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Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2011-05-31.

Permanent Link: http://ufdc.ufl.edu/UFE0024260/00001

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Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2011-05-31.
Physical Description: Book
Language: english
Creator: Chandrasekharan, Vijay
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Electronic Thesis or Dissertation

Notes

Statement of Responsibility: by Vijay Chandrasekharan.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Sheplak, Mark.
Electronic Access: INACCESSIBLE UNTIL 2011-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024260:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024260/00001

Material Information

Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2011-05-31.
Physical Description: Book
Language: english
Creator: Chandrasekharan, Vijay
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Statement of Responsibility: by Vijay Chandrasekharan.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Sheplak, Mark.
Electronic Access: INACCESSIBLE UNTIL 2011-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024260:00001


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FinancialsupportforthisworkwasprovidedbyNASALangleyResearchCenterandtheUniversityofFlorida.IthankmyadvisorProf.MarkSheplakforhisroleasamentor,bothtechnicallyandotherwise.Iwouldspeciallyliketothankhimforprovidingtherightkindofguidanceandtheintellectualfreedomtorealizemyresearchinterests.IamalsogratefultomycommitteemembersDavidArnold,LouCattafesta,andNamHoKimfortheirtechnicalinputs,helpingmesucceedinthisproject.IamespeciallygratefultomanyofmyformerandpresentcolleaguesattheInter-disciplinaryMicrosystemsGroup(IMG).DavidMartinandKarthikKadirvelhelpedmeunderstandinterfacecircuitsforcapacitivesensorsthroughinsightfuldiscussions,evenwhentheywerepressedfortime.IprobablydonothavebetterwordstothankJeremySellsandJessicaMeloyfortheirhelpwiththisprojectatthedesignimplementationandtestingstage.ThenumeroushoursofdiscussionsandtheeortsofJeremyandJessicareallybroughtoutthebestinusasateamandnotjustmyeorts.IwouldliketothankBenjaminGrinandBrianHomeijerforallthosehoursofbrainstormingandtechnicaldiscussions.Itwasthebesttimeingraduateschoolintermsofthelearningprocessthatwestartedtogetherbackin2003.MyspecialthankstoMattWilliamsforhiswillingnesstohelpnumeroustimes,mayitbeonmechanicsorproofreadingmydissertation.IamgratefultoJohnGrinforhishelpwiththeexperimentalsetup.IthankBrandonBertolucciforhistechnical,aswellashisphotographicassistance.IthankSaraHomeijer,SheetalShetye,NaigangWang,andJanhaviAgashefortheirhelpwithSEMimages.IamalsogratefultoallofthestudentsatIMG.Honestly,IlearntagreatdealfromdiscussionswithfellowgraduatestudentsatIMGthanfromdiscussionswithfaculty.LearningprocesswasneveraseasyandeectivebeforeandperhapsmaynotbethesameafterIMG.IwouldliketothankthefacultyatIMGforfacilitatingsucharesearchenvironmentinthegroup.IamthankfultoKenReedatTMRengineeringforhisexcellentandtimelymachiningworkforsensorpackages.OthertechnicalassistancewasprovidedattheUniversityof 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 15 CHAPTER 1INTRODUCTION .................................. 17 1.1WallShearStressandBoundaryLayers .................... 18 1.1.1LaminarBoundaryLayer ........................ 21 1.1.2TurbulentBoundaryLayer ....................... 22 1.1.3SmallScalesinTurbulenceandSensorRequirements ........ 26 1.2ResearchObjectives ............................... 33 1.3DissertationOrganization ........................... 33 2BACKGROUND ................................... 35 2.1ShearMeasurementTechniques ........................ 35 2.1.1IndirectMeasurementTechniques ................... 35 2.1.2DirectMeasurementTechniques .................... 37 2.2Conclusion .................................... 50 3DEVICEMODELING ................................ 52 3.1Quasi-StaticModel ............................... 52 3.1.1SmallDeectionAnalysis ........................ 54 3.1.2LargeDeectionAnalysis ........................ 57 3.1.3ElectrostaticBehavior .......................... 58 3.2DynamicModeling ............................... 82 3.2.1LumpedParameters ........................... 83 3.2.2ElectromechanicalTransduction .................... 84 3.2.3EquivalentSensorCircuit ........................ 89 3.3HigherOrderEectsandNoise ........................ 95 3.3.1FringingFields .............................. 95 3.3.2ParasiticCapacitance .......................... 97 3.3.3NoiseModel ............................... 97 4DESIGNOPTIMIZATION ............................. 101 4.1SequentialQuadraticProgramming-SQP ................... 101 4.2Optimization .................................. 103 4.2.1SensorPerformanceRequirements ................... 103 6

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.............................. 104 4.3.1ObjectiveandDesignVariables .................... 104 4.3.2Constraints ............................... 107 4.3.3DesignSpecications .......................... 110 4.4ResultsandDiscussion ............................. 110 4.4.1SensitivityAnalysis ........................... 114 4.5Conclusion .................................... 119 5DEVICEFABRICATIONANDPACKAGING ................... 121 5.1FabricationProcess ............................... 122 5.1.1FloatingElementTrench ........................ 123 5.1.2SeedlessElectroplatingofNickel .................... 124 5.1.3BacksideRelease ............................ 126 5.1.4DieSeparation .............................. 126 5.1.5TheMEMSShearStressSensor .................... 128 5.2Packaging .................................... 129 6EXPERIMENTALCHARACTERIZATION .................... 131 6.1ExperimentalSetup ............................... 131 6.1.1ImpedanceMeasurements ........................ 131 6.1.2MeanShearStressMeasurement .................... 133 6.1.3DynamicMeasurements ......................... 137 6.1.4NoiseMeasurement ........................... 141 6.2ResultsandDiscussion ............................. 142 6.2.1ImpedanceMeasurements ........................ 142 6.2.2MeanShearStressCharacteristics ................... 144 6.2.3DynamicCharacteristics ........................ 148 6.2.4NoiseCharacteristics .......................... 156 6.2.5ExperimentalUncertaintyEstimation ................. 159 6.3Summary .................................... 160 7CONCLUSIONSANDFUTUREWORK ...................... 162 7.1Conclusions ................................... 162 7.2Non-IdealitiesinSensorDesignandCharacterization ............ 164 7.2.1DesignAspects ............................. 164 7.2.2CharacterizationAspects ........................ 168 7.3RecommendationsforFutureSensorDesigns ................. 175 APPENDIX AMECHANICALANALYSIS ............................. 178 A.1BeamDeection ................................. 178 A.1.1GoverningEquation ........................... 178 A.1.2SmallDeectionAnalysis ........................ 181 7

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........................ 183 A.2LumpedParameters .............................. 192 A.2.1LumpedMass .............................. 193 A.2.2LumpedCompliance .......................... 195 BTWOPORTELEMENTMODELING ....................... 198 B.1TwoPortModels ................................ 198 B.2LinearConservativeTransducers ........................ 199 CSHEARSTRESSINPWTWITHREFLECTIONS ................ 205 DPROCESSTRAVELERANDPACKAGINGDETAILS .............. 209 REFERENCES ....................................... 214 BIOGRAPHICALSKETCH ................................ 221 8

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Table page 1-1Laminarboundarylayerparameters[8]. ...................... 22 3-1Sensorgeometryforeectivetetherlengthcalculation. .............. 66 4-1Lowerandupperboundsfordesignvariablesandowspecications. ...... 110 4-2Constantparametersforoptimization. ....................... 110 4-3CapacitiveshearstressoptimizationresultsforCp+Ci=2:2+0:3pF. ..... 112 4-4Tolerancechartforvariablesandconstants. .................... 119 4-5DesignsensitivitybasedonMonteCarlo(MC)simulation. ............ 119 6-1Measurementsettingsforstaticcalibrationintheowcell. ............ 145 6-2SensitivityestimatesusingMonteCarlotechniqueonthemeanshearstressmea-surementdata. .................................... 148 6-3MeasurementsettingsofB&KPULSEMulti-AnalyzerSystem(Type3109)fordynamiccalibrationinthePWT. .......................... 149 6-4SpectrumanalyzersettingsfornoisemeasurementinadoubleFaradaycage. .. 156 6-5Integratedvoltagenoiseoorofthesensoratdierentfrequencyranges. .... 158 6-6Comparisonofpredictedandmeasuredsensorperformanceparameters. ..... 161 7-1Comparisonofmeasuredsensorperformancewithpreviouswork. ........ 163 9

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Figure page 1-1Velocityproleofviscousuidowoveraatplate. ............... 19 1-2Schematicofa2Dboundarylayerforowoveraatplateshowingboththelaminarandturbulentregions(adaptedfrom[8]). ................ 22 1-3Nearwall,meanvelocityinaturbulentboundarylayerorlawofthewall(adaptedfrom[11]). ...................................... 25 1-4SchematicofenergydistributioninturbulenceasafunctionofwavenumberatsucientlyhighRe(adaptedfrom[13]). ...................... 28 1-5Reynoldsnumberfordierentightregimesatsealevel(adaptedfrom[15]). 29 1-6VariationofKolmogorovlengthandtimescaleswithowReynoldsnumber(x=1m;U1=50m=s). .................................. 30 1-7Variationofshearstressandcorrespondingshearforceonsensorarea(Asensor=2)withowReynoldsnumber(x=1m;U1=50m=s). .............. 31 2-1Simplied2-dschematicshowingtheplanview(top)andcross-section(bottom)ofatypicaloatingelementshearstresssensorstructure(adaptedfromreviewbyNaughtonandSheplak[6]). ........................... 38 2-2Schematicofcrosssectionofpolyimidecapacitiveshearstresssensorwithdier-entialcapacitivereadoutschemeusingintegrateddepletionmodeP-MOStran-sistors(adaptedfromSchmidtetal.[37]). ..................... 40 2-3Schematicofa)foldedbeamwithlateralcapacitivechangeandb)forcefeedbackcapacitivestructure(adaptedfromPanetal.[38]). ................ 41 2-4Topviewofcantileverbasedoatingelementcapacitiveshearsensor(adaptedfromZheetal.[44]). ................................. 43 2-5Crosssectionalschematicofa)twodiodeandb)'split'diode(3diodes)opticalshutterbasedmicromachinedshearstresssensor(adaptedfromPadmanabhan'swork[21]). ...................................... 44 2-6Conceptualschematicshowingbottomandsideviewoffabry-perotoatingele-mentshearstresssensor(adaptedfrom[52]). .................... 46 2-7SchematicoftheopticalMoireshearstresssensor.TheshiftoftheampliedMoirepatternisimagedusingamicroscopeattachedtoaCCDcamera(withpermissionfromauthor[54]). ............................ 47 10

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............................ 48 2-93Dschematicofasideimplantedpiezoresistiveshearstresssensor.Theimplantsaresuchthattwotethersareintensionwhiletwoareincompressionforhighersensitivity(withpermissionfromauthor[60]). .................. 49 3-1Schematicofgeometryofthedierential,capacitiveshearstresssensor. .... 53 3-2Simpliedmechanicalmodeloftheoatingelementstructure. .......... 55 3-3Schematicofparallelplatecapacitoranalogoustooverlappingcombngers. .. 59 3-4Simpliedschematicofindividuala)tether,b)combnger,andc)oatingelementcapacitances,respectively. .............................. 63 3-5Schematicofnon-uniformtethercapacitorswithprimaryandsecondarygaps. 64 3-6Dierentialcapacitancesensingmodelforsensingcapacitorswiththeasymmetricgaps. ......................................... 67 3-7Simpliedclassicationofcapacitiveinterfaceelectroniccircuits. ........ 69 3-8Simpliedsensorcircuitrywithachargeamplieralongwithnoisesources. .. 72 3-9Simpliedsensorcircuitrywithavoltageamplieralongwithnoisesources. .. 74 3-10Simplieddierentialcapacitancesensecircuitryusingsynchronousmodulationanddemodulationtechniquewithavoltageamplier. .............. 75 3-11Schematicindicatingspectraofthesensoroutputateachstageofthesensorcircuitry. ....................................... 77 3-12Schematicofcapacitivetransductionschemeusingasinglesensecapacitance. .. 86 3-13Schematicofcapacitivetransductionschemeusingdierentialcapacitivesensingtechnique. ....................................... 87 3-14Schematicofequivalentcircuitforthedierentialcapacitiveshearstresssensor. 89 3-15Resonantmodesofoatingelementstructure. ................... 91 3-16Asymmetriccombngerstructuretoestimateeectoffringingelectricelds. 96 3-17Simpliedschematicofsensorcircuitrywithnoisesources. ............ 98 4-1Schematicofoperatingspaceofthesensor. ..................... 103 4-2SensitivityofoptimizedMDStodesignvariablesforDesign1. .......... 115 11

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............................. 118 4-4SensitivityofoptimizedMDS,S;overall,andfmintotolerancesinvariablesinconstantsofDesign1usingMonteCarlosimulation. ............... 120 5-1Schematicshowingtheplanviewofasingledieoftheproposedshearstresssensoranditssectionviewindicatingvariouslayers. ................... 121 5-2Stepbystepfabricationprocess. ........................... 122 5-3SEMimagesofcombngerstructureetchedusingDRIE,giving3Dperpective. 124 5-4SEMimageofacleavedwafersampledepicitinga)verticalsidewallsandb)uniformlyplatednickel. ............................... 127 5-5MicroscopicimageofareleasedoatingelementsensorstructurefromDesign3(Table 4.4 ). ...................................... 128 5-6A2mm2mmsenseelementona5mm5mmsensordie. .......... 128 5-7Schematicofsensorpackageforshearstresscharacterization. ........... 129 5-8Photographsofsensorpackagedona30mm30mmPCB. ........... 130 6-1Schematicshowingthedie-levelimpedancemeasurementsetupforthesensor. 132 6-2Schematicshowingthemeanshearstress/staticcalibrationsetupusingPoiseuilleowina2-Dchannel. ................................ 133 6-3Schematicwithopticalimageofsensordie(5mm5mm)indicatingoat-ingelement,contactpads,andinterfacecircuit(voltagebuer)formeanshearcharacterization. ................................... 134 6-4Schematicofthebiasingcircuitschemetocontrolphaseandamplitudeofbiasingsignalstonulloutoutputfromapotentialmismatchinsensorcapacitances. .. 137 6-5Aschematicofthedynamiccalibrationsetupformeasuringshearsensitivityusingrigidterminationwiththesensorlocatedatpressurenodeandvelocitymaximum. 139 6-6Aschematicofthedynamiccalibrationsetupformeasuringpressuresensitivityusingnormalincidenceacousticwaves. ....................... 141 6-7Aschematicofthedynamiccalibrationsetupforshearstressmeasurementwithplaneprogressiveacousticwaves. .......................... 142 6-8AschematicofthenoisemeasurementsetupforthepackagedshearstresssensorusingadoubleFaradaycage. ............................ 143 6-9Capacitancefromimpedancemeasurementsonthepre-packagedsensordie. .. 144 12

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........................... 146 6-11Distributionofthemeasuredshearstresssensitivityduetouncertaintyinthemeasuredshearstressandsensoroutputvoltage. ................. 147 6-12Sensoroutputvoltageat1:128kHzatabiasvoltageof10V.Sensorisplacedataquarterwavelength(velocitymaxima)fromtherigidtermination. ..... 150 6-13Voltageoutputasafunctionofpressureat4:2kHzatdierentbiasvoltages. 151 6-14Linearsensoroutputvoltageasafunctionofshearstressat4:2kHzat3dierentbiasvoltages. ..................................... 152 6-15Sensoroutputvoltagenormalizedbybiasvoltageasafunctionofshearstressat4:2kHz. ....................................... 153 6-16Schematicshowingasetofoverlappingcombngersdeectionduetobothshear(s)andduetopressure(p). ............................ 154 6-17FrequencyresponseofsensoratVb=10Vusingin=0:5Paasthereferencesignaluptothetestinglimitof6:7kHz. ..................... 155 6-18MeasuredoutputreferrednoiseoorofthepackagedsensorinVrms=p ................................ 157 6-19Zoomedinplotoftheoutputreferrednoiseoorofthepackagedsensornear1kHzatdierentbiasvoltages. .......................... 157 7-1Schematicwiththetopviewandsectionofasensorbondpadshowingtheasso-ciatedelectricalimpedances. ............................. 165 7-2Schematicwiththetopviewandsectionofasensorbondpadshowingtheasso-ciatedelectricalimpedances. ............................. 166 7-3Eectivesensecapacitancevariationwithfrequencyasafunctionofsubstrateresistance. ....................................... 166 7-4Capacitancemeasurementdriftindicatedviatherstandlastmeasurementpriortoensembleaveraging. ................................ 169 7-5Driftinmeancapacitancewithsubsequentdcbiassweeps. ............ 170 7-6Magnitudeofsensortomicrophonetransferfunctionasafunctionofacousticpressure. ........................................ 171 7-7Phaseofsensortomicrophonetransferfunctionasafunctionofacousticpressure. 171 7-8Transferfunctionbetweenthesensorandthereferencemicrophonefornormalacousticincidence. .................................. 172 13

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............... 173 7-10ComparisonmeanshearstressmeasurementsatVb=2:5Vatdierentchannelwidths,h,intheowcell. .............................. 175 7-11MechanicalLEMforpressuresensitivityofthesensor. .............. 176 A-1Adeectedbeamwitharbitrarycurvature. ..................... 178 A-2Simpliedmechanicalmodeloftheoatingelementstructure. .......... 180 A-3Schematicofonehalfofaclamped-clampedbeamunderlargedeection. .... 189 B-1Generalrepresentationofanidealtwo-portelement. ............... 198 B-2Circuitrepresentationofthetransformerandgyrator. .............. 199 B-3Impedancetoimpedanceanalogyrepresentationofatwoportelement. .... 199 B-4Equivalentcircuitrepresentationofatransducerusingimpedanceanalogy. ... 202 B-5Schematicofaparallelplatecapacitivetransducer. ................ 202 B-6Circuitrepresentationofacapacitivetransducerforconstantchargebiasing. 204 C-1SetupforshearstressinPWTforageneralimpedancetermination. ...... 205 D-1PCBLayoutanddimensions. ............................ 212 D-2DrawingofLuciteplug. ............................... 213 14

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dyy=0;(1{1)wherewallistheshearstress,isthedynamicviscosity,uisthestreamwisevelocityoftheuid,andyisthecoordinatenormaltothewall.Figure 1-1 depictsthevelocitygradientduetoowoveraatplate.Thevelocityiszeroatthestationarywallduetotheno-slipboundarycondition,andreachesthefreestreamvelocity,U1,atanitedistancefromthewall.Thisviscousdominatedregionclosetothewallisknownasaboundarylayer.Usually, 18

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theboundarylayerthickness,,isdenedasthedistancefromthesurfacewherethelocaluidvelocityis99%ofthefreestreamvelocityi.e.,u=0:99U1[8].Skinfrictionmeasurementhasbeenofinterestsincetheearly1870s[9].Thereductionofskinfrictionisoneoftheimportantreasonsbehindwallshearstressmeasurement.Inadditiontoskinfriction,dragisalsocausedduetoowseparation.Flowseparationoccurswhentheadversepressuregradientovercomesthemomentumoftheow.Theregionofadversepressuregradientisalsoknownasthepressurerecoveryregion.Whentheowseparates,iteectivelyencountersadeformedbodyafterthepointofseparation.Thisresultsinanetintegratedpressureforceinthedirectionofow,causingpressuredrag.Fromaowcontrolperspective,separationdragisdetrimentalforanaircraftasitreducesliftanditsfueleciency.Thus,wallshearstressmeasurementisextremelyimportantforowcontrolwhereowseparationisdelayedusingavarietyofcontroltechniques[8].Fromthediscussionsofar,itisevidentthatthemeasurementofwallshearstresshaspracticalimplicationsandalsogivesinsightintotheowphysics.AsstatedbyHaritonidis,\Themeanstressisindicativeoftheoverallstateoftheowoveragivensurfacewhiletheuctuatingstressisafootprintoftheindividualprocessesthattransfermomentumtothewall"[10].Ingeneral,theviscousdiusionofowmomentumfromthewallduetoshearingeectsresultsinaboundarylayer[7].Thenon-dimensionalReynoldsnumber,Re,whichistheratioofinertialtoviscousforces,isusedtodescribetheextentofviscousdiusion,resulting 19

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;(1{2)whereisthedensityoftheuid,Uisatypicalvelocityscaleoftheow,andlisatypicallengthscaleofinterestintheow.TheReynoldsnumbermayalsobeinterpretedastheratioofviscousdiusiontoconvectivetimescalesintheow[8].Thus,highvaluesofReresultinthinboundarylayerswhilelowvaluesofReresultinthickerboundarylayersinbothlaminarandturbulentows.Inanincompressibleowoverathinbody,theboundarylayerowisusuallylaminarfor1000
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1{5 and 1{6 aregeneralinnatureandarevalidforbothlaminarandturbulentboundarylayersforincompressibleowoveratplates[8].Section 1.1.1 describesalaminarboundarylayer,whichisformedatrelativelylowReynoldsnumbers.Generalcharacteristicsofasteady,two-dimensionalboundarylayerandtheirrelevancetowallshearstressmeasurementsareprovided.InSection 1.1.2 ,turbulentboundarylayersformedathighReynoldsnumbersaredescribed.Section 1.1.3 providesadiscussiononrelevantlengthscalesandtimescalesessentialfortheunderstandingofturbulentowphysics.Theselengthandtimescalesaretranslatedlaterintosensordesignrequirements. 1-1 providestherelevantparametervaluesforalaminarboundarylayerusingtheBlasiussolution.TheserelationsmaybeusedtotheoreticallyvalidatethemeasurementaccuracyoftheshearstresssensorinaZPGlaminarboundarylayer. 21

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xp 1-2 isaschematic 22

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u; u0v0;(1{11)whereu0andv0aretheuctuatingcomponentsofthevelocities.ThebarinEquation 1{11 representsthetimeaverageatxedlocations,assumingastatisticallysteadybutinhomo-geneousoweld[11].Thevelocitiesdecomposedintermsofatimeaveragedmeananductuatingcomponents,arerepresentedasu= 23

