<%BANNER%>

Development of a MEMS Piezoelectric Microphone for Aeroacoustic Applications

Permanent Link: http://ufdc.ufl.edu/UFE0041968/00001

Material Information

Title: Development of a MEMS Piezoelectric Microphone for Aeroacoustic Applications
Physical Description: 1 online resource (260 p.)
Language: english
Creator: WILLIAMS,MATTHEW D
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

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

Notes

Abstract: Passenger expectations for a quiet flight experience coupled with concern about long-term noise exposure of flight crews drive aircraft manufacturers to reduce cabin noise in flight. During the aircraft component design or redesign process, aeroacousticians use advanced experimental techniques to help guide these noise-reduction efforts. Chief among their available tools are arrays, distributed collections of microphones that spatially sample pressure fluctuations. Different array configurations are deployed in flight tests on the exterior of aircraft, enabling characterization of the turbulent boundary layer, identification of noise sources, and/or assessment of the effectiveness of candidate noise-reduction technologies. The requirements for microphones used in aircraft fuselage arrays are demanding. They should be small, thin, and passive; respond linearly to a large maximum pressure; possess audio bandwidth and moderate noise floor; be robust to moisture and freezing; and exhibit stability to large variations in temperature and humidity. Microelectromechanical systems (MEMS) microphones show promise for meeting the stringent performance needs for this application at reduced cost, made possible using batch fabrication technology. This research study represents the first stage in the development of a microphone that meets these needs. The developed microphone utilized piezoelectric transduction via an integrated aluminum nitride layer in a thin-film composite diaphragm. A theoretical lumped element model and associated noise model of the complete microphone system was developed and utilized in a formal design-optimization process. Seven optimal microphone designs with 515-910 micron diaphragm diameters and 500 micron-thick substrate were fabricated using a variant of the film bulk acoustic resonator (FBAR) process at Avago Technologies. Laboratory test packaging was developed to enable thorough acoustic and electrical characterization of nine microphones. Measured performance was in line with sponsor specifications, including sensitivities in the range of 30-40 uV/Pa, minimum detectable pressures in the range of 75-80 dB(A), 70 Hz to greater than 20 kHz bandwidths, and maximum pressures up to 172 dB. With this performance in addition to their small size, these microphones were shown to be a viable enabling technology for the kind of low-cost, high resolution fuselage array measurements that aircraft designers covet.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by MATTHEW D WILLIAMS.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Sheplak, Mark.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-04-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0041968:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041968/00001

Material Information

Title: Development of a MEMS Piezoelectric Microphone for Aeroacoustic Applications
Physical Description: 1 online resource (260 p.)
Language: english
Creator: WILLIAMS,MATTHEW D
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

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

Notes

Abstract: Passenger expectations for a quiet flight experience coupled with concern about long-term noise exposure of flight crews drive aircraft manufacturers to reduce cabin noise in flight. During the aircraft component design or redesign process, aeroacousticians use advanced experimental techniques to help guide these noise-reduction efforts. Chief among their available tools are arrays, distributed collections of microphones that spatially sample pressure fluctuations. Different array configurations are deployed in flight tests on the exterior of aircraft, enabling characterization of the turbulent boundary layer, identification of noise sources, and/or assessment of the effectiveness of candidate noise-reduction technologies. The requirements for microphones used in aircraft fuselage arrays are demanding. They should be small, thin, and passive; respond linearly to a large maximum pressure; possess audio bandwidth and moderate noise floor; be robust to moisture and freezing; and exhibit stability to large variations in temperature and humidity. Microelectromechanical systems (MEMS) microphones show promise for meeting the stringent performance needs for this application at reduced cost, made possible using batch fabrication technology. This research study represents the first stage in the development of a microphone that meets these needs. The developed microphone utilized piezoelectric transduction via an integrated aluminum nitride layer in a thin-film composite diaphragm. A theoretical lumped element model and associated noise model of the complete microphone system was developed and utilized in a formal design-optimization process. Seven optimal microphone designs with 515-910 micron diaphragm diameters and 500 micron-thick substrate were fabricated using a variant of the film bulk acoustic resonator (FBAR) process at Avago Technologies. Laboratory test packaging was developed to enable thorough acoustic and electrical characterization of nine microphones. Measured performance was in line with sponsor specifications, including sensitivities in the range of 30-40 uV/Pa, minimum detectable pressures in the range of 75-80 dB(A), 70 Hz to greater than 20 kHz bandwidths, and maximum pressures up to 172 dB. With this performance in addition to their small size, these microphones were shown to be a viable enabling technology for the kind of low-cost, high resolution fuselage array measurements that aircraft designers covet.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by MATTHEW D WILLIAMS.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Sheplak, Mark.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-04-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0041968:00001


This item has the following downloads:


Full Text

PAGE 1

DEVELOPMENTOFAMEMSPIEZOELECTRICMICROPHONEFOR AEROACOUSTICAPPLICATIONS By MATTHEWD.WILLIAMS ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

PAGE 2

c r 2011MatthewD.Williams 2

PAGE 3

Tomywife,Laura,whocamewithmetoGainesville forfouryearsbutstuckwithmeforsix 3

PAGE 4

ACKNOWLEDGMENTS TheInterdisciplinaryMicrosystemsGroup(IMG)atUniversi tyofFloridahasbeen anoutstandingplacetoearntwograduatedegrees,andthere aremanypeopletothank. Myadvisor,MarkSheplak,deservestremendouspraiseforth eincredibleresearchgroup heputtogetherandnowmaintainstogetherwithDavidArnold, LouCattafesta,Toshi Nishida,HughFan,HuikaieXieandYKYoon.Overthelastsixyears, Markhaspushed mewellbeyondanyimaginedlimitationsIhadwhenIarrived, andhehasdoneitwitha mixofbluster,compassion,acumen,andgenerositythatisu niqueonlytohim.Iwillowe MarkimmenselyforanyfuturesuccessthatIenjoy.Forayoun gfatherlikemyself,hehas alsobeenaterricrolemodel. IhavebenetedsignicantlyfrommycontactwiththeotherI MGprofessorsaswell, mostnotablyDavidArnoldandLouCattafesta,whoareatoncet remendousresearchers, teachers,andmen.Theybothservedasmembersofmycommitte eanditwasapleasure workingwiththeminmanydierentcapacities.DavidArnoldt aughtme,whetherhe knowsitornot,aboutvision;Iadmirehisuniqueabilitytoc utthroughtheweeds.I aspiretoLouCattafesta'slevelofprecisioninallthatIdo Ihaveenjoyedmanyfruitfulconversationswithmyothercom mitteemembers, Nam-HoKimandBhavaniSankar,aswell.Bothhavealwaysbeenex tremelyhelpful andcordial,andIthankthemwholeheartedlyforalloftheir support.IalsooweDavid Nortonadebtofthanksforservingonmycommitteeandforgran ting,asassociatedean, additionalrexibilityinmyfundingsituationformynalse mester. IenteredgraduateschoolwithaNationalScienceFoundation GraduateFellowship forwhichIamexceedinglygrateful,notjustforthefunding itsuppliedbutforthe doorsthatitopened.BoeingCorporationwasthesponsorfor mydissertationwork;I owethemforthefundingtheyprovidedandfortheprivilegeo fworkingonaproblem ofsuchimportancetothem.JimUnderbrinkofBoeingalwayske ptawatchfuleyeon myprogress,anditwasourclosecontactlateintheprojectt hatreallysolidiedmy 4

PAGE 5

understandingofthebigpicture.Ibenetedimmenselyfrom workingwithhimand cannotthankJimenoughforbeingsogivingofhistimeandsow illingtoteach.His commitmenttoimprovethetechnologyofaeroacousticmeasu rementsisinspiring. MycolleagueswithinIMGdeservehighpraise.BenGrinhasb eenamentortome sincethemomentIsteppedontheUniversityofFloridacampus .Icanonlyhopethat Ihavecontributedafractionasmuchtohisdevelopmentashe hastomine.Myother seniorcolleagueswhohavesincegoneontoindustry,VijayCh andrasekharanandBrian Homeijer,werealwaystremendouslysupportiveaswell.Fina lly,JessMeloyiseasilythe mostsimultaneouslyhelpfulandknowledgeablepersonIhav eeverknown;Ioermy sincerestapologiestoherforsoregularlyaskingforherci rcuitexpertise. Abondisformedbetweengraduatestudentswhoworkontheirp roposalsor dissertationsatthesametime,andsoitiswithfondmemorie sthatIwilllookback onmytimeinthetrencheswithAlexPhippsinthesummerof2008 andJeremySellsand DrewWetzelinthespringof2011.Iwillnotsoonforgetourmu tualsupport(orallthe work). ThecombinedsocialandintellectualaspectofIMGcannotbe ignored,andsoitisin thatspiritthatIthankBrandonBertolucci,ChrisBahr,Dyl anAlexander,DavidMills, ErinPatrick,NikZawodny,JessicaSockwell,MiguelPalavic cini,MatiasOyarzun,and honoraryIMGerRichardParker.Whetherat80'snight,afoot balltailgate,happyhour, orafrisbeegame,Ihavebeenprivilegedtosharetheircompa ny. Ihaveworkedwithmanyoutstandingundergradraduatesonth isprojectwhodeserve recognition:TianyReagan,AnupParikh,AdamEcker,KalebEr win,andKyleHughes. Inparticular,itisTianyReagan'srelentlessnessthatha smostdirectlycontributedto thesuccessofthisproject.Herngerprintsarealloverthis dissertation. ThanksareduetoDavidMartin,OsvaldoBuccafusca,andAtul GoelatAvago Technologiesforalwaysworkingwithmeonthepiezoelectri cmicrophoneprojectingood 5

PAGE 6

faithandwithexpectationsforitssuccess.Theydeservemu chcreditfortheresultsthat wereachieved. Customerservicecontinuestodeclineintoday'sworld,but thepeopleatBrueland Kjr,Polytec,andTMREngineeringhavenotheard.JimWyatt andJoeChoualways camethroughmyanswerstomymicrophonequestionswhenthei rcompetitorsdidnot; ArendvonderLiethandJohnFoleyworkedtirelesslytoensure IMG'slaservibrometer systemstayedrunningatleastuntilIgraduated;andKenRee dalwaysturnedupwith high-qualitymechanicalpartsinrecordtime. Thanksareduetomyundergraduateadvisor,PaulJoseph,for turningmeonto researchintherstplace.Inaddition,myparentsDavidand AnnamadeallthatIhave accomplishedpossible.LongbeforeMarkSheplakwaspreach ingthewisdomofsettinghis studentsupforsuccessandgettingoutoftheway,myparents weredoingjustthatwith theirson. Thelatterpartsofgraduateschoolcanbehardonafamily,bu tmywifeLaurawasa rock.Wordscannotthankherenoughforthesacricesshemad etomakethisdissertation possible.Ididitallforherandourdaughter,Callahan. 6

PAGE 7

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................4 LISTOFTABLES .....................................11 LISTOFFIGURES ....................................13 ABSTRACT ........................................18 CHAPTER1INTRODUCTION ..................................20 1.1Motivation ....................................20 1.2ResearchObjectives ...............................25 1.3DissertationOverview .............................27 2MICROPHONEFUNDAMENTALS ........................29 2.1SoundandPseudoSound ...........................29 2.2TheRealitiesofMicrophoneDesign ......................31 2.3MicrophonePerformanceMetrics .......................36 2.3.1FrequencyResponseandSensitivity ..................37 2.3.2NoiseFloorandMinimumDetectablePressure ............38 2.3.3LinearityandMaximumPressure ...................42 2.3.4DynamicRange .............................44 2.3.5SummaryofMicrophonePerformanceMetrics ............44 2.4Summary ....................................45 3PRIORART .....................................46 3.1ReviewofMEMSPiezoelectricandAeroacousticMicrophon es .......46 3.2Summary ....................................58 4MEMSPIEZOELECTRICMICROPHONE ....................62 4.1Piezoelectricity .................................62 4.2DesignforFabrication .............................66 4.3Summary ....................................70 5MODELING .....................................71 5.1LumpedElementModelingOverview .....................71 5.2LumpedElementModelofaPiezoelectricMicrophone ...........73 5.2.1Elements .................................76 5.2.1.1Transduction .........................76 5.2.1.2Structuralelements ......................78 7

PAGE 8

5.2.1.3Acousticelements .......................80 5.2.1.4Electricalelements ......................83 5.2.2DiaphragmMechanicalModel .....................83 5.2.3FrequencyResponse ...........................88 5.2.3.1Sensor .............................90 5.2.3.2Actuator ............................92 5.2.4Electricalimpedance ..........................93 5.2.5Validation ................................94 5.2.5.1Diaphragmmodelvalidation .................95 5.2.5.2Lumpedelementmodelvalidation ..............97 5.3InterfaceCircuitry ...............................98 5.3.1VoltageAmplier ............................99 5.3.2ChargeAmplier ............................102 5.3.3NoiseModels ..............................104 5.3.3.1Noisemodelwithvoltageamplier .............105 5.3.3.2Noisemodelwithchargeamplier ..............108 5.3.4Selection .................................110 5.4Summary ....................................112 6OPTIMIZATION ...................................113 6.1DesignOverview ................................113 6.1.1DesignVariables .............................113 6.1.2Objective .................................115 6.2Formulation ...................................117 6.3Approach ....................................119 6.4ResultsandDiscussion .............................121 6.5Summary ....................................126 7REALIZATIONANDPACKAGING ........................128 7.1Realization ...................................128 7.1.1Geometry ................................128 7.1.2FabricationResults ...........................128 7.2Dicing ......................................130 7.2.1DicingProcess ..............................131 7.2.2DicingResults ..............................134 7.3Packaging ....................................134 7.4Summary ....................................140 8EXPERIMENTALCHARACTERIZATION ....................141 8.1ExperimentalSetup ...............................141 8.1.1DieSelectionSetup ...........................141 8.1.2DiaphragmTopographyMeasurementSetup .............144 8.1.3AcousticCharacterizationSetup ....................145 8.1.3.1Frequencyresponsemeasurementsetup ...........145 8

PAGE 9

8.1.3.2Linearitymeasurementsetup ................150 8.1.4ElectricalCharacterizationSetup ...................153 8.1.4.1Noiseroormeasurementsetup ...............154 8.1.4.2Impedancemeasurementsetup ...............156 8.1.4.3Parasiticcapacitanceextractionsetup ...........158 8.1.5ElectroacousticParameterExtraction .................159 8.1.5.1Complianceandmassmeasurementsetup .........160 8.1.5.2Frequencyresponsemeasurementsetup ...........165 8.1.5.3Eectivepiezoelectriccoecientmeasurementset up ....167 8.2ExperimentalResults ..............................168 8.2.1DieSelection ...............................168 8.2.2DiaphragmTopography .........................173 8.2.3AcousticCharacterization .......................175 8.2.3.1Frequencyresponse ......................175 8.2.3.2Linearity ............................176 8.2.4ElectricalCharacterization .......................180 8.2.4.1Noiseroor ...........................180 8.2.4.2Impedance ...........................183 8.2.4.3Parasiticcapacitanceextraction ...............184 8.2.5ElectroacousticParameterExtraction .................187 8.3Summary ....................................195 9CONCLUSION ....................................196 9.1RecommendationsforFuturePiezoelectricMicrophones ...........198 9.2RecommendationsforFutureWork ......................202 APPENDIX ADIAPHRAGMMECHANICALMODEL ......................204 A.1Strain-DisplacementRelations .........................205 A.2KirchhoHypothesis ..............................207 A.3EquationsofMotion ..............................208 A.4ConstitutiveEquation .............................214 A.5DisplacementDierentialEquationsofMotion ................217 A.6EquationsofEquilibrium ............................219 A.6.1Nonlinear ................................219 A.6.2Linear ..................................221 A.7ProblemSolutions ...............................222 A.7.1Linear ..................................223 A.7.1.1Generalsolutions .......................223 A.7.1.2Particularsolutions ......................224 A.7.1.3Innerregion:tension( x (1) > 0) ...............226 A.7.1.4Innerregion: x (1) =0 .....................226 A.7.1.5Innerregion:compression( x (1) < 0) ............227 9

PAGE 10

A.7.1.6Outerregion:tension( x (2) > 0) ...............228 A.7.1.7Outerregion: x (2) =0 ....................229 A.7.1.8Outerregion:compression( x (2) =0) ............230 A.7.2Nonlinear ................................231 A.8Closing ......................................234 BBOUNDARYCONDITIONINVESTIGATION ..................235 CUNCERTAINTYANALYSIS ............................237 C.1Approach ....................................237 C.2FrequencyResponseFunction .........................238 C.3NoiseFloor ...................................238 C.3.1Spectra ..................................239 C.3.2NarrowBand ..............................240 C.3.3Integrated ................................240 C.4Impedance ....................................240 C.5ParasiticCapacitanceExtraction .......................240 C.6ParameterExtraction ..............................241 DMATERIALPROPERTIES .............................242 REFERENCES .......................................243 BIOGRAPHICALSKETCH ................................260 10

PAGE 11

LISTOFTABLES Table page 1-1Fuselagearrayapplicationrequirements. ......................27 2-1Performancecharacteristicsofcommonaeroacousticmi crophones. ........45 3-1SummaryofMEMSmicrophones. ..........................59 4-1TypicalpropertiesofpiezoelectricmaterialsinMEMS. ..............66 5-1Geometricdimensionsofanexampledevice. ....................95 5-2Comparisonofvoltageandchargeampliertopologies ..............110 6-1Microphonedimensionsxedbythefabricationprocess. .............114 6-2Designvariablebounds. ...............................118 6-3Constantvaluesusedintheoptimization. .....................121 6-4Targetthin-lmresidualstresses. ..........................121 6-5Optimallayerthicknesses. ..............................124 6-6Optimizationresults. .................................124 7-1Designdimensions. .................................129 7-2Filmproperties. ....................................129 7-3Tapeandsubstratethicknesses. ...........................132 7-4Dicersettings. .....................................133 7-5Epoxydispensersettings. ..............................137 7-6Wirebondsettings. ..................................137 8-1Dieselectionlaservibrometersettings. .......................143 8-2Scanningwhitelightinterferometersoftwaresettings ...............145 8-3Settingsformicrophonefrequencyresponsemeasuremen tsinPULSE. ......148 8-4FrequencyresponsemeasurementsettingsusedatBoeing ............151 8-5Totalharmonicdistortionmeasurementsettingsusedat Boeing. .........153 8-6Noiseroormeasurementsettings. ..........................155 8-7Impedancemeasurementsettings. ..........................157 11

PAGE 12

8-8Pressurecouplermeasurementsettings. .......................163 8-9Settingsforsensitivitymeasurementofpressurecoupl ermicrophones. ......166 8-10Waferstatistics. ....................................172 8-11Pre-andpost-packagingLVmeasurements. .....................172 8-12Microphonefrequencyresponsecharacteristicsat1kHz inair. ..........176 8-13THDmeasurementsperformedatBoeingCorporation. ..............180 8-14Minimumdetectablepressuremetrics. .......................183 8-15Extractedelectricalparameters. ...........................185 8-16Open-circuitsensitivityestimates. ..........................187 8-17Extractedmechanoacousticparameters. ......................191 8-18Extractedelectroacousticparameters. ........................193 9-1RealizedMEMSpiezoelectricmicrophoneperformance. ..............197 9-2PerformancecharacteristicsofMEMSpiezoelectricmic rophone138-1-J3-F. ...199 C-1Parasiticcapacitanceextractionuncertainties. ...................241 D-1Propertiesofmicrophonediaphragmmaterials. ..................242 D-2Propertiesofgases. ..................................242 12

PAGE 13

LISTOFFIGURES Figure page 1-1Boeing777fuselageinstrumentedwithanarrayofmicrop hones. .........22 1-2AeroacousticphasedarraysdeployedaspartoftheQTD2pr ogram. .......23 2-1Force-displacementcharacteristicsforaperfectspri ng. ..............32 2-2Frequencyresponseofasecond-ordersystem. ...................33 2-3ConstitutivebehaviorforaDungspring. ....................35 2-4Variouscavitycongurations. ............................35 2-5Typicalaeroacousticmicrophonefrequencyresponse. ...............38 2-6Noisemodelforaresistor. .............................40 2-7Noisemodelforaresistorinparallelwithacapacitor. ..............40 2-8Low-passlteringofthermalnoise. .........................41 2-9VoltagenoisespectrumforanLTC6240amplier. .................41 2-10Idealandactualresponseofamicrophone. ....................43 2-11Operationalspaceofanaeroacousticmicrophone. ................45 3-1Piezoelectric(ZnO)microphonewithintegratedbuera mplier. .........47 3-2Piezoelectric(ZnO)microphoneutilizingmultiplecon centricelectrodes. .....48 3-3Piezoelectricmicrophoneutilizingaromaticpolyurea ...............48 3-4Piezoelectric(ZnO)microphonewithcantileversensin gelement. .........49 3-5CrosssectionoftherstaeroacousticMEMSmicrophone. ............50 3-6PiezoresistiveMEMSmicrophoneforaeroacousticmeasu rements. ........51 3-7Second-generationaeroacousticMEMSmicrophone. ................52 3-8Adual-backplatecapacitiveMEMSmicrophone. ..................53 3-9EarlyPZT-basedpiezoelectricmicrophone. .....................53 3-10Piezoelectric(ZnO)microphonewithtwoconcentricel ectrodes. .........54 3-11Measurement-gradeMEMScondensermicrophonedevelop edatBruelandKjr. 55 3-12Piezoelectric(PZT)microphoneforaeroacousticappl ications. ..........56 13

PAGE 14

3-13Top-viewofmicrophonestructuresfromFazzioetal.(2 007). ...........57 3-14Crosssectionofasecond-generationAlNdouble-cantil evermicrophone. .....58 4-1Venndiagramforpiezoelectric,pyroelectric,andferr oelectricmaterials. ....63 4-2FBAR-variantprocesslmstack. ..........................67 4-3Potentialcirculardiaphragmpiezoelectric/metallm stackcongurations. ...69 4-4Outlineoffabricationsteps. .............................70 5-1Illustrationoftheelectrical-mechanicalanalogy. ..................73 5-2Piezoelectricmicrophonestructure. ........................74 5-3Piezoelectricmicrophonelumpedelementmodel. .................75 5-4Two-portpiezoelectrictransductionelement. ...................77 5-5Laminatedcompositeplaterepresentationofthethinlmdiaphragm. ......84 5-6Derectionofaradiallynon-uniformcompositeplatewit hresidualstress. ....88 5-7Boundaryconditionsappliedtoaradiallynon-uniformp iezoelectriccomposite. .89 5-8Lumpedelementmodelwithcollectedimpedances. ................90 5-9Impedanceratiosappearingintheopencircuitfrequenc yresponseexpression. .91 5-10Comparisonofopen-circuitsensitivityexpressions. ................92 5-11Lumpedelementmodelofthepiezoelectricmicrophonea sanactuator. .....93 5-12Finiteelementmodelforvalidationexercise. ....................96 5-13AnalyticalandFEApredictionsof w inc (0)(pressureloadingcase). .......96 5-14RelativeerrorbetweenanalyticalandFEApredictions of w inc (0). ........97 5-15AnalyticalandFEApredictionsof w inc (0)(voltageloadingcase). ........97 5-16LumpedelementmodelandFEApredictionsoffrequencyr esponsefunction. .98 5-17Non-idealoperationalampliermodel. .......................100 5-18Lumpedelementmodelwithvoltageamplier. ...................100 5-19Non-idealchargeampliermodel. ..........................102 5-20Lumpedelementmodelwithchargeamplier. ..................103 5-21Noisemodelforthemicrophonewithvoltageampliercir cuitry. ........105 14

PAGE 15

5-22Output-referrednoiseroorforthemicrophonewithavo ltageamplier. ....107 5-23Noisemodelforthemicrophonewithchargeampliercirc uitry. .........108 5-24Output-referrednoiseroorforthemicrophonewithcha rgeamplier. ......109 6-1Cross-sectionofthepiezoelectricmicrophonewithnot abledimensions. .....114 6-2Paretofrontexample. .................................117 6-3Optimizationapproach. ...............................120 6-4ParetofrontassociatedwithminimizationofMDPandmax imizationofPMAX. 122 6-5Normalizeddesignvariablevaluesforeachoptimization .............123 6-6SensitivityofMDPto 10%perturbationsinthedesignvariables. .......125 6-7SensitivityofPMAXto 10%perturbationsinthedesignvariables. ......126 6-8SensitivityofMDPtoin-planestressvariations. ..................126 7-1WaferofpiezoelectricmicrophonesfabricatedatAvagoT echnologies. ......130 7-2Dicingbladeandsampleorientation. ........................131 7-3DicingprocessforMEMSpiezoelectricmicrophonedie. ..............133 7-4Micrographsofmicrophonedie(designsA-G). ...................135 7-5Explodedviewofthelaboratorytestpackage. ..................136 7-6Microphoneendcap. .................................137 7-7CloseupphotographofapackagedMEMSpiezoelectricmic rophone. ......138 7-8Voltageampliercircuitryincludedinthemicrophonep ackage. .........139 7-9Voltageampliercircuitboardlayout. .......................139 7-10Chargeampliercircuitdiagram. ..........................140 7-11CompletepackagedMEMSpiezoelectricmicrophone. ...............140 8-1Experimentalsetupfordieselection. ........................143 8-2Predictedfrequencyresponsemagnitudeinairandheliu m. ............147 8-3Planewavetubesetupforacousticcharacterization. ...............148 8-4Microphoneswitchingprocedure. ..........................149 8-5Innitetubemeasurementsetup. ..........................151 15

PAGE 16

8-6LinearitymeasurementsetupatBoeingCorporation. ...............153 8-7TripleFaradaycagesetupfornoiseroorcharacterizati on. ............155 8-8Noiseroormeasurementsspans,frequencyresolution,an daverages. ......156 8-9Impedancemeasurementsetupusingaprobestation. ...............157 8-10Pressurecouplerassembly. ..............................162 8-11Closeupdepictionofamicrophonedieinthepressureco uplersetup. ......163 8-12Experimentalsetupforextractionofacousticmassand compliance. ......164 8-13LaservibrometerscangridoverlayedondesignEmicrog raph. ..........164 8-14Experimentalsetupforpressurecouplercalibration. ................165 8-15Experimentalsetupformicrophonecalibrationinthep ressurecoupler. .....166 8-16Experimentalsetupforextractionofeectivepiezoel ectriccoecient. .....167 8-17Mapsofdicedsectionofwafer116(alldesigns). ..................168 8-18Mapsofdicedsectionofwafer138(alldesigns). ..................169 8-19Resonantfrequencymapsforwafer116. ......................169 8-20Centerdisplacementsensitivitymapsforwafer116. ................170 8-21Resonantfrequencymapsforwafer138. ......................171 8-22Centerdisplacementsensitivitymapsforwafer138. ................171 8-23Changesinresonantfrequencyanddisplacementsensit ivityduetopackaging. .173 8-24Staticderectionprolesofseveralmicrophonediaphr agms. ............174 8-25Staticderectiondierencesforpre-andpost-package dmicrophones. ......174 8-26Microphonefrequencyresponsesinhelium. ....................175 8-27Piezoelectricmicrophonefrequencyresponsefunctio nsatlowfrequencies. ....177 8-28Linearitymeasurements. ...............................178 8-29Linearitymeasurementsshowingunusualnonlinearbeh avior. ..........178 8-30THDmeasurements. .................................179 8-31Output-referrednoiseroors. ............................181 8-32Minimumdetectablepressurespectra. .......................182 16

PAGE 17

8-33Noiseroorspectrafor116-1-J7-A. .........................182 8-34AdmittancemeasurementsandtsformicrophoneB5-E. .............184 8-35Frequencyresponsefunctionofmicrophone116-1-J7-A. ..............186 8-36Parasiticcapacitanceextractionformicrophone1161-J7-A. ...........186 8-37Comparisonofpressureattestandreferencelocations inpressurecoupler. ...188 8-38Frequencyresponseofpiezoelectricmicrophonesinpr essurecoupler. ......189 8-39Displacementperpressureplots. ...........................190 8-40Displacementpervoltageplots. ...........................192 8-41Comparisonofmeasuredandtheoreticaltrendsforextr actedparameters. ....194 8-42Correctedfrequencyresponsemagnitudeofmicrophone sinpressurecoupler. ..195 9-1AMEMSpiezoelectricmicrophonedieonaplayingcard. .............196 A-1Laminatedcompositeplaterepresentationofthethin-l mdiaphragm. .....204 A-2Layercoordinatesforanarbitrarycompositelayup. ................216 B-1Finiteelementmodelforinvestigationofboundarycomp liancy. .........235 B-2DerectionprolesfromFEAwithclampedandcompliantbo undaryconditions. 236 B-3FEAresultsformodelswithclampedandcompliantbounda ryconditions. ...236 C-1Noisespectra95%condenceintervals. ......................239 C-2MDPspectra95%condenceintervals. ......................239 17

