1
DefectDiagnosisusinga Curr ent Ratio basedQuiescentSignalAnalysisModelfor Commercial Power Grids
Chintan Patel, Ernesto Staroswiecki, Smita Pawar, Dhruva Acharyya and Jim PlusquellicDepartment of CSEE, University of Maryland, Baltimore County
for JETTA fr om a special issue guest edited by Dr. Rafic Makki
Contact Author
Jim Plusquellic
UMBC, CSEE, ECS 212
1000 Hilltop Circle, Baltimore, MD - 21250
Email: [email protected]
Phone: 410-455-1349
Fax: 410-455-3969
DefectDiagnosisusinga Curr ent Ratio basedQuiescentSignalAnalysisModelfor Commercial Power Grids
Authors:
Chintan Patel, [email protected]
Ernesto Staroswiecki, [email protected]
Smita Pawar, [email protected]
Dhruva Acharyya, [email protected]
Jim Plusquellic, [email protected]
Addr ess:Department of CSEE, University of Maryland Baltimore County
1000 Hilltop Circle,Baltimore,
MD - 21250.
Abstract
QuiescentSignalAnalysis(QSA)is a novel electrical-test-baseddiagnostictechniquethat usesIDDQ measure-
mentsmadeat multiple chip supplypadsas a meansof locating shortingdefectsin the layout.Theuseof multiple
supplypadsreducestheadverseeffectsof leakagecurrentbyscalingthetotal leakagecurrentovermultiplemeasure-
ments.In previous work, a resistancemodelfor QSAwas developedand demonstratedon a small circuit. In this
paper, theweaknessesof theoriginal QSAmodelare identified,in thecontext of a productionpowergrid (PPG)and
probecard model,anda new modelis described.Thenew QSAalgorithmis developedfromtheanalysisof IDDQ con-
tour plots. A “family” of hyperbolacurvesis shownto be a goodfit to the contourcurves.Theparameters to the
hyperbolaequationsare derivedwith thehelpof insertedcalibration transistors.Simulationexperimentsare usedto
demonstrate the prediction accuracy of the method on a PPG.
Keywords:
IDDQ, IDDT, quiescent signal analysis, test, power grids
Abstract
QuiescentSignalAnalysis(QSA)is a novel electrical-test-baseddiagnostictechniquethat usesIDDQ measurements
madeat multiplechip supplypadsasa meansof locatingshortingdefectsin the layout.Theuseof multiplesupply
padsreducestheadverseeffectsof leakage currentby scalingthetotal leakage currentover multiplemeasurements.
In previouswork, a resistancemodelfor QSAwasdevelopedanddemonstratedon a smallcircuit. In this paper, the
weaknessesof theoriginal QSAmodelare identified,in thecontext of a productionpowergrid (PPG)andprobecard
model,anda new modelis described.Thenew QSAalgorithmis developedfromtheanalysisof IDDQ contourplots.A
“family” of hyperbolacurvesis shownto bea goodfit to thecontourcurves.Theparameters to thehyperbolaequa-
tionsarederivedwith thehelpof insertedcalibration transistors.Simulationexperimentsareusedto demonstratethe
prediction accuracy of the method on a PPG.
1.0 Introduction
IDDQ hasbeenamain-streamsupplementaltestingmethodfor defectdetectionfor morethanadecadewith many
companies.With theadventof thedeepsubmicrontechnologies,theuseof single-thresholdIDDQ techniqueresultsin
unacceptableyield loss.Settinganabsolutepass/fail thresholdfor IDDQ testinghasbecomeincreasinglydifficult due
to the increasingsubthresholdleakagecurrents[1]. Currentsignatures[2], delta-IDDQ [3] andratio-IDDQ [4] have
beenproposedasameansfor calibratingfor thesehighsubthresholdleakages.Thesetechniquesrely onaself-relative
or differentialanalysis,in which the averageIDDQ of eachdevice is factoredinto the pass/fail threshold.However,
these proposed forms of calibration are expected to become less effective over successive technology generations.
An alternative calibrationstrategy thatmayhave betterscalingpropertiesis to distributethetotal leakagecurrent
acrossa setof measurements.This is accomplishedby introducingprobinghardwarethatallows themeasurementof
IDDQ at eachof the supply ports.The methodproposedin this work, called QuiescentSignal Analysis (QSA), is
designedto exploit this typeof leakagecalibrationfor defectdetectionandasameansof providing informationabout
DefectDiagnosisusinga Curr ent Ratio basedQuiescentSignalAnalysisModelfor Commercial Power Grids
Chintan Patel, Ernesto Staroswiecki, Smita Pawar, Dhruva Acharyya and Jim Plusquellic
Department of CSEE, University of Maryland, Baltimore County
This work is supported by a Faculty Partnership Award from IBM’s Austin Center for Advanced Studies(ACAS) Program and by an NSF grant, award number 0098300.
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the defect’s location in the layout [5][6]. This latter diagnosticattribute of QSA may provide an alternative to
image-basedphysical failure analysisproceduresthat arechallengedby the increasingnumberof metal layersand
flip chip technology.
A resistance-baseddiagnosticmodelfor QSAwasdevelopedin previousworksandsimulationexperimentswere
usedto demonstratethe diagnosticcapabilitiesof the QSA methodon a small circuit [5][6]. In this paper, several
weaknessesof theresistance-basedmodelareuncoveredfrom simulationsof a productionpower grid (PPG).A cur-
rent-ratio-basedmodel is proposedand demonstratedto improve on defect localizationaccuracy of the original
method[7]. Thenew methodrequirestheinsertionof calibrationtransistors(CT), oneundereachof thesupplypads
in thedesign,thatpermit theshortingof thepower andgroundsupplyrails at pointscloseto thesubstrate.Thestate
of theCTsarecontrolledby scanchainflip-flops.TheIDDQs obtainedwhenoneof theCTsis turnedon areusedto
calibratethe IDDQs measuredundera failing IDDQ pattern.The calibrationtechniqueis shown to addressseveral
weaknessesof thepreviousmodelincludingnon-zeroprobecardresistanceandirregularsupplygrid topologies.Cur-
rent ratios,asopposedto absolutecurrents,areproposedasa meansof reducingthedependenceof the localization
algorithmon thevalueof thedefectcurrent.SPICEsimulationexperimentsdemonstratethat themaximumpredic-
tion error is 650 units in a 30,000 by 30,000 unit area.