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1{11 isatermintheReynolds-averagedmomentumequa-tionderivedfromtheNavierStokesequationbydecomposingvelocity,pressureandstressesintotheirrespectivemeananductuatingcomponents.Physically,thistermservesasthemechanismformomentumtransportduetomacroscopicmixinginaturbulentboundarylayer,similartoviscousdiusioninalaminarboundarylayer.Notethatthisistrueforwallunits,y+>5,whereasfory+5,viscousdiusioncontinuestodominate.Inthepast,vonKarman'sintegralapproachhasbeenusedtoobtainowparametersforaturbulentboundarylayerjustasithasbeenforlaminarboundarylayers.Inthisdissertation,itwillbeusedtoprovideroughscalinginformation.AnassumedvelocityprolesuggestedbyPrandtlbasedonpipeowtheoryis[8] U1y 1=7:(1{14)UsingEquation 1{5 andEquations 1{6 withtheassumedvelocityofEquation 1{14 resultsin[8] x0:16 1{15 agreewellwithpublishedatplatedata[8]andserveasroughestimatesforvalidatingmeanshearstressmeasurementsinturbulentows.Atypicalnearwallnon-dimensionalmeanvelocityproleofaturbulentboundarylayerisshowninFigure 1-3 .Thenearwallturbulentboundarylayerconsistsofthreedierentregions:theviscoussublayer,thebuerlayer,andtheinertialsublayer.Theportionoftheboundarylayerforwhichy+5iscalledtheviscoussublayer.Themeanowinthislayerscalesas[12,13] u+=f(y+)y+: 1{16 indicatesthatthemeanvelocityscaleslinearlywithy+.Similartoalaminarboundarylayer,themomentumtransportinthisregionisdominatedbyviscousdiusion.However,unlikethelaminarboundarylayer,theviscoussublayerstillpossessesuctuations 24

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duetoturbulenceintheow[11].Aspreviouslymentioned,thewallshearstressmea-suredinaturbulentboundarylayerthuspossessesbothmeananductuatingcomponents.Theviscoussublayerisalsoreferredtoasthelaminarsublayerduetothesimilarityofitspropertiesandthoseoflaminarboundarylayers[12].Aty+>50,itisfoundthat[12] u+=f(y+)1 1{17 showsthatintheinertialsublayer,themeanvelocityhasalogarithmicvariationwithy+.Thetransitionregionbetweentheviscousandtheinertialsublayersisknownasthebuerlayer.Thebuerlayerusuallyexistsbetween5y+30.Inthisregionthesheareectsaretransferredbyacombinationofboththeviscouseectsandturbulentexchanges. 25

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1{19 ,isusedwithasthelengthscaleinsteadof`torelatethelargescalemotionstotheKolmogorovscalesas u 3=4=Re3=4; u 1=2=Re1=2; uu 1=4=Re1=4: 1{15 1{21 1{22 ,and 1{23 ,theKolmogorovscalesintermsofthedistancefromtheleadingedge,x,ofaatplateandthecorrespondingReynoldsnumber,Rex,areestimatedviathe1=7thpowerlawasfollows:0:632xRe11=14x;T40x U1Re4=7x;

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Atypical1-Ddistributionofturbulentenergy,E(k),asafunctionofthewavenumber,k,isshowninFigure 1-4 .Theturbulentwavenumberkisdenedask=!=v2 ; 1{25 ,thefrequenciescorrespondingtothesestructuresscaleas,O(1).ThisenergygetsredistributedfromthelargescalestructurestothesmallscaleturbulentstructuresontheorderoftheKolmogorovscales()withfrequenciesthatscaleasO(1).AsRexin-creases,thesescalesgetsmaller,resultinginchallengingsensordesignevenwithmodernmicromachiningtechnology.Thus,shearstresssensorlengthscalesandbandwidthsareapplicationspecicwithstringentdesignrequirementswithincreasingvaluesofRe.ThedierentightregimesasafunctionofReynoldsnumberareshowninFigure 1-5 28

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ThechallengesinsensordesignforresolvingKolmogorovscalesmaybefurtherexplainedviathefollowingstudy.ConsiderowoveraatplatewithU1=50m=sandashearstresssensorofsensingarea,Asensor=2,placedatx=1mfromtheleadingedge.Forthisexample,anyvariationinReisduetochangeinthermodynamicparameters.FromEquations 1{5 and 1{15 ,w=1 2U21Cf=1 2U210:027 29

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Thecorrespondingshearforce,Fs,onthesensorisFs=wAsensor=w2: 1-6 .Figure 1-7 showsthevariationofwandFswithRex.Whiletheshearstresslevelsremainatmoderatevalues,thesensingarea(2)andthecorrespondingFsdropdramatically.Forexample,atRex1107,2mand1=T12kHz.Furthermore,forAsensor=24m2,Fs=16pNandw=4Pa,whichisthemaximummeasurableshearstress.ForagivenrangeofRexthevariationinisO(102)higherthanthevariationinT.Similarly,thevariationofFsisalsomuchhigherthanthechangeinw,makingshearstresssensingmorechallengingathigherRex.Nowconsiderameasurementwheretheturbulentshearstressisroughly40dBre1Pabelowthemeanshearstressof4Pawithapreferredsignaltonoiseratioof20dB.Thus,thesensorneedstohaveaminimumdetectablesignal(MDS)of4mPaoradynamicrangeof60dB.TheshearforcecorrespondingtothisMDSis16fN. 30

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Figure1-7.Variationofshearstressandcorrespondingshearforceonsensorarea(Asensor=2)withowReynoldsnumber(x=1m;U1=50m=s). ItisimportanttonotethattheKolmogorovscalesrepresentthesmallestuidstructuresinturbulence,butnotnecessarilythesmallestscalesofinterestfromthesensingperspective.Forinstance,inowcontrolapplications,shearstresssensorsmaybeusedasfeedbacksensorsfordragreduction.Numerousresearcharticlesonowcontrolforturbulentdragreductionsightturbulentstreaksasamajorcauseofskinfrictiondrag.Turbulentstreaksarealternatingspanwiseregionsoflowandhighspeeduidorientedinthestreamwisedirectionandarequiescentmostofthetime[16].AtlowRevalues,thestreaksarespaced100l+apart[16]andareroughly40l+wide[17],circumventingtheneedforthesensortoalwaysresolvetheKolmogorovscales.Forexample,formeasuringshearstressfromturbulentstreaksthatare40l+wide,letusassumethesensormaybeatmost20l+initsmaximumsensingdimension.Usingtheexampleinthepreviousparagraph,forw=4Pa,20l+200m,whichistwoordersofmagnitudegreaterthanthelengthscalesrequired 31

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2 .Padmanabhan[18]usedaoatingelementshearstresssensorwhilebothLofdahletal.[19]andAlfredssonetal.[20]usethermalshearstresssensors.Theirresearcheortshavesensorlengthscalesof4l+[21],5l+[19],and1020l+[20].Asstatedearlier,toaccuratelymeasuretheKolmogorovuctuations,thesensordimensionsmustbeequaltoorsmallerthanthesescales.SensorresolutionattheKolmogorovscalesessentiallyallowsonetousethesensorforawidevarietyofapplications.However,previouseortsindicatethatthesensordimensionsarerestrictedbysignalresolu-tionissuesandfabricationlimitations.NumerousdirectandindirectmicromachinedsensorshavebeenpreviouslydevelopedandwillbereviewedinChapter 2 withemphasisontheirstrengthsandlimitations.Floatingelementsensorsusedfordirectmeasurementofwallshearstressmayalsobesensitivetovibrationsandpressureuctuations.Iftheseundesirablesensitivitiesarenoteliminated,theshearstressmeasurementsmaybeerroneous.Themagnitudeoftheuctuatingpressureforcesisapproximatelytwoordersofmagnitudehigherthanfortheshearforces.Directnumericalsimulationstostudythewallpressureeectsshowthat,basedonfrequency,thewallpressureuctuationsare720dBhigherthanthestreamwisewallshearstressand1520dBhigherthanthespanwisecomponent[22].Thesensordesignshouldalsoensureminimumerrorsduetomisalignment,pressuregradientsandgaps[9]. 32

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2 and 3 ,respectively. 1 introducedwallshearstressandmotivationforitsmeasurement.Scalingstudiesandcorrespondingsensorre-quirementswerepresented.Chapter 2 isareviewofpreviousresearcheortstodevelopmicroscale,directwallshearstresssensors.Existingsensingtechniques,includingcapacitivetransductionusingcombngers,arediscussed.Chapter 3 presentsthesensor'smechanicalandelectrostaticmodel,electromechanicaltransduction,interfacecircuitry,andthenoisemodel.Mathematicalrepresentationsfornoiseandsensitivityarederivedtouseinadesignoptimizationstrategy.Chapter 4 containstheformulationoftheoptimizationproblemandanexplanationoftheoptimizationschemetobeimplemented.Optimizationresultsarepresentedtogetherwithsensitivityanalysisforthedesign.Chapter 5 providesdetailsofthefabricationstepsandthepackagingschemeforthesensor.Chapter 6 explainsthesensor 33

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7 providesconclusionsandgivesrecommendationsforthenextgenerationofcapacitiveshearstresssensors. 34

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1 ,shearstressmaybedirectlydeterminedfromthemeasurementofshearforceonaknownarea.However,othermethodologiesalsoexistandareusedinpracticetoestimatetheshearstressfrommeasuredowparameterssuchasvelocity,jouleanheatingrateetc.Thusbasedonthequantitymeasured,thetechniquesarebroadlyclassiedasdirect(measuresshearforce)orindirect(measuresotherquantitites)measurementtechniques.Shearstresssensorsmayalsobecategorizedasconventional(macroscale)ormicroscalesensors.Severalresearcheortsinthepasthavebeendirectedtowardsdevelopingbothdirectandindirectmicroscalesensors.Microscaledirectsensors,duetotheirdirectnatureandfavorablescalingtocaptureturbulentowphysics(seeChapter 1 ),arebettersuitedforquantitativetime-resolvedshearstressmeasurement.Thediscussioninthischaptermostlyconcentratesonpreviousresearchondirectmicromachinedsensors. 2.1.1IndirectMeasurementTechniquesIndirecttechniquesrelyonaknowncorrelationbetweenthemeasuredparameterandshearstresstoestimatethelatter.Heat/masstransferbaseddevices,surface/owobstacledevicesandvelocityprolemeasurementtechniquesareusedtoindirectlyestimatewallshearstress.ReviewsbyWinter[9]andHaritonidis[10]describethebenetsandlimitationsofthesemeasurementtechniques. 35

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36

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2-1 showsaschematicofaoatingelementsensor.Themotionoftheoatingelementistransducedintoaproportionalelectrical/opticalsignaltomeasuretheshearstress.Thesedevices,however,haveissuesoftheirownasrstexplainedbyWinter[9].Thefollowingproblemsaregenerallyassociatedwithconventionaloatingelementsensors[9,10]: 1. Minimumdetectableshearstress(MDSS)increaseswhenthesensorsizeisdecreasedtoimprovespatialresolution.Asmallsensorsizeresultsinsmallerintegratedshearforce(Figure 1-7 )andthusalowerstresssensitivityandcorrespondinglyhigherMDSS. 37

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2. Theeectofessentialgapsaroundtheoatingelement 3. Theeectofmisalignmentoftheoatingelementwithrespecttothesurroundingsurface 4. Forcesduetopressuregradients 5. Eectsofgravityandaccelerationwhenplacedonamovingobjectoronanon-levelsurface 6. MassivedevicesresultinpoortemporalresolutionNaughtonandSheplak[6],basedonpreviousresearcheorts,provideadetaileddis-cussionshowingthefavorablescalingofmicromachinedshearstresssensorsforquantitativetimeresolvedshearstressmeasurements.Theirndingsshowthatmicromachinedsensorsoerthepotentialforhightemporalandspatialresolution.Theytheoreticallyoeratleastve-ordersofmagnitudeimprovementinthesensitivitybandwidthproductandabouttwo 38

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1. MEMSsensorscanmeasureshearforcesoftheorder0:1nN.Thiscorrespondstoashearstressoforder10mPaforaoatingelementsizeof100m100m. 2. Sensorsfabricatedusingstandardmicromachiningtechnologydonotneedassemblyofsensorcomponents,eliminatingsomemisalignmentissuesalthoughsensorpackagingandinstallationonthetestsetupisstillpronetomisalignmenterrors. 3. MicromachiningallowsgapsofO(1m),renderingthesurfacehydraulicallysmoothexceptatveryhighRe[7]. 4. Threeordersofmagnitudereductioninscalecomparedtoconventionalsensorsgreatlyreducespressuregradienterrorsinmicromachinedsensors. 5. Thecross-axissensitivityofthesesensors,withrespecttoacceleration,isthreeordersofmagnitudesmallerthanconventionalsensorsbecauseofreducedsensormass. 6. Thermalexpansionerrorsaremitigatedduetomonolithicfabricationtechniquesbutcarefulpackagingisalsoimportanttoavoidtheseerrors.TohighlightthebenetsandchallengesfacedbyMEMSbasedshearstresssensors,thissectionwillfocusonpreviousresearch,specicallyonmicromachineddirectshearstresssensors.Basedonthetransductionschemeused,thesensorsarecategorizedascapacitive,optical,andpiezoresistive,respectively. 2-2 .Theoatingelementwas500m500m30mandthetethers 39

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were1000m5m30m.Thesensordemonstratedasensitivityof52V=Pa[37].Ithowever,sueredfromdriftduetomoistureontheorderofseveralV=min.Moisturehastwodierenteects,1)Moistureinduceshydroscopicin-planestressesthataectsthemechanicalsensitivityand2)Itchangesthedielectricpropertiesofthepolyimideresultingindrift[36].Ingeneral,anychargeaccumulationattheairdielectricinterfacewillresultindrift[6].Panetal.andHymanetal.developedcombnger-basedcapacitiveshearstresssensors[38{40].Theydevelopedtwosensordesigns,onebasedondierentialcapacitivemeasurement,andtheotherbasedonforcebalancingofmovingcapacitiveplatesusingafeedbacksignal.Figure 2-3 showsaschematicofbothsensors.Intherstdesigna 40

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foldedbeamstructuresupportashearstresssensingelement.Thefoldedbeamsprovidedtherequisiterestoringforce.Inthissensor,theoutputduetocapacitancechangewasundetectableduetohighparasiticattenuation.Hymanetal.attributetheparasiticstothedirectelectricalcontactbeingmadetothedevice[39].Opticallymeasureddeectioninaknownowresultedinamechanicalsensitivityof9Pa=mor0:11m=Pa.Thesecond 41

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2-4 .ThedierentialcapacitancechangeduetoelementdeectionismeasuredusinganotheshelfcircuitcomponentMS3110fromIrvineSensors.Flowcalibrationofthesensorresultedinahighnoiseoorof0:04Paandalargesensitivityof337mV=Paduetocomplianceofthelongbeam[44].Theauthorsexpressedconcernoversensormisalignmentinthechannelleadingtoerrors.Therewas13%uncertaintyinthemeasurement,whichwasattributedtothepotentialmisalignmenterrorsandchannelheightuncertainty.Therewereothersourcesofuncertaintyandscatterinthedata,whicharestillbeinginvestigated[44].Thedynamicbehaviorofthesensorwasnotreported.Achargeamplicationschemeisusedintheinterfacecircuitry.Usingthisschemerendersthesensorsensitivityindependentoftheparasiticsinthesystembut,thenoiseoorissensitivetoparasitics(seeSection 3.1.3.3 ).Thisexplainsthehighnoiseoorofthesensor.Furthermore,allcapacitivesensorsinherentlyarehighimpedancedevicesandarethereforesusceptibletoelectromagneticinterference(EMI). 42

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McCarthyetal.andTiliakosetal.haveattemptedtodevelopasimilarsensorusingdierentialcapacitancemeasurementandforcefeedbackforhighshearstressapplications(10Pa10kPa).Theyhoweverdidnotperformanexperimentalcharacterizationandtheirworkisthereforenotdiscussedanyfurther[45{47].Similarly,Desaietal.designedandfabricatedanovelMEMSstructurefor2-Dshearstressmeasurementusingthewaferthicknessfortheoatingelement[48].Thesensorproposedtouseadierentialcapacitivetransduction.Thoughthesensorwasfabricated,noexperimentalcharacterizationhasbeenreported. 2-5 .Four 43

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tethers500mlongand10mwidesuspendedtheoatingelement1:2moverasiliconsubstrate.Thetethersandoatingelementwere7mthick.Acoherentuniformlightsourcelocatedabovethesensorilluminatestheexposedareaofthetwophotodiodes,result-inginadierentialphotocurrent.Thephotocurrentisideallyzerowhenthereisnomotionastheilluminatedareasofthetwophotodiodesarethesamebydesign.Inthepresenceofaforceduetoshearstress,thelateralsensormotionincreasestheilluminatedareaofonephotodiodewhiletheareafortheotherdecreasesbythesameamount,resultinginadif-ferentialphotocurrent.Thedierentialphotocurrentisproportionaltoboththemagnitudeanddirectionoftheshearstress.Intensitygradientsacrosstheoatingelementresultedin 44

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2-6 .The1:5mm1:5mm20mmembraneprovidestherequisiterestoringforce.Theoatingelementwas200m200minlengthandwidth.Thereectingportionoftheoatingelementwas400mhigh.Withnogaps,thissensorcircumventedowdisturbanceissues,wastrulyushmounted,andcouldalsobeusedindierentuids.Twoopticalbersalignedorthogonaltoeachotherallow2-Dshearstressmeasurements,thoughthisabilitywasnotdemonstratedexperimentally.ThedetectiontechniquewasalsoimmunetoEMI.Thestaticcalibrationresultsindicateasensitivityof0:65Pa=nmor1:54nm=Paandtemperaturesensitivityof3:4nm=K[52].Thetemperaturesensitivitywasthushigherthantheshearstresssensitivity.Thesensoralsohadahighnoiseoorof23nm,whichcorrespondstoashearstressof1:3Pa.Thedynamicresponse 45

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ofthesensorwasnotreported,butthelargeheightoftheoatingelementsuggeststhatthesensorwouldhavealargemassandthereforealowerbandwidth.Thedynamicrangeofthesensorwasalsonotreported.Thesensormayalsohavebeensensitivetopressureuctuationsandvibrationsofthemembrane,supportingtheoatingelement.Horowitzetal.designedanopticaloatingelementshearstresssensorthatusedanalignedwaferbonding/thinbackprocess[53,54].Figure 2-7 showsaschematicofthesen-sor.Thesensorhasopticalgratingsonthebacksideoftheoatingelementandonthetopsideofatransparentpyrexwafer.ThegratingsformageometricMoirepatternthatampliesthemechanicalmotionoftheoatingelement.TheMoirepatterndisplacement 46

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wasmeasuredusingalinescanCCDcameraandimagedusingamicroscope.Bydesign,thesensorisinsensitivetooutofplanemotionduetopressure,vibrationetc.sincethemotionproducesnochangeintheMoirefringepattern.Likemostopticalsensors,thissensorisalsoinsensitivetoEMI.Drawbacksofthissensorincludeabulkyopticalsetupandtheas-sociatedpackagingrequirements,restrictingitsusetoanopticalbench.Thesensorhadastaticmechanicalsensitivityof0:26m=Pawithalinearresponseuptothetestinglimitof1:3Pa.Thedynamictestingindicatedaresonantfrequencyof1:7kHz,andanoiseoorof6:2mPa=p 47

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2-8 .Thetetherswerealignedalongthedirectionoftheowtowithstandthehighshearstresses.Theoatingelementwas120m140m5mwhilethetetherswere30m10m5m,respectively.Thesensorhadbacksideelectricalcontactswhichrepresentedanimportantlandmarkintermsofachievingtrulyushmountednon-opticalsensorsforshearstressmeasurement.Sheplaketal.notedthatuseofanisotropicwetetchingprocessforthebacksidecontactresultedinlargediesizes[23]. Figure2-8.3Dschematicofapiezoresistivesensorwithtopsidetetherimplantsorientedinthedirectionofowforhighshearstressmeasurements(adaptedfromNgetal.andGoldebergetal.[56,57]). Barlianetal.studiedsidetetherimplantsandadditionaltop-sideimplantstoalsosenseoutofplanemotioninaoatingelementshearstresssensor[58,59].Theypresentedmechanicalcharacterization,temperatureanddopingeects,andtheeectsofannealingonnoiseforasingleimplantedcantileverbeam(6000m400m15m)[59].However,Barlian'sinitialowcharacterizationresultswerepreliminaryandhadsignicantscatterintheirmeasurements,warrantingfurtherinvestigation[58]. 48

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Lietal.usedsideimplantedpiezoresistorsonthetethersoftheoatingelementsensor.AschematicofasideimplantedoatingelementsensorisshowninFigure 2-9 [60].Theypresentedpreliminaryshearstresscharacterizationresultsfora1mm2oatingelementwithtethers1mmlongand30mwide.Theoatingelementandtetherswere50mthick.ThesensorwascalibratedusingaStokeslayerexcitationtogenerateshearstressinaplanewavetube[51].Thesensorwaslinearuptoameasuredshearstressof2Paandpossessedanormalizedsensitivityof2:83V=V=Pa.Thenoiseoorwas11:4mPaat1kHzandthetemperaturesensitivitywas0:42mV=C[61].Theupperendofthedynamicrangewaslimitedbythetestinglimitof2Pa[61].Thedynamicresponseofthesensorindicatedaatfrequencyresponseuptothetestinglimitof6:7kHz.Thissensordemonstratedpromisingresultsandoersarobustsensorforshearstressmeasurements.Highsensitivitytoin-planemotionduetosidewallimplants,lowsensitivitytooutofplanemotionbothduetogeometryandasimplereadoutscheme,arethebenetsofthissensordesign.Ina 49

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3 )remainaconcernforcapacitivesensors.However,theseveralbenetsofcapacitivesensorsoeropportunitiestodesignaroundtheirlimitations. 50

