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Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.

Permanent Link: http://ufdc.ufl.edu/UFE0017526/00001

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Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.
Physical Description: Book
Language: english
Creator: Martin, David T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer 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 David T Martin.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Nishida, Toshikazu.
Local: Co-adviser: Sheplak, Mark.
Electronic Access: INACCESSIBLE UNTIL 2008-02-29

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0017526:00001

Permanent Link: http://ufdc.ufl.edu/UFE0017526/00001

Material Information

Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.
Physical Description: Book
Language: english
Creator: Martin, David T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer 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 David T Martin.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Nishida, Toshikazu.
Local: Co-adviser: Sheplak, Mark.
Electronic Access: INACCESSIBLE UNTIL 2008-02-29

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0017526:00001


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FinancialsupportforthisworkhasbeenprovidedbyaNationalScienceFoundationgrantandbySandiaNationalLaboratories.Ithankmyadvisors,ToshiNishidaandMarkSheplak,fortheirmanyhelpfultechnicaldiscussions,aswellastheircareerandpersonaladvice.Iamalsogratefultomycommitteemembers,LouCattafestaandRobFox,fortheirassistanceinthesuccessofthisproject.IamespeciallygratefultomymanycolleaguesintheInterdisciplinaryMi-crosystemsGroup.KarthikKadirvalandJianLiuworkedcloselywiththisprojectandeachaddedtheirinsightandhelpfulcontributions.IamgratefultoRobertDieme,BenGrin,StephenHorowitz,BrianHomeijer,andToddSchultz,fortheirassistanceintheexperimentalsetups.IthankBrandonBertolucciforhistechnicalassistance,aswellashisphotographicandartisticassistancealongwithTai-anChen.IamalsogratefultoallofthestudentsatIMGfortheirengagingtechnicaldiscussions,friendship,andcomradery.IamparticularlythankfulfortheexcellentmachiningworkperformedbyKenReedatTMRengineering.IalsoacknowledgeKeckPathammavong,ofEngent,Inc.,forhisskillinwirebondingthepackageddevices.OthertechnicalassistancewasprovidedattheUniversityofFloridabyProf.Ho-BunChanandhisstudentCoreyStamboughwiththesupercriticalrelease,AlOgdenwithpackaging,andStephenTedeschiinpreparingtheSEMimages.PeteLoeppert,fromKnowlesAcoustics,isacknowledgedforprovidingampliersusedaspartofthiswork.TheElectricalandComputerEngineeringdepartmentalstaisthankedfortheirkindassistance.Ithankmyparents,EdandNorineMartin,fortheirsupportandguidance.Theyinstilledinmeagoodworkethicandperseverance;withoutwhich,thisprojectwouldnothavebeenassuccessful.Aboveall,Iamgratefultomywife,Erica,forhernever-endingpatienceandsupport. iii

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page ACKNOWLEDGMENTS ............................. iii LISTOFTABLES ................................. vii LISTOFFIGURES ................................ ix ABSTRACT .................................... xv CHAPTER 1INTRODUCTION .............................. 1 1.1Motivation ................................ 1 1.2ResearchObjectives ........................... 6 1.3DissertationOverview ......................... 8 2BACKGROUND ............................... 9 2.1PrinciplesofMicrophoneOperation .................. 9 2.2TransductionMechanismsandScaling ................ 15 2.2.1IntroductiontoElectromechanicalTransducers ........ 15 2.2.2IntroductiontoPiezoelectricMicrophones ........... 17 2.2.3IntroductiontoPiezoresistiveMicrophones .......... 20 2.2.4IntroductiontoOpticalMicrophones ............. 23 2.2.5IntroductiontoCapacitiveMicrophones ............ 25 2.2.6ScalingSummary ........................ 29 2.3PreviousMEMSMicrophones ..................... 30 2.3.1LiteratureReviewofPiezoelectricMicrophones ........ 30 2.3.2LiteratureReviewofPiezoresistiveMicrophones ....... 34 2.3.3LiteratureReviewofOpticalMicrophones .......... 39 2.3.4LiteratureReviewofCapacitiveMicrophones ......... 42 2.3.5LiteratureReviewSummary .................. 55 3MICROPHONEMODELING ........................ 57 3.1Quasi-StaticModeling ......................... 57 3.1.1DiaphragmModel ........................ 58 3.1.2ElectrostaticModel ....................... 62 3.1.3Non-linearStaticElectromechanicalAnalysis ......... 89 3.2LumpedElementModeling ....................... 98 3.2.1Diaphragm ............................ 101 3.2.2MicrophoneStructure ...................... 104 3.2.3ElectrostaticTransduction ................... 106 3.2.4ElectrostaticCompliance .................... 114 3.2.5CompleteLumpedElementModel ............... 117 iv

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................ 120 3.2.7Quasi-StaticPull-In ....................... 128 3.3NoiseModel ............................... 133 3.3.1MicrophoneNoise ........................ 133 3.3.2InterfaceCircuitNoise ..................... 137 4DESIGNANDTHEORETICALPERFORMANCE ............ 141 4.1MicrophoneDesign ........................... 141 4.1.1MicrophoneStructure ...................... 141 4.1.2DiaphragmDesign ........................ 144 4.1.3BackplateDesign ........................ 145 4.1.4MicrophoneDesignSummary ................. 147 4.2PredictedMicrophonePerformance .................. 147 4.2.1Sensitivity ............................ 148 4.2.2FrequencyResponse ....................... 149 4.2.3NoiseFloor ........................... 152 4.2.4Pull-inVoltage .......................... 154 4.3Summary ................................ 155 5DEVICEFABRICATION .......................... 156 5.1ProcessFlow .............................. 156 5.1.1SUMMiTVProcessSteps ................... 156 5.1.2Post-SUMMiTVProcessSteps ................ 157 5.2MetallizationandWireBondingIssues ................ 161 6RESULTSANDDISCUSSION ....................... 163 6.1RealizedMEMSMicrophone ...................... 163 6.2MicrophonePackaging ......................... 166 6.2.1InterfaceCircuits ........................ 167 6.2.2PrintedCircuitBoard ...................... 168 6.2.3FinalPackage .......................... 169 6.3ExperimentalSetup ........................... 171 6.3.1AcousticExperimentalSetup .................. 171 6.3.2LaserVibrometer ........................ 176 6.3.3FaradayCage .......................... 177 6.4ExperimentalResults .......................... 178 6.4.1LinearityandTotalHarmonicDistortion ........... 178 6.4.2FrequencyResponse ....................... 185 6.4.3ResonantFrequency ....................... 188 6.4.4NoiseFloor ........................... 191 6.4.5Discussion ............................ 194 v

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.................. 196 7.1Conclusions ............................... 196 7.2RecommendationsforFutureWork .................. 197 7.3RecommendationsforFutureMicrophoneDesigns .......... 199 APPENDIX ALUMPEDELEMENTMODELOFACLAMPEDCIRCULARPLATE 201 A.1LumpedCompliance .......................... 201 A.2LumpedMass .............................. 203 BMICROPHONEFREQUENCYRESPONSE ................ 205 CUNCERTAINTYANALYSISOFMICROPHONEPERFORMANCE .. 209 C.1TheoreticalSensitivityUncertainty .................. 209 C.1.1MicrophonewithChargeAmplier .............. 209 C.1.2MicrophonewithVoltageAmplier .............. 211 C.2TheoreticalResonantFrequencyUncertainty ............. 213 C.3TheoreticalNoiseFloorUncertainty .................. 214 C.4ExperimentalSensitivityUncertainty ................. 214 DOVERVIEWOFTHESUMMiTVPROCESS ............... 216 BIOGRAPHICALSKETCH ............................ 234 vi

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Table page 1-1Comparisonofaudioandaeroacousticmicrophonespecications ..... 2 1-2SpecicationsofseveralBruelandKjrmicrophones ........... 3 1-3Designgoalsforanaeroacousticmicrophone ................ 7 2-1ScalingpropertiesofMEMSmicrophones. ................. 30 2-2SummaryofthespecicationsofpiezoelectricMEMSmicrophones. ... 32 2-3SummaryofpiezoresistiveMEMSmicrophones .............. 36 2-4SummaryofopticalMEMSmicrophones .................. 39 2-5SummaryofpreviouscapacitiveMEMSmicrophones. ........... 42 2-6ComparisonoftheBruelandKjrMEMSmicrophonetonon-MEMSBruelandKjrmicrophones ......................... 52 2-7ComparisonofpreviousaeroacousticMEMSmicrophonesandtheBruelandKjr4138traditionalcondensermicrophone. ............. 56 3-1Summaryoftheoreticallinearsensitivityofcondensermicrophones. ... 88 3-2Summaryoftheelectrostaticforceactingonthediaphragmofcapacitivemicrophones. ................................. 89 3-3Lumpedelementmodelingconjugatepowervariables. ........... 99 3-4Lumpedelementsforvariousenergydomains. ............... 99 3-5Expressionsfortheacousticlumpedelementsofthemicrophone. ..... 119 4-1Microphonephysicalproperties ....................... 148 4-2Acousticlumpedelementvaluesforthedesignedmicrophone. ...... 151 4-3Frequencyresponseparameters ....................... 151 4-4Summaryofspecicationsforthedesignedmicrophone .......... 155 6-1Summaryofthelinearityresultsforthemicrophoneswithvoltageampli-ers ...................................... 182 6-2Summaryofairgapandparasiticcapacitanceestimatesforthemicro-phonestestedwithvoltageampliers. .................... 183 6-3Summaryofthelinearityresultsforthemicrophoneswithchargeampli-ers ...................................... 185 vii

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................ 189 6-5Summaryofthenoisemeasurementresultsformicrophonestestedwithvoltageampliers. .............................. 192 6-6Summaryofthenoisemeasurementresultsformicrophonestestedwithchargeampliers. ............................... 193 6-7Summaryofthemeasurementresultsforallmicrophones ........ 194 6-8Minimumdetectablesignalexpressedinvariousequivalentunits. ..... 195 6-9Comparisonofthedesigneddual-backplatecapacitivemicrophonetotheB&K4138condensermicrophoneandpreviousaeroacousticMEMSmi-crophones. ................................... 195 D-1ProcessdataasreportedbySandiaNationalLaboratoriesfortheSUMMiTVprocess. .................................... 217 viii

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Figure page 1-1Relationshipbetweenthewavelengthoftheacousticpressure,microphonesize,anddiraction. ............................. 3 1-2Theimportantdimensionsforacousticarrays. .............. 5 2-1Schematicrepresentationofagenericmicrophone. ............ 9 2-2Illustrationoftheoperationofagenericmicrophone.Theincidentpres-surecausesadiaphragmdeectionwhichproducesanoutputvoltage. 10 2-3Typicalfrequencyresponseofanunder-dampedmicrophoneshowingthekeyfeatures. ................................. 11 2-4Thetypicalactual({)andideal(--)responseofamicrophonetovaryingincidentpressureamplitudes. ........................ 12 2-5Thetypicalpowerspectraldensityofthenoiseoorforamicrophone. 14 2-6Typicalcrosssectionsofpiezoelectricmicrophones.Thedetailsoftheelectrodegeometryandventchannelarenotshown. ............ 18 2-7Typicalcrosssectionsofpiezoresistivemicrophones. ............ 20 2-8Equivalentcircuitofapiezoresistivemicrophonewithfouractivepiezo-resistorsinaWheatstonebridgeconguration. .............. 22 2-9Typicalcrosssectionsofber-opticlevermicrophones. .......... 23 2-10Typicalcrosssectionsofsingle-backplatecapacitiveMEMSmicrophones. 26 2-11CrosssectionsofadierentialMEMScapacitivemicrophone. ....... 27 2-12TimelineshowingmilestonesinpiezoelectricMEMSmicrophonedevel-opment. .................................... 31 2-13TimelineshowingmilestonesinpiezoresistiveMEMSmicrophonedevel-opment. .................................... 35 2-14TimelineshowingmilestonesinopticalMEMSmicrophonedevelopment. 39 2-15TimelineshowingmilestonesincapacitiveMEMSmicrophonedevelop-ment. ..................................... 45 2-16SacricialmicromachiningprocessowusedbyScheeperetal. ...... 46 2-17CrosssectionofthemicrophonedevelopedbyKuhnelandHess ..... 48 2-18IntegratedcircuitryusedbyPedersenetal. ................. 50 ix

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...................................... 53 3-1Crosssectionofthedual-backplatecapacitivemicrophoneshowingthekeycomponents. ............................... 58 3-2Schematicoftheidealizedcirculardiaphragm. .............. 59 3-3Normalizeddeectionofaclampedcircularplate. ............ 61 3-4Non-lineardiaphragmdeectioncomparedtolineardeection. ..... 62 3-5Modelofatwoplateelectrostatictransducer. ............... 63 3-6Electricalmodelofasingle-backplatecondensermicrophonewithacon-stantvoltage. ................................. 68 3-7Simpliedcircuitofasingle-backplatecondensermicrophoneandachargeamplier. ................................... 71 3-8Electricalmodelofasingle-backplatecondensermicrophonewithvolt-agesourceappliedthroughalargeresistor. ................ 74 3-9Simpliedcircuitofasingle-backplatecondensermicrophoneandavolt-ageamplier. ................................. 75 3-10Circuitmodelofasingle-backplatemicrophoneandavoltageamplierwithparasitics. ................................ 76 3-11Dual-backplatecondensermicrophonewithdirectlyconnectedbiasvolt-ages. ..................................... 79 3-12Electricalmodelofadual-backplatecapacitivemicrophonewithachargeamplier. ................................... 81 3-13Dual-backplatecondensermicrophonebiasedwithvoltagesourcescon-nectedthroughalargeresistor. ....................... 83 3-14Simpliedcircuitofadual-backplatemicrophoneandavoltageampli-er. ...................................... 84 3-15Modelofthetopcapacitorwithanon-uniformairgap. ......... 90 3-16Capacitanceofthetopbackplateaspredictedbythenon-uniformgapmodelandtheparallelplatemodelasafunctionofdiaphragmdisplace-ment. ..................................... 92 3-17Capacitanceofthetopbackplateaspredictedbythenon-uniformgapmodelandeectiveareaapproximationmodel. .............. 94 x

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........................ 96 3-19Non-linearityofboththesingle-backplateanddual-backplatemicrophonesforvaryingvaluesofparasiticcapacitance ................. 98 3-20SymbolforthetransformerLEMelement. ................. 101 3-21Springandpistonmodelforadistributeddiaphragm. .......... 102 3-22Schematicdiaphragmofthedual-backplatemicrophoneshowinghowvariousfeaturesofthestructurearemodeled. ............... 104 3-23Cross-sectionofthesingle-backplatecapacitivemicrophoneshowingrele-vantparametersforthetransformerdiscussion.Theoutputiseitherthechargeorvoltageonthediaphragm. .................... 107 3-24Transformermodelingthetransductionfromtheacousticdomaintotheelectricaldomainforasinglecapacitorbiasedwithaconstantvoltage. 108 3-25Transformermodelingthetransductionfromtheacousticdomaintotheelectricaldomainforasinglecapacitorbiasedwithaconstantcharge. .. 109 3-26Cross-sectionofthedual-backplatecapacitivemicrophoneshowingrele-vantparametersforthetransformerdiscussion.Theoutputiseitherthechargeorvoltageonthediaphragm. .................... 110 3-27Transformermodelforthedual-backplatecapacitivemicrophonebiasedwithaconstantvoltage. ........................... 111 3-28Transformermodelforthedual-backplatecapacitivemicrophonebiasedwithaconstantcharge. ............................ 113 3-29Comparisonofthediaphragmrestoringforcetotheelectrostaticforce. 117 3-30Schematicdiaphragmofthedual-backplatemicrophoneshowinglumpedelementsincludedintheLEM. ....................... 118 3-31Lumpedelementmodelofthedual-backplatecondensermicrophone. .. 118 3-32Frequencyresponseofadual-backplatemicrophoneexampleintermsofpd=pinaspredictedbytheLEM. ....................... 121 3-33Lowfrequencyequivalentcircuitofthedual-backplatemicrophone. ... 122 3-34Highfrequencyequivalentcircuitofthedual-backplatemicrophone. .. 123 3-35Equivalentcircuitofthedual-backplatemicrophoneformid-rangefre-quencies. ................................... 124 xi

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.............................. 125 3-37Frequencyresponseofthenormalizedairgapdistanceshowingtheef-fectsofacompliantbackplate. ........................ 127 3-38Single-backplatecapacitivemicrophoneschematicshowingtherelevantforcesforquasi-staticpull-in. ........................ 129 3-39Acousticnoisemodelofthemicrophone. ................. 134 3-40AcousticnoisemodelforReff. ....................... 135 3-41AcousticnoisemodelforRv. ........................ 136 3-42Theoreticalnoisecontributionsofamicrophoneexamplereferredtothepressureacrossthediaphragm. ....................... 137 3-43Noisemodelofthechargeampliercoupledtothemicrophone. ..... 138 3-44Noisemodelofthevoltageampliercoupledtothemicrophone. .... 139 4-1Crosssectionofthedesigneddual-backplatemicrophone. ........ 142 4-2Microphone3-Dview. ............................ 142 4-3Detailsoftheanchorsandelectricalconnectionsareshown. ....... 143 4-4Simpliedmodelofthediaphragmandtopbackplate .......... 145 4-5Crosssectionoftheventchannel.Therearetwocomponents:oneisinparallelwiththetopbackplateandtheotherconnectstothecavity. .. 150 4-6Frequencyresponseofthedesigneddual-backplatemicrophoneaspre-dictedbytheLEM. .............................. 152 4-7Theoreticalnoisecontributionsofthemicrophonereferredtothepres-sureacrossthediaphragm. ......................... 153 4-8TheoreticaloutputvoltagenoisePSDofthemicrophonewithachargeamplier. ................................... 153 4-9TheoreticaloutputnoisePSDofthemicrophonewithavoltageampli-er. ...................................... 154 5-1ProcessstepsofthemicrophonefabricationthroughthecompletionoftheSUMMiTVprocess. ........................... 157 5-2FabricationstepsperformedafterthecompletionoftheSUMMiTVpro-cess. ...................................... 159 xii

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... 163 6-2Photographshowingthetopofthemicrophone. .............. 164 6-3SEMimageshowingthethreelayersofthemicrophone. .......... 165 6-4SEMimageofacross-sectionviewofanunreleasedmicrophonedie. ... 165 6-5SEMimageoftheelectricalconnectiontothediaphragm. ........ 166 6-6Schematicdiagramofthemicrophonepackage. ............... 166 6-7PhotographoftheSiSonicTMmicrophoneamplier. ............ 167 6-8Photographofthechargeampliercircuitboard. ............. 168 6-9Picturesoftheprintedcircuitboardusedinthemicrophonepackage. .. 169 6-10Photographofthemicrophoneembeddedintheprintedcircuitboard. .. 170 6-11Photographoftheassembledmicrophonepackage. ............. 170 6-12Largeplanewavetubeexperimentalsetup. ................ 172 6-13Theoreticalmagnituderesponseofthedual-backplatemicrophoneinair(-)andhelium(--). .............................. 173 6-14MagnituderesponseoftwoB&K4138condensermicrophonesinair. .. 173 6-15MagnituderesponseoftwoB&K4138condensermicrophonesinhelium. 174 6-16GraphicdescriptionofTHDmethodology. ................. 175 6-17Experimentalsetuptodeterminetheresonantfrequencyofthemicro-phone. ..................................... 176 6-18TypicalpressurerecordedbyreferencemicrophoneforLVmeasurement. 177 6-19Faradaycageexperimentalsetupfornoisemeasurements. ........ 178 6-20Outputvoltagevs.pressureforvoltageampliermicrophonesboundedbythetheoreticalsensitivityestimate. ................... 179 6-21Sensitivityvs.pressureforvoltageampliermicrophones. ........ 180 6-22Outputvoltagevs.pressureforvoltageampliermicrophonesbiasedwith2:0V. .................................... 180 6-23Sensitivityvs.pressureforvoltageampliermicrophonesbiasedwith2:0V. .................................... 181 6-24Theoreticalnon-linearityforadual-backplatecondensermicrophone. .. 181 xiii

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.................................... 182 6-26Outputvoltagevs.pressureforchargeampliermicrophones,boundedbythetheoreticalestimate. ......................... 184 6-27Sensitivityvs.pressureforchargeampliermicrophones. ......... 184 6-28Totalharmonicdistortionforchargeampliermicrophones. ....... 185 6-29Magnituderesponseforvoltageampliermicrophonesextendingto25kHz,boundedbythetheoreticalestimate. .................... 186 6-30Magnituderesponseforvoltageampliermicrophonesupto20kHz. .. 186 6-31Phaseresponseforvoltageampliermicrophones. ............. 187 6-32Magnituderesponseforchargeampliermicrophoneswithminimalripple. 187 6-33Phaseresponseforchargeampliermicrophoneswithminimalripple. .. 188 6-34Magnituderesponseforallchargeampliermicrophones,boundedbythetheoreticalestimate. ........................... 188 6-35Phaseresponseforallchargeampliermicrophones. ............ 189 6-36FFTofthevelocitymeasuredbythelaservibrometerresultingfromanacousticimpulse. ............................... 190 6-37MeasuredoutputPSDnoiseforthevoltageampliermicrophones. ... 191 6-38Inputreferrednoiseforthevoltageampliermicrophones. ........ 192 6-39MeasuredoutputPSDnoiseforthechargeampliermicrophones. .... 193 6-40Inputreferrednoiseforthechargeampliermicrophones. ......... 193 B-1Lumpedelementmodelofthedual-backplatemicrophoneshowingrele-vantimpedancesandvolumevelocities. .................. 205 C-1Illustrationofsensitivitydataanalysis. ................... 215 D-1CrosssectionoftheSUMMiTVprocess. .................. 216 xiv

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Amicrophoneisaninstrumentthatmeasuresanacousticsignalandgeneratesanelectricaloutput.Microphoneshavemanycommonapplicationsrangingfromuseincellularphonesandcomputerstohighqualitystudiomicrophonesformusicrecording.However,thereisalessfamiliarapplicationformicrophones:microphonesareutilizedbycommercialaircraftmanufacturerstoassistinthedevelopmentofquietaircraft.Communitiessurroundingairportsobjecttotheloudnoisesproducedbyapproachinganddepartingaircraft.Therefore,strictregulationsexisttolimitthenoiseradiatedbycommercialaircraft.Toreducethenoiseradiationofairframesandjetengines,aircraftmanufacturersperformrigoroustestingduringthedevelopmentandqualicationoftheirproducts.Themicrophonesusedforthesemeasurementshavespecicationsthatdiergreatlyfromacommonaudiomicrophone. Theindustryreliesonexpensivenon-MEMSmicrophonesforaeroacousticmeasurements.Todate,therehavebeenmanyMEMSmicrophonesdeveloped;someareevensuccessfulcommercialproducts.However,themajorityaretargetedforaudioapplications.TheexistingaeroacousticMEMSmicrophonesshowpromise;however,theperformancemustbeimprovedtocompetewithexistingnon-MEMSmicrophones.ThegoalofthisresearchistodesignaMEMS-based xv

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ThisstudydetailsathoroughreviewofpreviousMEMSmicrophonesandidentieswhicharemostsuitableforaeroacousticmeasurements.Furthermore,thespecicopportunitiesforimprovementarediscussed.Athoroughdevelopmentofthetheoryofoperationforcapacitivemicrophonesispresented.Usingthistheoreticalframework,thedesignofanaeroacousticcapacitiveMEMSmicrophoneispresented.ThemicrophoneisfabricatedusingtheSUMMiTVprocessatSandiaNationalLaboratories.MultiplemicrophonesaretestedandtheresultsindicatethedesignedmicrophonecomparesfavorablytopreviousaeroacousticMEMSmicrophones. xvi

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Inaneorttoreducetheimpactofairportsandairtravelonlocalcommuni-ties,theFederalAviationAdministration(FAA)hasregulatedthelevelofnoisethataircraftmayradiate.TheUSCodeofFederalRegulationsspeciesteststhatacommercialaircraftmustpassforitsairworthinesscertication.Requirementsarespeciedforthreegeneralclassesofaircraftandarebrokendownfurtherbyweight.Theregulationsforeachclassofaircraftspecifythemaximumallowableeectiveperceivednoiselevel(EPNL).TheEPNListhemeasurednoiselevelcorrectedforatmosphericconditions,thedurationofthesounds,andthespecicoperatingconditionsofthejetengine(s).Forexample,foranaircraftweighing617;200poundsormore,themoststringentrequirementlimitsthenoiseduringapproachto105EPNdB[ 1 ]. Tomeettheserequirements,thenoiseradiationofanaircraftmustbeconsid-eredduringitsdesign.Todesignquieteraircraft,itisimportanttolocalizeandunderstandthesourcesofnoisegeneration.Thebehaviorofairframesandjeten-ginescanbestudiedbyconductingmeasurementsonscalemodelsinawindtunnelwhereconditionsarewellcontrolled[ 2 ].Aeroacousticmeasurementsareperformedtoquantifythesoundeldandtoprovideinsightintonoisegenerationmechanismssothatthenoisecanbereducedtoacceptablelevels.Akeycomponentinanyaeroacousticmeasurementsetupisthemicrophone.Theperformancecharacter-isticsoftheselectedmicrophonegreatlyimpactsthesuccessofthemeasurementsandthequalityoftheresults.Someofthecharacteristicsofthemicrophonetoconsiderarethedynamicrange,sensitivity,bandwidth,stability,size,andcost[ 2 ]. 1

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Amicrophonemustmeetseveralrequirementstobesuitableforaeroacous-ticmeasurements.Table 1-1 summarizeshowtherequirementsforaeroacousticmeasurementsdierfromaudiomeasurements.Thespecicationsforaudiomicro-phonesaremainlydrivenbythecapabilitiesofthehumanear.Thiscomparisonisanexample;inpractice,themicrophonespecicationsaredrivenbyaparticularapplication. Table1-1.Comparisonofaudioandaeroacousticmicrophonespecications MaxPressure120dB160dBBandwidth20Hz-20kHz20Hz-100kHzNoiseFloor20dB26dBSizeO(5mm)O(500m) 2 ].However,themicrophonefrequencyresponseshouldextenddowntoatleast20Hzforgeneralcharacterization.Thereareseveralcommercialmicrophonescurrentlyavailablethatmeetsomeoftherequiredspecications[ 3 ].ThespecicationsofthesemicrophonesaresummarizedinTable 1-2 Thebandwidthrequirementshaveimplicationsforthesizeofthemicrophone.WhenthewavelengthoftheincidentpressureislargecomparedtothesizeofthemicrophoneasshowninFigure 1-1(a) ,thepresenceofthemicrophonedoesnot

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Table1-2.SpecicationsofseveralBruelandKjrmicrophones Specication419049394138 Diameter12:7mm(1=2in)6:35mm(1=4in)3:18mm(1=8in)MaxPressure148dB164dB168dBBandwidth3Hz-20kHz4Hz-100kHz6:5Hz-140kHzNoiseFloor17dB=p disturbthesoundeld.However,athigherfrequencies,thewavelength,,oftheacousticwavebecomessmalleraccordingtotherelation=c=f;wherecistheisentropicspeedofsound.Whenthewavelengthisontheorderofthesizeofthemicrophone,theincidentpressureisscattered(orreected)oofthemicrophone,andanon-uniformsoundpressureiscreatedonthediaphragm[ 4 ].Thiseect,knownasdiraction,isshownschematicallyinFigure 1-1(b) .Themicrophonemeasuresthesumoftheundisturbedpressureeldandtheadditionalpressuregeneratedbydiractionduetothepresenceofthemicrophone. Wavelengthlargerthanthesizeofthemicrophone:minimaldiraction (b) Wavelengthsmallerthanthesizeofthemicrophone:diraction 5 ].

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Thedegreetowhichthemicrophonedisturbsthesoundelddependsontheangleofincidenceofthesoundwave.Iftheangleofincidenceisknown,thenthemicrophonecanbedesignedtocompensateforthediraction[ 6 7 ].Therefore,inordertohavediractionfree-measurementsingeneral, wherekisthewavenumbergivenbyk=!=c,andaisthemicrophoneradius.Theisentropicspeedofsoundisc,and!istheangularfrequency.Foramaximumfrequencyof89:6kHz,kaisequalto1withamicrophoneradiusof600m.Therefore,themicrophoneradiusshouldbelessthan600m[ 4 ]. Thelowerlimitoftheinputdynamicrangeisdeterminedbythecombinednoiseofthemicrophoneandinterfacecircuitry,andtheoverallsensitivity.Itisdesirabletohaveloweroutputnoiseandahighsensitivityforalowminimumdetectablesignal.Inasensorsystem,thereareotherconsiderationssuchastheresolutionofthedataacquisitionsystem.Ifananalog-to-digitalconverter(ADC)isused,thedynamicrangeisapproximately6dBforeachbitofresolution[ 8 ].Thusa16bitADChasadynamicrangeofapproximately96dBanda12bitADChasadynamicrangeof72dB. Tolocalizethenoisesource,acousticarraysareoftenused.Anacousticarrayconsistsofalargenumberofmicrophonesarrangedinaspecicgeometry.Throughtheuseofbeamformingsignalprocessing,whereappropriateweightsanddelaysareappliedtoeachmicrophone,aselectivespatialresponsecanbeachieved.Thisallowstheacousticarraytoeectivelylistentoaparticularregioninspace[ 2 ].Duetothelargenumbersofmicrophonesutilizedintypicalacousticarrays,MEMSmicrophonesareattractiveduetothepotentialadvantagesofbatchfabrication.Thisenablesthepossibilityofagreatlyreducedcostpermicrophonecomparedtotraditionalnon-MEMSmicrophones;whichcanexceed$2000permicrophone[ 9 ].

