpA:174Appendix185References1.
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Instrum.
Meas.
40(4),770–772(1991)186AppendixAbouttheAuthorAmirZjajoreceivedtheM.
Sc.
andDICdegreesfromtheImperialCollegeLondon,London,U.
K.
,in2000andthePh.
D.
degreefromEindhovenUniversityofTechnology,Eindhoven,TheNetherlandsin2010,allinelectricalengineering.
In2000,hejoinedPhilipsResearchLaboratoriesasamemberoftheresearchstaffintheMixed-SignalCircuitsandSystemsGroup.
From2006until2009,hewaswithCorporateResearchofNXPSemiconductorsasaseniorresearchscientist.
In2009,hejoinedDelftUniversityofTechnologyasaFacultymemberintheCircuitandSystemsGroup.
Dr.
Zjajohaspublishedmorethan70papersinreferencedjournalsandconferenceproceedings,andholdsmorethan10USpatentsorpatentspending.
HeistheauthorofthebookLow-VoltageHigh-ResolutionA/DConverters:Design,TestandCalibration(Springer,2011,Chinesetranslation,2012).
HeservesasamemberofTechnicalProgramCommitteeofIEEEDesign,AutomationandTestinEuropeConference,IEEEInternationalSymposiumonCircuitsandSystemsandIEEEInternationalMixed-SignalCircuits,SensorsandSystemsWorkshop.
Hisresearchinterestsincludemixed-signalcircuitdesign,signalintegrityandtimingandyieldoptimization.
A.
Zjajo,StochasticProcessVariationinDeep-SubmicronCMOS,SpringerSeriesinAdvancedMicroelectronics48,DOI:10.
1007/978-94-007-7781-1,SpringerScience+BusinessMediaDordrecht2014187IndexAAcquisitiontime,139Analogtodigitalconverter,6,11,117,131,132,135,137–139,141,143–145,183Autocorrelationfunction,175BBand-limiting,7Bartels-Stewartalgorithm,58,71boostingtechnique,129CCalibration,3,7,11,123,124,132,135,136,141–144,152,156Channelleakage,1,38Chipmultiprocessor,102Choleskydecomposition,93Choleskyfactor,34,36,58,59,71,80,98,107Chopping,126Circuitsimulation,23,168Circuityield,18,37Clockperiod,39Coarseconverter,96,131,132,141Comparator,74,76,77,80,124,126,133,141Comparingrandomvariables,20,22,182ComplementaryMOS,1–7,13,17,30,37,39,43–45,75,77,117,118,125,131,137,146,149,151,153,154Computeraideddesign(CAD),23,67Continuousrandomvariable,20,22,42,182Continuous-timelter,41,56,58,80,151Continuous-timeintegrator,56Corneranalysis,17Correlationcoefcient,43function,20,22,57ofdeviceparameters,1,3,18,27,136spatial,27Courant-Friedrichs-Lewynumber,96Covariance,20,21,27,32,39,56,57,61,67,80,84,92–94,110,151,171,175,176,184Crank-Nicolsonscheme,95,96Criticaldimension,38,137Cross-coupledlatch,124Cumulativedistributionfunction,18Cumulativeprobability,184DDesignfortestability,171Devicetolerances,13,43,151Deviceundertest,121,135,137,182Detector,96,120,121,123,124Die-levelprocessmonitor,120,121,123,124,134,137Differentialalgebraicequations,24Differentialnon-linearity,137,139,140,185Digitaltoanalogconverter,136Dirichletboundarycondition,88Discreterandomvariable,6,42,110Discrete-timelter,71,80,151Discrete-timeintagrator,71,92Distortion,6,8,68,119,134,135,144,145Distributionacrossspatialscales,22arbitrary,110ofdevicecharacteristics,1ofdeviceparameters,1,3,18,27ofdiscreterandomvariable,110ofnoisemargins,38ofthresholdvoltage,1upperboundon,8withstrongcorrelations,13,143Drain-inducedbarrierlowering,7Dual-residueprocessing,132A.
Zjajo,StochasticProcessVariationinDeep-SubmicronCMOS,SpringerSeriesinAdvancedMicroelectronics48,DOI:10.