It is notpossibleto evaluatetheQSA algorithmon theentire80,000by 80,000unit areaof thePPGusingSPICE
dueto the largesizeof theR model.Instead,a specializedpower grid simulationenginecalledALSIM is used[8].
The anomaliesin the grid’s structurein this larger areaincreasethe maximum prediction error to 1,340 units.
Although the predictionaccuracy is goodfor mostcases,an alternative “lookup table” approach(in contrastto the
hyperbola-basedapproach)is likely to be moreaccuratefor irregular grid regionsor configurations.The enhanced
simulationcapabilitiesof ALSIM enablethis strategy, aloneor in combinationwith the hyperbola-basedapproach
described in this paper.
Theremainderof this paperis organizedasfollows.Section2.0describesrelatedwork. Section3.0givesa brief
descriptionof theoriginal resistance-basedQSAtechnique,identifiesits weaknessesanddescribesthebasisof anew
model.Section4.0presentsthedetailsof thecurrent-ratio-basedQSA model.Section5.0givesexperimentalresults.
Section 6.0 gives our conclusions and areas of future research.
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2.0 Background
Severaldiagnosticmethodshavebeenproposedbasedon IDDQ measurements.In general,thesemethodsproduce
a list of candidatefaultsfrom a setof observedtestfailuresusinga fault dictionary. Thelikelihoodof eachcandidate
fault canbe determinedby several statisticalalgorithms.For example,signatureanalysisusesthe Dempster-Shafer
theory, which is basedon Bayesianstatisticsof subjective probability [9]. Delta-IDDQ makesuseof theconceptsof
differential currentprobabilisticsignaturesand maximumlikelihood estimation[10]. Although thesemethodsare
designedto improve theselectionof fault candidates,in many cases,they areunableto generatea singlecandidate.
Otherdifficultiesof thesemethodsincludetheeffort involvedin building thefault dictionaryandthetime requiredto
generate the fault candidates from the large fault dictionary using tester data.
TheQSA procedurecanhelpprunethecandidatelist producedby IDDQ andothervoltagebaseddiagnosticalgo-
rithms.Thephysical layout informationgeneratedby our methodcanbeusedwith informationthatmapsthelogical
faultsin thecandidatelists to positionsin thelayout.In addition,it maybepossibleto usethe(x,y) locationinforma-
tion provided by QSA asa meansof reducingthe searchspacefor likely candidatesin the original fault dictionary
procedure. This can reduce the processing time and space requirements significantly.
3.0 QSA Models
QSAanalyzesasetof IDDQ measurements,eachobtainedfrom individualsupplypads,to predictthelocationof a
shortingdefect.Theresistive elementof thepower grid causesthecurrentdrawn by thedefectto benon-uniformly
distributedto eachof the supplypads.In particular, the defectdraws the largestfraction of its currentfrom supply
padstopologically “nearby”. The sameis true of the leakagecurrents.However, only the leakagecurrentsin the
vicinity of the defectcontribute to the measuredcurrentin thesepads.The smallerbackgroundleakagecomponent
improvestheaccuracy of thedefectcurrentmeasurement.As describedin previousworks,QSAalsoproposestheuse
of regression analysis as a means of eliminating the remaining leakage component from the measured values [5][6].
3.1 The Resistance-based QSA Model
The fraction of the defectcurrentprovided by eachof the padsin the region of the defectis proportionalto the
equivalent resistancebetweenthe defectsite andeachof the pads.The differencesin thesevaluescanbe usedto
localizethedefectusinga methodbasedon triangulation.For example,Figure1 shows a shortingdefectin anequiv-
alentresistancemodelof a simplepower grid. Here,Req0 throughReq3 representtheequivalentresistancesbetween
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eachof the supplypads,Padi, andthe defectsite shown in the centerof the figure.The following setof equations
describetherelationshipbetweenthepower supplybranchcurrents,I0 throughI3, andVdef, thevoltageat thedefect
site.
In Eq.1, theIsarethemeasuredIDDQs.TheRps representtheprobecard’s resistances,whichweassumearevery
smallwith respectto theReqandcanbeignored(thisassumptionis addressedbelow). This leavestheReqandVdefas
unknowns.Withoutadditionalinformation,it is notpossibleto solve theseequationssincethereare4 equationsand5
unknowns.However, for thepurposeof diagnosis,only therelationshipsbetweentheReqareneeded.Relative equiv-
alent resistances can be computed with respect to a reference equivalent resistance, Reqj, as given by Eq. 2.
Under the condition that Rp << Req (otherwisethe modelshown if Figure1 is not complete),it is possibleto
obtainanaccuratepredictionof thedefect’s locationby solving thecircle expressionsgiven in Eq. 3 for a common
point of intersection given byx andy.
Theparametershi andki representthex andy coordinatesof thecenterof the ith circle.Thethreecircleequations
arerelatedto correspondingequationsfrom thesetdescribedby Eq.2 throughtheReq. Here,Reqj is assumedto be1.0
andReqaandReqbarecomputedfrom Eq.2 usingtheIDDQ measurements.Parameterm is usedto maptheresistances
given on the left in Eq. 3 to distances in the layout.
Thechoiceof thesupplypadsto beusedin thetriangulationprocedureis basedon two criteria.First, thesupply
I i Reqi Rpi+( )× VDD V–def
= for i = 0,1,2,3 (1)
I i Reqi Rpi+( )× I j Reqj Rpj+( )×=
Reqi
I jI i---- Reqj×
I jI i---- Rpj× Rpi–+=
with
(2)
i j≠
solving for Reqi in terms of Reqj gives
m Reqj× x h j–( )2 y k j–( )2+=
m Reqa× x ha–( )2 y ka–( )2+=
m Reqb× x hb–( )2 y kb–( )2+=
(3)
Figure 1. Equivalent resistance model of the power grid with a shorting defect.
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padsaresortedaccordingto themagnitudeof their correspondingIDDQ. Thesupplypad,j, with the largestIDDQ is
selectedfollowed by two orthogonallyadjacentsupplypads,a andb, to pad j sourcingthe next two largestvalues.