PAGE 18

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DEVELOPMENTOFAMEMSPIEZOELECTRICMICROPHONEFOR AEROACOUSTICAPPLICATIONS By MatthewD.Williams May2011 Chair:MarkSheplakMajor:MechanicalEngineering Passengerexpectationsforaquietrightexperiencecouple dwithconcernabout long-termnoiseexposureofrightcrewsdriveaircraftmanu facturerstoreducecabinnoise inright.Duringtheaircraftcomponentdesignorredesignp rocess,aeroacousticiansuse advancedexperimentaltechniquestohelpguidethesenoise -reductioneorts.Chief amongtheiravailabletoolsarearrays,distributedcollec tionsofmicrophonesthat spatiallysamplepressureructuations.Dierentarraycon gurationsaredeployedin righttestsontheexteriorofaircraft,enablingcharacter izationoftheturbulentboundary layer,identicationofnoisesources,and/orassessmento ftheeectivenessofcandidate noise-reductiontechnologies. Therequirementsformicrophonesusedinaircraftfuselage arraysaredemanding. Theyshouldbesmall,thin,andpassive;respondlinearlyto alargemaximumpressure; possessaudiobandwidthandmoderatenoiseroor;berobustt omoistureandfreezing; andexhibitstabilitytolargevariationsintemperaturean dhumidity.Microelectromechanical systems(MEMS)microphonesshowpromiseformeetingthestr ingentperformanceneeds forthisapplicationatreducedcost,madepossibleusingba tchfabricationtechnology. Thisresearchstudyrepresentstherststageinthedevelop mentofamicrophonethat meetstheseneeds. Thedevelopedmicrophoneutilizedpiezoelectrictransduc tionviaanintegrated aluminumnitridelayerinathin-lmcompositediaphragm.A theoreticallumpedelement 18

PAGE 19

modelandassociatednoisemodelofthecompletemicrophone systemwasdeveloped andutilizedinaformaldesign-optimizationprocess.Seve noptimalmicrophonedesigns with515-910 mdiaphragmdiametersand500 m-thicksubstratewerefabricatedusing avariantofthelmbulkacousticresonator(FBAR)processat AvagoTechnologies. Laboratorytestpackagingwasdevelopedtoenablethorough acousticandelectrical characterizationofninemicrophones.Measuredperforman cewasinlinewithsponsor specications,includingsensitivitiesintherangeof3040 V = Pa,minimumdetectable pressuresintherangeof75-80dB(A),70Hztogreaterthan20kHz bandwidths,and maximumpressuresupto172dB.Withthisperformanceinaddi tiontotheirsmallsize, thesemicrophoneswereshowntobeaviableenablingtechnol ogyforthekindoflow-cost, highresolutionfuselagearraymeasurementsthataircraft designerscovet. 19

PAGE 20

CHAPTER1 INTRODUCTION Microphonesareamongthemostfundamentalofphysicaltool sintheaeroacoustician's toolboxforlocating,understanding,andultimatelyreduc ingnoisesourcesinaircraft.The expenseofmeasurement-gradeaeroacousticmicrophonessu itableforhighpressure levelmeasurementsplacesrestrictionsoneventhemostric hlyfundedaeroacoustician's experimentalplans.Sizealsoremainsanissueinsomeappli cations.Optionsareneeded, andanewclassofhigh-performance,reduced-sizemicropho nesmanufacturedusing low-costbatchfabricationtechnologymaybetheanswer.Th egoalofthisresearchis developmentanddemonstrationofjustsuchamicrophone. Thischapteropenswiththemotivationforthedevelopmento famicroelectromechanical systems(MEMS)-basedaeroacousticmicrophone.Researcho bjectivesandcontributions arethengiven,followedbyanoutlinefortheremainderofth isstudy. 1.1Motivation Withtheworldwideairlinereetestimatedtodoubleinthene xt15years[ 1 ],aircraft manufacturersincreasinglyfaceregulatoryandmarketdri venpressurestoreduceaircraft noise.Prolongedexposuretoaircraftnoise|arecognizedf ormofpollution|inareas surroundingairportsisknowntohaveadverseeectsonanim albehaviorandcanlead toincreaseinbloodpressure,stress,andfatigueinhumans [ 2 ].IntheUnitedStates,the FederalAviationAdministration(FAA)dictatesnoisestandard sthataircraftmustmeet inordertoreceiveairworthinesscerticationintermsofe ectiveperceivednoiselevel (EPNL).TheEPNLofanaircraftisameasureofthesubjectiveim pactofitsnoiseon humans,takingintoaccountthesoundlevel,frequencycont ent,andduration[ 3 ].Noise standardsalsocontinuetogrowmorestringentabroad[ 1 ]. Passengerexpectationsforaquietrightexperience[ 4 ]coupledwithconcernabout long-termnoiseexposureofrightcrews[ 5 ]alsodriveaircraftmanufacturerstoreduce cabinnoiseinright.Cabinnoisehastraditionallybeenlim itedusinginsulatingpanels 20

PAGE 21

andskindampersonthefuselage.Practicalrestrictionson thesizeandweightofthese thinpanelslimittheireectivenessinreducinglow-frequ ency(long-wavelength)noise[ 4 ]. Treatingthenoiseatitssourceisapromisingmethodforred uctionoflow-frequencynoise withweightsavingscomparedtoinsulatingpanels. Duringtheaircraftcomponentdesignprocess,aeroacousti ciansuseadvanced experimentaltechniquestohelpguidenoise-reductioneo rts.Chiefamongtheiravailable toolsaremicrophonearrays,distributedcollectionsofmi crophonesthatspatiallysample pressureructuations.Dierentarrayswithdierentpurpo sesaredeployed:dynamic pressurearrayscapturehydrodynamicpressureructuation sassociatedwithaturbulent boundarylayer(inadditiontoanyincidentacousticructua tions),whileaeroacoustic phasedarraysareusedtoresolvenoisesources. In2005{2006theQuietTechnologyDemonstrator2(QTD2)pro grambrought togetheraconsortiumofaerospaceindustryleadersforase riesofteststoevaluate noise-reductiontechnologies.Agoalofthetestswastodet erminetheeectivenessof variousengineinletandexhaustcongurationsatreducing noisetransmittedtothecabin orradiatedtothecommunitybelow.Onenoisesourcethatrec eivedparticularattention wasshockcellnoise 1 ,\amajorcomponentofaftinteriorcabinnoise"atcruiseco nditions thatpropagatesaftoftheengine[ 6 ].Adynamicpressurearraydeployedinrighttestsis picturedinFigure 1-1 andwascomposedof84microphones.Thearrayenabledspectr al mappingofpressureructuationsassociatedwithboundaryl ayerandshockcellnoisealong thefuselage,comparisonoflevelsbeforeandafterenginet reatments,andidenticationof axialfuselagelocationssubjectedtothehighestshockcel lnoiselevels[ 7 ].Asimilararray wasdeployedforwardoftheenginesforcharacterizationof buzzsawnoise 2 [ 6 ]. 1 Shockcellnoiseis\generatedbytheinteractionbetweenth edownstream-propagatingturbulence structuresandthequasi-periodicshockcellsinthejetplu me"[ 6 ]. 2 Buzzsawnoiseis\multiple-pure-tonenoisegeneratedbyhi gh-speedturbofansunderconditionsof supersonicfantipspeeds"[ 8 ]. 21

PAGE 22

Microphonearray Figure1-1.Boeing777fuselageinstrumentedwithanarrayo fmicrophones.[Courtesy BoeingCorporation] Aeroacousticphasedarraysenableothersophisticatednois e-assessmentcapabilities viaanimportantfamilyofprocessingtechniquesknownasbe amformingalgorithms. Theseschemesallowaeroacousticianstoselectively\list en"toregionsinspace.Mapsof theacousticalpowerreachingthearrayfromaselectedspat ialregioncanbegenerated, andacousticiansusethisinformationtolocatenoisesourc esortojustifyexperimental/numerical studiesofspecicnoisegenerationmechanisms.Inadditio n,arraymeasurementsobtained fromdierenttestcongurationscanbecomparedtoassesst heeectivenessofnoise treatments. Figure 1-2A showslinearandellipticphasedarrayscomposedof132and1 81 microphones,respectively,deployedaspartofthestatice nginetestcomponentofthe 22

PAGE 23

= Lineararray @ @I Ellipticarray Engine A B Figure1-2.Aeroacousticphasedarraysdeployedaspartofth eQTD2program[ 9 ].A) Linearandellipticphasedarrayslocatedaftofanaircraft engine.B)Relative soundpowerlevelmapcreatedviabeamforming.[CourtesyBo eing Corporation] QTD2program[ 9 ].Staticenginetests,withtheirlowercostandcomplexity compared torighttests,enableamorecomprehensiveassessmentofno isereductiontechnologies viainclusionofmoreenginecongurationsandinstrumenta tion.Theellipticarrayin Figure 1-2A wasdesignedtoenablediscriminationbetweenfanandcores ourcesofengine noise,whilethelineararraywasusedprimarilytoidentify noisesourcesalongthejetaxis. Anexamplemapoftherelativesoundpressurelevelsassociat edwithaparticularengine conguration,foundviabeamformingwiththeellipticarra y,isshowninFigure 1-2B Arrayperformanceisafunctionofthenumberandarrangement ofmicrophonesthat compriseit,inadditiontotheindividualmicrophonechara cteristics.Adynamicpressure arrayforturbulentboundarylayermeasurementsmusthavea dequatelysmallsensors withhighbandwidthandclosespacinginordertoresolvethe smallestlengthandtime scalesofinterestintherow.Tworelevantrepresentativel engthscalesaretheKolmogorov lengthscaleandviscouslengthscale[ 10 ].TheratiooftheKolmogorovmicroscale tothe boundarylayerthickness ,forexample,scalesas[ 11 12 ] Re 3 = 4 ; (1{1) 23

PAGE 24

whereRe = u= istheeddyReynoldsnumberthatcharacterizestheturbulen tboundary layerand isthekinematicviscosity.Theeddyvelocity u andboundarylayerthickness serveasthevelocityandlengthscalesinRe ,respectively.Dynamicpressurearraydesign forturbulentboundarylayersthusbecomesmorechallengin gastheReynoldsnumber increases[ 13 ]. Phasedarraysusedforbeamformingalsohavestringentrequ irementsoftheir own.Developmentsinaperiodicphasedarraydesign[ 14 ]havehelpedtorelaxthe sensor-to-sensorspacingandchannel-countrequirements ,buttheneedforhigherchannel countsatlowercostremains.Thedynamicrangeofaphasedar ray,forexample,improves withthenumberofmicrophones[ 14 15 ].Inabookchapterhewroteonphasedarray measurementsinwindtunnels,JamesUnderbrinkofBoeingCor poration|aforemost expertinaeroacousticphasedarraytechnology|wrotethis ofhisexperiencesdesigning phasedarrays:\Indozensofphasedarraytests,nomatterho wmanymeasurement channelswereavailable,morewouldhavealwaysbeenbetter "[ 14 ].Achievinghigh channelcountsisparticularlychallengingforhighfreque ncyarrays,inwhichsmall aperturesareusedinordertoaccuratelycapturedirective sources.Small-aperturearrays withhighchannelcountsrequiresmallsensors. Limitationsexistinthedeploymentofhigh-channel-count arrays,includingthecost perchannel,datacollectionandstoragecapabilities,and compatibilitywithexisting testfacilities[ 15 ].Inaddition,microphonessuitableforuseinaeroacousti carray measurementsmustoftenmeetdemandingrequirements,incl udingsensingofhigh soundpressurelevels( > 160dB)withlowdistortion( < 3%)andhighsensitivitystability (hundredthsofadB).Dependingonthescaleofthetest,larg ebandwidths(upto90kHz for1/8scale[ 14 ])mayalsobenecessary.Measurement-gradesensorsthatme etthese criteriaareexpensive,oftencostingupwardsof$2k.Withu navoidableequipmentloss inaeroacoustictesting,wheremeasurementsmaybedoneinh ighpressurewindtunnels, 24

PAGE 25

outdoors,orinfull-scalerighttests,thelargeinitialin vestmentgiveswaytosignicant recurringcostsaswell. MEMSmicrophonesshowpromiseformeetingthestringentper formanceneeds ofaeroacousticapplicationsatreducedcost,madepossibl eusingbatchfabrication technology[ 16 { 21 ].Atreducedcostperchannel,higherdensityarrayswithbe tter performancebecomepossible.Inaddition,thereisanobvio usrelationshipbetween sensorcostandtheneedfortime-consumingprotectivemeas ures;madecheapenough ( < $50/channel),\disposable"sensorswouldeliminatedozen sofman-hoursfrommoderate sensor-count(50{100)testsorevenmorefromverylargeins tallations. Perhapsmostimportantly,thesmallsizeofMEMSmicrophone spositionthem asanenablingtechnologyformoreadvancedmeasurements,p articularlyinfull-scale righttestswheresensorsmustbeextremelythinandrobust. OnereasontheKulite microphonearrayontheBoeing777fuselageinFigure 1-1 aresparselydistributed |otherthancostconstraints|isbecausesensorlocationsm ustbecarefullychosen toavoidrowdisturbancescausedbyupstreamsensorsaecti ngthosedownstream. Withthesesensordensityrestrictions,deployedarraysha venotyetbeensucientfor beamforming[ 22 ].Thinnersensorsrequiringsmallerpackagingmaybemored ensely packed,enablingbothhigher-resolutionmapsoftheructua tingpressureeldonthe fuselageandeventually,beamformingofin-rightdatatoid entifydominantnoisesources foractual|notsimulated|rightconditions. 1.2ResearchObjectives Thegoalofthisresearchisthedesign,fabrication,andcha racterizationofaMEMS microphoneappropriateforuseinaeroacousticarrays.Amon gtheapplicationareasare ryoverarrays[ 23 { 25 ],staticenginetestarrays[ 9 26 27 ],andfuselagearrays[ 4 7 22 ], eachwithitsownsetofrequirements.Theprimaryapplicati onforthisworkisthe fuselagearray;staticenginetestarrays,withlessstring entspecicationsinmanyrespects, areviewedasasecondaryapplication. 25

PAGE 26

Thedemandingsetofrequirementsforfuselagearraymicrop honesmayonlybemet bycarefulengineeringdecisionsevenintheearlydesignst ages.Toovercomefuselage instrumentationchallengesalreadydiscussed,size|part icularlythickness|isextremely important;onlymicrofabricatedsensorsarecapableofach ievingthesmallsizesneeded. Themicrophonesmustberobust,particularlytomoisture.M icrophoneswithlow complexitythatfullyleverageexistingdataacquisitione quipmentarehighlydesirable forrighttestsatremotelocationsinvolvingthousandsofs ensors.Specically,lowpower consumption,characteristicofpassivesensorsinwhichon lyinterfaceelectronicsneedbe powered,enablestheuseofcompactdataacquisitionsystem swithintegratedstandard 4mAconstant-currentsources.Amongthetransductionmecha nismsavailablefor microfabricatedmicrophonesarecapacitive,piezoresist ive,optical,andpiezoelectric,but onlypiezoelectrictransductionoerstherightmixofrobu stness,simplicity,performance, andpassivity.AreviewofMEMSmicrophonesfromtheacademi cliteratureinChapter 3 showsthepromiseofpiezoelectricmicrophonesformeeting fuselagearrayapplication requirements. Theprojectsponsor,BoeingCorporation,specieddesignr equirementsforthe fuselagearrayapplicationthatarefoundinTable 1-1 .Theserequirementswerederived fromthesponsor'sdesiretomeetorexceedexistingmeasure mentcapabilities.The currentsensorinuse,theKuliteLQ-1-750-25SG,isacustom -packagedversionof thecommercially-availableKuliteLQlineofpressuretran sducers.Itsperformance characteristics,asprovidedbyBoeing,arecollectedaswe llinTable 1-1 3 Perhaps themostdicultcompetingspecicationsinTable 1-1 tobemetarethemaximum pressureof172dB(400timesthethresholdofpainforhumans )andminimumdetectable pressureof93dBoverallsoundpressurelevel(OASPL).There lationshipbetweenthese 3 Denitionsoftheimportantmicrophoneperformancemetric sarefoundinChapter 2 26

PAGE 27

specicationsandavarietyofotherdesigntrade-osaredi scussedatlengthinChapters 5 and 6 Table1-1.Fuselagearrayapplicationrequirements. MetricMEMSRequirementKuliteLQ-1-750-25SG Sensingelementsize 1 : 9mm864 864 m 2 Sensitivity500 V = Pa y 1 : 1 V = Pa Minimumdetectablepressure 48 : 5dB z 48 : 5dB z 93dBOASPL # 93dBOASPL # Maximumpressure 172dB172dB Bandwidth20Hz{20kHz x < 20Hz{20kHz+ Packagedthickness0 : 05in0 : 07in y Withon-boardgain z 1Hzbincenteredat1kHz # 20Hz{20kHz 3%distortion x 2dB Thescopeofthisstudyisthedesign,fabrication,andlabor atorycharacterization ofapiezoelectricMEMSmicrophonethatreachesthedesigns pecicationsofTable 1-1 Anumberofadditionalneeds,includingstabilityoverawid erangeoftemperatures ( 60 Fto150 F),robustnesstotheharshhigh-altitudeenvironmentandm oisture,and ultra-thinpackaging,falloutsideofthisscope.Theseite msrepresentfutureresearchand developmentwork. 1.3DissertationOverview ThischapterestablishedtheneedforanaeroacousticMEMSm icrophonesuitablefor aeroacousticarraymeasurements.Designgoalsweredened formicrophonedeployment infull-scaleright-testfuselagearrays.InChapter 2 microphonefundamentalsand performancemetricsaredened,theninChapter 3 previousworkintheareaofMEMS microphonesisreviewed.Thechoiceofpiezoelectricmater ial,fabricationprocess,and basicmicrophonegeometryareaddressedinChapter 4 .Asystem-levellumpedelement modelandanovelpiezoelectriccompositeplatemodelarede velopedinChapter 5 and thenusedfordesignoptimizationinChapter 6 .InChapter 7 ,thefabricationresults andpackagingprocessarediscussed.Chapter 8 presentscharacterizationandparameter 27

PAGE 28

extractionresultsfortherealizedpiezoelectricMEMSmic rophones,andChapter 9 concludeswithnalobservationsandsuggestionsforfutur ework. 28

PAGE 29

CHAPTER2 MICROPHONEFUNDAMENTALS Thischaptercoversthefundamentalsofmicrophones.First ,theconceptsofsound andpseudosoundareintroduced.Next,therealitiesofdesig ningamicrophonetosense sound,includinginherentlimitationsinphysicalsystems andcommoncharacteristics,are discussed.Thevariousperformancemetricsformicrophone sarethenaddressed.Atthe conclusionofthechapter,microphoneperformanceissumma rizedinaholisticwayin termsofsoundpressureandfrequency. 2.1SoundandPseudoSound Awave,asdenedbyBlackstock[ 28 ],is\adisturbanceordeviationfroma pre-existingcondition."Soundwaves,inparticular,area disturbanceinpressure.This pressuredisturbanceischaracterizedviathepressuredec omposition P = p 0 + p; (2{1) where P istheinstantaneousabsolutepressure, p 0 isthestaticpressure,and p isthe ructuatingpressure.Thisructuationisknownastheacoust icpressureandisreported inunitsofPascal(Pa).Soundwavespropagateaslongitudin alwavesviaamolecular collisionprocess,inwhichindividualparticlemotionocc ursinthesamedirectionaswave propagation[ 28 ]. Theeldofaeroacousticsisconcernedwiththegenerationo fsoundbymovingrows andthepropagationofsoundfromthem.Inthestudyofaerody namicallygenerated sound,isitimportanttodistinguishsound,whichpropagat esasawaveandisa compressibility-basedphenomenon,frompseudosound,whi chdecaysrapidlyaway fromitssourceandishydrodynamicinnature[ 29 30 ].Bothsoundandpseudosound arepresentintheturbulentboundarylayerassociatedwith rowoveranaircraftfuselage. Pseudosounddoesnotpropagateinairawayfromtheairplane butcantransmittothe 29

PAGE 30

interiorviainducedvibrationonthefuselageskin.Asaresu lt,pseudosounddoesnot contributetogroundlevelnoise,butdoesplayaroleincabi nnoise[ 30 ]. Becausesoundpressuresvaryoverawiderange,theyarequan tiedonalogarithmic scale.Soundpressurelevel(SPL)isdenedinunitsofdecib els(dB)as[ 28 31 ] SPL=20log 10 p rms p ref ; (2{2) where p rms isthermspressureleveland p ref isareferencepressure.Inair,itisstandard for p ref tobetakenas20 Pa,theapproximatethresholdofhearinginthe1{4kHzrange foryoungpersons[ 28 ].Typicalsoundpressurelevelsthereforevaryfrom0dB(at the thresholdofhearing)to120dB(atthethresholdofpain)[ 28 ].Soundpressurelevels associatedwith,forexample,aircraftenginescanexceedt histhresholdbyordersof magnitude. Giventhehumanear'snonlinearandfrequencydependentbeh avior,various psychoacousticmeasuresofsoundareusedtoquantifynoise levelsinahuman-oriented way.Frequency-weightingisoftenusedtoobtainsoundpres surelevelsthatmore accuratelyrerecthumanjudgementsofloudness.Threesuch schemesareknownasA-, B-,andC-weighting,witheachaccountingforfrequency-de pendenthearingcharacteristics inhumansatdierentsoundpressurelevels.A-weightingisa ppropriateforthelowest soundpressurelevelsanditsuseisthemostprevalent.Soun dpressurelevelsthat havebeenweightedaretraditionallydenotedindB(A),etc.As soundenergymaybe distributedoverabroadrangeoffrequencies,integrating soundpressureoverfrequency (usuallytherangeofhumanhearing)producesanotherusefu lmeasure,theoverallsound pressurelevel(OASPL).TheOASPLmayalsobeobtainedfromafr equency-weighted spectrum.Theeectiveperceivednoiselevel(EPNL),mentio nedbrieryinChapter 1 isanoverallsoundpressuremetricusedforaircraftcerti cationthataccountsfor frequency/tonalcontentandduration[ 32 33 ]. 30

PAGE 31

2.2TheRealitiesofMicrophoneDesign Atransducerisadevicethatusesaninputinoneenergydomai ntoproducea correspondingoutputinanotherenergydomain.Amicrophon eisaparticularkind oftransducerthatconvertsaninputacousticsignalintoan outputelectricalsignal. Toperformthisconversion,themicrophonepossessesamech anicalelement,usuallya diaphragm,thatdisplacesunderanincidentacousticpress urewave.Anelectromechanical transductionmechanismservestoeitherconvertthismecha nicalreactiontoanoutput electricalsignaloruseittomodulateanexistingelectric alsignal. Theidealmechanicalelementforthiselectromechanicalsy stemisalinear,massless spring,i.e.onethatobeystheconstitutiverelationship f a ( t )= kx ( t ) ; (2{3) where f a istheappliedforce(input)analogoustopressure, k isthespringstiness,and x isthedisplacement(output)analogoustoanelectricalsig nal.Becausethespringis perfectlylinear,thisrelationshipcontinuestoholdrega rdlessofthemagnitudeofthe input f a .Thefrequencyresponseofthisideal,masslessspringis[ 34 ] X ( f ) F a ( f ) = 1 k ; (2{4) where X and F a aretheFouriertransformsof x and f a .Regardlessoftheexcitation frequency f ,theinput F a andoutput X arerelatedbytheconstant1 =k (thegainfactor) andarealwaysperfectlyinphase(zerophasefactor).Thepe rfectspringthusresponds toaninputofanymagnitudeatanyfrequencywithperfectde lity.Theseresponse characteristicsarererectedinFigure 2-1 .Ifamasslessspringbyitselfcouldserveas amicrophone,itcoulddetectthequiestwhisperortheloude stexplosionatinfrasonic, sonic,orultrasonicfrequenciesandreproduceitperfectl y. Mechanicalsystemsintherealworldnecessarilypossessma ssaswellasdamping, soitshouldcomeasnosurprisethatthefrequencyresponseo fareal\spring"diers 31

PAGE 32

0 1 0 1 Displacement, xForce, f 0 1 0 1 =k X F a 0 1 0 Frequency, f \ X F a Figure2-1.Force-displacementcharacteristicsforaperf ectspring. markedlyfromtheidealspring.Thegoverningequationfora representativesingledegree offreedommass( m )-spring( k )-damper( b )systemis m x + b x + kx = f a ; (2{5) whereeach symbolizesdierentiationwithrespecttotime, d=dt .Equation 2{5 isthe classicalequationforasecond-ordersystem.Thefrequenc yresponsefunctionisthen X ( f ) F a ( f ) = 1 =k 1 f f n 2 + j 2 f f n ; (2{6) wherethenaturalfrequency f n =1 = 2 p k=m andthedampingratio = b= 2 m! n = b= 4 mf n [ 34 ].Thefrequencyresponsefunctionofthemass-spring-damp ersystemisnow afunctionoffrequencyasshowninFigure 2-2 forvariousvaluesofthedampingratio. Anunder-damped( < 1)second-ordersystemhasamaximumgainattheresonanceor dampednaturalfrequency, f r = f n p 1 2 2 .Ifthissystemaloneservedasamicrophone, thesignalcomponentswithfrequenciesnear f r wouldbeampliedconsiderablycompared tothoseatotherfrequenciesandtheoriginalsignalcouldn otberecoveredexactly withoutaccurateknowledgeoftheentirefrequencyrespons efunction.Figure 2-2 also showsthatunder-dampedsystemshaveexcellentphaserespo nseoverawidefrequency 32

PAGE 33

range,butasthedampingratioisincreased,signicantpha selagintheoutputresults. Whenworkingwithrealmechanicalsystemsthatbehavethisw ay,anengineermust decidewhatkindofgainandphaseerrorareacceptableandov erwhatfrequencyrange theyareachievable. 10 3 10 2 10 1 10 0 10 1 10 2 10 1 10 0 10 1 Norm.Mag., k X F a =0 : 001 =0 : 1 =1 10 3 10 2 10 1 10 0 10 1 100 0 NormalizedFrequency, f f nPhase, \ X F a [ ]Figure2-2.Frequencyresponseofasecond-ordersystem. Aperfectlylinearspring|evenonethataccountsformassan ddamping|also doesnotexist,asphysicalsystemsrespondlinearlyatbest overalimitedrangeofinputs. Theelasticlimitisawell-knownthresholdbeyondwhichman ymaterialstransitionfrom linearelastictononlinearplasticbehavior.However,inma nymechanicalsystems,the linear/nonlinearthresholdisactuallydictatedbytheons etofgeometricnonlinearity, whichoccurswhendisplacementsbecomesucientlylargeth attheirrelationshiptostrain isnolongerapproximatelylinear.TheDungspringisawell -knownsingle-degree-of-freedom representationofageometricallynonlinearmechanicalsy stem,anditisgovernedbythe 33

PAGE 34

equation m x + b x + k 1 x + k 3 x 3 = f a : (2{7) Forsucientlysmallvaluesoftheinput f a (correspondingtoasucientlysmalloutput x ),thenonlineartermdoesnotsignicantlycontribute.Nonl inearspring-hardening behavior( k 3 > 0)isshowninFigure 2-3 togetherwiththelinearizationabout x =0. Inputwaveform f a ( t )isincreasinglydistortedattheoutput x ( t )asitsamplitudeexceeds theapproximatelylinearregionofthesensitivitycurvein Figure 2-3 Consider,forexample,theinput-outputrelationshipexpr essedasaTaylorseriesover alimiteddomainas[ 35 ] x ( t )= b 1 f a ( t )+ b 3 f 3 a ( t ) ; (2{8) wherea f 2 a termisnotincludedsuchthat x isanoddfunctionof f a .Foraninput f a ( t )= a 1 sin( !t ),theoutputbecomes,aftermakinguseoftrigonometricide ntities, x ( t )= a 1 b 1 + 3 4 a 31 b 3 sin( !t ) 1 4 a 31 b 3 sin(3 !t ) : (2{9) Duetothenonlinearinput-outputrelationship,therespon se x containsasignal componentatfrequency3 despitethepresenceofonlyasignalcomponentatfrequency attheinput.Thisnonlinearphenomenonisincontrasttotha tofanidealized linearsystem,forwhichmagnitudeandphaseoftheinputsig nalaremodiedbutthe frequenciesoftheinputsignalarepreserved[ 34 ].Itisthusimportantforamicrophone designertoknowtherangeofinputsforwhichtheassumption oflinearityisvalid. Inordertopromoteapressuredierenceacrossthemechanic alsensingelementof amicrophoneinanacousticeld,acousticpropagationbetw eenthefrontandbackof thesensingelementmustbeimpeded.Ingeneral,thesensing element(forexample,a diaphragm)issuspendedoverabackcavity,withonesideexp osedtotheacousticeld andtheotherexposedtothecavity.Thecompositionoftheba ckcavitymustthenbe determined;obviouschoicesarethatitcanbesealedatvacu umorcontainaruid.Forthe 34

PAGE 35

0 0 S Actual Ideal Force, f aDisplacement, xFigure2-3.ConstitutivebehaviorforaDungspring. p atm p =0 A p = p ( 8 ;T ) p atm B P P Pq Vent p = p atm p atm C Figure2-4.Variouscavitycongurations.A)Vacuumsealed. B)Fluidisolated.C) Vented. latter,theruidcanbeisolatedfromorventedtothemeasure mentmedium.Eachofthese congurationsareshowninFigure 2-4 Thereareconsequencestoeachofthesechoices.Avacuum-se aledcavityasin Figure 2-4A enablesmeasurementofstaticpressurechanges,butasacon sequenceleaves thediaphragmalwayssubjectedtoatmosphericpressureloa ding.Acousticsignalsthen causethediaphragmtooscillateaboutastatically-derect edconguration.Inorderfor thisstaticderectiontonotexceedtheapproximatelylinea rregimeofoperation,the diaphragmmustbeverystiandthuslesssensitivetoacoust icperturbations,which eveninhighSPLaeroacousticapplicationsareoneormoreor dersofmagnitudesmaller thantheequivalent194dBatmosphericpressure.Alternativ ely,themicrophonecanbe 35