1007/978-94-007-7781-1,SpringerScience+BusinessMediaDordrecht2014189Dynamiclatch,74,77,80,122Dynamicrange,4,6,7,10,135Dynamicvoltage-frequencyscaling,102EEffectivechannellength,31,160Effectivenumberofbits,135,183Eigenvaluedecomposition,36Energyoptimization,18,37,45,47,50,150Estimator,45,60,66,109,134,182Euler-Maruyamascheme,63–65Expectation-maximization,118,127,146,152ExtendedKalmanlter,93,109FFastfouriertransform,139,145Figureofmerit,101Fineconverter,132,141Fittingparameter,22,39Forgettingfactor,137Frequencymeasurements,4GGain-bandwidthproduct,6Galerkinmethod,22,85,106,109Gatelength,1,7,17,27,149Gatewidthvariability,9,119Gaussianmixturemodel,128Gradient-searchmethod,41Gramian,34–36,43,45,98,99,108,112,151HHammarlingmethod,36,108Heatsource,14,84,87,89Heuristicapproach,14,40,56Hotcarriereffect,7IIncidencematrix,25,62Integralnon-linearity,144Integratedcircuit,1,2,13,14,17,24,49,56,57,60,85,91,110,117,133,149,150,157Integrator,56Interfacecircuit,4,120,123Interpolation,30,32,86,97Intra-die,3,118,160Itostochasticdifferentialequations,14,18,24,26,32,49,57,62,65JJacobian,26,29,61,65,110,111,112,167,168,170KKalmanlter,84,85,92,93,109,112,151Karhunen-Loeveexpansion,20,22,39Kirchhoffcurrentlaw(KCL),6,60,155Kogge-Stoneadder,45,48,49,150LLeastmeansquare,43,92,182Leastsignicantbit,139,182Linewidthvariation,6Lossfunction,134Lyapunovequations,35,36,43,58,59MMatching,4,6,10,160,162,166Manufacturingvariations,22,91Matrix,25,26,32,34–37,43,56–59Maximumlikelihood,66,71,127,128Meansquareerror,43,92,182Measurementcorrectionfactor,22Milsteinscheme,64,65,79,80,150Mobility,8,136,137,140,161,163,164Mobilityreduction,161Modelorderreduction,14,33,34,43,44,85,95,98,108,109,112,151Modiednodalanalysis,24,32Momentestimation,20,27,33Monte-Carloanalysis,172MOSFET,1,3,4,137,152NNegativebiastemperatureinstability,13,83,117Newton'smethod,19,24Neyman-Pearsoncriticalregionnode,177,178Nodalanalysis,25,32,167Noiseexcessfactor,22,97,143margins,6,75simulation,55,70Non-stationaryrandomprocess,153Normalcumulativedistributionfunction,18NormaldistributionCentrallimittheorem,33Normalrandomvariable,24,26190IndexOOffset,5–7,10,73,121,124–127,132,134,137,138,141–143,160,164,165,181,182Operationaltransconductanceamplier,126Optimizationdeterministic,18,37sensitivity-driven,37stochastic,18,49,150Ordinarydifferentialequations,24,57,65,85,89PParametervector,65,66,127,128,169,170Parameterspace,23,28,49,128,150Parametricfunctions,13,20,176Parametricyield,23Parametricyieldlossimpactofgatelengthvariability,17impactofgatelengthvariation,17,20,27impactofpowervariability,155Parametricyieldmetric,23Parametricyieldoptimization,23Partialdifferentialequations,24,85Powerdynamic,3,92,103,117,124static,49,150Powermanagementblock,103Printedcircuitboard,134Probabilitydensityfunction,26,40,62,92,93,135,169,182Probabilitydistribution,18Processcontrolmonitor,119Processvariation,1,3,10–14,17,18,25,27–29,31,32,37,38,42,43Processwindowprogrammablegainamplier,73Processingelements,85,86,90,91Proportionaltoabsolutetemperature,125,126Pseudo-noisesequence,79QQuadraticprogramming,41,130Qualityfactor,69Quantizer,132,135,182,183RRandomerror,10Randomdopantuctuation,2,4,12Randomfunction,25,57,169Randomgatelengthvariation,3,10,17,37,160Randomintrachipvariability,3,10,20,22,33,162Randomprocess,20,22,31Randomsampling,180Randomvariability,20,21,39,63,110,182Randomvariables,20–22,39,63,110Randomvector,25,26,40,42,469Randomtelegraphnoise,2,4,152,153Reliability,10,13,41,63,83,84,117,149,152,154RepresentationsofrandomvariableResiduals,43,44,61,88,89,107,108,144Riccatiequation,98,107,108,112,151Runge-Kuttamethod,14,57,85,86Runtime,30,48,84,90,92,100–103,111SSchurdecomposition,58,71Sensors,3,84,90–92,109,110,112,137,141,151Short-channeleffects,1,38Signaltonoiseanddistortion,183Signaltonoiseratio,8,9,144,145Signaltonoiseplusdistortionratio,145Signicancelevel,178,179Singularvaluedecomposition,34,99Spatialcorrelation,10,22,27Spatialdistribution,18,20,31,33,39,42,43,47,109,110,140,162,163,165,166,184Spuriousfreedynamicrange,135,144,145Standarddeviation,18,20,31,33,39,42,43,47,110,140Staticlatch,27,77Stationaryrandomprocess,153Statisticaltiminganalysis,14,18,27,29,32,33,42,49,150Steepestdescentmethod,136Stochasticdifferentialequations,14,18,24,26,32,49,56,62,150Stochasticprocess,1,20,33,57,175Supportvectormachine,118,129,146,152Surfacepotentialbasedmodels,118,129,146,152,142Switchedcapacitor,67,71–73Systemonchip,3Systematicdrift,136Systematicimpactoflayout,127Systematicspatialvariation,20Systematicvariability,20Index191TTaylorseries,70,77,84,93,95Thermalmanagement,4,13,83,90,102,110,117,149,151Temperaturemonitor,125,126,132,140Temperaturevariability,14,155Testcontrolblock,120,123,141Teststructures,137Thresholdvoltage,1,3,11,17–19,21,22,31,37–19,47,60,117,119,127,136,137,140,149,152,155,157Thresholdvoltagebasedmodels,19,60Timetodigitalconverter,118,146,155,156Tolerance,13,23,49,107,142,169,172,176Totalharmonicdistortion,135Transconductor,68Transientanalysis,26,29,32,56Transistormodel,24,29,42,49,150,157,158Truncatedbalancedrealization,34,108,109,112,151UUnbiasedestimator,179Unscentedkalmanlter,14,93,109Unscentedtransform,93,109VVariablegainamplier,42,71,73–75,80,151Verylarge-scaleintegratedcircuit,91,92,117,127,132Voltagevariability,2,3,12,17WWafer,3,20,118,119,139,166Wienerprocess,57,63–65Within-die,4,91Worst-caseanalysis,23,127YYield,1,2,6,10,13,14,18,19,23,24,32,37,38ZZero-crossing,135192Index