Notethatthismodelis basedon two simplifying assumptions:auniformresistance-to-distancemappingfunctionand
negligible valuesfor Rp. A uniform resistance-to-distancemappingfunctionis usedto describepower gridsin which
the equivalent resistance and Euclidean distance between any two points on the grid are proportional.
An exampleapplicationof this triangulation-basedmethodis shown in Figure2. Threedottedcirclesareshown
whosecentersaredefinedby thepositionsof thePad1, Pad2 andPad3. Theradii arelabeledwith theappropriateReq
valuesasgiven in Eq. 3. For example,Pad3 definesthecenterof thecircle with smallestradius,i.e., it is thesupply
padwith thelargestIDDQ. Its radiusis labeledwith Reqj in thefigure.Theinitial radii of thethreecirclesarethenmul-
tiplied by a commonfactor, m, to a commonpoint of intersection.This point is labeledas“PredictedDefectLoca-
tion” in the figure to contrast it with the “Actual Defect Location”.
3.2 Weaknesses of the Resistance-based Model
Unfortunately, theassumptionsof theresistance-basedmodelarenotvalid in many situations.Here,it is assumed
thattheRpsaresmallrelative to theReq. Underthisassumption,themeasuredIDDQsarerelatedto theReqasgivenby
Eq. 4 (derived from Eq. 2).
Therefore,theresistance-basedQSA modelassumesthatthecurrentratiosareinverselyproportionalto theresis-
tanceratios.If thevaluesof Rps aresimilar to or largerthantheReq, thentherelationshipgivenby Eq.4 is weakened
and the accuracy of the triangulation approach is correspondingly reduced.
In thenext section,we presenta morecompleteequivalentresistancemodelof theCUT thatbetterrepresentsan
actualprobecardmodelin which theRps aresignificant.Thenew modelrequiresadditionalinformationin orderto
solve for unknownssuchastheReq andRp. A new QSA methodis proposedthatobtainsthis informationfrom cali-
Reqi
I jI i---- Reqj×≅ or
ReqiReqj-----------
I jI i----≅
If these terms are negligible then
(4)
Reqi
I jI i---- Reqj×
I jI i---- Rpj× Rpi–+=
Figure 2. Triangulation under r esistance model.
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bration transistorsmeasurements.However, it shouldbe notedat this point that large valuesof Rp will adversely
affect the precisionrequiredin the measurementof the IDDQs underany proposedstrategy. This follows from a
numericalanalysisof Eq. 2, that shows the convergenceof all currentratiosto the ratiosdefinedby the Rps asthe
magnitude of theRps are increased to and above theReq.
Anotherweaknessof the resistance-basedQSA modelis with regard to theuniform resistance-to-distancemap-
ping function. Most supply topologiesarepoorly modeledasuniform. In previous work, we proposeda mapping
function basedon resistancecontoursto dealwith complicatedirregular topologies[5]. In this work, we proposea
secondstrategy basedon theuseof a currentratio lookup-table.Both techniquesrequireresistanceandcurrentpro-
files of thegrid to bederived in advancethroughsimulations,andshouldbeavoided,if possible,in casesinvolving
more regular topologies.
The topologyof the PPGunderinvestigation in this work fits betweenthe totally regular and totally irregular
extremes.The mappingfunction is not strictly uniform but, becausethe physical structureof the grid is regular in
many places,it is possibleto modeltheresistanceperunit distancebetweeneachpairingof supplypadsusinga con-
stant.Thenew hyperbola-basedQSA methoddescribedin this paperis ableto calibratefor this typeof power grid
resistance-to-distanceprofile usingmeasureddataonly. Therefore,it providesa simpleralternative to a lookup-table
approach.
3.3 The PPG’s Physical Characteristics
Figure3 shows the80,000by 80,000unit layoutof thePPG.ThePPGinterfacesto a setof externalpower sup-
pliesthroughanareaarrayof VDD andGND C4 pads.A C4 padis a solderbumpfor anareaarrayI/O scheme.The
PPGhas64 VDD C4sand210GND C4s(not shown in Figure3). The64 VDD C4sdivide thePPGinto 49 different
regionscalledQuads.ALSIM simulationsexperimentswererun on theentirePPG.However, dueto spaceandtime
constraints,it wasnotpossibleto runSPICEsimulationsontheentirePPG.Rather, aportionof thePPGconsistingof
9 quadswassimulatedusingSPICE.Thisportionconsistsof thelower left 9 Quadsasshown in Figure3, andis sub-
sequently referred to as the Q9. The Q9 occupies a 30,000 by 30,000 unit area.
Figure 4. Layout details of the PPG.
Figure 3. Layout of the PPG.
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In orderto deriveanelectricalmodelof thePPG,wefocusedouranalysisontheportionshown in thelower left of
Figure3 identifiedastheQuad.Figure4(a)expandson this view by showing a moredetaileddiagramof this 10,000
by 10,000unit region. This is again expandedin Figure4(b) which shows a stacked four metallayerconfiguration,
with m1 andm3 runningvertically andm2 andm4 runninghorizontally. TheC4sareconnectedto wide runnersof
verticalm5, shown in the top portionof Figure4(a),thatare,in turn, connectedto them1-m4grid. In eachlayerof
metal,theVDD andGND railsalternate.In theverticaldirection,eachm1rail is separatedby adistanceof 432units.
The alternatingvertical VDD and GND rails are connectedtogetherusing alternatinghorizontal metal runners.
Stackedcontactsareplacedat theappropriatecrossingsof thehorizontalandvertical rails. Thegrid is fairly regular
exceptin the region labeled“irregular region” in theupperright cornerof Figure4(a).Them1 in this region of the
layout varies from the regular pattern shown in Figure 4(b).
TheR modelof theQuadwasobtainedfrom anextractionscriptwhichusesprocessparametersfrom theTSMC’s
0.25µm process[11]. 1Ω resistanceswereinsertedbetweenthepower suppliesandtheR modelof thegrid to model
thetesterpower supply(s)andprobecardcontactresistancesto thechip.Althoughour simulationmodeluses1Ω for
all probecardresistances,theanalyticalmodelthatwederivebelow accommodatesamorerealisticprobecardmodel
in which probe card resistanceis different from one pad to another. The combinedresistancenetwork contains
approximately 27,000 resistors.