PAGE 36

operatedaboutthenonlinearly-derectedoperatingpoint, butsensitivitybecomeshighly dependentonatmosphericpressureanddynamicrangeislike lysacriced.Forallofthese reasons,thevacuum-sealedcavitycongurationofFigure 2-4A istypicallyonlyutilizedas anabsolutestaticpressuresensorandnotasamicrophone. Meanwhile,aruidmediuminsideacavityactsasanadditiona lspringandthus hasitsownimpactontheoveralldynamicsofthesystem[ 28 ].Theconguration ofFigure 2-4B |inwhichthereferencepressureisset|enablesmeasuremen tof dierentialstaticanddynamicpressureandistypicalofdy namicpressuresensors.One downsideisthatunintendedchangesinthereferencepressu reimpactthemeasurement. Forexample,atzeropressurethereissensitivitytotemper aturechangeinthecavityruid duetoexpansion,particularlyifthecavityissealed. Microphonesareusuallyvented|thecavityisconnectedtot heambientenvironment byathinchannelasinFigure 2-4C |toavoidtheeectsofstaticpressure.Thechannel allowsstaticpressureequilibrationbetweenthefrontand backofthediaphragm,butmore rapidpressurechangesassociatedwithacousticwavesstil lcausethediaphragmtovibrate [ 36 ].Asaresult,aventedmicrophoneislessresponsivetosound wavesbelowacertain designfrequency.Inaddition,sincethecavityisconnecte dtotheoperatingenvironment, itislledwiththeassociatedgas(usuallyair). Thus,microphonesgenerallysharethetraitsshowninthecr osssectionofFigure 2-4C : adiaphragm(thetypicalmechanicalsensingelement);acav ity,whichisolatesthefront andbackofthediaphragmandprovidesroomforittoderect;a ndavent,whichallows staticpressureequilibrationbetweenthefrontandbackof thediaphragm.Atransduction mechanism(notshown)isresponsibleforproducingelectri caloutput. 2.3MicrophonePerformanceMetrics InSection 2.2 ,therealitiesofmicrophonedesignwereaddressedfromthe perspective ofaclassicalsecond-ordersystem.Commonfeaturesofmicr ophonesandtheirrolesin 36

PAGE 37

determiningmicrophoneperformancewereestablished.Int hissection,thevariousmetrics usedtocharacterizetheperformanceofamicrophonearedis cussedinturn. 2.3.1FrequencyResponseandSensitivity Thetypicalfrequencyresponseofanunder-dampedaeroacou sticmicrophoneisshown inFigure 2-5 .Theregionofthefrequencyresponsethatisapproximately constantis knownastheratbandanditscorrespondingmagnitudevaluei scalledthesensitivity, S .ThesensitivityhasunitsofV = Pa(oroftendBre1V = Pa)andrelatesoutputvoltage toinputpressureforfrequenciesintheratband.Microphon emanufacturersquotethe sensitivityonspecicationsheetsataparticularrat-ban dfrequency;forDanishcompany BruelandKjr,aprominentsupplierofmeasurementqualit ymicrophones,thisisusually 250 Hz [ 31 ].Thetotalfrequencyrangeoverwhichthefrequencyrespon seisequaltothis sensitivitytowithinsometolerance,usually 3dB(orsometimes 2dB),isknownas thebandwidth[ 31 ].Thelowerendofthebandwidthat f 3 dB isthe cut-onfrequency while f +3 dB isthe cut-ofrequency .Theventstructure,transductionmechanism,and/or interfaceelectronicsdictatethelowfrequencyresponseo fthemicrophone,andthusthe cut-onfrequency.Theresonancebehaviorofthediaphragm( ortheroll-oforoverdamped microphones)dictatesthecut-ofrequency.Althoughonlyt herst(orfundamental) resonanceisshowninFigure 2-5 ,microphonesinrealityexhibitaninnitenumberof additionalresonancesbecausetheyarecontinuoussystemw ithinnitedegreesoffreedom [ 37 ]. AlsoillustratedinFigure 2-5 ,thephaseofanidealmicrophoneintheratband iszero,meaningthereisnolagbetweeninputandoutput.Inc ommercialcondenser microphones,thedampingisoftentunedtoreducetheresona ntpeaktowithinthe 3dB limitsoreliminateitentirely,whichextendsthebandwidt hbutcausesearlyphaseroll-o asdiscussedinSection 2.2 (Figure 2-2 )[ 31 ]. Itwouldseemthatachievingahighmicrophonesensitivityi saprimarydesigngoal. Increasingthesensitivity,afterall,ensuresahigher(an dpresumablyeasiertomeasure) 37

PAGE 38

10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 20 10 0 10 20 3dB +3dB f 3dB f +3dB Bandwidth Frequency[Hz]NormalizedMagnitude[dB] 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 180 90 0 90 180 Frequency[Hz]Phase[ ]Figure2-5.Typicalaeroacousticmicrophonefrequencyres ponse(magnitudenormalized byrat-bandsensitivityandphase). outputsignalforthesameinputsignal.However,amplicati onoftheoutputsignalcan achievemuchthesameeect.Inthenextsection,itwillbesh ownthatwhileahigh sensitivityisbenecial,itisnotofprimaryimportance.2.3.2NoiseFloorandMinimumDetectablePressure Noise,inageneralsense,istheoutputsignalofadeviceinth eabsenceofan intendedinput.Noisemaybeclassiedasintrinsicnoise,at rulyrandomoutputin theabsenceofinput,andextrinsicnoise,whichisduetopic kupofunwantedexternal signals.Inamicrophone,aninputpressurethatyieldsanou tputvoltagelowerthan thenoiseofthemicrophone(thenoiseroor)cannotbeeasily detected;amicrophone's minimumdetectablepressureisthereforedenedasthepres surethatproducesanoutput signalequivalenttothenoiseroor. 38

PAGE 39

Themostcommonintrinsicnoisesourceisthermalnoise,whi chispresentinelectrical andmechanical/acousticsystemsinthermodynamicequilib rium.Intheelectricaldomain, thisformofnoiseiscalledJohnsonorNyquistnoiseandisdue torandomthermal motionofchargecarriers[ 38 39 ];themechanical/acousticanalogisBrownianmotion, therandomthermalmotionofparticles[ 40 ].Theructuation-dissipationtheorem[ 41 ] establishestherelationshipbetweenthermalnoiseanddis sipationinasystem.Gabrielson summarizestheructuation-dissipationtheoremthusly[ 42 ]:\Ifthereisapathbywhich energycanleaveasystem,thenthereisalsoaroutebywhichm olecular-thermalmotion inthesurroundingscanintroduceructuationsintothatsys tem."Asaresult,anysource ofdissipationisalsoasourceofnoise[ 42 ].Thermalnoisehasuniformpoweratall frequencies 1 andisconvenientlydenedintermsofpowerspectraldensit y(PSD)as [ 39 43 ] S n =4 k B TR; (2{10) where k B istheBoltzmannconstant, T isthetemperature,and R isthedissipationor damping.Foranelectricalsystem, R isinunitsofnandthus S n isinunitsofV 2 = Hz; theuseofEquation 2{10 inotherenergydomainsisdiscussedfurtherinSection 5.3.3 Anequivalentnoisemodelforaresistorconsistentwithther uctuation-dissipation theoremisshowninFigure 2-6 .Here,a\noisy"resistorhasbeenreplacedwithaperfect noiselessresistorinserieswithanoisesource v n withspectraldensityfunctiondenedin Equation 2{10 Equation 2{10 impliesthatthermalnoisealwaysincreaseswithdissipati on;thisis onlypartiallytrue.Inreality,theplacementofthedissip ativeelementinthecircuitplays arole.Takingaresistorinparallelwithacapacitorasanex ampleandmeasuringoutput 1 Inreality,thermalnoisehasuniformnoisepoweratfrequen ciesforwhich hf=k B T 1,where h isPlanck'sconstant.Thisconditionholdstoapproximatel ythemicrowave band[ 39 ]. 39

PAGE 40

v n R + v o Figure2-6.Noisemodelforaresistor. v n R C + v o Figure2-7.Noisemodelforaresistorinparallelwithacapac itor. noisevoltageacrossthecapacitor,asinFigure 2-7 ,alowpasslterisformed.Asaresult, theshuntcapacitanceactuallyservestoattenuatethenois eathighfrequencies.As R increases,theltercutofrequency( f c =1 = 2 RC )iscorrespondinglyreducedandnoise powerisshiftedtolowerandlowerfrequencies,asillustra tedinFigure 2-8 .Thisformof thermalnoiseissometimescalled k B T=C noisebecausewhentheoutputnoisePSDis integratedoveraninnitebandwidth,thesquaredrmsoutpu tnoisevoltageisequalto k B T=C [ 39 ].Theconceptof k B T=C noiseisshowntobeimportantinthecontextofa piezoelectricmicrophoneinChapter 5 Non-equilibriumnoisesourcesalsoexistinsolidstatedevi ceswhendirectcurrent ispresent(forexample,inoperationalampliers).Onesuc hnoisesource,rickernoise, hasaninversefrequencydependenceandisoftencalled1 =f noise .Itisdominantat lowfrequencies,butatasucientlyhighfrequency,called the cornerfrequency ,thermal noise[ 43 ]becomesdominant.Inthecontextofmicrophones,forexamp le,1 =f noiseis presentinpiezoresistivemicrophones[ 38 ]andiscommonininterfaceelectronicsused inmicrophones.Figure 2-9 showsthetransitionfrom1 =f noisetothermalnoiseforthe voltagenoiseoftheLTC6240amplier[ 44 ]utilizedinthisstudy(seeChapter 7 ). 40

PAGE 41

10 3 10 2 10 1 10 0 10 1 10 1 10 0 10 1 10 2 10 3 R = R 0 R =10 R 0 R =100 R 0 f c 0 = 1 2 R 0 C NormalizedFrequency, f=f c 0OutputPSD/4 k B TR 0Figure2-8.Low-passlteringofthermalnoise. 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 17 10 16 10 15 10 14 10 13 CornerFrequency 1 =f Noise ThermalNoise Frequency[Hz]NoisePSD[V 2 = Hz]Figure2-9.VoltagenoisespectrumforanLTC6240amplier[ 44 ]. Extrinsicnoiseisaltogetherdierent,inthatitoriginat esexternaltothesensor andistypicallydeterministicinnature[ 45 ].Avoidanceofpickupofomnipresent electromagneticsignalsradiatedfromeverydayelectroni cs(at50Hzto60Hzand harmonics)isimportantforanaudiosensorandcanbeaparti cularchallengeforsensors withhighelectricalimpedance[ 46 ].Ingeneral,theimpactofextrinsicnoisecanbe mitigatedatthepackage-levelusingcarefulcircuitlayou tandshieldingtechniques[ 43 ], thoughshieldingofmicroscalesensorsbecomesmoredicul tatlowfrequencieswhen theskindepthofelectromagneticradiationbecomeslargea ndthickerconductiveshields becomenecessary[ 45 ]. 41

PAGE 42

Theminimumdetectablepressureistheinput-referrednois eofamicrophone integratedoverabandwidthofinterest, p min = s Z f 2 f 1 S v o ( f ) j S j 2 df; (2{11) where S v o ( f )istheoutput-referrednoisePSD[V 2 = Hz]and S isthemicrophonefrequency responsefunction.Minimumdetectablepressureisoftenre portedasaSPL,i.e. MDP=20log 10 p min p ref : (2{12) Equation 2{11 clarieswhysensitivityaloneisnottheprimarydesignmet ricofinterest. Althoughhighsensitivitynaturallyleadstoalowminimumde tectablepressure,thenoise characteristicsofthemicrophoneanditsassociatedelect ronicsalsoplayanimportant role. Severalvariationsoftheminimumdetectablepressuremetr icexistwithdierent physicalandpsychoacousticfocuses.Integrationoverana rrowbandwidthinEquation 2{11 yields\narrowbandMDP";foranaeroacousticmicrophone,t heintegrationiscommonly overa1Hzbincenteredat1kHz.Thisnarrowbanddenitionprov idesinformationatan importantfrequencytowhichhumansoundsensitivityishig h[ 28 ]andiseasytocompare andcompute.However,itsayslittleabouttheoverallmicrop honenoisecharacteristics. IntegrationoverthebandwidthofthedeviceinEquation 2{11 (e.g.theaudioband), meanwhile,givestheminimumdetectablebroadbandrmspres surelevel.Inthiscase, MDPisreportedinunitsofdBOASPL(overallsoundpressurele vel).Finally,itisalso commonforthenoisespectrumtobeA-weightedinordertomimi ctheoverallhuman soundperception;MDPisthengiveninunitsofdB(A).2.3.3LinearityandMaximumPressure ItwasestablishedinSection 2.2 thataperfectlylinearmechanicalsensingelement doesnotexist.Asaresult,theactualresponseofamicrophon ecanonlybeapproximated aslinearforsucientlysmallpressureinputs.Whenthepre ssurebecomes\large," 42

PAGE 43

0 0 S Actual Ideal Pressure, p [Pa]Voltage, v [V]Figure2-10.Idealandactualresponseofamicrophone.higher-ordereects,oftengeometricnonlinearityofthed iaphragmortransduction nonlinearities,becomeimportant.Thetypicalcharacteri sticsofanactualmicrophone responsearecomparedtotheideallinearresponseinFigure 2-10 .Thelocalslopesofthe linescorrespondtotheidealandactualmicrophonesensiti vity. Waveformdistortionisalwayspresentinreal,nonlinearsy stems.Asdiscussedin Section 2.2 ,aninputwaveform A sin( !t )doesnotemergefromanonlinearsystem purelyasanoutput B sin( !t + );theoutputsignalalsocontainsfrequenciesatinteger multiplesofthefundamentalfrequency,calledharmonics. Intypicalnomenclature,a signalcomponentwithfrequency n! isreferredtoasthe n thharmonic.Theassumption oflinearityimpliesthatthepowerdistributedtothesecon dandhigherharmonicsis negligiblysmallwithrespecttotherst. Toquantifytheextentofnonlinearityintheresponseofami crophoneforaparticular inputpressurelevel,thetotalharmonicdistortionmetric isused.Manyvariantsonthis metricexist[ 35 47 48 ]andthusgreatcaremustbetakenwhenitisusedtocompare dierentmicrophones.Thedenitionoftotalharmonicdist ortionusedhereis[ 47 49 ], THD= vuuut 1 P n =2 v o 2 ( f n ) v o 2 ( f 1 ) 100% ; (2{13) 43

PAGE 44

whichrepresentstheratioofthermsoutputvoltageinallhi gherharmonics( f n n = 2 ::: 1 )tothatintherstforasingletoneinputpressuresignalat f 1 .Themaximum pressure p max foramicrophoneisthepressureatwhichtheTHDreachesapres cribed value(often3-10%).Themaximumpressuremaybereportedin unitsofPaordBwith thenomenclaturePMAXusedforthelattercase.2.3.4DynamicRange Together,MDPandPMAXdenetheoperatingpressurerangefor amicrophone, calledthe dynamicrange .ItisdenedinunitsofdBas DR=20log 10 p max p min =PMAX MDP : (2{14) BecausethereareseveralvariationsonthedenitionofMDP ,thedynamicrangemetric isoftenwrittenasarangeoftwonumbers(e.g.MDP{PMAX)rathe rthanindB.When Equation 2{14 isused,clarifyinglanguageisoftenincluded. 2.3.5SummaryofMicrophonePerformanceMetrics Microphoneperformancecanbecondensedintotheconceptof anoperational\space" inthefrequencyandpressuredomains,picturedinFigure 2-11 .Theboundsofthis \space,"arerelatedtoeachoftheperformancemetricsdisc ussedinSection 2.3.1 {Section 2.3.3 Notethatalthoughthe\space"isshowninFigure 2-11 asrectangleforsimplicity,both MDPandPMAXareingeneralfrequencydependent. Toprovidecontextforeachofthepresentedperformancemet rics,theproperties ofwell-knownaeroacousticmicrophonesfromBruelandKj r(B&K)andKulite arecollectedinTable 2-1 .Allofthesemicrophonesarehigh-frequencyinstruments appropriateformodel-scalemeasurements.Sensitivities oftheBruelandKjr4138and 4938pressure-eldmicrophones(1/8"and1/4"diameters,r espectively)areontheorder of1mV = Pa,whilethesmallerKulitemicrophone(.093")hasalowers ensitivityonthe orderof1 V = Pa.Asaresult,theKulitemicrophonealsohasasignicantly highernoise 44

PAGE 45

SoundPressure SoundPressureFrequency Frequency Operational\Space" ofMicrophone p max p min VoltageSensitivityDynamicRangeBandwidth f 3dB f +3dB Figure2-11.Operationalspaceofanaeroacousticmicropho ne. Table2-1.Performancecharacteristicsofcommonaeroacou sticmicrophones. MetricB&K4138[ 50 ]B&K4938[ 51 ]KuliteMIC-093[ 52 ] Sensitivity[mV = Pa]11.6.004 Bandwidth6 : 5Hz{140kHz y 4Hz{70kHz y < 125kHz # MDP[dB]5230 z 100 z PMAX[dB]168 172 194 DynamicRange[dB]11614294 y 2dB # Resonantfrequency z A-weighted 3%distortion roor(100dB(A))comparedtotheBruelandKjrmicrophones. Everymicrophonein Table 2-1 possessesPMAX > 160dB. 2.4Summary Thefundamentalsofmicrophones,includingthephysicalst ructureandperformance metrics,wereaddressedinthischapter.Knowledgeofthese topicssetsthestagefora reviewofthestate-of-the-artofMEMSmicrophonesinChapt er 3 andmicrophonedesign inChapters 5 and 6 45

PAGE 46

CHAPTER3 PRIORART Inthischapter,areviewofrealizedmicroelectromechanic alsystems(MEMS) microphonesprovidescontextforthedevelopmenteortsof thisstudy.Theliterature onMEMSmicrophonesisextensive,withmosteortsfocusedo nmicrophonesfor consumeraudioapplications.Therequirementsassociated withaudiomicrophones diersignicantlyfromthoseofanaeroacousticmeasureme ntmicrophone.Intheformer applicationarea,theminimumdetectablepressurerequire mentsareparticularlystringent (usually < 30dB(A)),whilerequirementsforbandwidth(10{15kHz)andma ximum pressure(typically < 120dB)arelessimportant.Maximumpressureandbandwidth requirementsformicrophonestargetedataeroacousticmea surementsvarywiththe specicmeasurement,sometimesreachingorexceeding160d Band100kHz,respectively. Thenoiseroor,meanwhile,islesscriticalthanforaudiomi crophones. ThereviewinthischapterisrestrictedtoMEMSmicrophones utilizingpiezoelectric transductionandMEMSmicrophonestargetedataeroacousti capplications.MEMS microphonesoftheseclassicationsformaportraitofthes tate-of-the-artfromwhichthe piezoelectricmicrophonedevelopedinthisstudyemerges. AgeneralreviewofMEMS microphoneswaswrittenbyScheeper(1994)[ 53 ]andmorerecentbutunpublishedreviews werecompletedbyMartin(2007)[ 21 ]andHomeijer(2008)[ 54 ]. 3.1ReviewofMEMSPiezoelectricandAeroacousticMicropho nes Therstmicrofabricatedpiezoelectricmicrophone,depic tedinFigure 3-1 ,was developedbyRoyeretal.(1983)[ 55 ].Itwascomposedofasputteredzincoxide(ZnO) layeratopathincircularsilicondiaphragm.Somedevicesf eaturedanintegratedon-chip MOSbueramplier,thoughthehighestsensitivityof250 V = Pawasreportedfora non-integrateddevice. In1987,Kimetal.[ 56 ]oftheBerkeleyIntegratedSensorCenterpresentedthe secondpiezoelectricMEMSmicrophonefabricatedusingZnO thinlm,thistimeon 46

PAGE 47

A B Figure3-1.Piezoelectric(ZnO)microphonewithintegrate dbueramplier[ 55 ].A) Structure.B)Layercomposition.[ReprintedfromSensorsa ndActuators,vol 4,Royeretal.,ZnOonSiIntegratedAcousticSensor,pgs.357 {362,Copyright 1983,withpermissionfromElsevier.] asiliconnitridediaphragm.Siliconnitridewascitedasha vingmoreeasilycontrolled stressandthicknessthansilicon.The3mm 3mm 2 msquarediaphragmfeatured multipleconcentricsegmentedaluminumtopelectrodes,as showninFigure 3-2A andpolysiliconbottomelectrodes.Theobtainedfrequency responsewasnotrat;the sensitivitywas50 V = Patowithin9dBfrom4kHzto20kHz.Apatentwasissuedin 1988[ 57 ].Later,throughapartnershipwithOrbitSemiconductor,K imetal.(1989) [ 58 ]wereabletointegratethesamebasicmicrophonedesignwit hacomplementary metal-oxide-semiconductor(CMOS)amplieron-chip.In19 91[ 59 ],anumberof improvementsweremadetothemicrophonedesignthatresult edinafactorof5 improvementinsensitivity,thougharatfrequencyrespons ewasstillnotobtained. Thecross-sectionofthemicrophonewithintegratedampli erfromthatworkisfoundin Figure 3-2B A1988GermanlanguagedissertationbyFranz[ 60 ],ofDarmstadtUniversityof Technology,featuredapiezoelectricmicrophonedesignut ilizingaluminumnitride(AlN). Thisworkwasnotpublished,butaccordingtoSchellinetal. ,alsofromDarmstadt University,themicrophonehadasensitivityof25 V = Pa[ 61 ].Thoseauthorsintroduced apiezoelectricmicrophoneoftheirownin1992(showninFig ure 3-3 ),whichusedthe organiclmaromaticpolyureaasthepiezoelectric.Amaxim umsensitivityof126 V = Pa 47

PAGE 48

A B Figure3-2.Piezoelectric(ZnO)microphoneutilizingmult ipleconcentricelectrodes [ 56 58 59 ].A)Multipleconcentricelectrodeconguration[ 56 ].[ c r 1987 IEEE.Reprinted,withpermission,fromKimetal.,IC-Proce ssedPiezoelectric Microphone,IEEEElectronDeviceLetters,Oct.1987.]B)Cr oss-sectional view,includingintegratedamplier[ 59 ].[ c r 1991IEEE.Reprinted,with permission,fromKimetal.,ImprovedIC-CompatiblePiezoe lectric MicrophoneandCMOSProcess,Proceedingsof1991Internati onalConference onSolid-StateSensorsandActuators,Jun.1991.] Figure3-3.Piezoelectricmicrophoneutilizingaromaticp olyurea[ 61 ].[ c r 1992IEEE. Reprinted,withpermission,fromSchellinetal.,SiliconS ubminiature MicrophoneswithOrganicPiezoelectricLayers:Fabricati onandAcoustical Behaviour,IEEETransactionsonElectricalInsulation,Aug .1992.] wasachieved(thoughthetypicalresponsewas4 V = Pato30 V = Pa).Themicrophone exhibitedanon-ratfrequencyresponseduetoadampedmecha nicalresonanceinthe audioband.Asecondincarnationofthemicrophonein1994[ 62 ]featuredanotherorganic lm,P(VDF/TrFE),asthepiezoelectric.Animprovedsensitiv ityof150 V = Pawas achievedbutthefrequencyresponsewasstillnotratinthea udioband. 48

PAGE 49

Figure3-4.Piezoelectric(ZnO)microphonewithcantileve rsensingelement[ 64 ].[ c r 1996 IEEE.Reprinted,withpermission,fromLeeetal.,Piezoele ctricCantilever MicrophoneandMicrospeaker,JournalofMicroelectromech anicalSystems, Dec.1996.] In1993,Riedetal.oftheBerkeleySensorandActuatorCenter extendedthework ofKim[ 56 { 59 ].Thenewiteration[ 63 ]madeuseofa2 : 5mm 2 : 5mm 3 : 5 msilicon nitridestructurallayerwithimprovedstresscontrol.Thi slayerwasdesignedtobethick relativetootherdiaphragmlayers,whichwerefabricateda tcorporatepartnerOrbit Semiconductorandnotcontrolledforstresses.ZnOwasagai nusedasthepiezoelectric andlarge-scaleintegratedCMOScircuitswereincludedonchip.Aratfrequencyresponse wasobtainedfrom100Hz{18kHz,withasensitivityof0 : 92mV = Pa.In1996,Leeet al.[ 64 ]ofthesameresearchgrouppresentedanewpiezoelectricmi crophonewith ZnOonalowpressurechemicalvapordeposition(LPCVD),lowstresssiliconnitride cantileversensingelement,picturedinFigure 3-4 .Theenhancedcomplianceofthis \cantileverdiaphragm"resultedinahighsensitivityof30 mV = Pa.However,withthe morecompliantdiaphragmalsocamealowresonantfrequency andaresultingbandwidth ofonly100Hzto890Hz.Alateriteration[ 65 ]improvedthebandwidthto1 : 8kHzwhile maintainingthesamesensitivity. 49

PAGE 50

Figure3-5.CrosssectionoftherstaeroacousticMEMSmicr ophone[ 17 ].[Reprintedwith permissionoftheAmericanInstituteofAeronauticsandAstron autics.] In1998,Sheplaketal.[ 16 17 ]introducedtherstMEMSmicrophonedesigned specicallyforaeroacousticsapplications(Figure 3-5 ).Themicrophoneincludedfour dielectricallyisolatedpiezoresistorsontopofa210 mdiameter,0 : 15 mthicksilicon nitridediaphragmforsensingofdiaphragmderection.Lump edelementmodelingwas usedtopredictperformance.Asensitivityof2 : 24 V = Pa = Vwasmeasuredtowithin 3dBfrom200Hztothetestinglimitof6kHz,thoughthefrequenc yresponsewas predictedtoberatupto300kHz.Alinearresponsewasobtaine duptothetestinglimit of155dB.Thedevicenoiseroorwas92dB = p Hzat250Hz. In1999,Naguibetal.[ 66 67 ]introducedtwosquarediaphragm(510 mto710 m onaside)piezoresistivemicrophonedesignsforuseinmeas uringjetscreechnoise. Sensitivitiesof1 : 2mV = Pa = Vto1 : 8mV = Pa = Vweremeasuredoverafrequencyrangeof 1 : 5kHz{5kHz.Thedynamicrangewasnotreported.In2002,Huanget al.[ 68 ]improved theperformanceofthe710 mdesignthroughtheuseofanimprovedfabricationprocess. Thenewmicrophone,forwhichadepictionisfoundinFigure 3-6 ,yieldedthehighest reportedmaximumlinearpressureforaMEMSmicrophoneyetr eportedintheliterature, 174dB.Theauthorswereonlyabletoconrmaratfrequencyre sponseupto10kHz becauseoftestinglimitations. 50

PAGE 51

Figure3-6.PiezoresistiveMEMSmicrophoneforaeroacoust icmeasurements[ 68 ]. [Reprinted,withpermission,fromHuangetal.,ASiliconMic romachined MicrophoneforFluidMechanicsResearch,JournalofMicrom echanicsand Microengineering,2002.] Startingin2001,researchersattheInterdisciplinaryMic rosystemsGroup(IMG) attheUniversityofFloridapresentedanumberofMEMSmicrop honesspecically designedforaeroacousticmeasurementpurposes.In2001,Ar noldetal.[ 18 ]madeseveral modicationstothepiezoresistivemicrophonedesignofSh eplaketal.[ 16 17 ]inorder toimproveperformance,particularlytheMDP:thedevicewa senlargedinordertolimit misalignmenteects;noiseinthepiezoresistorswasreduc edviaareductioninresistance andtheuseofhigherqualitywafers;ahigherdopingconcent rationwasusedforthe piezoresistors;andnally,aplasma-enhancedchemicalva pordeposition(PECVD)silicon nitridepassivationlayerwasaddedtoprotectthedevicefr ommoistureandreducedrift. AmicrographofthemicrophoneisshowninFigure 3-7 .TheMDPwasindeedloweredto 52dB(1Hzbincenteredat1kHz)andalinearresponsewasmeasur eduptothetesting limitof160dB,thoughthesensitivitywasreducedbynearly afactorof3.Thefrequency responseofthismicrophonewaslatercharacterizeduptove ryhighfrequenciesatBoeing Corporation;itshowedaratresponsetowithin 1dBoutto100kHz[ 19 ]. AeroacousticmicrophonesweredevelopedatIMGutilizingot hertransduction methodsaswell.In2004,Kadirveletal.[ 70 ]describedthedesign,fabrication,and testingofanintensity-modulatedopticalMEMSmicrophone .Theintensitynoisein thelightsourcecontributedtoahighMDPof70dBandthedevi cewaslinearonlyto 132dB.In2007,Martinetal.[ 71 72 ]discussedadual-backplatecapacitiveMEMS 51