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51

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4 ).Mechanicalandelectricalmodelsdevelopedinthischapterareessentialcomponentsfortheoptimizationstudy.Thischapterisorganizedintothreemajorsections.InSection 3.1 ,anexistingquasi-staticdeectionmodelpresentedin[36,64]isextendedtoincludetheeectofcombngers.Expressionsformechanicaldeectionsarepresentedusingbothlinearandnonlineartheories.Anovelasymmetricdierentialcapacitivesensingschemeisproposed.Electrostaticbehaviorofthesensorisexplained.Ashortcomparisonofcapacitiveinterfacecircuitsisgiven,followedbyintegrationofthechosencircuitrywiththedevicemodel.InSection 3.2 ,thedynamiccharacteristicsoftheshearstresssensorareanalyticallystudied.Lumpedelementmodelingisemployedtoestimatethedynamiccharacteristicsofthesensor.InSection 3.3 ,higherordereectssuchasthefringingeldsandparasiticcapacitanceeectsareexplained.Thenoiseofthesystemisalsodiscussed. 2.1.2 .Arectangularproofmass,suspendedoverasmallcavitybyfourcomplianttethers,formstheoatingelement.Aowacrosstheoatingelementsurfaceexertsashearforce,whichproducesaproportionallateraldeection.Thetethersactasspringsthatrestoretheoatingelementtoitsmean/reference 52

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3-1 ,depictsthecombngers,tethers,andtheoatingelement. Figure3-1.Schematicofgeometryofthedierential,capacitiveshearstresssensor. Adeectionchangesthecapacitancebetweenthesurroundingsubstrateandthetethers,oatingelementandbetweencombngers,resultinginaproportionalchangeinvoltage(Section 3.1.3.1 ).Theresultingchangeinvoltageisdetectedusingavoltagebuer(seeSection 3.1.3.3 ).Foranalysispurposes,thetethersaremodeledasclampedbeamsand 53

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3-2 ,eachpairoftethersformaclamped-clampedbeam,withapointloadPappliedatcenterbytheshearstress,w,ontheoatingelement.Auniformlydistributedload,Q,alongthelengthofthetethersaccountsfortheshearforceonthetethers[37].Twosuchbeamssharetheloadappliedontheoatingelementbecauseofthegeometricsymmetry.ThusasinglepairoftethersthatformthebeamexperienceaforceduetotheshearstressgivenasP=wWeLe 2[N] (3{1)andQ=wWt;[N=m] (3{2) 54

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whereWeisthewidthoftheoatingelement,Leisthelengthoftheoatingelement,Nisthenumberofcombngersontheoatingelement,Wfisthewidthofeachcombnger,andLfisthelengthofeachcombnger.Thegoverningdierentialequationisgivenby 2:(3{3)whereEistheYoung'smodulusofelasticity,Mxisthemomentalongthelengthofthebeam,Iisthemomentofinertiaaboutthey-axis,andwisthein-planedeectioninthedirectionoftheappliedload.Forbothsmallandlargedeections,theslopedw(x) 55

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dxx=0=0;(zeroslope)anddw dxx=Lt=0:(symmetry) (3{5)TheEulerBernoullitheoryandsmallbeamdeectionsinresponsetoanappliedforceareassumedwhilesolvingEquation 3{4 .Thetheoryhypothesizesthatastraightlinetransversetotheneutralaxisremainsstraight,inextensible,andnormalbeforeandafterdeformation[65].Assumingpurebendingandsmalldeectionsallowsthenonlinearextensionalstrainalongthelengthofthebeamtobeneglected.ThedetailedderivationofthemodelispresentedinAppendix A .Thelineardeectionusingthistheoryis, 3.2 ,thiscenterdeectionisusedtoestimatethelumpedparameters.Themaximumdeectionatthecenterofthebeam(x=Lt)is 56

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3{7 ,therearethreetermscontributingtothedeection:thersttermisduetotheoatingelement,thesecondtermisduetothecombngersandthenaloneisduetothetethers. 3.1.2.1 )andananalyticalsolutiontechnique(Section 3.1.2.2 ).Thevalidityofboththesesolutiontechniqueshavebeenveriedpreviouslyusingniteelementanalysisandcomparedwiththelinearquasi-staticsolutionin[64]. A providesthedetailedderivationofthesolutionusingthismethod.Theexpressionfortheoatingelementdeectionbasedonthismethodis 4 Wt2!=wWeLe 3{8 isnonlinearandissolvedusingnumericalsolutiontechniques.Unlikethepreviousanalyticaltechnique,whichyieldedasolutionfordeectionasafunctionofthepositionalongthebeam,theenergymethodisusedonlytosolveforthecentraldeectionwithanassumedshapefunctionormodeshape.ThecenterdeectionisthequantityofinterestbecausethepermissiblenonlineardeectionisdenedwithreferencetothelinearcentraldeectioninEquation 3{7 (seeSection 4.3.2.2 ).Theterm, 4 Wt2!;(3{9)isaconsequenceofbeamstieningduetodeformationalongtheneutralaxis.ThesolutionapproachesthelinearsolutioninEquation 3{7 forsmalldeectionsi.e.,=Wt<<1. 57

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A )usingthissolutiontechniqueis, @x2dx: 3{10 isprovidedinAppendix A 58

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Figure3-3.Schematicofparallelplatecapacitoranalogoustooverlappingcombngers. 3-3 .Thetransducerconsistsoftwoconductingparallelplatesseparatedbyadielectricmediumsuchasair.Oneplateisxedandtheotherisfreetomove,suchthatthegapbetweentheplatesmaychange.ThecapacitancebetweentheplatesisC="A=g; 3{13 and 3{14 tohold: 1. 59

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Theelectriceld,EEisnormaltotheplates.Anappliedforceresultsinachangeingap(t);thetimevaryingcapacitanceisthengivenby 3{15 intothisexpressiongivesthevoltage, (3{17)anddEp=Fed+QdV(voltagevariation), (3{18)respectively.Ifnoadditionalchargeorvoltageisappliedtothecapacitorelectrodes,achangeinstoredpotentialenergyiseectedbythemotion,ofthemovableplate.Consequently,theelectrostaticforceisgivenas, 2Q(t)2 2C(t)V(t)2: 3{21 and 3{20 intoEquation 3{19 resultsin d1 2Q(t)2 d1 2C(t)V(t)2:(3{22) 60

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3{15 intotheaboveexpressionresultsin,Fe=Q(t)2 g02: 3{23 indicatesthatwhenthereisaconstantchargeacrossthecapacitor,Feisindependentoftheplatemotion.Thisavoidstheelectrostaticpull-ininstabilityfortheconstantvoltagecaseinEquation 3{24 asapproachesg0[66].Arestoringforceduetothemechanicalcomplianceofthestructureopposesthedeec-tionofthemovableplate.ThemechanicalforceFm(t)intermsofthecompliance,Cme,isexpressedas 3{16 3{23 3{24 ,and 3{25 ,thecharacteristicelectrostaticequationsforthesystemarewrittenasV(t)=Q(t) {z }ChargeControl=(t) g02| {z }VoltageControl: 3{26 and 3{27 indicatesthatvoltage,V(t),andforce,F(t),arecoupled.Inaddition,thevoltageandtheforceonthecapacitorarenonlinearlyrelated.Thesystemthereforeneedstobelinearized.ThelinearizationofthesystemandtheapproachfortwoportmodelingisexplainedinAppendix B .Inthenextsection,atwoportmodeloftheentiresensorstructureisillustratedbasedonthederivationsinAppendix B 61

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B isderivedtoillustratetheelectrostatictransductionmechanism.ConsiderFigure 3-1 andFigure 3-4 ,wheretherearepairsofcapacitancesformedbydierentgapsoneithersideofthesensorstructureasfollows: 2"TtL0 2"TtL0 3-5 .Sincetheelectrodesurfacesaremetalized,auniformsurfacechargedensity,,is 62

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assumedonthetethersandthesurroundingsubstrate.Bydenition,thecapacitanceis[67],Z+EEds=Q=C; 63

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withconstantpermittivity,",EEisnormalatbothconductorsurfaces,whichsimpliesEquation 3{31 toZ+EEds= "Z+ds=Q=C: 3{32 ,thecapacitanceformedbythiselementisdC=dQ "g(x)R0dg: 3{33 isrewrittenasdC="dA g(x): 64

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3{6 forw(x)andsolvingtheintegral,Cng=LtZ0"Tt 3{36 isnumericallyevaluatedinMAPLE.Thechangeincapacitanceduetothenon-uniformgapvariationiscomputedusingCng=CngCtether; g0: 3{37 and 3{39 arenumericallycomparedforafewgeometriesforsmalldeectionsusingMAPLE.Table 3-1 showsthegeometryandparametersforthisstudy.TheresultsinTable 3-1 indicatethatforsmalldeections,CparallelisapproximatelytwicethevalueofCng.Foreaseofmodeling,thisdierenceisaccountedasaneectivetetherlength,Lteff,allowingthetetherstobemodeledasparallelplatecapacitors.Theeectivetetherlengthistherefore,LteffLt 65

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Shearstressw10PaTetherLengthLt1000mTetherWidthWt10mFloatingElementLengthLe1000mFloatingElementWidthWe1000mTetherThicknessTt45mNumberofcombngersN100CombFingerWidthWf4mCombFingerlengthLf150mInitialgapfortethercapacitanceg03:5mDielectricPermittivity"8:851012FFloatingElementYoung'sModulusEy160MPaCparallel6:637fFCng13:65fF 66

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3{28 3{30 3{41 ,and 3{42 inEquations 3{43 and 3{44 areC0=C10=C20=(N1) 2"TtL0 2"TtL0 2Lf 2Lf 3-6 ). Figure3-6.Dierentialcapacitancesensingmodelforsensingcapacitorswiththeasymmetricgaps. 67

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3{45 andisgivenasLeff=3Lteff+Le+(N1) 2Lf+Lteff+(N1) 2Lfg01 3{46 into 3{45 ,thenominalcapacitanceissimpliedandrewrittenasC10=C20="TtLeff 3-1 .Themeasurementrequirementsforthesensorinclude,measuringdcandacchangeincapacitanceduetoshearstress.TypicalsensorparametersthatinuencetheinterfacecircuitdesignareC00:41:5pF,C325fF.Thesensordesignstargetamaximumshearstressof10Pa.Ahighsensorresolutionenablesmeasurementofweakturbulentmotionandmaximizesdynamicrange.Thisrequirestheinterfacecircuittohavealownoiseoor. 68

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3-7 showstheblockdiagramillustratingtheclassicationofinterfacecircuitry.Thisworkfocusesonopen-looptechniqueswithanalogelectroniccomponents. Figure3-7.Simpliedclassicationofcapacitiveinterfaceelectroniccircuits. Beforedescribingthedetailsoftheproposedsensorcircuitry,existingopen-loop,analoginterfacecircuitcongurationswillbediscussed.Themostcommonlyusedanalog,open-loop 69

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3{16 tobedirectlyproportional.Anychangeincapacitancewithaconstantchargeorvoltageappliedtothecapacitorproducesaproportionalvoltageorchargeoutput,respectively.Aninterfacecircuitcanthereforebedesignedtosenseeitherthechargeorthevoltageacrossthecapacitor.Constantvoltagebiasingwithachargeamplierandconstantchargeschemeinconjunctionwithavoltageamplierarethetwotechniquesgenerallyusedincapacitivesensors,resultinginanoutputvoltage. 3.1.3.1 andalsoshowninFigure 3-1 .Thissensorgeom-etryenablesmeasurementofadierentialcapacitancechange.Bothsingleanddierentialcapacitancesensingschemeshavebeenimplementedforsensorspreviously.Martin[72]pro-videsacomparisonofsingleanddierentialcapacitancesensingschemesusingbothcharge(constantvoltagebias)andvoltage(constantchargebias)ampliers.Anappropriatebias-ingschemeforanelectrostatictransducerisimportantfromtheinterfacecircuitandoverallperformanceperspective.Itisthereforerelevanttoperformacomparativestudyofchargevsvoltageampliersandsinglevsdierentialcapacitancemeasurementschemes.Kadirvel[73]andMartin[72]provideinsightintotheaspectsofdierentbiasingtechniques.Inthepastseveralotherresearchershavealsostudiedtheinterfacecircuitschemesusingchargeandvoltageampliers[69{71].Inthissection,acomparativestudyoftheconstantchargeandconstantvoltagebiasingschemesispresented. 70

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3{24 ),eventuallyleadingtopullin.Theuseofconstantchargebiasingincombinationwithavoltageamplieravoidsthisproblem.Theconstantchargebiasingcreatesaconstantelectriceldacrossthegapindependentoftheplatemotion(seeEquation 3{23 ),preventingelectrostaticpull-in[66].ThestudyconductedbyKadirvel[73]revealsseveraldetailsaboutsensitivityandnoiseinchargeandvoltageampliers.Thesensitivityofchargeampliersisindependentoftheparasiticcapacitance,whileparasiticshaveapronouncedeectonthesensitivityofvoltageampliers.Theadverseeectofparasiticsinvoltageampliersincreasessignicantlywithadropinsensorcapacitance.However,withanincreaseinparasiticcapacitance,thenoiseincreasesinchargeampliers.Thestudyalsorevealedthatnoiseperformanceinbothchargeandvoltageampliersisbetterwhentheyareoptimizedforlowcurrentnoise. w: 3-8 showsatypicalchargeampliercircuitwithnoisesources.WhiledecreasingCfincreasesthesensitivity,italsoincreasesthecut-onfrequencyofthecircuit.Thecut-on 71

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frequencyis 2RfCf;(3{49)whereRfisthefeedbackresistorthatsetsthedcoperatingpoint.LoweringCfalsolowersthebandwidthandincreasesthenoiseinchargeampliers[73].Thiscanbeseenfromtheexpressionforthetotalnoisepowerspectraldensity(PSD)attheoutputofachargeamplier,givenasSv0chgamp=Sva1+Ctot 72

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3-9 showsatypicalvoltageampliercircuitincludingdominantnoisesources.Anycapacitanceattheinputoftheamplier,exceptthesensorcapacitance,formsapotentialdividerreducingthevoltageattheinputoftheamplier.TheexpressionsforsensitivityandnoiseforthevoltageampliercircuitareSvamp=C1C2 w 3.2.3 ).Alargersensorcapacitanceshouldthereforehelpmitigatetheilleectsofparasitics.However,theratio,C C0,shouldnotdropsignicantlybecauseitresultsinthelossofsensitivity.ThiswillbeillustratedmathematicallywhileexplainingtheexpressionforsensitivityoftheproposedsensorinSection 3.2.3 .ThecombinationofthebiasresistorRbandthetotalcapacitanceCtot(includingparasitics),formahighpasslter.Thecut-onfrequency,1 2RbCtotcanbeloweredusingalargebiasresistor.However,thiswillincreasethenoisecontributionattheoutput.Thesimplicityofthevoltageampliercircuitmakesitanattractiveoptionforcapacitivesensors.ThereaderisreferredtoKadirvel'swork[73]oninterfacecircuitsforadetailedunderstandingofthetradeosinvolvedinusingchargeandvoltageampliers.Martin[74]reportedbetterresultsusingavoltageampliercomparedtoachargeampli-erwithotheshelfcomponentsforadualbackplatecapacitivemicrophonewithnominalcapacitancessimilartothesensordevelopedinthiswork.Thesamevoltageamplierwillbeusedfortheproposedsensor.Desirabletraitsofshearstresssensorsincludetheirability 73

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tomeasureboththemeananductuatingshearstresscomponents.Intheproposedsensor,meananductuatingshearstresscomponentswouldcorrespondtomeananductuatingchangesincapacitancerespectively.Asstatedpreviously,bothchargeandvoltageamplierspossessacharacteristiccut-onfrequency,1=2RC,makingthemunsuitableformeasuringdc(mean)capacitancechanges.ThesynchronousMOD-DMODschemeenablesthemeasurementofstaticcapacitancechangesormakingmeanmeasurements.Thistechniquehasbeenusedwidelyinthepastforcapacitiveinertialsensors[69,70].ThefollowingsectionexplainstheMOD-DMODschemefordierentialcapacitancemeasurementusingavoltageamplier.Theinterfacecircuitisexplainedandderivationsfortheoutputvoltageofthesystemareprovided.Forthederivationspresented,eachcapacitorisassumedtobeaparallelplatecapacitor. 3-10 .Eachsensecapacitorisrepresentedasaparallelcombinationoftwocapacitorswhichchangeintheoppositesense.Thecapacitorsinparallelcorrespondtothegaps,g01andg02thatchangeinaoppositesensewiththesensormotion.Aunitygainvoltagefollowerisusedtorepresentthevoltageamplier.Thenon-invertinginputisconnectedtothecommonelectrodeofthecapacitor(theoatingelementinthiscase).Theothertwoelectrodesofthecapacitors 74

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arebiaseddirectlyusingsinusoidalvoltagesignals,+vcandvcrespectively.Forsimilarimplementationsforinertialsensors,squarewavesaregenerallyused[75].However,squarewavesinherentlyhaveharmoniccontentwhichmayinterferewiththemeasurementsignal.Squarewavesalsopossesslesspowerinthefundamentalfrequencycomparedtosinewavesatagivenamplitude,loweringthesensitivity(Section 3.2.3 ).Cpistheparasiticcapacitanceduetoonboardconnectionlinesandwirebonds.Ciistheinputcapacitanceoftheamplier.ThebiasresistorRbsetsthedcoperatingpointoftheamplier.ThecombinationofthetotalcapacitanceattheinputoftheamplierCtotandRbformahigh-passlterwithacut-onfrequency,1=2RbCtot.Thechargeatthecommonnoderemainsapproximatelyconstantprovidedtheacbiasvoltagefrequencyisgreaterthanthecut-onfrequency,i.e.,fcuton>1=2RbCtot[66].Theacbiasvoltagesonthetwocapacitorsareequalinamplitudeandfrequency,butoppositeinphase.Thus,intheabsenceofaphysicalexcitation,i.e.,nochangeincapacitance,thevoltageatthecommonnodeisideallyzero(C1C2).Incaseofmismatchedcapacitors,thenominaloutputvoltagewillhaveanamplitudeproportionaltothemismatchatthebiasingfrequency,restrictingthedynamicrangeofthesensor.Anappropriatephaseadjustmentcircuitrymaybeusedtonulltheoutputresultingfromaninitialmismatchinnominalcapacitances(seeChapter 6 ).Ashearstressinputcausesachangeincapacitanceresultinginavoltagechangeatthemiddlenode.Theamplier 75

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B ).Foreaseofunderstanding,Figure 3-11 showsasimpleschematicshowingtheexpectedsensoroutputspectrumateverystageofthecircuitrypresented.Thebiasingresultsinamodulatedoutputattheamplieroutput,wherethemeanchangeincapacitanceisatthecarrierfrequencyandthedynamicchangeisobservedatthesidebandsofthemodulatedoutput.Thereisanadditionalnoisecomponentfromtheamplier.Themodulatedoutputisthendemodulatedandlowpasslteredtoextracttheoriginalmodulating/inputsignal.Therewillbeadditionaldcosetvoltagesandnoisecontributionfromthedemodulatorandthelowpasslter,whicharenotshowninthegure.Especially,thenet1=fnoisecomponentattheoutput,atlowfrequencies,needsconsiderationtoensuresucientsignalresolution.Thederivationsthatfollowprovideabetterunderstandingofthebenetsofthemodulationtechniquetomeasurestaticcapacitancechangeandhencethemeanshearstress.TheinstantaneouscapacitancesforthesensorareC1="Tt"3Lteff+Le+(N1) 2Lf 2Lf 2Lf 2Lf 76

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where,vc=Vacsin(!ct)arethesinusoidalbiasvoltagesthatactasthecarriersignalsforthemodulationprocess.ThebiasvoltagefrequencyischosentobehigherthanthebandwidthofthesensorwhileensuringthatnochargedissipatesthroughRb.Consequently,thebiasresistorisessentiallytreatedasanopencircuitforthevoltageanalysis.Thecombinationofsenseandparasiticcapacitancesformapotentialdivider,withbiasvoltagesourcesconnectedtooneelectrodeofeachofthesensecapacitors.Theanalysisassumesonesensecapacitorisbiasedatatime,followedbysuperpositiontoobtainthenetoutputvoltage. 77

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3{59 and 3{62 .Theresultis,vin=vinC1+vinC1andvin=vcC1C2 3-10 hasaunitygain;i.e.,G=1.Theequationshowever,arederivedforanamplierwitharbitrarygain.Equations 3{63 and 3{64 verifythattheinput,andhencetheoutputofthe 78

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3{53 and 3{54 ,Equation 3{64 canbesimpliedandrewrittenas,vout1=Gvc0@3Lteff+Le+(N1) 2Lfhg21g11 2Lfhg22g12 3Lteff+Le+(N1) 2Lfhg21+g11 2Lfhg22+g12 3{45 .Thus,thechangesincapacitancesarealsoequalinmagnitudebutoppositeindirection.ConsideraoatingelementmotionsuchthatC1increasesandC2decreases,resultinging11=g01;g12=g02+;g21=g01+;andg22=g02: 3.2.3 ).SubstitutingEquations 3{66 and 3{45 intoEquation 3{65 gives 2Lf 3Lteff+Le+(N1) 2Lfg201 2Lf 3Lteff+Le+(N1) 2Lfg01 g01:(3{67)Theterm,Hgap=1Lteff+(N1) 2Lf 3Lteff+Le+(N1) 2Lfg201 2Lf 3Lteff+Le+(N1) 2Lfg01 3{67 representsanattenuationtermduetothepresenceofthesecondarygapg02whichpartiallycancelsoutthecapacitancechangeduetotheprimarygapg01.Thisis 79