3.4 The Quad’s Electrical Characteristics
Figure4(b) alsoshows a setof currentsourcesthatwereinsertedindividually in a sequenceof simulationsasa
meansof evaluatingtheelectricalbehavior of theresistancemodelat theVDD C4s.Thecurrentsources,whichmodel
thepresenceof a shortingdefect,wereplacedat regularintervalsbetweenm1 VDD andGND runners.An equivalent
resistancemodelof theQuadis shown in Figure5 with oneof thecurrentsourcesinserted.Thefour grid equivalent
resistances,Req, in theuppercenterportionof thefigurearethesourceof resistancevariationasseenfrom thepower
supplies,asthecurrentsourceis movedin the layout.Thestrengthof thecorrespondenceof theseresistancesto the
positionof thedefectdeterminestheaccuracy of the triangulationprocedureusedin QSA. It is thereforeprudentto
evaluate this relationship for the Quad.
Thereareseveral significantdifferencesbetweenthis modelandthe modelshown in Figure1. First, underthe
assumptionthatthevaluesof theRp arenon-zero,thegrid resistancesbetweentheC4s,e.g.R01 shown on thetop left
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of Figure5, areneededin any completeequivalentresistanceexpressionsuchasthatgivenby Eq.2. Second,theReq
areactuallythreedimensionalin natureandcanbemodeledasRz andRxy asshown on theright sideof Figure5. Rz
adversely impacts the accuracy of the triangulation procedure for the same reasons given earlier forRp.
A third limitation in our original resistance-basedmodelis relatedto the resistanceprofile thatcharacterizesthe
PPGunderinvestigation in this research.Our analysisrevealsthat the variationin equivalentresistanceover small
verticalintervalsof theQuad,e.g.,alongtheinterval betweentwo contactpointsin m1, is onorderwith thevariation
acrossthe entireQuad.For example,the segmentlengthgiven betweenpointsA andB in Figure4(b) is approxi-
mately630units.Usingthem1 resistanceparameterfor TSMC’s 0.25µm processyieldsa valueof 5.6Ω. Therefore,
in m1alone,theresistancevariesfrom 0Ω at thecontactto 5.6||5.6= 2.8Ω in thecenter. On theotherhand,theaver-
ageresistancefrom thecenterof theQuad(shown in Figure4(a))to any of theVDDs (distanceof ~7,000units)is less
than6Ω. The increasingwidth of the metal runnersfrom m1 to m5 is responsiblefor theseresistanceto distance
anomalies.
In orderto gain insightinto otheralternativediagnosticstrategies,wefirst derivedtheprofilesof thenetwork vari-
ablesincludingReq, Vdef (thevoltageat thedefectsite)andtheIDDQsat theVDD C4s.Theprofileswerederivedfrom
the resultsof 2,600SPICEsimulationexperimentsof the Quad.In eachsimulation,a 20mA currentsourcewas
placedbetweenm1 VDD andGND rails at differentlocationsin thelayout.Figure6 shows thecurvesfor Req0andI0
(at C40) andVdef (thecurrentsource’s terminalvoltageat theconnectionpoint on them1 VDD rail) for a setof simu-
lationsrunalongthelinesidentifiedasx-sliceandy-slicein Figure4(a).TheReq0valueswerecomputedusingEq.5.
It is clearfrom thesegraphsthatthevariationsin Req0andVdefalongthey dimensionaresignificantlylargerthan
Req0
VDD Vdef–( )I 0
-----------------------------------=
VDD = 2.5VVdef = voltage at the defect siteI0 = current through VDD0
(5)
where,
Figure 5. Equivalent resistance model of the Quad.
Figure 6. Network variable plots for sources along x-slice (top) and y-slice (bottom) lines of Figure 4(a).
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thosealongthex dimension.In contrast,thecurrentsarewell behavedalongeitherdimension.Thestaggeredarrange-
mentof VDD andGND grids,asshown in Figure4(b), causesthe total resistancebetweenVDD andGND to change
slowly acrossthegrid, throughtheexchangeof nearlyequalresistancefragmentsbetweentheVDD andGND grids.
This keepsthecurrentswell behavedwhile theresistancesto, andvoltagesat, thedefectsiteoscillateinverselywith
each other.
3.5 Contour Profiles of the Quad
Anotherusefulview of thebehavior of thesenetwork variablesis throughcontourplots.A line within a contour
plot is definedastheparametervaluesover which thevalueof the function remainsconstant.Contoursareparticu-
larly usefulwhendatais to befit to a function.Figures7 and8 show theequivalentresistanceandcurrentcontoursof
theQuadfor VDD0 (only every 3rd contourcurve is shown.) Thex andy axescorrespondto the(x,y) coordinatesof
theQuadasshown in Figure4(a).Thejaggednatureof thecurvesasshown in Figure8 modelsabandwhosewidth is
definedby thevertical line segmentsin thecurves.It is clearthat theequivalentresistancecontourplot is difficult to
make useof. Thesameis trueof theVdef contourplot (not shown). In contrast,thecurrentcontoursareelliptical in
shape,(exceptfor a region in theupperright handcorner, identifiedas“irregularregion” givenearlierin referenceto
Figure 4(a)). Similar patterns are present in the current contour plots of the other VDDs.
Therefore,a diagnosticmethodbasedon currentsis likely to yield the bestresults.However, unlike equivalent
resistance,thedisadvantageof usingthecurrentsdirectly is thedependency that is createdbetweenthecontoursand
themagnitudeof thedefect’s shortingcurrent.Currentratiosareanalternative thatreducethis dependency sincedif-
ferent values of defect current are reflected as the same ratio in the C4 IDDQs.
Thecontourplot for I0/I1 is shown in Figure9. Like theI0 contourplot, thecontourlinesarewell behaved.How-
ever, theelliptical curvescharacterizingthe I0 plot now appearashyperbolacurves,particularlyin theregion to the
left of x=5000.Thesetor “f amily” of hyperbolasis centeredat themidpointbetweenthepositionof C40 (lower left)
andC41 (upperleft). Thecontourcurvethatpassesthroughthismidpoint(y=5000)onthey axisis nearlylinearalong
Figure 7. Req0contours of Quad.
Figure 8. I0 contours of Quad.
Figure 9. I0/I1 contours of Quad.