PAGE 52

X X X X X X X X X Xz Tapered piezoresistor X X X X X X Xz Arc piezoresistor X X X X Xz Vent channel 9 Ventport 9 Diaphragm Figure3-7.Second-generationaeroacousticMEMSmicropho ne[ 18 ].[Reprintedfrom[ 69 ] withpermissionfromauthor.] microphonedesign,depictedinFigure 3-8 .Thedualbackplatesformedtwocapacitors withthemicrophonediaphragm,allowingasensitivity-inc reasingdierentialcapacitance measurement.ThemicrophonewasfabricatedusingtheSandi aUltra-planar,Multi-level MEMSTechnology5(SUMMiTV)fabricationprocessandtheinter faceelectronics includedano-packagechargeamplier.Theauthorsreport edexcellentagreement betweenlumpedelementmodelpredictionsandexperiment.A dynamicrangeof 41dBto164dBandbandwidthof300Hz{20kHzweremeasured,with theupperend ofthebandwidthlimitedbytestingcapabilities.Laterimp rovementsinpackagingand interfaceelectronics(namelytheuseofalow-noisevoltag eamplierinsteadofacharge amplier)resultedinasignicantreductionofMDPto22 : 7dB.Thesensitivitywasalso reducedto166 V = Pa[ 73 ]. In2002,Zhangetal.[ 74 ]usedleadzirconatetitanate(PZT)inaMEMSmicrophone forthersttime.Thesensitivityofthecantilever-basedm icrophoneswerereportedto varyfrom10mV = Pato40mV = Paforsquaregeometries200 mto2mmonaside,though nodetailsweregivenofthemeasurementsetup.Lateriterat ionsfromZhaoetal.[ 75 76 ] movedawayfromthecantilevergeometrytothatpicturedinF igure 3-9 ,withsquare diaphragmsfrom600 mto1mmonaside.Theyachievedaremarkablyratfrequency responsefrom10Hzto20kHz,withasensitivityof38mV = Pa. 52

PAGE 53

Figure3-8.Adual-backplatecapacitiveMEMSmicrophone[ 71 ].[ c r 2007IEEE. Reprinted,withpermission,fromMartinetal.,AMicromach ined Dual-BackplateCapacitiveMicrophoneforAeroacousticsMe asurements, JournalofMicroelectromechanicalSystems,Dec.2007.] nr nnnn r n r n n n Figure3-9.EarlyPZT-basedpiezoelectricmicrophone(ada ptedfromZhaoetal.2003 [ 75 76 ]). Thenextpiezoelectricmicrophone,picturedinFigure 3-10 ,waspresentedbyKoet al.in2003[ 77 ].ThesquarediaphragmwasformedfromZnOsandwichedbetwe entwo concentricsegmentedaluminumelectrodesonLPCVDsiliconn itride.Themicrophone hadasensitivityofapproximately30 V = Paandaresonanceat7 : 3kHz. Inordertoavoidresidualstressissuesomnipresentinsili connitridediaphragms,Niu andKim[ 78 ]proposedanovelbimorphstructurein2003.Thelmstackwa scomposed 53

PAGE 54

Figure3-10.Piezoelectric(ZnO)microphonewithtwoconce ntricelectrodes[ 77 ]. [ReprintedfromSensorsandActuatorsA,vol.103,Koetal.,Mi cromachined PiezoelectricMembraneAcousticDevice,pgs.130{134,Copy right2003,with permissionfromElsevier.] ofZnO,aluminum(Al)electrodes,andparyleneDasthestruct urallayer.Concentric segmentedelectrodeswerealsousedasin[ 56 58 59 ].Asensitivityof520 V = Pawas achieved,whichwasanimprovementover[ 56 58 ]butnot[ 59 ]. In2003,eortsatBruelandKjr[ 79 ]yieldedameasurementqualityMEMS condensermicrophone,depictedinFigure 3-13 .Thedesigngoalforthemicwastoachieve a1/4"measurementmicrophonewithnoisecharacteristicsn earthatofatraditional 1/2"BruelandKjr4134microphone(18dB(A)).Anumberofst abilityissueswere alsoconsidered,includingsensitivitytotemperature,re lativehumidity,andstatic pressure.Thedesignfeatureda1 : 95mmoctagonalLPCVDsiliconnitridediaphragm withchrome/goldelectrodesmountedinatitaniumhousing. Themicrophonehada dynamicrangeof23dB(A){141dBandbandwidthof251Hz{20kHz. In2004,Polcawich[ 80 ]shareddevelopmentdetailsforaPZTMEMSmicrophone targetedforuseinaMEMSphotoacousticspectrometerorrem oteacousticsensor.The circulardiaphragmdiametersrangedfrom500 mto2000 mandwerefabricatedin designswith80%PZTcoverageinthecenterofthediaphragma nd20%PZTcoverageon theoutsideedgeofthediaphragm.Sensitivitiesof97 : 9nV = Pato920nV = Pawerereported andresonantfrequencieswere O (100kHz).Noindicationofthebandwidthordynamic rangewasgivenforthesemicrophones. 54

PAGE 55

Figure3-11.Measurement-gradeMEMScondensermicrophone developedatBrueland Kjr[ 79 ].[ c r 2003IEEE.Reprinted,withpermission,fromScheeperetal. ,A NewMeasurementMicrophoneBasedonMEMSTechnology,Journa lof MicroelectromechanicalSystems,Dec.2003.] Alsoin2004,Hillenbrandetal.[ 81 ]suggestedtheuseofchargedcellularpolypropylene (commerciallyknownasVHD40)asthepiezoelectricmateriali namicrophone.They presentedresultsfortwostructuresformedfromsingleand velayergluedstacksof metallizedVHD40,whichhadsensitivitiesof2mV = Paand10 : 5mV = Pa,respectively. Atotalharmonicdistortionofonly1%at164dBwasreported, thoughitisnotclear towhichofthetwodesignsthemeasurementappliedorhowthe measurementwas performed.Thislargemaximumpressure,inadditiontoalar gereportedbandwidthupto nearly140kHzforthesinglelmdesign,makesthismicrophon epotentiallyappealingfor useinaeroacousticapplications.However,themicrophonew asnotbatchfabricatedand concernsaboutthetemperaturestabilityofthechargedlm werenoted. Horowitzetal.(2007)[ 20 ]ofIMGintroducedtherst|andpriortothisstudy, only|piezoelectricMEMSmicrophonedesignedspecically foraeroacousticapplications. Thecircularmicrophonediaphragmwasapiezoelectric(PZT )unimorphwithanannular 55

PAGE 56

A B Figure3-12.Piezoelectric(PZT)microphoneforaeroacous ticapplications[ 20 ].[Reprinted withpermissionfromS.Horowitzetal.,Developmentofamicr omachined piezoelectricmicrophoneforaeroacousticsapplications ,Journalofthe AcousticalSocietyofAmerica,vol.122,pp.34283436,Dec.20 07. c r 2007, AcousticSocietyofAmerica.] piezoelectriclmstackonasiliconlayer,asshowninFigur e 3-12 .Lumpedelement modelingwasusedtopredictdeviceperformance.Thereport edmaximumpressureof 169dBexceededthedesigngoalof160dB.Thefrequencyrespo nsecouldnotbemeasured beyond6 : 7kHzduetoequipmentlimitations,thoughtheresonantfrequ encyofthe microphonediaphragmwasfoundviaalaservibrometermeasu rementtobe59kHz. Thissuggestedausablebandwidthnearlysucientfor1/4sc alemodelaeroacoustic measurementapplications. Alsoin2007,Fazzioetal.ofAvagoTechnologies[ 82 ]describedseveralmicrophones producedusingavariantoftheFBAR(lmbulkacousticresona tor)process.Thecircular diaphragmwascomposedofAlNwithmolybdenum(Mo)electrode sinoneofthree congurations:annularring,innerdisk,oracombinationo fthetwo.Fewperformance specicationswereincludedinthepaper.Thefabricationp rocesswasoutlinedseparately byLamersandFazzio[ 83 ]. LeeandLee(2008)presentedaZnOmicrophoneutilizingacir culardiaphragmwith annularZnO/Molmstack.Limitedcharacterizationworkwa sconducted.Thefrequency responsewasnotrat,withthesensitivityvaryingfromarou nd1 V = Paat400Hzto 56

PAGE 57

A B Figure3-13.Top-viewofmicrophonestructuresfromFazzio etal.(2007)[ 84 ].A)Annular outerelectrode.B)Combinedcircularinnerandannularout erelectrodes.[ c r 2003IEEE.Reprinted,withpermission,fromFazzioetal.,D esignand PerformanceofAluminumNitridePiezoelectricMicrophones, 14th InternationalConferenceonSolid-StateSensors,Actuator s,and Microsystems,Jun.2007.] around100 V = Paat10kHz.Theresonancewasreportedtobe54 : 8kHz,sothesourceof thevariationwasnotclear. Inhis2010doctoraldissertationfromtheUniversityofMich igan,RobertLittrell describedtwogenerationsofpiezoelectricmicrophonesba sedondouble-layeredAlN/Mo cantilevers[ 85 ].Thegoaloftheworkwastodemonstratealownoisepiezoele ctric microphone.Intherstgeneration,anarrayof20cantileve rswereusedasthemicrophone sensingelements,withmodelpredictionsleadingtothesel ectionof0 : 5 mAlNlayers. Theresultingmicrophoneswerefoundtohavehigherthanexp ectednoiseroor(58dB(A)) duetopoorlmqualityandthushighdielectricloss.Inaddi tion,stressinthethin lmsresultedinslightlycurvedcantileversthatreducedt heventresistance.Thesecond generation,withcross-sectionshowninFigure 3-14 ,featuredthickerAlNlayers(1 m) forwhichbetterlmqualitywasknowntobeachievable,modi cationstothefabrication processthatenabledindividualpatterningofAlNandMo,and reductionofthenumber ofcantileversto2(395 mlongby790 mwide)inaneorttoreducethegaparound themandthusincreasetheacousticresistance.Atestedmic rophonehadasensitivity of1 : 82mV = Pa,minimumdetectablepressureof37dB(A),and3%THDat128dB .The non-standardmethodusedtocalculateTHDinvolvedsummatio nofharmonicamplitudes ratherthanharmonicpowers[ 47 86 87 ]andthusthedistortionat128dBwaslikely 57

PAGE 58

nr r Figure3-14.Crosssectionofasecond-generationAlNdouble -cantilevermicrophone (adaptedfromLittrell2010[ 85 ].) over-predicted.SomestressremaininginthebottomMolaye rresultedincurvatureofthe cantileversensingelementsandthusalowventresistancet hatnecessitatedpackagingthe microphoneswithlargecavityvolumes.Packagedinthisway ,thefrequencyresponsewas showntoberatatleastto8kHz,nearwhichtheplanewavetubec alibrationprocedure brokedownduetocut-onofnon-planaracousticmodes.Themi crophonewasreportedto havearesonantfrequencyof18kHz. Table 3-1 providesachronologicalsummaryofpiezoelectricMEMSmic rophones discussedinthissection,includingperformancedatanotg iveninthetext.Duetothe myriadwaysresearcherspresentdata,performancespecic ationscollectedinTable 3-1 oftenrequiredinterpretation;consultationoftheorigin alsourceisthusnecessaryinorder tojudgethetrueperformanceofanyparticularmicrophone. 3.2Summary SincetherstMEMSpiezoelectricandaeroacousticmicroph onesin1983and1998, respectively,signicantprogresshasbeenachieved.MEMS piezoelectricmicrophones havebeenfabricatedusinganumberofdierentmaterials(Z nO,AlN,polyurea,PZT, etc.)andgeometries(squareandcircularmembranes,novel electrodecongurations,etc.). DevelopmentofMEMSaeroacousticmicrophoneshasincluded theuseofpiezoresistive, capacitive,optical,andpiezoelectrictransductiontech niquesandhasseensteadyincreases indynamicrangeandbandwidth.Sucientevidenceexiststo suggestthatmeetingthe aggressiveperformancespecicationsforafuselagemicro phoneisachievable. 58

PAGE 59

Table3-1.SummaryofMEMSmicrophones. AuthorTransduction Method SensingElementSize SensitivityDynamic Range Bandwidth(Predicted) Royeretal.1983[ 55 ] Piezoelectric(ZnO) 1 : 5mm 30 m250 V = Pa66dB{ N/R 10Hz{10kHz(0 : 1Hz-10kHz) Kimetal.1987[ 56 ] Piezoelectric(ZnO) 3mm y 3 : 6 m 0 : 5mV = Pa72dB{ N/R 20Hz{5kHz Franz1988[ 60 ] Piezoelectric(AlN) 0 : 72mm 2 1 m # 25 V = Pa # 68dB(A) # { N/R N/R{45kHz # Kimetal.1989[ 58 ] Piezoelectric(ZnO) 2mm y 1 : 4 m 80 V = PaN/R3kHz{30kHz Kimetal.1991[ 59 ] Piezoelectric(ZnO) 3 : 04mm y 3 m 1000 V = Pa50dB(A){ N/R 200Hz{16kHz Schellinetal.1992[ 61 ] Polyurea0 : 8mm y 1 m 4{30 V = PaN/R100Hz{20kHz Riedetal.1993[ 63 ] Piezoelectric(ZnO) 2 : 5mm y 3 : 5 m 920 V = Pa57dB(A){ N/R 100Hz{18kHz Schellinetal.1994[ 62 ] Piezoelectric(P(VDF/TrFE)) 1mm y 3 m150mV = Pa60dB(A){ N/R 50Hz{16kHz. Leeetal.1996[ 64 ] Piezoelectric(ZnO) 2mm z 4 : 5 m 3mV = PaN/R100Hz{890Hz Leeetal.1998[ 65 ] Piezoelectric(ZnO) 2mm z 1.5{4 : 7 m 30mV = PaN/R50Hz{1 : 8kHz Sheplaketal.1998[ 16 17 ] Piezoresistive105 m 0 : 15 m 2 : 24 V = Pa V92dB x { 155dB 200Hz{6kHz(100Hz{300kHz) Naguibetal.1999[ 66 67 ] Piezoresistive510 m y 0 : 4 m 1 : 2 V = Pa VN/R1 : 5kHz{5kHz Arnoldetal.2001[ 18 ] Piezoresistive500 m 1 m 0 : 6 V = Pa V52dB x { 160dB 1kHz{20kHz(10Hz{40kHz) Huangetal.2002[ 68 ] Piezoresistive710 m y 0 : 38 m 1 : 1mV = Pa V53dB x { 174dB 100Hz{10kHz Scheeperetal.2003[ 79 ] Capacitive1 : 9mm 0 : 5 m 22 : 4mV = Pa23dB(A){ 141dB 251Hz{20kHz Koetal.2003[ 77 ] Piezoelectric(ZnO) 3mm y 3 m30 V = PaN/R1kHz{7 : 3kHz Niuetal.2003[ 78 ] Piezoelectric(ZnO) 3mm y 3 : 2 m 520 V = PaN/R100Hz{3kHz Zhaoetal.2003[ 75 ] Piezoelectric(PZT) 0.6{1mm y N/R 38mV = PaN/R10Hz{20kHz Kadirveletal.2004[ 70 ] Optical500 m 1 m 0 : 5mV = Pa70dB x { 132dB 300Hz{6 : 5kHz (1Hz{100kHz) Polcawich2004[ 80 ] Piezoelectric(PZT) 250 m{1mm 2 : 18 m 97 : 9{ 920nV = Pa N/RN/R Radiusofcirculardiaphragm. y Sidelengthofsquarediaphragm. z Sidelengthofcantilever. x 1 Hz bin. # Perreferences[ 62 88 ] 59

PAGE 60

Table 3-1 .Continued. AuthorTransduction Method SensingElementSize SensitivityDynamic Range Bandwidth(Predicted) Hillenbrandetal.2004[ 81 ] Piezoelectric(VHD40) 0 : 3cm 2 area 55 m 2 : 2mV = Pa37dB(A){ 164dB 20Hz{140kHz 0 : 3cm 2 area 275 m 10 : 5mV = Pa26dB(A){ 164dB 20Hz{28kHz Martinetal.2007[ 71 72 89 ] Capacitive230 m 2 : 25 m 390mV = Pa41dB x { 164dB 300Hz{20kHz Horowitzetal.2007[ 20 ] Piezoelectric(PZT) 900 m 3 m 1 : 66 V = Pa35 : 7dB x (95 : 3dB(A)) {169dB 100Hz{6 : 7kHz (100Hz{50kHz) Fazzioetal.2007[ 82 ] Piezoelectric(AlN) 350 m 1 : 44 m N/R60dB{ 155dB 1kHz-6kHz Leeetal.2008[ 90 ] Piezoelectric(ZnO) 1mm 1 m1 Vto100 VN/R < 1kHz Martinetal.2008[ 73 ] Capacitive230 m 2 : 25 m 166 V = Pa22 : 7dB x { 164dB 300Hz{20kHz Littrell2010[ 85 ] Piezoelectric(AlN) 0 : 62mm 2 { 2 : 3 m 1 : 82mV = Pa37dB(A){ 128dB 50Hz{8kHz(18 : 4kHz) Radiusofcirculardiaphragm. x 1 Hz bin. { 2cantilevers ReferringtoTable 3-1 ,theperformanceofseveralpriormicrophonedesignsappro ach thebenchmarkssetforthisstudy.ThemicrophonesofMartin etal.[ 21 72 73 ]and Hillenbrandetal.[ 81 ]possessedstrongperformancebutthefundamentalability ofthe underlyingtechnologiestowithstandtheharshhigh-altit udeenvironmentonanairplane fuselageisquestionable.InthecaseofMartinetal.,theca pacitivetransductionmethod ishighlysusceptibletofailurefrommoistureshortingthe electrodes,inadditiontonot beingapassivetechnology.Meanwhile,thechargedcellula rpolypropylenelmutilizedby Hillenbrandetal.hasnotedtemperaturestabilityproblems thatrequirefurthermaterial development[ 81 ];itwasalsonotbatch-fabricated[ 91 ]. ThemicrophonesofHuangetal.[ 68 ]andHorowitzetal.[ 20 ]alsocomecloseto meetingthedesigngoalsforthisworkwithbetterpromiseof robustness.Theformerhas thehighestreportedmaximumlinearpressure(174dB)butal soacorrespondinglyhigh noiseroor(53dB = p Hz);meanwhile,thelatterhasalowernoiseroor(47 : 8dB = p Hz) butthemaximumpressure(169dB)obtainedfrommeasurement sisalsolowerthan 60

PAGE 61

desired.Neitherfrequencyresponsewasexperimentallycon rmedtoberatoverthe entireaudiorange.Kulitemicrophonesalreadydeployedin fuselagearraysutilizethe samepiezoresistivetechnologyasHuangetal.[ 68 ],andthustheprimaryadvantageofthe piezoelectricmicrophoneofHorowitzetal.[ 20 ]isitspassivity. ThemicrophoneofHorowitzetal.[ 20 ]provedapiezoelectricMEMSmicrophone isaviabletechnologyforobtainingperformancenearthatn eededfordeploymentina fuselagearray.However,improvementsarerequiredtomeett heperformanceobjectives inChapter 1 .Themicrophoneof[ 20 ]wasnotdesignedusingoptimizationtechniques andthusitisunlikelythatperformancewasmaximized.Inth isstudy,asystem-level modelofthepiezoelectricfuselagemicrophoneisdevelope dinChapter 5 andusedto produceoptimalmicrophonedesignsinChapter 6 .Inaddition,thechoiceofgeometry andmaterials|particularlythepiezoelectricmaterial|p rovideadditionalavenuesfor improvedperformance.Thesechoicesareaddressedinthene xtchapter,Chapter 4 61

PAGE 62

CHAPTER4 MEMSPIEZOELECTRICMICROPHONE ItwasdeterminedinSection 1.2 andconrmedintheliteraturereviewofChapter 3 thatamicrophoneutilizingpiezoelectrictransductionwa sthebestchoiceforaircraft fuselagearrayapplications.Inthischapter,materialand fabricationissuesarediscussed. First,thepiezoelectriceectisreviewed,andpossiblepi ezoelectricmaterialchoices fortheMEMSmicrophonearecompared.Next,thecommercialfa bricationprocess usedtoproducethepiezoelectricmicrophoneisdescribeda ndacompatiblegeometryis established. 4.1Piezoelectricity Piezoelectricmaterialsarethosewhichexhibitcouplingb etweenstrainandelectric eld.Achangeinelectriceldresultingfromstraininthem aterialisreferredtoasthe directpiezoelectriceect;meanwhile,thespontaneousst rainingofamaterialwhichresults fromanexternallyappliedelectriceldiscalledtheconve rsepiezoelectriceect[ 92 ]. Thepresenceofthepiezoelectriceectinamaterialisinti matelytiedtoitscrystal structure.Onlycrystalswhichlackacenterofsymmetry,ca llednoncentrosymmetric crystals,maybepiezoelectric.Twentyoutof21noncentros ymmetriccrystalclasses exhibitpiezoelectricity[ 92 93 ].Fromthose,somearepolarandpossessanetdipole momentintheunstrainedstate.Thesepolarcrystalsalsoex hibitpyroelectricity,the couplingoftemperatureandelectriceld.Materialsinwhi chtheorientationofthe polarizationmaybechangedunderapplicationofanexterna lelectriceld,andforwhich thechangeremainsafterremovaloftheeld,arecalledferr oelectricmaterials[ 93 94 ]. SeeFigure 4-1 foraVenndiagramshowingtheinterrelationshipsbetweenp iezoelectric, pyroelectric,andferroelectricmaterials. Manyferroelectricmaterialsarepolycrystallineandthus donotexhibitpiezoelectricity onthemacroscale.Therandomorientationofcrystalsmeans thematerialsareisotropic fromaconstitutivelawperspective.Thesematerialsmaybe madepiezoelectricallyactive 62

PAGE 63

Piezoelectric Pyroelectric Ferroelectric Figure4-1.Venndiagramforpiezoelectric,pyroelectric, andferroelectricmaterials. throughaprocesscalledpoling.Inthisprocess,themateri alsareheated,andastrong externalelectriceldisappliedthatcausesthepolarizat iondirectionwithinthematerials toreorient.Afterreductionoftemperatureandremovalofth eexternaleld,thenew polarizationorientationremainsandthepiezoelectrice ectismacroscopicallyactive. Duringthisprocess,symmetryinthedirectionofpolingisd estroyedandtheresulting materialistransverselyisotropic[ 92 ].Thepoleddirectionisreferredtoasthe3-direction inthelocalmaterialcoordinatesystem. Theconstitutiveequationsoflinearpiezoelectricityare givenin\compressedmatrix" (or\abbreviatedsubscript")forminthenomenclatureofth eIEEEstandardas[ 95 96 ] D i = d iq T q + Tij E j (4{1) and S p = s Epq T q + d jp E j ; (4{2) where i;j =1 ::: 3and p;q =1 ::: 6.Intheseequations, D i arethethreecomponents oftheelectricdisplacement, d iq arethe18piezoelectricstrainconstants, T q arethesix mechanicalstresses, Tij arethesixelectricalpermittivities, E j arethethreecomponents oftheelectriceld, S p arethesixengineering(nottensoral)strains,and s Epq arethe 36elasticcompliancecomponents.Symmetryconsideration sforamaterialreducethe numberofindependent d iq S p ,and s Epq .Thesuperscripts T and E indicatethatthe materialpropertiesmustbemeasuredunderconditionsofco nstantstressandelectric 63

PAGE 64

eld,respectively[ 96 ].Intheabsenceofpiezoelectriccoupling,all d coecientsare zero,andtheconstitutiveequationsreducetothoseforpur elydielectric[ 97 ]andlinear elastic[ 92 98 ]materials,respectively.Asecondconvenientformofthel inearconstitutive equationsis[ 92 95 96 ] D i = e iq S q + Sij E j (4{3) and T p = c Epq S q e Tjp E j ; (4{4) where c Epq = s E 1 pq (4{5) arethe36elasticstinesscomponents, e iq = d ip c Epq (4{6) arethe18piezoelectricstressconstants[ 96 ],and Sij = Tij e iq d jq (4{7) arethepermittivitycomponentsmeasuredunderconstantst rain[ 96 ]. Piezoelectricmaterialsusedinmicrosystemsincludelead zirconiumtitanate(PZT), zincoxide(ZnO),aluminumnitride(AlN),aromaticpolyurea, polyvinylideneruoride (PVDF),andothers.PZTisthemostpopularduetolargepiezoe lectriccoecients.Itis aferroelectricmaterialthatrequirespolingandisavaila bleinpolycrystalline,textured, andepitaxialthinlms[ 93 ]. AlNandZnOoersignicantlylowerpiezoelectriccoecient sthanPZT,but theirdielectricpropertiesstillmakethemattractivemat erialsforsomeapplications.In bendingmodesensorsthatutilizethe d 31 coecient,forinstance,agureofmeritisthe 64

PAGE 65

piezoelectric\ g "coecient 1 [ 94 ], g 31 = d 31 33 ;r 0 ; (4{8) whichisrepresentativeoftheopencircuitelectriceldpe rappliedmechanicalstress. BecauseAlNandZnOpossesssignicantlylowerpermittiviti esthanPZT,their g 31 coecientsareactuallybothsuperior. AlNandZnOarebothpyroelectric,butnotferroelectric,and thuscannotbe poled[ 93 94 ];theymustinsteadbeorientedappropriatelyduringdepos ition.ZnO hastraditionallybeenthematerialofchoiceforMEMSpiezo electricmicrophones(see Section 3.1 )duetothedicultyofdepositingAlNlms[ 99 ]andthusbetteravailability ofZnO[ 100 ].Thesituationhasbeenrectiedwiththeadventofmodernm agnetron sputteringtools[ 100 { 102 ],andithasbeenrecognizedthatAlNholdsseveraladvantage s overZnO.Zincisafast-diusingionthatpresentsproblems forintegrationwithsilicon semiconductorprocessing[ 93 100 ].Inaddition,withabandgapof3eV,ZnOisreallya semiconductorandthereisalwaysriskthatinadvertentdop ingcoulddegradeitsdielectric properties,i.e.resultinahighdielectricloss[ 93 100 103 ].Itisthusdiculttoobtain ZnOlmswithhighresistivity[ 100 ]. Thedissipationqualitiesofapiezoelectricareanimporta ntconsiderationwhen comparingmaterialchoicesforpiezoelectricsensors.Low resistivity(highdielectricloss)is especiallycripplinginlow-frequencysensors,forinstan cebelow10kHz[ 100 ].Lossinessis typicallycharacterizedviatheelectriclosstangent[ 97 ], tan = !" 33 ;r 0 ; (4{9) where istheelectricalconductivityand istheradialfrequency.Thelosstangent representstheratioofdissipatedpowertostoredpowerina dielectric.Thepresenceof 1 Comparethistotheexpressionforopencircuitsensitivity forthepieozelectricmicrophone,Equation 5{35 65

PAGE 66

dissipationintroducesnoise,andthustan playsaroleindeterminingthesignal-to-noise ratioofpiezoelectricsensors.Agureofmeritis[ 100 ], d 31 ( s E11 + s E12 ) p 0 33 ;r tan ; (4{10) whichrepresentstheintrinsicsignal-to-noiseratioofth ematerial.AlNoutperformsZnO inthisregard,withbothalowerlosstangentandhigherintr insicsignal-to-noiseratio.A comparisonoftypicalmaterialpropertiesandthediscusse dguresofmeritforPZT,AlN, andZnOaregiveninTable 4-1 Table4-1.Typicalpropertiesofpiezoelectricmaterialsi nMEMS. y PZT z AlNZnO Properties E [GPa] 76345127 [10 3 kg = m 3 ][ 20 ]7.73.265.6 d 31 [pm = V]-274-2-5.4 d 33 [pm = V]593511.7 e 31 [C = m 2 ] -6.5-0.6-0.6 e 33 [C = m 2 ]23.31.61.3 33 ;r 14708.59.2 tan [ 93 100 ]0.01{0.030.0030.01{0.1 FiguresofMerit j g 31 j [V = m = Pa]0.0180.0270.07 j d 31 j = s E11 + s E12 p 0 33 ;r tan [10 5 Pa 1 = 2 ]11.7{20.321.44.3{13.5 y Materialpropertiesdrawnfrom[ 104 ]unlessotherwisenoted z PZT-5H[ 92 96 ] AlNwasselectedasthematerialforuseinthepiezoelectricm icrophoneduetoits relativelyhigh g 31 coecientandbestsignal-to-noiseratioamongthecommont hin-lm piezoelectricmaterials.Inthenextsection,thecommerci alprocessusedtofabricatethe microphonesandtheroleitplayedinearlydesignchoicesis discussed. 4.2DesignforFabrication Apartnershipwasformedwith AvagoTechnologies ofFortCollins,COforfabrication ofthepiezoelectricMEMSmicrophoneusingavariantofthei rlmbulkacousticresonator (FBAR)process[ 82 83 105 ].AnFBARisanelectromechanicallterthatutilizes theresonanceofbulkacousticwavesexcitedinathinpiezoe lectriclm[ 106 ].FBAR 66

PAGE 67

Figure4-2.FBAR-variantprocesslmstack.duplexersandltersforcellularphoneshavebeenproduced involumeusingtheFBAR processsince2002[ 101 ]andAvagoTechnologiesremainstheworld'sonly\highvolum e producerofthin-lmAlNproducts"[ 83 ].Theyarebyfarthemostsuccessfulpiezoelectric MEMSproductsonthemarket[ 103 ]. AdepictionofthelmstackusedinthemodiedFBARprocessof AvagoTechnologies isshowninFigure 4-2 .Itiscomposedofpassivation,electrode,piezoelectric, and structurallayers;allbutthelatterarecomponentsofthes tandardFBARprocess.A keyreasonAlNwaschosenoverZnOintheoriginalFBARprocessd evelopmentwas becauseofitsbettersemiconductorprocesscompatibility [ 101 ].Theelectrodematerialin thelmstackofFigure 4-2 ,molybdenum(Mo),wassubsequentlyselectedbecauseitwas asti,low-lossacousticmaterialwithhighconductivitya ndcompatibleetchchemistry withAlN[ 101 106 ].AsproprietaryfeaturesoftheFBAR-variantprocess,thema terials usedinthepassivationandstructurallayersarenotdisclo sed.Adeepreactiveionetch (DRIE)formsacavityunderneaththediaphragminFigure 4-2 Theperformanceofmicrophoneswiththinlmdiaphragmsise xtremelysensitive tolmstress,whiletherstpriorityinFBARfabricationisp iezoelectriclmthickness uniformity[ 83 ].LeveragingtheFBARprocess,therefore,requiresrenewed attentionto lmstress.Thinlmsaresusceptibletodevelopingbothint rinsicandextrinsicresidual 67