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Amicrophoneshouldmeetadditionalspecicationsforuseinanacousticarray.ThephysicaldimensionsoftwomicrophonesinanacousticarrayareshowninFigure 1-2 .Toavoidspatialaliasinginregularlyspacedarrays,twoadjacentsensorsshouldbecloserthanonehalfofawavelength.Asseeninthegure,thedimensionsofthepackagemustbesmallenoughtomeetthemicrophonespacingrequirements.Therefore,toavoidaliasingat89:6kHz,themicrophonesshouldbeatmost1:9mmapart. Figure1-2.Theimportantdimensionsforacousticarrays. Anotherconsiderationforacousticarraysisthephaseandamplitudematchingbetweenmicrophones.Therelativephasebetweenchannelsiscrucialinformationinbeamformingalgorithms.Mismatchbetweenmicrophonechannelscancausesteeringvectorerrors.Thereforeitisadvantageoustohavemicrophonesthatarephaseandamplitudematched.Howeveritisimportanttonotethatevenwhenthemicrophonesarephasematched,otherpartsofthesignalpathmayalsohavephaseerror,suchastheinterfacecircuitryandADC[ 2 10 ]. Microphonecostisanothersignicantfactor,especiallybecauseofthelargenumberofsensorsusedinacousticarrays,typicallynumberinginthe100s[ 11 ].A

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reductioninthe\perchannel"costofanarraycaneithersignicantlyreducethetotalcostofthearray,orallowtheuseofadditionalsensors.Thespatialresolutionofthearrayisproportionaltothediameterofthearray;however,theminimummicrophonespacingdeterminestheupperfrequencylimit[ 2 ].Therefore,itisadvantageoustousealargenumberofmicrophonesinanacousticarray.Thus,themicrophoneshouldalsobedesignedsuchthatitsmanufacturingandpackagingresultinalowcost. Microelectromechanicalsystems(MEMS)technologyhasthepotentialtomeetalloftheaboverequirements.MEMSmicrophonesareconstructedusinglithog-raphybasedfabricationtechniquessimilartothoseusedtofabricateintegratedcircuits[ 12 ].Thereforeasmalldevicesizeiseasilyattainable.Obtainingphaseandamplitudematchingisalsopossible.Howeverthisalsodependsonspecicchar-acteristicsofthedevicefabricationandpackaging,suchasawellcontrolledventchannel[ 10 ].Inaddition,itiscost-eectivetofabricateMEMSdevicesinlargenumbers.Multiplewafersareprocessedineachlot,andhundredstothousandsofdevicesareoneachwafer.Therefore,MEMSsensorsfabricatedinsucientvolumehavethepotentialforagreatlyreducedcostovertraditionalsensors[ 9 ]. 1.1 andaresummarizedinTable 1-3 .ThepreviousresearchinMEMSmicrophonesisdiscussedindetailinChapter 2 .SeveralpreviousMEMSmicrophoneshavebeendesignedforaeroacousticapplications.ThemostnotabledevicesarethepiezoresistivemicrophonepresentedbyArnoldetal.[ 13 ],thepiezoelectricmicrophonedevelopedbyHorowitzetal.[ 14 ],andthecapacitivemicrophonedevelopedbyScheeperetal.[ 15 ].However,thebenchmark

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foraeroacousticmicrophonesistheconventional(non-MEMS)BruelandKjr4138condensermicrophone. Table1-3.Designgoalsforanaeroacousticmicrophone Maximumpressure160dBBandwidth20Hz-90kHzNoiseoor26dB=p 2.3 ,itisclearthatcapacitivemicrophonesingeneralhaveshownhighsensi-tivitiesandlownoiseoors.Singlebackplatemicrophoneshavelimitationswithregardstopull-ininstabilityandlinearity,furtherdiscussedinChapter 3 .How-ever,theadvantagesofnegativefeedbackcanbeleveragedwithadual-backplatecapacitivemicrophone;givingthepotentialforincreasedstability,bandwidth,andlinearity. Forthisdissertation,adual-backplatecapacitivemicrophonehasbeende-signed,fabricated,andcharacterized.ThemicrophonehasbeenfabricatedusingtheSUMMiTVprocessatSandiaNationalLaboratories[ 16 ].Adetailedlumpedelementmodelhasbeendevelopedtomodelthemicrophonedynamics.Ithasalsobeenextendedtodevelopanoisemodelofthemicrophoneandinterfacecircuitry.Atotalof10microphoneshavebeencharacterized;7withavoltageamplier,and3withachargeamplier.Thisworkhighlightsthedierencesbetweentheseampliertopologieswithregardstoalow-capacitanceMEMSmicrophone. Thecontributionsofthisworkareasfollows:

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1 introducedandmotivatedthetopicofthisdissertation.ThebackgroundinformationrelevanttothisworkisdiscussedinChapter 2 .Themodelingisdis-cussedinChapter 3 .Thisincludestheformulationofalumpedelementmodelforthedual-backplatemicrophonestructureaswellasanoisemodel.ThedesignandtheoreticalperformanceofthemicrophoneispresentedinChapter 4 .Chapter 5 de-scribesthedetailsofthedevicefabrication.TheexperimentalresultsarediscussedinChapter 6 .Finally,concludingremarksandsuggestionsforfutureworkaregiveninChapter 7 Supportingdetailsforthisdissertationareincludedinasetofappendices.Appendix A givesthedetailsofofthelumpedelementmodelforaclampedcircularplate.AdetailedderivationofthepredictedfrequencyresponseisgiveninAppendix B .TheuncertainlyanalysisforthetheoreticalmicrophoneperformanceispresentedinAppendix C .Finally,thedetailsoftheSUMMiTVprocessarediscussedinAppendix D

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Inthischapter,thebasicprinciplesofoperationforamicrophonearedis-cussed.Thekeyguresofmeritareexplainedanddened.Next,thecommonlyusedtransductionmechanismsforMEMSmicrophonesarepresented,includingthescalingrelationshipsbetweendevicesizeandperformance.Finally,areviewofpreviousMEMSmicrophonesisgiven. 2-1 showsaschematicrepresentationofagenericmicrophone.Theacousticenergyexistsintheformofanincidentpressurewave.Informationiscontainedintheamplitude,frequency,andphaseofthepressurewave. Figure2-1.Schematicrepresentationofagenericmicrophone. Mostmicrophonessharesomecommontraitswitheachother.Theyhaveadiaphragm,orcantileverbeam,thatisexposedtotheincidentsoundpressure.Thesoundpressureactsonthediaphragmandcausesittodeect,asshowninFigure 2-2 .Thedeectionisdetectedbyatransductionmechanismandtypicallyanelectricaloutputisgenerated.Microphonesalsohaveaventchanneltoprovidepressureequalizationtothecavity.Itwillbeshownlater,inSection 3.2 ,thatthisventchannelcausesthemicrophonetoonlyrespondtotimevaryingpressures.Thisdistinguishesamicrophonefromanabsolutepressuresensorthatcanmeasurestaticpressures. 9

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Figure2-2.Illustrationoftheoperationofagenericmicrophone.Theincidentpres-surecausesadiaphragmdeectionwhichproducesanoutputvoltage. Alinearmicrophonesubjecttoasinusoidalincidentpressure,Pin(!),withamplitudePinandfrequency!,hasanoutputofthefollowingform, whereVo(!)andPin(!)aretheFouriertransformsoftherespectivetimeseriessignals[ 17 ].ThetermH(!)isthefrequencyresponsefunctionandiswrittenintermsofitsmagnitude,jH(j!)j,andphase,\H(!),asfollows Themagnitudeofthetransferfunctiondescribesthesensitivity,S,ofthemicro-phoneasafunctionoffrequency.Similarly,thephaseofthefrequencyresponseisthephaseshift,,ofthemicrophoneasafunctionoffrequency. Inorderforthemicrophoneoutputtoaccuratelyrepresentthespectralcontentoftheacousticinput,itisnecessaryforamicrophonetohaveaatfrequencyresponse,suchthatthesensitivitydoesnotvarywithfrequency.Furthermore,theidealmicrophonehaszerophaseshift.Atypicalfrequencyresponseforanunder-dampedmicrophoneisgiveninFigure 2-3 .Therangeoffrequenciesforwhichthemagnituderesponseofthemicrophoneisattowithinagiventolerance,suchas3dB,givestheusablefrequencyrangeofthemicrophone.Thisfrequency

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rangeisknownastheatbandregionofthefrequencyresponseormoresimplythebandwidthofthemicrophone.Whenasensitivityofamicrophoneisreported,itisimpliedthatthisisthesensitivityintheatbandregion. Figure2-3.Typicalfrequencyresponseofanunder-dampedmicrophoneshowingthekeyfeatures. AlsoshowninFigure 2-3 isthecut-onfrequencyandresonantfrequency.Thelow-frequencyresponseofthemicrophoneisdominatedbytheventchannelandthecavityvolume.Thesetwoelementsofthemicrophonecreatearstorderhigh-passlterandacorrespondingcut-onfrequency.Belowthecut-onfrequency,themagnituderesponsehasaslopeof20dBperdecade.Itisalsopossibleforthelowfrequencyresponseofthemicrophonetobedominatedbytheinterfaceelectronics.Athighfrequencies,thefrequencyresponseofthemicrophoneisdominatedbytheresonantfrequencyandthedampingofthemicrophone.Thediaphragmhasamechanicalresonancethatisafunctionofitscomplianceandmass.Thedamping

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inthemicrophonestructuredeterminestheshapeofthefrequencyresponseneartheresonance.Anunder-dampedsystemwillhaveadistinctresonantpeakasshowninFigure 2-3 ,whileanover-dampedsystemwillnothavesuchapeak.However,inafree-eldmeasurement,scatteringeectsofthemicrophonestructuremayaecttheshapeofthefrequencyresponseneartheresonantfrequencypossiblyextendingthebandwidth.Amicrophonecanbedesignedforaknownacousticeld,i.e.pressure-eldorfree-eld,tomaximizethebandwidth[ 6 ].Apressure-eldiswherethesoundpressurehasthesamemagnitudeandphaseatanypoint.Conversely,afree-eldiswhereacousticwavespropagatefreely;typicallyplanewaveswithadeterminedpropagationdirectionareassumed[ 18 ].Abovetheresonantfrequency,thefrequencyresponsehasaslopeof40dBperdecade. Figure2-4.Thetypicalactual({)andideal(--)responseofamicrophonetovaryingincidentpressureamplitudes. Aswaspreviouslydiscussed,thefrequencyresponseofthemicrophonedescribeshowthesensitivityvarieswithfrequency,similarly,thelinearityofthemicrophonedescribeshowthemagnitudeofthemicrophoneoutputvarieswiththeamplitudeoftheincidentpressure.ShowninFigure 2-4 istheideal(--)andactual({)responseofatypicalmicrophonetoasingle-tonepressurewithavaryingamplitude.Ideally,theoutputvoltagevarieslinearlywiththeamplitudeoftheincidentpressure.However,inpractice,varioussourcesofnon-linearity

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limittheusefulmaximumpressure.Asseeninthegure,theactualresponseofamicrophonedeviatesfromtheideallinearresponseaboveamaximumpressure.Typically,themaximumpressureforwhichthemicrophoneisconsideredlinearisthepressureatwhichthenonlinearresponsediersfromtheideallinearresponsebymorethan3%.Sinceanon-linearresponseresultsindistortionsintheoutput,thenon-linearityofthemicrophonecanbeexpressedintermsofthetotalharmonicdistortion(THD)inthefrequencydomain.TheTHDisdenedastheratioofthetotalpowerinallthehigherharmonics(n>2)tothepowerinthefundamentalfrequencyasfollows[ 17 ] THD=1Pn=2p2(!n) Anotherparameterofinterestformicrophonesisthenoiseoor.Themi-crophonenoise,alongwiththenoisecontributionsfromtheinterfacecircuitry,denesthelowerendofthedynamicrangesinceitistheoutputofthemicrophonewhennoinputisapplied[ 19 ].Thislowerlimitoftheinputdynamicrangeistheminimumdetectablesignal(MDS). Noiseistypicallyexpressedintermsofapowerspectraldensity(PSD);forexample,ithastheunitsofV2=Hzforelectricalnoise.ThereforethetotalnoisepowerdependsonthePSDintegratedoverthebandwidthofinterest[ 19 ].AtypicalnoisePSDisshowninFigure 2-5 .Inthisexample,theoutputreferredvoltagenoiseofamicrophoneincludescontributionsfromthemicrophoneitselfaswellastheinterfaceelectronics. Systemsinthermodynamicequilibriumexhibitthermalnoiseproportionaltothedissipationpresentinthesystem[ 20 ].ThermalnoiseisalsoreferredtoaswhitenoisebecausethePSDisatforallfrequencies.Anadditionalsourceofnoiseisickernoise,ormorecommonlyreferredtoas1/fnoisebecausethenoisePSDisinverselyproportionaltofrequency.ThisnoisesourceisonlypresentwhenaDC

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Figure2-5.Thetypicalpowerspectraldensityofthenoiseoorforamicrophone. currentisowingandistypicallyseeninpiezoresistorsandinterfaceelectronics.Theickernoisetypicallydominatesatlowerfrequencies.ThecornerfrequencyisthefrequencyatwhichthenoisePSDoftheickernoiseequalsthethermalnoise[ 19 ].Thedynamicresponseofthemicrophonecanshapethecontributionofindividualnoisesources.Thismayresultintheatthermalnoiseofaresistorhavinganon-atspectralshapeattheoutputofthemicrophone. Aspreviouslystated,thetotalnoisepowerdependsonthemeasurementband-width.Therefore,itiscommonforthenoiseoorofamicrophonetobegivenforaparticularbandwidth.Forexample,thenoisecanbegivenataspecicfrequencyforanarrowbandwidth,suchasthenoiseat1kHzina1Hzbandwidth.Con-versely,thenoisecanbeintegratedoveraspeciedbandwidth,suchas20Hzto20kHzforaudiomicrophones.AnothercommonmetricforaudiomicrophonesistheA-weightednoise,denoteddBA.Here,thenoisespectrumispassedthroughalterapproximatingtheresponseofthehumanear,thenthenoisespectrumisintegratedandconvertedtodB[ 5 ].Thismetricisnotappropriateforaeroacousticmicrophones,sinceboththebandwidthofthislterandtheweightingarenotrelevant.Amoreusefulnoisegureofmeritforaeroacousticmicrophonesisthenoiseinanarrowbandwidth,suchas1Hzor1=3octave,ataparticularfrequency

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suchas1kHz,becausethemicrophonesignalsareoftensampledandanalyzedinthefrequencydomain.Thusthebandwidthfforthenoisepoweristhebinwidthinthefrequencydomain.Ideally,thenoisespectrumoverthefullmicrophonebandwidthisgiven. Therearesomegeneralscalingrelationshipsthatthenoiseoorofamicro-phoneexhibits.Forexample,astheresonantfrequencyandmaximumpressureincrease,thesensitivitytendstodecrease.Ingeneral,astheresonantfrequencyincreases,thenoiseoorwillincrease.Furthermore,asthemaximumpressureincreases,thenoiseoorwillalsoincrease.Thiswillinturnincreasetheinputrefereedpressurenoiseofthemicrophone.TheserelationshipswillbecomeevidentbystudyingpreviousMEMSmicrophonesinSection 2.3 2.2.6 .AdetailedderivationoftheoperationofcapacitivemicrophoneswillbegiveninChapter 3 .BeforediscussingthedetailsofthefourprincipletransductionschemesforMEMSmicrophones,thegeneralpropertiesofelectromechanicaltransducerswillbediscussed.

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conservativevs.non-conservative,reciprocalvs.non-reciprocal,anddirectvs.indirect[ 21 ]. Alineartransducerisdesirabletoensuremeasurementshavehighspectraldelity.Therearevariousnon-linearitiesthatexistintransducersthatcanlimitthedynamicrangeofthedevices.Forexample,thepressureinduceddiaphragmdeectioncanbecomenon-linearforlargedisplacements[ 22 ].Furthermore,thetransductionfromamechanicaldisplacementtoanelectricaloutputcanalsobenon-linear.Thisisthecaseforcapacitivetransducers.However,thetransducercanbelinearizedaboutapointforasmallregionofoperation.Forcapacitivetransducers,thiscanbeaccomplishedviaabiasvoltageorcharge[ 21 ]. Transducerscanalsobeenergyconserving,i.e.energyisnotlostduringthetransductionfromoneenergydomaintoanother.Examplesofenergy-conservingtransducersincludemagnetic,piezoelectric,andcapacitive.Opticalandpiezoresis-tivetransducersareexamplesofnon-energyconservingdevices. Areciprocaltransduceriscapableofbi-directionaloperationbetweentwoenergydomains.Forexample,areciprocaltransducerconvertsasignalfromoneenergydomaintotheelectricaldomain;furthermore,itconvertsasignalfromtheelectricaldomaintotherstenergydomain.Intherstmodeofoperation,thetransducerisoperatingasasensor;whileinthesecondmodeofoperation,thetransducerisoperatingasanactuator.Anelectrostaticmicrophoneisanexampleofareciprocaltransducer.Itcanoperateasamicrophoneandconvertanacousticsignaltoanelectricalsignal.Furthermore,itcanoperateasanactuatorandconvertanelectricalsignalintoanacousticsignal[ 21 ]. Anotherpropertyofelectromechanicaltransducersiswhetherthetransductionisdirectorindirect.Inadirecttransducer,thereisadirectrelationshipbetweenoneenergydomainandanother.Anexampleisanelectrodynamictransducer.Themotionofaconductorinamagneticeldresultsinaninducedvoltageasgivenby

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Lenz'Law[ 23 ].Conversely,acurrentthroughaconductorinamagneticeldwillresultinaforceontheconductor[ 23 ].Anelectrostatictransducer,however,isanindirecttransducer. TherearefourcommontransductionschemesusedforMEMSmicrophones.Thesearethepiezoelectric,piezoresistive,optical,andcapacitivetransductionmechanisms.ThesearediscussedindetailinsectionsSection 2.2.2 throughSection 2.2.5 24 ]. Thelinearpiezoelectricconstitutiveequationsexpressedintermsofthestressanddisplacementare[ 25 ] whereDiandEkaretheelectricdisplacementandelectriceldwhoseunitsare[C=m2]and[V=m],respectively.Similarly,TklandSijarethemechanicalstressandstrainwithunitsof[Pa]and[m=m],respectively.ThemechanicalcomplianceforaconstantelectriceldissEijkl,and"Tikistheelectricpermittivityforaconstantstress.Thepiezoelectriccoecient,dkp,quantiesthepiezoelectricresponseforagivenstrainorappliedelectriceld[ 24 ].Thesubscriptsdenotethecomponentofeachvariableinaparticulardirection.Inabendingmodepiezoelectricmicrophone,therelevantpiezoelectriccoecientisd31wheretheelectricdisplacementinthe

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`3'directionisrelatedtoamechanicalstraininthe`1'direction.Conversely,athicknessmodetransducerreliesonthed33piezoelectriccoecient. Therearetwobasiccongurationsofpiezoelectricmicrophonesthathavebeendevelopedinthepast:(1)deviceswiththindiaphragmsor(2)deviceswithacantileverbeamasillustratedinFigure 2-6 .Thetopgureshowsatypicalpiezoelectricmicrophonewithadiaphragm.Ontopofthisdiaphragmisastackconsistingofalowerelectrode,apiezoelectriclm,andatopelectrode.Thisisasimpliedcrosssectioninwhichinsulationlayersandthedetailsoftheelectrodegeometryarenotshown.Thebottomgureshowsacrosssectionofapiezoelectricmicrophonethatutilizesacantileverbeam.Thistypeofmicrophonehasasimilarelectrode/piezoelectricstack,howeveritislocatedattheclampedendofthebeambecausethestressesareconcentratedinthisregion. Diaphragmcongurationusedin[ 26 ] (b) Cantileverbeamcongurationusedin[ 27 ] Thereareavarietyofmaterialsthatcanbeusedastheactivepiezoelectricelement.Themostcommonlyusedpiezoelectricthinlmformicrophonesiszincoxide,ZnO.Othermaterialsthatcanbeusedareleadzirconatetitanate,

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PZT,aluminumnitride,AlN,andaromaticpolyurea[ 14 ].Therearevarioustradeoswhenselectingthepiezoelectricmaterialsuchasthemagnitudeofthepiezoelectriccoecient,thelmstability,relativepermittivity,andcompatibilitywithotherprocesses.Forexample,AlNisfullycompatiblewithacomplimentarymetaloxidesemiconductor(CMOS)processbuthasarelativelylowpiezoelectriccoecientcomparedtoPZT,whichcannotbeusedinaCMOSprocess[ 14 ].Anotherparameterofinterestforpiezoelectricmicrophonesisthegcoecient,whichconsidersthebehaviorofpiezoelectriccomposites[ 28 ]. Piezoelectricmicrophonescanbeoperatedinoneoftwoways:theycanbeoperatedinavoltagemodewheretheoutputvoltageisampliedbyavoltageamplier;ortheycanbeoperatedinachargemodewheretheoutputchargeisconvertedtoavoltagebyachargeamplier.Theadvantageofusingthechargemodeisthattheoverallsensitivityisnotaectedbyparasiticcapacitance,forexamplebychangingthecablelength[ 29 ]. WhendiscussingMEMSdevicesandcomparingtotraditionaldevices,itisimportanttostudyhowthedeviceperformancescalesasthedevicesizeisreduced.Thesensitivityisproportionaltothestressinthediaphragm,whichisproportionalto(a=h)2foraplate[ 30 ].AsseenfromEquation 2{5 ,thestresscreatesanelectricdisplacement.Thevoltageacrossthepiezoelectricelementisproportionaltothethicknessofthepiezoelectricmaterial,hpe.Therefore,iftheaspectratioofthediaphragmremainsxed,thestressinthediaphragmwillnotchangeasthedimensionsarereduced.However,ifthepiezoelectricthicknessisreduced,thesensitivitywillbelower;assumingthediaphragmstressisindependentofthepiezoelectricstress.Thebandwidthofthemicrophoneisdominatedbytheresonantfrequencyofthediaphragmwhichisproportionaltoh=a2,therefore,asthesizeofthemicrophoneisreduced,thebandwidthincreases[ 21 ];thisassumesthatthediaphragmismodeledasaplate.Themicrophonecontributestothe

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noiseoorduetointernalresistancewhichgivesrisetop 19 ],howeverdominantnoisesourcesforpiezoelectricdevicesaretypicallytheinterfacecircuitryandenvironmentalinterferencefromsourcessuchaspowerlines[ 29 ]. ij whereisthepiezoresistancetensorandTisthestresstensor.ThiseectwasrstobservedinsiliconbySmithin1954[ 31 ]. Junctionisolatedpiezoresistorsusedin[ 32 ] (b) Dielectricallyisolatedpiezoresistorsusedin[ 33 ]

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Thepiezoresistanceeectofsiliconcanbeleveragedtocreateamicrophone.Figure 2-7 showsacrosssectionoftypicalcongurationsforpiezoresistivemicro-phones.ThestructureofthepiezoresistivemicrophoneissimilartothatofthepiezoelectricmicrophoneinFigure 2-6(a) inthattheybothhaveadiaphragm,substrate,andacavity.Theuniquefeatureofapiezoresistivemicrophoneisthatithasoneormorepiezoresistorsthatarestressedwhenthediaphragmdeects.Thesearetypicallylocatedneartheedgeofthediaphragmbecausethestressesareconcentratedinthisregion.Astheincidentpressuredeectsthediaphragm,stressescausetheresistivityofthepiezoresistortochange.ThepiezoresistorscanbeembeddedinthediaphragmasinFigure 2-7(a) ;inthiscase,thepiezoresistorsareisolatedfromthediaphragmbyareversebiasedpnjunction.AnalternativeistodielectricallyisolatethepiezoresistorsfromthediaphragmasisshowninFigure 2-7(b) Thereareseveralwaystoarrangethepiezoresistors.ThemostcommonistousefouractiveresistorsconnectedinaWheatstonebridgeconguration,asshowninFigure 2-8 .Here,thefourresistorsallhavethesamenominalvalue,R.Theyarearrangedandsizedsuchthatwhenthediaphragmdeects,tworesistorsincreasebyRandtwodecreasebyR.Theoutputofthemicrophoneisadierentialvoltagethatisgivenby RVB:(2{7) Thesensitivityscalingofthepiezoresistivemicrophoneissimilartothatofapiezoelectricmicrophone.Thestressinthediaphragmisproportionalto(a=h)2[ 30 ].Thisstresscreatesachangeinresistancethroughthepiezoresistivetransductioncoecients.Thus,thesensitivitywillnotbereducedastheareaisreducedaslongastheaspectratioremainsthesame.Thesensitivityalsoscaleswithbiasvoltage,thusahighbiasvoltageisdesirable.However,themaximumbiasvoltageislimitedbypowerdissipation,heating,andelectro-migration[ 34 ].

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Figure2-8.Equivalentcircuitofapiezoresistivemicrophonewithfouractivepiezo-resistorsinaWheatstonebridgeconguration. Furthermore,increasingthebiasvoltageincreasesthe1=fnoiseofthemicrophone.Thethermalnoiseisproportionaltop 19 ].Thebandwidthofthemicrophoneisdominatedbytheresonantfrequencyofthediaphragm,whichscalesash=a2;thusasthediaphragmsizeisreduced,thebandwidthwillincrease[ 21 ]. Theperformanceofpiezoresistivemicrophonesisaectedbytemperature.Forexample,asthetemperatureincreases,thermalnoiseincreases.Furthermore,athighertemperatures,theleakagecurrentinjunctionisolateddevicesincreases.Astheoperatingtemperatureofthedevicevaries,thesensitivitycanexhibittemperaturedrift.Thecoecientsareafunctionoftemperature,whichimpactsthesensitivity.Howeverthiscanbecompensatedthroughtheuseofcircuitry[ 35 ]. Piezoresistivemicrophoneshavetheadvantageofnotbeingaectedbyparasiticcapacitance.Therelativelylowoutputresistancethatistypicalofpiezoresistivemicrophonesallowstheuseofinstrumentationampliers,asusedbyArnoldetal.[ 13 ],withoutgreatconcernfortheinputcapacitanceoftheamplier.Duetomismatchinthenominalvaluesofthepiezoresistors,thedierentialoutputofafull-bridgedeviceistypicallypassedthroughahighpasslterbeforeamplication.

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36 ]. Singleberbundleforincidentandreectedlight[ 37 ] (b) Twoseparatewaveguidesforincidentandreectedlight[ 38 ] TherearetwocongurationsofintensitymodulationbasedmicrophonesthathavebeenusedinMEMSbasedopticalmicrophones.Schematicsofber-opticlevercongurationsareshowninFigure 2-9 .Ineachcase,alightsourceisarrangedsuchthatitisincidentonadiaphragm.Asthediaphragmmoves,theamountofrecoveredlightismodulated.Figure 2-9(a) showsacongurationwheretheincidentandreectedlightareinthesameberbundle.Thesedevicesaretypicallycomprisedofaberbundlethatisusedinconjunctionwithamicrofabricateddiaphragm.MicrophonesofthetypeshowninFigure 2-9(b) haveseparatepathsfortheincidentandreectedlight.Theytypicallyhaveamicrofabricatedwaveguidethatisplacedincloseproximitytoadiaphragm.

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Therearesomeadvantagestousinganopticalmicrophone.First,electronicsarenotneededatthemeasurementlocation,thusopticalmicrophonesareinsen-sitivetoelectromagneticinterferenceanddonotemitelectromagneticradiation.Furthermore,opticalmicrophonescanbedeployedinharshenvironmentsthatarenotsuitableforelectronics[ 38 ]. However,thereareseveraldrawbackstotheopticalmicrophone.Opticalmicrophonesrequireanexternalreferencelightsource.Furthermore,theoutputvoltageofthemicrophonemaybesensitivetouctuationsinthereferencelightsource.Thus,averystablereferencelightsourceisnecessaryoradditionalcircuitrymustbeincludedtocompensatefoructuationsinthereferencelightsource.Thepackagingisdicultbecausethewaveguidesandthediaphragmmustbecarefullyaligned.Furthermore,thepackagingmustprotectthealignmentfromenvironmentalvibrations[ 39 ]. Toconverttheopticalsignaltoanelectricalsignal,optoelectronicsareanecessarycomponentofanopticalmicrophonesetup;typicallyaphotodiodeisused.Thephotodiodecanbeasignicantsourceofnoiseduetoshotnoise[ 19 ].Othernoisesourcesincludethermalradiationofthemembraneandopticalbers,aswellasrandompressureuctuationsonthediaphragm;however,thesenoisesourcesaretypicallynotdominantforopticalmicrophones[ 40 ]. Thesensitivityofanopticalmicrophoneisproportionaltothediaphragmdeection,ratherthanthebendingstressaswasthecaseforthepiezoelectricandpiezoresistivemicrophones.Thedeectionofaclampedcircularplateisproportionaltoa4=h3[ 30 ].Thiscanbefactoredintotwoterms,(a=h)2(A=h);whereAisthesurfaceareaofthediaphragm.Thersttermremainsconstantiftheaspectratioisxed,whilesecondtermwilldecreaseasthemicrophonedimensionsarereduced.Therefore,iftheaspectratioremainsconstant,thesensitivitywillbereducedasthedevicesizeisreduced.Thebandwidthwillbe

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limitedbytheresonantfrequencyofthediaphragm,whichscalesash=a2foraplate[ 21 ].Thus,thebandwidthwillincreaseastheradiusisreduced. 2-10(a) {Figure 2-10(f) .Capacitivemicrophonessharesomecommonfeatureswiththeprevioustypesofmicrophones,suchasadiaphragmandcavity,howeverthereareseveraldistinctfeaturesthatareuniquetocapacitivemicrophones.Inadditiontoadiaphragm,thecapacitivemicrophonealsohasaporousbackplateseparatedfromthediaphragmbyanairgap.Thebackplateholesallowtheincidentpressuretopassthroughthebackplateanddeectthediaphragm.Sucientbackplateholesmustbeusedorthebandwidthofthemicrophonewillsuerbecausethemicrophonewillbecomeover-damped.Thebackplateholesmayalsobeusedtotunethedampingtomaximizethebandwidthforfree-eldmicrophones[ 18 ]. TherearemanypossiblecongurationsforthebackplateinMEMScapacitivemicrophones.Forexample,thebackplatecanbeperforatedwithalargenumberofholesandbelocatedabovethediaphragmasshowninFigure 2-10(b) ,orthebackplatecanbelocatedbeneaththediaphragmandhaveasmallnumberofholesasshowninFigure 2-10(a) Thesingle-backplateMEMScapacitivemicrophonecanbemodiedbytheadditionofanotherplateasshowninFigure 2-11(a) .Thedual-backplatemicrophonehastwobackplates,oneoneithersideofadiaphragm.Historically,thistypeofdierentialelectrostatictransducerwasknownasapush-pulldevice[ 49 ].Theearliestusesofthistopologywasforelectrostaticloudspeakersdatingbacktoa1924GermanpatentissuedtoH.Riegger[ 50 ].

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Microphonewithabackplatecon-tainingasmallnumberoflargeholesbelowthediaphragm[ 41 ] (b) Microphonewithahighlyperforatedbackplateabovethediaphragm[ 42 ] (c) Microphonewithacorrugateddia-phragm[ 43 ] (d) Microphonewiththebackplatebe-lowthediaphragmandapatternedsubstrateabovethediaphragm[ 44 ]. (e) Microphonewithasolidbackplateandsmallcavity[ 45 ] (f) Microphonewithahighlyper-foratedbackplatebelowthediaphragm[ 46 ] ThistypeofdevicewasproposedforMEMSmicrophonesbyBayetal.[ 51 ]anddevelopedbyRombachetal.[ 47 52 53 ]andMartinetal.[ 54 ].Thedual-backplatemicrophonehasseveraladvantagesoverthesingle-backplatestructure.Ithasthepotentialforuptotwicethesensitivity,ahigherbiasvoltagefurtherincreasingthesensitivity,andincreasedlinearityassumingcomparablematerialsandgeometrytoacorrespondingsingle-backplatemicrophone.Thedual-backplatemicrophonecanalsobeoperatedinclosedloopwithsymmetricelectrostaticforcesactingonthediaphragm.Similarly,thedualdiaphragmmicrophonehasonebackplatewithadiaphragmoneitherside,asshowninFigure 2-11(b) .This

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Dual-backplatecapacitiveMEMSmicrophone[ 47 ]. (b) Dual-diaphragmcapacitiveMEMSmicrophone[ 48 ]. typeofmicrophonewasproposedbyBayetal.[ 48 ].Whilethisstructureisnotwellsuitedtoforcefeedbackapplications,itdoesoerthepotentialforincreasedsensitivityandlinearity.Thisstructurecanalsobehermeticallysealedtoreducetheimpactsoftheenvironmentonthedeviceperformance. Theplatesinacapacitivemicrophoneareconductive,thusoneortwocapac-itorsareformeddependingonthetypeofcapacitivemicrophone.Thecapacitorscanbeapproximatedbyaparallelplatecapacitor,whichhasacapacitanceof d;(2{8) whereAisthesurfacearea,0isthepermittivity,anddisthedistancebetweentheplates[ 23 ].Inair,thepermittivityisassumedtobethepermittivityoffreespaceinavacuum.Whenthemicrophoneisexposedtoanincidentsoundpressure,thediaphragmdeects.Thisdeectioncausesthemagnitudeofthecapacitancetochange.Varioustypesofinterfacecircuitrycanbeusedtodetectthecapacitancechange[ 12 ]. Therearetwogeneralclassesofcapacitivemicrophones:condenserandelectret.Condensermicrophonesarebiasedwithanexternalvoltagesource,whileelectretsarebiasedwithaxedpermanentcharge.Thexedchargeistypicallyimplantedintoathindielectriclayeronthebackplate[ 55 ].Electretmicrophoneshavetheadvantageofnotbeingsusceptibletoelectrostaticpull-in.However,the

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fabricationismoredicultbecauseastableembeddedchargemustbeproduced.Electretmicrophonesaretypicallyusedinlow-powerandportableapplicationssuchassoundlevelmeters[ 18 ]. Theperformanceofthecapacitivemicrophonedoesnotscaleasfavorablyassomeoftheothertypesofmicrophones.ThederivationofthebackgroundmaterialusedforthescalinganalysiswillbegiveninChapter 3 .Thesensitivitydependsonboththecomplianceofthediaphragmandtheelectriceldintheairgap[ 55 ].Therefore,thesensitivityisproportionaltotheelectriceld,VB=g,theaspectratioofthediaphragm,(a=h)2,andtheratioofthediaphragmareatothediaphragmthickness,(A=h).ThesurfaceareaofthediaphragmwithradiusaisgivenbyA,histhediaphragmthickness,VBisthebiasvoltage,andgisthegapthickness.Therefore,thesensitivitywillbereducedastheareaisreduced,eveniftheaspectratioiskeptconstant.Iftheelectriceld,VB=g,remainsconstant,thiscomponentofthesensitivitywillnotbeaectedbyscaling.However,thereisanupperlimittothebiasvoltagethatcanbeusedwithcapacitivemicrophonesduetoelectrostaticcollapseofthediaphragm.Thispull-involtage1isproportionaltog3=2[ 12 ].Thus,electriceldwillscaleasg1=2andwillbenegativelyaectedbyareductioninmicrophonesize. Anotherissueforcapacitivemicrophonesisthemagnitudeofthecapacitance.Asthedeviceisscaleddown,thecapacitancedecreases,thiscanleadtolossesduetoparasiticcapacitances.Inaddition,thekT=Cnoise,thetotalnoiseacrossthecapacitorintegratedforallfrequencies,willincrease[ 19 ].However,thisisnotasignicantissueifthemicrophoneisoperatedinasmallbandwidth;asisthecasewhenthemicrophoneissampledandanalyzedinthefrequencydomain.The 3.2.7

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bandwidth,however,hasthepotentialtoincreaseasthemicrophonedimensionsarereduced.Theresonantfrequencyisproportionaltoh=a2.However,theacousticresistanceofthebackplateholesisproportionalto1=g3and1=Nh,whereNhisthenumberofholes.Thus,theresistancewillincrease;ifitislargeenoughtoexcessivelydampthemicrophone,theoverallbandwidthcouldbereduced[ 21 ]. Thenoiseoorforcapacitivemicrophonescontainscontributionsfromthethermomechanicalnoiseofthesensorandnoisesourcesfromtheinterfaceelec-tronics[ 6 ].Incapacitivemicrophones,theventresistancehasthepotentialtodominatethelowfrequencynoise;especiallyinhigh-sensitivitydevices[ 18 ].Noisecontributionsfromtheinterfacecircuitalsodependonthetypeofinterfacecircuitchosen;thiswillbediscussedfurtherinSection 3.3 2-1 .Thescalingofthesensitivity,bandwidth,andthegain-bandwidth(GBW)productaregivenforpiezoelectric,piezoresistive,optical,andcapacitivemicrophones.TheGBWproductcomparisonillustrateshowtheoverallmicrophoneperformancescales.Assumingthediaphragmaspectratioandthepiezoelectricthicknesstodiaphragmthicknessratiobothremainconstant,thesensitivityofthepiezoelectricmicrophonewilldecreaseasthemicrophonedimensionsarereducedwhilethebandwidthwillincrease.TheGBWproductwillremainunchanged.Similarly,theGBWproductoftheopticalmicrophonewillalsoremainunchangedasthemicrophonedimensionsarereduced.Theoverallperformanceofthepiezoresistivemicrophonewillincreasewhiletheperformanceofthecapacitivemicrophonewilldecrease. Thereareseveralissuesasthesensorsbecomeverysmall.Asthediaphragmradiusisreduced,thethicknessmustbecomeverysmalltomaintaintheaspectra-tio;thiscouldposefabricationproblems.Additionally,asthediaphragmthickness

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isreduced,thethicknessofthepiezoresistorsbecomesverysmall.Thislimitshowtheminimumvalueofeachresistor,whichaectsnoiseperformance.Capacitivemicrophonesalsohaveadditionalissuesatsmallscales;thebackplateresistancecanbecomelarge,eectivelyloweringthebandwidth,andthecapacitanceofthedeviceisreducedwhichcancauselossesduetoparasiticcapacitances. Table2-1.ScalingpropertiesofMEMSmicrophones. Piezoelectrica2 a2hpe a2VB ha2 a2a2 ha2 a2VB 2-2 showsasummaryofpublishedpiezoelectricMEMSmicrophones.AtimelineofmilestonesinthedevelopmentofpiezoelectricMEMSmicrophonesisgiveninFigure 2-12 TherstmicrofabricatedmicrophonewaspresentedbyRoyeretal.[ 26 ]in1983.Thisdeviceuseda3mthicklayerofZnOsputteredontopofa30mthickcirculardiaphragmwitharadiusof1:5mm.Aluminumwasusedfortheelectrodes.Asensitivityof250V=Paandabandwidthfrom10Hzto10kHzwasreported.Thesensitivityvariedby5dBoverthisrange.Thenoiseoorwasmeasuredtobe73dBA.