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aline to thecenterof theQuad(shown by the‘dot’ in thecenterof thefigure).Thiscurvedefinesthepointsin thelay-
out thatareexpectedto produceanI0/I1 currentratio closelyapproximatedby 1.0.TheI0/I1 increaseto a maximum
in the lower left corner. ThemaximumI0/I1 currentratio is largely determinedby theRz componentof resistanceat
C40 andRp0. As anexample,theI0/I1 maximumfor theQuadis 1.55andtheI0/I2 maximumis 1.84.Thesemaximum
currentratioscanbedeterminedexperimentallyusinga simpletestcircuit. We describethis testcircuit andits other
benefits after we derive the analytical model for the new QSA procedure.
4.0 The Current Ratio Model for QSA
Thedensityof thecontourcurvesin thelower left quarterof Figure9, i.e. theregionwith x andy coordinatesless
than5000,is higherthanthedensityin otherregionsof theQuad.For example,thenumberof contourcurvesbelow
they=5000is 10 while thenumberabove this point is 6. Therefore,the I0/I1 andI0/I2 currentratiosareexpectedto
provide thebestresolutionfor defectsthatoccurin this region.UndertheassumptionthattheC4swith largestIDDQs
areclosestto thedefectsite,it is straightforwardto identify therelevantregionandto computetheappropriatecurrent
ratios from the measureddata.(This assumptionis later removed.) The morechallengingproblemis to determine
how to use these ratios to identify the location of the defect in the selected region.
Themoststraightforwardmethodis to usethemeasuredratiosto selecttwo contourcurves.For example,Figures
10and11show theI0/I1 andI0/I2 contourcurvesobtainedfrom simulationsperformedonthelower left quarterof the
Quad(only every othercurve is shown). Thepoint of intersectionof two curves,onefrom eachfigure,identifiesthe
positionof the defectin the layout.This is the generalideabehindthe lookup-tablemethodreferredto above. The
drawbackof this methodis the large numberof simulationsthat areneeded(onefor eachcandidatepositionin the
layout)to build thetable.Powergrid simulatorssuchasALSIM makethispracticalandweexpectthisapproachto be
useful for irregular grid topologies. However, a simpler method is possible in many situations.
An alternative strategy is to derive a functionthatapproximatesthecontourcurvesusingthemeasuredquantities,
i.e. thecurrentratios,asparameters.As notedabove, thecurrentratiocontourcurvesaresimilar in shapeto hyperbo-
las.Figures10 and11 show a setof curvesderivedfrom hyperbolassuperimposedon thecontourcurvesfor illustra-
Figure 10. I0/I1 contours (jagged) and hyperbolas (smooth curves) for lower left quadrant of Quad.
Figure 11. I0/I2 contours (jagged) and hyperbolas (smooth curves) for lower left quadrant of Quad.
13
tion. In orderto realizethis mapping,it is necessaryto derive expressionsfor the hyperbolaparameters.Eq. 6 and
Figure12 defineandillustrate“horizontally-oriented”hyperbolas,suchasthoseshown in Figure11. Thearrows on
the right of Figure 11 illustrate the region in which these curves are represented in Figure 12.
Figure12portraystheroleof thea andb parametersin agraphanddefinesanadditionalparameter, c, thatis used
to definetherelationshipamongthesetsor “f amilies” of hyperbolasin Figures10 and11.A family of hyperbolasis
definedasasetthatshareacommoncenterandfocus.Theh andk parametersin Eq.6 definethecenterof thehyper-
bolas.Thecentersof thehyperbolacurvesshown in Figures10 and11 areidentifiedat (h,k)=(0,5000)and(5000,0),
respectively, andrepresentthemidpointbetweenthefoci. Thefoci of thehyperbolasaregivenby F1 andF2 in Figure
12. These points represent the (x,y) coordinates of the C4 VDDs.
Thea andb parametersof thehyperbolasneedto bedefinedin termsof thecurrentratios.Fortunately, thenature
of thecontoursdefinedby thegrid allow analternative formulationof Eq. 6 asgivenby Eq. 7. Here,b2 is replaced
with (c2 - a2). Sincec is fixedfor all hyperbolasin thefamily asthedistancebetweentheir center, (h,k), andthecoor-
dinates of the C4 supply pad, this makesb dependent ona. Therefore, onlya needs to be defined.
Fromthediagramshown in Figure12,a definesthepointof intersectionof theI0/I2 hyperbolawith thehorizontal
line definedbetweenthecenter(h,k) = (5000,0)andC40 (F1 in thefigure).Therefore,a variesfrom 0, at thecenter, to
L/2 atC40, whereL is definedasthedistancebetweenC40 andC42 (10,000for theQuad).Thecurrentratiosatpoints
alongthis line increasefrom 1.0 at the centerto the maximumcurrentratio, e.g.1.84 for I0/I2 in the Quad.If this
maximumcurrentratio is known, thenthefunctionthatdefinesa canbederivedfrom thelumpedR modelshown in
Figure 13 as follows.
x h–( )2
a2
------------------- y k–( )2
b2
-------------------– 1= (6)
x h–( )2
a2
------------------- y k–( )2
c2
a2
–-------------------– 1= (7)
Figure 12. Definitions of the hyperbola parameters
Figure 13. Lumped R model for the hyperbola a parameter.
14
Eqs.8 and9 give theexpressionsfor thecurrentratio β02 anda without proof. ReqT (total resistance)is equalto
thesumof theequivalentresistances,i.e. Req0+ Req2, betweenthedefectsiteandeachof the two C4s.Rp0 andRp2
are the probe card resistances at C40 and C42, respectively.
4.1 Calibration Transistors
As pointedoutearlier, Req0andReq2, andthereforeReqT, cannotbeobtainedin thedefectivechip.However, under
thespecialcasewherethedefectshown in Figure13 is positionedon a line betweenC40 andC42, we canobtaina
closeapproximationof ReqT experimentally. This is accomplishedby insertinga calibrationtransistor(CT0) under
C40, asshown in Figure14. Thesourceanddrainof theCT0 connectto VDD andGND in m1 andprovide a way to
conditionallyshortthesenodestogether. By positioningtheCT0 directly underC40 (at thelowestresistanceposition
from m1 to C40), the maximumcurrentratio, β02(CT0)= I0(CT0)/I2(CT0), canbe obtained.This is accomplishedby
placingthechip into astatethatdoesnotprovoke thedefectandturningonCT0 usingthescanchainflip-flop driving
its gate.