PAGE 68

stresses.Thermalexpansionmismatchbetweenlms,substr ate,andpackageleadto thermalstress,themostcommonextrinsicstress.Intrinsi cstressescanbecausedbya varietyoffactors,includinglatticemismatch,impuritie s,volumechangeprocesses(e.g. phasetransformationoroutgassing),oratomsbeingtrappe dinhigh-energycongurations [ 107 108 ].Theamountofstresscontrolvariesbyprocess;sputterde position(commonfor AlN),forinstance,isacomplexprocessthatdoesaordsomere xibilitytotailorstresses. Customizationofthestressstateisachievedviaadjustmen tofbiaspower,argonpressure, sputteringgasmass,temperature,and/ordepositionrate[ 108 ].AvagoTechnologiesisable toadjustlmdepositionparameterstotargetalargerangeo flmstresses. TheFBAR-variantlmstackprovidedsomerexibilityinthese lectionofmicrophone geometry.WithaDRIEstepalreadyintegratedintotheFBAR-v ariantprocess|as opposedtoananisotropicreleaseetch|thediaphragmwasno tlimitedtorectangular geometry[ 90 109 ].Acirculardiaphragmoersseveraladvantagesoverrecta ngular geometries:itiseasiertomodel/designsincerst-modevi brationscanbereducedtoa 1-Daxisymmetricproblem,asopposedtoa2-Dprobleminther ectangularcase;itlends itselftosimplerelectrodecongurationsovertherectang ularcase,inwhichhighstress regionsarenotuniformlydistributedalongtheboundary[ 90 ];andthecirculargeometry doesnotinherentlyincludelifetime-reducingstressconc entrations. Clampedcirculardiaphragmspossesshighstress/strainre gionsbothalongthe circumferenceandatthecenter.Thecongurationofthepie zoelectric/electrodelms onthestructurallayerthereforepresentsanotherdesignc hoice[ 110 ].Asinglestackof piezoelectric/electrodelmspresentstheleastcomplexi ty;twosuchcongurationsare showninFigure 4-3 .Figure 4-3A showsthepiezoelectric/electrodestackinthemiddleof thediaphragm,calledherethe\centraldisc"conguration ,whileFigure 4-3B showsthe stackasan\outerannulus."Concernsaboutthecontributio nofelectrodetracesrunning overthediaphragmintheformercase,particularlytheirpo tentialcontributiontodevice 68

PAGE 69

A B Figure4-3.Potentialcirculardiaphragmpiezoelectric/m etallmstackcongurations.A) Centraldisc.B)Outerannulus. stinessandparasiticcapacitanceandthepossibilitytha ttheycouldpromoteasymmetric modalvibrations[ 20 111 ],ledtothechoiceoftheannularconguration. AlthoughexactFBARprocessdetailsareproprietary,agenera loutlineoffabrication stepsforamicrophonestructurewaspublishedbyAvagoTechn ologies[ 82 83 ]andis summarizedinFigure 4-4 .Theprocessinvolvedbothsurfaceandbulkmicromachining startingwitha675 mthick,150mm(6")siliconwafer(Figure 4-4A ).First,ashallow cavitywasetchedandlledwithsacricialmaterial,which servedtodenethediaphragm diameterandsetanetchstopforsubsequentbacksideproces sing.Thewafersurface wasthinnedto500 mandplanarizedviachemical-mechanicalpolishing(CMP)a sin Figure 4-4B .Thestructural,metal,piezoelectric,andpassivationla yers,inadditionto thebondpads,werethendepositedandpatternedinasetofpr oprietaryprocesssteps (Figure 4-4C ).Whatisknownfromtheopenliteratureaboutthelmdeposi tionis thatAlNistypicallysputter-deposited[ 100 { 103 ],oftenatlowtemperatures( < 200 C). PossibleetchchemistriesforAlNandMoincludechlorineand ruorinegas,respectively [ 101 ].Projectionstep-and-repeatphotolithography[ 43 ]wasusedtorepeatthesame10 10patternofmicrophones(withdie2mmonaside)overtheent irewafer.ADRIEfrom thebacksideofthewaferformedthebackcavity(Figure 4-4D )andthesacricialmaterial wasremovedtoreleasethediaphragm(Figure 4-4E ). 69

PAGE 70

ABeginwithabaresiliconwafer. Sacricialmaterial BEtchcavityinsiliconwaferanddepositsacricial material.PerformCMP. Structural AlN H Hj Passivation ) Mo + CDepositandpatternlms. DDRIEthroughbacksideandstoponsacricial layer. EReleasethediaphragmviaremovalofsacricial layer. Figure4-4.Outlineoffabricationsteps. 4.3Summary Inthischapter,materialandfabrication-relatedissuesr elevanttothepiezoelectric MEMSmicrophonewerediscussed.First,therelativemerits ofcommonthin-lm piezoelectricmaterialswerereviewed,andthematerialch oiceofAlNwasestablished. Next,amicrophonegeometryutilizinganannularpiezoelect riclmstackandcompatible withtheselectedfabricationprocesswaschosen.Finally, theproprietaryFBAR-variant fabricationprocesswasdescribedviareferencetoopenlit eratureonthesubject.Withthe microphonegeometryandcompositionrmlyestablished,th enextchapterfocusesonthe developmentofamodeltopredictitsperformance. 70

PAGE 71

CHAPTER5 MODELING Inthischapter,amultiple-energy-domaindynamicmodelof thepiezoelectric microphoneisdevelopedthatallowscomputationofperform ancemetricssuchas sensitivity,bandwidth,andminimumdetectablepressure. First,anoverviewofthe lumpedelementmodelingtechniqueisgiven.Next,anoverall lumpedelementmodelof themicrophoneisintroducedandpredictivemodelsforitsc omponentpartsarediscussed inturn.Withthelumpedelementmodelestablished,several importantquantities, includingtheopencircuitfrequencyresponsefunctionoft hemicrophoneandtheoverall electricalimpedance,arederived.Theneedforinterfacee lectronicsandtheirimpacton systemresponseisthenaddressed.Finally,twoarchitectu resforinterfaceelectronics|a voltageamplierandchargeamplier|areintegratedwitht helumpedelementmodel andtheirrelativemeritsarediscussedintermsofsensitiv ityandminimumdetectable pressure. 5.1LumpedElementModelingOverview Apiezoelectricmicrophoneconvertsenergybetweentheaco usticandelectrical energydomains.Accuratepredictionofitsbehaviorrequire sphysics-basedmodelsthat capturetheunderlyingtransportofenergy.Unfortunately, exactanalyticalsolutionsto governingdierentialequationscouplingmultipleenergy domainsarerarelyavailable [ 112 ].Numericalsolutionstotheseequationsusingtechniquess uchastheniteelement methodarepossible,butareoftencomputationallyintensi veanddonotreadilyprovide physicalinsight[ 43 ].Therefore,acompromiseindelityinexchangeforecien tand physicallyinsightfulmodelsiswarranted;thisistheover ridingreasonfortheuseofthe lumpedelementmodelingtechnique. Whenthewavelengthofaphysicalphenomenonbeingmeasured ismuchgreaterthan thecharacteristiclengthscale( L )ofthesensoritself,spatialandtemporalvariations ofthephysicalphenomenonmaybedecoupled[ 113 ].Amicrophone,forinstance,which 71

PAGE 72

hasasensingelementmuchsmallerthanthewavelengthofinc identacousticwavessees adistributionofpressurewhichisessentiallyuniform.Alt houghthepressurecontinues tochangewithtime,thechangesareeectivelyfelteverywh ereonthemicrophone diaphragmatthesameinstant[ 35 ].Foranacousticsignalatafrequencyof20kHz, =17mminairandthediaphragmdiametermustbemuchlessthan Underthecondition L ,thedistributedenergystoragemechanismsofthetrue systemmaybelumpedintoequivalentenergystorageelement s,called\lumpedelements." Inthemechanicaldomain,thismeansadistributedsystemwi thinnitedegreesof freedommayberepresentedbyanequivalentsingle-degreeof-freedommass-spring-damper system.Generalizedkineticenergyisstoredinalumpedmas s,generalizedpotential energyisstoredinalumpedcompliance(inverseofspringst iness),andenergyis dissipatedinalumpeddamper.Thelumpedelementsmaybefou ndviaatruncated seriesexpansionofthecompleximpedance[ 28 ].Alternatively,theymaybefoundfrom equatingtheenergystorageofthetruesystem(usingthesta ticsolutiontoapproximate thedynamicone)withtheenergystorageintheideallumpede lements.Theresulting single-degree-of-freedomrepresentationisthenvalidup toandjustbeyondtherst resonantfrequencyofthetruesystem[ 114 ]. Convenientanalogiesexistbetweenthemechanical/acoust icdomainsandthe electricaldomain.Amass-spring-dampersystemmayberepr esentedbyanequivalent LCRcircuitinwhichaninductorisanalogoustoamass,acapa citanceisanalogousto acompliance,andaresistorisanalogoustoadamper[ 43 ].Thisanalogyisillustratedin Figure 5-1 .Theconjugatepowervariablesintheelectricaldomain,vo ltage(an\eort variable")andcurrent(a\rowvariable"),arethenanalogo ustoforceandvelocityin themechanicaldomainorpressureandvolumevelocityinthe acousticdomain.These so-calledconjugatepowervariablesmaybedenedineachen ergydomain[ 43 ]. Thecircuitanalogy,inconjunctionwithlumpedelementmod eling,maybe employedtoproduceasystem-levelmodelinwhichthelumped elements,includingthose 72

PAGE 73

M R k=1/C F x A F x M R C B Figure5-1.Illustrationoftheelectrical-mechanicalana logy.A)Mass-spring-damper system.B)Inductor-capacitor-resistorcircuit. representativeofdierentenergydomains,areallinterco nnectedinawaythatcaptures theenergyexchangeofcomponentsinthetruesystem.Techni quesdevelopedforcircuit analysisthenbecomeavailable,aswellastheintuitiveund erstandingofcircuitdiagrams thatmanyengineersshare.Theendresultisaninsightful,e cient,andacceptably accuratemodelofamultiple-energy-domainsystem. Thelumpedelementmodelingtechniqueandequivalentcircu itrepresentationshave historicallybeenusedintheeldofelectroacoustics[ 35 114 115 ]andhavefoundusein thedesignofmicroelectromechanicalsystems(MEMS)trans ducers[ 17 20 21 116 ].The techniqueisutilizedinthisstudytoperformmodel-basedd esign. 5.2LumpedElementModelofaPiezoelectricMicrophone Inthissection,asystem-levellumpedelementmodelofthep iezoelectricMEMS microphoneisproduced.ThedevelopmentissimilartoHorowi tzetal.(2007)[ 20 ],with extensionstotheunderlyingmechanicalmodeling.Thecros ssectionofthepiezoelectric microphonestructureconsideredinthisstudyisshowninFi gure 5-2 .Itincludesa diaphragm,whichderectsunderanincidentacousticpressu re;acavity,whichallowsthe diaphragmtomove;andavent,whichconnectsthecavitytoth eambientenvironment andtherebyeliminatessensitivitytostaticpressurechan ges.Asthediaphragmderects, straininthepiezoelectriclayeryieldsanelectricelddu etothedirectpiezoelectriceect. Theelectriceldissensedasavoltagedierenceacrossthe electrodes|thisisthe microphoneoutput. 73

PAGE 74

Inageneraldesignsetting,muchrexibilityexistsinthese lectionofpiezoelectric microphonegeometryandlmstackcomposition,butapiezoe lectricmicrophone diaphragmmustatminimumincludeasetofelectrodesandapi ezoelectriclayer.The microphoneinthisstudyiscomposedofthin-lmmaterialsd ictatedbytheFBAR-variant fabricationprocessasdescribedinSection 4.2 ;theseincludeanaluminumnitride(AlN) piezoelectriclayer,molybdenum(Mo)electrodes,andstru cturalandpassivationlayersfor whichtheproprietarymaterialchoicesarenotdisclosed. + + nr n = = = r r !""r r Figure5-2.Piezoelectricmicrophonestructure. Lumpedelementmodelingwasintroducedasanecientandsim pletechniquefor estimatingthebehavioroftransducers.InFigure 5-2 ,thelumpedelementsthatrepresent eachofthemicrophone'smajorcomponentsareidentiedbas edontheunderlyingphysics ofeach.Toavoidconfusion,theconventionusedhereisthat therstsubscriptonan elementstandsforthedomain( a cousticor e lectrical)andthesecondsubscriptprovides identication.Forinstance, M ad isthelumpeddiaphragmmassintheacousticdomain. EachoftheelementsfoundinFigure 5-2 areconnectedinanequivalentcircuit asshowninFigure 5-3 .Thediaphragmismodeledasalumpedmassandcompliance, M ad and C ad ,respectively[ 113 ].Dampingisincludedasaresistance, R ad ,thataccounts forlossmechanismssuchasthermoelasticdissipation[ 117 ]andanchor/supportloss [ 118 ].Couplingbetweenthediaphragmandtheaironthefreeside ismodeledusinga 74

PAGE 75

C ac M ac R av R ad + R ad;rad M ad + M ad;rad C ad p a :1 C eb C eo R ep R es + v o Figure5-3.Piezoelectricmicrophonelumpedelementmodel radiationmass M ad;rad andresistance R ad;rad thatareconnectedinserieswith C ac and M ad becausetheyallexperiencethesamevolumevelocity.Theca vityismodeledasa massandcompliance, M ac and C ac ,respectively.Theventisrepresentedasaresistance R av .JustasinFigure 5-2 ,theplacementof R av inFigure 5-3 providesanalternatepath forvolumevelocity,andthusapressuredrop,betweentheam bientenvironmentandthe backcavity.Couplingbetweentheacousticalandelectrica ldomainsiscapturedusing atransformerwithturnsratio a .Electricalelementsareconnectedontherightside ofFigure 5-3 andincludethesensecapacitance C eb ,theparasiticcapacitancedueto electrodeoverhangbeyondthediaphragm C eo ,piezoelectriclossresistance R ep ,andseries resistance R es .The\ b "in C eb standsfor\blocked,"meaningitisthecapacitancethat remainswhenthepiezoelectricisblockedfrommotion(andt husnovolumevelocityrows intothetransformer).Theelement R es representstheresistanceofanyleadsorwireson orconnectingtothemicrophone.Themicrophoneoutputisth evoltage v o Althoughtheimpedancesofeachcomponentofthemicrophonea reintroduced hereascombinationsofmasses/inductances,compliances/ capacitances,anddampers/ resistances,itissucientatthisstagetorecognizethate achcomponentpossessesan impedanceandthattheformoftheimpedanceisdictatedbyth eassociatedphysics.The originsofeachimpedancearediscussedinthenextsections 75

PAGE 76

5.2.1Elements Inthissection,thevariouselementsincludedinthelumped elementmodelforthe piezoelectricMEMSmicrophonearediscussedinturn.First ,modelingofthepiezoelectric transductionisdiscussed.Next,structuralelementsthatr epresentthediaphragmare denedandtheassociatedunderlyingdiaphragmmechanical modelisoutlined.Acoustical andelectricalelementsarethenexamined.5.2.1.1Transduction Modelingthetransductionofthepiezoelectricmicrophone requiresaknowledge oftheconstitutivebehaviorofpiezoelectricmaterials.T he3-Dconstitutiveequations [ 95 96 ]werediscussedinSection 4.1 andmaybewrittencompactly,denotingvectorsand matriceswithboldsymbols,as 8><>: D S 9>=>; = 264 T d d T s E 375 8><>: E T 9>=>; : (5{1) The1-D,time-harmonicequivalentofEquation 5{1 is[ 20 113 ] 8><>: I q 9>=>; = 264 j!C ef j!d a j!d a j!C ad 375 8><>: V p 9>=>; ; (5{2) where I iscurrent, q = j! 8 isvolumevelocity, 8 isvolumedisplacement, V is voltage,and p ispressure.Thequantities C ef d a C ad a ,and C eb allserveasconstitutive propertiesofthepiezoelectricandaredenedinturninthe comingparagraphs.Itmay beshownusingcircuitanalysistechniquesthatatransform erinthecongurationof Figure 5-4 isequivalenttoEquation 5{2 givenappropriatedenitionsfor a and C eb Thus,justasEquation 5{2 couplestheacousticandelectricaldomain,sotoodoesthe two-portelectroacousticcircuitelementofFigure 5-4 .Asusedinthelumpedelement modelofFigure 5-3 ,thiselementcouplesthediaphragm/cavity/ventresponse tothe electricalresponseofthepiezoelectric. 76

PAGE 77

a :1 C ad q + p C eb I + V Figure5-4.Two-portpiezoelectrictransductionelement. Thecapacitanceoftheannularlmstackiscomposedofasens ecapacitancethat contributestothetransductionandaparasiticcapacitanc eduetoelectrodeoverhang beyondthestresseddiaphragmregion.Theelectricalfreec apacitance C ef ,i.e.the capacitanceobservedwhenthediaphragmisfreetomove,iss implytheparallelplate capacitancebetweentheelectrodes[ 35 ], C ef = A e h p : (5{3) Here, istheabsolutepermittivityofthepiezoelectriclayer, A e = ( a 22 a 21 )isthe electrodearea,and h p isthedistancetheelectrodesareseparatedbythepiezoele ctric layer.Itisrelatedtotheelectricalblockedcapacitance C eb as[ 35 113 ] C eb = 1 k 2 C ef ; (5{4) where k istheelectromechanicalcouplingfactordenedfrom k 2 = d 2a C ef C ad ; (5{5) whichisrepresentativeoftheeciencyofenergyconversio nfromonedomaintotheother, thoughlossesarenotaccountedfor[ 35 ].Thediaphragmcompliance C ad isdenedfrom Equation 5{2 as C ad = 8j V =0 p ; (5{6) sothediaphragmcomplianceinthetransductionrepresenta tionofFigure 5-4 isthe volumedisplacementperpressureundershort-circuitcond itions,calledtheshort-circuit 77

PAGE 78

compliance[ 35 ].Calculationof C ad fromEquation 5{6 isrelatedtothestructuralmodel ofthediaphragmthatisdiscussedfurtherinSection 5.2.1.2 Twodenitionsfortheacousticpiezoelectriccoecient d a maybeextractedfrom Equation 5{2 .Therstis d a = Q j V =0 p ; (5{7) where Q istheelectriccharge,whichisrelatedtothecurrentas I = j!Q .Thesecond denitionis d a = 8j p =0 V : (5{8) ThechoiceofwhichofEquations 5{7 to 5{8 touseforcalculationof d a isdictatedbythe easeofcalculatingthequantities Q or 8 fromamechanicalmodelofthediaphragm; theirequality,orreciprocity,isimpliedfromthelinearp iezoelectricconstitutiverelations. Finally,theturnsratioofthetransformer(ortransductio nfactor)isdenedas[ 20 113 ] a = d a C ad : (5{9) 5.2.1.2Structuralelements Usingthelumpedelementmethod,theelectromechanicalbeha viorofthediaphragm iscapturedinaseriesofelements.Thedistributedmassand complianceofthediaphragm arecollected,or\lumped"intoanacousticmass M ad andcompliance C ad thattogetherin serieswiththeacousticdamping R ad formtheimpedanceofthediaphragm.Takenalone, theseelementsaresucienttorepresentthediaphragmasas ingle-degree-of-freedom system.However,thepiezoelectrictransductionmechanism isintegrateddirectlywith thediaphragmandtheeectivepiezoelectriccoecient d a isalsodependentonthe diaphragm'selectromechanicalbehavior.Todeterminethe valuesoftheseelementsfora givendiaphragmconguration,predictivecapabilitiesar eneeded. First,however,itisexpedienttodeneeachoftheelements undertheassumption thatapredictionforthestatictransversediaphragmdispl acement, w ( r ),duetoa 78

PAGE 79

pressureorvoltageinputisavailable.With w ( r )known,thevolumedisplacedbythe diaphragmisdenedasitsareaintegral,i.e. 8 = Z a 2 0 w ( r )2 rdr: (5{10) Equation 5{10 maybeusedtocomputetheacousticcompliance C ad ortheacoustic piezoelectriccoecient d a perEquations 5{6 and 5{8 Thelumpedmassofthediaphragmintheacousticdomainisfou ndfromequating thekineticenergyofthelumpedmasstotheactual,distribu tedkineticenergyofthe diaphragm.Thisequalityisgivenas 1 2 M ad ( j! 8j V =0 ) 2 = 1 2 Z 8 [ j!w ( r ) j V =0 ] 2 d 8 ; (5{11) wherethevolumevelocity q andactualplatevelocity_ w ( r )areassumedtimeharmonic. Thestipulationthat V =0ismadebecausepressure,notvoltage,istheeortvariab lefor thiselement.SolvingEquation 5{11 whilemakinguseofEquation 5{6 yields M ad = a 2 R 0 A w ( r ) j 2V =0 2 rdr 8j 2V =0 ; (5{12) where A [kg = m 2 ]istheaerialdensityofthediaphragm, A ( r )= Z z t z b ( r;z ) dz; (5{13) and z b and z t arethetopandbottom z -coordinatesofthediaphragm,respectively. MakinguseofEquation 5{6 M ad isequivalentlywrittenas[ 20 113 ] M ad = 2 C 2 ad a 2 Z 0 A w ( r ) j V =0 p 2 rdr; (5{14) whichthoughawkwardlysuggesting M ad isdirectlydependentoncomplianceand pressure,isconvenientforperformingcalculations. 79

PAGE 80

Finally,thelumpedresistance R ad isrelatedtotheclassicaldampingcoecient as [ 34 ] R ad =2 r M ad C ad : (5{15) Thedampingcoecientisusuallydeterminedexperimentall ybecauseofthedicultyof bothpredictingwhatdampingmechanismsareimportantandm odelingtheireects.In thisstudy, =0 : 03|representativeofanobservedvalueforasimilardevice [ 119 ]|is assumed. Withthelumpedelementsassociatedwiththediaphragmden edandtheneedfor predictionof w ( r )motivated,Section 5.2.2 detailsthemodelimplementationforthis study.First,however,theremaininglumpedelementsarede ned. 5.2.1.3Acousticelements Inthissection,lumpedelementscapturingtheimpactofthe presenceofruidin andaroundthemicrophonearedened.Theseincludeimpedan cesassociatedwithruid externaltothemicrophone, R ad;rad and M ad;rad ,ruidwithinthebackcavity, C ac and M ac andruidinthevent, R av .Ineachoftheseelements,thegasdensity 0 andisentropic speedofsound c 0 appearregularly,inadditiontotheacousticwavenumber k = !=c 0 Theproduct 0 c 0 isknownasthecharacteristicimpedanceoftheruidmedium, Z 0 Thediaphragmre-radiatessoundtothesurroundingruidasi tvibrates,andthis interactionwiththeruidimpactsthediaphragmdynamics.T heso-calledRayleigh integral[ 28 ]governstherelationshipbetweenthevibrationsofa\pist on"inarigidbae (representativeofthemicrophonediaphragm)andtheradia tedpressureeld.Itmaybe solvednumericallyforanarbitrarypistonmodalvibration ,butintheinterestofsimplicity andcomputationaleciency,theclassicalsolutionforari gidcircularpistonmoving withuniformvelocityisleveragedtopredicttheeectofth eruidonthediaphragm. Thediaphragmandarigidcircularpistonasradiatorsaresi milarincharacter,withthe fundamentaldierencebeingthatthepistonmovesasarigid bodywithasinglevelocity, whilethediaphragmdoesnot.Theacousticradiationimpeda nceofarigidcircularpiston 80

PAGE 81

withanundeterminedeectiveradius|notequaltotheradiu softhecirculardiaphragm |is[ 28 ] Z = Z 0 a 2eff 1 2 J 1 (2 ka eff ) 2 ka eff + j 2 K 1 (2 ka eff ) 2 ka eff ; (5{16) where J 1 istherst-orderBesselfunctionoftherstkindand K 1 istherst-orderStruve function. Tondtheeectiveradius a eff ,thevolumevelocityofthediaphragm, q = j! 8j V =0 ; (5{17) isequatedtothevolumevelocityofanequivalentcircularp istonmovingwiththecenter velocityofthediaphragm, q = j!w (0) j V =0 a 2eff : (5{18) Solvingfor a eff thenyields a eff = s 1 8j V =0 w (0) j V =0 ; (5{19) fromwhichtheeectivearea A eff = a 2eff mayalsobecalculated.Foragivendiaphragm geometry,acircularpistonofradius a eff andcorrespondingarea A eff thereforeproduces thesamevolumedisplacementandshouldhavethesameapprox imateradiativeproperties. Inthelow-frequencyapproximation( ka eff 1),obtainedbyperformingaMaclaurin seriesexpansionofEquation 5{16 anddroppingtermsoforder( ka eff ) 3 andhigher,the radiationimpedanceofairreducestoamass[ 28 ], M a;rad = 8 0 3 2 a eff ; (5{20) andaresistance, R a;rad = 0 2 2 c 0 : (5{21) Thesequantitiescapturetheeectsofairparticlesmoving togetherwiththediaphragm andthelossofacousticenergyintothesurroundingmedium. Thislow-frequency approximationofEquation 5{16 isvalidtowithin5%uptoapproximately ka =0 : 43. 81

PAGE 82

Theruidinthecavitybehindthediaphragmalsoimpactsitsd ynamics.Thecavity impedanceisderivedfromtheclassicalacousticssolution fortheacousticimpedanceofa rigid-walledtubewitharigidtermination[ 28 ], Z ac = j Z 0 A c cot( kd c ) ; (5{22) where A c isthecavityareaand d c isthecavitydepth.Whentheacousticwavelength ismuchlessthanthelengthofthetube,atruncatedseriesex pansionyieldsanacoustic compliance, C ac = 8 c 0 c 20 ; (5{23) where 8 c = d c A c isthecavityvolume,andanacousticmass, M ac = 0 8 c 3 A 2c : (5{24) For kd c 0 : 3,thecontributionof M ac tothecavityimpedanceislessthan3%ofthe contributionof C ac anditmaybeneglected.However,itisretainedbecauseitsin clusion addslittleadditionalcomplicationtothemodel.TheFBAR-v ariantprocessmakesuseof siliconwafersthatare500 mthickfollowingthechemical-mechanicalpolishstep,yie lding d c =500 m.Asanexample,forthiscavityat20kHz, kd c =0 : 18. Finally,therowthroughtheventchannelismodeledasfully developed,pressure drivenrowbetweentwoparallelsurfaces[ 43 120 ].Thecanonicalventstructurehasa length L v andarectangularcrosssectionofheight h v andwidth b v ,with b v h v .This thinchannelrunsfromthecavityunderneaththediaphragma ndemergestopsidethrough acircularholeinthelmstack.Theimpedanceoftheventiss implytheresistance,[ 43 ] R av = 12 L v b v h 3v ; (5{25) where istheviscosityoftheruid.FortheFBAR-variantfabricatio nprocess, L v = 50 m, h v =2 m,and b v =25 m. 82

PAGE 83

5.2.1.4Electricalelements Electricalelementsfoundinthelumpedelementmodelrepre sentthecapacitance ofthepiezoelectriclmstack( C eb ),aparasiticcapacitanceassociatedwithelectrode overhangpastthediaphragm( C eo ),theresistanceofthepiezoelectric( R ep ),and theresistanceassociatedwithleads( R es ).Theelectricalblockedcapacitance C eb was addressedinSection 5.2.1.1 aspartofthetransductionmodel. Theelectrodesandpiezoelectricoverhangslightlypastth efreediaphragmregion, actingasaparasiticcapacitance.Usingtheparallelplatec apacitanceformulafor predictivepurposes,theresultis C eo = A o h p ; (5{26) wheretheelectrodeoverhangarea A o = ( a 23 a 22 ). Apotentialdierencegeneratedacrossapiezoelectriccan notremainindenitelydue tochargeleakageacrossit.Thiseectisaccountedforinth elumpedelementmodelusing thepiezoelectriclossresistance, R ep .Itisfoundviathewell-knownrelationshipbetween resistanceandthematerialpropertyresistivity( p forthepiezoelectric)[ 97 ], R ep = p h p A e : (5{27) Evenintheabsenceofavent,thepresenceof R ep precludesamicrophoneoutputvoltage v o whenastaticpressureactsonthediaphragm. Theseriesresistance R es representsleadsandwirebondsconnectingthemicrophone toexternalcircuitry.Itwasestimatedfromimpedancemeas urementsofearlyprototype devices(withtypicalleadgeometriesfortheFBAR-variantp rocess)tobeapproximately 4kn.Theimpactofthiselementisgenerallynegligiblebuti tisincludedforcompleteness. 5.2.2DiaphragmMechanicalModel AsestablishedinSection 5.2.1.2 ,displacementpredictionsforapiezoelectric microphonediaphragmunderpressureandvoltageloadingar eneededinordertocalculate severallumpedelements,including C ad and M ad ,inadditiontotheeectivepiezoelectric 83