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3-6 andFigure 3-10 .Asg01=g02!0,Hgap!1i.e.,noattenuation.Onthecontrary,asg01=g02!1,Hgap!0i.e.,100%attenuation.Ifg01=g02>0,thedierentialschemestillholds,butthetwogapsswitchfunctionalities,whereg01actsasthesecondarygapandg02actsastheprimarygap.Theterm,Hc1=2C0 3{67 representstheattenuationduetoparasitics[63].However,theeectofelectromechanicaltransductionisnotaccountedyetandwillbeincorporatedinthenextsection.Theeectsofattenuationandthedesignconsiderationsthatarerequiredtomini-mizethemarealsodiscussedlater.SubstitutingEquations 3{68 and 3{69 inEquation 3{67 ,andsettingbiasvoltage,vc=Vacsin(!ct),gives g01Vacsin(!ct):(3{70)Equation 3{70 representsthesensoroutputvoltageamplitudethatismodulatedbythecarriersignal.Themodulatedoutputsignalissubsequentlydemodulatedusingamultiplier(demodulator)toretrievetheoriginalsignal.Duringthedemodulationprocess,theoutputfromthevoltageamplierismultipliedwithareferencesignalhavingthesamefrequency,!c,asthesinusoidalbiasvoltages.As-sumingareferencevoltage,vref=Vrefsin(!ct),theoutputofthemultiplieris 3{70 and 3{71 andsubstitutingforvrefresultsinvout=HgapHc1(t) UVacVrefsin2(!ct)orvout=HgapHc1(t) 80

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3{73 fordeectionwithEquation 3{72 resultsinvout=VacVrefHgapHc1G (3{74)orvout=VacVrefHgapHc1G 3{75 givesthenetdemodulatedvoltageoutputfromthemultiplier.Thissignalhasseveralfrequencycomponents,adccomponent,m,alowfrequencycomponentfromtheow,tsin(!t+),andhighfrequencycomponentsduetothecarriersignal,mcos(2!ct)tsin(!t+)cos(2!ct).Thecomponentsofinterestarethedcandthelowfrequencysignals,correspondingtothemeananductuatingpartsoftheow,respectively.Thehigherfrequencycomponentsarethereforelow-passlteredafterthedemodulationprocess.Thelow-passlteredoutputvoltageisthengivenas 81

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82

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A .Whenadeectionoccursduetoshearforceonthesensor,thetethersactasthesprings(compliance),storingpotentialenergy.Thedistributedmassofthetethersandtherigidmassoftheoatingelementwithcombngers,lumpedtothecenter,possesskineticenergy.Since,majorityofthetransductionoccursattheoatingelement(x=Lt),thecenterde-ection,=w(Lt),seemstobethenaturalchoiceforlumpedparameterestimation.Therearetwodierentsourcesofdissipation:elasticdampingduetointernalfrictionanduidicdampingduetoviscouseectsunderthesuspendedoatingelementstructure[42].Alinearsystemisrequiredtoensurespectraldelityofthesensor.Theenergyandtheco-energyofalinearsystemareequal[41].Therefore,thelinearmechanical(Equation 3{6 )isusedtocomputethelumpedmassandcomplianceoftheoatingelementstructureinsteadofthenonlinearmodel. 2Mmef02;(3{77)whereprepresentsmomentumandf0representsow.Thetotallumpedmassisthesumoftheeectivemassofthetetherslumpedaboutthecenter,andthemassoftheoating 83

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A ,thelumpedmechanicalmassofthesensoris 35WtLt 35WtLt 105WtLt 105NWfLf 315WtLt 1+NWfLf 22 A ,thecomplianceofthetethersisgivenas 4ETtLt 15WtLt B .Thetwoportmodel,transductionfactor,andtheelectromechanical 84

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B ,thesensortwo-portmodelusingasinglesensorcapacitance,C10,isexpressedas, 3{81 ,CMOandCEBinEquation B{30 havebeenreplacedwiththeCmeandC10,respectively.Therefore,CMO=Cme(openmechanicalcompliance) (3{82)andCEB=C10=C20:(blockedelectricalcapacitance) (3{83)Similarly,theelectromechanicaltransductionfactoris'0=Vb 85

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Equation 3{85 representstheeectivegap,whichaccountsforthelossinsensitivityduetogapattenuationdiscussedearlierusingEquation 3{68 .Theelectromechanicalcouplingfactoris2=V2b 3-12 showsthetransductionmechanismforacapacitivetransduceringeneral,andtheproposedshearstresssensorinparticularwithasingleendedcapacitance(C10).ThefreeelectricalcapacitanceC010isdenedasC010=C10 3{87 ,theeectoftheelectromechanicalcouplinghasbeenincorporated.Thevoltage,v0isrelatedtotheeectiveshearforce,fe,actingontheoatingelementsensorthroughtheturnsratioasfollows,v0='0fe: 3{84 intheexpressionabove,thevoltage,vinattheinputoftheinterfacecircuitryisvin=C010 86

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SubstitutingEquations 3{88 and 3{84 intotheequationabovegivesvin=C010 3{90 representsthevoltageavailableattheinputoftheinterfacecircuitryi.e.,amplier(idealamplier).Sofar,asinglesensecapacitancewasusedtoillustratethetransductionprocessandanexpressionfortheamplierinputvoltagewasobtained.Next,theamplierinputvoltageusingboththesensecapacitorsforthedierentialcapacitancesensingschemeisdescribed.Figure 3-13 showsthedierentialtransductionschemefortheoatingelementsensor.ThesameprincipleusedinderivingEquation 3{90 isusedtoobtainthenetoutputvoltageforthedierentialcapacitancescheme.Onesensecapacitorisconsideredatatimewhiletheotheractsasparasitic.Subsequently,superpositionisusedtoobtainthetotal 87

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3{90 ,theindividualamplierinputvoltagesarevinC1=C010 3{91 and 3{92 asvin=C010+C020 3{87 andtheeectivegapofEquation 3{85 intoEquation 3{93 ,givesvin=C10+C20 3{93 includestheeectofelectromechanicalcouplingintheparasiticattenuationterm(seeEquation 3{69 ).RecognizingthatthenominalcapacitancesarethesamefromEquation 3{45 thenewparasiticattenuationtermisHc=2C0 3{69 i.e.,thefreeelectricalcapaci-tance(C010)approachestheblockedelectricalcapacitance(C10).Inthenextsection,anequivalentcircuitisprovidedusinglumpedelementsfromSection 3.2.1 andthetransduc-tionschemedevelopedinthissection.ThederivedtransducedvoltageiscomparedwiththeexpressioninEquation 3{67 tofurtherexplaintheeectofelectromechanicaltransductionviatheelectromechanicalcouplingfactor,2(seeEquation 3{86 ). 88

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3.2.2 linksthemechanicalandtheelectricalsystemofthesensor.Inthissection,thecompleteequivalentcircuitofthesensorispresentedasshowninFigure 3-14 .Theresonantfrequency(bandwidth)andthestaticsensitivityofthesensorisestimatedfromthefrequencyresponseofthisequivalentcircuit.AsdiscussedinSection 3.1.3.3 ,theoutputvoltageisbueredusingahighimpedancevoltagefollower.Ideally,theinniteinputimpedanceofthevoltagefollowerallowstoassumeanopencircuitconditionafterthetransduction.Furthermore,weakelectromechanicalcou-pling(21)strengthensthisassumption.Thus,loadingofthesysteminFigure 3-14 duetotheinterfacecircuitryandparasitics,isneglected.Thefrequencyresponseisthereforede-terminedusingtheopencircuitcongurationandisdominatedbythemechanicalresponseofthesystem.Although,theloadingduetoparasiticsisneglectedforthetransduction,subsequentparasiticattenuation(Hc)ofthetransducedsignalisimportanttodeterminetheoverallsensorsensitivity.Itisnoteworthythattheopencircuitassumptionleadstoa 89

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3-14 isasimplesecondordersystem(springmassdamperorLCR)andisgovernedbythedierentialequation, dt2+Rd dt+k;(3{97)wherefshearrepresentstheshearforce,Mmerepresentsthelumpedmass,krepresentstheeectivestinessofthetethers,andRrepresentstheeectivedampinginthesystem.Thestiness,kisk=1 3{97 issolvedusingFouriertransform,resultinginthefollowingfrequencyresponsefunction: 2r 90

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3-15 Mode1(fr=5:177kHz) (b) Mode2(fr=14:39kHz) (c) Mode3(fr=25:08kHz) (d) Mode4(fr=46:91kHz) (e) Mode5(fr=140:9kHz) (f) Mode6(fr=141:0kHz) Next,theexpressionfortheoutputvoltageandstaticsensitivityofthesensorisderived.Considertheeectiveforce,feusedinthethetransductionprocess.ThisforceissameasthelastterminEquation 3{97 .ThisisalsoevidentfromacomparisonofEquation 3{25 91

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,andthetransferfunctionorgainfactor,G(!),betweenfeandfshear,G(!)=fe Cme; 3{96 giventotheinterfacecircuitryisobtainedbysubstitutingEquation 3{102 togivevin=HgapHc g01Vb: g01Vacsin(!ct): 3{70 ,whichdidnotaccountfortheelectromechanicalcouplingbetweenelectricalandmechanicaldomains.ComparisonwithEquation 3{70 showsthatalltermsarethesameexceptthetermHcinEquation 3{104 ,whichaccountsfortheelectromechanicalcouplinginthesystem.Similarly,theeectofelectromechanicalcouplingisincorporatedinthedemodulatedoutputsignalinEquation 3{76 ,rewrittenwithHcinsteadofHc1asvout=VacVrefHgapHcG 92

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{z }1HgapHc| {z }2G {z }3m=w {z }4: 3{107 onthesensitivity.Pro-ceedingtermbyterm,term1indicatesthatthestaticsensitivityisdirectlyproportionaltotheamplitudeproductofthebiasvoltage(Vac)andthereferencevoltage(Vref)suppliedtothemultiplier.TheupperlimitofVacisdeterminedfromthepull-inlimitorthedielectricbreakdownlimitofair,withabuiltinfactorofsafety.Usuallythepull-inconstraintdomi-nates(seeChapter 4 ).Vrefislimitedbythemaximuminputvoltagethatcanbeappliedtothemultiplier.WhilethesensitivitymaybeimprovedbyincreasingVacandVref,thesourcenoisecontributionmayalsoincreaseandshouldbeaccountedforwhilechoosingthevaluesofVacandVref.Term2capturestwoattenuatingeectsonthesensitivity:theeectofthesecondarygapg02viaHgapandthesensitivityscalinginrelationtothenominalsensorcapacitance,C0,andparasiticcapacitances,Cp+CiviaHc.Equation 3{68 indicatesthatHgap<1,loweringthesensitivity.Hgapincreasesasg02increasesincomparisontog01.Itisevidentthattheconditiong01g02needstobesatisedtominimizeattenuationfromthisterm.Higherg02,however,lowersthenumberofcombngersandthusthetotalcapacitance,whichisneededtoovercomeparasitics.Thus,thereisatrade-obetweenthenumberofcombngersandg02. 93

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3{107 istheratiooftheclosedloopgain,G,oftheampliertotwicethescalingfactor,U,ofthemultiplier.Thus,toincreasesensitivity,ahighgainamplifyingstageandamultiplierwithalowscalingfactorarefavorable.Furthermore,ahighgainamplifyingstagehelpstoboostthesignaltonoiseratio(SNR),minimizingtheeectofnoiseaddedinthesubsequentcircuitry.ThevalueofgainGisagainlimitedbythemaximumallowablevoltageinputtothemultiplier.Term4inEquation 3{107 representsthecontributionfromtheactualsensitivityofthesensorviathetransductionitself(seeEquation 3{108 ).Thenumerator(m=w)representsthemechanicalsensitivityofthesensor.Theratiom=g01representsasimpliedformofthechangeincapacitancetotheoriginalcapacitanceorthechangeingaptotheoriginalgap.Thecombinedratiothusrepresentsthesimplestformofsensitivity,whichispercentagechangeincapacitanceperunitshearstress.Ideally,thisvalueshouldbeaslargeaspossibleforthesensordesign;however,bandwidthandlinearityrequirementsconstraintheupperlimitofthisratio.Insummary,thedynamicmodelingsectiondiscussedlumpedelementmodelingandthisconceptwasusedtocomputelumpedelementsfromthedistributedmechanicalsystem. 94

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4 95

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3-16 showstheelectriceldlinesinthegapsg01andg02forasinglecombngersimulatedin3D,usingFEAinCOMSOL. Figure3-16.Asymmetriccombngerstructuretoestimateeectoffringingelectricelds. Thecombngersaresimulatedbecausetheircontributiondominatesthesensornominalcapacitanceinmostdesigns(Table 4.4 )andduetotheirasymmetricgaps.Combngersmadeofperfectelectricalconductorsareassumedforthissimulationandthedielectricmediumisair("r=1).Thevariousgeometricdimensionsforthisanalysisare,g01=3:5m,g02=20m,Wf=4m,andLf=170m.Thecolorgradientoftheeldlinesindicatesthevariationofelectricpotentialinthegaps.Typicallyforthecombnger,thesensorfringingeldcapacitanceis10%ofthenominalcapacitancei.e.,Cfr C00:1.Theratioofchangeinfringingeldcapacitancetothetotalchangeincapacitanceis13%i.e.,Cfr 96

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3.1.3.3 ,parasiticcapacitancehasadetrimentaleectonthesensitivityofthesensor.Anaccurateestimateoftheparasiticcapacitancesisthusessentialtoaccuratelypredictdeviceperformancecharacteristics.Theoreticalmodelsarenotpresentlyavailabletoobtainsuchanestimate.SomesourcesofparasiticsinatypicalMEMSsensorarepolysilicon/metallines,wirebonds,PCBtracesandBNCcables.Thecontributionofthesesourcesmayvaryfromsensortosensor,basedonlengthofmetallines,qualityofwirebondsetc.Martin[72]comparessensitivitiesofadualbackplatemicrophone,usingbothachargeamplierandavoltageamplier.Thesensitivitymeasuredusingachargeamplierisindependentofparasiticcapacitances,butisinuencedbyparasiticswhenmeasuredwithavoltageamplier.Thus,acomparisonofthesensitivitiesmeasuredusingthetwobiasingschemeshelpsinisolatingtheassociatedparasiticcapacitanceofthesensor.SimilartoMartin'swork,thesensordevelopedinthisdissertationalsousestheSiSonicTMmicrophoneamplier,courtesyKnowlesAcoustics[76].ItusesthesamedierentialcapacitancemeasurementschemeasusedinMartin'ssensor.Martin'sworkshowsexperimentalvalueofparasiticcapacitancesvaryfrom0:92pFto2:23pFfromtestsconductedonsevendierentmicrophones.Theproposedsensorhassimilarinterfacecircuitry/sensorpackagingandthesameamplierasMartin'smicrophone.Thehighervalueofparasiticcapacitance,2:23pF,shouldthereforeprovideaconservativeestimateofparasiticsfordesignpurposes. 4 .Foragivenbandwidth,theMDSismathematicallydened 97

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3-10 .Avoltageamplierampliesthemodulatedsignal,theoutputofwhichissuppliedtotheinputofthemultiplier(demodulator).Ifanamplierwithasucientclosedloopgainisused,thenoisecontributionfromthelaterstagesmaybeneglectedprovidedtheyhaveacomparableorlowernoiseoor.Therefore,inthenoisemodeltheinputreferrednoiseoftheamplierisanalyzedwithoutthecontributionofthesubsequentstages. Figure3-17.Simpliedschematicofsensorcircuitrywithnoisesources. InFigure 3-17 threenoisesourcesareconsidered:thermalnoiseofthebiasresistorRb,theinputreferredvoltagenoiseSva,andthecurrentnoiseSiaoftheamplier.Thenoisesourcesareassumedtobeuncorrelated,allowingsuperpositiontocomputethetotalnoiseattheoutput.Thenoisesourcesarepresentedintermsoftheirspectraldensityandmustbeintegratedoverthenoisebandwidthtoobtainthetotalrmsnoisevoltageattheamplieroutput.Asinglenoisesourceisconsideredatatimeandthecorrespondingoutputnoiseis 98

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3{107 andEquation 3{115 ,theMDSforthesensorisMDS=Snoise;in 99

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3{115 andEquations 3{111 showthatthecontributionofthecurrentnoiseincreasesasthetotalcapacitance,CTdecreasesduetosmalldevicecapacitance.Henceitisdesirabletohaveminimumcontributionfromthistermformicroscaleelectrostaticsensors.Moreover,notheoretical/empiricalmodelsexistfortheaccurateestimationofparasiticsthatcontributetoCT.Thustohaveapredictablenoiseestimate,itisdesirabletouseanamplierdesignedforaverylowcurrentnoise. 100

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3 areusedtoformulatemathematicalfunctionsandconstraintsoftheresultingnonlinearconstrainedoptimizationproblem.Sequentialquadraticprogramming(SQP),whichisoneoftheseveralavailableoptimizationtechniques,isusedforoptimization.TheconstrainedoptimizationproblemisimplementedusingtheoptimizationtoolboxinMATLAB.Specically,theSQPfunc-tionfminconinMATLABusesagradientdescentbasedmethodtooptimizenonlinearlyconstrainedproblems.Section 4.1 givesabriefoverviewoftheoptimizationtechniqueused,itsbenets,anditsshortcomings.Section 4.2 explainsthedesignrequirements,identiesthedesignvari-ables,objectivefunction,andtheconstraints.Section 4.4 providesoptimizationresultsandsensitivityanalysis. 101

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4{1 accountsfortheconstraintsinagivenoptimizationproblemandirepresentstherelationbetweentheobjectiveandconstraintfunctionswhentheconstraintsareactive.Forinstance,ifnoconstraintsarepresentorareinactive(i=0),theLagrangiansimplyreducestotheobjectivefunction.OptimalityisensuredbysolvingforisuchthattheKuhn-Tuckerconditionsaresatised[77].TheKuhn-TuckerconditionsarenecessaryconditionsforoptimalityandarestatedasrL(x;t;)=rfnPi=1irGi=0;iGi=0fori=1;;ni0fori=1;;n: 4{2 showsthattheoptimizationalgorithmreliesonthereducinggradientsofboththeobjectiveandconstraintfunctionsandsolvesforisuchthatrL=0.Consequently,basedontheinitialvaluesofdesignvariables,thesolvermayarriveatalocalminimuminsteadoftheglobalminimumoftheobjectiveinthedesignspace.Repeatingtheoptimizationforseveraldierentinitialconditionsprovidesgreaterpossibilityofarrivingattheglobalminimum[77].Thesecondandthirdconditionsarefortheconstraints,whichindicateiftheconstraintsaresatisedwhilebeingactiveorinactiveforoptimalityi.e.,Gi=0andi6=0orGi<0andi=0.Thebenetsofthisoptimizationtechniqueincludefasterconvergence,easyimplementa-tion,andhighaccuracy.DrawbacksofSQParethatitrequiressmoothanalyticalfunctionstocomputegradientsandisnotaglobaloptimizationalgorithm. 102

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4.2.1SensorPerformanceRequirementsSensorsingeneralneedtosatisfyseveraltargetspecicationslikelargedynamicrangeandbandwidth,inadditiontohighsensitivityandlownoiseoor.Thesetermsandtheirdesirablevaluesarebrieyexplainedinthissection.TheoperatingspaceforasensormayberepresentedbyaplotasshowninFigure 4-1 .Theoperatingspaceisboundbythelimitsofthesensorperformance.Ideally,agooddesignisaimedatmaximizingthesensoroperatingspace.Theextentsoftheoperatingspaceareboundbythedynamicrangeandbandwidthofthesensor. Figure4-1.Schematicofoperatingspaceofthesensor. Thedynamicrangeofasensorisdenedasthemaximumrangeofthephysicalinputthatthesensorcanreliablymeasure.Theupperendofthedynamicrangeisdetermined 103

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3{109 thustranslatesintomaximizingtheoverallsensitivitywhileminimizingthenoiseoorofthesensor. 4.3.1ObjectiveandDesignVariablesThegoalofthisoptimizationistominimizetheminimumdetectablesignal(shearstress)(MDS)forthedierentialcapacitiveshearstresssensor.IntheexpressionfortheoutputgivenbyEquation 3{105 ,thetermVref 104

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3.3.3 ,themechanicalthermaldampinginthesystemisassumedtobenegligible.Asaresult,theelectronicsdominatetheoverallnoiseoor.TheSiSonicamplierusedinthisimplementationhasameasuredinputreferredvoltagenoisepowerspectraldensityof41016V2 3{115 ).Also,itmayneedanexternalbiasresistor,whosenoisecontributionvarieswiththesensorcapacitance.Withthepresentnoisespecicationsfortheamplier,thenoiseoorreducestoitsvoltagenoiseandisrepresentedusingacurvetasSo=41016(1+395=f): 4{6 .Note,theoverallsensitivityincludesthenon-idealitiesandattenuationfactors(HcandHgap)duetoparasiticsandasymmetricgaps.TheLEMdevelopedinChapter 3 isusedtoestimatethefrequencyresponseofthecoupledelectromechanicalsystem.Theoatingelement,tethers,andcombngershavethesamethickness,Tt,whichisdecidedbythemateriallayerthicknessduring 105

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1. BiasvoltageamplitudeVac TetherlengthLt TetherwidthWt ElementlengthLe ElementwidthWe TetherthicknessTt CombngeroverlapL0 CombngerwidthWf Primarygapg01 Secondarygapg02Thisresultsin10designvariablesrepresentedasavector,!X=(Vac;Lt;Wt;Le;We;Tt;L0;Wf;g01;g02): (2Wf+g01+g02): 106

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1 thegoalistodesignasensortomeasuremax10Pa.Thecharacteristicviscouslengthscale,l+,forthisshearstressisapproximately5:3m.Thespatialresolutionissetbythemaximumsensorsizewhichisrestrictedto400l+viaupperbounds(UB)foraconservativedesigntodemonstrateproofofconcept.Threedierentscalesareconsidered100l+,200l+,and400l+,respectively.Thelowerbounds(LB)aresetbythesensingareaneededtohaveareasonablesignaltonoiseratiosansattenuation.Nominalcapacitance,fabricationandhydraulicsmoothnessrequirementsareusedtodeterminetheboundsonthegapsaroundthesensor.Thelargestgapisroughly3:8l+,thusensuringthathydraulicsmoothnessismaintained.Theupper(UB)andlower(LB)boundsonthedesignvariablesare(Table 4-1 )LB(Vac;Lt;Wt;Le;We;Tt;L0;Wf;g01;g02)UB: 107