Themeasuredvaluesof I0(CT0)andI2(CT0) resolveseveralissuesrelatedto theapplicationof this technique.First,
β02(CT0)allowsEq.8 to besolvedundertheboundaryconditionm=0. If thesameprocessis repeatedusingacalibra-
tion transistorCT2, positionedunderC42, thenEq. 8 canbesolvedundera secondboundarycondition,m=L, using
β02(CT2). With threeequations,ReqT, Rp0, andRp2 in Eq. 8 cannow be eliminated,allowing a to be expressedasa
functionof themeasuredcurrentratios,β02, β02(CT0)andβ02(CT2). This is possiblebecausethevaluesof ReqT, Rp0,
(8)
β02 =I 0I 2-----
ReqT L m–( )× L Rp2×+
m ReqT× L R× p0+----------------------------------------------------------------=
and aL2--- m–=
aL2---
LReqT-------------
Rp2 β02 Rp1×( )– ReqT+
1 β02+( )---------------------------------------------------------------
–=
Substituting and solving fora yields
(9)
Figure 14. Calibration Transistor and controlling scan-chain FF.
15
andRp2 arenearlyinvariantacrossthethreetests.Eq.10and11givestheexpressionsfor a andb in termsof thecur-
rent ratios derived from C40 and C42 CT tests.
Thus,a nicefeatureof this calibrationtechniqueis that it is independentof theRp, which arelikely to vary from
touch-down to touch-down of the probe card.
A secondproblemaddressedby the CTs is relatedto the proceduredescribedin Section3.1. Pad selectionis
accomplishedby sortingtheIDDQ valuesandidentifying thepadwith the largestIDDQ asthe“primary” pad(padj).
Two (of thefour) orthogonallyadjacentsupplypadsto padj arethenselectedfrom thetopof thesortedlist. Unfortu-
nately, this algorithmfails to selectthepadssurroundingthedefectundercertainconditions.For example,Figure15
shows a portionof thesupplygrid with 9 C4s.Thedefectis locatedin theupperleft Quadandtherefore,thealgo-
rithm shouldselectC43 asthe“primary” padandC41 andC47 astheorthogonallyadjacentpads.However, if Reqa>
Reqb, thesortedlist placesC45 above C41 andthealgorithmincorrectlyselectsC45. This typeof resistanceanomaly
can occur, for example, if the power grid mesh is denser between C43 and C45 than it is between C43 and C41.
TheCT datacanbeusedto instrumentamorerobustpadselectionalgorithm.Thecurrentratioβ31(CT3)= I3(CT3)/
I1(CT3)obtainedby turningontheCT underC43 givestheupperboundonthecurrentratiobetweenC43 andC41. The
currentratiocomputedunderthecircuit statewith thedefectprovoked,β31, is necessarilylessthantheβ31(CT3), since
β31(CT3)is themaximumratio.Therefore,animprovedalgorithmselectsthecorrectsecondarypads,e.g.C41 instead
mL β02 CT0( ) 1 β+ 02 CT2( )( ) β02– 1 β+ 02 CT2( )( )( )
1 β+ 02( ) β02 CT0( ) β02 CT2( )–( )--------------------------------------------------------------------------------------------------------------------------=
with,β02= current ratio I0/I2 at state with defect provokedβ02(CT0)= current ratio I0(CT0)/I2(CT0) with CT0 on
L= distance between two adjacent C4 VDDs
aL2--- m–= (10)
β02(CT2)= current ratio I0(CT2)/I2(CT2)with CT2 on
b c2
a2
–L2---
2a
2–= =
(11)
Figure 15. Anomalies in complex grids.
16
of C45, by using CT ratio data.
It is alsopossiblein somegrid configurationsthatthelargestIDDQ is not drawn from thepadthatis closestto the
defectsite.In this case,theexisting algorithmdoesnot selectthecorrect“primary” pad.We arecurrentlyinvestigat-
ing the use of CT data to solve this problem, and hope to describe a solution in a future work.
A third problemaddressedby theCTs is relatedto theassumptionthat theunity currentratio line (the1-line or
centerfor thehyperbolas)is positionedmidwaybetweentheC4s.This is only truefor simplegrids(suchastheQuad
shown in Figure4) if theRp areequal.If theRp arenot equal,Eq.10 canbeusedto derive theoffset,c’, of the1-line
by settingβ02 to 1 and simplifying.
A similar shift occursin morecomplex grids, suchas that shown in Figure15, but for a reasonrelatedto the
degreeof symmetryin theC4ssurroundinga region. For example,thebottomportionof thegrid in Figure15 con-
tainsa row of threeC4s,C40, C42 andC44. The1-line in the lower left Quadis shown skewedto theright from the
midpointgivenby L/2. Theasymmetryin theC4ssurroundingthis region,e.g.C4s0, 1 and6 ontheleft andC4s2-5,
7 and8 on theright, areresponsiblefor this shift. We arecurrentlyevaluatingmorecomplex circuit modelssuchas
the oneshown in Figure5 asa meansof formulatingan expressionthat accountsfor this shift. Experimentally, we
determined that Eq. 12 yields a good approximation of the offset,c’, of 1-lines for Quads within the PPG.
4.2 Leakage Current
Oneelementthat we haven’t addressedis the impactof leakagecurrent.A secondcalibrationmethodwaspro-
posedin previouswork to dealwith leakage[5]. Althoughcalibratingfor leakageis clearlyanimportantissue,wedo
not focus on it in this work becauseof spacelimitations. The limited numberof experimentsconductedthus far
involving leakageindicatethat it hasonly a small impacton the accuracy of the predictions.The sameis true for
experimentsconductedusingdifferentvaluesof defectcurrent.Currentratiosarenaturallyrobust to thesevariables
but a quantitative analysis of their impact remains to be determined and will be addressed in a future work.
4.3 The QSA Procedure
Theprocedureto localizeadefectfollows from thediscussiongivenin theprevioussection.Onceachip is identi-
c′ L2--- β02 CT0( )β02 CT2( )( )= (12)
17
fied asdefective,e.g.from a Stuck-Ator IDDQ go-nogotest,thefollowing testsareperformedundertheQSA proce-
dure.