PAGE 84

coecient d a .Inthissection,thepriorartformodelingofsuchstructur esissummarized andthemodelimplementationusedinthisstudyisdescribed .Themajorityofmodel development,however,isfoundinAppendix A Themicrophonediaphragmismadeupofcompositelayers,and thusitsharessome commoncharacteristicswithmacroscalelaminatedcomposi tes.Modelingofcomposite laminatesiswell-developed,andanappropriatetheoryfor modelingofhighaspect-ratio, thin-lmcompositessuchasthemicrophonediaphragmisthe classicallaminatedplate theory(CLPT)[ 121 122 ].ThesimpliedgeometricalrepresentationofFigure 5-5 shows thediaphragmasacircularlaminatedcompositeplatewitha nintegratedpiezoelectric layerandstepdiscontinuityat r = a 1 .Incommonvernacular,thediaphragmofFigure 5-5 isof\unimorph" 1 geometry,meaningthereisasinglepiezoelectriclayer[ 123 ].Two commonunimorphcirculardiaphragmcongurationsweresho wninFigure 4-3 n r Figure5-5.Laminatedcompositeplaterepresentationofth ethin-lmdiaphragmunder pressureandvoltageloading. Theliteratureonpiezoelectriccompositeplates,evennar rowedtounimorphsof circulargeometry,isextensive.Althoughunimorphsmaycon tainpiezoelectricand 1 Similarly,theterm\bimorph"referstoastructurewithtwo piezoelectriclayers,andsoon[ 123 ]. 84

PAGE 85

structurallayersofequalradii,thosewithradiallynonun iformlayercompositionsas inFigure 5-5 areofthemostinterestinthisstudy.Analyticalinvestigat ionsofthis geometryappeartohaverootsintheRussianliteraturewith AntonyakandVassergiser (1982)[ 124 ],whopresentedastaticmodelofasimply-supportedtwo-la yercircular unimorphtransducerinwhichtheradiusofthepiezoelectri clayerwaslessthanthat ofthestructurallayer.Thegoverningequationsweresolve dpiecewiseoneithersideof thestepdiscontinuity,withmatchingconditionsonmoment sanddisplacementsapplied atit.Simply-supportedboundaryconditionswereused.Aneq uivalentelectroacoustic circuitwasusedtoexaminethevariationofsensitivityand electromechanicalcoupling coecientwithchangesinthicknessandradiusratios.Evse ichiketal.[ 125 ]performed asimilarstudyin1991,butsolvedthetimeharmonicgoverni ngequations.Theimpacts ofclamped,free,andhingedboundaryconditionswerediscu ssed.ChangandDu(2001) [ 126 ]investigatedessentiallythesameproblembutalsoformal lydeterminedoptimized congurationsforlargeelectromechanicalcouplingfacto randstaticderection. Astaticmodelofaclampedpiezoelectriccircularplatewit hradiallynonuniform layerstogetherwithatwo-portelectroacousticequivalen tcircuitrepresentationwas developedinaseriesofconferenceandjournalpapersfromt heInterdisciplinary MicrosystemsGroupattheUniversityofFlorida[ 113 127 128 ]intheyears2002{2006. InPrasadetal.(2002,2006)[ 113 128 ],acompact,closed-formsolutionwasoeredfor theproblemofaclampedcentraldiscunimorph.Layercompos itionwasgeneralizedin theprovidedsolutionviauseofthestinessmatrices A B ,and D ,thoughtheouter regionwasrestrictedtosymmetriclayups.Thetwo-portele ctroacousticequivalentcircuit developedhadthesameformutilizedbyAntonyakandVassergi ser[ 124 ].Themodel wasvalidatedexperimentallyandwithniteelementanalys is[ 113 ].Anotherversionof themodelpresentedinWangetal.(2002)[ 127 ]includedin-planeresidualstressasan input,motivatedbyitssignicantimpactinmicrofabricat edstructures.Validationagainst nonlinearniteelementanalysiswasprovided. 85

PAGE 86

In2003,LiandChen[ 129 ]foundthederectionproleofasimply-supported unimorphwithinner-discactuatorandbondlayer.Later,se veralpapersfromagroup attheUniversityofPittsburghaddressedcircularpiezoele ctricunimorphs.In2005, Kimet.al.[ 130 ]presentedmodelsforacircularunimorphwithuniformpiez oelectric andstructural-layerthicknessesbuttwodierentelectro decongurations.Intherst conguration,theelectrodesfullycoveredthepiezoelect riclayer;inthesecond,the electrodesweresegmentedintoinnerandouterregionswith reversedpolarization.In 2006,Moetal.[ 131 ]investigatedatwolayerunimorphwithclamped,simplysup ported, andelasticedgeconditions.Bothradiallyuniformandnonu niformlayercompositions werediscussed.Theauthorsfocusedonthevariationofdere ctionproleswithanumber ofparameters,includingthickness,radius,andelasticmo dulusratiosofthepiezoelectric tostructurallayer.Experimentalvericationwasalsogiv en.Thenextyear,thesame authorsmodiedthemodelwithasegmentedelectrodecongu ration[ 130 ]toinclude elasticallyrestrainededgeconditions.Experimentalver icationofthemodelwasprovided [ 132 ]. DeshpandeandSaggere(2007)[ 133 ]providedageneralizedmodelforpredictionof thedisplacementsofacircularpiezoelectricplatewithas ingleradialdiscontinuity.The easewithwhicharibitrarylayercongurationscouldbeinc ludedviaavoidanceofearly simplicationstothe A B ,and D stinessmatriceswasemphasized.Finiteelementand experimentalvericationweregivenforarangeofvoltagea ndpressureloadings.Papilaet al.(2008)[ 134 ]providedasimilarlygeneralformulationforacircularpi ezoelectricplate withtworadialdiscontinuities. Otherpapersacknowledgedfortheircontributiontocompos itepiezoelectricsensors andactuators|notjustforcirculargeometries|includeth oseofLee[ 135 136 ]and Reddy[ 137 ].Eachcontainsdiscussionofsensorandactuatorformsfor thegoverning piezoelectricplateequations. 86

PAGE 87

In-planeresidualstressesarenearlyomnipresentbyprodu ctsofmicrofabrication processesandoftendominatethebehaviorofthin-lmmecha nicalstructures[ 20 ]. Predictingtheimpactofstressondiaphragmperformanceis thusextremelyimportant, andonlythemodelofWangetal.[ 127 ]soughttoincludetheseeects. Asaresult,thisstudyutilizesextendedversionsofthatmod el,includingbothlinear andnonlinearformulations.Thelinearmodelwasextendedt oincludearbitrarylm stacks.Thenonlinearversionofthemodelwasbasedonthevo nKarmanplatetheoryand wasdevelopedtoassessthetransitionfromlineartononlin earresponseofthemicrophone diaphragm.Inbothlinearandnonlinearcases,residualstr essesaretakentobeknown inputsforthemechanicalmodel.Theirpresencegivesriset oastatictransversederection evenintheabsenceofanappliedpressureorvoltage,asshow ninFigure 5-6A .Itisthe incrementalderectionaboutthisstaticprole|duetoappl icationofpressureorvoltage |thatcharacterizestheresponseofthemicrophone.Increm entalderectiondueto pressureloadingisisillustratedinFigure 5-6B .Mathematically,theinitial,incremental, andtotalderectionarerelatedas w inc ( r )= w tot ( r ) w ini ( r ) : (5{28) Here,theinitialderection, w ini (= w j V & p =0 ),ispurelyduetoresidualstresses;the incrementalderection, w inc isduetopressureorvoltageloading;andthetotalderectio n, w tot (= w j V j p 6 =0 ),isduetobothresidualstressandexternalloading.InSec tion 5.2.1.2 ,the diaphragmderection w alwaysreferstotheincrementaldisplacement, w inc Themodelwasderivedusingthesametwo-domainsolutionmet hodologythatis prevalentintheliterature,withthegoverningequationso ftheCLPTsolvedoneither sideoftheradialdiscontinuityandmatchedviaboundaryco nditionsattheinterface. Figure 5-7 depictstheideaoftheboundarymatchingprocess,whereat r = a 1 the displacements,inadditiontotheforceandmomentresultan ts N r and M r associated witheachdomain(0
PAGE 88

wini(r) A p wini(r) winc(r) B Figure5-6.Derectionofaradiallynon-uniformcompositep latewithresidualstress.A) Initialderection, w ini ( r ).B)Incrementalderectionduetopressureloading, w inc ( r ). piezoelectriciscommunicatedviaequivalentpiezoelectr icforceandmomentresultants, N p and M p ,appearingintheseinterfacematchingconditions.Loadin goftheplate includesbothauniformpressureandlayer-wisevoltagedi erencesasoriginallydepicted inFigure 5-5 Adetailedderivationofthelinearandnonlinearpiezoelec triccompositeplatemodels arefoundinAppendix A .Solutionmethodologiesarealsogiveninbothcases;theli near modelissolvedusingasemi-analyticalapproachwherecons tantsofintegrationarefound numericallyratherthanexplicitly,whilethenonlinearmo delisformulatedforsolutionvia aboundaryvalueproblemsolverpackage,forexample bvp4c inMATLAB[ 138 ]. 5.2.3FrequencyResponse Withalloftheindividuallumpedelementsdened,theequiv alentcircuitmodelof Figure 5-3 iscomplete.Usingstandardcircuitanalysistechniques,th ismodelmaybe 88

PAGE 89

nnr nnnr n nnr M (1) r N (1) r M (2) r N (2) r a 1 a 2 Figure5-7.Boundaryconditionsappliedtoaradiallynon-u niformpiezoelectriccomposite plate. probedtodeterminethemicrophonefrequencyresponsefunc tion, H m ( f ).Simplication ofthemicrophonefrequencyresponsefunctionenablesadir ectestimateoftherat-band sensitivity, S .Withminoralterations,theactuatorsensitivitymayalso becalculated. Thesequantitiesareinvestigatedinturninthefollowings ub-sections. First,however,collectingimpedancestogetherfacilitat esthecircuitanalysis.Dening Z ac = j!M ac + 1 j!C ac ; (5{29) Z ad = j! ( M ad + M ad;rad )+ R ad + R ad;rad + 1 j!C ad ; (5{30) and Z ep = R ep 1+ j!R ep ( C eb + C eo ) ; (5{31) condensesthemathsubstantially.Here, Z ac issimplytheseriescombinationofthecavity complianceandmass, Z ad collectsallofthediaphragmandradiationimpedancesin series,and Z ep capturestheparallelcombinationof C eb C ea ,and R ep .Makinguseofthese denitions,thecondensedequivalentcircuitforthemicro phonelumpedelementmodelin Figure 5-8 results. 89

PAGE 90

R av Z ac Z ad p a :1 Z ep R es + v o Figure5-8.Lumpedelementmodelwithcollectedimpedances 5.2.3.1Sensor UtilizingFigure 5-8 ,theopen-circuitoutputvoltage v o isrelatedtotheinputpressure p viacircuitanalysisas H m;oc ( f )= v o p = 1 = a 1+ Z ac R av 1+ Z ad Z ep 2a + Z ac Z ep 2a ; (5{32) whichistheopen-circuitfrequencyresponsefunctionfort hemicrophone.Figure 5-9 showsthetypicalmagnitudeassociatedwitheachoftheimpe danceratiosappearingin Equation 5{32 2 Thecut-onbehaviorisdictatedbythecavity/ventcombinat ionofthe Z ac =R av term,whichisonlygreaterthanorcomparabletounityatlow frequencies.Over theremainingfrequencyrange,the Z ad =Z ep termdominatesallothers.Thecapacitive componentsofEquation 5{32 dominateintheratband.Eliminatingtheinductiveand resistiveimpedancecomponentsyieldsanestimationofthe rat-bandsensitivity, S oc = a 2a + ( C eb + C eo ) C ad 1+ C ad C ac : (5{33) Thecavityisideallyfarmorecompliantthanthediaphragms uchthatitdoesnothave anappreciableeectonthemicrophonesensitivity,asinFi gure 5-9 .Akeysimplifying 2 RefertoTable 5-1 fortheexampledevicegeometryandAppendix D formaterialproperties. 90

PAGE 91

10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 8 10 4 10 0 10 4 10 8 Frequency[Hz]Magnitude Denom.ofEqn. 5{32 Z ac =R av Z ad =Z ep 2a Z ac =Z ep 2a Figure5-9.Impedanceratiosappearingintheopencircuitf requencyresponseexpression, Equation 5{32 assumptionisthus C ad C ac 1 : (5{34) EmployingthisapproximationandmakinguseofEquations 5{3 and 5{9 ,Equation 5{35 is furthersimpliedto S oc d a C ef + C eo : (5{35) Thisextremelysimpleexpressionshowsthattheopencircui tmicrophonesensitivityin theratbandis|togoodapproximation{onlyafunctionofthe eectivepiezoelectric coecient,theparallelplatecapacitanceofthepiezoelec triclmstack,andthesmall parasiticcapacitanceassociatedwithelectrodeoverhang beyondthediaphragm.Ideally, C eo C ef issatisedand C eo doesnotplayarole,either.Oneperhapssurprisingfeature ofEquation 5{35 isthatthediaphragmcompliance, C ad ,doesnotappearexplicitly; however,themechanicalbehaviorofthediaphragmisstillv erymuchcapturedwithinthe eectivepiezoelectriccoecient, d a Acomparisonoftheexpressionsforsensitivity,Equations 5{33 and 5{35 ,withthe overallopen-circuitfrequencyresponsefunction,Equati on 5{32 ,isshowninFigure 5-10 3 Agreementisexcellentintheratband,withEquation 5{35 slightlyover-predictingthe 3 RefertoTable 5-1 fortheexampledevicegeometry. 91

PAGE 92

ratbandsensitivityontheorderofafewpercentduetonegle ctofcavitycompliance.As plotted,Equations 5{33 and 5{35 falldirectlyontopofeachother. 10 1 10 2 10 3 10 4 10 5 10 6 100 80 Frequency[Hz]Magnitude[dBre1V = Pa] Equation 5{32 Equation 5{33 Equation 5{35 Figure5-10.Comparisonofopen-circuitsensitivityexpre ssionsandthefullopen-circuit frequencyresponseofthelumped-elementmodel. 5.2.3.2Actuator Becauseitisfareasiertoapplyaknownvoltagetothephysic alpiezoelectric microphonethanaknownpressure,interrogatingthemicrop honeinitsreciprocal resonatormodecanprovideusefulinformation.Itisinstru ctive,then,toconsiderin themodelingstagehowtheactuatorresponsecomparestothe sensorresponse.The equivalentpiezoelectricactuatorhasbeenaddressedprev iously[ 139 ],andtheassociated lumpedelementmodel,withvoltagesourceaddedontheelect ricalside,isshownin Figure 5-11 .Interrogatingthismodel,thevolumedisplacement 8 (= q=j! )throughthe diaphragmlegofthecircuitperappliedvoltage v is H a ( f )= 8 v = a =j! Z ep 2a 1+ R es Z ep Z ep 2a + Z ad + Z ac Z av Z ac + Z av : (5{36) Althoughactuatorsaretypicallyoperatedatresonance,pro bingtheratbandactuator responseisusefulinthecontextofevaluatingdevicestose rveasmicrophones.Inthe 92

PAGE 93

Z ac R av Z ad q a :1 Z ep R es v Figure5-11.Lumpedelementmodelofthepiezoelectricmicr ophoneasanactuator. ratband,capacitiveelementscontinuetodominate,giving S a = 8 v = a C ad 1+ C ad C ac : (5{37) AgainundertheassumptionofEquation 5{34 andemployingEquation 5{9 ,theendresult issimply S a d a : (5{38) Comparingthisexpressiontothatfortheopencircuitsensi tivity,Equation 5{32 ,onesees thattheyarebothproportionalto d a .Thisimpliesthattheactuatorresponseprovides somemeasureoftheexpectedsensorresponse.Thisideaisre visitedinChapter 8 inthe contextofmicrophoneselection.5.2.4Electricalimpedance Themicrophone'selectricalimpedancecanimpactcircuitd esignchoicesandthus havingapredictionisimportant.Interrogatingthecircui tinFigure 5-11 ,theequivalent electricalimpedanceseenbythevoltagesourceis Z eq =( Z acv + Z ad ) k Z ep + R es ; (5{39) orcollectingterms[ 140 ], Z eq = R es + Z ep 1 1+ ; (5{40) 93

PAGE 94

where = Z ep 2a Z acv + Z ad : (5{41) Assumingthecavityisverycompliant(Equation 5{34 ), Z eq intheratbandreducesto Z eq = R es + R ep 1+ j!R ep ( C ef + C eo ) : (5{42) 5.2.5Validation Themodelspresentedinthischapter|boththediaphragmmod elaloneandthe completelumpedelementmodel|werevalidatedusingtheni teelementmethod, acomputationaltechniqueusedtosolveboundaryvaluespro blems.Finiteelement modelscangenerallycapturemoreoftheunderlyingphysics ofaproblemthananalytical models,whichoftenrequiresignicantsimplifyingassump tionstobemadetractable.The improveddelityofniteelementmodelingcomeswiththeco stofincreasedcomputation timeassociatedwithsolvinglargesystemsofequations. TheniteelementmodelwascreatedandsimulatedinABAQUSv6. 8-2usingthe basicgeometryofFigure 5-5 andtheassociatedgeometricdimensionsofTable 5-1 ,shown toscaleinFigure 5-12 .MaterialpropertiesarefoundinAppendix D exceptthefullAlN stinessandpiezoelectricmatrices,whichweredrawnfrom Tsubouchietal.(1985)[ 141 ]. BoundaryconditionsarepicturedinFigure 5-12B andincludearollerconditiononthe diaphragmedgeat r = a 3 (toallowfreeexpansionofthelminthethickness-directi on) andfullyclampedconditionsalongthebottomdiaphragmedg e, a 2 r a 3 .Asecond modelinAppendix A.8 comparestheuseofthisboundaryconditionwithoneincludi ng thesiliconsubstrate.Theelectricalboundaryconditionf orthebottompiezoelectric surfacewaszeroelectricpotential,andanequationconstr aintproducedanequipotential topsurfacetosimulatethetopelectrode.Remaining(free) surfacesweresubjectto defaultnaturalboundaryconditionsofzerotraction[ 142 ]andzeronormalcomponent ofelectricruxdensity[ 97 ],respectively.Nodampingwasappliedinthemodel.The geometrywasmeshedwith52kbilinearaxisymmetriccontinu umelementsapproximately 94

PAGE 95

0 : 125 monasideoftypesCAX4E 4 forthepiezoelectriclayerandCAX4 5 otherwise.A close-upviewofthemeshisshowninFigure 5-12C Table5-1.Geometricdimensionsofanexampledevice. y DimensionSymbolValue[ m] ThicknessesPassivation h pass 0.14 TopMoElectrode h e;top 0.15 PiezoelectricLayer(AlN) h p 1 BottomMoElectrode h e;bot 0.6 StructuralLayer h struct 2 RadiiInner a 1 306 Outer a 2 345 Outerwithoverhang a 3 348 y DesignD(seeChapter 6 ) Ineachmodelrun,thestructurewasrstallowedtoequilibr atefromtheresidual stress,appliedviathe*INITIALCONDITIONScommand,inastatic generalstepwith geometricnonlinearityincluded(NLGEOMon).Afterward,var iousstepswereperformed dependingonthenatureofthevalidationexercise.Eachoft heseisdiscussedinthe followingsubsections.5.2.5.1Diaphragmmodelvalidation Thediaphragmmodelisrequiredtoprovideaccuratepredict ionsof w ( r )fromwhich elementssuchas C ad M ad ,and d a arecalculatedforinputtothelumpedelementmodel. Simulationswerecompletedforrangesofbothpressureandv oltageloadingtoassessthe accuracyofthemodel. Fromthenonlinearlyderectedbasestate,arangeofpressur eandvoltageinputswere sweptinageometricallynonlinearstaticgeneralstep.Pre ssurewassimplyappliedasa uniformloadoverthetopofthediaphragm,withvaluesrangi ngfrom100dBtobeyond 4 CAX4E:4-nodebilinearaxisymmetriccontinuumelementwit helectricpotentialdegreeoffreedom 5 CAX4:4-nodebilinearaxisymmetriccontinuumelement 95

PAGE 96

@ @R Piezoelectric lmstack Axisofsymmetry( r =0) A ClampedBC RollerBC H Hj Electrode surface Pressureload B C Figure5-12.Finiteelementmodelforvalidationexercise. A)Geometrytoscale.B) Zoomed-inviewofannularpiezoelectriclmstackandbound aryconditions. B)Zoomed-inviewofmeshedannularpiezoelectriclmstack 180dB.Theresultsofthesimulationarecomparedtotheline arandnonlineardiaphragm modelsinFigure 5-13 intermsofincrementalcenterderection( w inc (0)).Agreement withthenonlinearmodelisexcellentovertheentirerangeo finputs,whilethereissome deviationfromthelinearmodel,asexpected,atveryhighso undpressurelevels.The relativeerrorbetweenthetwomodelsandtheniteelementm odelisalsoshownin Figure 5-14 ,witherrorverynearlyzeroouttopressurelevelsapproach ing170dBforthe linearmodel. 100120140160180 10 3 10 1 10 1 Pressure[dBre20 Pa]w inc (0)[ m] LinearModel NonlinearModel FEA Figure5-13.AnalyticalandFEApredictionsof w inc (0)(pressureloadingcase). 96

PAGE 97

100120140160180 0 20 40 60 Pressure[dBre20 Pa]RelativeErrorin w inc (0)[%] LinearModel NonlinearModel Figure5-14.RelativeerrorbetweenanalyticalandFEApred ictionsof w inc (0)(pressure loadingcase). Inasecondmodelrun,variousvaluesofappliedvoltagewere alsosweptinastatic generalstep.Anelectricpotentialwasappliedtoareferenc enodeandtheequation constraintenforcedanequipotentialtoppiezoelectricsu rface.Theresultsfromthenite elementandanalyticalmodelsarecomparedinFigure 5-15 ,whichshowsthatallthree modelsagreeclosely(from3%to7%relativeerror). 012345 0 2 4 6 8 Voltage[V]w inc (0)[nm] LinearModel NonlinearModel FEA Figure5-15.AnalyticalandFEApredictionsof w inc (0)(voltageloadingcase). 5.2.5.2Lumpedelementmodelvalidation Withthediaphragmmodelindependentlyveried,thefreque ncyresponsefunction ofthemicrophone|sanssomephysics|wasfoundvianiteele mentmodelingand comparedtothelumpedelementmodelprediction.Properlyc apturingtheacoustics wouldrequireafullthree-dimensionalmodel(fortheventg eometry)andsimulation 97

PAGE 98

offreespaceonthediaphragmexterior.Withthevalidityof theacousticelements (particularlythecavityandradiationimpedances)well-e stablished[ 28 35 36 ],thenite elementmodelvalidationwasperformedpurelytoprovetheq ualityofpredictionsfor theelectromechanicalelements.Essentially,then,thise xercisefurthervalidatedthe piezocompositeplatemodelandalsothelumpedelementmode lingapproachforpredicting microphonediaphragmdynamics. Asteady-statedynamics(direct)stepwasusedtondtheste ady-stateharmonic responseofthediaphragmtopressureloading.Thisstepwas alinearperturbation procedurethatcalculatedthediaphragmresponsedirectly fromthemass,damping,and stinessmatricesofthesystem[ 143 ].Theresponsewasevaluatedat150logarithmically spacedfrequencypointsfrom0 : 01Hzto350kHz.TheresultsareshowninFigure 5-16 Withtheacousticsnotincludedintheniteelementmodel,t hecut-onwasnotpredicted, buttheratbandresponsesagreedtowithin0 : 05dB(0 : 6%)andtheresonantfrequencies werealsowell-matched.Solutionofthissteptookontheord erof10minutestosolve usingtheniteelementmodelcomparedtosecondsusingthel umpedelementmodel. 10 0 10 1 10 2 10 3 10 4 10 5 10 6 140 120 100 80 60 Frequency[Hz]j H m;oc ( f ) j [dBre1V = Pa] LEM(Equation 5{32 ) FEA Figure5-16.LumpedelementmodelandFEApredictionsoffre quencyresponsefunction. 5.3InterfaceCircuitry InSection 5.2 ,anequivalentcircuitmodeloftheentirepiezoelectricmi crophone wasusedtopredictitsopencircuitsensitivity.Unfortunat ely,theactofmeasuringthe 98

PAGE 99

outputvoltageofthemicrophonecircuitnecessarilyloads it,andthechangeinoutput voltagecanbesubstantialiftheloadimpedanceisnotsigni cantlyhigherthanthesource impedance[ 144 ].Alow-capacitance(singlepF)piezoelectricmicrophone caneasilyhave electricalimpedancecomparabletothetypicalinputimped anceofadataacquisition system(DAQ)(1Mn-10Gn)intheaudiofrequencyrange.Asares ult,themicrophone byitselfcannotbeconnecteddirectlytoaDAQwithoutexper iencinganapparentchange insensitivity.Avarietyofcircuitarchitecturesexistfo rtransformingtheapparentsource impedanceofthemicrophone.Twosucharchitectures|avolt ageamplierandcharge amplier|areaddressedinSections 5.3.1 and 5.3.2 Unfortunately,connectinganidealoperationalamplierco ngurationtothe microphonedoesnotcompletethestory.Wirebondsandtrace srunningfromthephysical microphonetotheamplierintroduceparasiticcapacitanc e.Internaltransistorsatthe amplierinputalsocontributeaniteinputcapacitance[ 145 ].Forstabilitypurposes, theamplierrequiresagroundpathfordccurrentrow.Theim pactoftheseadditional impedancesareaddressedforboththevoltageandchargeamp liercases. 5.3.1VoltageAmplier Onewaytoalleviatetheproblemofsourceloadingistouseav oltageamplier,which producesanoutputvoltageproportionaltoinputvoltage[ 146 ]whilealsoprovidingalow outputimpedancefortheentiremicrophone/ampliersyste m.Avoltageamplierwith unitygainisknownasa buer or voltage-follower .Themodeloftheoperationalamplier accountingforparasiticcapacitance C ep ,amplierinputcapacitance C ea ,andamplier biasresistance R ea isshowninFigure 5-17 .Fromthismodel,thenewimpedance, Z ea = R ea 1+ j!R ea ( C ea + C ep ) ; (5{43) isdened.However,earlytestsofprototypepiezoelectricm icrophonesindicatedthey couldbeoperatedinastablemannerwiththedielectricloss ofthepiezoelectricservingas thedcgroundpathinplaceofabiasresistor.Asaresult, R ea isnotutilizedinthisstudy 99

PAGE 100

(making Z ea purelycapacitive),thoughitiscarriedthroughforcomple teness.Thevoltage ampliercircuitisshownconnectedtothemicrophonecircu itinFigure 5-18 + v R ea C ea C ep v + v o A + v Z ea v + v o B Figure5-17.Non-idealoperationalampliermodel.A)Operat ionalamplierwith parasiticcapacitancesandbiasresistor.B)Operationala mplier representationwithequivalentimpedance. Beforeevenbeginningacircuitanalysis,onecanimmediate lyintuitthattheparallel combinationof Z ep and Z ea (assumingherethat R es isnegligibleincomparison)alters thelowfrequencyRCcutooriginallyassociatedwithonly Z ep .Thepresenceofabias resistortendstoraisethebreakfrequency,whiletheadded capacitancetendstolowerit. p R av Z ac Z ad a :1 Z ep R es + v o + Z ea Figure5-18.Lumpedelementmodelwithvoltageamplier. 100

PAGE 101

AnalyzingthecircuitofFigure 5-18 ,thefrequencyresponsefunctionofthecomplete systemisfoundtobe H m;va ( f )= v o p = 1 = a 1+ Z ac R av 1+ Z ad Z ep 2a + Z ac Z ep 2a + 1 Z ea 2a 1+ Z ac R av Z ad + R es 2a 1+ Z ad Z ep 2a + Z ac 1+ R es Z ep : (5{44) Equation 5{44 isacomplicatedexpressionthatdoesnotprovidereadyinsi ght,butagain simplicationsareeasilymade.Takingcapacitiveelement sasdominantintheratband, thefrequencyresponseofthemicrophone/voltageamplier congurationis S va = a 2a + 1 C ad + 1 C ac ( C eb + C eo + C ep + C ea ) : (5{45) Again,employingtheapproximation C ac C ad andmakinguseofEquations 5{3 and 5{9 S va d a C ef + C eo + C ep + C ea ; (5{46) whichintermsoftheopen-circuitsensitivitybecomes S va = S oc C ef + C eo C et ; (5{47) where C et = C ef + C eo + C ep + C ea (5{48) isthetotalcapacitance.Therepercussionsofusingthevol tageamplierarenowclear. FromEquation 5{47 ,onecanseethattheopencircuitsensitivityisattenuated bythe factor( C ef + C eo ) =C et ,whichis always lessthanunity.Theproblemiscompoundedfor sensorswithlowcapacitance,forwhichtheparasiticcapac itancesaremorelikelytobeof similarorderto C ef ;attenuationofthesensitivityinthiscasecanbesignica nt. 101