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Figure2-12.TimelineshowingmilestonesinpiezoelectricMEMSmicrophonedevel-opment. TheBerkeleySensorandActuatorCenterhasbeenactiveinpiezoelectricMEMSmicrophoneresearch.TheirrsteortwaspublishedbyKimetal.[ 56 ]in1987.Thisdevicecomprisedofa2mthick2mm2mmsquarediaphragmwithaZnOpiezoelectriclm.Laterdeviceswerereportedin1989[ 57 ]and1991[ 58 ].BothofthesedevicesalsousedaZnOpiezoelectriclayeratopasiliconnitridediaphragm.However,forthesedevices,aCMOSvoltageamplierwasfabricatedonthesamedieasthemicrophones.Inaneorttoimprovetheperformanceovertherstdevice,animprovedprocesstocontrolthediaphragmstresswasused,aswellasalargerdiaphragmandactivearea.However,neitherofthesemicrophonesexhibitedaatfrequencyresponse,althoughthesecondgenerationmicrophonehasthelowestreportednoiseoorforpiezoelectricMEMSmicrophones. In1992,Schellinetal.[ 59 ]reportedapiezoelectricmicrophoneutilizingapolymerlmfortheactivematerial.Polyureawasusedforthepiezoelectricmaterialbecauseithasalargerpiezoelectriccoecientthanaluminumnitrideandzincoxide.Ad31piezoelectriccoecientof5:7pC=N7:0pC=Nwasreportedbytheauthors.Althoughamaterialwithahigherpiezoelectriccoecientwasused,thesensitivitywasmuchlowerthanthatreportedpreviouslybyKimetal[ 58 ].Furthermore,thebandwidthwasnotat;itvariedbyabout12dBoverthebandwidthofthemicrophone.

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Table2-2.SummaryofthespecicationsofpiezoelectricMEMSmicrophones. Royeretal.1983[ 26 ] 1:5mm*30m 250V=Pa 10Hz-10kHz(0:1Hz-10kHz) Kimetal.1987[ 56 ] 2mmy2m 0:5mV=Pa 20Hz-5kHz 57 ] 2mmy1:4m 80V=Pa 3kHz-30kHz 58 ] 3:04mmy2:0m 1000V=Pa 200Hz-16kHz 59 ] 0:8mmy1:0m 4000V=Pa-30V=Pa 100Hz-20kHz 60 ] 2:5mmy3:5m 920V=Pa 100Hz-18kHz 27 ] 2mmz4:5m 38mV=Pa 100Hz-890Hz 61 ] 2mmz1:5m 30mV=Pa 50Hz-1:8kHz 62 ] 3mmy3:0m 30V=Pa 1kHz-7:3kHz 63 ] 3mmy3:2m 520V=Pa 100Hz-3kHz 64 ] 1mmyN/R PZT 38mV=Pa 10Hz-20kHz 65 ] 0:3cm2area55m 2:2mV=Pa 10:5mV=Pa 28kHz 14 ] 900m*3:0m 0:75V=Pa Riedetal.publishedresultsforanotherpiezoelectricmicrophonein1993[ 60 ].Thismicrophonehasmuchbetterperformancethanthepreviouspiezoelectricmicrophones.Again,asquaresiliconnitridediaphragmwasusedwithZnO.Furtherimprovementstotheprocessweremadetocontrolthestressinthenitridelayer.ZnOwasusedforthepiezoelectricmaterial.Thismicrophonedemonstratedaatfrequencyresponsefrom100Hzuptoneartheresonantfrequencyof18:3kHz.In1996,Leeetal.[ 27 ]ofthesameresearchgroupreported

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workonacantilevermicrophone.ThisdevicehasacrosssectionasshowninFigure 2-6(b) .Animportantrequirementforacantilevermicrophoneisthatthestressesmustbecontrolledsothatthecantileverwillnotcurl.Toavoidcurling,alowstresssiliconnitridelayerissandwichedbetweentwolayersofsiliconnitridewith150MPaoftensilestress.Whilethisdevicehadahighsensitivityof38mV=Pa,thebandwidthwaslimitedto890Hz.Thisdevicecanalsobeusedasamicrospeaker;atresonance,asoundpressurelevelof100dBwasproducedintoacouplerwithavolumeof2cm3froma6Vpeak(Vp)inputat4:8kHz.AnimproveddevicewaspublishedbyLeeetal.in1998[ 61 ]withasensitivityof30mV=Paandabandwidthof1:8kHz. In2003,piezoelectricmicrophoneswerereportedbyKoetal.[ 62 ],andNiuandKim[ 63 ].ThedevicedevelopedbyKoetal.isapiezoelectricmicro-phone/microspeaker.Operatedasamicrophone,ithasalowsensitivityandfairlylowbandwidthextendingto7:3kHz.Whenoperatedasaspeaker,asoundpressureof284mPawasachievedatadistanceof1cmatthesecondresonantfre-quencyof13:3kHzwithadrivevoltageof15Vp.NiuandKimutilizedparylene-D,amaterialwithlowstressandstinesscomparedtosiliconnitride,inabimorphcongurationinanattempttocreateamicrophonewithahighsensitivity.Thedevicedisplayedasensitivityofabout520V=Paandabandwidthlimitedto3kHz.AnotherdevicewaspresentedbyZhaoetal.[ 64 ]in2003.ThismicrophoneutilizedPZTforthepiezoelectricmaterial.Asensitivityof38mV=Pawasachievedoverabandwidthextendingto20kHz. In2004,Hillenbrandetal.publishedresultsfortwopiezoelectricmicrophonedesignsthathaveseveralspecicationsthatareattractiveforaeroacousticmea-surements[ 65 ].Thisdeviceusesacellularpolypropylene(VHD40)lmforthepiezoelectriclayer.Deviceswerefabricatedwithasingle55mlmandvelmsconnectedinseries.Thesinglelmdeviceachievedadynamicrangeof37dBAto

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164dBwithatheoreticalresonantfrequencyof140kHz.Theauthors,however,didnotdiscussthepotentialaeroacousticapplicationsofthemicrophone. TheonlypiezoelectricMEMSmicrophonedevelopedtomeetthespecicationsforaeroacousticmeasurementswaspresentedbyHorowitzetal.in2005[ 14 ].ThisdeviceusesthepiezoelectricmaterialPZTatopasilicondiaphragm.Themeasureddynamicrangeextendsfrom47:8dB=p Therehasbeengoodprogressinthedevelopmentofpiezoelectricmicrophones.Earlydevicessueredfrompoorstresscontrolinthediaphragmswhichresultedinnon-atfrequencyresponses.Laterdevicesimprovedinthisrespect,however,somehadverysmallbandwidths.ThemicrophonedevelopedbyRiedetal.[ 60 ]wasthersttoshowaatfrequencyresponse.Forallofthemicrophones,excepttheworkofHorowitzetal.andHillenbrandetal.,thedynamicrangewasnotquantied.Duetoitssmallersize,Horowitzetal.reportthemostsuitablepiezoelectricmicrophoneforaeroacousticmeasurements.. 2-3 showsasummaryofthekeyspecicationsforpiezoresistiveMEMSmicrophones.AtimelineofmilestonesinthedevelopmentofpiezoresistiveMEMSmicrophonesisgiveninFigure 2-13 Therstuseofapiezoresistivematerialtocreateamicrophonewasin1957byBurns[ 76 ].Itwasconstructedofamacro-scalealuminumsquarediaphragmwithan8in:sidelength.Acantilevertransferredthediaphragmmotiontoapiezoresistivebimorphcantileverthatconsistsoftwo0:016in:thickslabsofn-type

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Figure2-13.TimelineshowingmilestonesinpiezoresistiveMEMSmicrophonedevelopment. germanium.WhilethisdevicewasnotaMEMSmicrophone,itdemonstratedthatsemiconductorpiezoresistivematerialscanbeusedtocreatetransducers. In1969,siliconwasleveragedtocreateaMEMSmicrophonewithsinglecrystallinep-typesiliconpiezoresistorsjunctionisolatedfromthen-typesubstrate[ 66 ].ThisdeviceusestwopiezoresistorsaspartofRC-oscillators.Astheresistancechangesduetotheincidentpressure,theoscillationfrequencyismodulated.Theauthorsnotedtheneedtopassivitythesurfaceofthedevicetoavoiddriftduetohumidity.Alayerofsiliconnitridewasusedforthispurpose. ApiezoresistiveMEMSmicrophonewaspresentedin1992bySchellinetal.[ 33 ].Thismicrophoneconsistsofasquarediaphragmwithfourp-typepolysiliconpiezoresistorsdielectricallyisolatedfromasilicondiaphragm.Severaldevicesweremadewithvaryingdopingconcentrations;theresistanceofthesedevicesvariedfrom300to21k.Withabiasvoltageof6V,thesensitivitywas4:2V=PaV.Thefrequencyresponsebetween100Hzand5kHzvariedby3dB,witharesonantfrequencyat10kHz. In1994,Kalvesenetal.[ 67 ]presentedamicrophoneformeasurementsinturbulentgasows.Thisdeviceusestwoactivep-typepolysiliconpiezoresistorsdielectricallyisolatedfromasquarepolysilicondiaphragmwitha100msidelength.Twoadditionalpolysiliconresistorsarecreatedon-dietocompletethe

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Table2-3.SummaryofpiezoresistiveMEMSmicrophones Peakeetal1967[ 66 ] N/Ry N/R N/R Schellinetal1992[ 33 ] 1:0mm*1m 100Hz-5kHz 67 ] 100m*0:4m 10Hz-10kHz(2mHz-1MHz) Kalvestinetal1995[ 68 69 ] 300m*0:4m 10Hz-10kHz(10Hz-0:9MHz) Schellinetal1995[ 32 ] 1mm*1:3m 70 71 ] 105my0:15m Naguibetal1999[ 72 73 ] 510m*0:4m :18V=PaV-1:0V=PaV 1kHz-5:5kHz 1kHz-5:5kHz 13 ] 500my1:0m Huangetal2002[ 74 ] 710m*0:38m 75 ] N/R1:0m 100Hz-8kHz Wheatstonebridge.Thisdevicehadahighnoiseoorof96dBA.Thisdevicewasalsotherstwithanintegratedcavityandventchannel.Resultsforasecondgen-erationdevicewerepresentedin1995[ 68 69 ].Thisdeviceagainusesdielectricallyisolatedpolysiliconpiezoresistors,howeverthediaphragmsizewasincreasedto300minsidelength.Whilethisdevicehasa6dBlowernoiseoorthantherstdevice,italsohasalowersensitivity.Infact,bothmicrophones,withsensitivitiesof0:09V=PaVand0:03V=PaVrespectively,havelowsensitivities.Thiswasattributedtothesmallcavitybeneaththediaphragmwhichacousticallysti-enedthediaphragm.Inaddition,theresonantfrequencyofthesedeviceswasnear1MHz.Thisistoohighforturbulentgasowmeasurements,whichtheauthors

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reportashavingnegligiblefrequencycontentabove10kHz.Alowerresonantfrequencythroughamorecompliantdiaphragmwouldincreasethesensitivity. In1995,Schellinetal.[ 32 ]reportedonasecondpiezoresistivemicrophone.Thisdevicehasfouractivep-typepiezoresistorscreatedbyimplantingann-wellwithboron.Thediaphragmthicknesswasincreasedto1:3mandwithabiasvoltageof8V,thesensitivitywas3:8V=PaV.Thenoiseoorofthemicrophoneis61dBA,whiletheupperupperlimitofthedynamicrangeis128dB.Theauthorsconcludedthatthesensitivitycouldbeincreasedbyreducingthein-planestressinthediaphragmandbyanimprovedpiezoresistordesign. TheInterdisciplinaryMicrosystemsGroupattheUniversityofFloridahasbeenactiveinthedevelopmentofaeroacousticmicrophones,includingpiezoresistivemicrophones.Theirrstdevice,publishedbySheplaketal.[ 70 71 ],hasa1500Athickcirculardiaphragmwith210mdiameter.Thedeviceusesfourp-typesingle-crystallinesiliconpiezoresistorsdielectricallyisolatedfromthediaphragm.Acompactintegratedwindingventchannelandcavitywasusedtogiveawelldenedcutonfrequency,whichisestimatedtobe100Hz.Thepredictedbandwidthex-tendsupto300kHz,anditisexperimentallyveriedtobeatupto6kHz.Thedevicehasadynamicrangeof92dB=p 13 ].Dierencesbetweenthismicrophoneandthepreviousmicrophoneincludealargerdiaphragm,lowerresistancepiezoresistors,andasiliconnitridepassivationlayertominimizedrift.Thisdeviceachieved40dBreductioninnoiseoor;however,ithasalowersensitivityandpredictedresonantfrequency(150kHz).ThismicrophonealsodemonstratesthepotentialforexcellentmatchingbetweendevicesthatMEMStechnologyoers;thesensitivityandphaseresponseofeightdevicesvariedby0:1dBand0:2,respectively. AnothergroupofresearchersfromMichiganStateUniversity,IllinoisInstituteofTechnology,andtheUniversityofMichiganhavebeencollaboratingonthe

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developmentofpiezoresistiveMEMSmicrophones.In1999,Naguibetal.[ 72 73 ]presentedresultsfortwomicrophonedesignswithtwodiaphragmsizes.Thesedeviceshavefourdielectricallyisolatedsingle-crystallinesiliconp-typepiezoresistors.Limitedresultsforthesemicrophoneswerepresented;theyhadasensitivityupto10V=Pa.AthirdmicrophonewasreportedbyHuangetal.[ 74 ]in2002.Thisdevicehasasimilargeometrytothepreviousmicrophone,howeveranimprovedprocesstofabricatethepiezoresistorswasusedtolowerthenoiseoor.Thisdeviceusesdielectricallyisolatedpoly-crystallinesiliconpiezoresistors.Thedynamicrangeisfrom54dB=p In2004,Lietal.[ 75 ]presentedapiezoresistivemicrophonewithelectronicsintegratedonthesamedie.Fourpolysiliconresistorswereplacedattheedgeofa1mthicksiliconnitridediaphragm.Thediaphragmareawasnotreported.Theampliedsensitivityis10V=PaVwithabiasvoltageof5Vandthenoiseoorisapproximately34dB=p Therehavebeenseveralpiezoresistivemicrophonedesignsthatmeetoneormoreoftherequirementsforaeroacousticmeasurements.Themicrophonepre-sentedbyArnoldetal.[ 13 ]isthebestpiezoresistivemicrophoneforaeroacousticmeasurementstodate.Ithassucientbandwidthanddynamicrange.Otherde-vices,suchastheworkbySheplaketal.[ 71 ]andHuangetal.[ 74 ]showedalinearresponseuptosoundpressurelevelsapproachingorexceeding160dB,howevertheysueredfromahighnoiseoororinsucientbandwidth,respectively.Earlypiezoresistivedevicesingeneralwereplaguedbyhighnoiseoors.However,morerecentdeviceshaveshownthatitispossibletofabricatepiezoresistivemicrophones

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withlowernoiseoors.Theseimprovementsaremainlyduetoimprovedresistorgeometrydesignandimprovedprocessows. 2-14 andthereportedspecicationsoftheopticalmicrophonesaregiveninTable 2-4 Figure2-14.TimelineshowingmilestonesinopticalMEMSmicrophonedevelop-ment. Table2-4.SummaryofopticalMEMSmicrophones Abeysingheetal2001[ 77 ] N/R 0:017V=Pa Kadirveletal2004[ 37 ] 500m*1:0m Leeetal2004[ 78 ] 100m*1:0m N/R Halletal2005[ 79 ] 2:1mmz3:3m N/R-4kHz 80 ] 800m*1:5m N/R-20kHz 81 ] 800mz5m N/R N/R-2kHz In1985(andagaininlateryears),Gartheetal.proposedusingmicromachin-ingtechnologyinthedevelopmentofintegratedopticalmicrophones[ 38 82 83 ].

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The1985devicehasanintegratedwaveguidechipfabricatedusingpolymethylmethacrylate(PMMA)placedincloseproximitytoadiaphragm,asshowninFig-ure 2-9(b) .Forthisrstdevice,ratherthanusingamembrane,amovablemirrorwasused.Themirror'spositionwascontrolledbyamicro-positioningdevice.Thecouplingratio,ameasureoftherecoveredlight,variedasexpectedversusthemir-ror'sdistance;however,therearenoresultsintermsofmicrophonespecications.Inaddition,resultswerepresentedforanon-MEMSopticalmicrophone.Thisdeviceutilizedabackplatewithavariableair-gaptocontrolthedamping.Thefrequencyresponseextendsfrom50Hzto18kHz.Anoiseoorof38dBAwasachievedwithareferenceopticalsignal. TheoreticalworkonanintegratedopticalmicrophonewaspublishedbyGreywall[ 84 ]in1999.AmicrophonestructuresimilartoFigure 2-9(a) wasdiscussed.Greywallfocusedonatheoreticalcomparisonofacondensermicrophoneandanopticalmicrophone.Theauthorconcludesthattheopticalmicrophonecanhaveasensitivitycomparabletothecondensermicrophone. In2001,Abeysingheetal.[ 77 ]presentedresultsforanopticalMEMSpressuresensor.ThisdevicehasastructuresimilartothatshowninFigure 2-9(a) .Thediaphragmissilicon,towhichaborosilicatemultimodeberwasanodicallybonded.Beforethebond,acavitywascreatedintheberbundlebyetchingthebercore.Thesensitivityofthismicrophoneis0:017V=Paandithasalinearresponseupto347kPa,or204dB.Thebandwidthandnoiseoorwerenotreported,however.Withitsverylowsensitivityandhighoperatingpressures,thepressuresensorisnotwellsuitedforuseasamicrophone. AnaeroacousticopticalMEMSmicrophonewaspresentedbyKadirveletal.[ 37 ]in2004.ThisdevicehasastructureasshowninFigure 2-9(a) .Themicrophoneconsistsofa1mthicksiliconnitridecirculardiaphragm,500minradius,thatisdepositedontopofasiliconsubstrate.Aluminumisdepositedon

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topofthesiliconnitridetoincreasethediaphragm'sreectivity.Aberbundlewaspositionedinthecavitybeneaththediaphragm.Thedeviceshowsanoverallsensitivityof0:5mV=Pa,anoiseoorof70dB=p In2004,Leeetal.[ 78 ]publishedresultsforanopticalmicrophonewithin-tegratedphotodetectors.Thisdevicehasa200mdiameter1mthickcircularaluminumdiaphragm.Thesensitivityofthismicrophoneis0:5A=Pa.AseconddeviceisreportedbyHalletal.in2005[ 79 ].Thisdeviceisunique,inthatafullyintegratedopticalmicrophoneisproposed.Averticalcavitysurfaceemittinglaser(VCSEL)isusedforthelightsource;howeverforthispaper,thelightsourcewasnotintegratedwiththemicrophone.Thelasershinesthroughasubstratethatcontainsaholeforthelasertopassthroughandthephotodetectors.Thephotodetectorsreceivethereectedlightfromadiaphragmonathirdsubstrate.Furthermore,adiractiongratingisfabricatedunderthediaphragm.Thefabri-cateddevicecontainedasquarediaphragmwithasidelengthof2:1mmandathickness3:3m.Althoughtheresonantfrequencywasmeasuredtobe44:8kHzinavacuum,thebandwidthislimitedto4kHzduetoacousticdamping.Thedia-phragmhasasensitivityof44A=PaandanA-weightednoiseoorof2:4102A.Thesensitivitiesandnoiseoorsforthesedeviceswerenotreportedintermsofanincidentpressure;rathertheyarereportedwithrespecttothediaphragmdisplacement. Alsoin2005,Bucaroetal.[ 80 ]discussedtheirworkonanopticalmicrophonedesign.Thedevicehasadiameterof1:6mmandathicknessof1:5m.TheMEMSdiaphragmisconstructedusingbulkmicromachiningonanSOIwaferwitha1:5mthickdevicelayer.Thedevicehasasensitivityofapproximately

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25mV=Paandaresonantfrequencyofapproximately20kHz.Thenoiseoorofthismicrophoneis30:6dB=p Songetal.reportedontheirworkonanopticalmicrophonein2005[ 81 85 ].Thisdeviceconsistsofa800um800m,5msquarereectiveplateandamulti-modeber.Thismicrophonehasausablebandwidthupto2kHz,howeverthedynamicrangeandnoiseoorwerenotinvestigated. AcommercialopticalmicrophoneisproducedbyOptoacoustics,Ltd.,foruseinMRImachines.TheopticalmicrophoneiswellsuitedforusenearMRIequipmentbecauseofthehighmagneticeldspresent.Themicrophonecanbedesignedwithoutanymetalcomponents.Thus,itisnotinuencedbythemagneticeld.Thisdeviceoperatesuptoafrequencya15kHzandasoundpressurelevelof140dB.Ithasasensitivityof1:5mV=Pa. 2-5 andatimelineofmilestonesinthedevelopmentofcapacitiveMEMSmicrophonesisgiveninFigure 2-15 Table2-5.SummaryofpreviouscapacitiveMEMSmicrophones. Hohmetal.1984[ 41 ] 8:0mm13m 100Hz-7:5kHz 86 87 ] 3:0mm2:5m 100Hz-15kHz 88 ] N/R1:5m 100Hz-15kHz 89 ] 0:8mm:25m 200Hz-20kHz 44 ] 2mm5m 13mV=Pa 500Hz-2kHz 6:1mV=Pa 100Hz-5kHz

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Table2-5.Continued 2mm8m 1:4mV=Pa 500Hz-20kHz 90 ] 2mm5:1m 91 ] 2mm1m 40Hz 92 ] 2mm1m 100Hz-10kHz 93 ] 0:8mm:25m 100Hz-20kHz 45 ] 500m1m N/R 0:4mV=Pa N/R-20kHz N/R 2mV=Pa N/R-7kHz N/R 3:5mV=Pa N/R-2:5kHz N/R 2:4mV=Pa N/R-10kHz 42 ] 1:8mm8m 300Hz-13kHz 94 ] 1:8mmN/R N/R N/R 5-10V N/R N/R 5-10V 43 95 ] 1mm1:2m 100Hz-9kHz 96 ] 2mm0:5m 100Hz-10kHz 97 ] 1mmz0:5m 200Hz-10kHz 98 ] 1:6mm0:9m 100Hz-15kHz 100Hz-15kHz 99 ] 2:2mm1:1m 234Hz=Pax 46 ] 2:6mm2m 100Hz-10kHz 100 ] 2:2mm1:1m 101 ] 0:4mmz0:75m 150Hz-10kHz

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Table2-5.Continued Torkkelietal.2000[ 102 ] 1mm0:8m 10Hz-12kHz 47 52 53 ] 2mm0:49m 1:5V 103 ] 1mm1:2m 100Hz-19kHz 104 ] 1:2mm0:4m 10pF 105 ] 1mm620nm 15 ] 1:95mmz0:5m 106 ] 320mN/R N/R 1pF 1:4kHz=Pax 100Hz-6kHz 107 ] (70m190m)0:4m 7:3mV=Pa 0:1Hz-100kHz 54 ] 0:23mmz2:0m 41 ]in1984.ThismicrophonehasaMylardiaphragmsuspendedaboveabackplatewithonelargehole,similartoFigure 2-10(a) .Thisdeviceusessilicondioxidedepositedonthebackplateforthechargedlayer,chargedtoabout350V;thechargedensitywasnotreported.Limitedcharacterizationindicatesabandwidthfrom100Hzto7kHzwithin3dB. Later,twootherelectretMEMSmicrophonesweredeveloped.In1989,Sprenkelsetal.[ 86 87 ]publishedresultsforanelectretmicrophone.ThegeometryofthisdeviceissimilartoFigure 2-10(a) ,however,thebackplatemakescontactwiththecenterofthediaphragm.Thisdeviceexhibitedaatbandwidthto15kHzwitha5%variationbetweendevices.Also,in1989,Murphyetal.[ 88 ]

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Figure2-15.TimelineshowingmilestonesincapacitiveMEMSmicrophonedevelop-ment. publishedresultsforanotherelectretmicrophonewitharesonantfrequencyofabout15kHz.Thechargedensitieswerenotreportedforeitherofthesetwomicrophones. TheearlycapacitiveMEMSmicrophones,whileutilizingmicromachiningtechnology,stillrequiredmanualassemblysteps.ThemicrophonedevelopedbyHohmetal.[ 89 ]in1989improvedonpreviousdesignsbyfabricatingthediaphragmandbackplateusingonlymicromachining.ThisdevicewastherstMEMScondensermicrophone.However,thebackplateanddiaphragmcomponentsstillwerejoinedtogetherbyhand.Twowaferswereusedforthefabricationofthediaphragmandbackplate,andthetwopartsweregluedtogether.ThestructureofthismicrophoneissimilartoFigure 2-10(d) ,withapatternedsubstrateabovethediaphragm.Whilethesedeviceswerefunctioningmicrophones,thepatternedsubstrateabovethediaphragmhasthepotentialtocausenegativescatteringeects.Itisbettertohaveaushmounteddiaphragm.Hohmetal.fabricatedmicrophoneswithvaryingdiaphragmstressbycontrollingbytheion-implantationdose.Forabiasvoltageof28V,thesensitivitiesrangedfrom0:2mV=Pato

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4:3mV=Paandthebandwidthsrangedfrom2kHzforthemostsensitivedevice,andto20kHzfortheleastsensitivedevice.Thesedeviceswerecharacterizedwithvoltageampliers. BergqvistandRudolf[ 44 ]publishedtherstcapacitivemicrophonetobefabricatedwithouthandassembly,usingonlymicromachining,in1990.Twowafersareusedforthefabricationofthediaphragmandbackplate,thentheyarejoinedtogetherwithananodicbond.Inthesameyear,aseconddevicewaspublished[ 90 ].BothofthesemicrophoneshaveastructuresimilartoFigure 2-10(d) andbothusedvoltageampliers.Theseconddeviceimprovedontheperformanceoftherstbyreducingtheairgap,thusincreasingthesensitivityandcapacitance,andincreasingthenumberofacousticholes,whichincreasedthebandwidth.ThiswastherstMEMScapacitivemicrophonewithahighlyperforatedbackplate. Figure2-16.SacricialmicromachiningprocessowusedbyScheeperetal.[ 91 ]. ThepreviousMEMScapacitivemicrophoneshavebeenfabricatedsuchthatthediaphragmandbackplateareformedonseparatedwafersandthenjoinedtogether.In1991,Scheeperetal.[ 91 ]presentedacondensermicrophonefabricatedusingsacricialmicromachining.Themicrophonefabrication,showninFigure 2-16

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usesaluminumasthesacriciallayerwhichsupportsthediaphragmandisthensubsequentlyetched.Whilethisdevicewasthersttousesacricialprocessingtofabricateacondensermicrophone,themicrophoneitselfhaslimitedusebecauseofthelimitedbandwidth.In1992,Scheeperetal.[ 92 ]improvedonthepreviousdesign.Thisdeviceagainusedsacricialmicromachining.Toimproveonthepreviousdevice,alargenumberofbackplateholeswereused;thestructureofthemicrophonenowresemblesFigure 2-10(b) .Furthermore,theairgapthicknesswasincreasedtoreducetheacousticdamping.Thismicrophonehadaatbandwidthoutto10kHz. Duetothenegativeaspectsofcapacitivemicrophonescaling,severalinnova-tivemicrophonegeometrieshavebeendeveloped.Theseeitherfocusonreducingtheacousticresistanceoftheairgap,orreducingthein-planestressofthedia-phragmtoincreasethesensitivityofthemicrophone. In1992,KuhnelandHess[ 93 ]publishedresultsforacondensermicrophonewithastructuredbackplate.Across-sectionofthisdeviceisshowninFigure 2-17 .Thismicrophoneachievedabandwidthupto20kHz.Groveswereplacedinthebackplatetoreducetheresistanceintheairgap.Therfore,inadditiontoreducingthebackplateresistancebyincreasingthenumberofholes[ 92 ],thisdevicedemonstratesthatastructuredbackplatecanalsobeusedtoreducethebackplateresistance.However,thestructuredbackplateresultsinanon-uniformairgap;thiswillaecttheelectrostaticbehaviorofthemicrophone.Howevertheauthorsdidnotdiscussthisbehavior. Atechniqueusedtoincreasethediaphragmcompliance,andthusthesen-sitivity,istorelievediaphragmstressusingacorrugateddiaphragm.Scheeperetal.[ 108 ]producedtherstMEMScorrugateddiaphragmin1994.TherstmicrophonetoutilizeacorrugateddiaphragmwaspublishedbyZouetal.[ 43 95 ]

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Figure2-17.CrosssectionofthemicrophonedevelopedbyKuhnelandHess[ 93 ]. in1996.CorrugatedmicrophoneshaveacrosssectionsimilartothatinFigure 2-10(c) .Thisdevicehasa0:4mthickdiaphragmwith8mdeepcorrugations. OtherdevicesthatusedacorrugateddiaphragmincludetheworkofCunning-hamandBernstein[ 97 ]in1997,Lietal.[ 103 ]in2001,andKressmanetal.[ 105 ]in2002.ThemicrophonedesignedbyCunninghamandBernsteinhasa0:5mthickdiaphragmwith1mdeepcorrugations,whilethedevicedesignedbyLietal.hasa1:2mthickdiaphragmwithasingle300mdeepcorrugation.ThedevicereportedbyKressmanetal.hasa0:62mthickdiaphragmwith1:2mdeepcorrugations.Thisdeviceisalsotherstcorrugatedelectretmicrophone.Whilethecorrugationsincreasethecomplianceofthediaphragm,whichincreasesthesensitivity;thefabricationprocessbecomesmorecomplex. AseriesofmicrophoneswithvariousdiaphragmareasandairgapthicknesseswerefabricatedbyBourouinaetal.[ 45 ].Thisworkdemonstratesthescalingattributesofcapacitivemicrophones.ThesemicrophonesallhavethegeometryshowninFigure 2-10(e) withasolidbackplateandasmallcavity.Theedgelengthofthesquarediaphragmrangedfrom500mto1mandtheairgapthicknessrangedfrom5mto7:5m.Fromtheirresults,itisclearlyseenthatincreasingthediaphragmsizeincreasessensitivityanddecreasesbandwidth.Inaddition,increasingtheairgapthicknesslowersthesensitivityandincreasesthebandwidth. In1994,BergqvistandGobet[ 42 ]reportedonacapacitivemicrophonefabricatedusingsurfacemicromachiningandelectroplating.Tofabricatethedevice,

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rstasacriciallayerofphotoresistisdeposited.Copperisthenelectroplatedontopoftheresisttoformthebackplate.Thesiliconwaferisetchedfromthebackside;theetchistimedsuchthat8mofthewaferremainswhentheetchiscompleted.Thissiliconlayerformsthediaphragm.ThisdevicehasacrosssectionsimilartoFigure 2-10(b) .Boththecapacitanceandsensitivitywerelowerthanexpected.Thiswasattributedtothebucklingofthebackplatewhichincreasedtheairgaptoabout5m. Therstcapacitivemicrophonewithon-chipcircuitrywaspresentedbyBernsteinandBorenstein[ 94 ]in1996.Thisdeviceutilizedelectroplatingtofabricatethebackplate,similartotheprocessowusedbyBergqvistandGobet[ 42 ].Theon-chipinterfacecircuitrywasaJFETbuerwithaneectiveinputcapacitanceof0:5pF. Pedersenetal.publishedresultsforseveralmicrophoneswithon-chipinter-facecircuitry.Theirrstdevice,publishedin1997[ 98 ],didnotincludeon-chipcircuitry.However,thismicrophonewasthebasisfortheirfuturework.ItusedpolyimideforboththediaphragmandbackplateandthemicrophonegeometryissimilartothatofFigure 2-10(b) .Ametallizationconsistingofchromiumandgoldwasusedtoproduceanelectricallyconductivediaphragmandbackplate.Thisdeviceexhibitedabandwidthupto15kHzandanoiseoorof34dB.In1998,Pedersenetal.reportedonanewmicrophonewithintegratedelectronics.Themicrophonegeometrywassimilartothepreviousdesign,howeverithaslargerdimensions.AschematicoftheinterfacecircuitisshowninFigure 2-18 .Themi-crophoneisavariablecapacitorinanoscillator;asthecapacitancevaluechanges,thefrequencyofoscillationchanges.Whilethisdeviceusedauniquedetectionscheme,ithadahighnoiselevelforanaudiomicrophoneof60dBA. Pedersenetal.[ 100 ],in1998,improvedtheperformanceoftheirpreviousmicrophonedesign[ 99 ]byintegratingthesamemicrophonewithadierent

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Figure2-18.IntegratedcircuitryusedbyPedersenetal.[ 99 ]. interfacecircuitthanpreviouslyused.Forthisdevice,anintegratedDC-DCconvertergeneratesthebiasvoltageforthemicrophoneandthemicrophoneoutputisbueredbyanintegratedsourcefollower.ThisdevicehadthelowestnoiseoorofthethreemicrophonesdevelopedbyPedersenetal.;itwas27dBA. Schaferetal.[ 101 ],ofKnowlesAcoustics,developedacondensermicrophoneforhearingaidusein1998.Thisdeviceisuniqueinthatitusesadiaphragmthatissupportedinthemiddle,ratherthanclampedattheperimeter.Thisyieldsacompliancenearlyvetimeslargerthananedgeclampeddiaphragmofthesamedimensions.Corrugationsareusedtorelievethein-planestressofthediaphragm.ThismicrophonealsofeaturesanintegratedchargepumptogeneratethebiasvoltageaswellasanintegratedCMOSbuer.Thesensitivityofthemicrophoneis14mV=Paforabiasvoltageof12V.Thebandwidthisatfromabout150Hzto10kHzandtheresonantfrequencyisaround17kHz.TheA-weightednoiseisaslowas28dBA.Thisdeviceusedauniquediaphragmarrangementandiswellsuitedforahearingaid. KnowlesAcousticsproducesseveralcommercialMEMScapacitivemicro-phones.Ratherthattheon-chipbuerdescribedinthe1988paper[ 101 ],thesearehybridpackagedwitho-chipvoltagebuersforaloweroverallmanufacturingcost.Congurationsareavailablewitheitherunitygainoragainof20dB.AnexampleistheSP0101[ 109 ],whichhasasensitivityof7:9mV=Pa,abandwidthof10kHz,

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andanoiseoorof35dBA.Allofthesemicrophones,with10%distortionat115dB,areforhighvolume,lowcostaudioapplications.ThecommercialSiSonicmicrophoneswerealsoreportedin2006byLeoppertandLee[ 110 ]. In2000,Torkkelietal.[ 102 ]publishedresultsforacondensermicrophonethatusedpolysilicondopedwithboronatalevelof1019cm3forboththediaphragmandthebackplate.Theannealingtemperaturewasadjustedforeachlayertoachievethedesiredstress.Thebandwidthisatfrom10Hzto12kHz,andthenoiseooris33:5dBA.Initially,thesensitivitywaslowerandthenoiseoorwashigher;anincreaseinthecavityvolumefrom0:8mm3to110mm3improvedtheseparameters.Thisdeviceusedachargeamplierfortheinterfacecircuitry. ThemajorityofthepreviousMEMScapacitivemicrophoneshavebeende-signedforaudioapplications,e.g.hearingaids.In2003,Scheeperetal.[ 15 ],ofBruelandKjr,developedaMEMS-basedmeasurementmicrophonethatcanbeusedforsoundpressurelevelswellinexcessof120dB.Thisdevicehasa0:5mthickdiaphragmwitharadiusof1:95mm;theoctagonaldiaphragmisapproxi-matedasbeingcircular.Thediaphragmisjoinedtothebackplatewaferwithanairgapof20m.Thismicrophoneispackagedinarobustmetalshell,however,duetothegeometryandfabricationofthemicrophone,thediaphragmisnotushwiththetopsurfaceofthesensor.ThecrosssectionissimilartoFigure 2-10(d) ,howeverthebacksideisenclosedbythepackage.Withabiasvoltageof200V,asensitivityof22mV=Pawasachieved.Themicrophonehasabandwidthupto20kHz;thelowerlimitofthebandwidthwasnotreported,howeverthefre-quencyresponseextendstoaslowas251Hz.Themostdistinguishingfeatureofthemicrophoneisitsdynamicrange,whichextendsfrom23dBAto141dB.AcomparisonofthisdevicetootherBruelandKjrmicrophonesisgiveninTable 2-6