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3.2 ,themechanicalthermaldampinginthesystemisneglected.Therefore,fhighisdenedastheundampednaturalfrequency,fresofthesystemforestimationpurposes.Thus,thebandwidthconstraintmaybesimplywrittenasfminfres: 3%; 3{8 andisthelineardeectionobtainedfromEquation 3{7 108

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g0124C0 212

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4-1 .TheconstantsandmaterialpropertiesusedfortheoptimizationareshowninTable 4-2 Table4-1.Lowerandupperboundsfordesignvariablesandowspecications. 5251001000104010020001002000455051504203:5153:520105&10 ElectricalPermittivityofFreeSpace"08:8541012F=mRelativePermittivityofAir"r1:005FloatingElementYoung'sModulusEy168GPaDensityofFloatingElement2330kg=m3AmplierInputCapacitanceCi0:3pFParasiticCapacitanceCp2:2pF 110

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4-3 .Thebiasvoltage,Vacisrestrictedeitherbyitsupperboundorbytheactivepull-inconstraintforallthedesigns.Thetetherlength,LtalwayshitsitsUBtendingtomaximizethesensitivitybyincreasingthetethercompliance.However,thetetherwidthisnotalwaysatitsLBbecauseofthebandwidthandtheactivepull-inconstraint.Thus,thebandwidthismaximizedandthepull-ininstabilityisavoided,but,atthecostofsensitivity.ThisisbetterunderstoodfromtherelationsSensitivityCme; 3{107 ).Thus,thedetrimentaleectofthepull-involtageistwo-fold,itlimitsVacandCme,loweringthesensitivity.TheoptimizeralwaysmaximizesthesensorsensingareaviaLeandWetoincreasetheavailableshearforce,improvingsensitivity.Thetetherthicknessisalwayspushedtoitslowerbound.Equation 3{80 indicates,Cme/1 111

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3 viathegapattenuationfactor,HgapinEquation 3{68 .Largeg02alsodecreasesthenumberofcombngersN,loweringnominalcapacitance.Thus,atradeoexistsbetweennominalcapacitanceneededtomitigateparasiticattenuation,andlargerg02requiredtolowerHgap.Insomecases,thecombngerwidth,Wfisalsomaximized.Increasedwidthcon-tributestoincreaseinsensingareaanddecreaseinnumberofcombngersN.Therefore,thickerngersresultinlowernominalcapacitance.Similartothediscussioninthepreviousparagraph,thereistradeobetweenimprovedsensitivityandlowerC0i.e.,higherparasiticattenuation,resultingfromwiderngers. Table4-3.CapacitiveshearstressoptimizationresultsforCp+Ci=2:2+0:3pF. 16:46 TetherLengthLt(m) 1000 1000 1000 1000 23 15 15 ElementLengthLe(m) 1000 500 1000 1000 1000 500 1000 1000

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45 45 45 45 150 150 150 150 170 170 170 170 5:1 20 4.0 20 3.5 3.5 3.5 3.5 20 20 31 15 71 31 ResonantFrequency/Bandwidthf(kHz) 5 5 5 5 5 5 10 SensitivitySoverall(dBre1V=Pa) SensitivitySoverall(mV=Pa) 35:60 19:20 9:360 7:040 18:80 MDSat1kHz(Pa) 0:562 1:040 2:140 2:840 1:060 AttenuationduetoCp+Ci;Hc(dB)

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0:18 0:18 0:18 0:18 0:18 FloatingElementCapacitanceCE(pF) 0:23 0:11 0:06 0:11 0:11 CombFingerCapacitanceCf(pF) 1:16 0:30 0:14 0:73 0:30 CapacitanceC0(pF) 1:56 0:59 0:38 1:03 0:59 FullScaleDeectionmax(nm) 60 60 50 10 120 CapacitanceChangeC(fF) 22:6 8:73 5:22 3:55 17:5 PercentageChange(C=C0)% 1:45 1:48 1:37 0:35 2:95 PullinVoltageVPI(V) 26.9 19.4 4.4 ispresented.TheMDSandthevariablesarebothnormalizedbytheirrespectiveoptimumvalues.Thevariablesareperturbed20%oneithersideoftheiroptimumvalue.Figure 4-2 showsthesensitivityofnormalizedMDStothetennormalizeddesignvariables.However,someofthesesolutionsmaybeinvalidbecausetheconstraintsmaybeviolatedandarestudiedseparately.TheresultsindicatetheMDSishighlysensitivetovariationintetherdimensions,LtandWt.Thedesignhasminimumsensitivitytothesecondarygap,g02andthecombnger 114

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widthWf.TheMDSvariationwitheachindividualvariablesandconstraintsisalsostudied.OnlyonedesignvariableisperturbedatatimetounderstandtheeectontheMDS.Theregionoftheplotthatisnotshadedorhatchedindicatesthefeasibleregion.Theshadedportionindicatestheregionwheretheconstraintisviolatedwhereasthehatchedportionindicatestheinfeasibleregionsetbythebounds.Figure 4-3 showstheresultsforeachdesignvariable.ConsidertheresultinFigure 4-3(b) forLt,forwhichtheoptimalMDShasthehighestsensitivity.TheoptimalMDSislimitedbytheupperboundofLt.AnyfurtherincreasewouldhaveloweredtheMDSbutagainrestrictedsimultaneouslybythepull-inandresonantfrequencyconstraints.Thepresentedanalysisprovidesinsightintotheexpectedvariationsintheobjectiveandconstraintfunctionsresultingfromchangeintheoptimizedvariables.However,thisstudyislimitedtochangeinonevariableatatimeandasstatedearliermayresultinviolationofconstraintsforothervariables.Theexpectedchangeinsensorperformance(Design1)duetosimultaneousvariationsinthedesignvariablesandparametersofinterestisalsostudiedstatistically,usingaMonte 115

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116

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117

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Carlo(MC)simulation.Inthissimulation,aGaussiandistributionisassumedforeachparameterofinterest.Theapproximatetolerancesanduncertainties,thatmaybeexpectedinthevariablesandconstants,areusedtobuildtheGaussianprole.Table 4-4 showsthetolerancechartforthevariousparametersusedinthisstudy.TheperformanceparametersofinterestforthisanalysisareMDS,overallsensitivity(S;overall),andbandwidth(fmin).Thesimulationuses50000samplesforeachinput/designparameterofinterest.ThevariationsinMDS,S;overall,andfminarerepresentedintheformofhistogramsafteroutlierrejectioninFigure 4-4 .Thedistributionisslightlyskewedtothenonlinearityinthesystemandduetocorrelationbetweenthevariables.Forinstance,whengapg01increasesg02proportionallydecreasesandviceversa.However,forthisanalysisallvariableswereassumedtobeuncorrelated.Forsmallperturbationstheseeectsarenotpronounced,resultinginagaussiandistribution.TheverticalredlineintheguresindicatestheoptimizedvaluefromTable 4.4 .ThemeanandstandarddeviationforeachofthedesignparametersaresummarizedinTable 4-5 .Thetableshowsthatdespitenitetolerances 118

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Table4-4.Tolerancechartforvariablesandconstants. BiasVoltageVac0:2VTetherLengthLt5mTetherWidthWt0:5mFloatingElementLengthLe10mFloatingElementWidthWe10mTetherThicknessTt1mInitialFingerOverlapL05%CombFingerWidthWf5%PrimaryCombSeparationg0115%SecondaryCombSeparationg021mDielectricPermittivity10%DensityofFloatingElement(Si)y10%FloatingElementYoung'sModulusE10%AmplierInputCapacitanceCi10%ParasiticCapacitanceCp10% MDS(Pa)0:5620:5280:2920:765S;overall(mV=Pa)35:6034:9220:7049:13fmin(kHz)5:004:964:285:65 119

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MDS (b) Bandwidth(fmin) formulationi.e.,designvariables,objectivefunction,constraints,andvariablebounds.Theoptimizedresultsforvedierentdesignswerepresented.Lastly,asensitivityanalysiswasperformedtostudytheeectofvariationoftheoptimizeddesignvariablesontheMDSandconstraintsforDesign1.AMonteCarlosimulationwasperformedtoinvestigatetheeectonsensorperformanceduetosimultaneousvariationindesignvariablesandconstants.Theoptimizationprovidedbetterinsightintothedesignphysics.Dominantdesignvariablesthatinuencethedesignmostareidentiedeasilyviathesensitivityanalysis. 120

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5-1 showstheschematicofasingledieoftheshearstresssensor. Figure5-1.Schematicshowingtheplanviewofasingledieoftheproposedshearstresssensoranditssectionviewindicatingvariouslayers. 121

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5-2 .First,theoatingelementstructureisdenedusingDRIE.Next,athinlayerofnickeliselectroplatedonthedevicelayer.Afronttobackalignmentstepdenesthemaskforthebacksideetch.ThisisfollowedbybacksideDRIEwiththeBOXlayerservingasanetchstop.Lastly,theoatingelementstructureisreleasedbywetetchingtheBOXlayer. Figure5-2.Stepbystepfabricationprocess. Themotivationforelectroplatingonthesiliconisbrieyexplainedherebeforeillustrat-ingthefabricationprocess.Thehighlydopedmicromachineddevicelayerenablesseedlesselectroplatingontheentireexposedsensorsiliconsurface.Seedlesselectroplatingeliminatesseveralstepsassociatedwithseededelectroplating,signicantlysimplifyingthefabricationprocess.Electroplatingeliminatesunwantedchargeaccumulationonthedielectricnative 122

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D 3 .Parallelwallsenableauniformelectriceldbetweenthecombngers,whichisthetheoreticalbasisofthesensingschemeanalyzedinChapter 3 .Figure 5-3 showsscanningelectronmicroscopy(SEM)imagesindicatingthequalityoftheetchandtheasymmetricgaps(Chapter 3 )oftheoatingelementstructure. 123

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5-4 showsbothverticalsidewallsanduniformityoftheplatedmetal. 126

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doublesidedheatsensitivetape.Usingadicingsaw,thewaferisdicedalongthedicinglinesetchedontothewaferintheinitialDRIEstep.Thewaferisthenplacedonahotplateat80Ctoremovethetape,resultinginindividualdie.EachindividualdieisimmersedinBOEfor20mintoreleasethesensors.ThesensorsareimmediatelyimmersedinmethanolandthensupercriticallydriedinCO2toavoidstiction.Figure 5-5 showsanopticalimageofafullyprocessed,releasedoatingelementsensordie,etchedontheSOIwafer. 127

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5-6 showsthesensordieplacedonaFloridaquartertogetaperspectiveofthesizeofthesensor. Figure5-5.MicroscopicimageofareleasedoatingelementsensorstructurefromDesign3(Table 4.4 ). Figure5-6.A2mm2mmsenseelementona5mm5mmsensordie. 128

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5-7 showstheschematicshowingtheoverallpackagingschemeforthesensor.Thecontactstotheboardaremadeusinggoldwirebonds.TheKnowlesSiSonicamplierisplacedonthebacksideofthePCBincloseproximitytothesensortominimizeparasiticattenuationarisingfromtheconnectiontracesonthePCB(Figure 5-8 ).SMBconnectorsonthebacksideofthePCBallowelectricalcontacttothesensorforbiasing,inputandoutputsignals,alsoshowninFigure 5-8 .ThedetaileddrawingsofthePCBandLuciteplugareprovidedinAppendix D alongwithvendordetails. Figure5-7.Schematicofsensorpackageforshearstresscharacterization. 129

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130

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3 andChapter 5 ,isexperimentallycharacterized.Design1inTable 4-3 isexperi-mentallystudiedforimpedance,static,dynamic,andnoisecharacteristics.Theexperimentalsetupforthesemeasurementsisdescribed.Experimentalresultsanddiscussionsthatfollowgivephysicalinsightintothesensorperformancecharacteristics. 6-1 .Themeasurementsetupconsistsoffourprobes,twoofwhich(HpandHc)areconnectedtoonebondpad/electrodeandtheothertwo(LpandLc)areconnectedtotheotherbondpad/electrode.HpandHcstandforhighpotentialandhighcurrentandLpandLcstandforlowcurrentandlowpotential,respectively.Theimpedanceofthesensorismeasuredusing 131

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anHP4294Aimpedanceanalyzer,whichusesanI-VmethodwhereZsensor=V I=VHpLp 132

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6-2 showstheschematicofthesensorcalibrationsetupforstatic/meanshearstressinputs.TwodimensionalPoiseuilleowisassumedinthechannel,330mmlong,100mmwide,and1mminheight,whichmaybevariedifnecessary.Theexpressionofshearstressforsteady,fullydeveloped,2-Dowinachannelis[82]w=h dx; 133

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12dP dxh2; 2V: ; 6{4 is2:9Pa,M0:06andRe442.ThisvalidatesthePoiseuilleowassumption,whichistrueifRe<1400andM<0:3[82]. Figure6-3.Schematicwithopticalimageofsensordie(5mm5mm)indicatingoat-ingelement,contactpads,andinterfacecircuit(voltagebuer)formeanshearcharacterization. 134

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6-3 showsapictureofafabricatedsensorandaschematicoftheinterfacecircuitconnectionsforthismeasurement.Forstaticshearstresscharacterization,theowratethroughthechannelisvariedandsensoroutputvoltageismeasured.Achangeinowratealtersthewallshearstressviaachangeinthepressuregradientalongthelengthofthechannel.TheowrateisvariedelectronicallyusinganAALBORGGFC47massowcontroller,andthecorrespondingpressuredierentialandsensoroutputvoltagearerecorded.AKeithley2400sourcemeterservesasthevoltagesourcetocontroltheowcontroller.ThepressuredierentialismeasuredusingtheHeiseST-2Hpressuregaugeusingadierentialpressuremodule(50"H2O).AnSR560lownoiseamplierwithunitygainbandpassltersthesensoroutput(aroundthebiasingfrequency)fornoisereduction.ThisoutputisthenfedtoaDAQcard.PressuredropfromtheHeisepressuregaugeandthesensoroutputvoltageareacquiredusingLabViewonaPC.Equation 6{4 andtherecordeddataareusedtoestimatethesensor'smeanshearstresssensitivity. 3.1.3.3 ,isnotimplementedduetoissueslikedcosetsanddriftassociatedwiththedemodulationcircuitry.Onlythemodulationschemeisimplementedformeanshearstressmeasurements.Thissectionexplainsthereasonsandthepracticalimplementationofthecurrentbiasingscheme.ThecongurationshowninFigure 6-3 resultsinamodulatedvoltageattheamplierinput.Ashearinducedmeancapacitancechangeproducesacorrespondingchangeinthisvoltageamplitude.ThemaximumvoltagethatmaybeappliedtotheinputoftheSiSonicvoltagebuerislimitedto100mV.Withoutanappliedshearstress,ideallymatchedsensecapacitorsandout-of-phase(180)biasvoltagesresultinnovoltageattheinputnode(Section 3.1.3.3 )[42].Mathematically,thismaybeexpressedas(Equation 3{64 )vin=Vacsin(!t)C1C2 135

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6{11 mayberewrittenasvinmismatch=C1Vac1C2Vac2 6{12 ,thenvin=0i.e.,withnoshearstressinput,thenominaloutputvoltageofthebiasedsensor,withmismatchedcapacitors,iszero.Toachieve=1,abiasingcircuitryisrequiredthatcanindependentlycontrolbothamplitudeandphaseofthetwobiasvoltagesignals.Note,thatthemismatchisapracticalaspectwhichdependsonfabricationprocessandisusuallyquantiedviaelectricalcharacterization.Acustommadebiasingcircuitisdesignedandimplemented,whichconsistsoffourAD4898operationalampliers,showninFigure 6-4 ,withtherequiredphaseadjustmentandbiasingcapabilities.Asinusoidalsignalfromafunctiongeneratorisinputtotwoop-amps,whichserveasphaseadjustcircuits.Theyareconnectedinanallpasslterconguration[85],havingvariablecapacitors,Cvar,toindependentlyadjustthephaseoftheiroutputs.Theoutputsofthetwolterop-ampsareinputtothevoltagebuerandtheinvertingop-ampwithanadjustablegain(=R2 136

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voltageatthemiddlenodeviaphaseandamplitudeadjustment.Thishelpstousehigherbiasvoltages,improvingsensitivityanddynamicrange. 6-3 ,butwithdcbiasingforthesensingcapacitors.Aknownoscillatingshearstressinputisgeneratedusingacousticplanewavesinaduct[51].Theoscillatingacousticeldinconjunctionwiththeno-slipboundaryconditionattheductwallresultsinanoscillatingvelocitygradient,generatingafrequency-dependentshearstress.Thisenablesatheoreticalestimateofthewallshearstressiftheacousticpressureisknownatagivenaxiallocation 137

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cej(!tkx0)tanhar !; 6{13 isa1-Dshearstresssolutionandmaybeusedfor2-Dacousticeldsfor>2[86].The1-DsolutioninEquation 6{13 isusedinthisdissertationbecausethelowestfrequencyformeasurementsinthePWTis1kHz,whichcorrespondsto=520.Therearethreedierentmeasurementsetups,eachofwhichvaryintermsofthetermi-nationusedinthePWTandthesensorlocation.TherstsetupusesarigidterminationattheendofthePWTforshearstressmeasurements,thesecondsetuphasthesensormountedattheendfornormalacousticincidenceforpressuremeasurement,andthelastsetupusesananechoicterminationforcombinedshearandpressuremeasurements.Theremaybeacousticreections,whichvaryasafunctionofthePWTtermination.Forstandingacousticplanewaves,whichhasreectedwaves,theinputshearstressisinstanding=1 !ejkdsRejkds C-1 ).Appendix C providesthetheoreticalderivationforthedynamicwallshearstressbasedonacousticreectionsfromthetermination. 6-5 .PlanewavesaregeneratedbyaBMS4590Pcompressiondriver(speaker), 138

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mountedatoneendofthePWT,whiletheotherendisttedwitharigidtermination.ThePWTconsistsofa1"1"crosssectionsquareduct,withacut-onfrequencyof6:7kHzforhigherordermodesinair,abovewhichEquation 6{13 isnolongervalid.Thesensorbeingtestedandareferencemicrophone(B&K4138,1=8")areushmountedatknownpositionsalongthelengthofthetubeandtheacousticfrequencyischosensuchthatthesensorisatavelocitymaximaandthemicrophoneisatapressuremaxima.AB&KPULSEMulti-AnalyzerSystem(Type3109)actsasthemicrophonepowersupply,dataacquisitionunit,andsignalgeneratorforthecompressiondriver.Foraperfectlyrigidtermination(R=1),theacousticwaveisreectedinphasewiththeincidentwave,resultinginapressuremaxima(doubling)andavelocityminimaatthetermination[87].Thereferencemicrophoneisplacedadjacent(<3mmaway)totherigidtermination(pressuremaxima)withoutappreciablephaseerrorsatthetestingfrequencies(6:6kHzmaximum).Atapressurenodethesensorresponseisdominatedbyshearstress,whichisafunctionofthevelocitygradient.Anothermicrophone,locatedatthesame 139

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1.1.3 ).Theout-of-planesensitivityofthesensorisaresultofamismatch(non-idealcase)inthesensornominalcapacitance.Forout-of-planemotionbothsensorcapacitorschangeinthesamesensei.e.,increaseordecreasesimultaneously.Thusthedierentialoutputvoltageiszeroformatchedcapacitors.However,acapacitancemismatchresultsinaproportionaloutputvoltage.ThismaybesimplyillustratedusingtheexampleinSection 6.1.2.1 .ConsiderC2=C1,where<1,andwithdcbiasvoltages,Vb.Considerthedynamicout-of-planemotionissinusoidali.e.,sin(!1t).TheresultinginputvoltagetotheamplierduetosinusoidalvariationinthecapacitanceandbasedonEquation 6{11 isvinmismatchp=Vb(1)C1sin(!1t) 6-6 ). 6{13 )isapproximately142:7dBor273Pa.Thispressureisovertwoordersof 140

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magnitudehigherthantheshearstressvalue.LetSshearandSpressurebetherespectiveshearandpressuresensitivitiesandletVshear=SshearwandVpressure=Spressurepbetherespectivevoltagesduetoshearandpressureinputs.Now,ifSshear100Spressure,thenforthecurrentexample,VshearVpressure.Thusthepressuresensitivitymaybetwoordersofmagnitudesmallerthanshearsensitivitybut,ifthepressureinputistwoordersofmagnitudehigherthantheshearinput,thepressurecontributiontothetotaloutputisthesameasthatfromshear.Toinvestigatethis,thePWTisttedwithananechoictermination(a30:7"longber-glasswedge),whichminimizesacousticreectionstoensureplanepropagatingwavesintheduct.Thecompositesensitivity(shearandpressure)andthefrequencyresponseofthesensorarestudiedwiththissetup,showninFigure 6-7 .ThesetupinSection 6.1.3.1 isnotusedtostudythefrequencyresponseofthesensorbecausethexedsensorlocation(nodalposition)forpressureminimainthestandingwavepatternlimitsthenumberofacousticfrequencies. 141

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isat1106V=p 6-8 .TheoutputoftheSiSonicbuerismeasuredusingaSR785spectrumanalyzer.Thedetailsofthesourcesofnoiseandtheexperimentalsetupforthismeasurementcanbefoundin[88].Thisexperimentisrepeatedbothwith(totalnoise)andwithoutthesensor(setupnoise)toisolatethesensornoisefromthatofthemeasurementsystem. 142

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acrossC1andC2toestimatetheirnominalcapacitances.Thedcbiasvoltageacrossthecapacitorsissweptfrom018Vwitha400mV(peak)sourcesignalat50kHz.Thebandwidthforthismeasurementissetto5,whichpreciselycontrolsthesweepparameter(dcbiasvoltage).Theaveraginginthesystemisturnedoandthemeasurementisrepeated31timeswith100pointsforeachmeasurement.Ensembleaveragesareusedtoestimatethemeanandstandarddeviationofthemeasuredcapacitanceasafunctionofbiasvoltage.Fig-ure 6-9 showstheresultsfromtheimpedancemeasurements.TheresultsaresignicantlydierentfrompredictedcapacitancevalueinChapter 4 duetolargeparasiticcapacitances.Thepredictedcapacitanceis1:56pFwhilethemeasuredvaluesareapproximately14:8pFand16:3pFforC10+p1andC20+p2,respectively.Theparasiticcapacitanceisformedbe-tweenthehighlydopeddevicelayerandtheoatzonebulksiliconsubstrateseparatedbythedielectricBOXlayer.ThedierencebetweenC10+p1andC20+p2isroughly1:5pF,which 143