PAGE 102

+ v + C ea C ep v v o C efb R efb A + v + Z ea v v o Z efb B Figure5-19.Non-idealchargeampliermodel.A)Operational amplierwithparasitic capacitances.B)Operationalamplierrepresentationwit hequivalent impedance. 5.3.2ChargeAmplier Chargeampliersareso-namedbecausetheyproduceanoutpu tvoltageproportional totheinputcharge[ 146 ].Theyarepopularampliersforcomparabletechnologiest o thepiezoelectricmicrophone,e.g.piezoelectricacceler ometers[ 146 147 ].Amodelofthe operationalamplierandthenon-idealitiesthataccompan yitisshowninFigure 5-19 ThechargeampliercircuittopologyisshowninFigure 5-20 ,wherethefeedback impedance Z efb connectedtotheinvertingterminalisaparallelcombinati onofafeedback resistorandcapacitor, R efb and C efb ,respectively.Thisimpedanceintroducesanew low-frequency RC cutothatmustbetunedtoavoidcuttingintothebandwidtho f thesensor.However,withthenon-invertingterminalservin gasadcpathtoground, theimpedance Z ea isonlycapacitive(i.e. C ep + C ea ).Performingcircuitanalysison Figure 5-20 ,themicrophonefrequencyresponsefunctionisfoundtobe H m;ca ( f )= Z efb a Z ad Z ac R es 1 Z acv + 1 Z ad 2a Z ad + 1 R es + 1 Z ep 2a ; (5{49) 102

PAGE 103

p R av Z ac Z ad a :1 Z ep R es + Z efb v o Z ea Figure5-20.Lumpedelementmodelwithchargeamplier.where 1 Z acv = 1 Z ac + 1 R av : (5{50) Intheratband,Equation 5{49 simplybecomes S ca = 2a =C efb 1 C ad + 1 C ac : (5{51) Assumingagainthat C ac C ad S ca d a C efb ; (5{52) whichcanberewrittenintermsofopencircuitsensitivitya s S ca = S oc C ef + C eo C efb : (5{53) Equation 5{53 revealsthatthechargeampliergainfactoristheratiooft heelectrical freecapacitancetothefeedbackcapacitanceandthattheph aseisshifted180 .The choiceof C efb |sometimescalleda\rangecapacitor"[ 148 ]|grantsadesignerthe latitudetotunethesensitivityoftheentiremicrophone/a mpliersystem.Inaddition, parasiticcapacitancesplaynorolebecausetheyarevirtua llygrounded[ 147 ]. 103

PAGE 104

5.3.3NoiseModels Inthissection,noisemodelsaredevelopedforboththevolt ageandchargeamplier circuittopologies.Ultimately,thegoalofthenoisemodels istopredicttheoutputnoise PSDassociatedwiththemicrophone/circuitrycombination .Theminimumdetectable pressure(MDP)iscalculablefromtheresultviaEquation 2{11 or 2{12 NoisehasbeenpreviouslydiscussedinSection 2.3.2 .Intheelectricaldomain,thermal noiseisproportionaltotheresistanceandtemperature.In termsofpowerspectral density,thenoisefromanelectricalresistor R e isgivenas[ 43 ] S v R e =4 k B TR e (5{54) or S i R e = 4 k B T R e (5{55) inunitsof[V 2 = Hz]and[A 2 = Hz],respectively.Thesuperscripts v and i denotewhether S R e denesasourceofvoltageorcurrentnoise.Similarly,inth eacousticdomain,the noisecontributionofadissipativeelementintermsofpowe rspectraldensityis S p R a =4 k B TR a (5{56) or S q R a = 4 k B T R a (5{57) inunitsof[Pa 2 = Hz]and[m 3 = s = Hz],respectively.Thesuperscripts p and q indicate whether S R a denesasourceofnoiseintermsofpressureorvolumeveloci ty,respectively. Tondtheoutputnoiseofthecircuit,allsourcesarerstre movedandnoisesources, denedbyEquations 5{54 or 5{55 intheelectricaldomainandEquations 5{56 or 5{57 intheacousticdomain,areaddedatthesiteofeachresistor /dissipator.Eortsources (superscript v and p )areaddedinserieswiththeresistors/dissipators,while rowsources ( i and q )areaddedinparallel[ 40 42 ].Undertheassumptionthatthenoisesources areuncorrelated,themethodofsuperpositionofsourcesis usedtondthetotalpower spectraldensityattheoutputduetoallnoisesources[ 40 43 ].Thevoltageandcharge ampliercircuitarchitecturesaretreatedinturninthefo llowingsubsections. 104

PAGE 105

5.3.3.1Noisemodelwithvoltageamplier Thenoisemodelforthemicrophone/voltageampliercombin ationisfoundin Figure 5-21 .Thesubscriptofeachsourceindicatestheresistorwithwh ichitisassociated. Thechoiceofusinganeortsourceinseriesorarowsourcein parallelwitheach resistanceispurelyoneofconvenience.Additionalnoiseso urcesareaddedforthe amplieratitsinput[ 39 ]thatrepresenttheinput-referrednoiseassociatedwithi nternal transistorsandresistors[ 149 ].Thesecharacteristicsareknownaprioribasedonthechoi ce ofamplier. Z acv S q R av S p R ad Z ad a :1 Z ep S i R ep S v R es R es S i R ea + S v o Z ea S v a S i a Figure5-21.Noisemodelforthemicrophonewithvoltageampl iercircuitry. BasedonFigure 5-21 ,theoutputPSD[V 2 = Hz]is S v o = S v o;R av + S v o;R ad + S v o;R ep + S v o;R es + S v o;R ea + S v o;a ; (5{58) i.e.thesummationoftheoutput-referrednoiseofeachindi vidualsource.Fromcircuit analysis,theindividualoutputnoisecontributionsare S v o;R av = Z acv A ( Z acv + Z ad ) 1 Z ep + 1 Z ea + R es Z ep Z ea + 2A 1+ R es Z ea 2 4 k B T R av ; (5{59) 105

PAGE 106

S v o;R ad + R a;rad = A ( Z acv + Z ad ) 1 Z ep + 1 Z ea + R es Z ep Z ea + 2A 1+ R es Z ea 2 4 k B T ( R ad + R ad;rad ) ; (5{60) S v o;R ep = ( Z eq R es ) Z ea Z eq + Z ea 2 4 k B T R ep ; (5{61) S v o;R es = Z ea Z eq + Z ea 2 4 k B TR es ; (5{62) S v o;R ea = Z eq Z ea Z eq + Z ea 2 4 k B T R ea ; (5{63) and S v o;amp = Z eq Z ea Z eq + Z ea 2 S i a + S v a : (5{64) NoteinEquation 5{64 thatthecurrentnoise S i a ismultipliedbytheparallel combinationofthemicrophoneoutputimpedance Z eq and Z ea ;lowampliercurrent noiseisthereforeveryimportantforhighimpedancedevice s.ThepiezoelectricMEMS microphone,byvirtueofitssmallexpectedcapacitance,is justsuchadevice. Figure 5-22 showsaplotofoutput-referrednoisecontributionsfromea chnoise source,with R ea neglectedasestablishedinSection 5.3.1 .Thesameexamplegeometry usedforvalidationinSection 5.2.5 isusedhere.Ampliernoisecharacteristicswere takenfromtheLinearTechnologiesLTC6240amplier,alownoiseamplierwitha 3pFinputcapacitanceandinput-referredvoltageandcurre ntnoiseroorsof7nV = p Hz and0 : 56fA = p Hz,respectively[ 44 ].Thenoiseassociatedwith R ep dominatesatlow frequenciesuntilitgiveswaytoampliercurrentnoisenea r10kHz.Thevoltagenoise contribution,inthiscase,iswellbelowthecurrentnoisec ontribution.Meanwhile,the combinedacousticnoisecontributioniscompletelyinsign icantcomparedtotheelectrical noise. Thenoiseassociatedwith R ep andtheampliercurrentnoiseareclearlydominant intheexampleofFigure 5-22 .However,noisecharacteristicsofdierentampliersare sucientlyvariablethat S v a alsowarrantscontinuedinclusioninthenoisemodel.Takin g 106

PAGE 107

10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 22 10 20 10 18 10 16 10 14 10 12 Frequency[Hz]NoisePSD[V 2 = Hz] Acoustic R ep R es S v a S i a Total Figure5-22.Output-referrednoiseroorforthemicrophone withavoltageamplier. justthenoiseassociatedwith R ep andtheampliernoiseoftheamplierasdominant,the noiseroorofthemicrophonewiththevoltageamplierarchi tecturecanbeapproximated intheratbandas S v o 1 !C et 2 S i a + 4 k B T R ep + S v a : (5{65) NotethatperFigure 5-22 ,thereissomeerrorassociatedwithEquation 5{65 inthe current/voltage-noisedominantregion,wherethesumcont ributionofothernoisesources becomessignicant.MakinguseofEquations 2{11 and 5{46 ,theminimumdetectable pressureisthen p min vuuuut Z f 2 f 1 1 j!C et 2 S i a + 4 k B T R ep + S v a C ef + C eo C et S oc 2 df; (5{66) whichaftermakinguseofEquation 5{35 ,becomes p min vuut Z f 2 f 1 S i a + 4 k B T R ep ( !d a ) 2 + S v a C 2 et d 2a # df: (5{67) SeveralimportantconclusionsemergefromEquation 5{67 .First,increasing d a decreases p min .Thisfollowsnaturallyfromknowledgeofthefactthatincr easingsensitivity decreases p min ,with S oc = d a = ( C ef + C eo )fromEquation 5{35 .Followingthislogic, theinverserelationshipbetween S oc and C ef wouldalsoseemtosuggestthatalow capacitancedevicewouldyieldalower p min .However,Equation 5{67 showsthatthisis 107

PAGE 108

Z acv S q R av S p R ad Z ad a :1 Z ep S i R ep S v R es R es + Z efb S v o S i R efb Z ea S v a S i a Figure5-23.Noisemodelforthemicrophonewithchargeampli ercircuitry. notalwaystrue; C ef playsnoroleinthenoiseroorwhenthedominantcontributor sare R ep and S i a .Whenthedominantcontributoris S v a ,alowtotalcapacitance C et isdesirable. Finally,notethatthersttermrollsoas1 =! 2 ;thisresultsinattenuationofnoisedueto R ep athighfrequencies,butcurrentnoisePSDinampliersofte nincreasesas 2 5.3.3.2Noisemodelwithchargeamplier Thenoisemodelassociatedwiththechargeamplierarchite ctureisshownin Figure 5-23 .Fromthismodel,thetotaloutputnoisePSDisthus S v o = S v o;R av + S v o;R ad + R ad;rad + S v o;R es + S v o;R ep + S v o;R efb + S v o;amp ; (5{68) wheretheindividualnoisecontributionsare S v o;R av = a Z acv Z efb ( Z ad + Z acv ) 1+ R es Z ep + 2a R es 2 4 k B T R av ; (5{69) S v o;R ad = Z efb a ( Z ad + Z acv ) 1+ R es Z ep + 2a R es 2 4 k B T ( R ad + R ad;rad ) ; (5{70) S v o;R ep = Z efb ( Z eq R es ) Z eq 2 4 k B T R ep ; (5{71) S v o;R es = Z efb Z eq 2 4 k B TR es ; (5{72) 108

PAGE 109

S v o;R efb = j Z efb j 2 4 k B T R efb ; (5{73) and S v o;amp = 1+ Z efb Z eq k Z ea 2 S v amp + j Z efb j 2 S i amp ; (5{74) whererecall Z eq istheelectricalimpedanceofthemicrophone,introducedi nSection 5.2.4 Clearly,oneimportantconclusionfromthenoisemodelisth at Z efb guresprominently ineachofEquations 5{68 to 5{74 .Inaddition,onlythevoltagenoiseisimpactedbythe presenceofparasitics. Individualnoisesourcesassociatedwiththemicrophonean dchargeamplier architectureareshowninFigure 5-24 .TheexampleamplierwastakenastheTexas InstrumentsOPA129,withanassumedinputcapacitanceof3pF andmanufacturer-supplied input-referredvoltageandcurrentnoiseroorsof15nV = p Hzand0 : 1fA = p Hz,respectively. Thefeedbackimpedanceswerechosenas C efb = C ef + C eo 8pF(unitygain)and R fb =2Gn(cut-oat10Hz).Withthisconguration,thedominantn oisesource isagainseentobe R ep atlowfrequencies,whileampliervoltagenoisedominates beyondthecornerfrequencyatapproximately10kHz.Thefeed backresistance R efb alsoshowspotentialofcontributingifchosenasalowerval ue.Again,theacousticnoiseis inconsequential. 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 20 10 18 10 16 10 14 10 12 10 10 Frequency[Hz]NoisePSD[V 2 = Hz] Acoustic R ep R es R efb S v a S i a Total Figure5-24.Output-referrednoiseroorforthemicrophone withchargeamplier. Muchlatitudeexistsintheselectionof R efb ,andtheamplier,sonoiseassociated witheachofthem,togetherwiththeever-dominantnoisesou rce R ep ,areincludedin 109

PAGE 110

simplicationstotheoverallnoiseroor.Intheratband, S v o simpliesto S v o 1+ C et C efb 2 S v a + 1 !C efb 2 S i a +4 k B T 1 R ep + 1 R efb : (5{75) Notefromthisequationthattheresistornoisecanbevieweda soriginatingfroman equivalentresistor, R ep k R efb .Theminimumdetectablepressurethenfollows,aftersome simplication,as p min vuuut Z f 2 f 1 24 S i a +4 k B T 1 R ep + 1 R efb ( !d a ) 2 + ( C efb + C et ) 2 S v a d 2a 35 df: (5{76) Again,increasing d a decreases p min .Althoughparasiticsdonotimpactthesensitivityin thechargeampliercase,Equation 5{76 showsthattheystilltendtoincrease p min when S v a isimportant.Thetermcontaining S i a and R ep k R efb rolls-oas1 =! 2 ,butagain, S i a tendstoincreaseas 2 5.3.4Selection Table 5-2 containsasummaryofthetwomainperformancecharacterist icsofthe microphoneandinterfacecircuitryaddressedinSections 5.3.1 to 5.3.3 :sensitivityand minimumdetectablepressure.Inthevoltageampliercase, thetheoreticalopen-circuit sensitivityisalwaysattenuatedbyparasiticcapacitance s,whileinthechargeamplier casethesensitivityisnotaectedbyparasitics.Inthecha rgeampliercase,adesigner haslatitudetoattenuateorgainthesensitivityviathecho iceoffeedbackcapacitoras well.Table5-2.Comparisonofvoltageandchargeampliertopolo giesforusewitha piezoelectricmicrophone. Sensitivity( S )Minimumdetectablepressure( p min ) VoltageAmplier C ef C et S oc s R f 2 f 1 S i a + 4 k B T R ep ( !d a ) 2 + S v a C 2 et d 2a df ChargeAmplier C ef C efb S oc vuut R f 2 f 1 S i a +4 k B T 1 R ep + 1 R efb ( !d a ) 2 + ( C efb + C et ) 2 S v a d 2a # df 110

PAGE 111

Comparingtheminimumdetectablepressuresforthetwoampl iercongurations term-by-term,theampliercurrentnoisecontributioniss eentobethesamefor both,assumingbothampliershaveequivalentcurrentnois echaracteristics.Atbest, theadditionalnoisefromthebiasresistorinthechargeamp casecanbemitigated bychoosing R efb R ep .Thenalvoltagenoisetermiswherethetwoaretruly dierentiated;assumingequivalentampliervoltagenois einbothcongurations,the totalcontributiontotheminimumdetectablepressurefrom thechargeampcircuitwill alwaysbehigherduetotheappearanceof C efb inthenumerator. Foramicrophonewithveryhighgain( C efb C ef ),theaddedvoltagenoiseof thechargeampcanbeminimized,but C efb cannotbedecreasedwithoutbound.The feedbackimpedanceintroducesanadditionalcut-onfreque ncy, f c =1 = 2 R efb C efb As C efb decreases, f c increasesandthebandwidthofthemicrophonecanbereduced Compensatingwithalarger R efb isnotalwaysstraightforward[ 149 ].Thereisthusa delicatebalancebetweengain,cut-o,andnoiseinthechar geamplierarchitecture. Theprimaryadvantageofchargeampliersisthatthemicrop honesensitivityisnot dependentonparasiticcapacitance.Parasiticcapacitanc eisintroduced,forexample,by wirebonds,traces,orcablesbetweenthesensorandtheinte rfaceelectronics.Charge ampliers,then,arepopularbecausetheycanbelocatedrem otelyfromtheactualsensor; changesincableortracelengths(andtheassociatedchange inparasiticcapacitance)do notaectthesensitivityorrequiresubsequentrecalibrat ion[ 146 ].Meanwhile,avoltage ampliermustbelocatedclosetothesensortominimizethea ttenuationinsensitivity. Deployingthousandsofmicrophonesontheexteriorofanair craftdemandsthe utmostinsimplicity.Collocatingthemicrophoneandsigna lconditioningcircuitryin asinglepackageyieldsacompactandcompletesensorsystem thatcanbeconnected directlytoaDAQwithoutregardforadditionalcircuitry.E veninthelaboratorysetting, theampliermaybelocatedincloseproximitytothemicroph one.Thevoltageamplier 111

PAGE 112

istheappropriatechoiceforsuchacase.Inaddition,there lativesimplicityofthevoltage amplierconguration,withitslowpartcountandfewertra de-ostoassess,isattractive. Asaresult,thevoltageamplierwaschosenastheinterfacec ircuitforthisstudy. ThemajorityofmeasurementspresentedinChapter 8 arespecictothevoltageamplier case.Measurementsforonemicrophoneinstrumentedwithac hargeamplier|for comparisonofsensitivityandtoestimateparasiticcapaci tances|arepresentedin Section 8.2.4.3 5.4Summary Inthischapter,modelsfortheperformanceofapiezoelectr icmicrophonehavebeen developed,includingalumpedelementmodel,adiaphragmme chanicalmodel,andnoise models.Inthenextchapter,thedevelopedmodelsareusedin astructuraloptimization formulationtodeterminethegeometrythatdeliversoptima lmicrophoneperformance. 112

PAGE 113

CHAPTER6 OPTIMIZATION Thischapterisconcernedwithchoosingmicrophonedimensi onswithinconstraints suchthatthe\best"performanceisobtained;thisprocessi sknownasoptimization[ 150 ]. ThelumpedelementmodeldevelopedinChapter 5 providespredictionsofmicrophone performanceandaidsintuitiveunderstandingofdesigntra deos.Theintuitiveselection ofa\best"designinthepresenceofmanydesignvariablesan dconstraints,however,is dicult.Thelowcomputationalcostassociatedwiththelum pedelementmodelmakesit ideallysuitedforintegrationwithanoptimizationalgori thmthatsystematicallyidenties the\best"design.Inthischapter,anoverviewofthedesign optimizationproblemisrst given,includingdiscussionofgeometricdimensionsavail ableforselectionandperformance characteristicstobeextremized.Next,theoptimizationpr oblemisformallydenedand theapproachforsolvingitisoutlined.Finally,theresult softheoptimizationprocessare discussed. 6.1DesignOverview 6.1.1DesignVariables Theuseofacommercialfoundryprocesstofabricatedevices leveragessignicant engineeringinvestmentbutalsoplacesconstraintsonavai lablegeometries.Witha compatiblegeometryestablished,animportantearlystepi nthedesignprocessis thusidenticationofdesignvariables.Figure 6-1 showsacross-sectionalviewofthe piezoelectricmicrophone|asdictatedbythelmbulkacous ticresonator(FBAR) variantprocessdiscussedatlengthinSection 4.2 |withimportantdimensionslabeled. Freedimensionsmayserveasdesignvariablesforthestruct uraloptimization problem,whilexeddimensions,denotedinFigure 6-1 witha symbol,maynot.The cavitydepth d c issetbythewaferthickness.Thediaphragmoverlap a 0 andundercut a c arestandardfeaturesoftheFBAR-variantfabricationproce ss,asisthepassivation 113

PAGE 114

D D D n r Figure6-1.Cross-sectionofthepiezoelectricmicrophone withnotabledimensionstobe considered;thosedenotedwith arexedbythefabricationprocess. layerthickness h pass .Thevaluesassociatedwiththesexeddimensionsandother snot showninFigure 6-1 arecollectedinTable 6-1 Table6-1.Microphonedimensionsxedbythefabricationpr ocess. DimensionValue mDescription a o 3Widthofdiaphragmoverhang a c 35Widthofdiaphragmundercut h pass 0.14Thicknessofpassivationlayer d c 500Cavitydepth L v 50Ventlength h v 2Ventheight b v 25Ventwidth Meanwhile,several\free"dimensionsremainwhosevaluesm aybeselectedwithin boundsestablishedbythefabricationprocess,includingt helmthicknessesand diaphragmradii.Therearethus7designvariablesintotal: theinnerradius, a 1 ;the widthoftheannularpiezoelectriclmstack, a ;andthelmthicknessesassociated withthetopelectrode,piezoelectric,bottomelectrode,a ndstructurelayers, h e;top h p h e;bot ,and h struct ,respectively.Thedimension a isusedinplaceof a 2 tospecifythe outerradiusofthediaphragm(i.e. a 2 = a 1 + a )becauseitmakesselectionofthe twodimensionsindependent;using a 2 asadesignvariablerequiresenforcementofthe condition a 2 a 1 .Notealsothatthecavityradius a c issetbyselectionofthediaphragm 114

PAGE 115

radii,asfromFigure 6-1 a c = a 1 + a a c : (6{1) 6.1.2Objective Theextremizationofaperformancemeasuresubjecttocerta inconstraintsisthe purposeofoptimization.Determininganoptimaldesignrs trequirestheappropriate measure(s)ofwhatconstitutes\best"performance|called theobjectivefunction(s)| tobeidentied.Theconceptoftheoperational\space"inth efrequencyandpressure domainswasintroducedinChapter 2 intermsofthemicrophonebandwidthanddynamic range.Maximizingthis\space"subjecttotheneedsofthepa rticularapplicationisone wayofapproachingmicrophonedesign.Atminimum,aMEMSpie zoelectricmicrophone designmustbeidentiedthatpreciselymeetsallsponsorpe rformancespecications (Section 1.2 ).Therstquestiontobeansweredintheoptimizationproce ssisthus whetherornotthespeciedperformanceisachievablewithi nthedesignspaceestablished bythefabricationprocess,basegeometry,materialchoice s,etc.Beyondthat,the questionstobeansweredarewhetherperformancecanbeimpr ovedbeyondthegiven specicationsandwhatadditionalperformancegainsaremo stbenecial. Microphonebandwidthexceedingtheaudiorange(20Hz{20kHz) isnotbenecial inanyfull-scaleaeroacousticmeasurementapplication,i ncludingthefuselagearray application.Althoughadditionalbandwidthcouldenableth emicrophonetobeleveraged tomodel-scaleapplications,examiningthedesigntrade-o sforfull-scaleandmodel-scale measurementswasnotafocusofthisstudy.Exceedingthespe cieddynamicrange, meanwhile,hasanobviousbenetinthetargetfuselagearra yapplication:loweringMDP improvesmeasurementresolution. 1 Inadditiontoimprovingperformanceinthetarget application,exceedingspecicationsonMDPcouldenablet hemicrophonetobeleveraged 1 LoweringMDPimprovesmeasurementresolutionuptothelimi tsoftheassociateddataacquisition system. 115

PAGE 116

directlytootherfull-scaleapplications,suchasryovera rrays.Minimumdetectablesignal ingeneralhasbeenestablishedasakeycomparativegureof meritforsensors[ 151 152 ]. Exceedingthespeciedmaximumpressurelevelof172dB|the highestpressure levelofpracticalinterestinaeroacousticmeasurementso faircraft|doesnotyieldsimilar benets.However,thedesigntrade-obetweenthespeciedP MAXandobtainable MDPisoffundamentalimportanceforthepresentdesigneor t;intheeventthat speciedperformanceforthesetwoquantitiesisnotachiev able,knowledgeofthe trade-osdrivesspecicationrevisionsordesignspacemo dications.Tostudythe trade-os,extremizationofbothMDPandPMAXweretakenasop timizationobjectives. Theresultingoptimizationformulationisknownasa multicriteria or multiobjective optimization[ 150 153 ]. Duetocompetitionamongobjectivefunctions,multiobject iveoptimizationproblems arecharacterizedbythenon-existenceofauniquesolution .Forexample,anynumberof minimumvaluesforMDPmaybeachievablegivensacricesint hemaximumattainable valueofPMAX.Withoutadecision-makertoexpresspreference ,asetofmathematically equivalentsolutionsknownasthePareto-optimalsetemerg es[ 153 ].Asolutionissaid tobe Paretooptimal iftheselectionofanyothersetofdesignvariablevaluesre sultsin allobjectivefunctionsremainingunchangedoratleastone getting\worse"[ 150 ].An exampleofasetofPareto-optimalsolutions|oftencalleda Paretofront |isshownin Figure 6-2 ,wheremaximizationofPMAXandminimizationofMDParetaken asthetwo objectives.Inthisgure,designsA,B,andCarePareto-opti malbutDisnot.Similarly, Papilaetal.(2006)[ 152 ]foundPareto-optimalsolutionsassociatedwithsimultan eous maximizationofsensitivityandminimizationofelectroni cnoiseforapiezoresistive microphone. AlgorithmsexistforndingthesetofPareto-optimalsoluti onsdirectly[ 153 154 ]. However,morecommonly-availablesingle-objectiveoptimi zationsoftwaretoolsmaybe usedtondtheParetofrontviasolutionofasequenceofcons trainedsingle-objective 116

PAGE 117

n rn Figure6-2.Paretofrontexample.problems.Usingthisapproach,oneobjectiveisextremizedw hiletheotheristreatedasa constraint[ 150 ].TheconstraintisvariedoverarangeofvaluesuntilthePa retofrontis resolved.Thisisknownasthe -constraintmethod[ 153 ]andisusedintheoptimization approachforthepiezoelectricmicrophone,discussedfurt herinSection 6.3 6.2Formulation Inthissection,theoptimizationproblemisformalized.Th eobjectivefunction,design variables,bounds,andconstraintsarealldenedanddiscu ssed. Theobjectiveoftheoptimizationis min X f obj ( X )=MDP ; (6{2) wherethenarrow-banddenitionofMDPisselectedforthiss tudy,i.e.MDPevaluated fora1Hzbinwidthcenteredat1kHz.Theassociateddesignvari ablesare X = f a 1 ; a;h etop ;h p ;h ebot ;h struct g (6{3) subjecttobounds(orsideconstraints) LB X UB : (6{4) Specicvaluesof LB and UB setbytheFBAR-variantprocessarefoundinTable 6-2 Geometrical,fabrication,modeling,andperformancecons traintsareallpresentinthe optimizationproblem.Manyfabricationconstraintsarere rectedintheboundsplacedon 117

PAGE 118

Table6-2.Designvariablebounds. XLB [ m] UB [ m] a 1 5600 a 5600 h e;top 0.10.2 h p 0.31 h e;bot 0.20.6 h struct 12 eachdesignvariable,whileotherconstraintsaredependen tonmultipledesignvariables. Theseareclassiedaslinearornonlinearconstraintsdepe ndingontheirfunctional dependenceonthedesignvariables.Thereare3linearconst raintsand1nonlinear constraint.Theconstraintsare:1.Themicrophonediaphragmmustbesucientlythinsuchtha ttheKirchhoplate theoryusedinthediaphragmmechanicalmodelremainsappli cable.Thethinness ofthediaphragmwasquantiedviatheaspectratio, AR ( a=h ),forboththeinner (0 r a 1 )andouter( a 1 r a 1 + a )regionsofthediaphragm.Theconstraints are a 1 AR ( h pass + h struct )(6{5) and a AR ( h pass + h e;top + h p + h e;bot + h struct ) : (6{6) AR waschosentobe10[ 121 ]. 2 2.Afabricationconstraintonthemaximumradiuswasmorere strictivethanthe sensingelementsizerequirementofSection 1.2 : a 1 + a 600 m : (6{7) 3.Afabricationconstraintwasalsoplacedontheminimumra dius: a 1 + a 250 m : (6{8) 4.Thesolenonlinearconstraintwasonthemaximumpressure ;thepressureatwhich totalharmonicdistortion(THD)reached3%wasrequiredtome etorexceed172dB 2 Aplateisgenerallydenedas\astructuralelementwithpla nformdimensionsthatarelargecomparedtoitsthickness"[ 121 ].Thespecicminimumrelationshipbetweenthesedimensio nsisnotprecisely prescribed,thoughaspectratiosof10{20arecommonlycite d[ 37 121 ]. 118