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Table2-6.ComparisonoftheBruelandKjrMEMSmicrophonetonon-MEMSBruelandKjrmicrophones Diameter12:7mm(1=2in)6:35mm(1=4in)3:18mm(1=8in)3:9mmMaxPressure160dB164dB168dB141dBBandwidth4Hz-20kHz4Hz-100kHz6:5Hz-140kHz251-20kHzNoiseFloor18dBA29:5dBA55dBA23dBA 106 ]developedamicrophonefabricatedusingastandardCMOSprocess.Thediaphragmwasfabricatedusinganarrayofsix320m320msquarediaphragms.Eachdiaphragmconsistsofawindingpatternofmetalandoxidetocreateameshmembrane.Thesubstrateisusedasthebackplate.ThegeometryofeachmicrophoneissimilartoFigure 2-10(a) .Afrequencymodulationtypeofinterfacecircuitryisusedwherethemicrophoneisthevariablecapacitorinanoscillator.Thesignalistransmittedo-chipviaanFMsignalandisrecoveredwithanFMreceiver.Thesensitivityis1:4kHz=Paandthebandwidthwasattowithin3dBovertherangefrom100Hzto6kHz.Boththebandwidthandthesensitivityofthisdevicewerenotclosetothepredictedvalues.ThiswasprimarilyduetotheuncertaintiesofthemechanicalpropertiesinaCMOSprocess. Hansenetal.[ 107 ]publishedresultsforawidebandwidthcapacitivemicro-phonein2004.ThisisauniquedevicethatisbasedonaRFdetectionscheme.Ratherthanusingthetypicalsingle-backplatestructure,asmallsealedstructurewasusedasshowninFigure 2-10(e) ,withnovent.Adiaphragmwassuspendedoverthesubstratecreatingasmallsealedvolume.Alargenumberofthesede-viceswereconnectedinalinetoformatransmissionline.Asthecapacitancechanged,thephasespeedofthetransmissionlinechanged.Thedevicestructure

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andequivalentcircuitareshowninFigure 2-19 .Each0:4mthickdiaphragmis70m190minarea;45ofthedevicesareconnectedinaRFtransmissionlinewithatotalactivecapacitanceof3:56pF.Asensitivityof7:3mV=Pawasachievedoverabandwidthof0:1Hzto100kHz.Onedrawbackofthisdeviceisitsrelativelyhighnoiseoorat63:6dBA.Thisdevicehastheadvantageofbeingsealedtoprotectitfromtheenvironment.Thelargebandwidthofthismicrophoneiswellsuitedtoaeroacousticmeasurements.However,noinformationisgivenregardingthethemicrophone'smaximumpressureorlinearity. Figure2-19.SchematicandequivalentcircuitofthemicrophonedesignedbyHansenetal.[ 107 ]. In2006,Pedersen[ 111 ]presentedresultsforacapacitivemicrophonedesignedforhighfrequencyapplications.ThismicrophoneisderivedfromtheKnowlesSiSonicTMmicrophone;howeverthedesignismodiedtoextendthebandwidth.Themicrophonehasasensitivityof398V=Paandaninputreferrednoiseoorof22dB=p

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2-11 showstwopossiblecongurationsforadierentialcapacitivemicrophone.Therehasbeenalimitedamountofpreviousresearchintothesetypesofcondensermicrophones. In1996,Bayetal.[ 48 ]investigatedadual-diaphragmcapacitivemicrophone.Atheoreticalanalysisofthedual-diaphragmmicrophonewasdiscussed.Inaddi-tion,asuggestedprocessowwasgiven.Accordingtotheauthors,theadvantagesofthistypeofmicrophoneareanincreasedsensitivityandanimmunitytodustandmoisturesinceitishermeticallysealed.In1999,Bayetal.[ 51 ]publishedatheoreticaldiscussiononadual-backplatemicrophone.Duetoseveralproblemsthatwereforeseenwiththedoublediaphragmmicrophone,suchascomplicatedprocessingandsensitivitytobarometricpressure,theauthorsfocusedtheiref-fortsonadual-backplatemicrophone.Thedependenceofthesensitivityofthemicrophonetobiasvoltage,in-planediaphragmstress,diaphragmthickness,andairgapthicknesswereinvestigated.Forimplementationofforce-feedback,theuseofasigma-deltaconverterwassuggestedasitprovidesadigitaloutput;howeveracomparisonofforcefeedbackinterfacecircuittopologieswasnotgiven.WhilethesetwostudiesbyBayetal.providesomeusefulinsightintotheadvantagesanddisadvantagesofdierentialcapacitivemicrophones,actualdeviceshaveyettobefabricated. Therstdual-backplatemicrophonetobefabricatedandsuccessfullytestedwaspresentedbyRombachetal.[ 47 52 53 ]in1999.Thelowerbackplateisacompositeconsistingofa0:85mthicklayerofsiliconnitrideand0:4mthicklayerofborondopedpolysilicon.Thelowerbackplatehasatotalstressof180MPa.A0:9mlayerofsilicondioxideisdepositedastherstsacriciallayer.Thenastackof0:045mofsiliconnitride,0:4mofpolysilicon,and

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0:045mofsiliconnitridearedepositedtocreateadiaphragmwithastressof45MPa.Asecond0:9mlayerofsilicondioxideisdepositedtoformthesecondsacriciallayer.Thetopbackplateconsistsof3mofpolysiliconwithastressof220MPa.Thetopbackplateispatternedfromthetopofthewaferusingtraditionalphotolithography.Thebottombackplateispatternedafteralloftheupperlayershavebeendepositedbyetchingfromthebackside.Thisremovestheneedforchemicalmechanicalpolishingbecausetherearenofeaturesinthelowerbackplatewhenthediaphragmisdeposited.Thediaphragmisasquarewitha2mmsidelength.Thecharacterizationwasconductedonthewafer-scalewiththemicrophonesactuatedfromthebacksideofthewafer.Alownoisevoltageamplierwasusedfortheinterfacecircuitry.Withabiasvoltageof1:5V,asensitivityof13mV=Pawasachieved.Thefrequencyresponseisatfrom10Hztoabout20kHz.Thenoiseooris22:5dBAandtheupperlimitofthedynamicrangeis118dB.Thisdevicewastherstsuccessfuldual-backplatemicrophone.Itsperformanceiswellmatchedtoaudioapplications,howeverfurtherdevelopmentisneededtoproduceadevicesuitableforaeroacousticmeasurements. 2-7 summarizestheresultsforthehighestperformingaeroacousticmicrophonesreportedtodateforeachtransductionscheme(excludingthedual-backplateaeroacousticmicrophone). ThepiezoelectricmicrophonedevelopedbyHorowitzetal.[ 14 ]canmeasurethehighestmaximumpressure.Furthermore,basedonthistableithasthelargest

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Table2-7.ComparisonofpreviousaeroacousticMEMSmicrophonesandtheBruelandKjr4138traditionalcondensermicrophone. 3 ]Capacitive1:6mm168dB20dB*6:5Hz{140kHzArnoldetal.[ 13 ]Piezoresistive500m160dB52dB*10Hz{100kHzyScheeperetal.[ 15 ]Capacitive1:95mm141dB23dBA251Hz{20kHzHorowitzetal.[ 112 ]Piezoelectric900m169dB48dB*100Hz{50:8kHzzPedersen[ 111 ]Capacitive180m140dB22dB*50Hz{75kHzz dynamicrange;however,thenoiseoorforthecapacitivemicrophoneisreportedindBA,whichhasahighervaluethandB=p

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Inthischapter,adetailedmodelofacapacitivemicrophoneispresented.Theanalysisisbrokenintothreemajorsections. InSection 3.1 ,thequasi-staticbehaviorofthemicrophoneisdiscussed.Thisincludesthediaphragmbehavior,electrostaticbehavior,andnally,adiscussiononthenon-linearpropertiesofthemicrophone. InSection 3.2 ,thedynamicbehaviorofthemicrophoneisstudied.Thisisfacilitatedusingthelumpedelementmodelingtechnique.Anequivalentcircuitmodelofthemicrophoneisdeveloped.Thisisusedtopredictthefrequencyresponseandidentifykeyfeaturesthatimpactthemicrophoneresponse.Thisisfollowedbyadiscussionofelectrostaticpull-in;aphenomenonofparticularinterestforcapacitivemicrophones. Finally,inSection 3.3 ,anoisemodelisdeveloped.First,thederivedlumpedelementmodelisusedtoestimatethenoisegeneratedbythemicrophoneitself.Thisisfollowedbyaninvestigationofthenoiseduetointerfacecircuitry. Acrosssectionofthedual-backplatemicrophoneisshowninFigure 3-1 .Themajorelementsofthemicrophonearethediaphragm,topbackplate,bottombackplate,airgaps,backplateholes,cavity,andventchannel.Thediaphragmislocatedbetweenthetwobackplatesandtheyareseparatedbytwoairgaps.The 57

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backplateshaveholestoallowtheacousticpressuretodeectthediaphragm.Acavityiscreatedbeneaththemicrophonestructure.Theventchannelequalizesthepressureinthecavitytotheambientpressure. Figure3-1.Crosssectionofthedual-backplatecapacitivemicrophoneshowingthekeycomponents. 4 ] Themicrophoneisconstructedsuchthatthepressureinthecavityremainsconstantandisequaltotheambientpressure.1Thusthepressurebelowthediaphragmequalsp0,whilethepressureabovethediaphragmwillequalP.Therefore,thenetpressureactingonthediaphragmisequaltotheacousticpressureperturbation,p. 3.2

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AschematicofthediaphragmisshowninFigure 3-2 .Forthisanalysis,thediaphragmisassumedtobehomogeneous,axisymmetric,andlinearlyelastic.Furthermore,thediaphragmisassumedtohaveaperfectlyclampedboundaryconditionaroundtheperimeterofthediaphragmandtohavezeroresidualin-planestress.Thediaphragmhasaradiusaandathicknessh.ItisassumedthattheYoung'sModulus,E,andPoisson'sratio,,arebothknownandthediaphragmisloadedwithauniformpressure,p. Figure3-2.Schematicoftheidealizedcirculardiaphragm. 16 ].Asthemagnitudeofthediaphragmdeectionincreases,thestrainintheneutralaxiswillincrease.However,forsmalldeections,theneutralaxisstraincanbeneglected. Thegeneralgoverningdierentialequationforthesmalldisplacementsolutionforaclampedcircularplateisgivenby[ 22 ] D;(3{2) wherewisthetransversedeectionoftheplateandDistheexuralrigidityoftheplate;whichisgivenby

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Byassuminganaxisymmetricsolutionwherethedeectiononlydependsontheradialcoordinater,Equation 3{2 canbesimpliedto 1 drrd dr1 drrdw dr=p D:(3{4) Thedeectionoftheclampedcircularplateissubjecttofourboundaryconditions.First,thecenterdeectionisnite;second,thedeectionattheclampedboundaryiszero;third,theslopeoftheplateattheclampedboundaryiszero;andforth,theslopeoftheplateiszeroatthecenter.Respectively,thesecanbewrittenmathematicallyas dr(0)=0;andBC4:dw dr(a)=0:(3{5) BysolvingEquation 3{4 withtheboundaryconditionsgiveninEquation 3{5 ,thefollowingexpressionfortheplatedeectionisobtained: a22:(3{6) SubstitutingtheexpressionfortheexuralrigidityfromEquation 3{3 intoEqua-tion 3{6 givesanexpressionforthedeectioncompletelyintermsoftheplategeometryandmaterialparameters, 16Eh31r a22:(3{7) Thisgivesthedeectionofthediaphragmforsmalldisplacements.Thedeectionshapeisgivenbytheterm1(r=a)22,whilethecenterdisplacementoftheplateisgivenby

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ThenormalizeddiaphragmdeectionisshowninFigure 3-3 .Themaximumdeec-tionisinthecenterofthediaphragmandhasamagnitudegivenbyEquation 3{8 .Theslopeofthedeectioniszeroatthediaphragmcenterandattheclampedboundaries. Figure3-3.Normalizeddeectionofaclampedcircularplate. 22 ]. Thelargedeectionoftheplateisapproximatedby[ 22 ] 1+0:488w(0)2 Thusthelargedeectionisessentiallythesmalldeection,givenbyEquation 3{8 ,scaledbythefactor1+0:488w(0)2

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nonlinearityintothediaphragmdeectionwhenthemicrophoneisexposedtohighincidentpressures. ThebehaviorofthenonlineardeectionofaclampedcircularplateisshowninFigure 3-4 .Forsmallpressures,uptothedottedlineinthegure,theideallineardeectionandnon-lineardeectionareapproximatelyequal.However,athigherpressures,thecubicnonlinearityeectivelystiensthediaphragmandthenonlineardeectionissmallerthantheidealdeection.Furthermore,inthefrequencydomain,thenonlinearityproducesharmonicdistortion. Figure3-4.Non-lineardiaphragmdeectioncomparedtolineardeection.

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49 ],andmorerecentlybyRossi[ 21 ],amongothers.BeforediscussingthedetailedderivationsoftheelectrostaticmicrophoneinSection 3.1.2.2 andSection 3.1.2.3 ,abriefoverviewisgivenrstwithoutderivation. 3-5 .Thisdeviceconsistsoftwoparallelconductingplates.Oneisassumedtobexed,andtheotherismovable.Acapacitanceexistsbetweenthetwoplates,givenbyC=0A=x;whereAisthesurfaceareaandxisthedistancebetweenthetwoplates.AsindicatedinFigure 3-5 ,theplatesareadistancex0apartwhenthesystemisatrest.Themoveableplatemovesadistancex0asaresultofthenetforceappliedtotheplate.TheequilibruimcapacitanceisC=0A=x0andthetime-varyingcapacitanceisgivenby Figure3-5.Modelofatwoplateelectrostatictransducer.

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ThevoltagebetweentheterminalsofthecapacitorisgivenbyV(t)=Q(t)/C(t),whereQ(t)isthechargeontheplate.SubstitutinginEquation 3{10 ,thevoltageiswrittenas Thevoltagebetweentheplatesresultsinanelectrostaticforcegivenby[ 21 ] 20A 2Q2(t) Atthispoint,noassumptionshavebeenmaderegardingthemethodofapplyingthevoltageorchargetothecapacitor.TheforcemaybewritteneitherintermsofthevoltageV(t)orthechargeQ(t),astheyareequivalent.TheelectrostaticforceisattractivebetweenthetwoplatesandisnegativeforthesignconventionshowninFigure 3-5 Inaphysicaltransducer,amechanicalrestoringforceispresent.Asthemoveableplatedeects,anassociatedspringprovidesaforcetoopposethedeection[ 21 ].Writtenintermsofthemechanicalcompliance,Cm,theforceis Thecharacteristicelectrostaticequationsgoverningthissystemarewrittenas[ 21 ] and 20A 2Q2(t) Thesegeneralequationsdescribethebehaviorofaparallelplateelectrostatictransducer.Itisevidentthatthevoltageandforcearecoupled;thatis,theyare

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dependentoneachother.Furthermore,boththevoltageandforcearenon-linear,ingeneral.Thevoltageisproportionaltox0(t)Q(t)andtheforceisproportionaltoV2(t).Inaddition,thereisapotentialsingularityintheforceasx0(t)approachesx0. Atthispoint,RossilinearizedEquation 3{14 andEquation 3{15 byassumingsmallperturbationsviapolarization[ 21 ].Thevoltageandchargewereassumedtobecomposedofameanandtimevaryingcomponent.Thesmallperturbationassumptionimpliesthatx0(t)x0,v0(t)V0,andq0(t)Q0.However,forthemicrophonedesigner,itisusefultoconsiderphysicalimplementationsofacapaci-tivemicrophoneandtheconstraintstheyimposeonthecoupledelectromechanicalequations(Equation 3{14 andEquation 3{15 ). 3{14 andEquation 3{15 canbesimplied. First,theconstantvoltagecaseisconsidered.Inthiscase,V(t)becomesV0.Furthermore,Q(t)isexpressedasV0C(t).Foramicrophonewithconstantvoltage,Equation 3{14 isrewrittensuchthattheoutputisthechargeQ(t).Thusthecoupledelectrostaticequationsbecome x0x0(t);(3{16) and 20A

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Hence,theoutputchargeisnon-linearintermsofx0(t).Furthermore,theforcehasasingularityatx0(t)=x0,resultinginanpull-ininstability[ 49 ].Electrostaticpull-iniscausedbytheincreasingnetforceasthediaphragmapproachesthebackplate;itisexaminedinmoredetailinSection 3.2.7 .Physically,thislimitsthemaximumbiasvoltage,V0,thatmaybeappliedtothemicrophone. ThebehaviordescribedbyEquation 3{16 andEquation 3{17 appliestocondensermicrophonesintworegimes.Therstiswhenthebiasvoltageisdirectlyappliedtoacondensermicrophone,asisthecasewhenachargeamplierisused.Theamplierconvertstheoutputchargetoavoltageandmaintainsaconstantpotentialacrossthecapacitor.Thesecondregimeoccurswhenacondensermicrophoneisbiasedthroughalargeresistor,suchaswithavoltageamplier,andtheDCbehaviorofthemicrophoneisconsidered.ThelargeresistorpreventsthechargefromchangingduetoACinputs.TheDCbiaspointofthemicrophoneissetbytheexternalbiasvoltage.Inthisbiascondition,themicrophoneissusceptibletopull-incausedbytheexternalbiasvoltage[ 49 ]. Thesecondgeneralcaseofaconstantchargeisnowconsidered.Thisoccursforanelectretmicrophonewithanembeddedcharge[ 21 ]andforacondensermicrophonebiasedthroughalargeresistorwhensubjecttoACinputs[ 49 ].Inthiscase,theoutputisthevoltage,V(t),acrosstheterminalsofthemicrophone.Thusthecharacteristicelectrostaticequationsbecome and 2Q20 Fortheconstantchargecase,theoutputvoltageislinearwithrespecttox0(t).Additionally,theforceisalsolinearwithx0(t)andthereisnotasingularityintheforce.Thus,aslongasthechargeremainsconstant,pull-inwillnotoccur[ 49 ].

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Thisintroductionpresentedthebasicprinciplesofmodelingelectrostatictransducers.TheclassicalrepresentationinRossi[ 21 ]wasappliedtophysicalmicrophoneimplementations.Severalkeyissueswereidentied,suchaspull-ininstabilityandnon-linearities.Tofurtheranalyzetheelectrostaticbehaviorofcapacitivemicrophones,theyarenowexaminedinadesign-orientedmethodologygivingkeyphysicalinsight.Integraltothisanalysisistheconsiderationandimpli-cationsoftheinterfacecircuitry.Thisanalysisislimitedtocondensermicrophones,althoughtheconstantchargeresultscanbeappliedtoelectretmicrophones.First,thesingle-backplatecondensermicrophoneisdiscussed.Then,theresultsareappliedtothedual-backplatecondensermicrophone.Eachmicrophonetypeisconsideredwithbothachargeamplierandavoltageamplier. Foreachofthefourcases,theanalysisfollowsthemethodologylistedbelow: 1. Denethegeometry. 2. Derivethecapacitance,charge,andvoltageonthecapacitor(s). 3. Introducedetailspertainingtotheinterfacecircuitry. 4. Derivetheoutputvoltageofthemicrophone. 5. Derivetheelectrostaticforcebetweenthediaphragmandbackplate(s). Eachcapacitoristreatedasaparallelplatecapacitor.Fortheinitialdiscus-sion,theareaofeachcapacitorissimplyassumedtobethephysicalareaoftheplates.TheeectsofthediaphragmcurvatureontheelectrostaticbehaviorareinvestigatedinSection 3.1.3 .InadditiontotheelectroacousticstextbooksbyHunt[ 49 ]andRossi[ 21 ],additionalbackgroundoncondensermicrophonesisavailableintheliterature[ 113 { 115 ].

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chargeamplierisusedfortheinterfacecircuitry.Avoltageamplieristypicallyusedwiththethesecondcase. 3-6 .Thesingle-backplatecondensermicrophoneconsistsofaxedbackplateseparatedfromamovablediaphragmbyanominalgapdistanceg0.Asshowninthegure,thediaphragmmovesbyadistanceg0;thustheairgapbetweenthebackplateandthediaphragm,g,isgivenby ThebackplateanddiaphragmarebothcircularwitharadiusaandasurfaceareaofA=a2.Thediaphragmisassumedtomoveasarigidpistonwithadisplacementequaltothecenterdeection,w(0).Therefore,thecapacitorisalwaystreatedasaparallelplatecapacitor. Figure3-6.Electricalmodelofasingle-backplatecondensermicrophonewithaconstantvoltage. Avoltage,VB,isappliedbetweenthediaphragmandbackplateandthecapacitanceoftheparallelplatecapacitorisgivenby[ 23 ] g:(3{21)

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SubstitutingEquation 3{20 intoEquation 3{21 resultsinthefollowingexpressionforthecapacitance g0g0:(3{22) Thechangeincapacitanceduetothegapchangeisthephysicalphenomenonthatresultsinthemicrophoneoutput.Thecapacitancemayalsobeexpressedintermsofameancapacitance,C10,andachangeincapacitance,C1,as whereC10is g0:(3{24) TondC1,rsttheexpressionforthetotalcapacitance,givenbyEquation 3{22 ,andthemeancapacitance,givenbyEquation 3{24 ,aresubstitutedintoEquation 3{23 ;thus g0g0=0A g0+C1:(3{25)SolvingforC1resultsin C1=g0 g0g0:(3{26) ThechangeincapacitanceissimplythecapacitanceC1,givenbyEquation 3{22 ,scaledbythefactor(g0=g0). Thechangeincapacitance,givenbyEquation 3{26 ,isnon-linearintermsofthegapchange.However,ifg0isassumedtobesmallcomparedtog0,alinearexpressionforC1canbefound.ThelinearizationofEquation 3{26 beginswith C1=g0 g0g0g0+g0 whichisrewrittenas C1=0A g0g0g0+g02

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Theg02termsarenegligibleandthereforedropped.Thus,Equation 3{28 issimpliedto C1=g0 g0:(3{29) Thisexpressionisnowlinearintermsofg0.ComparingEquation 3{29 toEqua-tion 3{26 ,thefactor(g0=g0)nowscalesthemeancapacitance,C10.Thelinearizedcapacitancebecomes g01+g0 Thechargeonthecapacitor,Q1,givenbyQ1=C1VB,isexpandedto g0g0VB:(3{31) ThechargemayalsobeexpressedintermsofC1, FromEquation 3{32 ,ameancharge,Q10andachangeinchange,Q1aredened; g0VB(3{33) and Q1=C1VB:(3{34) SubstitutingEquation 3{24 andEquation 3{29 intoEquation 3{32 ,thenalexpressionforthelinearizedchargeonthecapacitoris Tondtheoutputvoltageofasingle-backplatecondensermicrophonewithaconstantvoltage,theinterfacecircuitymustalsobeconsidered.ShowninFigure 3-7 isasinglebackplatemicrophoneconnectedtoachargeamplier.ThedetailsoftheDCbiascircuitryarenotshown.Oneplateofthemicrophoneisbiasedwith

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aDCvoltageVB,whiletheotherplateisconnectedtotheinvertinginputofanoperationalamplier.Thenon-invertinginputisgrounded.Afeedbackcapacitor,Cf,isplacedinthefeedbackpathfromtheoutputtothenegativeinput.Duetonegativefeedback,theotherplateofthemicrophoneisheldatsmallsignalground[ 116 ]. Figure3-7.Simpliedcircuitofasingle-backplatecondensermicrophoneandachargeamplier. Thechargeamplierstorestheinputcharge,Qin,onthefeedbackcapacitortogenerateanoutputvoltageequalto Theinputchargeisthetimevaryingcomponentofthechargeonthecapacitor,Q1,givenbyEquation 3{34 .Therefore,theoutputvoltageofthemicrophoneiswrittenas SubstitutingEquation 3{29 intoEquation 3{37 givesthelinearizedoutputvoltage, Tondthenalexpressionfortheoutputvoltage,thecenterdeectionofthediaphragmgivenbyEquation 3{8 issubstitutedforg0.Thereforetheoutput

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voltageofthesinglebackplatecapacitivemicrophonewithaconstantvoltageisgivenby However,itisnotedthatthisexpressionfortheoutputvoltagedoesnotconsidertheeectsoftherestofthemicrophonestructureincludingthecavity.ThisisfurtherinvestigatedinSection 3.2 Itisalsousefultostudytheelectrostaticforceonthediaphragm.AsshowninFigure 3-6 ,thebackplateanddiaphragmaredirectlybiasedwithaconstantvolt-age.Thisvoltagecreatesanelectrostaticforcethattendstomovethediaphragmtowardsthebackplateasindicatedinthegure. Theelectriceldintheairgapbetweenthediaphragmandbackplateisgivenby[ 23 ] Theelectrostaticenergydensity,ue[J=m3],intheairgapisgivenby 20E2;(3{41) therefore,thetotalenergyis where8istheintegrationvolumebetweenthetwoplates.Assumingaparallelplatecapacitorandneglectingfringingelds,thisvolumeisequaltoAg;thusueisconstantoverthevolume.SubstitutingEquation 3{40 andEquation 3{41 intoEquation 3{42 gives[ 23 ] whichisrewrittenas 2C1VB2:(3{44)

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Fortwoparallelplatesbiasedwithaconstantvoltage,theelectrostaticforceisgivenby BysubstitutingEquation 3{44 intoEquation 3{45 andrecallingthatC1isafunctionofg0asgiveninEquation 3{22 ,theelectrostaticforceiswrittenas 2VB2dC1 Thusthenalequationfortheelectrostaticforceforaconstantvoltageisgivenby 2VB20A TheelectrostaticforcegiveninEquation 3{47 isinthepositivexdirection,whichcausesthediaphragmtodeectuptowardsthebackplate.Furthermore,theelectrostaticforceincreasesasthediaphragmmovesclosertothebackplate,approachinginnityasthegapbecomesverysmall.Withoutarestoringforce,thediaphragmwouldalwayscollapseintothebackplate.InSection 3.2.7 ,therelationshipbetweentherestoringforceofthediaphragmandtheelectrostaticforceisinvestigated.Inaddition,theforceisproportionalto1=g0g02,whichintroducesanonlinearityifthedeection,g0islarge.However,atlargedeections,theaccuracyofthismodelwouldsuerbecausethetwoplatesarenolongerparallel.Amoregeneralmodelaccountingforanon-parallelplatecapacitorisinvestigatedbyPedersen[ 117 ]

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essentiallyconstant[ 49 ].AmodelofthemicrophoneinthiscongurationisshowninFigure 3-8 Figure3-8.Electricalmodelofasingle-backplatecondensermicrophonewithvolt-agesourceappliedthroughalargeresistor. ThegapandcapacitancearegivenbyEquation 3{20 andEquation 3{22 ,respectively.Thevoltageacrossthecapacitor,V1,isgivenbyV1=QB=C1.Thus,V1is Unlikethenonlinearchargefromtheconstantvoltageanalysis,giveninEqua-tion 3{31 ,thevoltageacrossthecapacitorbiasedwithaconstantchargeislinearwithrespecttog0.Equation 3{48 maybedenedintermsofthenominalcapaci-tance,C10,andthechangeincapacitance,C1,suchthat Figure 3-9 showsthemicrophoneconnectedtoavoltagebuer.Oneplateofthemicrophoneisconnectedtothevoltagesource,whiletheotherisconnectedtotheamplier.Thissecondplateisalsoconnectedtogroundthroughalargeresistor.ThehighpasslterformedbyRbandC1forcestheDCvoltageonthesecondplatetoequalzero,butforfrequenciesabove1=(2RbC10),thevoltageonthisplateisfreetochange.

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Figure3-9.Simpliedcircuitofasingle-backplatecondensermicrophoneandavoltageamplier. ThisbiasarrangementresultsinaxedDCchargeonthecapacitorgivenby Therefore,thevoltageacrossthecapacitor,V1,isgivenby ByexpressingV1asV1=VBVout,whereVoutistheoutputvoltageofthemicrophone(andthebueramplier),Equation 3{51 isrewrittenas ByexpandingandsimplifyingEquation 3{52 ,thefollowingexpressionfortheoutputvoltageisobtained, SubstitutingEquation 3{26 andEquation 3{22 intoEquation 3{53 yields fortheoutputvoltage.Thisexpressionislinearwithg0andlinearizationisnotnecessary.SubstitutingEquation 3{8 intoEquation 3{54 resultsinthenal

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expressionfortheoutputvoltage, However,inaphysicalimplementationofasingle-backplatecondensermicro-phoneandvoltageamplier,theeectsofparasiticcapacitanceandtheamplierinputcapacitancemustbeconsidered.Figure 3-10 showsthesingle-backplatemi-crophonewithabiasresistorandvoltagebuerasbefore;however,nowaparasiticcapacitance,Cp,andaninputcapacitance,Ci,havebeenadded. Figure3-10.Circuitmodelofasingle-backplatemicrophoneandavoltageamplierwithparasitics. Forthisconguration,thebiaschargeisstillgivenbyEquation 3{50 .How-ever,whenthevoltageattheinputtotheamplierchanges,thereischargesharingbetweenthemicrophonecapacitanceandtheparasiticandinputcapacitances.Therefore,Equation 3{52 becomes {z }=(C10+C1)(VBVout)| {z }+(Cp+Ci)(0Vout)| {z }initialchargeonchargelostchargemicrophonetosharing;(3{56) theinitialcharge,QB,thechargeonthemicrophonecapacitorandthechargelostduetochargesharingareshown.SolvingEquation 3{56 forVoutgivestheoutputvoltageforaconstantbiascharge,

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NotethatwhenCpandCiarepresent,thechargeonthemicrophonecapacitorisnotconstant.However,thetotalchargeinthesystemisconstant.ComparingEquation 3{57 toEquation 3{53 givesanattenuationfactor,HC,denedas Thisresultquantiesthesignallossduetotheparasiticandinputcapac-itancesandagreeswithpreviouslypublishedanalysis[ 55 ].Theterm,Hc,isafunctionofthetopcapacitancevalue,C1.Therefore,Hcisnotconstantasthepressureloadingvariesonthemicrophone.Thisintroducesanon-linearityintotheoutputvoltagebecausetheoutputvoltageisnolongerlinearlyproportionaltotheincidentpressure.Ingeneral,asthetotalparasiticcapacitanceincreases,thenon-linearityincreases.ThisbehaviorisfurtherexploredinSection 3.1.3 Next,theelectrostaticforcebetweenthebackplateanddiaphragmiscon-sidered.Forthisanalysis,theeectsoftheparasiticandinputcapacitancesareassumedtobenegligiblesothatthechargeonthecapacitorremainsconstant.Theelectriceldbetweenthebackplateandthediaphragmwhenbiasedwithaconstantchargeisgivenby[ 23 ] Theelectrostaticenergydensity,ue,isgivenbyEquation 3{41 forthiscaseaswell.BysubstitutingEquation 3{59 intoEquation 3{41 andEquation 3{42 ,thefollowingexpressionfortheelectrostaticenergyisobtained, Theintegrationvolume,8,istheregionwheretheelectriceldexistsandisequaltoAg.Assumingthebackplateanddiaphragmareparallel,theelectrostatic

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energyisgivenby[ 23 ] Theelectrostaticforcebetweentwoconductorswithaconstantchargeisgivenby[ 23 ] BysubstitutingEquation 3{61 intoEquation 3{62 ,thefollowingexpressionfortheelectrostaticforceisfound, dg01 where,dg0=dg.Solvingthisexpressiongivesthenalresultfortheelectrostaticforcebetweentwoparallelplateswithaconstantcharge, Unliketheelectrostaticforcewithaconstantvoltage,thisforceisconstantregard-lessofthediaphragm'sposition. 3-11 .Itisassumedthatallthreeplateshavethesameradius,a,andsurfaceareagivenbyA=a2.Furthermore,thenominalairgapbetweenthediaphragmandeachbackplateisassumedtobeequaltog0.Thediaphragmisperturbedadistanceg0,whichcauses

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thetopairgaptobereducedandthebottomairgaptobeincreased;theairgapsaregivenby and Figure3-11.Dual-backplatecondensermicrophonewithdirectlyconnectedbiasvoltages. Thederivationoftheoutputvoltageofthedual-backplatecapacitivemicro-phonewithaconstantvoltagefollowsthemethodusedforthesingle-backplatemicrophone.Thetopandbottomcapacitances,respectively,are g0g0;(3{67) and g0+g0:(3{68) SimilartoC1,C2isexpressedintermsofameancapacitance,C20,andachangeincapacitance,C2,suchthat

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Thechangeinthebottomcapacitanceisfoundinthesamemannerasthatforthetopcapacitance.Equation 3{69 isrewrittenas g0+g0=0A g0+C2:(3{70)SolvingthisforC2resultsin C2=g0 g0+g0:(3{71) SimilartoC1,Equation 3{71 isthechangeincapacitanceC2,scaledbythefactor(g0=g0).However,thechangeinC2hastheoppositesignofthechangeinC1.Thisdierentialcapacitancechangeisakeyfeatureofthedual-backplatemicrophone.Furthermore,thelinearizedchangeinbottomcapacitanceis C2=g0 g0:(3{72) Thechargeonthetwocapacitors,C1andC2,is and Thedual-backplatecondensermicrophoneisnowconsideredwithachargeamplierasshowninFigure 3-12 .Thetopbackplateisbiasedwith+VBandthebottombackplateisbiasedwithVB.Thediaphragmisconnectedtotheinputofthechargeamplierandisheldatsmallsignalground.TheDCvoltagelevelofthediaphragmissetbyabiasresistornotshowninthegure. Asbefore,thegainofthechargeamplierisgivenbyEquation 3{36 .Therearetwocomponentstotheoutputvoltage;onefromeachcapacitor.Theinputcharge,Q,isthesumofQ1andQ2.ThesearefoundfromEquation 3{73 andEquation 3{74 ,usingthelinearizedformofthecapacitancechanges;andaregiven

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Figure3-12.Electricalmodelofadual-backplatecapacitivemicrophonewithachargeamplier. by Q1=VBC10g0 and Q2=VBC20g0 Assumingthattheareasandnominalgapsarethesameforeachcapacitor,C10=C20andC1=C2.Thus,thetotalinputchargeistwicethatofthesingle-backplatecapacitivemicrophone.Furthermore,theoutputvoltage,whichisgivenby isalsotwicethatofthesingle-backplatecapacitivemicrophone(Equation 3{38 ).Bysubstitutingthediaphragmdeection,givenbyEquation 3{8 ,forg0,thenalexpressionfortheoutputvoltageofadual-backplatecapacitivemicrophonebiasedwithaconstantvoltageisdetermined; Next,theelectrostaticforceactingonthediaphragmisconsidered.Therearetwocomponentstotheforce,onefromthetopbackplateandonefromthebottom

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backplate.Theelectrostaticforceduetothetopbackplate,Fe1,andtheforceduetothebottombackplate,Fe2,arefoundinthesamemannerusedtoderiveEquation 3{46 asfollows, 2VB2dC1 and 2VB2dC2 EvaluatingtheseexpressionsusingEquation 3{67 andEquation 3{68 gives 2VB20A and 2VB20A Thetwoelectrostaticforcesactinoppositedirections.Thisarisesbecausetheelectrostaticforcebetweenthediaphragmandeachbackplateisattractive;thusthediaphragmispulledtowardsthebackplate.Therefore,Fe1isdirectedinthepositivexdirection,andFe2isdirectedinthenegativexdirection.ThetotalelectrostaticforceactingonthediaphragmisequaltoFe1+Fe2, 2VB20A 2VB20A furthersimplicationyields Theelectrostaticforceforthedual-backplatecapacitivemicrophonewithaconstantvoltageislessthanthatofthesingle-backplatemicrophone,althoughtheforcestillapproachesinnityasthediaphragmmovestowardseitherbackplate.Whenthediaphragmisintherestposition,themagnitudeoftheelectrostaticforceisequaltozero.Whenthediaphragmisperturbed,theelectrostaticforceis

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directedtowardsthenearestbackplate;thusifg0ispositive,Feisdirectedinthepositivexdirectionandifg0isnegative,Feisdirectedinthenegativexdirection. 3-13 ,apositivevoltageisappliedtothetopbackplateandanegativevoltageisappliedtothebottombackplate.Thisresultsinoppositechargesonthetwocapacitors. Figure3-13.Dual-backplatecondensermicrophonebiasedwithvoltagesourcesconnectedthroughalargeresistor. ThetopandbottomairgapdistancesaregivenbyEquation 3{65 andEquation 3{66 ,andthetopandbottomcapacitancesaregivenbyEquation 3{67 andEquation 3{68 ,respectively.Thevoltagesacrosseachcapacitorare and Aswasthecasewiththesingle-backplatecapacitivemicrophone,thevoltagesgiveninEquation 3{85 andEquation 3{86 varylinearlywithg0.