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isapproximatelysameasthepredictedsensorcapacitance.Basedonthefabricationpro-cessandthesensorgeometrysuchadierenceisunlikelyinthesensorcapacitanceitself,suggestingthatthedierencemustbeduetotheunexpectedlylargeparasiticcapacitance.Furthermore,asuspecteddriftduetotheprobestationwasalimitationinobtaininganaccuratequantitativemeasurementofC10+p1andC20+p2.Thisrestrictedfurthercharacter-izationtoestimateC10andC20andalsotoquantifytheirmismatch.Thedetailsofthemeasurementdrift,unexpectedparasiticcapacitance,andtheconsequentnon-idealitiesinthesensorperformancearediscussedinSection 7.2.2.1 andSection 7.2.1.1 144

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6-1 .Thenumberofaveragesandthesamplingtimesarechosensuchthat,foreachowratesetting,thepressureandsensorvoltagemeasurementsapproximatelyendatthesametime. Table6-1.Measurementsettingsforstaticcalibrationintheowcell. 6-10 showsthesensitivityplotforthismeanshearmeasurement.Theinitialvoltageduetothecapacitivemismatchis 145

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6.2.5 ).Alinearcurvetonthedatasetisusedtoestimatethesensorsensitivity(R2=0:9958).Thesensorhasalinearsensitivityof0:94mV=Pauptothetestinglimitof2:9Pa. Figure6-10.Linearsensoroutputvoltageasafunctionofmeanshearstressatbiasvoltageamplitudeof2:5Vat10kHz. Theplotindicatesalargeuncertaintyintheshearstressvaluecomparedtotheoutputvoltagesbutthetisbasedonlyonthemeanvaluesofshearstressandsensoroutputvoltage.Suchlargeerrorvalueswarrantfurtheranalysistodeducetherangeofsensitivitieswithinthemeasureduncertaintylimits.ThisisachievedviaaMonteCarlosimulationassumingaGaussianprolefortheuncertaintyateachpoint.Eachmeasurementpointisrandomlyperturbedfromitsmeanvaluewithinitserrorboundsinbothshearstressandoutputvoltage.Alineartontheresultingcurveisusedtoestimatethesensitivity.Thisprocedureisrepeated50000timestobuildadistributionoftheoverallshearstresssensitivityrepresentedintheformofahistogram.Thecorrespondingdistributionofthenormalizedresidualsvalues(R2)foreachtarealsoplotted.Figure 6-11 showsthehistogramsforthesensitivityandR2.ThesensitivityandR2valuesandthe95%condenceintervalsfromthis 146

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6-2 .The2DscatterofthesensitivityandR2indicatesthatthesensitivityfromthemeasurementistowardsanextremeofthedistributionwherethetisbetterwithR2=0:9958.However,theperturbationsaboutthemeanmeasuredvaluesresultinasensitivitythatismuchdierent(1:02mV=Pa)butwithlowermeanqualityoft(R2=0:8288).Theellipsoidalshapeofthescatterplotprovidessomeintuitionintothiseect.Animportantassumptionforthisanalysisisthattheperturbationsforeachdatapointareuncorrelated,whichisreasonablesinceeachmeasureddatapointshouldbeastatisticallyindependentvalue. Sensitivity (b) JointHistogram (d) 2DScatter 147

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Sensitivity(mV=Pa)1:0170:9971:037NormalizedSensitivity(mV=V=Pa)0:4070:3990:415NormalizedResidual(R2)0:82880:77510:8825 6.1.3 ),thesensorcharacterizationinthePWTinvolvesthreedierentsetups.Dynamicshearsensitivityresultsareobtainedusingtherigidtermi-nationinthePWT.PressuresensitivityresultsareillustratedwiththesensormountedattheendofthePWTfornormalacousticincidence.Thefrequencyresponseandcombinedsensitivity(shear+pressure)results,usingtheanechoictermination,areprovided.Duringeachmeasurement,therecordedquantitiesareasfollows:therespectiveautopowerspectraldensitiesofthesensorandthereferencemicrophoneoutputsandthetransferfunctionandcoherencebetweenthesensorandthereferencemicrophonesignal.Forallsensitivityesti-mates,themeasuredautospectraoftherespectivesensorandreferencemicrophonesignalsareused.Forthefrequencyresponsemeasurement,thecomputedsensitivitiesandthetrans-ferfunction(sensortomicrophone)areused.ThemeasurementsettingswerekeptthesameforalldynamicmeasurementsandaresummarizedinTable 6-3 .Thenormalizedrandomerrorfortheautospectrumestimatesis3:2%.Thenormalizedrandomerrorsforthemag-nitudeandphaseofthefrequencyresponsefunctionareeach0:04%(ReferSection 6.2.5 ). 6-5 ),wherethesensorisplacedataquarterwavelength(velocitymaxima)awayfrom 148

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6-12 showsthedynamicshearsensitivityofthesensor.Thesensitivityofthesensorisdeducedusingalinearcurvetonthedataset(R2=0:99995).Theestimatedsensorsensitivityis7:66mV=PaatdcbiasofVb=10V.Duringthismeasurement,theSPLatthesensoraxiallocationwas40dBlowerthanthemaximumpressuremeasuredbythereferencemicrophoneatthetermination.Thenormalizedsensorsensitivitiesfromboththemeananddynamicshearstressmea-surementsare0:407mV=V=Paand0:766mV=V=Pa,respectively.Thedierencebetweenthetwovaluesisroughly47%,whichmaybeattributedtonon-idealitiesinthesensorandinthemeasurementsetupseveralofwhicharediscussedinChapter 7 149

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6-13 ).Thefrequency(4:2kHz)ischosenforcomparisonwiththecombinedsensitivitymea-surementsthatarepresentedsubsequently.TheSPLrangefortheexperimentis80150dBvariedinstepsof5dB,tobeconsistentwiththeshearsensitivitytests.Theminimalvaria-tionoftheoutputatlowSPLisbecausethesensoroutputvoltageisbelowthenoiseoorofthemeasurementsystem.ThepressuresensitivityofthesensoratVb=10Vis4:8V=Pa(R2=0:9976),whichisabout1000timessmallerthantheshearsensitivityofthesen-sor(Figure 6-5 andFigure 6-10 ).AcomparisonofthedynamicandshearsensitivitiesatVb=10Vresultsinapressurerejection,Hp,ofapproximately64dBforcomparableforcinginshearandpressure,whichiscomputedasHp=20log(Sshear=Spressure): 150

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6{14 .Sincetherearenopressurenodeswithprogressiveacousticwaves(withweakreections),thesensorencountersbothshearandpressure.Theexperimentsareconductedatthreedierentdcbiasvoltages,5V,8V,and10V,atafrequencyof4:2kHz.Thisfrequencyischosentoensurehighsignaltonoiseratio(forshearstress)evenatlowSPLs,whichislimitedbythedrivingcapabilityofthecompressiondriver(FromEquation 6{13 ,p0p 6-14 .Notethatalthoughthevoltagemeasurementisplottedasafunctionofshearstress,crossaxis(out-of-plane)sensorsensitivityduetopressurealsocontributestotheoutputvoltage.Inallthreecases,thesensorexhibitsalinearsensitivity(R2min=0:9995)uptothetesting 151

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limitof1:1Pa.Themeasuredsensitivitiesare23mV=Pa,19mV=Pa,and11mV=Pa,respectivelyatVb=10V,8V,and5V.Thecombinedsensorsensitivityisdirectlyproportionaltotheappliedbiasvoltageasexpected(Equation 4{3 ).Aplotofthenormalizedsensitivity(outputvoltagepervoltagebias)indicatesthesameviathecollapseinthedatafordierentbiasvoltages(Figure 6-15 ).Thenormalizedsensitivitiesagreecloselytowithin6%andare2:268mV=V=Pa,2:316mV=V=Pa,and2:196mV=V=Pa,atVb=10V,8V,and5V,respectively.Comparisonofthenormalizedsensitivityestimatesfromtherigid(2:268mV=V=Pa)andanechoictermination(0:766mV=V=Pa)experimentsindicatessignicantdierence(66%)inthesensorperformanceinthepresenceoflargepressuresignals.Note,unliketheanechoiccase,fortherigidterminationthesensorisprimarilyforcedbyshearstressgiventhatitisatapressurenodeandvelocitymaximum.Themonitoredpressureatthisnodeis40dBortwoordersofmagnitudebelowthepeakacousticpressure.Also,withacombinedinputhavinglargepressuresignalsinadditiontoshear,thesensorshearsensitivitymaydropfromitsnominalvalue.Forexample,considerthesensorhasaniteout-of-planedeection.Thisreducestheoverlappingareaofthecapacitors,which 152

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lowersthesensitivitytoin-planemotion,showngraphicallyinFigure 6-16 .Furthermore,theeectivein-planecompliancemaybelowersincethedeected(out-of-plane)structureoersahigherstinesstoin-planemotion.Thestieningofthedevicemaynotbeanissueforsmallout-of-planedeections,butforlargedeections(largepressures)thismaysignicantlyalterdeviceperformance.Thismayneedfurthertheoreticalandexperimentalanalysisforbetterunderstandingofthesensorresponsetocombinedshearandpressureinputs.ThefrequencyresponseofthesensorismeasuredforVb=10V,withtheanechoicterminationinplace(Figure 6-7 ).Aspreviouslystatedarigidterminationmaynotbeusedforthisexperimentsincethepressurenodesintheresultingstandingwavepatternarelimited.Thisrestrictsthedierentdiscreetfrequenciesavailablefortestingthefrequencyresponse.Thesensorandthereferencemicrophoneareplacedatthesameaxiallocationalongthelengthofthetube.Theexpressionofthefrequencyresponsenormalizedbytheinputshearstressis[86]H(f)=out @V; 153

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whereV(f)isthesensoroutputvoltagecorrespondingtotheknownshearstressinput,in,whichistheoreticallyestimatedusingEquation 6{13 .Theterm@=@Vistheinverseoftheat-bandsensitivitymagnitudeofthesensor.Thereisanadditionalphasecomponentinthetransferfunctionbetweenthesensorandmicrophone.Thiscomponentchangesasafunctionoftheterminationused,SPL,andtherespectivelocationsofthesensorandreferencemicrophoneintheacousticeld(Section 7.2.2.2 ).Correctingforthisphase,,Equation 6{17 mayberewrittenasH(f)=out @Vej: 154

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6-17 showsthemagnitudeandphaseofthemeasuredfrequencyresponsefunction(FRF)conductedatVb=10Vforatheoreticalshearstressinputmagnitudeof0:5Pa.TheSPLisapproxi-mately140dB2dBovertherangeoftestingfrequencies.AsinglefrequencyisexcitedatatimeandtheSPLisadjustedtomaintainaconstantshearstress(theoretical). Figure6-17.FrequencyresponseofsensoratVb=10Vusingin=0:5Paasthereferencesignaluptothetestinglimitof6:7kHz. Asexpectedfromthedesign,theFRFshowsaclearsecond-ordersystemresponsewithaatbandregionandaresonancefrequencyof6:2kHz.Thequalityfactor,Q,isapproximately79.Assumingasecondordersystem,thistranslatesintoadampingratio, 155

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7.2.2.2 6-4 Table6-4.SpectrumanalyzersettingsfornoisemeasurementinadoubleFara-daycage. Span(Hz)100400160012800Numberofsamples/FFTlines800800800800Binwidth(Hz)0:1250:5216Numberoflinearspectralaverages3024012004800ChannelcouplingACACACACWindowingfunctionHanningHanningHanningHanningWindowoverlap(%)75757575

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Figure6-19.Zoomedinplotoftheoutputreferrednoiseoorofthepackagedsensornear1kHzatdierentbiasvoltages. 157

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6-18 .Thehighestmeasuredthermalnoisepowerspectraldensityis114nV=p 6-19 showsazoomedinplotofthesensornoiseooraround1kHz.Themeasurednoisepowerspectraldensityincreaseswiththeappliedbiasvoltage.TwopotentialreasonsexistforsuchadependenceonVb.First,thenoisecontributionfromthebiassourceitselfmayincreasewithitsoutputvoltage,raisingthesensornoiseoor.Second,thenominalcapacitancesC1andC2maychangewiththeappliedbias.Ateachbiasvoltagetheoatingelementsettlesatanewmeanposition,whichisaresultofelectromechanicalcouplingandtheinitialnominalcapacitancemismatch.Thenoiseattheamplierinputisthusafunctionoftheresultingimpedancedividerformedbythesensingcapacitors.Table 6-5 showstheintegratednoiseoorofthesensorovertheentirerangeofmeasurementandoverspecicfrequencyrangesofinterest.Thisisrelevantfortimedomainshearstressmeasurementsforrealtimeowcontrolapplications. Table6-5.Integratedvoltagenoiseoorofthesensoratdier-entfrequencyranges. 15:6mHz12:8kHz(totalmeasurednoise)1:5mV10Hz10kHz27:3V100Hz10kHz18:2V1kHz10kHz10:2V 4{5 usingthedynamicsensitivityatVb=10V,is14:9Pa(f=1kHz@1Hzbin).Thiscorrespondstoamechanicaldeection=0:17pmandacapacitancechangeC=75zF,whicharecomputedusingthedesignvaluesforthegeometryandassumingparallelplatecapacitances.Asdiscussedpreviously,thequantitativeaccuracyofjH(f)j(shearoutputperunitshearinputasafunctionoffrequency)isquestionableduetothecombinedshearandpressureresponse.However,anaccurateestimateofjH(f)j,ifavailable,maybeusedtoobtainanMDSspectrumin 158

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1r; 159

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6-6 providesacomparisonofthepredictedandmeasuredvaluesforthesensor.Thebandwidthanddynamicrangeestimatesareincloseagreementwhilethesensitivity,noiseoor,andMDSestimatesaresignicantlydierent.Thedierenceinpredictedandmeasuredsensorperformancehasseveralpossibleexplanations.Alargeparasiticcapacitanceexistsbetweenthesensorelectrodesduetotheelectricalpaththroughtheoatzonesiliconsubstrate,loweringsensitivity.Thecapacitancemismatchandthenoisecontributionofthebiasingsourcesincreasestheoverallnoiseoorofthesystem.Thesetwoeectsaredirectly 160

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NormalizedSensitivitySoverall(mV=V=Pa) 1:424 0:407(mean)/0:766(dynamic) Bandwidth(kHz) 5 6:2 NoiseFloor(nVrms=p 20 114@Vb=10V 0:562 14:9 DynamicRangeatVb=10Vandmax=1:9Pa(dB) 107 102 7 161

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2 ,whichhighlightedtheircapabilitiesandlimitations.Usingsomelessonslearntfrompreviousresearch,threekeyareaswereidentiedandfocusingonthesecontributedtotheoverallimprovementinsensorperformance.Thesekeyareas,whichalsogivethemaincontributionsofthiswork,areasimpleandnovelsensorstructure,aneectiveandpotentiallyinexpensivefabricationprocess,andasystemleveloptimizationofthedesign.Thesensorwasdesignedtohaveanasymmetriccombngerstructuretoformcapacitors.Whileachievingarelativelyushsurface,thisdesignallowstheuseofasinglemateriallayertodenethesensorstructureandelectricalcontactpads.Additionalcapacitancebetweenthestationarydevicesubstrateandboththeoatingelementandtethers,wasalsoutilizedtoimprovesensitivity.Thegeometryofthesensorandadierentialsensingschemehelpintheattenuationofanyoutputduetoout-of-planemotionresultingfrompressure,vibration,etc. 162

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7-1 providesacomparisonofthesensorperformancewithsomepreviouswork.ThecomparisonindicatesthatthepresentworkoutperformsthepreviousbestbythreeordersofmagnitudeinMDSandatleastanorderofmagnitudeindynamicrange. Table7-1.Comparisonofmeasuredsensorperformancewithpreviouswork. PresentWork2:02:01:914:91066:27:66Zhe[44]3:23:20:160:040:531337Padmanabhan[50]0:50:5101:410316y320Schmidt[37]0:50:5130:01y100:47

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7-1 showsaschematicofasinglebondpadtoexplainthesourcesofparasiticcapacitance.Duringfabrication,onlytheparallelplatecapacitance,CP,betweenthebondpadandthestationarysubstratearounditwasconsidered.Floatzone(FZ)bulksiliconwasusedintheSOI,assumingthattheelectricalpaththroughthissiliconwouldbenegligiblysmall.However,thisresultedinaniteresistivepaththroughthesilicon.Asaresult,thecapacitanceduetothedielectricSiO2betweenthehighlyconductivedevicelayerandtheweaklyconductivebulksubstrateaddstotheparasiticcapacitance.ThiscapacitanceduetothestationarysubstrateisCoxSandthatfromthebondpadisCoxBP. 164

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Thecircuitmodelfortheentiresensor,includingtheadditionalparasiticimpedances,isshowninFigure 7-2 .Asanexample,asingleendedsensecapacitancevaluebetweenpoints1and2issimulatedasafunctionoffrequencyforsubstrateresistance,RFZ=10KandRFZ=1M.Theparallelplatecapacitancemodelisusedforcomputingthevariouscapacitancevalues.ThesubstrateandbondpadareasserveasparallelplatesseparatedbythedielectricBOXthickness.ThesimulatedresultsareshowninFigure 7-3 .Theplotindicatesthatasthesubstrateresistanceorthefrequencyincreases,theadditionalparasiticcapacitancefromthesubstratevanishesandthecapacitancereachesamuchlowervalue(CP).Thus,bychoosingahighresistivitysubstrateandahighfrequencybiassignaltheadverseeectsofparasiticsubstratecapacitancemaybemitigated,improvingoverallsensitivity.ThisanalysiswasbasedonconstantvaluesofCoxSandCoxBP.Inreality,thesecapaci-tancevaluesaretimedependentandformeanshearstressmeasurements,theymayvary 165

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Figure7-3.Eectivesensecapacitancevariationwithfrequencyasafunctionofsubstrateresistance. withtheappliedsinusoidalbiasvoltages[90].Thesecapacitorsaresimilartometaloxidesemiconductor(MOS)gatecapacitors.Inthepresentdevice,thehighlyconductivedevicelayer,theBOX,andtheoatzonebulksubstrateformtheMOScapacitors.Thecapaci-tancevaluedependsonthedepletionlayerthicknesswhichvarieswiththebiasvoltagevalue.Adelta-depletionmodelisusedtoexplainthevoltagedependenceoftheMOScapacitor. 166

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7{1 indicatesthattheparasiticMOScapacitancevariesnonlinearlywithanappliedsinusoidalbiasvoltage.ThustheattenuationfactorHc(Equa-tion 3{95 ),whichisafunctionoftheparasiticcapacitance,alsoinducesanonlinearvariationinthesensorsensitivity.Therefore,theoverallsensorsensitivity,S;overall,alsovarieswithanapplieductuatingbiasvoltage.Thismayinhibittheuseofthesensorforsimultaneousmeananddynamicshearstressmeasurementswithsinusoidalbiasvoltages.Givenallthesedierentissues,usinganinsulatingsubstrateinsteadofoatzonesiliconwilleliminatetheparasiticsubstratecapacitance(MOS)issues,improvingoverallsensorperformance. 167

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2p 6.2.1 ,anaccurateestimateofthedierencebetweenC10andC20wasnotachievedduetodriftassociatedwiththeprobestation.Figure 7-4 showsthedriftinthemeasurementofoneofthesensorcapacitors.Theplotshowstherstandthelastmeasurementsweepoutofthe31sweepsofthedcbiasvoltageforensembleaveraging.ThetrendinthethedriftisalsoindicatedinFigure 7-5 ,whichisaplotofthemeancapacitance 168

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Figure7-4.Capacitancemeasurementdriftindicatedviatherstandlastmeasurementpriortoensembleaveraging. However,longermeasurementsindicatethatthedriftcontinuesovertime.Thedriftisabout5fFoverthecourseof31biasvoltagesweeps,whichisroughly22%ofthepredictedfullscalecapacitancechange.Suchahighdriftisthereforealimitationineectivelyquantifyingthenominalcapacitanceandthechangeincapacitanceasafunctionofbiasvoltage.Potentialcausesofdriftintheprobestationinclude,chargebuildupontheprobesandpoorcontactoftheprobetipswiththesensorbondpads.Alackofastandardsubstratewasalsoalimitationinthecompleteunderstandingofthenatureandsourceofthedrift. 6.1.3 ).Thedynamicsensitivitywasdeterminedbasedontheautospectralestimates 169

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ofthemicrophoneandsensorsignalswhilemaintaininghighcoherencevaluesbetweenthesesignals.Aconstantsensitivityimpliesthatthetransferfunction,Hms,betweenthesensorandthemicrophonesignalshouldbeconstant.However,contrarytoexpectation,Hms,varieswiththeSPL.Figure 7-6 andFigure 7-7 showthetransferfunctionasafunctionofSPLforthecalibrationswiththerigidandanechoicterminations,respectively,atVb=10V.Themaximumvariationinthemagnitude,jHmsj,overtherangeofSPLsis221nVfortherigidterminationwhileitis2:99Vfortheanechoictermination.Similarly,themaximumvariationinthephase,ms,is2:5forarigidterminationanditis11fortheanechoictermination.SincethesamemicrophonewasusedforthereferenceSPLmeasurements,thisvariationispotentiallyafunctionoftheterminationattheendofthetubeandtheSPLatthesensorlocation.Incontrasttotheanechoicterminationcase,fortherigidtermination,theSPLis40dBlowerthanthepeakSPLatthereferencemicrophone.ThisintuitivelysuggeststhatlocalSPLatthesensorlocationsignicantlyaectsthesensoroutput.WhilethehighSPLvariationmaybeattributedtotheacousticnonlinearity,thelowSPLbehaviorneedsfurtherinvestigation. 170

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Figure7-7.Phaseofsensortomicrophonetransferfunctionasafunctionofacousticpressure. 171