• First, thechip is setto a statethatprovokesthedefectandtheindividual IDDQ valuesaremeasured.TheC4 pad,
j, sourcingthelargestIDDQ andtwo orthogonallyadjacentC4pads,x andy, areidentifiedasdescribedin Section
4.1. The current ratiosβjx andβjy are computed.
• Thechip is thenput into a statethatdoesn’t provoke thedefect.TheCT for the jth padis turnedon andthecur-
rent ratiosβjx(CTj) andβjy(CTj) are computed.
• Similarly, the currentratios βjx(CTx) and βjy(CTy) are computedfrom measurementsmadewith CTx and CTy
turnedon. Eq. 12 givestheoffsetsneededto derive the two centersof thehyperbolas,(h’,k)x and(h,k’)y, along
the x- and y-dimension, respectively, from padj.
• Eq. 10 is then used to deriveax anday parameters using Lx = 2*c’x and Ly = 2*c’y for L.
• Thebx andby parameters are computed using Eq. 11.
• Thesetwo pairsof a andb parametersdefineboththepositionandshapeof onehyperbolafrom eachof thetwo
families,e.g.asillustratedin Figure16 usingthehyperbolacurvesfrom Figures10 and11. The intersectionof
these two hyperbolae gives the predicted location of the defect.
The algorithm,asstated,requiresa changein the stateof the CUT after the first setof IDDQ measurementsare
made.Therefore,the contribution of leakageto the currentsmeasuredwith the CTs turnedon is different thanthe
contribution underthe statewith the shortingdefectprovoked. The vector-to-vector leakagevariation is likely to
adverselyaffect theaccuracy of thepredictions.An alternative testprocedurethatdoesnot changetheCUT’s stateis
to performtheCT testswith thedefectprovoked.ThecurrentsmeasuredundertheCT testscanbe“adjusted”by sub-
tractingthe currentsmeasuredunderthe defectprovoking test.Even thoughthe presenceof the defect’s currentis
likely to changetheequivalentresistancesof theCUT undertheCT tests,we expecttheerrorintroducedby this type
of procedureto besmallerthantheerror introducedundertestscenariosin which thevector-to-vectorleakagevaria-
tion is large.
Figure 16. Example prediction using the hyperbola curves from Figures 10 and 11.
18
5.0 Experimental Results
This algorithmwasappliedto thedataobtainedfrom 200SPICEsimulationsof the30,000by 30,000unit region
of thePPGreferredto asQ9 in Figure3. A threedimensionalerrormapplotting thepredictionerroragainstthe(x,y)
coordinateof theinserteddefect(modeledusingacurrentsource)is shown in Figure17.Thepredictionerroris com-
putedasthedirecteddistancebetweenthepredictedlocationandtheactuallocationof thedefect.Theaverageand
worst case prediction errors are 215 and 650 units, respectively.
Thesizeof thesimulationmodelfor theentirePPGshown in Figure3 madeit impossibleto performSPICEsim-
ulationson it. Instead,thePPGwassimulatedusinga specializedpower grid simulationenginecalledALSIM. The
predictionerror map from 500 ALSIM simulationsis shown in Figure18. The averageandworst caseprediction
errorsare410 and1,340units, respectively. The increasein predictionerror is largely dueto the moresignificant
anomalies in the grid’s structure over the larger region defined by the entire PPG.
6.0 Conclusions
The weaknessesof our previously derived resistance-basedQuiescentSignalAnalysismodelareaddressedin a
new current-ratio-basedtechnique.Calibrationtransistorsareproposedto reducethe adverseeffectsof probecard
resistancevariationsonthepredictionaccuracy of thenew QSAtechnique.Thecalibrationtransistordatais alsoused
to accountfor power grid resistancevariationsfrom oneregion to the next andasymmetricalor irregular arrange-
ments in the positions of the power supply pads.
Thecurrentratio contoursderived throughSPICEsimulationsof a commercialpower grid areshown to bewell
approximatedby “f amilies” of hyperbolacurves.An analyticalframework is derivedthatallows themeasuredIDDQ
data to be translated to physical (x,y) layout coordinates, that represent the position of the defect.
Although the analyticalmodel that we presentin this work accountsfor testerenvironmentvariablessuchas
probecardresistancevariations,thesimulationdatawasderivedfrom a simplermodel.For example,theprobecard
resistancewasheldconstantat1Ω ateverysupplypad,20mAwasusedfor defectcurrents,andleakagecurrentswere
not included.As pointedout,currentratiosarenaturallyrobustto variationsin defectcurrentandthecalibrationtran-
sistorsin combinationwith regressionanalysisare expectedto be effective in dealingwith leakages.Simulation
Figure 18. Prediction error map for the entire PPG.
Figure 17. Prediction error map of the Q9.
19
experiments are currently underway to verify these hypotheses.
Thelast issuethat remainsto beexploredis theeffectivenessof this techniqueon othertypesof grid topologies.
For significantly irregular grids, we expecta lookup-tableapproachto be moreaccuratethan the hyperbola-based
technique.We areinvestigating theuseof power grid simulatorssuchasALSIM asa meansof makingthis typeof
approach practical.
Acknowledgments
We thank Dr. Anne Gattiker and Dr. Sani Nassif at IBM Austin Research Lab for their support of this research.
References
[1] T.W.Williams, R.H.Dennard,R.Kapur,M.R.Mercer& W.Maly, “I DDQ test:SensitivityAnalysisof Scaling”, ITC, 1996,
pp.786-792.
[2] A.E.Gattiker and W.Maly, “Current Signatures”, VTS, 1996, pp.112-117.
[3] C. Thibeault, “On the Comparison of Delta IDDQ and IDDQ Test”, VTS, 1999, pp. 143-150.
[4] PeterMaxwell, PeteO’Neill, RobAitken, RolandDudley,NealJaarsma,Minh Quach,Don Wiseman,"CurrentRatios:A
self-Scaling Technique for Production IDDQ Testing", ITC, 1999, pp.738-746.
[5] Jim Plusquellic,“IC DiagnosisUsing Multiple SupplyPadIDDQs”, DesignandTest,Vol. 18, No. 1, Jan/Feb2001,pp.
50-61.