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Similartotheanalysisofthesingle-backplatemicrophonewithconstantcharge,thecapacitancesaredenedintermsofanominalcapacitanceandachangeincapacitance.Thus,Equation 3{85 andEquation 3{86 become and whereC1isgivenbyEquation 3{26 andC2isgivenbyEquation 3{71 ThemicrophoneisconsideredwithavoltageamplierasshowninFigure 3-14 ;themicrophoneisrepresentedbyC1andC2.ThetopandbottombackplatesarebiasedwithvoltagesVBandVB,respectively.Thebiasresistor,RbholdstheDCvoltageofthediaphragmat0V.However,forfrequenciesabovethecut-onfrequency,1=2Rb(C10+C20),thevoltageofthemiddleplateisfreetochangewhilethechargeoneachmicrophonecapacitorremainsconstant(neglectingtheeectoftheparasiticandinputcapacitances). Figure3-14.Simpliedcircuitofadual-backplatemicrophoneandavoltageampli-er. ThechargestoredoneachcapacitorduetothebiasvoltageisQ=C0VB,thusthevoltage,V1,acrossthetopcapacitorandthevoltage,V2,acrossthebottomcapacitorare

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and Tondtheoutputvoltage,theinitialandnalchargeoneachcapacitorisanalyzed.Theprincipleofsuperpositionisused,whereeachvoltagesourceisconsideredindividually.Thustherearetwocomponentstotheoutputvoltage.Therefore,onlyoneofthecapacitorshasaninitialchargeforeachcomponent. Forthedual-backplatemicrophone,onedevicecapacitoractsasaparasiticcapacitorfortheotherdevicecapacitor.Therefore,thereischargesharingbetweenthetwocapacitors,evenwithoutconsideringtheeectsofparasiticandinputcapacitances.Whilethechargeoneachcapacitormaynotbestrictlyconstant,thetotalchargeinthesystemisconstant.Generally,theinitialandnalchargeforeachcapacitorisnotthesame.However,forthecaseofequalmeancapacitances,equalcapacitancechanges,andnoparasiticandinputcapacitance,thechargeoneachcapacitorisconstant. Fornow,theeectsoftheparasiticandinputcapacitancesareneglected.Firsttheinitialchargefromthetopcapacitorisanalyzed;thus,thevoltagesourceforC2,(VB),issetto0.TheinitialchargeisequaltoQ10=C10VBandisequatedtothenalchargeasshown wherethevoltageV2isequalto0Vout.Note,thatthechargefromC1isfreetomovetoC2.Indeed,itmusttosatisfytherelationQ=CVforeachcapacitor.Therefore,Equation 3{91 canbeexpandedto andsolvingforVoutduetoC1gives

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Similarly,theinitialchargeonthebottomcapacitor,Q2=C20VB,isequatedtothenalchargewiththevoltagesourceforC1setto0, wherethevoltageV1=0Vout.Equation 3{94 isre-writtenas TheoutputvoltageduetoC2is ThetotaloutputvoltageisthesumofEquation 3{93 andEquation 3{96 BysubstitutingEquation 3{26 ,Equation 3{71 ,Equation 3{67 ,andEquation 3{68 intoEquation 3{97 ,theoutputvoltageissimpliedto ThenalexpressionfortheoutputvoltageisfoundbysubstitutingEquation 3{8 intoEquation 3{98 Consideringtheeectsoftheparasiticcapacitance,Cp,andtheinputcapaci-tance,Ci,Equation 3{97 becomes

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ComparingtheoutputvoltagegiveninEquation 3{100 tothatinEquation 3{97 ,givestheattenuationfactor,Hc,forthedual-backplatemicrophoneas Similartothesingle-backplatemicrophonewithavoltageamplier,theoutputofthedual-backplatemicrophonepackagedwithavoltageamplierislinearwhentheeectsofparasiticcapacitancesarenotconsidered.TheimpactofHconthelinearityisinvestigatedinSection 3.1.3 Nowtheelectrostaticforceactingonthediaphragmisconsidered.Thereareagaintwocomponentstotheelectrostaticforce;onefromthetopbackplateandasecondfromthebottombackplate.Thetwoelectrostaticforces, and arefoundusingthesamemethodusedtoderiveEquation 3{64 .Thesetwoforcesareequalinmagnitudeandoppositeinsign.Theforceduetothetopbackplate,Fe1,isdirectedinthepositivexdirectionandtheforceduetothebottombackplateisdirectedinthenegativexdirection.ThetotalforceactingonthediaphragmisthesumoftheforcesgiveninEquation 3{102 andEquation 3{103 ,thusthetotalelectrostaticforceiszeroaslongasthechargeremainsconstant, 3-1 liststhesensitivitiesofboththesingle-backplateanddual-backplate

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microphonesbiasedwithaconstantchargeandaconstantvoltage,assumingparallelplatemotion.Thesensitivityofthedual-backplatecondensermicrophoneistwicethatofthesingle-backplatemicrophonewhenthebiasvoltageisapplieddirectly. However,forcondensermicrophonesbiasedthroughalargeresistor,asisthecasewhenthemicrophoneisusedwithavoltageamplier,thesensitivityofthesingle-backplateanddual-backplatemicrophonesarethesame.ThiscanbeunderstoodphysicallybycomparingEquation 3{53 toEquation 3{93 andEquation 3{96 .Allthreeequationshavethesamebasicform;theratioofachangeincapacitancetothefullcapacitancevalue,scaledbyabiasvoltage.Theoutputofthedual-backplatemicrophoneisthesumoftwocomponents:Equation 3{93 andEquation 3{96 .Assumingthatthegeometryofthetopandbottomcapacitorsarethesame,thesetwocomponentsoftheoutputvoltagearehalfofthevalueoftheoutputvoltageofthesingle-backplatemicrophone,giveninEquation 3{53 .Thisisbecausethedenominatorforthedual-backplatemicrophone,C1+C2,istwiceasbigasthedenominatorofthesingle-backplatemicrophone,C1.Thustheoutputofthedual-backplatemicrophoneisthesumoftwocomponents,eachonehalfthemagnitudeofthatofthesinglebackplatemicrophone. Table3-1.Summaryoftheoreticallinearsensitivityofcondensermicrophones. Singlebackplate 3-2 .Theelectrostaticforceactingonthediaphragmissimilarforboththesingle-backplatemicrophone

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anddual-backplatemicrophonedirectlyconnectedvoltagesources.Whilethemagnitudeislessforthedual-backplatemicrophone,bothforcesapproachinnityasthediaphragmnearsabackplate.Fortheconstantchargecase,theforceisconstantforthesingle-backplatemicrophoneanditisequaltozeroforthedual-backplatemicrophone. Table3-2.Summaryoftheelectrostaticforceactingonthediaphragmofcapacitivemicrophones. Singlebackplate 2VB20A 2VB20Ag0g0 49 ].Thisisnotthecasephysicallybecausethederivationoftheelectrostaticforceassumedparallelplatecapacitors;thisassumptionisnotvalidasthediaphragmapproachesabackplate.Thedynamic,non-linearpull-inanalysisisbeyondthescopeofthisdiscussion,furtherdiscussioncanbefoundin[ 118 ].

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discussed.ThiswaspreviouslyinvestigatedbyWarrenetal.[ 113 114 ]andDonketal.[ 115 ].Thefollowinganalysisexpandsontheirwork. ConsiderthegeometryshowninFigure 3-15 ;nowtheairgapisnolongerassumedtobeuniform.Thedeectionisassumedtobesmallsuchthatthechargedensityonthesurfaceoftheplatesisconstant;furthermore,thebackplateholesareneglected. Figure3-15.Modelofthetopcapacitorwithanon-uniformairgap. Thecapacitanceisdirectlyfoundfromthefollowing[ 23 ], C:(3{105) FortwoconductorswithaninnitesimalareaofdAandwithoppositesurfacechargedensitiesofmagnitude,theelectriceld,E,betweenthetwoconductorsis

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0Z0g(r)ds=dQ 0g(r):(3{106) BynotingthatdQ=dA,Equation 3{106 isrewrittenas g(r):(3{107) Thetotalcapacitanceisfoundbyintegratingovertheentireareaoftheconductors.Thegapdistance,g(r),isequaltothenominalgapdistance,g0,minustheplatedeectiongivenbyEquation 3{7 .Therefore,thecapacitanceisfoundbyevaluatingthefollowingintegral, g0w(0)h1r a2i2dr:(3{108) ThisintegralwassolvednumericallyusingMathCad;ithastwosolutions:oneforapositivediaphragmdeectionandoneforanegativedeection, q q ThecapacitancegiveninEquation 3{109 isthemeancapacitancemultipliedbyascalefactor.Whenthediaphragmdeectioniszero,thescalefactorisone;thereforethecapacitanceisexactlythatpredictedbythesimpleparallelplatemodel.Forthisconditionofzerodeection,theplatesareindeedparallel.Thenon-uniformgapandtheparallelplatecapacitancesareplottedinFigure 3-16

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versusthediaphragmdeection.Thechangeincapacitanceforthenon-uniformgapcapacitancemodelislessthanthatpredictedbytheparallelplatemodel. Figure3-16.Capacitanceofthetopbackplateaspredictedbythenon-uniformgapmodelandtheparallelplatemodelasafunctionofdiaphragmdisplacement. Thebehaviorofthenon-linearcapacitanceisofparticularinterestforsmalldisplacements,asthisisrelatedtothenominalsensitivityofthecapacitivemicrophone.TheexpressionforthecapacitancegiveninEquation 3{109 issimpliedusingaTaylorseriesexpansion,assumingthedeectionissmall[ 119 ].Thequantityp arctanhx=x+x3 and arctanx=xx3 respectively.ThisTaylorapproximationhaslessthan1%errorwhenthedia-phragmcenterdisplacementislessthan21%ofthenominalgap.Thex3termappearsinbothEquation 3{110 andEquation 3{111 withoppositesigns,there-foreasingleexpressionforthecapacitanceisfoundthatisvalidforbothpositiveandnegativediaphragmdeection.UsingEquation 3{110 ,Equation 3{111 ,and

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Equation 3{109 ,andbywritingw(0)asg0,thecapacitanceisapproximatedas g01+1 3g0 BycomparingEquation 3{112 toEquation 3{30 ,thechangeincapacitanceisonethirdofthatpredictedbytheparallelplatemodel.Therefore,thechangesinthetopandbottomcapacitance,givenbyEquation 3{26 andEquation 3{71 respectively,arerewrittenas C1eff=1 3g0 g0g0:(3{113) and C2eff=1 3g0 g0+g0:(3{114) Furthermore,thetotalcapacitanceofthetopandbottomcapacitorsbecomes g0+1 3g0 g0g0:(3{115) and g01 3g0 g0+g0:(3{116) LinearapproximationsofEquation 3{115 andEquation 3{116 are 3g0 and 3g0 Furthermore,thelinearchangesincapacitanceare C1effL=1 3g0 g0:(3{119) and C2effL=1 3g0 g0:(3{120)

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Theonethirdfactorisalsopresentintheeectivearea(Equation 3{137 developedinSection 3.2.1 );thusthecapacitancesgiveninEquation 3{119 andEquation 3{120 arerewrittenas C1effL=g0 and C2effL=g0 Thepreviouslyderivedresultsforthecapacitivemicrophonesensitivityareaccurateiftheeectiveareaisusedratherthanthephysicalarea.Thisagreeswiththeresultsobtainedby[ 115 ]. InFigure 3-17 ,thetopcapacitancepredictedbythenon-uniformgapmodel(Equation 3{109 )iscomparedtothecapacitancepredictedbythenon-linearuniform-gapmodel(Equation 3{115 )andthelinearuniform-gapmodel(Equa-tion 3{117 ).Bothuniform-gapmodelsareusedwiththeeectivearea. Figure3-17.Capacitanceofthetopbackplateaspredictedbythenon-uniformgapmodelandeectiveareaapproximationmodel. Theeectiveareamodelbridgestheparallelplatemodelandthenon-uniformgapmodel.Itmaybeachievedbyeitherassumingsmalldeectionforthefullnon-uniformgapmodelorbyusingtheeectivearea,Aeff,intheparallelplatemodel.Allmodelsarewellmatchedforsmalldeections.Therefore,thelinearcapacitance

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withtheeectiveareaisusedtoaccuratelypredictthemicrophonebehaviorforsmalldeections.Inaddition,ithastheadvantageofasimpleexpressionthatprovidesscalinginformation. 120 ]. Aspartofthepreviouselectrostaticdiscussion,thesourcesofelectrostaticnon-linearitywereidentied.Formicrophonesbiaseddirectlywithavoltagesource,anon-linearityisintroducedintotheCquantitiesbythegog0terminthedenominator;asshowninEquation 3{26 andEquation 3{71 .Itwasalsoshownthatmicrophonesbiasedthroughalargeresistordidnothavenon-linearity,intheabsenceofparasiticcapacitance. However,whenthemodelsaremodiedtoincludetheeectivearea,non-linearitiesareintroducedintothevoltageampliercases.Considerthesingle-backplatecapacitivemicrophonebiasedwithaconstantcharge.TheoutputvoltageisgivenbyEquation 3{53 .SubstitutingEquation 3{113 andEquation 3{115 intoEquation 3{53 ,theoutputvoltagebecomes 3g0 g0g0 g0+1 3g0 g0g0VB: Thisissimpliedto 3g0 3g0

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Therefore,theoutputvoltageisnolongerlinearlyproportionaltog0.Thenon-linearityintheoutputvoltageisdenedas %NL=VoutLVoutNL Theimpactofthenon-uniformgapontheoutputvoltagenon-linearitycanbeseenbycomparingFigure 3-18(b) toFigure 3-18(a) .Thenon-linearityforthechargeampliercasesisunchangedbythenon-uniformgapmodel.Conversely,thenon-linearityforthevoltageampliercasesisincreased. Idealparallelplatemodel. (b) ParallelplatemodelwithAeff Fullnon-uniformgapmodel. (d) Non-uniformgapwithnon-lineardeec-tion

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AsshowninFigure 3-17 ,theeectiveareaparallelplatemodelover-predictsthecapacitancechangecomparedtothenon-uniformgapmodel.Furthermore,thenon-linearityislessforthefourmicrophonecaseswhenthenon-uniformgapmodelisusedtopredictthemicrophoneoutput,asshowninFigure 3-18(c) .Theeectoftheelectricalnon-linearityistocausethechangeincapacitancetobecomegreaterforhigherincidentpressures.However,themechanicalnon-linearitycausesthediaphragmdeectiontobelessathigherpressures.Thesetwonon-linearitiesopposeoneanother;thusthenon-linearityforthefourmicrophonecasesislowestwhentheeectofthemechanicalnon-linearityisconsidered,asshowninFigure 3-18(d) Anotherpointofinterestisthatthenon-linearityofthedual-backplatemicrophoneismuchlowerthanthesingle-backplatemicrophone,regardlessofthetypeofinterfacecircuitryused.Physically,thisisduetothedierentialnatureofthedualbackplatemicrophone.Theidealcapacitancechangesareofoppositesign.Furthermore,thenon-linearitiesforthecapacitancechangesareofoppositesign;thustheyopposeeachother. Ideally,boththesingle-backplatemicrophoneandthedual-backplatemi-crophonedonothaveanyelectrostaticnon-linearitywhenusedwithavoltageamplier.However,previouslyitwasshownthatthenon-uniformgapintroducesanon-linearity.Anothersourceofelectrostaticnon-linearityformicrophonesusedwithvoltageampliersistheparasiticcapacitance.Theoutputvoltagevs.pressurenon-linearityofasingle-backplatemicrophoneanddual-backplatemicrophoneareshowninFigure 3-19(a) andFigure 3-19(b) ,respectively.Theeectsofthenon-lineardeectionareneglectedtoisolatetheeectoftheparasiticcapacitance.Withoutaparasiticcapacitance,thenon-linearityofthevoltageamplierislessthanthatofthechargeamplier.Astheparasiticcapacitancebecomeslarger,thenon-linearityapproachesthatofthechargeampliercases.

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Non-linearityforthesingle-backplatemicrophone. (b) Non-linearityforthedual-backplatemicrophone. 21 ]. Toconstructthelumpedelementmodel,themechanicalandacousticproper-tiesofthemicrophonearerepresentedbyequivalentcircuitelements.Eachenergydomainisrepresentedbyapairofconjugatepowervariables;whichingeneralareeortandow.Animpedanceanalogyisused,suchthattheimpedancerelatestheeort,e,totheow,f,viae=Zf.Forexample,intheelectricaldomain,theeortvariableisvoltageandtheowvariableiscurrent.Ineachenergydomain,theproductoftheeortandowvariablesispower.Asummaryoftheconjugate

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powervariablesforthemechanical,acoustic,andelectricaldomainsaregiveninTable 3-3 Table3-3.Lumpedelementmodelingconjugatepowervariables. MechanicalForce[N]Velocitym sPosition[m]AcousticPressureN m2VolumetricFlowhm3 3-4 .Forexample,intheelectricaldomain,potentialenergyisstoredinacapacitor;whileinthemechanicaldomain,potentialenergyisstoredinaspring.However,itiscommontorefertothemechanicalgeneralizedcapacitanceintermsofacomplianceratherthanaspring;thecomplianceisgivenby1=k. Table3-4.Lumpedelementsforvariousenergydomains. MechanicalRmNs mCmm NMm[kg]AcousticRaNs m5Cahm5 m4iElectricalR[]C[F]L[H]

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Arelationshipbetweenamechanicalimpedanceandanequivalentacousticimpedancemaybefound.Ageneralimpedance,Z,isdenedas f;(3{126) whereeandfaregeneralizedeortandow,respectively.Inthemechanicaldo-main,force,F,istheeortvariableandvelocity,v,istheowvariable.Thereforethemechanicalimpedanceis v:(3{127) Similarly,intheacousticdomaintheeortvariableispressure,P,andtheowvariableisvolumetricow,Q;thus Q:(3{128) Aneectivearea,AeffcanbedenedsuchthatP=F=AeffandQ=vAeff.Physically,theeectiveareamaintainscontinuityofvolumevelocitybetweentheacousticandmechanicaldomain.BysubstitutingtheseexpressionsforPandQintoEquation 3{128 ,theacousticimpedanceiswrittenas vAeff2=Zm Tomodelthetransductionbetweenenergydomains,anidealtransformerisused.ThecircuitmodelofatransformerisshowninFigure 3-20 .Theidealtrans-formertransformspowerfromoneenergydomaintoasecondenergydomain.Therelationshipbetweentheeortandowforthetwoenergydomainsisdescribedby[ 21 ]

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wherenistheturnsratioofthetransformer.TheorientationofpositivepolaritiesanddirectionofpositiveowsareshowninFigure 3-20 Figure3-20.SymbolforthetransformerLEMelement. Thelumpedelementsforthemicrophonearefoundbyisolatingvariousaspectsofthemicrophoneandanalyzingenergystorageanddissipation.First,eachelementisfoundandthenassembledtoconstructthecompleteequivalentcircuitmodelofthemicrophone.Analysisoftheequivalentcircuityieldsclosed-formestimatesofvariousperformancemetrics,includingthebandwidthandpull-involtage.Theseclosed-formsolutionsprovideinsightintothescalingofthemicrophoneperformancemetricsaswellasprovidingasetofdesignequations. Anamingconventionisusedforthelumpedelementsthroughoutthisdis-sertation.AlumpedelementiswrittenasZm;nwhereZiseitheraresistance,capacitance,orinductance,mistheenergydomain,andnisanabbreviationfortheelement.Forexample,Ca;cavistheacousticcavitycompliance. 3-21

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ofthepistonisnottheareaofthediaphragm,ratheritischosentomaintaincontinuityofvolumevelocitybetweenthephysicaldiaphragmandpistonmodel.Thepressureonthepistonisassumedtobeuniformandthelumpedelementswillbefoundsuchthatthepistondeectionforagivenpressureisequaltothecenterdeectionofthediaphragmforthesamepressure. Figure3-21.Springandpistonmodelforadistributeddiaphragm. A .Thelumpedmechanicalcomplianceoftheplate,Cm;p,isgiveninEquation A{11 andrepeatedhere Thestorageofkineticenergyassociatedwiththeplatemotionisrepresentedbyalumpedmass.Thelumpedmechanicalmassoftheplate,Mm;p,isgiveninEquation A{17 andrepeatedhere 21 ].Ingeneral,thevolumevelocity,Q,isgivenby

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Forapistonwithuniformvelocity,thevolumevelocityisequalto SubstitutingEquation A{14 intoEquation 3{133 givesthefollowingexpressionforthevolumevelocityofthediaphragm,Q=2v(0)aZ01r a22rdr; whichisevaluatedto ComparingEquation 3{136 toEquation 3{134 ,theeectiveareaoftheclampedcircularplateis whichisonethirdofthephysicalareaoftheplate. Theeectiveareaisusedtorelatethemechanicallumpedelementsoftheplatetotheacousticequivalent.Itisalsousedtomodeltheelectrostaticbehaviorofthemicrophone.AsdiscussedinSection 3.1.3 ,theuseoftheeectiveareatoestimatethechangeincapacitanceasthediaphragmdeectsresultsinamoreaccuratemodel. 3{129 .Thelumpedacousticcomplianceoftheplateis 16Eh3;(3{138) whilethelumpedacousticmassis

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3-22 showsthemicrophonestructureandhowvariousaspectsaremodeled.Theventgeometryinthegureisfordiscussiononly;therearemanypossibleventchannelcongurations.Thereareresistivelossesduetotheairgaps[ 121 ]andthebackplateholes[ 122 ].Thecavitiesformedbetweenthediaphragmandeachbackplateandthelargecavitybeneaththethreeplatesactasanacousticcompliance.Inaddition,themassofthecavityairmaybesignicant[ 4 ].Theventchannelismodeledasauidicresistance[ 123 ].Eachoftheseelementsarerstdiscussedinfurtherdetail;thenthecompletelumpedelementmodelofthemicrophoneisconstructed. Figure3-22.Schematicdiaphragmofthedual-backplatemicrophoneshowinghowvariousfeaturesofthestructurearemodeled. 121 ] wherenhisthenumberofbackplateholesandairistheviscosityofair.ThetermAristheratioofthetotalareaofthebackplateholestotheareaofthebackplate

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andB(Ar)isdenedas 4ln1 8+1 2Ar1 8Ar2:(3{141) Thesecondcomponentofthebackplateresistanceisduetoviscouslossesasairpassesthroughthebackplateholes.Theacousticresistancemodelingthiseectis[ 122 ] whereabpandhbparetheradiusandthicknessofthebackplate,respectively.Thisexpressionfortheresistanceissimpliedbysubstitutingnha2h=a2bpforArasfollows, whereahistheradiusofeachbackplatehole. ThetotalresistanceintroducedbythebackplateandairgapisthesumofEquation 3{140 andEquation 3{143 .Bothresistancesareproportionaltotheviscosityofairandinverselyproportionaltothenumberofbackplateholes.TheresistancegiveninEquation 3{140 isalsoinverselyproportiontotheairgapcubed,g30.ThisresistanceissignicantasthedevicedimensionsarereducedtotheMEMSscale.ThesecondresistancegiveninEquation 3{143 isinverselyproportionaltotheradiusofeachbackplateholeraisedtothefourthpower,a4h.Thissuggeststhattheresistanceislesswithasmallernumberoflargerholes;however,thistypeofbackplatedesignnegativelyeectstheelectrostaticperformanceofthedevice. 4 ] 0c20:(3{144)

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Thecavitycomplianceisproportionaltothevolumeofthecavity.Therefore,fortheshallowcavitiesbetweenthediaphragmandthetwobackplates,thecompliancehereisnegligible;i.e.theairdoesnotcompressinthesesmallcavities.Thisisdesirableforthedual-backplatemicrophonesothatthefullincidentpressuredeectsthediaphragm. Themotionoftheairinthecavityalsohaskineticenergythatisrepresentedbyanacousticmass,givenby[ 4 ] whereAcavisthecrosssectionalareaofthecavity. 123 ] whereLeffistheeectivelengthofthechannelandDisthehydraulicdiameter.Theeectivechannellengthconsidersfeaturesofthechannelsuchasbends,whichmakethechannelbehaveasthoughitislongerthanthephysicallength. 3.1.2 ,wherethebehaviorofthesingle-backplateanddual-backplatecondensermicrophonewasdiscussedfortwobiasconditions.Separateequationsfortheoutputwereneededforeachcase.Inthissection,thetransformermodelisderivedforthesingle-backplatemicrophoneforbothbiasingconditions.Itisshownthatthemodelisthesameforbothcases.Then,thetransformermodelisextendedtothedual-backplate

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microphone.ThederivationsforthetransformerparametersutilizepreviousderivedresultsfromSection 3.1.2 3-23 .Thediaphragmisrepresentedbyitscompliance,Ca;d,andthenominalcapacitancebetweenthediaphragmandbackplateisC10.Themeancapacitanceisusedbecausealinearcircuitmodelisbeingdeveloped. Figure3-23.Cross-sectionofthesingle-backplatecapacitivemicrophoneshowingrelevantparametersforthetransformerdiscussion.Theoutputiseitherthechargeorvoltageonthediaphragm. 3-24 .Theinputpressureresultsinapressure,pd,thatactsonthediaphragm.Thispressureresultsinanoutputcharge,Q1.Thetransformerconvertsthepressureacrossthediaphragm,pd,toavoltage,vo.Thisvoltageisnotaphysicalvoltageonthemicrophone;rather,itrepresentsthetransduction.ThenalelectricaloutputisthechangeinchargeonthecapacitorC10.Fortheconstantvoltagecase,therightterminalofC10isconnectedtosmallsignalground.However,itcouldhaveanon-zeroDCvoltagelevelthatwouldbeconsideredinthenetbiasvoltageonthemicrophone.

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Figure3-24.Transformermodelingthetransductionfromtheacousticdomaintotheelectricaldomainforasinglecapacitorbiasedwithaconstantvoltage. FromEquation 3{130 ,thevoltage,vo,is Thechargeonthecapacitorduetothesmallsignalvoltagevoisthechangeincharge,Q1,fromthenominalcharge.Thischangeinchargeistheoutputchargeforthecapacitivemicrophonebiasedwithaconstantvoltage.Thus, Q1=C10vo(3{148) EquatingtheQfromEquation 3{34 ,usingthechangeincapacitancegiveninEquation 3{119 ,toEquation 3{148 resultsin 3g0 whichissimpliedto 3g0 SubstitutingEquation 3{147 forv0andEquation 3{8 forg0,theturnsratio,n,becomes 3VB

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The1=3factoraccountsforthediaphragmcurvatureandthephysicaldiaphragmradius,ad,isusedinthismodel. 3-25 .TheoutputisnowthevoltageattherightterminalofthecapacitorC10.ThemicrophoneisloadedbytheparallelcombinationofCpandCi. Figure3-25.Transformermodelingthetransductionfromtheacousticdomaintotheelectricaldomainforasinglecapacitorbiasedwithaconstantcharge. TheoutputvoltageinFigure 3-25 is Theoutputvoltageforasingle-backplatemicrophonewithaconstantchargebiaswaspreviouslygiveninEquation 3{57 ,whichisrewrittenas usingthemeancapacitancevalueC10tolinearizetheexpression.SubstitutingEquation 3{147 forvoandEquation 3{119 forC1andequatingEquation 3{152 andEquation 3{153 yields 3g0

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SubstitutingEquation 3{8 forg0,theturnsratioforthecondensermicrophonebiasedthroughalargeresistoris 3VB Thisisidenticaltotheturnsratiofortheconstantvoltagecase,giveninEqua-tion 3{151 ;thusthesametransformermodelisusedforeitherbiasingscheme. 3-26 .Thediaphragmisrepresentedbyitscompliance,Ca;d.ThenominalcapacitancebetweenthediaphragmandtopbackplateisC10andthenominalcapacitancebetweenthebottombackplateandthediaphragmisC20.Themeancapacitancesareusedbecausealinearcircuitmodelisbeingdeveloped. Figure3-26.Cross-sectionofthedual-backplatecapacitivemicrophoneshowingrel-evantparametersforthetransformerdiscussion.Theoutputiseitherthechargeorvoltageonthediaphragm. 3-27 .Thismodeldiersfromhistoricaltransformermodelsfordierentialelectrostatictransducersinthattwotransformersareused.Previousmodelshaveusedasinglecenter-tappedtransformer[ 49 ].However,thismodelpresentedhereispreferredbecauseit

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clearlyidentiesthecontributionofthesecondactivecapacitorandissuitableforbothmethodsofapplyingthebiasvoltagetothemicrophone.Thecenter-tappedtransformermodelwasonlyusedwiththevoltagebiasappliedthrougharesistor.Furthermore,thedouble-transformermodelgivesphysicalinsightintotherelationshipbetweensingle-backplateanddouble-backplateelectrostatictransducers. Figure3-27.Transformermodelforthedual-backplatecapacitivemicrophonebi-asedwithaconstantvoltage. Twotransformersareincludedinthismodeltorepresentthetransductionofboththetopandbottomcapacitors.TheybothareconnectedacrossCp,thusthepressurepdisontheacousticsideofthetwotransformers.Thepressurepdisconvertedtothevoltagesvo1andvo2throughthetwotransformers.Tocompletethetransformermodelforthedual-backplatecondensermicrophone,n1andn2mustbedetermined. Theoutputofthedual-backplatecapacitivemicrophonebiasedwithaconstantvoltageisQ1+Q1,asshowninFigure 3-27 .ThesechargeswerefoundinSection 3.1.2.3 tobeCVB,wherethecapacitancechangesaregivenbyEquation 3{119 andEquation 3{120 ,respectively.BysubstitutingEquation 3{8 for

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Q1=pd and Q2=pd Tondn1andn2,thechargeresultingfromvo1andvo2isconsidered.Thechargeduetovo1is Q1=C10n1pd;(3{158) similarly,thechargeduetovo2is Q2=C10n2pd:(3{159) ComparingEquation 3{158 toEquation 3{156 andEquation 3{159 toEquation 3{157 ,itcanbeseenthatn1isequalton2;furthermore, 3VB Again,thediaphragmcurvatureisfactoredintothismodel. 3-28 .Aswasthecaseforthesingle-backplatemicrophone,theoutputisloadedbytheparallelcombinationofCpandCi. Theoutputduetovo1andvo2is and

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Figure3-28.Transformermodelforthedual-backplatecapacitivemicrophonebi-asedwithaconstantcharge. respectively.Theoutputcontributionsfromthetopandbottomcapacitorsfoundusingelectrostaticanalysis,giveninEquation 3{93 andEquation 3{96 ,canbeexpressedas and ComparingEquation 3{163 toEquation 3{161 andEquation 3{164 toEquation 3{162 ,theturnsratiosforthedual-backplatemicrophoneareequaltothoseforthesingle-backplatemicrophone. 3VB Thecurvatureofthediaphragmdeectionisrepresentedinthismodelbythe1=3factor.Thisaccountsfortheeectiveareaofthediaphragm.Asinglemodelisusedtomodelanycondensermicrophone.Ifthemicrophoneisadual-backplatedevice,twotransformersaresimplyincludedinthemodel.Itisalsonotedthatthe

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turnsratioforthetransformerisnotdirectlyrelatedtothedevicecapacitances.Rather,theturnsratioisafunctionofthenominalgapdistance,g0,andthediaphragmpropertiesincludingtheradius,ad. 12 ]. Theelectrostaticcomplianceisrelevantforthediscussionofthemicrophonedynamics.Theconstantvoltagecasecorrespondstocondensermicrophonesbiaseddirectlywithavoltagesource.Theconstantchargecasecorrespondstocondensermicrophonesbiasedthroughalargeresistor.Forhighfrequencydiaphragmmotions,thechargeonthecondensermicrophoneisconstant. Tondtheelectrostaticcompliance,rst,theelectrostaticforceisidentied.Theelectrostaticcomplianceisgivenby 1 Thiscomplianceisfoundforthesingle-backplateanddual-backplatecondensermicrophones. 3.1.2.2 .Theseresultsareusedtodeterminetheelectrostaticcompliance.