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7-8 showsthevariationofthetransferfunctionwiththeinputSPLatVb=10V. Figure7-8.Transferfunctionbetweenthesensorandthereferencemicrophonefornormalacousticincidence. 172

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Figure7-9.Comparisonofcombined(shearandpressure)andpressuretransferfunctionsmeasuredbetweensensorandreferencemicrophone. Next,thecombinedpressureandsheartransferfunctionmeasuredwiththeanechoicterminationiscomparedwiththepressuretransferfunctionmeasuredusingnormalacousticincidenceshowninFigure 7-9 .Ahigher,combinedpressure-sheartransferfunctionindicatesthatdespitesimilarandhigherpressureforces,thesensorremainsmoresensitivetoshearinthefrequencyrangeofmeasurementthanitistopurepressure.Thescatterinthepressuredataisduetothepoorperformanceofthesensorforpressureinputs.Note,thatthesensorwasnotdesignedforpressuremeasurementsandreliespurelyonthecapacitancemismatch(ideallyzero)forthismeasurement.Similarqualitativenatureofthetwomagnitudere-sponsesmaybeduetotheshapingoftheoveralltransferfunctionbythepressureresponse 173

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h; 7-10 ).Themeasurementisdonetwiceforh=0:5mmandcomparedwiththeresultsforh=1mm.Therespectivesensitivitiesare0:932mV=Paand0:870mV=Paforh=0:5mmand0:879mV=Paforh=1mm.Thevariabilityofestimatedsensitivitiesforh=0:5mmishigherthanthedierencebetweensensitivitiesforh=1mmandh=0:5mm.Thesemeasurementsarethereforenotenoughtoquantifytheerrorsduetopressuregradienteects.Thisvariability 174

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6-4 ).Thisproblemalsoneedstobeaddressedfromthecircuitsperspective. Figure7-10.ComparisonmeanshearstressmeasurementsatVb=2:5Vatdierentchannelwidths,h,intheowcell. 7.2 frombothdesignandcharacterizationstandpoints.Thissectioncombinesthemandrecommendsanapproachforthenextgenerationofthedirectdierentialcapacitiveshearstresssensors.Forthesensordesign,thepressureresponseneedstobeaccountedforinthemodel.Thiswillincludetheeectsofthecavitycomplianceandthedamping/ventresistancetoventthecavitytotheambientpressure.Thiswillconsiderablyimprovethepressuresensitivityissues.AsimpleLEMforthesensorforpressureforcingisrepresentedinFigure 7-11 ,withthevariousmechanicallumpedelementsinpressureareindicatedwiththesubscript 175

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Figure7-11.MechanicalLEMforpressuresensitivityofthesensor. ofmagnitudehigherthantheshearforcing.Thismayyieldsomeinterestingresults.Thedesignwithcombinationofcavitystieningandcut-onfrequencymayenablethinnertethersimprovingin-planesensitivitywhileresultinginsucientpressurerejection.However,thetrade-owouldbelowernominalcapacitance.Fromthecircuitsstandpoint,theoptimizationmayincludethecurrentnoiseoftheamplierandthethermalnoiseofthebiasresistor.Thisallowstounderstandthetrade-oinvolvedinusingdierentampliersinsteadoftheSiSonic.Furthermore,thedemodulationandbiasingcircuitsmaybeincludedaspartoftheoptimizationtodevelopapredictivetoolfortheoverallsystemperformance,insteadofpiecewiseanalysis.Thesensormaybefabricatedbywaferbondingasiliconsubstratetoapyrexwafer.Thiswilleliminatetheparasiticsubstratecapacitanceissuesaltogether.Usingathin-backoftheoatingelementpriortobondingtothepyrexmayenablehigherbandwidthsbyloweringthemechanicalmass.Backsideelectricalcontactsusingelectronicthroughwafer 176

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177

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A.1.1GoverningEquationThegoverningdierentialequationisderivedbeforeexplainingthesolutionprocedure.Thecurvatureofabeamisrstdeterminedtoformulatethegoverningequations.ConsideranarbitrarycurveasshowninFigure A-1 FigureA-1.Adeectedbeamwitharbitrarycurvature. Thelengthofthearcforsuchacurveis dx2dx:(A{1) 178

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ds: A{2 isrewrittenas1 ds: dx: A{1 thedierentiallength,dsalongthecurveisds=s dx2dx: ds=d' dxdx ds: A{5 and A{4 intoEquation A{6 resultsin 1 ds=d dxtan1dy dx0@1 dx21A=d2y/dx2 dx23=2:(A{7)Nowconsiderabeamunderpurebending.Thecurvatureofsuchabeamis[92]1 EI; 179

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A{7 and A{8 givesMx A{9 isthegoverningequationforthedeectionofabeamduetopurebending.Forbothsmallandlargedeections,theslopedw(x) FigureA-2.Simpliedmechanicalmodeloftheoatingelementstructure. 180

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A-2 .ThisissameasFigure 3-2 ,whichisrepeatedhereforconvenience.TheassumptionsforbeamdeectionwerepreviouslystatedinSec-tion 3.1.1 andthereforenotrepeatedhere.Inthesimpliedmechanicalmodel,fortheshearstresssensor,eachpairoftethersformaclamped-clampedbeam,withapointloadPappliedatcenterbytheshearstress,w,ontheoatingelement.Auniformlydistributedload,Q,alongthelengthofthetethersaccountsfortheshearforceonthetethers.TwosuchbeamssharetheloadappliedontheoatingelementbecauseofthegeometricsymmetryasshowninFigure A-2 .Thus,theshearforceeachtetherpairthatformbeamsisP=wWeLe 2; (A{13)Duetosymmetrythesolutionisobtainedbysolvingforhalfthebeamlength.Theboundaryconditionsforaclamped-clampedbeamaremathematicallystatedasw(x=0)=0;(nodeection)dw dxx=0=0;(zeroslope)anddw dxx=Lt=0:(symmetry) (A{14) 3{4 .Thetheoryhypothesizesthatastraightlinetransversetotheneutralaxisremainsstraight,inextensible,andnormalbeforeandafterdeformation[65].Withtheseassumptions,thelinearsolutionforsmallbeamdeectionisobtained. 181

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A{16 in A{10 givesd2w(x) A{22 resultsinMA=RALt A{15 into A{23 andsimplifyingitgivesMA=PLt 182

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A{15 A{20 A{21 ,and A{23 intoEquation A{19 andgivenasw(x)=1 A{11 and A{12 ,thesimpliedexpressionforthedeectionis A{26 isfurthersimpliedusingEquation A{27 togivew(x)=w 3.1.2.1 )andananalyticalsolutiontech-nique(Section 3.1.2.2 )areillustrated.First,anexpressionforthetotalstraininthebeam 183

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@x2 @x+1 2@w @x2: @x+1 2@w @x2!dx=[u(x)]2Lt0| {z }0(clampedboundary)+1 22LtZ0@w @x2dx=1 22LtZ0@w @x2dx:(A{33)Thusthestrainintermsofthetransversedeectionduetostretchingofthebeamis"0xx=Lts 4Lt2LtZ0@w @x2dx: A{30 A{31 ,and A{34 ,thetotalstraininthebeamisrewrittenas"t=z@2w @x2+1 4Lt2LtZ0@w @x2dx: 184

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m3=Nm m3=N m2m m; A{40 mayberewrittenas~Wnormal="Z0E"d"=1 2E"2=1 2": 185

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2: A{41 into A{43 gives,W=TtZ0Wt=2ZWt=22LtZ01 2"tdxdydz: A{44 and A{45 intoEquation A{36 resultsinU=TtZ0Wt=2ZWt=22LtZ01 2E"2tdxdydz(P+QLt): A{11 and A{12 ,theaboveequationisrewrittenasU=TtZ0Wt=2ZWt=22LtZ01 2E"2tdxdydz| {z }I1w {z }I2: A{47 .InthesolutionprocedureusingtheRayleigh-Ritzmethod,atrialfunctionforthedeectionisusedtoevaluateI1,whichisw= 186

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dx= dx2=2 A{35 A{49 ,and A{50 togive 4Lt2LtZ0 Ltsin(Ltx) A{47 isevaluated,usingtheexpressionfor"fromEquation A{51 asfollows, 2TtZ0Wt 187

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A{52 into A{47 ,thetotalpotentialenergyofthesystemisexpressedasU=ETt24W3t {z }I1w {z }I2: d=0: A{53 intheaboveequationandknowingthat4 4resultsin1+3 4 Wt2!=wWeLe A{55 givestheexpressionforthedeectionusingenergymethodunderlargedeection.Theaccuracyofthesolutionishoweverlimitedbythechoiceoftrialfunction. A{9 andstillretainstheassumptiond2w dx21.TheschematictotherightinFigure A-3 representsthefreebodydiagramofthebodyofonehalfoftheclamped-clampedbeamshownintheguretotheleft.Balancingmomentsatagivensectionatadistancexfromthefreeendgives,M(x)=P 188

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whereM0istheresistingmomentatx=0.ThegoverningequationforthismethodobtainedbysubstitutingEquation A{56 into A{9 ,resultinginEId2w(x) {z }2w(x)=Q {z }Xx2+P {z }Yx(Faw(0)+M0) {z }Z; A{57 ,theaxialforce,Fa,duetothelargedeectionisdetermined.TheaxialstrainunderlargedeectionissameasEquation A{34 withlimitschangedforonehalfofthebeamgivenas"0xx=Lts 2LtLtZ0@w @x2dx: A{61 andEquation A{59 givesFa=EWtTt @x2dx: 189

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A{57 issolved,whichhasahomogenousandaparticularsolutionwrittenasw=wg+wp: A{67 into A{66 gives2a2ax2+bx+c=Xx2+YxZ: 2;b=Y 2; 4 A{69 intoEquation A{67 giveswp(x)=X 2x2Y 2x+2Z2X 4: 190

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A{63 A{65 ,and A{71 isw(x)=C1cosh(x)+C2sinh(x)X 2x2Y 2x+2Z2X 4; 2xY 2: A{14 gives,dw(0) 2=0)C2=Y 3; 2cosh(Lt)2X 2LtY 2=0)C1=Y+2XLtYcosh(Lt) A{74 and A{75 intoEquation A{72 thedeectionisrewrittenas 3sinh(x)X 2x2Y 2x+2Z2X 4:(A{76)Now,usingvaluesofX,Y,Z,andfromEquation A{57 theaboveequationbecomes, 2Fa:(A{77)Recognizingthatw(x=0)=w(0),theresistingmomentM0isdeterminedandisexpressedasM0=Q 21 191

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A{78 into A{79 gives, A{80 istheexpressionfordeectionofthebeamunderlargedeections,whichincludestheeectofstrainintheneutralaxis.Thegradientofthedeectioniswrittenas dx=P FaxP 1. StartwithaninitialvalueofaxialforceFa Obtaintheeigenvaluegivenby=q Estimatethedeectionw(x)andthegradientdw dx CalculateFaagainusingtherelationFa=EWtTt @x2dx Repeatstep1to4untiljFn+1aFnaj=Fn+1a1e8 6. UseFatoobtainthemaximumcenterdeectionw(0) 3.2 inChap-ter 3 .Inthissection,thederivationsforlumpedmassandlumpedcomplianceoftheoatingelementsensorstructurearepresented. 192

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2Mlumpedf20; A{28 isused.Thevelocityorowvariablerepresentedintermsofthedeectionisv(x)=j!w(x): A{83 and A{84 ,v(x)=v(Lt) {z }dmv(x)dv(x): 193

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2dx:(A{88)CombiningEquation A{88 and A{85 gives 2dx=NiTtWt A{82 and A{89 givesMtme=4NiTtWt A{90 ,whichaccountsformassoffourtetherssuspendingtheoatingelement.ThetotallumpedmassoftheentireoatingelementisthesumofthetetherandtheoatingelementmassgivenasMme=Mtme+Melement=Mtme+Ni(WeLe+NWfLf)Tt: 194

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A{28 A{29 A{90 ,and A{91 andevaluatingtheintegralresultsin(solveinMAPLE) 35WtLt 35WtLt 105WtLt 105NWfLf 315WtLt 1+NWfLf 2w2(Lt) A{11 and A{12 andrepresentsadiracdeltafunction.ThusthepotentialenergyinEquation A{93 isrewrittenasWPE=LtZ0w(x)Z0Q+P A{11 and A{12 into A{95 givesthepotentialenergyexpressedintermsoftheshearstress,w,togiveWPE=LtZ0w(x)Z0wWt+WeLe 195

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A{96 requirestheforcingshearstresstobeexpressedintermsofthedistributedsystemtosolveforthetotalpotentialenergy.SubstitutingforwfromEquation A{28 ,Equation A{96 isexpressedas (3(LeWeLt+NWfLfLt)+8WtL2t)x2(2LeWe+2NWfLf+8WtLt)x3+2Wtx43775w(x)dw(x)dx=LtZ026644ETtW3tWt+WeLe (3(LeWeLt+NWfLfLt)+8WtL2t)x2(2LeWe+2NWfLf+8WtLt)x3+2Wtx43775w2(x) 2dx:(A{97)Thedistributeddeectionw(x)isexpressedintermsofthecentraldeectionw(Lt)byeliminatingtheshearstresswfromEquation A{28 and A{29 ,togivew(x)=w(Lt)(3LeWeLt+3NWfLfLt+8WtL2t)x2(2LeWe+2NWfLf+8WtLt)x3+2Wtx4 A{97 andEquation A{98 resultsin WeLe+2WtLt WeLe2#(3LeWeLt+3NWfLfLt+8WtL2t)x2(2LeWe+2NWfLf+8WtLt)x3+2Wtx4!dx=w2(Lt)ETtW4t 15WtLt 1+NWfLf 15WtLt A{99 ,written 196

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15WtLt 1+NWfLf A{93 and A{100 andsolvingforCmegivesthelumpedcomplianceoftheoatingelementsensor,expressedas 4ETtLt 15WtLt 197

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B-1 ,whereA&BandC&Dareconjugatepowervariables(eortandow).Bydenition,thepowertransferisalwaysintothetwo-portelementandhencemustbeconserved[42],PowerNET=ABCD=0orAB=CD: FigureB-1.Generalrepresentationofanidealtwo-portelement. TwotypesoflinearelementssatisfytheconditioninEquation B{1 .Oneisatransformerandtheotherisagyrator,representedasfollows:Transformer B-2 .ThetransformermodelinEquation B{2 representsthetransductionofeort/owinonedomaintocorre-spondingeort/owinadierentenergydomainviathetransductionfactor,n.ThegyratormodelinEquation B{3 relateseortinoneenergydomaintoowinadierentdomain.Forexample,electrodynamicspeakersaremodeledusingagyratorwhereelectricalcurrent 198

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(B)throughacoilinamagneticeldproducesaproportionalactuationforce(D)viaelec-trodynamiccoupling.Atransformermodelisapplicableforacapacitivemicrophonewherepressure(A)istransducedintoaproportionalvoltage(D)duetodiaphragmdeection.Twoportnetworksalsoallowrepresentationusinganimpedanceanalogyoranadmit-tanceanalogybasedondenitionofA;B;C;andDinEquation B{1 .Foranimpedanceanalogytheacrossvariables,A&Dareeortvariables.ForadmittanceanalogytheacrossvariablesA&Dareowvariables.Thisdenitionusuallydependsonthespecictransduc-tionproblem.Asaconvention,intheimpedanceanalogyaneort(eg:pressure,voltage,force)isanacrossvariableandaow(eg:current,volumetricowrate,velocity)isathroughvariable.Anexampleofanimpedancetoimpedanceanalogyforatwo-portelementisshowninFigure B-3 FigureB-3.Impedancetoimpedanceanalogyrepresentationofatwoportelement. 199

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B-3 withtheinstantaneousvaluesofeortvariablesexpressedintermsofowvariablesas,v=v(i;u)andf=f(i;u); B{4 ,dv=@v @iu=0di+@f @ui=0du: IU=0I+V UI=0U=ZEBI+TEMU; IU=0I+F UI=0U=TMEI+ZMOU: IU=0; 200

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IF=0; UI=0; UV=0; UI=0; IU=0; ZMO: 201

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FigureB-4.Equivalentcircuitrepresentationofatransducerusingimpedanceanalogy. FigureB-5.Schematicofaparallelplatecapacitivetransducer. B-5 .Thenominalgapbetweentheplatesisx0andthechangeingapduetoasens-ing/actuationforceisx0(t)suchthat,x=x0x0(t): 202

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x="0A x0x0(t)="0A x01x0(t) B{22 isthesumofthevoltageatrestandthatduetoelectromechanicalcoupling.Theelectrostaticforceforaconstantchargebetweentheplatesis[42],FE=1 2Q2 2Q2 B{23 and B{24 ,thetotalforceonthediaphragmis,F=FM+FE=x0(t) 2Q2 2Q2 B{22 and B{25 are,V(t)=Q0(t) B{26 and 203

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intermsoftheirFouriercomponentsresultsin,V=1 B{30 indicatesthetransducerisreciprocalbutindirectinnature.ComparingEquation B{30 and B{17 ,'0=T ZMO=V x0CMO: B-6 .Thisisconsistentwiththegeneralmodelingapproachofalinearconservativetransducerdiscussedintheprevioussection.UsingEquation B{19 ,thecomplianceCEFis,1 j!x02,1 (j!)2CEBCMO=V2 FigureB-6.Circuitrepresentationofacapacitivetransducerforconstantchargebiasing. 204

PAGE 205

C-1 FigureC-1.SetupforshearstressinPWTforageneralimpedancetermination. Assumeafullydevelopedowinaduct,drivenbyanoscillatingpressuregradient.Thegoverningdierentialequationis@u @t=1 @xej!t+@2u @y2: 205

PAGE 206

C{1 is[93]u(y;t)=jej!t @x26641coshyq 3775: cp01e(j!tkx) 3775: cp02e(j!t+kx) 3775: 206

PAGE 207

C{3 and C{5 ,respectively.SubstitutingEquations C{4 and C{6 inEquation C{7 resultsinu(x;y;t)=jej!t 3775@p1 dyy=a: C{8 ,wall=ej!t !tanhar !@p1 C{3 and C{5 giveswall=1 tanhar !p01ej(!tkx)p02ej(!t+kx): tanhar !p01ej(!tkl+kd)p02ej(!t+klkd); tanhar !0@p01ejkl| {z }P+ej(!t+kd)p02ejkl| {z }Pej(!tkd)1A; tanhar !P+ejkdP 207

PAGE 208

C{15 inEquation C{14 resultsin,wall=1 !P+ejkdRejkdej!t: !P+ejkdsRejkdsej!t: C{18 intermsofpmeasured,intoEquation C{17 ,wall=1 !ejkdsRejkds !ejkdsRejkds 208

PAGE 209

1. Wafercleaning (a) SC-1(IonStrip)-15:3:2H2O:H2O2:NH4OHat75Cfor10min SC-2(OrganicStrip)-16:3:1H2O:H2SO4:H2O2at75Cfor10min Oxi-clean-50:1H2O:HFatroomtemperaturefor30s Acetone/Methanol/DIwash 2. Etchondevicelayertodenesensorstructure (a) CoatwithHMDSfor5min Spinresist(1m,AZ1512)-spinat500rpm100rpm=s5s4000rpm100rpm=s50s Softbake60s(hotplate)at95C ExposeusingmaskFEM-EVG620Ch-A29:5mW=cm2for1s(hardcontact) (e) DevelopusingAZ300MIFfor60s Postexposurebake(oven)-95Cfor60min EtchusingSTS-DRIE-recipeVJ pol50,42cyclesandrecipeBAO2 VJ(avoidsfooting),20cycles (h) Acetone/Methanol/DIwashtocleanphotoresist 3. Nickelelectroplating (a) PiranhacleantoremoveorganicCHF3passivation-3:1H2SO4:H2O2at120Cfor10min

PAGE 210

OxideremovalusingBOE(6:1)for5min Immediate2-propanolimmersionforsurfacewettingandtopreventoxidation (d) ImmediateimmersioninTechnicNickelsulfamatesolutionat90F(Terminals-Nielectrodepositive,SOIwafernegative) (e) Beginplatingsimultaneouslywithimmersion-30sstrikewith125mA(nucle-ation),5minplatingwith30mA(nergrains),5minplatingwith7mA(verynegrains) (f) RinsewithDIwater 4. Backside/Cavityetch (a) CoatbacksidewithHMDSfor5min Spinresistonbackside(10m,AZ9260)-spinat500rpm100rpm=s5s2000rpm100rpm=s50s SpinAZ9260onhandlesiliconwafer-2000rpm1000rpm/sramprate (d) Softbake30min(oven)at95C FronttobackalignusingEVG620 (f) ExposeusingmaskCEM-EVGCh-B62mW=cm2for23s(hardcontact) (g) DevelopusingAZ300MIFfor3min Spinresist(5m,AZ9260)-spinat500rpm100rpm=s5s4000rpm100rpm=s50s PlaceSOIwaferfacedownonhandlewaferandapplygentlepressureforuniformcontact (j) Postexposurebake(Oven)-95Cfor60min EtchbacksidecavityusingSTS-DRIE-recipeVJ pol50,470cyclesandrecipeBAO2 VJ(avoidsfooting),130cycles (l) RemovePRmask-Acetone/Methanol/DIwash (m) SeparatefromhandlewaferusingS-46PRstripper-heat80Cfor15min RinseAcetone/Methanol/DI 5. Dieseparation 210