[6] ChintanPatelandJim Plusquellic,“A ProcessandTechnology-TolerantIDDQ Methodfor IC Diagnosis”,VTS, 2001,pp.
145-150.
[7] C.Patel,E.Staroswiecki,D. Acharyya,S.PawarandJ.Plusquellic,”A CurrentRatioModel for DefectDiagnosisusingQui-
escent Signal Analysis”, Workshop on Defect Based Testing, VTS, 2002.
[8] S. Nassif and J. Kozhaya,”Fast Power Grid Simulation”, DAC, 2000, pp. 156-161.
[9] ChristopherL.Hendersonand Jerry M.Soden,“SignatureAnalysis for IC Diagnosisand Failure Analysis”, ITC, 1997,
pp.310-318.
[10] C. Thibeault,L. Boisvert, “Diagnosismethodbasedon delta IDDQ probabilisticsignatures:Experimentalresults”, ITC,
1998, pp.1019-1026.
[11] MOSIS at http://www.mosis.edu/Technical/Testdata/tsmc-025-prm.html.
[12] ChintanPatel,ErnestoStaroswiecki,DhurvaAcharyya,SmitaPawar,andJimPlusquellic,“DiagnosisusingQuiescentSignal
AnalysisonaCommercialPowerGrid”, Acceptedfor publicationin InternationalSymposiumfor TestandFailureAnalysis,
2002.
Figure Captions
Figure 1: Equivalent resistance model of the power grid with a shorting defect.
Figure 2: Triangulation under r esistance model.
Figure 3: Layout of the PPG.
Figure 4: Layout details of the PPG.
Figure 5: Equivalent resistance model of the Quad.
Figure 6: Network variable plots for sources along x-slice (top) and y-slice (bottom) lines of Figure 4(a).
Figure 7: Req0contours of Quad.
Figure 8: I0 contours of Quad.
Figure 9: I0/I1 contours of Quad.
Figure 10: I0/I1 contours (jagged) and hyperbolas (smooth curves) for lower left quadrant of Quad.
Figure 11: I0/I2 contours (jagged) and hyperbolas (smooth curves) for lower left quadrant of Quad.
Figure 12: Definitions of the hyperbola parameters.
Figure 13: Lumped R model for the hyperbola a parameter.
Figure 14: Calibration Transistor and controlling scan-chain FF.
Figure 15: Anomalies in complex grids.
Figure 16: Example prediction using the hyperbola curves from Figures 10 and 11.
Figure 17: Prediction error map of the Q9.
Figure 18: Prediction error map for the entire PPG.
21
defect
Pad1
Req0
Req1Req3
Req2
Rp3
Rp2
Rp1
VDD
Supply Ring
Rdef
I1
I0
I3
I2
Vdeflocation
Rp0
on probe cardPad3
Pad0 Pad2
Figure 1
22
Pad1
ActualDefectLocation
DefectLocation
Predicted
scaled by m
Pad3
Pad2Pad0
Reqa
Reqb
Reqj
Figure 2
23
Quad Q9 C4 VDD Pads
Figure 3
24
Figure 4
VDD1
C4Pads
y slice
x slice
irregular region
(a)
M5
M5
VDD0 VDD2
VDD3Gnd2 Gnd5
Gnd1
Gnd0
Gnd4
Gnd3
Vdd
Gnd Rails
M1
M2
M3
M4
Contacts
(b)
B
A
Rails
Currentsources
+
+
+
+
+
+
+
++
+
+
+
25
Figure 5
VDD1
VDD0 VDD2
VDD3
Defect Current
GND3
GND5GND2
GND1
GND0
VDD Grid
GND Grid
Rp3
Rp2
Rp0
Rp1
Rpg5
Rpg4
Rpg3Rpg0
Rpg2
GND4
Rz
Rxy
+- +
-
+-
+-
+-
+-
+-
+-
+-
+-
Req0
Req1 Req3
Req2
Rpg1
C4s
Source: 20mA
R01
26
Figure 6
I0
Req0
10
00
0
50
00
Oh
ms
mA
mV
2.0
9.0
6.6
4.22.3
Vdef
0
2.4
10
00
0
50
00
Oh
ms
mA
mV
2.5
8.5
6.5
4.72.4
0
2.4
I0
Req0
Vdef
x-coordinate
y-coordinate
27
Figure 7
Mon Aug 5 01:04:20 2002
1900.0 2840.0 3780.0 4720.0 5660.0 6600.0 7540.0 8480.0 9420.0 10360.0 11300.0 12240.0 13180.0 14120.0 15060.0 16000.0 16940.0 17880.0 18820.0 19760.0 20700.01.50
2.24
2.98
3.72
4.46
5.20
5.94
6.68
7.42
8.16
8.90
9.64
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Kili13.34
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Figure 8
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Kili13.34
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Figure 9
Mon Aug 5 01:13:52 2002
1900.0 2840.0 3780.0 4720.0 5660.0 6600.0 7540.0 8480.0 9420.0 10360.0 11300.0 12240.0 13180.0 14120.0 15060.0 16000.0 16940.0 17880.0 18820.0 19760.0 20700.01.50
2.24
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Kili13.34
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17.78
18.52
19.26
20.00
20.74
21.48
22.22
22.96Kili
23.70
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Figure 10
Mon Aug 19 14:44:08 2002
2016.4 2507.6 2998.8 3490.1 3981.3 4472.5 4963.7 5454.9 5946.1 6437.3 6928.5 7419.7 7910.9 8402.1 8893.3 9384.5 9875.7 10366.9 10858.1 11349.3 11840.51687.09
2054.41
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10135.63
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Figure 11
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Figure 12
a
bF1 F2
cC40
C42
Horizontal Hyperbolas
50000 10000
center
(h, k)
33
Figure 13
Rp0 I0
VDD0+-
Req0
+- Vdef
+-
Req2
I2
VDD2
m
L
where,
L/2
a
C42
ReqT = Req0 + Req2
C40
Rp2
34
Figure 14
CalibrationGate controlledusing a scan-flop
C4
Transistor
VDDGND
35
Figure 15
C41
Defect’s
C40
C46
C43
C42
C47
C45
C44
C48
position
Reqa Reqb
L/2
1-line
36
Figure 16
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Figure 17
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Figure 18
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