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3{47 .Thustheelectro-staticcomplianceis 1 dg01 2VB20Aeff Tohavealinearcircuitelement,thecompliancemustnotdependonthepositionofthediaphragm,g0.AlinearapproximationofEquation 3{167 isfoundassumingg0issmall.Thisresultsinthemechanicalelectrostaticcompliance, whereg0istheequilibriumgapofthebiasedsinglebackplatecondensermicro-phone.Therefore,theacousticelectrostaticcomplianceofasingle-backplatecondensermicrophonewithaconstantvoltageis 3{64 .Thus,theelectrostaticcomplianceis 1 dg0QB2 Theelectrostaticforcedoesnotdependong0,thusthestinessiszero,andthecomplianceis Physically,thismeanstheelectrostaticcompliancedoesnotimpactthedynamicresponseofasingle-backplatecondensermicrophonewithaconstantcharge.

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3{84 .Thisyieldsanelectrostaticcomplianceasfollows, 1 dg0"2VB20Aeffg0g0 EvaluatingEquation 3{172 andassumingg0issmall,yields fortheacousticelectrostaticcompliance.Thiscomplianceisone-halfofthevalueforanequivalentsingle-backplatecondensermicrophone.Thusthesofteningimpactonthedynamicbehaviorislargerforasingle-backplatecondensermicrophone. 3{104 )isequaltozero.Thustheelectrostaticcomplianceis anddoesnotaectthedynamicbehaviorofthedual-backplatecondensermicro-phonewithaconstantcharge. 3-29 .Linearapproximationsoftheelectrostaticforcearegivenasdashedlines.Itisthislinearizedforcethatisrepresentedbytheelectrostaticcompliance.Duetothenon-linearnatureoftheelectrostaticforce,the

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electrostaticcomplianceisareasonableapproximationforsmalldeviationsfromtheequilibriumpoint. Figure3-29.Comparisonofthediaphragmrestoringforcetotheelectrostaticforce. ByreferringtoFigure 3-30 tovisualizetheowthroughthemicrophonestructure,thenalsimpliedlumpedelementmodelisconstructed;itisshowninFigure 3-31 .TheoutputiseitherQorVoutdependingonifthebiasvoltageisapplieddirectlyorthroughabiasresistor.TheLEMisthesameforeithercase,andtheinterfacecircuitryforcesthedesiredbiascondition. Theincidentpressurerstseesthetopbackplateandtheventchannel.Theowthroughthebackplatemayeitherpassthroughthebackplateholesordeectthebackplate;thereforethebackplatecomplianceandresistanceareconnectedinparallel.Becausethecavitiesbetweenthediaphragmandbackplatesarevery

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Figure3-30.Schematicdiaphragmofthedual-backplatemicrophoneshowinglumpedelementsincludedintheLEM. Figure3-31.Lumpedelementmodelofthedual-backplatecondensermicrophone. small,thecomplianceofthesecavitiesisneglected.Theowthendeectsthediaphragm;thusthediaphragmmassanddiaphragmcomplianceareconnectedinserieswiththebackplateimpedance.Thesamemodelthatwasusedforthetopbackplateisusedforthebottombackplate;thisisconnectedinserieswiththetopbackplateanddiaphragmbecauseallthreeelementshavethesameow.After

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passingthroughthebottombackplate,thevolumevelocitymayeithercompresstheairinthecavityorpassthroughtheventchannel.Theventchannelisinparalleltotheseriescombinationofthetopbackplate,diaphragm,andbottombackplate;nallythecavitycomplianceisconnectedsuchthatitcanbefedfromtheventchannelortheowthroughthebottombackplate.Forthedesignedmicrophone,thecavitymassismuchlessthanthediaphragmmass,thusthistermisneglected. AsummaryofthelumpedelementsusedisgiveninTable 3-5 .ThecomplianceCa;pisusedforCa;bp1,Ca;d,andCa;bp2.Similarly,theplatemassisusedforMa;d.Theresistanceofeachbackplate,Ra;bp1andRa;bp2,isgivenbyRa;g+Ra;h.Theturnsratiosofthetopandbottomtransformers,n1andn2respectively,arebothgivenbyn. Table3-5.Expressionsfortheacousticlumpedelementsofthemicrophone. 0c20Ra;vVentresistance128Leff 3a4 3-31 ,thediaphragmisrepresentedbytheacousticcomplianceCa;d.Thisrepresentsthediaphragmcomplianceandtheelectrostaticcompliance,if

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present;andisgivenby Theacousticdiaphragmcompliance,Ca;d,andtheelectrostaticcompliance,Ca;el,sharethesameow.Therefore,theyareinseries,asrepresentedbyCa;d.Forcaseswheretheelectrostaticcomplianceisnegligible,Ca;dsimplybecomesCa;d. B ThefrequencyresponseoftheequivalentcircuitmodelshowninFigure 3-31 isgiveninEquation B{12 ;repeatedhereforconvenience, whereZpistheimpedanceoftheseriescombinationofthediaphragmandbothbackplates.Neglectingthecomplianceofthebackplates,thesimpliedfrequencyresponsecanbeapproximatedbyEquation B{15 ;repeatedhere, InFigure 3-32 ,thefrequencyresponseofpd=pingiveninEquation B{15 isplotted.Thevaluesusedforthisexamplearechosentoaccentuatethenotablefeaturesofthefrequencyresponse.Thediaphragmmass(7104[kg=m4])andcompliance(1:51016[m5=N])wereselectedtogivearesonantfrequencynear

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20kHz.Thetotaldampingof14108[Ns=m5]waschosentogiveanunder-dampedresponse.Finally,thecavitycompliance(601016[m5=N])andventresistance(11013[Ns=m5])werechosentogiveacut-onfrequencynear10Hz. AsshowninFigure 3-32(a) ,theatbandofthefrequencyresponseisbetweenthecut-onfrequency,fcut,andtheresonantfrequency,f0.Theloadingoftheelectricaldomainthroughthetransformersisneglectedhere.ThiseectdependsonhowthemicrophoneisbiasedandisdiscussedfurtherinSection 3.2.4 .Thedynamicsofthemicrophonestructurewithoutbiasaregivenbelow. Magnituderesponseofpd=pinforanexamplemicrophone. (b) Phaseresponseofpd=pinforanexamplemicrophone.

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Forpropermicrophoneoperation,thebackplateresistancesmustbemuchsmallerthantheventresistance,thustheycanalsobeneglected.ThelowfrequencyequivalentcircuitisshowninFigure 3-33 Figure3-33.Lowfrequencyequivalentcircuitofthedual-backplatemicrophone. Theexpressionforpd=pinforthissimpliedcircuitisgivenby Thisisasingle-polehighpassfunctionwithacut-onfrequencyof 2Ra;vCa;cav+Ca;d;(3{179) andaslopeof+20dB=decadebelowfcut[ 124 ]. 3-34 forfrequenciesneartheresonantfrequency.Athigherfrequencies,theventresistanceiseectivelyanopencircuit,comparedtotheotherimpedances.Thus,themicrophoneisapproximatelyasimplesecondordersystemwithamassofMa;d,acomplianceofC1a;d+C1a;cav1,andaresistanceofRa;bp1+Ra;Rbp2.

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Figure3-34.Highfrequencyequivalentcircuitofthedual-backplatemicrophone. Thetransferfunction,pd=pin,forfrequenciesneartheresonanceofthemicro-phoneisgivenby Themicrophoneischaracterizedintermsofitsnaturalfrequency,!0,anddampingratio,[ 124 ].Thenaturalfrequencyis whilethedampingratioisgivenby Ifthemicrophoneislightlydamped,orismuchlessthanone,thefrequencyresponsehasamaximumat!0.However,asapproachesone,thepeakinthemagnituderesponsemovesawayfrom!0to Ifisgreaterthanone,thenthemicrophoneisoverdamped,andthefrequencyresponsedoesnothaveapeak.

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3-35 Figure3-35.Equivalentcircuitofthedual-backplatemicrophoneformid-rangefrequencies. Intheatbandregionofthefrequencyresponse,theequivalentcircuitofthemicrophoneisreducedtoacapacitivepressuredivider.Themicrophoneresponseintheatbandis Thus,theincidentpressureisattenuatedbythecavitycompliance.Toavoidattenuation,thecavityshouldbelargesuchthatCa;cav>>Ca;d.Thisallowsthefullincidentpressuretoactonthediaphragm. 3{78 orEquation 3{99 withaconstantvoltagebiasoraconstantchargebias,respectively.

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Intheseexpressionsfortheoutputvoltage,thepressurepisthepressureacrossthediaphragm,pd. Inderivingtheexpressionsforthemicrophoneoutputvoltage,itwasassumedthattherelationshipbetweenthepressureincidentonthediaphragmandthedia-phragmdeectionislinear,asgivenbyEquation 3{8 .Furthermore,aparallelplatecapacitorwasassumedinwhichthechangeintheairgapwassolelyduetothediaphragmdeection.However,ifthereisasignicantpressuredierentialacrossthebackplates,andifthebackplatesaresucientlycompliant,thebackplateswillhaveanon-negligibledeection.ShowninFigure 3-36 isasituationwheretheairgapchangesbothduetodiaphragmandbackplatemotion.Theairgapsofadual-backplatemicrophonecanbeexpressedas and whereg0d,g0bp1,andg0bp2arethedeectionofthediaphragm,topbackplate,andbottombackplate,respectively. Figure3-36.Schematicofanairgapchangingduetobothdiaphragmmotionandbackplatemotion.

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Thefrequencyresponseforpd=pingiveninEquation B{12 factorsintheeectofthebackplatecompliances;itisrepeatedhere, Thisexpressionforpd=pinissimilartothatgiveninEquation B{15 ,exceptthatZpisnowageneralimpedancethatincludestheeectsofthebackplatecompliance.Asimilarexpressionforpbp1=pinandpbp2=pincanalsobefoundusingEquation B{16 andEquation B{17 .Afterthefrequencyresponsehasbeendeterminedforthepressureacrossthediaphragmandeachbackplate,thefrequencyresponseforthetotalchangeintheairgapscanbefound. ThedeectionofaplatewithcomplianceCcanbefoundifthepressureacrossthecomplianceisknown.Intheelectricaldomain,thechargeonacapacitorisgivenbyQe=CV.Similarlyintheacousticdomain,theacousticdisplacement,qais Therefore,thedeectionisgivenby Aeff:(3{189) UsingEquation 3{189 ,Equation 3{185 ,andEquation 3{185 ,thefrequencyresponseforthechangeinthetwoairgapsis and Thefrequencyresponseofg0forthetopcapacitorofadualbackplatemi-crophoneisshowninFigure 3-37 fortwocases:(1)anidealbackplatewithzerocompliance;and(2)abackplatewithnitecompliance.Thecomplianceofthe

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Magnituderesponseofthenormalizedchangeintheairgapdistance. (b) Phaseresponseofthenormalizedchangeintheairgapdis-tance. backplateforthisexamplewaschosentomakethebackplatecornerfrequencyapproximately100Hzsothattheeectisclear.UsingthevaluesusedfortheexamplemicrophoneinSection 3.2.6 ,abackplatecompliancethatis1000thedia-phragmcomplianceisneededtoobservetheeectofnitebackplatecompliance. Atlowfrequencies,thetwofrequencyresponsecurvesarethesame,howeverathigherfrequenciestheydiverge.Themagnitudeofthetotalgapchange,g0,islesswhenthecomplianceofthebackplateissignicant;thisreducesthesensitivityathigherfrequencies.Therearealsoanomaliesinthephaseresponseathigherfrequencies.Itisclearlydesirabletodesignthemicrophonesuchthattheeectofthebackplatecomplianceonthemicrophone'sresponsetoanincidentpressure

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isnegligible.However,thecompliantbackplatestilldeectsduetoelectrostaticforces. 3.1.2 ,theelectrostaticforceactingonthediaphragmhasthepotentialtocausethediaphragmtocollapseintothebackplate[ 12 ].Thisphenomenonisknownaspull-in.Anunderstandingofthepull-inbehaviorofamicrophoneisnecessarytodetermineitsstabilityandallowableoperatingconditions.Thestudyofthepull-inofamicrophonecanbedividedintotwoclasses:(1)quasi-staticpull-in,wherethediaphragmisinitiallyatrest,theincidentpressureiszero,andtheelectricalbiasisconstant;and(2)dynamicpull-in,wherethediaphragmmotion,incidentpressure,andtimevaryingelectricalbiasconditionsareconsidered[ 125 ].Onlyquasi-staticpull-inisconsideredhere;thisgivesthelimitsforthemaximumDCbiasvaluesthatcanbeused.Furtherdiscussionondynamicpull-inforcondensermicrophonescanbefoundin[ 118 ].Duringuseofthemicrophone,however,alowerelectricalbiasmustbeusedtoensurestability.Thequasi-staticpull-inisexaminedforbothsingle-backplateanddual-backplatecondensermicrophones. Forelectrostaticpull-intooccur,theelectrostaticforcemustincreaseasthediaphragmapproachesthebackplate.Becausestaticpull-inisonlybeingconsideredhere,thisanalysisisvalidregardlessofhowthebiasvoltageisapplied. AsshowninFigure 3-38 ,therearetwoforcesactingonthediaphragm.Themechanicalforce,Fm,istherestoringforceofthediaphragmandtheelectricalforce,Fe,istheelectrostaticforceduetotheappliedbias.Forsingle-backplatemicrophones,thisforcehasonecomponent;while,fordual-backplatemicrophones,thisforceisthesumoftwoelectrostaticforces.Forallcases,themechanicalforceis

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Thisisre-writtenintermsoftheairgapdistance,g, Theanalysisofthequasi-staticpull-infollowstheworkofSenturia[ 12 ]. Figure3-38.Single-backplatecapacitivemicrophoneschematicshowingtherelevantforcesforquasi-staticpull-in. 3{47 .Thisisexpressedintermsofg,theinstantaneousairgapdistance, 2VB20A g2:(3{194) AscanbeseenfromEquation 3{193 andEquation 3{194 ,themechanicalandelectrostaticforcesopposeeachother.Theelectrostaticforceisdirectedtowardsthebackplateinthepositivexdirection.Themechanicalforceisalwaysdirectedtowardsthediaphragmrestposition;assumingapositivediaphragmdeection,themechanicalforceisdirectedinthenegativexdirection. Thenetforceactingonthediaphragmis 2VB20A g21 Whenthebiasvoltageisinitiallyappliedtothemicrophone,theelectrostaticforceisgreaterthatthemechanicalrestoringforce.Atequilibrium,thetwo

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forcesbalanceeachother,thusFnetiszero.Thestabilityoftheequilibriumpointisdeterminedbywhetherasmallperturbationofthediaphragmtowardsthebackplatecausesanincreaseinthenetforceoradecrease.UsingthecoordinatesysteminFigure 3-38 ,anincreaseinFnetcausesthediaphragmtoacceleratetowardsthebackplate;thustheequilibriumpointisunstable.However,ifFnetdecreases,thentheequilibriumpointisstable. Forthesingle-backplatecondensermicrophone,@Fnetisgivenby Therefore,ifthediaphragmisperturbedtowardsthebackplate,@g<0,thenthequantityinparenthesismustbepositivesuchthat@Fnet<0.Thus, 1 Agapdistance,gPI,isdenedatwhichtheequilibriumtransitionsfromstabletounstable.ThebiasvoltagethatresultsinthisequilibriumpositionisVPI,whichisthepull-involtage.Atthiscriticalcondition,Equation 3{197 becomes 1 BysubstitutingEquation 3{198 intoEquation 3{195 andevaluatingatgPIandVPI,thecriticalgapdistancewherethestable-to-unstabletransitionoccursisfoundtobe 3g0:(3{199) FromEquation 3{199 andEquation 3{198 thepull-involtageis ThebiasvoltagemustbelessthanVPItoavoidquasi-staticpull-in.

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3{193 andtheelectrostaticforceisgivenbyEquation 3{84 .However,theforcesarewrittenintermsofg0ratherthanthegapdistancegbecausetherearenowtwogaps.Theelectrostaticforcehasbeendenedsuchthatapositiveg0decreasesthetopairgapandincreasesthebottomairgap. Thenetforceactingonthediaphragmis Now,thechangeinFnetduetoachangeing0isexamined.Tohaveastablesystem,apositive@g0mustproduceanegative@Fnet.ThedierentialofEqua-tion 3{201 withrespecttog0is Thequantityinthebracketsmustbenegativetohaveastablemicrophone,thus 1 Atthevergeofpull-in,withanappliedvoltageofVPIandanequilibriumpositionofg0PI,Equation 3{202 isre-writtenas 1

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SubstitutingEquation 3{204 intoEquation 3{201 andevaluatingatg0PIandVPIresultsin Therefore,theequilibriumpositionisatg0=0;substitutingthisintoEquation 3{204 resultsintheexpressionforthepull-involtage, Theresultsareobtainedthroughthisanalysisusingseveralassumptions.Thediaphragmdeectionisassumedtobelinear.Furthermore,thecapacitorformedbythediaphragmandbackplateisassumedtoremainparallelasthediaphragmdeects;thisisclearlynotthecaseasthediaphragmapproachesthebackplate.Finally,thebackplateisassumedtoberigidandforthedualbackplatemicrophones,thetwocapacitorsareassumedtobeidentical;thismaynotbethecaseinthephysicalmicrophone.Therefore,thesepredictionsmaynotmatch

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experimentallydeterminedvaluesofthepull-involtage.However,theseresultsdoprovidephysicalinsightintothepull-inphenomenonaswellasscalinginformation.Thepull-inproblemisinvestigatedfurtherbyLiuetal.[ 118 ]. Toderivethenoisebehaviorofthemicrophone,variousnoisesourcesareanalyzed.Eachoftheseismodeledasapowerspectraldensity(PSD)havingtheunitsofpowerperHz;i.e[V2=Hz]or[Pa2=Hz].ThenoisePSDfromeachnoisesourceisreferredtoacommonpointandtherespectivePSDsaresummed. 3-39 .Inthismodel,thecomplianceofthebackplateshasbeenneglected.Furthermore,thedampingofthebackplatesaswellasanyinternalstructuraldampingofthediaphragmhavebeencombinedintoasingleequivalentacousticresistor,Ra;eff.EachresistorinFigure 3-39 hasathermomechanicalnoisesourcewhichmayberepresentedbyeitheravolumevelocitynoiseorapressurenoise3.ThenoiseduetoresistorRa;effisrepresentedbyapressurenoisePSD,SP;Reff.ThenoiseduetoRa;v,however,isrepresentedbyacurrentnoisePSD,SQ;Rv.ThePSDofthesesourcesis4kTRa;effand4kT=Ra;v,respectively[ 20 ].

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Figure3-39.Acousticnoisemodelofthemicrophone. Tondthenoisecontributionsoftheacousticdampingwithinthemicrophone,thenoisefromeachofthesourcesisreferredtotheportlabeledSd.Thepressureatthislocationforcesthediaphragmandgeneratestheoutputvoltageofthemicrophone.Forthenoiseanalysis,theincidentpressureissettozeroandreplacedwithashortcircuit.ThenoisefromeachresistorisfoundindividuallyandthetotalnoisePSDatSdisthesumofthetwonoisecontributions. ThecircuitshowninFigure 3-39 reducestothenoisemodelshowninFigure 3-40 whenonlySP;Reffisconsidered.ThecurrentnoisesourceSQ;Rvisreplacedwithanopencircuit.TopressurenoisePSDreferredtothediaphragmisfoundusingapressuredivider.ThemagnitudesquaredofthepressuregainfromSP;RefftoSdisusedbecausethephaseinformationisirrelevantforuncorrelatednoisesources.

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Thiscanberewrittenas whichgivesanexpressionforthenoisePSDduetotheresistorReffreferredtothediaphragm. Figure3-40.AcousticnoisemodelforReff. Consideringonlythenoiseduetotheventresistor,thenoisemodelisdrawnasinFigure 3-41 .ThevolumevelocityPSDthroughthediaphragmduetoSQRvisfoundusingavolumevelocitydivider, whereZdistheseriesimpedanceofthediaphragmmass,diaphragmcompliance,andeectiveresistance.Solvingfortheparallelimpedanceinthenumeratorandsimplifying,Equation 3{209 becomes

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Figure3-41.AcousticnoisemodelforRv. ThepressurenoiseacrossthediaphragmcapacitanceduetothevolumevelocitynoisegiveninEquation 3{210 isSP;dRv=jQd=sCa;dj2, ThetwonoisequantitiesgiveninEquation 3{208 andEquation 3{211 areplottedinFigure 3-42 foranexamplemicrophone.Whenreferringthenoisesourcestothediaphragm,thenoisesourcesareshapedbythedynamicsofthemicrophone,asrepresentedbytheLEM.ThenoiseduetoRa;effisatbetweenthecut-onfrequencyandresonantfrequencyofthemicrophone.However,thenoisePSDduetotheventresistorisproportionalto1=f2inthesamefrequencyrange.Thus,theventresistorhasthepotentialtobethedominantacousticnoisesourceforlowfrequencies. Atlowfrequencies,thepressurenoisePSDduetoRa;effapproachesSP;Reff;meaningthatthetotalnoiseduetoeectiveresistanceappearsacrossthedia-phragm.Similarly,atlowfrequencies,thevolumevelocitynoisePSDduetoRa;vapproachesthevalueofR2a;vSQ;Rv.Athighfrequenciesbothofthenoisesourcesareshapedbytheresonantpeakandrolloaftertheresonance.

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Figure3-42.Theoreticalnoisecontributionsofamicrophoneexamplereferredtothepressureacrossthediaphragm. 3.1.2 ,capacitivemicrophonesmaybeusedwithvarioustypesofinterfacecircuitry;inparticular,achargeamplieroravoltageamplier.Inthissection,thenoiseofbothcircuitsisdiscussed. 3-43 .Thechargeamplierconsistsofanoperationalamplierwithresistor,Rfb,andacapacitor,Cfb,inthefeedbackloop.Attheinputoftheamplierisabiasresistor,Rb,andacapacitor,Ctot,thatrepresentsthetotalcapacitanceattheinputtotheamplier.Thisincludesthedevicecapacitanceaswellasanyparasiticcapacitance.Theinternalamplierismodeledbyavoltagenoiseandacurrentnoise,SvaandSia,respectively.Thebiasresistorandfeedbackresistorarebothmodeledbycurrentnoisesources,SiRfbandSiRb,respectively. Thetotalnoiseattheoutputisthesumofthecontributionsfromeachofthenoisesourcesandisfoundusingsuperposition.Tosimplifytheexpressions,theimpedancesZfbandZiaredened.Thesearethefeedbackimpedanceandinput

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Figure3-43.Noisemodelofthechargeampliercoupledtothemicrophone. impedanceandaregivenby and respectively. TheoutputnoisePSDofthechargeamplierduetotheelectricalnoisesourcesis ThevoltagenoisePSDoftheamplierscaledbythefactorj1+Zfb=Zij2.ThecurrentnoisePSDoftheamplierandthetworesistorsaddedtogetherareallscaledbythemagnitudeofthefeedbackimpedancesquared.Atfrequenciesabove1=2CfbRfband1=2CtotRb,whichisthetypicallythefrequencyrangeofinterestforachargeamplier,Equation 3{214 issimpliedto

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ThehigherthecapacitanceCtot,thehigherthecontributionduetoSva.Whileahighdevicecapacitanceisbenecialforsensitivity,parasiticcapacitanceincreasesthenoiseattheoutputofthechargeamplier.ThecurrentnoisePSDisscaledbyafactorofj1=sCfbj2.Thisfrequency-shapesthecurrentnoisespectrumsuchthatthenoisePSDattheoutputisproportionalto1=f2.Furthermore,thisistwicetheslopeofelectronicicker(1/f)noise[ 19 ].Therefore,thecurrentnoiseisdominantatlowfrequencies. 3-44 .Thenoisesourcesincludetheamplierandbiasresistor.Similartothechargeamplier,thetotalimpedanceattheinputtotheamplierisZi,givenbyEquation 3{213 Figure3-44.Noisemodelofthevoltageampliercoupledtothemicrophone. ThevoltagenoisePSDattheoutputis Aswasthecasewiththechargeamplier,thecurrentnoisePSDfromtheamplierandbiasresistoraddandarescaledbythemagnitudeoftheinputimpedancesquared.AhighfrequencyapproximationtoEquation 3{214 is Therefore,thevoltageamplierhasthepotentialtoexhibitnoiseshapingduetothecapacitanceCtot,similartothe(1=f2)noiseseeninthechargeamplier.Inthis

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case,ahigherparasiticcapacitancewouldreducethecontributionofthecurrentnoisesourcesbyincreasingCtot.Furthermore,boththevoltagenoiseandcurrentnoisemustbeconsideredwhenselectingavoltageamplier.

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Inthepreviouschapter,thetheoryofoperationofthemicrophonewasdis-cussed.Usingthedevelopedtheory,thedesignofthedualbackplatecapacitivemicrophoneispresented.Thisisfollowedbythetheoreticalmicrophoneperfor-mance. 1-1 foranaeroacousticmicrophone.ThemicrophonewasdesignedspecicallytobefabricatedusingtheSUMMiTVprocessatSandiaNationalLaboratories[ 16 ];thisprocessisdescribedindetailinAppendix D .Theuseofthisprocessimposesseveralconstraintsonthedesign;specically,thethicknessandresidualstressofeachlayerarenotfreetobespecied.Inthissection,theuseoftheSUMMiTVprocessowtocreatethedual-backplatemicrophonestructureisdiscussedrst.Thenthedesignforthediaphragmandthebackplatesispresented. D-1 .ThepolysiliconlayersmustbepatternedtocreateadevicesimilarinstructuretothatshowninFigure 3-1 .Thesacricialoxidelayersareusedtosupportthepolysiliconlayersduringfabrication,butarethenremovedattheendoftheprocess;thustheycannotbeusedasstructurallayers. Therearethreepolysiliconlayersthatareusedforthemicrophonestructure:Poly4isusedforthetopbackplate;Poly3isusedforthediaphragm;andthePoly2/Poly1laminateisusedforthebottombackplate.ThetopairgapiscreatedfromSacOx4andthebottomairgapiscreatedfromSacOx3.Inthedual-backplatestructuredepictedinFigure 2-11(a) ,whichissimilartothemicrophonedevelopedbyRombachetal.[ 47 ],thediaphragmisseparatedfromthebackplatesbyinsulatinglayers.ThisisnotthebestarrangementfortheSUMMiTVprocess 141

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becauseonlysilicondioxidecouldbeusedfortheinsulatinglayersandthesilicondioxideisetchedduringrelease.Therefore,forthismicrophone,anchorsarecreatedsuchthatbothbackplatesandthediaphragmaresupportedasshowninFigure 4-1 .Inthisarrangement,thebackplatesanddiaphragmareessentiallyconcentricshells. Figure4-1.Crosssectionofthedesigneddual-backplatemicrophone. Figure4-2.Microphone3-Dview. ThemicrophoneisshownfromthetopinFigure 4-2 .Fromthisview,thediaphragmandbottombackplatearenotvisible.Thebondpadsforthediaphragmandbothbackplatesarelabeled.Theotherbondpadsmaybeusedtoconnecttothesubstrateandtonullouttheparasiticcapacitanceunderneaththethreeconnectionstothediaphragmandbackplates. AdditionaldetailsofthemicrophonestructureareshowninFigure 4-3 .Theanchorsforthebottombackplate,diaphragm,andtopbackplateareshownin

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Crosssectionshowingthedetailsoftheanchorstructure. (b) Crosssectionshowingthedetailsoftheelectricalconnectionfromthediaphragmtothebondpad. (c) Crosssectionshowingthedetailsoftheelectricalconnectionfromthebottombackplatetothebondpad. Figure 4-3(a) .Thepolysiliconlayersarestackedtoconstructtheanchorsforthediaphragmandbackplate.Tomakeelectricalconnectionstothebottombackplateanddiaphragm,tunnelsarefabricatedintheanchors.TheelectricalconnectiontothediaphragmisshowninFigure 4-3(b) .ThePoly0layerofpolysiliconconnectsthediaphragmanchortothebondpad;abovethisconnection,thetopbackplateanchordoesnotmakecontactwithPoly0inthisregion.AsimilargeometrywasusedfortheconnectiontothebottombackplateasshowninFigure 4-3(c) .Here,

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theanchorsforboththediaphragmandtopbackplatedonotextenddowntoPoly0. AftertheimplementationstrategyfortheSUMMiTVprocessisdeveloped,thespecicdimensionsofthemicrophonearethendesigned.DuetotheconstraintsoftheSUMMiTVprocess,theparametersthatarefreetobechosenarethediameterofthediaphragmandthebackplateholegeometry. 3.1.1 areusedhere. Thepercentnonlineardeectionisdenedas %NL=wL(0)wNL(0) wherewL(0)isthelinearcenterdeectionofthediaphragmandwNL(0)isthenonlinearcenterdeectionofthediaphragm.BysubstitutingEquation 3{8 andEquation 3{9 intoEquation 4{1 andsettingthepercentnonlinearityto3%,thefollowingisobtained, 11 1+0:488w(0)2 SolvingEquation 4{2 givesaconstraintfortheratioofthecenterdeectiontothediaphragmthickness, Thisresultisgeneralforahomogeneous,circular,stress-freeplate;whenthemagnitudeofthecenterdeectionisequalto25%ofthethicknessoftheplate,thedeectionis3%nonlinear.SubstitutingEquation 4{3 intoEquation 3{9 resultsinthefollowing 16Eh4=0:25:(4{4)

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ThematerialconstantsandE,andthediaphragmthicknessharexed;there-fore,thediaphragmradiusisfoundfromEquation 4{4 .UsingthevaluesgiveninTable 4-1 for,E,andh,thediaphragmradiusischosentobe230m;theaspectratioofthediaphragmis102. AswasdiscussedinSection 3.2.6 ,ifthebackplateistoocompliant,therearenegativeeectsinthefrequencyresponse.AsimpliedmodelofthetopbackplateandthediaphragmisshowninFigure 4-4 ;thisanalysisisgeneralforeitherbackplate.Thepressuredropacrossthebackplateiscomparedtothatacrossthediaphragm.Forthisanalysis,onlytheimpedanceofthebackplateandthediaphragmcomplianceareconsidered. Figure4-4.Simpliedmodelofthediaphragmandtopbackplate Thebackplateisrepresentedbyacomplianceandaresistanceinparallel;thebackplatemodelisinserieswiththecomplianceofthediaphragm.Asshowninthe

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gure,theuniformpressureloading,p,resultsinavolumevelocity,Q,throughthebackplateanddiaphragm.Thisequivalentcircuitcanbetreatedasa\pressure"divider,wherethepressureacrossthebackplateisgivenby Similarly,thepressureacrossthediaphragmisgivenby ThedeectionofacomplianceduetoauniformpressureloadisgiveninEqua-tion 3{189 ;thisdeectionisproportionaltothepressureacrossthecompliance.ItcanbeseenfromEquation 4{5 thatthepressureacrossthebackplateissmallaslongasZa;bpismuchlessthanZa;d.Therefore,ifthisconditionismet,thedeec-tionofthebackplatewillbenegligible.Theimpedanceofthebackplateisgivenby Aslongas!islessthan1=(Ra;bpCa;bp),thebackplateimpedanceapproximatelyequalsRa;bp.Therefore,thebackplatedeectionduetoanacousticpressureisnegligibleifthebackplateresistanceislessthantheimpedanceofthediaphragmandifthefrequencyofoperationislessthanthecriticalfrequencyforeachbackplate, 2Ra;bpCa;bp:(4{8) Thebackplatesmustbedesignedwithasucientnumberofholessothatfbpforeachbackplateiswellabovethedesiredfrequencyrangeofoperation. However,asstatedabove,thereareseveralothertrade-os.Iftherearetoomanybackplateholes,thebackplatecomplianceisgreaterthanthatpredictedassumingtherearenotanybackplateholes.Furthermore,asbackplateareaislostduetotheholes,thecapacitanceisreduced.However,iftheholeradiusissmall

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enough,fringingeldsreducethecapacitanceloss.Anotherconsiderationisthatthebackplateholesserveasthepathfortheetchanttoremovetheoxidefromtheairgaps;enoughholesofsucientradiusmustbeprovidedforthisetch. Thebackplategeometrythatwasfoundtobestbalancethesetradeoswhileprovidingasucientlysmallbackplateresistancewastousebackplateholeswitharadiusof5mwithatotalareaofapproximately22%ofthebackplatearea.Forthetopbackplate,557holeswereusedand367holeswereusedforthebottombackplate.Thisbackplateholecongurationresultedinanfbpforthetopandbottombackplatesof1:3MHzand3:3MHz,respectively. 4-1 3 andthedesignparametersgiveninTable 4-1 ,thepredictedmicrophoneperformancespecicationsarefound,includingtheuncertaintyintheseestimates. Theuncertaintyanalysisforthesensitivity,resonantfrequency,andnoiseoorisderivedinAppendix C .TheuncertaintyvaluesusedforthethicknessofthediaphragmandairgapsaretakenfromTable D-1 .SandiaNationalLaboratoriesreportsdimensionaluncertaintyof0:1mforlinewidths.Thistranslatesintotheuncertaintyinthediaphragmradius.Materialparametersareassumedtohavea10%variation.

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Table4-1.Microphonephysicalproperties SymbolPropertyValueUnits 3.1.2 givesthesensitivityfromthepressureonthediaphragm,pd,toanoutputvoltage.However,theattenuationduetothecavitycompliancealsoaectsthesensitivity,asgivenbyEquation 3{184 .Tosimplifytheexpressionsforthesensitivity,itisassumedthatthenominalcapacitancesofthetopandbottomcapacitorsareequalandthefrequencyofoperationisintheat-bandregion. Therefore,thesensitivityforadual-backplatecapacitivemicrophonewithaconstantvoltagebiasisgivenby 3VB

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Similarly,thesensitivitywithaconstantchargebiasis 3HcVB Thesensitivitiesmayalsobewrittenintermsofthetransformerturnsratios,n1andn2,as and ThesensitivityforthedesignedmicrophoneisfoundbyevaluatingtheseexpressionswiththevaluesgiveninTable 4-1 andabiasvoltageof9:3V(thevalueusedforexperimentation).Thepredictedsensitivityforthedesigneddual-backplatecapacitivemicrophonewithadirectlyappliedbiasvoltageanda1:5pFfeedbackcapacitoris wheretheuncertaintyiscalculatedusingEquation C{12 .Similarly,forthesamemicrophonewithbiasvoltageappliedthroughalargeresistorandatotalparasiticcapacitanceof1pF,thepredictedsensitivityis ThelargeuncertaintyinsensitivityisduetotheuseofchemicalmechanicalpolishingintheSUMMiTVprocess.Thisgivesavariationofupto25%inthethicknessoftheairgaps.Thisresultsinanuncertaintyofcloseto50%inthesensitivityestimates.