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LaydowndoublesidedheatsensitivetapeREVALPHA(120C)onbacksideofwafer(withoutairbubbles) (b) Stickwafer,faceuponahandlesiliconwaferforsupportduringdicing (c) LaydownsinglesidedheatsensitivetapeREVALPHA(80C)onfrontsideofwafer(withoutairbubbles) (d) Dicewaferusingdicingsaw-Tresser620-KulickeandSoabladeS1235-thick-ness(11:2mm) (e) Heatonhot-plateat80Ctopeelotapefromfrontsideofwafer (f) Heatonhot-plateat120Ctoseparatediefromtapeandhandlewafer 6. Dierelease (a) EtchBOXinBOE(6:1)for20min Immediateimmersioninmethanol (c) SupercriticaldryinCO2(Prof.Ho-BunChan'slab) 1. SensorpackaginginPCB (a) CutrecessinPCBforsensormounting(CNCrecipename)-PCBlayoutshowninFigure D-1 -madeatSierraProtoExpress (b) GlueSiSonictobacksideofPCBusingepoxy(Dualbond707)-cureat35Cfor10min Wirebondtobondpadsusingballsize4.(Prof.Ho-BunChan'slab) (d) Coverbondswithepoxy(Dualbond707)-cureat35Cfor10min SolderSMBconnectorsforelectricalcontactandllthroughviaswithsolder (f) Gluesensorinplacewithepoxy(Dualbond707)-cureat35Cfor10min Wirebondusingballsize6andforceat100.(Prof.Ho-BunChan'slab) 2. SensorPCBpackaginginplug (a) SensorPCBispresstinLuciteplug-drawinganddimensionsshowninFigure D-2 -madeatTMRengineering(KenReed) 211

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212

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213

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[1] E.I.Administration,Internationalenergyoutlook2008,DOE/EIA-0484.Washington,DC:U.S.DepartmentofEnergy,http://www.eia.doe.gov/oiaf/ieo/index.html,2008. [2] M.Gad-elHak,\Flowcontrol:Passive,active,andreactiveowmanagement."Cam-bridgeUniversityPress,2000,ch.10,pp.209{210. [3] E.I.Administration,\Petroleummarketingmonthly,doe/eia-0380."Washington,DC:U.S.DepartmentofEnergy,2008,p.5. [4] ||,\Worldcrudeoilprices."Washington,DC:U.S.DepartmentofEnergy,http://tonto.eia.doe.gov/dnav/pet/pet-pri-wco-k-w.htm,2009. [5] R.J.Stiles,\Introductiontoaeronautics:adesignperspective."Washington:AIAA,2004,ch.3,p.80. [6] J.W.NaughtonandM.Sheplak,\Moderndevelopmentinshearstressmeasurement,"ProgressinAerospaceSciences,vol.38,pp.515{570,2002. [7] H.Schlichting,\Boundary-layertheory."NewYork:McGraw-Hill,1979,ch.1,pp.6{9. [8] F.M.White,\Viscousuidow."NewYork:McGraw-Hill,1991,ch.4-6. [9] K.G.Winter,\Anoutlineofthetechniquesavailableforthemeasurementsofskinfrictioninturbulentboundarylayers,"ProgressinAerospaceSciences,vol.18,pp.1{57,1977. [10] J.H.Haritonidis,\Themeasurementofwallshearstress,"AdvancesinFluidMechanicsMeasurements,pp.229{261,1989. [11] H.TennekesandJ.L.Lumley,\Arstcourseinturbulence."Cambridge,MA:MITPress,1972,ch.1-5. [12] H.Ludwieg,\Instrumentformeasuringthewallshearingstressofturbulentboundarylayers,"NACA,Tech.Rep.NACATM1284,1950. [13] J.S.J.Mathieu,\Anintroductiontoturbulentow."Massachusetts:CambridgeUniv.Press,2000,ch.5. [14] D.B.SpaldingandS.W.Chi,\Thedragofacompressibleturbulentboundarylayeronasmoothatplatewithandwithoutheattransfer,"JournalofFluidMechanics,vol.18,no.1,pp.117{143,1964. [15] R.L.Panton,\Incompressibleow."NewYorkChichester:Wiley,1996,ch.18,p.526. [16] S.K.Robinson,\Coherentmotionsintheturbulentboundary-layer,"AnnualReviewofFluidMechanics,vol.23,pp.601{639,1991. 214

PAGE 215

R.RathnasinghamandK.Breuer,\Activecontrolofturbulentboundarylayers,"J.FluidMech.,vol.495,p.209233,2003. [18] A.Padmanabhan,H.Goldberg,K.D.Breuer,andM.A.Schmidt,\Awafer-bondedoating-elementshearstressmicrosensorwithopticalpositionsensingbyphotodiodes,"JournalofMicroelectromechanicalSystems,vol.5,no.4,pp.307{315,1996. [19] L.LofdahlandM.Gad-elHak,\MEMS-basedpressureandshearstresssensorsforturbulentows,"MeasurementScience&Technology,vol.10,no.8,pp.665{686,1999. [20] H.Alfredsson,A.V.Johansson,J.H.Haritonidis,andH.Eckelman,\Theuctuatingwall-shearstressandthevelocityeldintheviscoussublayer,"Phys.Fluids,vol.Vol.31,pp.1026{1033,1988. [21] A.Padmanabhan,\Siliconmicromachinedsensorsandsensorarraysforshear-stressmeasurementsinaerodynamicows,"PhDthesis,DepartmentofMechanicalEngineer-ing,MassachussetsInstituteofTechnology,Cambridge,MA,1997. [22] Z.W.Hu,C.L.Morfey,andN.D.Sandham,\Wallpressureandshearstressspectrafromdirectsimulationsofchannelow,"AIAAJournal,vol.44,no.7,pp.1541{1549,2006. [23] M.Sheplak,L.Cattafesta,T.Nishida,andC.McGinley,\MEMSshearstresssen-sors:Promiseandprogress,"in24thAIAAAerodynamicMeasurementTechnologyandGroundTestingConference,AIAA-2004-2606,Portland,Oregon,2004. [24] F.H.Clauser,\Turbulentboundarylayersinadversepressuregradients,"JournaloftheAeronauticalSciences,vol.21,no.2,pp.91{108,1954. [25] B.OudheusdenandJ.Huijsing,\Integratedowfrictionsensor,"SensorsandActuatorsA,vol.15,pp.135{144,1988. [26] E.Kalvesten,G.Stemme,C.Vieider,andL.Lofdahl,\Anintegratedpressure-owsensorforcorrelationmeasurementinturbulentgasow,"SensorsandActuatorsA,vol.52,pp.51{58,1996. [27] C.Liu,J.B.Huang,Z.J.Zhu,F.K.Jiang,S.Tung,Y.C.Tai,andC.M.Ho,\Micromachinedowshear-stresssensorbasedonthermaltransferprinciples,"JournalofMicroelectromechanicalSystems,vol.8,no.1,pp.90{99,1999. [28] M.Sheplak,V.Chandrasekaran,A.Cain,T.Nishida,andL.Cattafesta,\Characteri-zationofasilicon-micromachinedthermalshear-stresssensor,"AIAAJournal,vol.40,no.6,pp.1099{1104,2002. [29] T.vonPapen,H.Stees,H.D.Ngo,andE.Obermeier,\Amicrosurfacefenceprobefortheapplicationinowreversalareas,"SensorsandActuatorsA:Physical,vol.97-98,pp.264{270,2002. 215

PAGE 216

S.Groe,W.Schroder,andC.Brucker,\Nano-newtondragsensorbasedonexiblemicro-pillars,"MeasurementScienceandTechnology,vol.17,no.10,pp.2689{2697,2006. [31] C.Brucker,D.Bauer,andH.Chaves,\Dynamicresponseofmicro-pillarsensorsmea-suringuctuatingwall-shear-stress,"ExperimentsinFluids,vol.42,no.5,pp.737{749,2007. [32] S.GroeandW.Schroder,\Meanwall-shearstressmeasurementsusingthemicro-pillarshear-stresssensormps3,"MeasurementScienceandTechnology,vol.19,no.1,2008. [33] S.GroeandW.Schrder,\Dynamicwall-shearstressmeasurementsinturbulentpipeowusingthemicro-pillarsensormps3,"InternationalJournalofHeatandFluidFlow,vol.29,no.3,pp.830{840,2008. [34] D.Fourguette,D.Modarress,F.Taugwalder,D.Wilson,M.Koochesfahani,andM.Gharib,\Miniatureandmoemsowsensor,"AIAApaper,2001-2982,2001. [35] M.Gharib,D.Modarress,D.Fourguette,andD.Wilson,\Opticalmicrosensorsforuidowdiagnostics,"in40thAIAAAerospaceSciencesMeetingandExhibit,AIAA-2002-252,Reno,NV,2002. [36] M.A.Schmidt,\Microsensorsforthemeasurementofshearforcesinturbulentboundarylayers,"PhDthesis,DepartmentofMechanicalEngineering,MassachussetsInstituteofTechnology,1988. [37] M.A.Schmidt,R.Howe,S.D.Senturia,andJ.Haritonidis,\Designandcalibrationofamicromachinedoating-elementshear-stresssensor,"TransactionsofElectronDevices,vol.35,pp.750{757,1988. [38] T.Pan,D.Hyman,M.Mehregany,E.Reshotko,andB.Williams,\Characterizationofmicrofabricatedshearstresssensors,"Proceedingsof16thInternationalCongressonInstrumentationinAerospaceSimulationFacilities,WPAFB,1995. [39] D.Hyman,T.Pan,E.Reshotko,andM.Mehregany,\Microfabricatedshearstresssensors,part2:Testingandcalibration,"AIAAJournal,vol.37,no.1,pp.73{78,1999. [40] T.Pan,D.Hyman,andM.Mehregany,\Microfabricatedshearstresssensors,part1:designandfabrication,"AIAA,vol.37,no.No.1,pp.66{72,1999. [41] M.Rossi,\Acousticsandelectroacoustics."Norwood,MA:ArtechHouse,1988,ch.5. [42] S.D.Senturia,\Microsystemdesign."Boston:KluwerAcademicPublishers,2001,ch.6,16. [43] J.Zhe,K.R.Farmer,andV.Modi,\AMEMSdeviceformeasurementofskinfric-tionwithcapacitivesensing,"inProceedingsof2001MicroelectromechanicalSystemsConference,IEEECircuitsandSystemsSociety,Berkeley,CA,USA,2001,pp.4{7. 216

PAGE 217

J.Zhe,V.Modi,andK.R.Farmer,\Amicrofabricatedwallshear-stresssensorwithcapacitativesensing,"JournalofMicroelectromechanicalSystems,vol.14,no.1,pp.167{175,2005. [45] M.McCarthy,L.Frechette,V.Modi,andN.Tiliakos,\InitialdevelopmentofaMEMSwallshearstresssensorforpropulsionapplications,"inPropulsionMeasurementSen-sorDevelopmentWorkshopNASAMarshallSpaceFlightCenter,Huntsville,Alabama,2003,pp.1{10. [46] N.Tiliakos,G.Papadopoulos,V.Modi,A.O'Grady,M.McCarthy,L.Frechette,J.-P.Desbiens,andR.Larger,\MEMSshearstresssensorforhypersonicaeropropulsiontestandevaluation,"in2006AnnualITEATechnologyReview,2006. [47] N.Tiliakos,G.Papadopoulos,A.O.Grady,V.Modi,R.Larger,andL.Frechette,\AMEMS-basedshearstresssensorforhightemperatureapplications,"in46thAIAAAerospaceSciencesMeetingandExhibit,AIAA-2008-274.Reno,NV:AIAA,2008. [48] A.V.DesaiandM.A.Haque,\DesignandfabricationofadirectionsensitiveMEMSshearstresssensorwithhighspatialandtemporalresolution,"JournalofMicromechan-icsandMicroengineering,vol.14,no.12,p.1718,2004. [49] A.Padmanabhan,H.Goldberg,M.A.Schmidt,andK.D.Breuer,\Asiliconmicroma-chinedsensorforshearstressmeasurementsinaerodynamicows,"in34thAerospaceSciencesMeeetingandExhibit,AIAA96-0422,Reno,NV. [50] A.Padmanabhan,M.Sheplak,K.D.Breuer,andM.A.Schmidt,\Micromachinedsen-sorsforstaticanddynamicshearstressmeasurementsinaerodynamicows,"TechnicalDigest,Transducers'97,Chicago,IL,pp.137{140,1997. [51] M.Sheplak,A.Padmanabhan,M.A.Schmidt,andK.S.Breuer,\Dynamiccalibrationofashear-stresssensorusingstokes-layerexcitation,"AIAA,vol.39,no.5,pp.819{823,2001. [52] F.-G.TsengandC.-J.Lin,\PolymerMEMS-basedfabryperotshearstresssensor,"IEEESensorsJournal,vol.3,no.6,pp.812{817,2003. [53] S.Horowitz,T.Chen,V.Chandrasekaran,T.Nishida,L.Cattafesta,andM.Sheplak,\Amicromachinedgeometricmoireinterferometricoatingelementshearstresssensor,"in42ndAIAAAerospaceSciencesMeetingandExhibit,2004-1042.Reno,NV:AIAA,2004. [54] S.Horowitz,T.Chen,V.Chandrasekaran,K.Tedjojuwono,T.Nishida,andL.Cattafesta,\Awafer-bonded,oatingelementshear-stresssensor,"inTechnicalDi-gest,Solid-StateSensor,ActuatorandMicrosystemsWorkshop,HiltonHead,SC,2004,pp.13{18. 217

PAGE 218

K.-Y.Ng,\Aliquid-shear-stresssensorsusingwafer-bondingtechnology,"Mastersthe-sis,DepartmentofElectricalEngineeringandComputerScience,MassachussetsInsti-tuteofTechnology,1990. [56] K.-Y.Ng,J.Shajii,andM.A.Schmidt,\Aliquidshear-stresssensorfabricatedus-ingwaferbondingtechnology,"inInternationalConferenceonSolid-StateSensorsandActuators,Transducers'91,1991,pp.931{934. [57] H.Goldberg,K.Breuer,andM.Schmidt,\Asiliconwafer-bondingtechnologyformicrofabricatedshear-stresssensorswithbacksidecontacts,"TechnicalDigest,Solid-StateSensorandActuatorWorkshop,pp.111{115,1994. [58] A.A.Barlian,R.Narain,J.T.Li,C.E.Quance,A.C.Ho,V.Mukundan,andB.L.Pruitt,\PiezoresistiveMEMSunderwatershearstresssensors,"inInternationalCon-ferenceonMicroElectroMechanicalSystems,MEMS2006.Istanbul,Turkey:IEEE,2006,pp.626{629. [59] A.A.Barlian,S.J.Park,V.Mukundan,andB.L.Pruitt,\Designandcharacterizationofmicrofabricatedpiezoresistiveoatingelement-basedshearstresssensors,"SensorsandActuatorsa-Physical,vol.134,no.1,pp.77{87,2007. [60] Y.Li,V.Chandrasekharan,B.Bertolucci,T.Nishida,L.Cattafesta,D.P.Arnold,andM.Sheplak,\Alaterallyimplantedpiezoresistiveskin-frictionsensor,"inTechnicalDigest,Solid-StateSensors,Actuators,andMicrosystemsWorkshop,HiltonHead,SC,2008,pp.304{307. [61] Y.Li,V.Chandrasekharan,B.Bertolucci,T.Nishida,L.Cattafesta,andM.Sheplak,\AMEMSshearstresssensorforturbulencemeasurements,"in46thAIAAAerospaceSciencesMeetingandExhibit,AIAA-2008-269.Reno,NV:AIAA,2008. [62] R.R.Spencer,B.M.Fleischer,P.W.Barth,andJ.B.Angell,\Atheoretical-studyoftransducernoiseinpiezoresistiveandcapacitivesiliconpressuresensors,"IEEETrans-actionsOnElectronDevices,vol.35,no.8,pp.1289{1298,1988. [63] P.R.Scheeper,A.G.H.vanderDonk,W.Olthuis,andP.Bergveld,\Areviewofsiliconmicrophones,"SensorsandActuatorsA:Physical,vol.44,no.1,pp.1{11,1994. [64] Y.Li,M.Papila,T.Nishida,L.Cattafesta,andM.Sheplak,\Modelingandoptimizationofaside-implantedpiezoresistiveshearstresssensor,"inProceedingofSPIE13thAnnualInternationalSymposiumonSmartStructuresandMaterials,paper6174-7,SanDiego,CA,2006. [65] J.N.Reddy,\Theoryandanalysisofelasticplates."Philadelphia,PA:Taylor&Francis,1999,ch.1,pp.21{25. [66] F.V.Hunt,\Electroacoustics;theanalysisoftransduction,anditshistoricalback-ground."Cambridge,MA:HarvardUniversityPress,1954,ch.6. [67] R.K.Wangsness,\Electromagneticelds."NewYork:Wiley,1986,ch.6. 218

PAGE 219

L.K.Baxter,\Capacitivesensorsdesignandapplications."NewYork:IEEEPress,1997,ch.3,pp.42{43. [69] B.E.BoserandK.W.Markus,\Designofintegratedmems,"inDesigningLowPowerDigitalSystems,EmergingTechnologies(1996),1996,pp.207{232. [70] B.E.Boser,\Electronicsformicromachinedinertialsensors,"inInternationalConfer-enceonSolidStateSensorsandActuators,Transducers'97,Chicago,vol.2,1997,pp.1169{1172vol.2. [71] J.C.Lotters,W.Olthuis,P.H.Veltink,andP.Bergveld,\Asensitivedierentialcapacitancetovoltageconverterforsensorapplications,"IEEETransactionsonInstru-mentationandMeasurement,vol.48,no.1,pp.89{96,1999. [72] D.T.Martin,\Design,fabricationandcharacterizationofaMEMSdual-backplateca-pactivemicrophone,"PhDthesis,DepartmentofElectricalandComputerEngineering,UniversityofFlorida,2007. [73] K.Kadirvel,\ClosedloopinterfacecircuitsforcapacitivetransducerswithapplicationtoaMEMScapacitivemicrophone,"PhDthesis,DepartmentofElectricalandComputerEngineering,UniversityofFlorida,2007. [74] D.Martin,K.Kadirvel,T.Nishida,andM.Sheplak,\AninstrumentgradeMEMScon-densormicrophoneforaeroacousticmeasurements,"in46thAIAAAerospaceSciencesMeetingandExhibit,AIAA-2008-257.Reno,NV:AIAA,2008. [75] B.E.BoserandR.T.Howe,\Surfacemicromachinedaccelerometers,"IEEEJournalofSolid-StateCircuits,vol.31,no.3,pp.366{375,1996,0018-9200. [76] P.V.LoeppertandS.B.Lee,\Sisonictm-therstcommercializedmemsmicrophone,"inSolid-StateSensors,Actuators,andMicrosystemsWorkshop,HiltonHead,SC,2006,pp.27{30. [77] R.T.HaftkaandZ.Grdal,\Elementsofstructuraloptimization."Dordrecht;Boston:KluwerAcademicPublishers,1992,ch.5. [78] D.T.Martin,J.Liu,K.Kadirvel,R.M.Fox,M.Sheplak,andT.Nishida,\Amicroma-chineddual-backplatecapacitivemicrophoneforaeroacousticmeasurements,"JournalofMicroelectromechanicalSystems,vol.16,pp.1289{1302,2007. [79] L.D.V.Llona,H.V.Jansen,andM.C.Elwenspoek,\Seedlesselectroplatingonpat-ternedsilicon,"JournalofMicromechanicsandMicroengineering,vol.16,no.6,pp.S1{S6,2006. [80] M.SchlesingerandM.Paunovic,Modernelectroplating,4thed.,ser.ElectrochemicalSocietyseries.NewYork:Wiley,2000. [81] \http://www.technic.com/chm/nickelel.htm." 219

PAGE 220

R.W.FoxandA.T.McDonald,Introductiontouidmechanics,5thed.NewYork:J.Wiley,1998. [83] A.M.Cain,\Staticcharacterizationofamicromachinedthermalshearstresssen-sor,"Thesis(M.S.),DepartmentofElectricalandComputerEngineering,UniversityofFlorida,1999.,1999. [84] Y.Li,\Side-implantedpiezoresistiveshearstresssensorforturbulentboundarylayermeasurement,"AerospaceEngineeringthesis,Ph.D.,DepartmentofMechanicalandAerospaceEngineering,UniversityofFlorida,2008. [85] A.S.SedraandK.C.Smith,\Microelectroniccircuits."NewYork:OxfordUniversityPress,1997,ch.11. [86] V.Chandrasekaran,A.Cain,T.Nishida,L.N.Cattafesta,andM.Sheplak,\Dynamiccalibrationtechniqueforthermalshear-stresssensorswithmeanow,"ExperimentsinFluids,vol.39,no.1,pp.56{65,2005. [87] D.T.Blackstock,\Fundamentalsofphysicalacoustics."NewYork,NY:Wiley,2000,ch.4. [88] R.Dieme,G.Bosman,T.Nishida,andM.Sheplak,\Sourcesofexcessnoiseinsiliconpiezoresistivemicrophones,"JournaloftheAcousticalSocietyofAmerica,vol.119,no.5,pp.2710{2720,2006. [89] J.S.Bendat,\Nonlinearsystemanalysisandidenticationfromrandomdata."NewYork:Wiley,1990,ch.8,9. [90] G.Bosman,\Universityoforida,privatecommunication,"2009. [91] R.F.Pierret,\Fieldeectdevices,"inModularseriesonsolidstatedevices;v.4,2nded.Reading,Mass.:Addison-WesleyPub.Co.,1990,ch.2. [92] R.C.Hibbeler,\Mechanicsofmaterials."UpperSaddleRiver,N.J.:PrenticeHall,1997,ch.12,pp.575{579. [93] V.Chandrasekaran,A.Cain,T.Nishida,andM.Sheplak,\Dynamiccalibrationtech-niqueforthermalshearstresssensorswithvariablemeanow,"inAerospaceSciencesMeetingandExhibit,38th,AIAA-2000-508,Reno,NV,2000. 220

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VijayChandrasekharangrewupinNagpur,acitylocatedrightinthecenterofIndia.Afterhighschool,hewenttotheNationalInstituteofTechnologyKarnataka(NITK),Indiawherehereceivedhisbachelor'sinmechanicalengineeringin2002.Afterworkingforayear,hestartedgraduateschoolintheDepartmentofMechanicalandAerospaceEngineeringattheUniversityofFloridain2003.HereceivedhisMSin2006andiscurrentlycompletinghisdoctoraldegreeattheInterdisciplinaryMicrosystemsGroupworkingundertheguidanceofProf.MarkSheplak.Stemmingfromhisinclinationtowardsproductdevelopment,Vijay'sresearchinterestsincludemicromachined(MEMS)sensorandactuatordesign,modeling,fabricationandcharacterization. 221