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model.Thecut-onfrequency,resonantfrequency,dampingratio,andatbandattenuationarealsofound. Figure4-5.Crosssectionoftheventchannel.Therearetwocomponents:oneisinparallelwiththetopbackplateandtheotherconnectstothecavity. AllofthelumpedelementsarecomputedusingtheparametersgiveninTa-ble 4-1 excepttheventresistance.Theventchanneliscreatedbythechannelforelectricalconnectionstotheanchorsforthediaphragmandbottombackplate.Afterthemicrophoneisreleased,thesacricialoxideisremovedandthechannelisopened.ThegeometryoftheventchannelisshowninFigure 4-5 .Tworesistancesareshown:Ra;v1whichconnectstheoutsidetoapointbetweenthetopbackplateanddiaphragm;andRa;v2whichconnectstothecavity.Thewidthoftheventchannelis42mandtheheightis2m.Thehydraulicdiameterofthechannelisgivenby4A=P,whereAandParethecross-sectionalareaandperimeterofthechannel,respectively[ 123 ];forthisgeometry,theventchannelhasahydraulicdi-ameterof3:8m.Thereisnotasignicantresistanceabovethebottombackplateanchorbecausetherestrictedwidthoftheventchannelextendsonlytotheendofthediaphragmanchor.Thelengthoftherstpartofthechannelis24mandthelengthofthesecondpartis23m.TheresistanceRa;v1isinparallelwiththetopbackplate,thereforeitdoesnotcontributesignicantlytothefrequencyresponse.

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Table4-2.Acousticlumpedelementvaluesforthedesignedmicrophone. SymbolDescriptionValueUnits m4Ca;bp1Topbackplatecompliance2:911017m5 m5Ca;bp2Bottombackplatecompliance7:041018m5 m5Ca;cavCavitycompliance5:061016m5 m5n1Topturnsratio4:15104V Pan2Bottomturnsratio4:15104V Pa 4-2 andisshowninFigure 4-6 .Itcanbeseenthatthebackplatecompliancehasanegligibleeectonthemicrophonefrequencyresponse. Finally,thevaluesforthecut-onfrequency,dampingratio,dampedresonantfrequency,andatbandattenuationaregiveninTable 4-3 .Themicrophonehasapredictedbandwidthfrom4Hzto1739kHz.Itislightlydamped,withadampingratioof0:09.Thelowvariationinthepredictedresonantfrequencyisduetothefactthattheresonantfrequencydoesnotdependontheairgap. Table4-3.Frequencyresponseparameters DescriptionValue Cut-onfrequency4HzDampingratio0:09Resonantfrequency173kHzFlatbandattenuation0:97

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Normalizedmagnituderesponseofthedesignedmicrophone. (b) Phaseresponseofthedesignedmicrophone. 3.3 .Therearetwocomponentstothenoiseoor,thethermomechanicalnoiseofthemicrophoneandtheelectricalnoiseoftheinterfacecircuitry. ThePSDoftheacousticnoisesourcereferredtothediaphragmisplottedinFigure 4-7 .ThenoiseduetoRa;effandRa;visshown.Belowapproximately400Hz,thenoisefromtheventresistoristhelargerofthetwoacousticnoisesources. ThetheoreticaloutputvoltagenoisePSDofthemicrophonepackagedwithachargeamplierisshowninFigure 4-8 .Thecontributionsfromtheacousticnoisesources,biasresistor,andamplierareshown.Thechargeamplierconsidered

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Figure4-7.Theoreticalnoisecontributionsofthemicrophonereferredtothepres-sureacrossthediaphragm. hereisbasedontheTLE2071opampmanufacturedbyTexasInstruments.Thisamplierisconguredasachargeamplierbyplacinga1pFcapacitoranda2Gresistorinthefeedbackpath.Parasiticcapacitanceonthecircuitboardresultsinanetfeedbackcapacitanceof1:5pF.Theinputreferredvoltagenoiseoftheamplieris17nV=p Figure4-8.TheoreticaloutputvoltagenoisePSDofthemicrophonewithachargeamplier.

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Thetheoreticalnoiseoorofthemicrophonepackagedwithavoltageam-plierisshowninFigure 4-9 .TheselectedvoltageamplieristheSiSonicmicro-phoneamplier,courtesyofKnowlesAcoustics.Anoisemodelwasnotprovided,however,measurementsofthevoltageampliergiveavoltagenoisedensityof41016V2=Hzandacurrentnoisedensityof51031A2=Hz.Theacousticnoiseisbelowtheelectricalnoiseforthemajorityofthefrequencyrange,ex-ceptnearresonance;ascanbeseenfromFigure 4-9 .At1kHz,thepredictedoutputvoltagePSDforthemicrophonepackagedwiththevoltageamplieris2:11015V2=Hz.Thisisequivalenttoaninputreferrednoiseof20dB=p Figure4-9.TheoreticaloutputnoisePSDofthemicrophonewithavoltageampli-er. 4-1 andtheequationsinSection 3.2.7 .Forcomparison,thepull-involtagesofasingle-backplatecapacitivemicrophonewiththesamediaphragmandairgapdimensionsarecalculated. Thequasi-staticpull-involtageforthesingle-backplatecondensermicrophoneisVPI=31V.Thequasi-staticpull-involtageforthedual-backplatecondenser

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backplatemicrophoneis Asummaryofthespecicationsforthedesigneddual-backplatecondensermicrophoneisgiveninTable 4-4 alongwiththepredicteduncertainty. Table4-4.Summaryofspecicationsforthedesignedmicrophone Sensitivity(CA)370V=Pa190V=PaSensitivity(VA)220V=Pa100V=PaResonantfrequency173kHz9kHzNoiseoor(CA)36dB3:5dBNoiseoor(VA)20dB4:0dB

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Thefabricationofthedual-backplatecapacitivemicrophoneispresentedinthischapter.ThemicrophonefabricationutilizedboththeSUMMiTVprocessatSandiaNationalLaboratoriesandfacilitiesattheUniversityofFlorida.Theprocessowispresentedintwoparts;theSUMMiTVfabricationandthepostprocessing. D ,mainlyconsistsofthedepositionandpatterningofalternatinglayersofpolysiliconandsilicondioxide.Polysiliconisusedtoformthemicrophonestructure.Silicondioxideisatemporary,orsacricial,materialtosupportthevariouslayersofpolysiliconduringfabrication.AftercompletionoftheSUMMiTVprocess,aseriesofpostprocessingstepsareperformedtoreleasethedeviceandcompletethefabrication. 5-1 .Theprocessowisdepictedasaseriesofschematiccrosssectionsthroughthecenterofthemicrophone. TheSUMMiTVprocessbeginswitha6in:siliconwafer.AsshowninFigure 5-1(b) ,alayerofsilicondioxideandsiliconnitridearethendeposited.Theseinsulatethepolysiliconstructurefromthesiliconsubstrate.Inaddition,thesiliconnitrideisusedtoprovideadhesionforthepolysilicon.Therstlayerofpolysilicon,Poly0,isthendeposited.Thispolysiliconlayerisusedtoformabasefortheanchorsandforelectricalinterconnections. ThefabricationofthebottomairgapanddiaphragmareshowninFigure 5-1(d) andFigure 5-1(e) ,respectively.A2mlayerofsacricialoxideformsaspacer 156

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Beginwith650mthicksiliconsub-strate. (b) Depositinsulationlayersofsilicondioxideandsiliconnitride. (c) Deposittherstpolysiliconlayer. (d) Depositandpatternalayerofsacricialoxide. (e) Depositpolysiliconandpatterntoformbottombackplate. (f) Depositandpatternsacricialoxidethatwillcreatethebottomairgap. (g) Depositpolysiliconandpatterntoformdiaphragm. (h) Depositandpatternsacricialoxideforthetopairgap. (i) Depositpolysiliconandpatterntoformtopbackplate. betweenthePoly0electricalconnectionsandthenextlayerofpolysilicon.ThebottombackplateisformedbydepositingandpatterningPoly2.Thebottomairgapspaceisheldbythenextsacriciallayer,Sacox3.ThediaphragmisthenformedfromPoly3,asshowninFigure 5-1(g) .ThenalstepsoftheSUMMiTVprocesscreatethetopbackplate,asshowninFigure 5-1(h) andFigure 5-1(i)

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schematiccrosssections,areshowninFigure 5-2 .Tocompletethefabricationofthemicrophone,thefollowingstepsarecompleted:depositmetalforbondpads;etchthesiliconsubstrate;etchtheoxideandnitridelayers;andetchthesacricialoxide.AlthoughitwasnotshowninFigure 5-1 forsimplicity,eachofthelayersthatweredepositedonthefrontsurfaceofthesiliconwaferduringtheSUMMiTVprocesswerealsodepositedonthebacksideofthewafer.Theselayersareremovedbeforethemicrophonecavityisformed. ThedeviceswerereturnedfromSandiaNationalLaboratoriesasunreleaseddie.Tofacilitatethepostprocessing,ahandlewaferisusedtosupporttheindividualmicrophonedieduringprocessing.Toconstructthehandlewafer,AZ9260photoresistisspunona100mmPyrexwaferat4000RPM,resultinginathicknessofapproximately4m.A100mmsiliconwaferisplacedonthephotoresistcoveredPyrexwaferandpressureisappliedbyhandtojointhetwowaferstogether,asshowninFigure 5-2(a) .TheAZ9260photoresististhenspunonthetopofthesiliconwaferat2000RPM,resultinginathicknessofapproximately9m.ThephotoresistispatternedandacavityisetchedviaDRIEinthehandlewafertoholdthemicrophonedie,asdepictedinFigure 5-2(b) ThemicrophonedieismountedinaninvertedpositionintothehandlewaferusingCrystalbond509,athermoplasticpolymeradhesive,asshowninFigure 5-2(c) .Thethermoplasticadhesiveallowsthemicrophonedietobeinsertedandremovedbyelevatingthetemperatureoftheadhesive,yetitprovidesastrongbondduringhandling.Mechanicallappingisperformedtoremovethebacksidelayers.Aslurryiscreatedonthesurfaceofaglassplatebymixing5mgritpolishingpowderwithwater.Themicrophonedie,supportedbythehandlewafer,islightlypressedontotheslurrycoatedplateandmovedinagure-8pattern.Thisremovesthebacksideofthemicrophonedieuntilthesurfaceofthedieisushwiththesurfaceofthehandlewafer,atwhichpointthelappingiscomplete.Theremaining

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BeginbyjoiningasiliconwafertoaPyrexwaferwithphotoresist. (b) EtchacavityinthesilicontoholdthemicrophonedieusingDRIE. (c) Attachthemicrophonedietothehandlewafer. (d) Removebacksidelayersviamechanicallapping. (e) EtchthesiliconsubstrateofthemicrophonedieusingDRIE. (f) EtchthinoxidelayerusingBOEandasiliconnitrideetchusingRIE. (g) Completethedevicefabricationbyetchingthesacricialoxide.

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thicknessofthesiliconsubstrateonthemicrophonedieisapproximately600mandthedevicenowresemblesFigure 5-2(d) Photoresististhenspunonthetopsurfaceofthehandlewafer.AZ9260isusedwithaspinspeedof2000RPMforatimeof50s.Thisresultsina9mthicklayerofphotoresist.ThephotoresistispatternedtocreateaholeaboveeachmicrophonestructureusinganEVG620maskaligner.Front-to-backalignmentisusedbecausethecavityisetchedfromthebacksideandmustbealignedtothemicrophonefeaturesonthetopsideofthemicrophonedie.Thebottomportionofthehandlewaferneedstobeclearsothatthepatternedsurfaceofthemicrophonediecanbeseen,thereforePyrexisused.Then,thesubstrateofthemicrophonedieisetchedusingDRIE.AschematiccrosssectionofthedeviceandhandlewaferaftertheDRIEstepisshowninFigure 5-2(e) Figure 5-2(f) showsthedeviceafterthecompletionofthenexttwoprocesssteps.The0:63mlayerofthermaloxideisetchedusinga6:1bueredoxideetchfor15min.Amaskisnotneededforthisetchbecausethesiliconsubstrateactsasthemask.ThenitridelayerisetchedusingaUniAxisICPRIEdryetch.Theetchwasperformedatapressureof10mTorrandapowerof500W;thegassesSF6andO2wereusedwithowratesof50sccmand10sccm,respectively.Similartotheoxideetch,thisetchdoesnotrequireamask;thesubstrateactsasamaskforthenitridelayer.Althoughthenitrideetchattacksthesiliconsubstrate,theeectonthemicrophonestructureisnegligible. Aftercompletionoftheabovesteps,themicrophonedieisremovedfromthehandlewaferbysofteningtheadhesivebyapplyingheatandtheglueisremovedbysoakingthedieinacetone.Themicrophonestructureisreleasedwitha40min:etchina49%hydrouoricacidsolution;thisisfollowedbyarinseindeionizedwater.Ifthemicrophonewassimplyremovedfromthewaterandlefttodry,the

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surfacetensionofthewaterwouldpullthediaphragmandbackplatestogether,renderingthedeviceuseless;thisphenomenonistermedstiction. ThereareseveraltechniquesthathavebeendevelopedinthepasttoavoidstictionduringthereleaseofMEMSstructures.Achemicalcoatingcanbeappliedtothedevicesurfacestoreduceadhesion[ 126 ].Otherapproachesfocusonavoidingtheliquidevaporationthroughtheuseoffreeze-drying[ 127 ]orasupercriticalphasetransformation[ 128 ].Forthisprocess,supercriticaldryingisusedwithliquidCO2usingaBal-tecCPD030criticalpointdryer.First,thedeionizedwaterissubstitutedwithmethanol.Then,usingtheBal-tecCPD,themethanolissub-stitutedwithliquidCO2atapproximatelyatemperatureof12Candapressureof50bar.ThetemperatureisthenslowlyraiseduntiltheliquidCO2undergoesasupercriticalphasechangetoagas.Thistransitionavoidsthepossibilityofstic-tion.AfterthegaseousCO2isvented,theprocessingisnished.AschematiccrosssectionofthenisheddeviceisshowninFigure 5-2(g) Therstmetallizationattemptsutilizedgoldwithachromiumadhesionlayerbecausethesemetalsareresistancetohydrouoricacidetches[ 129 ].However,deviceswithaCr/Aumetallizationdidnotfunction.Furtherinvestigationshowed

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thatthepolysiliconwasattackedduringthereleaseetchwhenexposedgoldwaspresent.Avulnerable0:3mpolysiliconinterconnectwasdamaged,severingtheconnectiontothebottombackplate.Thiscatalyticbehaviorofgoldinteractingwithhydrouoricacidtoetchsiliconhasbeenpreviouslyreported[ 130 131 ]. Toavoidthisissue,aprocesstoreleasethemicrophonewithaluminumbondpadswasdeveloped.Thisissomewhatchallenging,asHFbasedetchantstypicallyattackaluminum[ 129 ].AcommercialetchantmanufacturedbyTransene,Inc.,SiloxVapoxIII,wasidentiedasasiliconetchthathasasucientselectivitybetweensilicondioxideandaluminum.However,thisetchanthasarelativelyslowetchrate,a2hr:etchwasneeded.Overthislongetchtime,ithadanon-negligiblereactionwithaluminum.Bubblesformedonthesurfaceofthealuminum,whichloweredtheyieldofsuccessfullyreleasedmicrophones.Furthermore,thealuminumsurfacewasdamagedduringtheetch,anditwasnotpossiblemakeasuccessfulwirebond. Thenalsolutionwastoleavethebondpadswithoutmetallization.Engent,Inc.inNorcross,GA,anexternalcompany,wasidentiedtosuccessfullywirebondtobarepolysiliconbondpadsusingagoldballbonder.Theprocessowthroughreleasegivesahighyieldwithapproximately90%ofthemicrophonesreleasingproperly.Theyieldisreducedsubstantiallyduringpackaging.About50%ofthemicrophonesweresuccessfullywirebonded.However,theyieldwasfurtherreducedduetodiaphragmbuckling.Thismostlikelyoccurredduetopackage-inducedstressorelectrostaticdischarge.Thenalyieldthroughpackagingwasapproximately33%.

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Thekeyresultsobtainedforthedual-backplatecapacitivemicrophonede-scribedinChapter 4 andChapter 5 arepresentedinthischapter.Firstthefabricatedmicrophoneisdescribedindetail.Thisisfollowedbyadiscussionofthemicrophonepackagingdevelopedforcharacterization.Theexperimentalsetupforeachofthemeasurementsisdescribedandthentheresultsarepresented. 5 .AphotographofthedieisshowninFigure 6-1 .Therearemultipledevicesoneachdie.Theseincludetwolargedual-backplatemicrophonesdesignedforaudioapplications.Resultswerenotobtainedfromthesedevicesduetofabricationissuesandtheyarenotreferredtothroughouttheremainderofthisdissertation.Therearealsotwosingle-backplatemicrophoneteststructuresconsistingofthediaphragmandeitherthetoporbottombackplate. Figure6-1.Photographofthemicdiewiththeindividualmicrophoneslabeled. Oneachdie,therearefourcompleteaeroacousticdual-backplatemicro-phones.InFigure 6-1 ,thesearelabeled1,2,3,and6,respectively.Throughout 163

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thischapter,individualmicrophonesarelabeledbyalettercorrespondingtoaparticulardie,andanumbercorrespondingtothespecicmicrophoneonthedie.Forexample,microphoneQ1isthemicrophoneinposition1ondieQ. Aclose-upviewofoneofthemicrophonesisshowninFigure 6-2 .Thetopbackplateandbackplateholesarevisible.Thebondpadsconnectingtothebottombackplate,diaphragm,andtopbackplatearelabeledonthedieinpoly0.Therearethreeadditionalbondpadsthatmaybeusedasguardconnectionstominimizetheeectofparasiticcapacitanceonthedie,similartothoseusedontheSiSonicTMmicrophone[ 110 ].However,thesewerenotusedforthemicrophonecharacterization.ThenalbondpadonthetopleftofFigure 6-2 providesaconnectiontothesubstrate;thiswasalsonotusedforthecharacterization. Figure6-2.Photographshowingthetopofthemicrophone. ThethreelayersofthemicrophonearevisibleinFigure 6-3 .Thisgureisanscanningelectronmicroscopy(SEM)imageofareleasedmicrophone.Thedevice

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hasbeensectionedwithafocusedionbeam(FIB),revealingthediaphragmandbottombackplate. Figure6-3.SEMimageshowingthethreelayersofthemicrophone. FurtherdetailsofthemicrophonestructureareshowninFigure 6-4 andFigure 6-5 .TheseguresareSEMimagesofanunreleasedmicrophone.ThediewassectionedusingadicingsawandaFIBwasusedtosmooththesurface. Figure6-4.SEMimageofacross-sectionviewofanunreleasedmicrophonedie.

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Figure 6-4 isacross-sectionviewshowingthetopbackplate,diaphragm,andbottombackplate.Figure 6-5 isaclose-upviewoftheanchorcross-sectionthatshowsthedetailsoftheelectricalconnectiontothediaphragm(shownschematicallyinFigure 4-3(b) ). Figure6-5.SEMimageoftheelectricalconnectiontothediaphragm. 6-6 Figure6-6.Schematicdiagramofthemicrophonepackage.

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Themicrophonepackageconsistsofaprintedcircuitboard(PCB)thatholdsthemicrophonedie.ThePCBisplacedinaLuciteplug.Thisplugisdesignedtotintotheacoustictesthardwareandgiveaushsurface.Electricalconnectiontothemicrophonepackageismadeviacopperwiresextendingoutofthepackagebase. 6-7 isaphotographoftheamplierdieshowingtherequiredconnections.ThisamplierisinterfacedtothemicrophoneviathePCBdiscussedinSection 6.2.2 Figure6-7.PhotographoftheSiSonicTMmicrophoneamplier. ThechargeamplierusedisbasedontheTLE2071,manufacturedbyTexasInstruments.Thiscircuitisrealizedonanexternalcircuitboard,showninFig-ure 6-8 .TheboardcontainstheampliercircuitaswellasBNCconnectorsforthemicrophonebiasvoltages.A3-wireconnectionismadebetweenthechargeamplierPCBandthemicrophonepackagethatprovideselectricalcontactforthetopbackplate,diaphragm,andbottombackplate.

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Figure6-8.Photographofthechargeampliercircuitboard. 6-9(a) .ArecesshasbeencutintothePCBthatallowsthemicrophonetobeushwiththePCBsurface.AchannelisprovidedaroundtheperimeterofthecavitytocontainexcessepoxywhenthemicrophonedieisaxedtothePCB.Goldwirebondsareusedtoconnectthemicrophonedietothecircuitboard.ElectricalconnectiontothePCBismadeviathrough-holesinwhichthecopperwiresareinsertedandsoldered. ThebacksideofthePCBcanbeconguredintwowaystosupporteitheravoltageamplierorachargeamplier.Formicrophonedieusedwithvoltageampliers,thebacksideofthePCBispopulatedasshowninFigure 6-9(b) .ThisincludesaxingfouramplierdietothePCBusingaconductiveepoxy.Theampliersubstrateisgroundedtopreventdriftwhichcanresultfromstraycharge.WirebondsareusedtoconnectthePCBtotheamplierdieandarecoveredin

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Topview. (b) Bottomview. protectiveepoxy.Alsoincludedinthecircuitryisa50resistorinserieswitheachamplieroutputanda0:1Fbypasscapacitorforeachamplier'spowersupply.Thetracefromthediaphragmtotheamplierinputismadeasshortaspossibletominimizeparasiticcapacitance. Formicrophonedieusedwithachargeamplier,theback-sideofthePCBisnotpopulatedwiththepreviouslydescribedvoltageampliercircuitry.Inthiscasethediaphragmisdirectlyconnectedtooneofthecopperposts.RatherthanhavethisconnectionpermanentlyonthePCBandloadthevoltageamplierwithadditionalparasiticcapacitance,awirebondismadetojumperthediaphragmtracetothePCBoutput. 6-10 .ThemicrophoneisushwiththePCBsurfaceandthewirebondsarecoveredinepoxy.ThecopperpostsarecutsuchthattheyterminatejustbelowthesurfaceofthePCB;thusthesolderheightisaslowaspossible. ThenalpackageassemblyisshowninFigure 6-11 .TheLucitePCBholderhasadimensionof0:75in:0:75in:atthefrontsurface.

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Figure6-10.Photographofthemicrophoneembeddedintheprintedcircuitboard. Figure6-11.Photographoftheassembledmicrophonepackage.

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6-12 ,anacousticdriverisplacedatoneendofthetube.Thedeviceundertest(DUT)andareferencemicrophoneareplacedattheotherendsuchthattheyareatnormalincidencetotheincidentpressure.Duetothegeometryofthetube,thereisafrequencybelowwhichonlyplanewavespropagate;i.e.thepressureisuniformacrossthecrosssectionofthePWT[ 4 ].ThisallowstheDUTandthereferencemicrophonetobeexposedtothesameincidentpressure.Thisfrequencyis6:7kHzforair. AlsoshowninFigure 6-12 isthesupportinghardwarefortheplanewavetubeexperiments.TheBruelandKjrPulsemulti-analyzersystemprovidesafunctiongeneratortodrivetheacousticdriverandacceptstheinputsignalsfromtheDUTandreferencemicrophone.ThePulsesystemalsoperformsthedataanalysisfunctionsandrecordsthedata.APCBPiezoelectronics377A51condensermicrophoneisusedforthereferencemicrophone.Thismicrophoneisusedduetoitshighmaximumpressure.Forconsistency,itisusedasthereference

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microphoneforallacousticexperiments.ThesignalsenttotheacousticdriverisrstampliedbyaCrownK1poweramplier.ThesoundeldintheplanewavetubeisproducedbyaBMS4590Pcompressiondriver. Figure6-12.Largeplanewavetubeexperimentalsetup. However,thespeedofsoundinheliumisapproximatelythreetimesthespeedofsoundinair;thusonlyplanewavesexistforfrequenciesuptoapproximately20kHz.Operatingthemicrophoneinheliumhaslittleimpactonitsperformance.Thecavitycompliancedecreases,thusthepredictedsensitivityisreducedby1%andtheresonantfrequencyisincreasedbylessthan1%.Thepredictedfrequencyresponseofthedesigneddual-backplatecondensermicrophoneisshowninFigure 6-13 foroperationinbothairandhelium. Thelinearityexperimentsareconductedinair,andthefrequencyresponsemeasurementsusehelium.ThemagnituderesponseoftwoBruelandKjrcon-densermicrophonesinairandheliumisshowninFigure 6-14 andFigure 6-15 ,respectively.Themagnituderesponseisapproximately1(0dB)whenonlyplanewavespropagate.Itcanbeseenthattheuseofheliumextendsthisrangetoapproximately20kHz. ThereferencemicrophoneandDUTareexcitedwitha1kHztoneforthelinearityandTHDmeasurements.ThePulsesystemisconguredtocomputeazoom-FFTofthesesignalswitha6:4kHzspanandacenterfrequencyof3:4kHz.

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Figure6-13.Theoreticalmagnituderesponseofthedual-backplatemicrophoneinair(-)andhelium(--). Figure6-14.MagnituderesponseoftwoB&K4138condensermicrophonesinair. Thisensuresthatallharmonicsupto6kHzarecaptured.A1600lineFFTisusedgivingafrequencyresolutionof4Hz.Theamplitudeoftheincidentpressureisincrementedtocharacterizethemicrophonesoverawiderangeofamplitudes;ateachmeasurementpoint,75averagesaretakenwith0%overlap. Forthefrequencyresponsetests,thegeneratorissettoaperiodicrandomsignalwithaspanof25:6kHz.TheFFTanalyzerisconguredaccordinglytoaspanof25:6kHzwithafof16Hz.Eachsetofrecordeddatapresentedbelowistheresultof400averages.

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Figure6-15.MagnituderesponseoftwoB&K4138condensermicrophonesinhe-lium. 21 ].ThusthemeasuredpowerinalltheharmonicscanbeusedtoestimatetheTHD. However,astheincidentpressurebecomeslarge,non-linearitiesaregener-atedinthetestsetup.Theacousticdriveroutputssignicantsoundpressureatharmonicfrequencieswhendrivenwithasingletone.Additionally,theacousticwavepropagationbecomesnon-linearathighsoundpressurelevels[ 4 ].There-fore,theTHDofthemicrophonemustbeestimatedinthepresenceofexternalnon-linearities. ToestimatethetotalharmonicdistortionduetotheDUTmicrophone,theharmoniccomponentsduetotheexperimentalsetuparesubtractedfromtheDUToutputsignal.Thereferencemicrophonemeasuresthetotalacousticpressureincludingtheharmoniccomponents.TheMEMSmicrophonesensitivityisusedto

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converttheharmonicpressurecomponentsmeasuredbythereferencemicrophonetoacorrespondingrmsoutputvoltage.ThisrmsoutputvoltageateachharmonicissubtractedfromthemeasuredvoltageateachharmonicasdepictedinFigure 6-16 .Mathematically,thisisdescribedas wherevmeasnisthemeasuredrmsvoltageatthenthharmonic,Sisthemicrophonesensitivity,prefnisthermspressuremeasuredbythereferencemicrophoneatthenthharmonic,andveffnistheextractedrmsvoltageatthenthharmonicusedtoestimatethetotalharmonicdistortion. Figure6-16.GraphicdescriptionofTHDmethodology. Severalassumptionsarenecessaryforthisanalysistobevalid.Firstthereferencemicrophonemustnotintroduceanynon-linearitiesinthesystem.Themicrophonesaretesteduptosoundpressurelevelsapproaching170dB;therefore,thechoiceofreferencemicrophoneiscrucial.ThePCB377A51condensermicro-phonehasa3%distortionlimitof192dB.Thus,thereferencemicrophoneissucientforthesemeasurements.Additionally,thetotalpressuremeasuredbythereferencemicrophonemustbethesameasthatsensedbytheDUTmicrophone.A

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1kHztoneisusedfortheTHDmeasurements.Therefore,therstveharmoniccomponentspropagateasplanewavesinthePWT.TheseveharmonicsareusedintheTHDcalculations.ItisassumedtheDUTgeneratesnegligibleharmonicdistortionabovethe5thharmonic. 6-17 .Thecenterholeisusedtomaximizethemeasuredvelocity.Thevelocityisinferredfromthereturnedopticalsignalbythevibrometercontroller. Figure6-17.Experimentalsetuptodeterminetheresonantfrequencyofthemicro-phone. Themicrophoneisexcitedwithanacousticimpulsegeneratedbyacapgun.ThepressureisrecordedbyaBruelandKjr4138referencemicrophoneplacedapproximately1infromtheDUTmicrophone.AtypicalmeasuredpressureisshowninFigure 6-18 .Therecordedpressureisshapedbyahighpasslterwitha50kHzcornerfrequencythatwasusedonboththereferencechannelandthevelocitychannel;themeasuredpressurenear170kHzisat.Thelaservibrometer

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isconguredtocomputetheFFTofthetimeseriessignalsovera500kHzbandwidthwitha156:25Hzbinwidth.Duetothenatureofthisexperiment,averagingisnotperformed.WhilethetestmayberepeatedmultipletimesandtheresultingFFT'saveraged,thesourcemaynotberepeatable.Thedatapresentedhereisfromsingle-shotmeasurements. Figure6-18.TypicalpressurerecordedbyreferencemicrophoneforLVmeasure-ment. 6-19 .ThemicrophoneandinterfacecircuitryareplacedinsidetwoconcentricFaradaycages.AFaradaycageattenuateselectromagneticwavesandreducestheamountofelectromagneticinterferenceinsidetheboxduetoexternalsources.TheuseoftwoFaradaycagesimprovestheelectromagneticinterferencereduction[ 132 ].TheoutputofthemicrophoneisampliedandthensenttoaStanfordResearchSystemsSR785spectrumanalyzer.Thespectrumanalyzermeasurestheoutputnoisepowerspectraldensity.Theinputreferredpressurenoiseiscalculatedusingthemeasuredmicrophonesensitivity. Thenoisespectrumismeasuredfrom10Hzto102:4kHzbycombiningmeasurementsofthreeseparatefrequencyranges.Therstrangedfrom20Hzto

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Figure6-19.Faradaycageexperimentalsetupfornoisemeasurements. 200Hzwithabinwidthof0:25Hzand2300averages.Thesecondrangespannedfrom200Hzto1:6kHzandhadabinwidthof2Hzand4000averages.Thenalfrequencyrangespannedfrom1:6kHzto102:4kHzwitha128Hzbinwidthand30000averages. Microphonesfromfourchipswerepackagedforthecharacterization.Threewerepackagedwithvoltageampliersyieldingatotalofsevendeviceswithavoltageamplier;thesedieareI,M,andQ.Theforthdie,O,wascharacterizedwiththechargeamplier;yieldingthreedevices.Unlessotherwisestated,allmicrophoneswerebiasedwith9:3Vusingalkalinebatteries. 6-20 .Theincidentpressurewasvariedfromaslowas43dBandincreasedtonear160dB.Astheincidentpressureapproaches

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160dB,themicrophoneoutputsaturates.Thisisduetosaturationofthevolt-ageamplier,whichhasamaximuminputvoltageof500mV.Themeasuredsensitivitiesfallwithintheestimatedsensitivityrange. Figure6-20.Outputvoltagevs.pressureforvoltageampliermicrophonesboundedbythetheoreticalsensitivityestimate. Tobettervisualizetheextentofthelinearrangeofoperation,themeasuredsensitivityisplottedversusincidentpressureinFigure 6-21 .Thesensitivitiesofthesevenmicrophonesarematchedtowithin2:1dB.TheaveragesensitivityofeachmicrophoneisgiveninTable 6-1 alongwitha95%condenceinterval.ThecondenceintervalforeachmicrophoneiscomputedusingthemethodologydescribedinSection C.4 Todeterminethemaximumlinearrangeofeachmicrophonewithoutinuencefromtheinputrangeofthevoltageamplier,thelinearitymeasurementswererepeatedwithabiasvoltageof2:0V.Thislowersthesensitivitybyafactorof2=9:3;thus,theamplierdoesnotsaturateatthehighestsoundpressurelevels.TheoutputvoltageversusincidentpressureforthiscaseisplottedinFigure 6-22 .Thesedataareshownonalinearscaletoaccentuatethedierenceinsensitivitycausedbythebiasvoltagereduction.

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Figure6-21.Sensitivityvs.pressureforvoltageampliermicrophones. Figure6-22.Outputvoltagevs.pressureforvoltageampliermicrophonesbiasedwith2:0V. Themicrophonesareseentodeviatefromthelineartrendabove2000Pa.Thesensitivityisplottedversusincidentpressureforthemicrophonesbiasedwith2:0VinFigure 6-23 .Theincreaseinsensitivityathigherpressuresismostlikelyduetothenon-linearelectrostaticbehaviorandagreeswiththetheoreticalpredicationsofthedevicenon-linearity.Thepredictednon-linearityofadual-backplatecapacitivemicrophoneforseveralconditionsisshowninFigure 6-24 .Theworst-casenon-linearityoccursfornon-matchedcapacitorsandwhenthe

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mechanicalnon-linearityisnegligible.Intherealizedmicrophone,thecapacitorsdohavemismatch.Furthermore,theonsetofmechanicalnon-linearitycouldbedelayedduetouncertaintiesinthemechanicalmodeloriftheairgapsaresmallerthanpredicted. Figure6-23.Sensitivityvs.pressureforvoltageampliermicrophonesbiasedwith2:0V. Figure6-24.Theoreticalnon-linearityforadual-backplatecondensermicrophone. Theestimatedtotalharmonicdistortionforthevoltageampmicrophonesbiasedwith2:0VisgiveninFigure 6-25 .Mostmicrophoneshavebetween3%to5%THDnear164dB.Thespecicresultsforeachvoltageampmicrophoneare

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giveninTable 6-1 .Themaximumpressurelistedistherstdatapointforwhichthenon-linearityisintherangeof3%to5%;thecorrespondingTHDisgiven. Figure6-25.Totalharmonicdistortionforvoltageampliermicrophonesbiasedwith2:0V. Table6-1.Summaryofthelinearityresultsforthemicrophoneswithvoltageampli-ers MicrophoneMeasuredsensitivityMaxPressureTHDatPmax I2178:30:3V=Pa163:7dB4:0%I3151:60:2V=Pa165:7dB5:1%M1168:10:3V=Pa163:7dB3:4%M2184:40:3V=Pa160:7dB2:1%Q1145:10:2V=Pa165:3dB4:7%Q2165:30:4V=Pa164:1dB4:3%Q3172:10:4V=Pa164:1dB4:7% Themeasuredsensitivityandcapacitanceofthemicrophonesmaybeusedtoestimatetheairgapsandparasiticcapacitanceofeachdevice.TheresultsarelistedinTable 6-2 .Themicrophoneparameterwiththehighestvariabilityistheairgapdistance.Therefore,theareaofeachcapacitorisassumedtobethedesignedvalue.Eachairgapisestimatedfromthemeasuredcapacitanceasfollows, Cmeas:(6{2)

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Theairgapsmaythenbeusedtopredictthemicrophonesensitivity.However,theparasiticcapacitanceisnotknown.Therefore,Equation 4{12 isusedtoestimatetheparasiticcapacitance,allowingfornon-matchedcapacitors.Theestimatedparasiticcapacitancerangesfrom0:92pFto2:23pF.Thisincludesthecontributionoftheamplierinputcapacitance,estimatedtobe0:3pF. Table6-2.Summaryofairgapandparasiticcapacitanceestimatesforthemicro-phonestestedwithvoltageampliers. 6-26 .Thetestedpressurerangeforthesemicrophonesextendsfromapproximately80dBtoabove165dB.Lowerpressureswerenotusedforthechargeampliermicrophonesbecausesucientcoherence(>0:9)wasnotobtainedforlowerpressures.Thisispartiallyduetothehighernoiseoorofthechargeampliercircuit.However,themaincauseisinterferencecausedby60Hzpowerlinepick-up.Thechargeampliercircuitismoresusceptibletothisinterferencebecauseitisseparatedfromthemicrophonebya12in:lengthofcable. Thethreechargeampliermicrophonesarematchedtowithin0:8dB.ThesensitivitiesofthesemicrophonesareplottedversustheincidentpressureinFig-ure 6-27 .Thechargeamplierhasasucientmaximuminputandoutputvoltagerangetonotimpactthedynamicrangeofthechargeampliermeasurement.The

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Figure6-26.Outputvoltagevs.pressureforchargeampliermicrophones,boundedbythetheoreticalestimate. totalharmonicdistortionforthechargeampliermicrophonemeasurementsisgiveninFigure 6-28 .Twoofthemicrophones,O3andO6,havedistortionoflessthan1%at164dB.Thethirdmicrophone,O1,has3:3%THDat166:5dB.However,asucientnumberofmicrophoneshasnotbeentestedtostatisticallyconsidertheseresults.Thesensitivitieswith95%condenceintervals,aswellastheTHDresults,aregiveninTable 6-3 Figure6-27.Sensitivityvs.pressureforchargeampliermicrophones.

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Figure6-28.Totalharmonicdistortionforchargeampliermicrophones. Table6-3.Summaryofthelinearityresultsforthemicrophoneswithchargeampli-ers MicrophoneMeasuredsensitivityMaxPressureTHDatPmax O13551V=Pa166:5dB3:3%O3360:40:2V=Pa166:5dB0:2%O6388:80:7V=Pa163:6dB0:03% 6-29 ,themagnituderesponseisplottedovertherange300Hzto25kHz.Whilehigherordermodespropagateatfrequenciesabove20kHz,themicrophonesqualitativelydemonstratearesponseupto25kHz.Thetheoreticalsensitivityestimateforthemicrophonewithavoltageamplierisalsoincludedinthisgure.Themagnituderesponseisplottedupto20kHzinFigure 6-30 tobettershowthematchingbetweendevices.ThephaseresponseforthesevenmicrophoneswithvoltageampliersisplottedinFigure 6-31 Thephaseiscenteredaround180becausethereisaninversionfrompressuretovoltage.Thephaseismatchedtowithin2formostofthefrequencyrange.Thedipinphasearound10kHzcoincideswithreducedoutputfromtheacoustic

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Figure6-29.Magnituderesponseforvoltageampliermicrophonesextendingto25kHz,boundedbythetheoreticalestimate. Figure6-30.Magnituderesponseforvoltageampliermicrophonesupto20kHz. driver.Above19kHz,thephasedeviatesfromthenominalvalueof180ashigherordermodesbegintopropagate. Themagnitudeandphaseresponsehasbeenmeasuredforthethreemicro-phonestestedwithachargeamplier.Whilethechargeamplierdoesnotlosesensitivityduetoparasiticcapacitance,thiscapacitancecanaectthefrequencyresponseofthemicrophone.Vibrationsinthecableresultinamodulationoftheparasiticcapacitanceandacorrespondingchargeinjectionintotheamplier.Thisphenomenonwasobservedinthechargeampliermeasurements.Itwaspartiallymitigatedbysecuringthecableasmuchaspossible;howevervibrationswerenot

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Figure6-31.Phaseresponseforvoltageampliermicrophones. completelyeliminated.Themagnitudeandphaseresponseforthechargeampli-erwiththeattestmeasuredfrequencyresponseareshowninFigure 6-32 andFigure 6-33 ,respectively.Thisdevicerespondedbesttosecuringthecabling. Figure6-32.Magnituderesponseforchargeampliermicrophoneswithminimalripple. ThemagnitudeandphaseresponseforallthreechargeampliermicrophonesaregiveninFigure 6-34 andFigure 6-35 ,respectively.Thecableandpackagevibrationresultsintheripplepresentinthefrequencyresponsemeasurements.Thephaseresponseforthesemicrophonesiscenteredaround0becausethereisnoinversionfromincidentpressuretooutputvoltage.

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Figure6-33.Phaseresponseforchargeampliermicrophoneswithminimalripple. Figure6-34.Magnituderesponseforallchargeampliermicrophones,boundedbythetheoreticalestimate. 6-36 .TheresonantfrequencyofthemicrophoneisassumedtobethefrequencyatwhichtheFFTis

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Figure6-35.Phaseresponseforallchargeampliermicrophones. maximum.Theaverageresonantfrequencyis158:0kHz;theresultsforthethreemicrophonesarelistedinTable 6-4 Table6-4.Summaryoftheresonantfrequencyresults. MicrophoneMeasuredresonantfrequency O1158:0kHzO3159:1kHzO6157:0kHz

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MicrophoneQ1 (b) MicrophoneQ3 (c) MicrophoneQ6

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4.2.3 ,thevoltageamplierisexpectedtogivelowernoisecomparedtothechargeamplier;thisisveriedexperimentally. TheoutputreferrednoisePSDofthesevenmicrophoneswithvoltageam-pliersisgiveninFigure 6-37 .Therearemanypeaksat60Hzandharmonicfrequencies.Thereisalsoaregionofincreasednoisebetween30Hzand70Hz.Duetotheexcessspikesinthespectrum,itisunclearifthisisnoisegeneratedbythemicrophone/ampliersystem,orifitisinterference.ThemeasuredPSDiscloselymatchedbetweensixofthemicrophones(I1,I2,M2,Q1,Q2,andQ3);thesemicrophoneshaveanaveragePSDat1kHzof1:961015V2=Hzwithastandarddeviationof0:101015V2=Hz.MicrophoneM1hasahighernoiselevelof2:681015V2=Hz. Figure6-37.MeasuredoutputPSDnoiseforthevoltageampliermicrophones. TheinputreferrednoiseiscomputedbydividingthesquarerootofthemeasuredPSDbythemicrophonesensitivity.TheinputreferrednoiseforthesevenvoltageampliermicrophonesisplottedinFigure 6-38 .TheinputreferredsoundpressurenoiseindBreferencedto20Paat1kHzrangesfrom21:8dB=p

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23:7dB=p 6-5 Figure6-38.Inputreferrednoiseforthevoltageampliermicrophones. Table6-5.Summaryofthenoisemeasurementresultsformicrophonestestedwithvoltageampliers. I21:94101521:8I31:91101523:2M12:68101523:6M22:06101521:8Q11:97101523:7Q21:82101522:2Q32:10101522:5 6-39 .TheaveragemeasuredPSDat1kHzis7:661013V2=Hz.TheinputreferrednoisespectrumofthethreemicrophonesisplottedinFigure 6-40 .Theaverageinputreferredsoundpressurenoiseat1kHzis41:5dB=p 6-6

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Figure6-39.MeasuredoutputPSDnoiseforthechargeampliermicrophones. Figure6-40.Inputreferrednoiseforthechargeampliermicrophones. Table6-6.Summaryofthenoisemeasurementresultsformicrophonestestedwithchargeampliers. O17:09101341:5O37:90101341:8O68:01101341:2

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6-7 .Thechargeamplierdegradesthenoiseperformanceandmakesthemicrophonesusceptibletovibration. Table6-7.Summaryofthemeasurementresultsforallmicrophones VoltageampChargeampSpecicationMeanvalueStandarddeviationMeanvalueStandarddeviation Sensitivity(V=Pa)1661436818Maxpressure(dB)1641:61661:7Noiseoor(dB=p 6-7 tothepredictionsgiveninTable 4-4 ,allmeasuredvaluesfallwithintheexpecteduncertaintyoftheestimatesexceptforthenoiseoorofthemicrophonewiththechargeamplierandtheresonantfrequency.Theupperendofthepredictedrangeofthechargeampliermicrophonenoiseooris39:5dB;whiletheaveragemeasuredvalueis41:5dB.ThemeasuredoutputPSDofthechargeamplierishigherthanpredictedbyafactorof3:3.Thiscouldpossiblybeduetohigherthanexpectedparasiticcapacitanceorampliercurrentnoise.Thelowerendofthepredictedresonantfrequencyis161kHz.Thisis3kHzfromthemeasuredresonantfrequencyof158dB.Thismaybemeasurementerror,orduetoaslightcompressivestressthatmaybepresentinthediaphragm.Thiswouldreducetheresonantfrequencyandincreasethesensitivity.Withthelargeuncertaintyinthepredictedmicrophonesensitivity,smalldeviations

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betweenthemeasuredsensitivityandtheidealpredictedsensitivityhavenosignicance.TheminimumdetectablesignalisreportedinTable 6-8 forvariousequivalentunits.AllguresexcepttheA-weightednoisearefora1Hzbinat1kHz. Table6-8.Minimumdetectablesignalexpressedinvariousequivalentunits. MDSValue A-weighted60:4dBAPressure(dB)22:7dBPressure(Pa)274PaForce15:1pNDisplacement75:4fmCapacitance17:9zF 6-9 .WhiletheperformanceoftheB&Kmicrophoneisnotexceeded,thedual-backplatemicrophoneexceedstheperformanceofallpreviousaeroacousticMEMSmicrophonesintermsofkeyspecications. Table6-9.Comparisonofthedesigneddual-backplatecapacitivemicrophonetotheB&K4138condensermicrophoneandpreviousaeroacous-ticMEMSmicrophones. 3 ]1:6mm168dB20dB6:5Hz{140kHzArnoldetal.[ 13 ]500m160dB52dB10Hz{100kHzyScheeperetal.[ 15 ]1:95mm141dB23dBA251Hz{20kHzHorowitzetal.[ 112 ]900m169dB48dB100Hz{50:8kHzzPedersen[ 111 ]180m140dB22dB50Hz{75kHzz

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Theresearchgoals,objectives,andkeyresultsaresummarizedinthischapter.Themajorcontributionsofthisresearcharehighlighted.Finally,suggestionsforimprovementsandextensionstothisworkareprovided. Previouslydevelopedmicrophonesforaeroacousticapplicationshavebeenstudied.Whilegreatstrideshavebeenmadeindevelopingsuchamicrophone;theliteraturereviewidentiesareasofneededimprovement|inparticularextendingthedynamicrange.ExistingMEMSmicrophoneseitherhaveasucientnoiseoorormaximumpressure.However,noMEMSmicrophonedevelopedtodateapproachesboththelowerandupperendoftheBruelandKjr4138'sdynamicrange.Withanoiseoorof22dB=p Inadditiontodevelopingtherstdual-backplatecapacitivemicrophoneforaeroacousticmeasurements,othercontributionsofthisresearchareasfollows.TheSUMMiTVMEMSfoundryprocessatSandiaNationalLaboratorieshasbeenappliedinanovelwaytoMEMSmicrophonedesign.Themulti-level,planarized 196

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polysiliconprocesshasbeenleveragedtocreateadual-backplatecapacitivemicrophone.Thegreatestweaknessofthisprocessforthemicrophoneisthevariabilityitintroducesintothesensitivity.However,thewellcontrolledatnessandstresswerecriticalforthesuccessofthisresearch. Theexperimentalcharacterizationgivesacomparisonbetweentheperformanceofvoltageamplierandchargeamplierinterfacecircuitsforalow-capacitanceMEMSmicrophone.Thevoltageamplierisclearlysuperiorintermsofdynamicrange,givinganoiseoor19dBlowerat1kHz.Alsodemonstratedwastheimportanceofgoodpackaging.Thedesignedpackagerobustlycontainedthevoltageamplier.Whilethechargeamplierwasexternaltothemicrophonepackageseparatedbyalengthofcable.Thisresultsinexcessive60Hzinterferenceduringmeasurementswhenthemicrophonewastestedwiththechargeamplier.Inaddition,thepackagedidnotadequatelysupportthecables.Thisresultedinvibrationcorruptingthechargeamplierfrequencyresponsemeasurements. Inadditiontothepresentedexperimentalcharacterization,adetailedthe-oreticalbackgroundhasbeendiscussed.Thesingle-backplatecondenseranddual-backplatecondensermicrophoneshavebeencomparedintermsofelectrostaticbehavior,dynamicrange,andstability.Bothtypesofmicrophoneshavebeenconsideredtheoreticallywithachargeamplierandavoltageamplier.Themicro-phonetheoryisapplicabletothedesignofacapacitivemicrophoneforarbitraryapplication,includingaudioandaeroacoustic.

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shouldbecharacterizedtoatleast100kHztoqualifythemicrophoneovertheen-tiredesiredfrequencyrange.Ideally,thefrequencyresponsecanbemeasuredoverthefullbandwidthofthemicrophone.Thereareseveralchallengestoovercometomakethismeasurementareality.Anacousticsourceisneededtogeneratesoundoverthisfrequencyrange.Typicalaudiodriversarelimitedtoaround20kHz.Whileultrasonicdriversexist,theyaretypicallydesignedforanarrowoperatingbandwidth.Inadditiontothesource,anexperimentalsetupmustbedevelopedtoproduceacontrolledsoundeld.At100kHz,thewavelengthofsoundinairis3:4mm;thusaplanewavetubewouldneedacrosssectionwitha1:7mmsidelength.Thisisimpractical,asareferencemicrophoneandapackagedMEMSmicrophonewillnottinthiscrosssection.Thereforeafreeeldmeasurementislikelytobethebestoption. Anotherareaforimprovementisthemeasurementofthetotalharmonicdistortion.Forthismeasurement,themicrophonewouldideallyhaveapuresinewaveincidentpressure.Thusanyharmonicsgeneratedwouldbeduetonon-linearitiesinthemicrophone.Duetolimitationsintheexperimentalsetup,excessivenon-linearitieswereproducedwhengeneratingsoundpressurelevelsapproaching160dB.However,evenifanidealsignalgenerator,amplier,andacousticdriverwerefound;non-linearitieswouldstillexistduetothepropagationofthehighamplitudesoundwave.Anexperimentalsetupmaybedevelopedtopre-distortthesignalsenttotheacousticdriver,suchthatthemicrophonereceivesapuresinewave.R.Holmandemonstratedusingafeed-forwardlooptopre-distortasignaltoachieveapuresinewaveforasyntheticjetactuator[ 133 ].ThistechniquecanbeappliedtotheacoustictestsetupfortheTHDcharacterization. Inorderforthedesignedmicrophonetobesuitableforuseintheeld,severalimprovementsarenecessarytothemicrophonedieandpackage.First,theprocessowwouldneedtobemodiedtoprovidebondpadmetallization.

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Thiswouldallowforamorerobustwirebond;improvingyieldandreducingthechanceoffailureafterpackaging.Furthermore,withmoreexibilityintheprocessow,adesignoptimizationcouldbeperformedpotentiallyimprovingthedeviceperformance.Thediecouldberedesignedwithonlyonemicrophoneperdie.Thishasthepotentialtoreducethediesizeto1mm2.Furthermore,interfacecircuitryforonemicrophonewouldbeneededperpackage.Thiswouldgreatlyreducethepackagesize.Finally,arobust,shieldedpackageisneededforreliabilityandimmunitytoelectromagneticinterference. Thereareseveralotherexperimentsthatcanprovidedadditionalinsightforthemicrophonebehaviorandimplementeddual-backplatestructure.Thelaservibrometermaybeusedtomeasurethemodeshapeofthediaphragm.Byposi-tioningthelaserinthecenterofthetopbackplateholesacrossthediameterofthediaphragm,23scanpointsmaybeobtained.Inaddition,theLVmaybeusedtomeasuretheresponseofthetopbackplateduetoacousticandelectricalexcitations.Anexperimentalsetuputilizingashakerandareferenceaccelerometermaybeusedtodeterminethesensitivityofthemicrophonetovibrationinputs.Finally,theheliumPWTsetupcouldbeimprovedtoincludeamethodformeasuringtheconcentrationofheliuminthetube.Thismayallowforimprovedrepeatabilityandcontroloftheexperiment. Inattemptstodevelopasuccessfulbondprocessforthepolysiliconbondpads,severaltechniqueswereattempted.Severalmetallizationstrategiesweretested.Thisinvolvedpatterningphotoresistoverthebondpads.Therewassome

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dicultyinobtaininganevencoatingofphotoresistextendingtothebondpads.Thiswasbecauseofcloseproximityofthebondpadstothedieedge.Thisissuewasobserved|toalesserextent|forthebacksidephotoresistfortheDRIE.Aminimumdistanceof500misrecommendedfromthedieedgetofeaturespatternedonindividualdie.Thebondpadspacingandsizewasmadesmalltoaccommodatethenumberofdevicesonthedie;100umsizepadswitha50mspacewereused.Whilethisissucientforautomatedbonding,itleaveslittleroomforerrorforlaboratoryassembly.Abondpadlayoutwithaminimum200msidelengthandpitchisrecommendedforeaseinbonding.Inaddition,thisallowsalternativebondingoptionssuchasepoxy.However,theseissuescanbemitigatedbyhavingarobustbondpadmetallization.Thepolysiliconbondpadsproveddiculttobondto.Anexternalcompanywasneededtoproduceareliablegoldballbondtothepolysiliconpads. Finally,itisrecommendedtoconsiderthecircuitryandpackagingearlyinthedesignprocess.Thenoiseoorachievedforthismicrophonewasenabledviaahighqualityamplier.Arobustpackagewillfacilitateexperimentalcharacterization.Thepackagingusedforthismicrophonewaseasytouse;allowingforquicktransitionfromonedevicetothenext.However,itdidnotprovidesucientshielding.Thepackagecouldbeimprovedbysurroundingthemicrophoneandwireswithametalshield.Ideallyashieldedcablewouldentertheshieldedpackagehousingandconnecttothemicrophoneinternally.However,aninsulatingexteriorprovidesexibilityintheexperimentalsetupandcanhelppreventgroundloops.

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Inthisappendix,themechanicallumpedparametersforaclampedcircularplatearefound.Thisincludesthestorageofpotentialenergyrepresentedbyalumpedcomplianceandthestorageofkineticenergyrepresentedbyalumpedmass.TheseresultsareusedinSection 3.2 todevelopthelumpedelementmodelofadual-backplatecondensermicrophone. 21 ].Thepotentialco-energymaybegenerallyexpressedas Thelimitsoftheintegrationarefrom0toF0,whichistheforcethatcorrespondstothenalcenterdeection,w(0).Rewritingintermsofuniformpressure,p,actingonanarea,A,thepotentialco-energyisgivenby ThedeectionoftheplateasafunctionofpressureisknownfromEquation 3{6 .Itisnotedthatthedeection,w,isalsoafunctionofr.ThepotentialenergystoredinaninnitesimalsectionoftheplateofareadAisgivenby 201

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Assuminganaxisymmetricdeection,theareadAcanbeexpressedas2rdr.Therefore,byintegratingoverr,thetotalpotentialco-energyis Now,Equation 3{6 issubstitutedinforw(p;r),whichgives a22dp2rdr:(A{5) Theintegraloverpcanbedirectlyevaluated,thusEquation A{5 becomes a22rdr:(A{6) Theintegraloverrisequaltoa2=6,thusthepotentialco-energyisequalto FromEquation 3{8 ,theincidentpressure,p0,atwhichthecenterdeectionisw(0),isexpressedas a4w(0):(A{8) BysubstitutingEquation A{8 intoEquation A{7 ,thepotentialco-energybecomes, whichissimpliedto 264D Aswasstatedearlier,foralinearspring,thepotentialenergyisequaltothepotentialco-energy.ThepotentialenergyinalumpedspringisgivenbyWPE=1 2kw(0)2,thereforethelumpedmechanicalcomplianceisgivenbyCm=

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21 21 ] 21 ].Thekineticenergyofamass,m,withavelocity,v,isgivenby1 2mv2.Fortheplatewithanon-uniformvelocity,thekineticenergyinainnitesimalmassis 2v2dm:(A{12) Therefore,thetotalkineticenergyintheplateisgivenby 2Zv(r)2dm:(A{13) Thevelocityatthecenterofthediaphragmisv(0)andthevelocityofthedia-phragmasafunctionofris a22:(A{14) Assumingahomogenousplatewithmassperunitarea0equaltoh,wheretheplatehasadensityandthicknessh,thedierentialmass,dm,iswrittenas02rdr.NowEquation A{13 becomesWKE=0v(0)2aZ01r a24rdr: Evaluatingtheintegralgivesthefollowingresultforthekineticenergyintheplate,WKE=v(0)2

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FromEquation A{16 ,thelumpedmechanicalmassoftheplateis[ 21 ]

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Inthisappendix,theapproximatefrequencyresponseofthedualbackplatemicrophonewillbederivedusinglumpedelementmodeling.TheequivalentcircuitofthemicrophoneisshowninFigure B-1 .TheexpressionsforthelumpedelementsaregiveninTable 3-5 FigureB-1.Lumpedelementmodelofthedual-backplatemicrophoneshowingrelevantimpedancesandvolumevelocities. Usingbasiccircuitanalysistechniques[ 124 ],thefrequencyresponseofthemicrophonewillbefound.TheelectricalrelationV=IRisanalogoustoP=QRintheacousticdomain,assuminganimpedanceanalogy[ 21 ].Thequantitypd=pinisfoundasafunctionoffrequency.Theoutputvoltageofthemicrophone,asgiveninEquation 3{78 orEquation 3{99 ,isproportionaltotothepressureactingonthediaphragm.ThispressureisgivenbypdasshowninFigure B-1 Thepressure,pd,isgivenby whereQpisthevolumevelocityowingthroughtheplatesofthemicrophone.Theimpedanceofthecomplianceisgivenby1=sCa;d,wheresisthecomplex 205

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frequencyintheLaplacedomain[ 17 ].ThevolumevelocityQpisfoundusingavolumevelocitydividerrelation, Thetotalvolumevelocityowingintothemicrophone,Qinisgivenby whereZtotisthetotalimpedanceofthemicrophone. Tondanexpressionforthefrequencyresponseofthemicrophone,thevaluesofthevariousimpedancesmustrstbefound.Firsttheimpedanceofthediaphragm,Za;d,andeachbackplate,Za;bp1andZa;bp2,arefound.Thediaphragmimpedanceistheseriescombinationofthediaphragmcompliance,mass,andradiationmass.Theimpedanceofeachbackplateistheparallelcombinationofthebackplatecomplianceandresistance.Thethreeimpedancesareasfollows: (B{4) (B{5) Thesymbolisusedtoindicatetheparallelcombinationoftwoelements.

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Thereforetheimpedanceoftheseriescombinationofthethreeplatesis Byndingacommondenominator,thethreetermscanbecombinedasfollows, Thetotalimpedanceofthemicrophoneisgivenby whichisre-writtenas BycombiningEquation B{2 ,Equation B{3 ,andEquation B{10 ,thevolumevelocityQpis Therefore,thefrequencyresponseofthemicrophoneisfoundbysubstitutingEquation B{11 intoEquation B{1 ThenalexpressionforthefrequencyresponseisexpressedcompletelyintermsoflumpedelementsbysubstitutingEquation B{8 intoEquation B{12 ,howeverthisistoolongtowriteouthere.Asimpliedexpressionisobtainedby

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assumingthecomplianceofeachbackplateisnegligible1.Now,Equation B{8 becomes SubstitutingEquation B{13 intoEquation B{12 resultsin ByexpandingEquation B{14 andcollectingtermsbypowersofs,thenalexpressionforthesimpliedfrequencyresponseofthedual-backplatemicrophoneis ThepressureacrosseachbackplatecanbefoundinamannersimilartoEquation B{1 .Thediaphragm,topbackplate,andbottombackplatesharethesameow,thusthepressureacrossthetopbackplateis andthepressureacrossthebottombackplateis DiscussionprovidingphysicalinsightfortheresultsobtainedinEquation B{14 andEquation B{15 isgiveninSection 3.2.6 3.2

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Theuncertaintyforthetheoreticalperformancemetricsisderivedinthisap-pendix.TheformulationspresentedhereutilizeresultsobtainedinChapter 3 andChapter 4 .Thesensitivityofthedual-backplatecapacitivemicrophoneisanalyzedforbothachargeamplierandvoltageamplierinterfacecircuit.Furthermore,thepredictionsfortheresonantfrequencyandnoiseoorareexplored.Thenalsectionofthisappendixdescribesthetechniqueusedtoestimatethemeasuredsensitivityand95%condenceinterval. First,considertheuncertaintyofaquantityGwhichisafunctionofnparameters,x1;x2;:::xn,suchthat Eachparameter,xi,isassumedtohaveanuncertaintysuchthatithasavariationofUxiforagivencondencelevel.ItfollowsthattheuncertaintyinGis[ 134 ] Xi@G @xiUxi2:(C{2) ThepreviouslyderivedexpressionsareanalyzedaccordingtoEquation C{2 todeterminetheuncertaintyinthetheoreticalmicrophoneperformance. 3.2.3 andsimpliedresultsforbothamicrophonewithachargeamplierandavoltageamplieraregiveninSection 4.2.1 4{9 ,whichisre-writtenexplicitlyintermsofboth 209

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thetopairgap,g10,andthebottomairgap,g20, 3VB 3VB Neglectingthecavitystieningandexpressingthesensitivityintermsofonlymaterialparametersandgeometryparameters,Equation C{3 becomes 3a6d Cf+1 3a6d Cf;(C{4) neglectingdierencesinthetopandbottomcapacitorarea. Equation C{4 isthesumoftwocomponents,S1caandS2ca;whereS1caisafunctionofg10andS2caisafunctionofg20.Theyareidenticalexpectfortheairgap.ThusthesensitivityofEquation C{4 tochangesinthevariousparametersissolvedforbyconsideringtherstpartseparately.Theresultsarethendirectlyappliedtothesecondpart,substitutingg20forg10. Thisanalysisfocusesonparametersinherenttothemicrophone.Thus,theuncertaintyofthreeparametersareconsidered:thediaphragmradius,ad,theexuralrigidity,D,andtheairgapg10.Forthisanalysis,itisassumedthatVBandCfaremeasuredtosucientaccuracytonotaecttheuncertaintyofthepredictedsensitivity. However,theexuralrigidity,D,isafunctionofseveralparameters.There-foreuncertaintyanalysismustrstbeperformedonit.Equation 3{3 givestheexpressionforD,whichisafunctionofthemodulusofelasticity,E,thediaphragmthickness,hd,andPoisson'sratio,.ThesensitivityofDtotheseparametersis @E=h3d @hd=3Eh2d

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and @=Eh3d ThetotaluncertaintyinDis @EUE+@D @hdUhd+@D @U1 2:(C{8) ThesensitivityofS1catochangesinadis Cf:(C{9) ThesensitivityofS1catovariationsinDis 3a6d Cf:(C{10) Finally,theS1cavarieswithg10as 3a6d Cf:(C{11) ThetotaluncertaintyinSca,isgivenby 2; wherethe@S2ca C{9 ,Equation C{10 ,andEquation C{11 4{10 ,is

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re-writtenintermsofmaterialandgeometryparameters, 3VB 1+g10 3VB 1+g20 SimilartoEquation C{4 ,Svaisthesumoftwocomponents,S1vaandS2va.Eachofthesecomponentsisafunctionofbothg10andg20;however,duetothesimilaritybetweenthem,@S2va Beginningwithad,thesensitivityofS1vatochangesinadis 3VB 1+g10 3VB ThesensitivityofS1vatoDis 3VB 1+g10 ThesensitivityofS1vatog10is 3VB 1+g10 3VB Finally,thesensitivityofS1vatog20 3VB UsingEquation C{14 ,Equation C{15 ,Equation C{16 ,andEquation C{17 ,thetotaluncertaintyinSvais 2;

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3{181 .Fortheresonantfrequencyuncertaintyanalysis,theelectrostaticcomplianceandcavitystieningareneglected.Thus,theresonantfrequencyissimpliedto Substitutingexpressionsfortheacousticmass(Equation 3{138 )andtheacousticcompliance(Equation 3{139 )intoEquation C{19 ,theresonantfrequencyisre-writtenas 80a4d(12) 2:(C{20) Thesensitivityof!0totheparameters,ad,,,E,andhd,areasfollows: 40a3d(12) 80a4d(12) 2; 160a4d(12) 80a4d(12) 2; 80a4d Eh2d9 80a4d(12) 2; 160a4d(12) 80a4d(12) 2; and@!0 80a4d(12) 80a4d(12) 2: Thenalexpressionfortheresonantfrequencyuncertaintyis 2:

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Assuminganoutputpowerspectraldensityofsvo,theinputreferrednoisedensityis whereSmicisthesensitivityofthemicrophone.ThesensitivityofpitoachangeinSmicis Therefore,theuncertaintyinpiis C-1 .Themeasuredsensitivityisplottedversuspressure.Inadditiontoestimatingthemeasuredsensitivity,acondenceintervalforthesensitivityisdesired. Beginningwiththetwodatapointsforthelowestpressures,theaveragesensitivity, 135 ] ^s2max=(N1)s2

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FigureC-1.Illustrationofsensitivitydataanalysis. wherethereareNsamplesand2N1;1=2denotestheChi-squaredistributionwithN1degreesoffreedomanda1condenceinterval.Foronlytwodatapoints,Equation C{30 givesalargeestimatedvariation.Ifthenextdatapointfallswithintherange x+^smax;(C{31)Nisincreasedandtheaboveprocedureisrepeated.ThisprocesscontinuesuntilthethenextmeasuredsensitivityfallsoutsidetherangegiveninEquation C{31 .Thesensitivityofthemicrophoneisgivenby x< x+stN1;=2

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Thedevicestructureofthemicrophonerequiresthreeindependentplanarconductinglayerstoformthetwobackplatesandthediaphragm.Sandia'sUltra-planarMulti-levelMEMSTechnology(SUMMiTV)processow[ 16 ]isagoodmatchforthisdevice.Ithasvelow-stresspolysiliconlayersandemployschemicalmechanicalpolishing(CMP)toachieveultra-atstructurallayers. Figure D-1 showsacrosssectionoftheSUMMiTVprocess.Theprocessbeginswiththegrowthof0:63mofthermaloxideandthedepositionof0:80mofLPCVDsiliconnitride.Theselayersprovideisolationfromthesiliconsubstrateaswellasananchorforthepolysilicon. FigureD-1.CrosssectionoftheSUMMiTVprocess. Theremainderoftheprocessconsistsofdepositingalternatinglayersofpolysiliconandsacricialoxide;allofthesacricialoxidelayersaredepositedusingLPCVD.A0:3mlayerofLPCVDpolysilicon,Poly0,isdeposited.ThisisfollowedbySacOx1,therstlayerofsacricialoxidewithathicknessof2m.ThenextthreelayersarePoly1,a1:0mLPCVDpolysiliconlayer;SacOx2,athin0:3mlayerofoxide;andPoly2,a1:5mlayerofLPCVDpolysilicon.Forthemicrophone,theentirelayerofSacOx2isremoved,thereforePoly1andPoly2combinetoformapolysiliconlayerwithatotalthicknessof2:5m.ThiscombinedpolysiliconlayerwillbereferredtoasPoly2.Thenextlayerofsacricial 216

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oxide,SacOx3isthendeposited.ThisoxidelayerisattenedusingCMP.Thisallowsthefollowinglayerofpolysilicontobeveryat,howeverthethicknessoftheoxideishighlyvariable;thethicknessesandtolerancesofallofthelayersisgiveninTable D-1 .Poly3,thenextlayerofPECVDpolysiliconisdepositedandhasathicknessof2:25m.Thenalsacricialoxidelayer,SacOx4,isdepositedandattenedusingCMP.Thisisfollowedbythenal2:25mlayerofPECVDpolysilicon,Poly4. TableD-1.ProcessdataasreportedbySandiaNationalLaboratoriesfortheSUMMiTVprocess. Poly00:29m0:01mPoly22:51m0:003mPoly32:27m0:01mPoly42:27m0:006mSacOx12:03m0:004mSacOx32:2m0:2mSacOx42:0m0:2m 16 ].Inaddition,thethicknessofthepolysiliconlayersiswellcontrolled,thusthecomplianceofthepolysiliconplatescanbeaccuratelypredicted. However,theSUMMiTVprocessdoeshavesomefeaturesthatarenotidealforadual-backplatecapacitivemicrophone.Thethreestructurallayersareallapproximatelythesamethickness.Therefore,assumingtheyhavesimilarradii,theywillhaveapproximatelythesamecompliance.Ideally,thebackplateswouldbe

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muchstierthanthediaphragm.Furthermore,thetoptwosacricialoxidelayershavevariablethicknesses;thiswillintroducesignicantuncertaintyinthepredictedmicrophoneresponse.Consideringthesetradeos,theSUMMiTVprocessisstillthebestchoiceforthefabricationofthismicrophone.

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DavidThomasMartinwasbornonNovember12,1979,inBethlehem,Penn-sylvania.HeattendedDoctorPhillip'sHighSchoolinOrlando,FL,graduatingin1997.HethenenrolledattheUniversityofFloridawherehereceivedhisbachelor'sdegreeinelectricalengineeringfromtheUniversityofFloridain2001.Duringhisnalsemesteranundergraduate,DavidjoinedtheInterdisciplinaryMicrosystemsGroupwhereheworkedonthepackagingofapiezoresistivemicrophonearray.InAugustof2001,DavidbeganhisgraduatestudiesattheUniversityofFloridawherehewasawardedafellowshipfromSandiaNationalLaboratories.InMayof2005,heearnedaMasterofSciencedegreeinelectricalengineering.DavidiscurrentlycompletinghisdoctoraldegreeattheUniversityofFlorida.Hisresearchinterestsincludemicroelectromechanicalsystems(MEMS)microphonedesign,packaging,andmicrophonecharacterization. 234