PAGE 119

perthedesignobjectivesinSection 1.2 .Withacomputationallyecientprediction methodfortotalharmonicdistortionofthemicrophoneunav ailable,aconstrainton staticnonlinearityofthediaphragmwasusedinstead.Fort hemaximumpressure p max ,thetotalcenterderectionofthediaphragmpredictedusin gthelinearand nonlinearmodels(seeAppendix A )wasrestrictedtobe 3%,i.e. w 0 ;l w 0 ;nl w 0 ;nl p = p max 0 : 03 ; (6{9) wheresubscript l indicatesthelinearmodelandsubscript nl indicatesthe nonlinearmodel.Althoughthequalityofthismeasureofnonl inearityasaprediction forTHDwasunknown,intuitionsuggestedthatTHDwouldtrends imilarly. Uncertaintyintheconstraintwaspartiallyaddressedinthe optimizationapproach, discussedinSection 6.3 Notethatnoconstraintsonbandwidthweredenedinordertom eetthe f 2dB targetssetoutinChapter 1 .Microphonesdesignedforhigh p max arenecessarilystiwith highresonantfrequencies,soitwasnotanticipatedthatsa tisfying f +2dB 20kHzwould beanissue.Noconstraintwasplacedon f 2dB (i.e. f 2dB 20Hz)outofconcernthat unreliablepredictionsforthisquantity,dominatedbyeit herthe R ep C eb or R av C ac break frequencies,woulddrivetheoptimization.Atthetimeofth eoptimization,predictions of R ep werebasedonimpedancemeasurementsofearlyprototypedev ices,buttherewas littlecondenceinthemeasurementquality.Meanwhile,th e R av predictionwasreliant onassumptionsalmostcertaintonotbesatised,forexampl efully-developedrowin theventchannel.Withtheventgeometryset,enforcing f 2dB 20Hzwouldleadthe optimizationalgorithmtoincreasethe R av C ac productviaenlargingthecavityradius, whichbyEquation 6{1 wouldleadtobiggerdiaphragmswithlowerstinessandlowe r achievable p max .Despitethelackofbandwidthconstraints,thebandwidtho ftheoptimal sensorwasassessedforadherencetothedesignrequirement sfollowingtheoptimization. 6.3Approach TheoptimizationproblemdenedinSection 6.2 isasingle-objectiveproblemwith bothlinearandnonlinearconstraints.Itwassolvedusingt he fmincon functionin MATLAB,whichisapplicabletononlinearconstrainedoptimi zationproblems.This 119

PAGE 120

functionusesasequentialquadraticprogramming(SQP)met hod[ 138 ]andthusisalocal optimizer[ 155 ].Inimplementingtheoptimizationusing fmincon ,theconstraintswere writtenas inequalitiesandnormalizedtobeof O (1).Similarly,thedesignvariables werescaledviatheirboundstovaryover[0 ; 1]. Theoptimizationapproachusingthe -constraintmethodisshowninFigure 6-3 First,astartingvalueofPMAXwasestablishedandtheoptimi zationwasrun.Witha feasiblesolutionfound,resultsweresaved.PMAXwasthenin crementedandtheprocess repeateduntilafeasiblesolutionwasnolongeravailable. UsingastartingPMAXvalue of160dBwithincrementationof0 : 5dB,theParetofrontwasobtainedforvaluesof PMAXleadinguptoandbeyondthetargetvalueof172dB.Amajor advantagetothis approachwastheabilitytoassessthesensitivityofMDPtou ncertaintyinPMAX,given aforementioneduncertaintyintheclosenessoftherelatio nshipbetweenthe3%static nonlinearityconstraintandtheactual3%distortionlimit nr n n n Figure6-3.Optimizationapproach. Thevaluesofconstantsusedintheoptimizationarefoundin Table 6-3 ,target residualstressessuppliedbyAvagoTechnologiesforeachof thethinlmsarefoundin Table 6-4 ,andthin-lmmaterialpropertiesarelocatedinAppendix D .InTable 6-3 ,the dampingratio wasestimatedfromasimilarpiezoelectricdevicedevelope dbyHorowitz [ 119 ].Thevalueofthepiezoelectricresistivity p andseriesresistance R es camefrom 120

PAGE 121

meanimpedancemeasurementsofseveralprototypemicropho nes.Thebiasresistor R ea wasdisregardedbecauseearlyexperimentsshowedthatitwa sunnecessaryforstable operationofthepiezoelectricmicrophonewithvoltageamp lier.Theamplierinput capacitance C ea andnoisecharacteristics S v a and S i a wereallobtainedfromthedatasheet forthechosenamplier,theLTC6240[ 44 ].Theresidualstresscharacteristicsofthe thin-lmstackfoundinTable 6-4 emergedfromsignicantprocessdevelopmenteorts atAvagoTechnologies,andtheinformationwasleveragedint heoptimizationtoenhance modelpredictions.Table6-3.Constantvaluesusedintheoptimization. Parameter Value [ 119 ] 0.03 R es 4 : 14kn p 22 : 8Mnm R ea 1 C ea [ 44 ] 3pF S v a [ 44 ] 7950Hz f +49 nV 2 = Hz y S i a [ 44 ] 1 : 27 10 6 Hz 2 f 2 4 : 85 10 5 Hz f +0 : 354 fA 2 = Hz y y Curvettodatain[ 44 ] Table6-4.Targetthin-lmresidualstresses. LayerResidualStress[MPa] Passivation 50 TopElectrode 150 Piezoelectric0BottomElectrode 100 Structural55 6.4ResultsandDiscussion Theoptimizationusingthe -constraintmethodyieldedtheParetofrontshownin Figure 6-4 .Inordertoincreasetheeectivepiezoelectriccoecient ,thegeneraltrendof theoptimizationalgorithmistoincreasethediaphragmout erradius( a 1 + a )asmuch 121

PAGE 122

aspossible(whilemakingminorchangestothepercentagepi ezoelectriccoverage, a=a 1 ) untilthenonlinearityconstraintbecomesactive.ForPMAX 165dB,themaximum radiusconstraintisactivatedandtheoptimizationalgori thmlosesitsprimarymethod ofreducingMDP.Asaresult,theattainableminimumvaluesof MDPareseentobe lesssensitivetothespeciedvalueofPMAXinthisregime.Me anwhile,therelationship betweenPMAXandMDPisseeminglylinearforPMAX 165dB,indicatingapower lawrelationshipbetween p max and p min .Notethatnofeasiblesolutionswerefoundfor PMAX > 174dB,beyondwhichtheminimumdiaphragmradiusconstrain tactivates. 160162164166168170172174 30 35 40 45 50 G F E D C B A Feasibleregion PMAX[dB]MDP[dB]Figure6-4.ParetofrontassociatedwithminimizationofMD Pandmaximizationof PMAX.Theshadedregionindicatesthetargetdesignspace. Designsselectedforfabricationareindicatedwithlabels A{GinFigure 6-4 .Designs A{Csatisedthedesigncriteria|withPMAX 172dBandMDP 48 : 5dB|andwere thusobviouschoices.Withthepossibilitythatthestaticn onlinearityconstraintwasa conservativepredictionfortotalharmonicdistortion,de signsD{G,whichdidnotreach thePMAX=172dBtarget,werealsoselectedinordertoprovidet hepossibilityofmeeting thePMAXtargetwithsuperiorMDPcomparedtodesignsA{C.Allof theselected designsfeaturedthesameoptimallmthicknessesandthusw ereabletobefabricated togetheronasinglewafer,eliminatingtheneedforseconda ryoptimizationtoconstrain thedesignstoasinglesetoflmthicknesses.Whenthisaddi tionalstepisrequired, performanceisinevitablysacricedforasubsetofdesigns 122

PAGE 123

Figure 6-5 showsvaluesfortheoptimaldesignvariablevalues, X i ( signifying \optimal"),normalizedto[0,1]viatheirindividualbound sandplottedversusPMAX. Bothelectrodethicknessesandthethicknessofthepiezoel ectriclayerwereconstantfor alldesigns,andthestructurallayerthicknesswasnearlys o.InthelowPMAXregime inwhichthemaximumradiusconstraintwasactive,theoptim izerturnstoreduction of h struct toreducediaphragmstinessandincrease d a .Ingeneral,theoptimization algorithmpushesthelmthicknessestotheirupperandlowe rboundstotunethe residualstressstatesuchthatthePMAXconstraintissatis able.Both a 1 and a wereheldrelativelyconstantforPMAX 165dBsincetheycouldnotbemadebigger; forPMAX > 165dB,themaximumradiusconstraintdeactivatesandtheop timization algorithmcontinuouslyreducesthediaphragmradius a 1 tostienthediaphragm. 160162164166168170172174 0 0 : 2 0 : 4 0 : 6 0 : 8 1 PMAX[dB]( X i LB i ) = ( UB i LB i ) a 1 a h struct h e;bot h p h e;top Figure6-5.Normalizeddesignvariablevaluesforeachoptim izationperformed,plotted againstPMAX. Thecommonlmthicknessessharedbythechosendesignsarec ollectedinTable 6-5 andtheradialdimensions(roundedtothenearest m)andperformancecharacteristicsof designsA{GarecollectedinTable 6-6 .DesignsA{CcorrespondingtoPMAXof174{172 dBweresubjecttothethinnessconstraintintheouterregio n,whichdictatedthat a equal AR timesthetotalthickness.Performingtheoptimizationsaf terdisablingthis constraintyieldednobetterthan0 : 1dBimprovementinMDP,soitwasnotasignicant performance-limitingfactor. 123

PAGE 124

Table6-5.Optimallayerthicknesses. SymbolValue[ m] h pass 0.14 y h e;top 0.1 z h p 1 x h e;bot 0.6 x h struct 2 x y Fixed z Atlowerbound x Atupperbound Table6-6.Optimizationresults. Design ABCDEFG PMAX[dB]174173172171170169168 a 1 [ m] y 219245274306338373412 a [ m] y 38 z 38 z 38 z 39404143 MDP[dB]48.146.545.043.341.840.338.7f 2dB [Hz]64666871737578 f +2dB [kHz]12911310089807265 S oc [dBre1V = Pa] 88 : 8 87 : 7 86 : 6 85 : 5 84 : 5 83 : 5 82 : 5 S va [dBre1V = Pa] 92 : 4 91 : 0 89 : 6 88 : 2 87 : 0 85 : 8 84 : 5 y Roundedtothenearest m z AR constraintactive Withnoconstraintplacedon f 2dB ,thismetricdidexceed20Hzforallofthe selecteddesigns.Furtherinvestigationrevealedthatitw asdominatedby R ep rather than R av .Onlyabout0 : 25%ofthetotaldesiredbandwidthdidnotmeetspecication s; givenaforementioneduncertaintyin R ep ,itwasdecidedtogoaheadwithfabrication. Meanwhile, f +2dB waswellabove20kHzasexpected,anddesignsA{Dwerepredicte dto possesssucientbandwidthforpotentialleveragingofthe microphonestomodel-scale applications. AnalyzingthesensitivityofMDPandPMAXtoperturbationsind esignvariablesor otherinputsyieldsadditionalinsightintotheresults.Fi gure 6-6 showshowperturbing 124

PAGE 125

optimaldimensionsassociatedwithdesignCby 10%aectedMDP. 3 Themost importantdesignvariableswere a 1 and a ,forwhicha10%variationyieldedapproximately 1{1.5dBchangeinMDP. 0 : 90 : 9511 : 051 : 1 1 0 1 X i =X iMDP-MDP [dB] a 1 a h struct h e;bot h p h e;top Figure6-6.SensitivityofMDPto 10%perturbationsinthedesignvariablesforDesign C.The x and y axesarereferencedtothevaluesofthedesignvariablesand MDP,respectively,attheoptimalsolution. Similarly,Figure 6-7 showsthesensitivityofPMAXtoperturbationsintheoptimal dimensionsfordesignC. 3 PMAXisseentobemoresensitivetothedesignvariables, mostnotably h struct a 1 ,and a inthatorder.PMAXisparticularlysensitiveto h struct becausethethickstructurallayer,withitstensilestress ,playsamajorroleinsettingthe overallstateofin-planestress.ComparingFigures 6-6 and 6-7 ,itisseenthatincreasing h struct 10%beyonditsoptimalvalueyieldsnearlya1dBimprovement inPMAXwith onlya0 : 3dBpenaltyinMDP.Thecalculusofmodifyinganyoftheother designvariables isnotasattractive,indicatingthatthe2 mupperboundon h struct isasignicant performance-inhibitor. Microphoneperformancemetricsarealsosensitivetouncer taintyinmodelinputs, mostnotablythoseforin-planestress.Tostudythissensit ivity,aMonteCarlosimulation wasperformedinwhichthestresseswereperturbedaboutthe irtargetvaluesusing 3 GiventhelinearnatureofFigure 6-6 andFigure 6-7 ,thesensitivitiesalsocouldhavebeencharacterizeddirectlyvialogarithmicderivatives[ 150 152 ]. 125

PAGE 126

0 : 90 : 9511 : 051 : 1 1 0 : 5 0 0 : 5 1 X i =X iPMAX-PMAX [dB] a 1 a h struct h e;bot h p h e;top Figure6-7.SensitivityofPMAXto 10%perturbationsinthedesignvariablesforDesign C.The x and y axesarereferencedtothevaluesofthedesignvariablesand PMAX,respectively,attheoptimalsolution. statisticssuppliedbyAvagoTechnologies.Theanalysiswas completedfortheMDP ofdesignCandresultsareshowninFigure 6-8 .Thesimulationmeanagreedwith thepredictedvalueof45dB,andthe95%condenceintervalw ascalculatedtobe 45 : 2dBto46 : 7dB.AsimilaranalysiscouldnotbecompletedforPMAXbecaus eof failuresintheiterativenonlinearsolverforalargeperce ntageofstressvaluesencountered duringtheMonteCarloiterations. 43 : 54444 : 54545 : 546 0 2 4 6 8 MDP[dBSPL]%OccurrenceFigure6-8.SensitivityofMDPtoin-planestressvariation sforDesignC,obtainedvia MonteCarlosimulation. 6.5Summary Inthischapter,theproblemofoptimizingtheperformanceo fthepiezoelectric microphoneintermsofdynamicrange(MDPandPMAX)wasdeneda ndexecuted. 126

PAGE 127

Sevendesigns(A{G)wereselectedforfabrication,with3(A{C )meetingorexceeding requirementsonMDPandPMAX.Optimizationtrendsandthesens itivityofbothMDP andPMAXtoperturbationsinthedesignvariableswerealsodi scussed.Thenextchapter addressestherealizationofthesemicrophonedesignsandp ackagingofthemicrophones forexperimentalcharacterization. 127

PAGE 128

CHAPTER7 REALIZATIONANDPACKAGING Thischapterfocusesontherealizationofindividuallypac kagedpiezoelectric microphones,bridgingthegapbetweenthetheoreticaldesi gnsofChapter 6 and experimentalcharacterizationinChapter 8 .First,theresultsofthefabricationprocess performedatAvagoTechnologiesarediscussed.Next,themeth oddevelopedtoseparate themicrophonedieisexplained.Finally,thelaboratoryte stpackagedevelopedspecically forthepiezoelectricmicrophonesisdescribed. 7.1Realization Thissectionfocusesonrealizationofthepiezoelectricmi crophones.Theas-fabricated microphonegeometriesareexplicitlygivenandfabricatio nresultsarediscussed. 7.1.1Geometry Optimalpiezoelectricmicrophonegeometrieswerefoundin Chapter 6 .Fromthose results,sevendierentgeometriescoveringaswathofdesi gnspaceweresubmittedfor fabrication.Withexpecteduncertaintiesinmodelpredict ionsandlmstresstargeting, fabricationofmultiplemicrophonegeometrieswasjudgedt oprovidethemostprobable, cost-sensitive,andschedule-eectivepathtomeetingper formancespecications.The diaphragmdimensionsfordesignslabeledA-G(inorderofinc reasingdiameter)arefound inTable 7-1 andtheircommonlmthicknessandstresstargetsareinTabl e 7-2 .After thefabricationlotwasstarted,Avagosuggestedbasedonrec entexperiencethatthetop electrodethickness h e;top bechangedfrom0 : 1 mto0 : 15 m.Themodelsconrmedthat sensitivityofMDPandPMAXtothisdesignvariablewaslow(re callFigures 6-6 and 6-7 ). Itwasthusdecidedtoacceptthechangein h e;top ,whichisrerectedinTable 7-2 7.1.2FabricationResults FabricationwasperformedatAvagoTechnologiesusingavari antoftheirlmbulk acousticresonator(FBAR)process[ 82 83 105 ].Thefabricationprocesswasaddressed 128

PAGE 129

Table7-1.Designdimensions. Design a 1 [ m] a [ m] a 2 [ m] A21938257B24538283C27438312D30639345E33840378F37341414G41243455 Table7-2.Filmproperties. LayerThickness[ m]StressTarget[MPa] StructuralLayer255BottomMo0.60AlN10TopMo0.15-150Passivation0.14-50 inSection 4.2 .AphotographofacompletedwaferisfoundinFigure 7-1 .Inall,Avago Technologiesdeliveredeight6"waferswithmicrophonedie 2mmonaside. AvagoTechnologiesprovidedthewaferswithlmstressinfor mation.Aftereachlm deposition,wafercurvaturewasmeasuredwithaTencorFlex usFLX5400andthelm stresswasestimatedfromthesemeasurementsusingStoney' sFormula[ 107 108 156 ], = E s t 2s 6 t f (1 ) 1 R 1 R 0 ; (7{1) where isthelmstress, E s ,and t s aretheYoung'sModulus,Poisson'sRatio,and thicknessofthesubstrate,respectively, t f isthelmthickness,and R 0 and R arethe radiiofcurvaturebeforeandafterlmdeposition,respect ively.Stoney'sFormulais awafer-levelstressestimationthatreliesonanumberofas sumptionsthatmaybe onlyapproximatelyvalid,suchastransverseisotropyofth esubstrate,uniformlm thickness,andhomogeneousstress,amongothers[ 107 ].Despitethecaveats,these estimatesrepresentedthebest-availablelmstressinfor mation. 129

PAGE 130

Figure7-1.Waferofpiezoelectricmicrophonesfabricated atAvagoTechnologies. ExperimentalresultsinChapter 8 arepresentedfordevicesfromtwowafers, identiedasnumbers116and138.Stressdataprovidedwasut ilizedtopredictmicrophone performanceforcomparisontocharacterizeddevicesinCha pter 8 .Devicesfromwafer116 werevisiblybuckled.Noneofthewafersdisplayedanyobviou svisualsignsoflarge cross-waferstressvariations. 7.2Dicing ProcessesfordicingfragileMEMSvaryandarehighlydepend entonwhetherornot theyrequiredirectcontactwiththeenvironment.MEMSacce lerometersoroscillators, forexample,canbeencapsulatedatthewaferlevelsuchthat theyareprotectedduring dicing.Unfortunately,MEMSmicrophonesmustbeexposedtot hemediumofacoustic propagationandthusdonotsharethisluxury.Inatradition aldicingsawoperation,the fragilethin-lmdiaphragmscanbedamagedbyvibration,de bris,orwaterpenetration. MethodsexisttoshepherdexposedMEMSstructuresthrought hedicingprocess, includingtheuseofpatternedorreleasabletapes[ 43 ],temporarybondingtoahandle wafer,delayingthereleaseetchuntilafterdicing[ 107 ],orchoosing\clean"dicingmethods 130

PAGE 131

suchasscribe/break[ 157 ]orlasercutting[ 158 ].Inthissection,anin-houseprocessfor dicingthemicrophonedieusingprotectivetapeisdescribe d. 7.2.1DicingProcess ThemostadvanceddicingoptionavailableatUF'sNanoscaleRe searchFacilityisan ADT7100DicingSaw,whichusesaphysicalbladeandassociate djetofdeionizedwater forcoolinganddebrisremoval.Figure 7-2 showsthebladedicingawafersample,with thewaterjetimpingingonthesampleoppositethecuttingdi rection.Protectionofthe micdiaphragmsisthusanecessity,butanysubstanceusedto protectthedevicesmustbe easilyremovablepost-dicewithoutdamagingthediaphragm s.Protectivetapes,suchas UVtapeorthermalreleasetapearesoundoptions.Intheproce ssdescribedherein,Nitto DenkoREVALPHAthermalreleasetape(No.3198M)wasusedfordia phragmprotection. Thistapeisdouble-sided,witharegularadhesiveononesid eandatemperaturesensitive adhesiveontheother.At120 C,thetemperaturesensitiveadhesivereleasescompletely Cutdirection X X X X X X X X Xz Waterjet ) Leadingedge P P P P Pq Trailingedge X X X X X X X X X Xy Sample J J J J^ Nickelresinblade Figure7-2.Dicingbladeandsampleorientation. Earlyexperimentsonprototype3mmdieusingsolelythermal releasetapetoprotect thediaphragmsprovedextremelysuccessful,withnearly10 0%yield.However,thesmaller 2mmdiedidnotprovidesucientareaforreliabletapeadhes ion,resultinginpeeling duringdicingandsignicantdieloss.Asaresult,anadditio nalprotectivepolyethylene tape(commonlyusedforsurfaceprotectionintheconstruct ionindustry)wasusedina 131

PAGE 132

moreelaborateprocess.Theentireprocesswasperformedon anADT7100DicingSaw equippedwithnickelresinblades. Themethodforusingtheprotectivetapesduringthedicingp rocessisdepictedin Figure 7-3 .Toreducethetimeandriskofeachdicerun,wafersections( orsamples)were individuallydiced.Theywereobtainedviadiamondscribin gandbreaking,withtypical piecescontaining3-6reticles.Thebacksideofthesamplew asrstaxedtomediumtack dicetape,usedformountingthesampleinthedicingmachine .Next,thethermalrelease tapewasappliedtothefrontside(diaphragmside)ofthesam ple,withonlytheprotective backingassociatedwiththethermalreleaseadhesiveremov ed.Thethermalreleasetape wasappliedevenwiththesampleedgesonthecutentryedges, butextendingothe sampleupto15mmonthetrailingedges(asshowninFigure 7-3A )toprovideadditional adhesionandprotectionfromthedicingmachine'swaterjet .Next,withoutremovingthe remainingplasticbackingonthebacksideofthethermalrel easetape,thepolyethylene tapewasappliedoverthecompletesampletoprovideadditio nalprotection,asshownin Figure 7-3A .Thesamplewasthendicedintherstdirection,withcutsex tendingslightly otheleadingandtrailingedgesofthesample.Thecomplete tapelayupforthisstep isshowninFigure 7-3B ,andthetapethicknesses,whichareimportantforsettingt he cutdepth,arecollectedinTable 7-3 .Importantdicingmachinesettingsarecollectedin Table 7-4 Table7-3.Tapeandsubstratethicknesses. MaterialThickness[ m] Substrate 500 Dicetape 130 Thermalreleasetape(withoutlaminates)160Thermalreleasetapelaminate75Polyethylenetape 70 Withcutscompletedinasingledirection,thesamplewasthe nremovedfromthe dicingmachineandthepolyethylenetapewassmoothlypeele dawayfromthethermal 132

PAGE 133

) Wafersample ) Thermalreleasetape ) Polyethylenetape X X X X X X X Xz Dice pattern 6 1 st Cut Direction 2 nd Cut Direction A A A AU Polyethylene tape A A AU Tape backing + Thermal releasetape + Wafer sample + Dice tape B A A AU Polyethylene tape B B BN Thermal releasetape r Wafer sample r Dice tape C Figure7-3.DicingprocessforMEMSpiezoelectricmicropho nedie.A)Aerialviewofdice processtapingtechnique.B)Cross-sectionalviewoftapin gtechniqueforrst directiondicecuts.C)Cross-sectionalviewoftapingtech niqueforsecond directiondicecuts. Table7-4.Dicersettings. ParameterSetting Spindlespeed30krpmEntryspeed0 : 5mm = s Cuttingspeed2mm = s Illumination(Coaxial/Oblique)11/53 releasetapeviatheplasticbackinglayer.Anewlayerofpol yethylenetapewasthen appliedasinFigure 7-3A toyieldthetapelayupofFigure 7-3C .Thesamplewasthen dicedintheseconddirection.Theresult,atthisstage,was thatthesamplehadbeen singulatedintoindividualdiewithsquaresofthermalrele asetapestillaxedonthe diaphragmside,whilestripsofthepolyethylenetaperemai nedontop.Withoutremoving 133

PAGE 134

thesamplefromthedicetape,thepolyethylenetapestripsw erethencarefullypeeled fromthethermalreleasetape.Diewereindividuallyremove dfromthedicetapeand placedonahotplateat120 C.Astheadhesivereleased,thetapebecameopaqueand often\poppedo"oftheindividualdie.Otherwise,therele asedtapewaseasilyremoved withtweezers. IndividualdiewerestoredingriddedGelStickyCarrierBox esfromMTICorporation. Thenamingconventionusedtorefertoaparticulardiewasba sedonitswaferoforigin, carrierboxnumber,gridlocationwithinthecarrierbox,an ddesignletter.Forexample, 138-1-E4-Dreferstoamicrophonedieoriginatingfromwafe r138,storedcarrierbox1at gridlocationE4,andofdesignD.7.2.2DicingResults A66mm 34mmsectionofwafer116and20mm 60mmsectionofwafer138 werediced.Wafer116wasdicedwithjustthermalreleasetap eforprotectionanda signicantnumberofdiewerebrokenatthetrailingedgesdu etothetapelosingadhesion duringthedicingoperation.Only58%yieldwasobtainedfor thissegmentofwafer116, withyieldcalculatedhereastheratioofunbrokendietothe totalnumberofdiewith releaseddiaphragms.Wafer138wasdicedusingtheprocessd escribedinSection 7.2.1 withsignicantlybetterresults(83%yield).Microphoned iethatmostfrequentlydid notsurvivethisapproachwerethosewiththelargestdiaphr agmsandthusthelowest non-diaphragmadhesionarea,designsE-G. Micrographsofindividualmicrophonedieofeachdesignare picturedfromsmallest (A)tolargest(G)inFigure 7-4 .Thesediewerefromwafer138andthusobtainedwith thedescribeddicingprocess.Theundoctoredmicrographss howlittleedgedamageor particulate. 7.3Packaging PackagingoftheMEMSpiezoelectricmicrophoneforitsinte ndedoperationin aeroacousticapplicationssuchasfuselagearraysorengin etestsrequiresasmall,thinand 134

PAGE 135

A B C D E F G Figure7-4.Micrographsofmicrophonedie(designsA-G).inexpensivesolution.Apackagethatmeetsalloftherequir ementsfordeploymentinthe elddemandssignicantdevelopmentandisbeyondthescope ofthisstudy.However, laboratorytestpackagingthatenablesseamlesstransitio noftheMEMSmicrophone intomultipletestsetupsamongtheresearchlaboratoryand projectsponsorisalsoan importantdevelopmentinitself.Thissectiondescribesth ecreationofalaboratory testpackagecompatiblewithcommontestxturesfor1/4"mi crophonesatboththe InterdisciplinaryMicrosystemsGroupandBoeingCorporat ion.Anin-depthlookatthe Boeingrush-mountadapterdesignscanbefoundin[ 14 ]. Theentirepackagewascomposedofstructuralandconnectiv itycomponentsas shownintheexplodedviewofFigure 7-5 .Themicrophonediewasepoxiedintoacircular printedcircuitboard|tobecalledthe\endcap"|whichwasi nturnconnectedtothe endofabrasstube.Alignmentwasaccomplishedviamatingali gnmentpinsandholeson thebrasstubeandendcap.Acircuitboardwithbueramplie rwashousedinsidethe brasstubeanditwasconnectedtothebacksideoftheendcapv iasolderedwires.Anylon sleevewasxedontheassembledbrasstubeandendcapviaset screwsinthethickest partofitsbaseandservedtoelectricallyisolatethebrass tubefromtestxtureswhile alsoensuringmountingrushness.Finally,heatshrinktubi ng(notshown)wasusedto 135

PAGE 136

stress-relievethewiresprotrudingfromthebrasstube.Br asstubesandnylonsleeveswere providedbyBoeingCorporation. + Nylonsleeve ) Brasstube Circuitboard B B B BM Endcap 6 Micdie : Wires Figure7-5.Explodedviewofthelaboratorytestpackage. Acloseuprenderingofthemicrophonedieintheendcapissho wninFigure 7-6A The0.3485"diameterendcap,atwo-layerprinted-circuitb oardlaidoutinNational Instruments'UltiboardsoftwareandfabricatedatSierraPr otoexpress(Sunnyvale,Ca), was0.093"thicktoaccommodatepost-millingofa500 mdeepmicrophonedierecess. Thedierecess,epoxywells,pinholes,andboardcut-outwer eallmilledaspost-processing stepsatUniversityofFloridausingaSherlineModel2000CNCm ill.Viasonthefrontside wereconnectedtosolderpadsonthebacksideforhookuptoin terfaceelectronics.An additionalviainoneepoxywellprovidedsubstrategroundi ngasaprecautionagainst hard-to-diagnoseissuesassociatedwitharoatingsubstra tepotential.Thefrontsideofthe endcapwasplatedwithsoftbondablegoldforeaseofwirebon ding.Figure 7-6B shows theprintedcircuitboardlayout. Themicrophonediewasepoxiedintotheendcapinatwostagep rocessusingan EFDUltimus2400PrecisionEpoxyDispenser.First,Ablebond8 4-1LMI[ 159 ](electrically conductivesilverepoxy)wasdispensedintheepoxywelltha tcontainedtheviaandthen thediewasplacedintherecess.Theepoxywascuredinatempe raturecontrolledovenat 150 Cfor1h.Next,CyberbondDualBond707[ 160 ]wasdispensedinbothepoxywells 136

PAGE 137

andcuredunderaUVlampfor24h.Duringcure,theDualBond707 becamesuciently ruidtoseepunderneaththemicrophonedieandeectivelyse althemicrophoneback cavity.WirebondsweremadewithaKulicke&Soa4124Series ManualBallBonding SystemandencapsulatedwithDowCorning3145RTVMIL-A-4614 6[ 161 ].Important settingsforepoxydispensationandwirebondingarefoundi nTable 7-5 andTable 7-6 respectively.Acompletedmicrophoneintheendcappackage isshowninFigure 7-7 @ @ @R Viaforsubstrate ground Epoxywells Q Q Q Qk @ @ @I Vias > Micdie A B Figure7-6.Microphoneendcap.A)Drawingshowingdieinplac e.B)Circuitboardlayout [ 162 ]. Table7-5.Epoxydispensersettings. Ablebond84-1LMI[ 159 ]Dualbond707[ 160 ]RTV[ 161 ] Pressure[psi]501960BackPressure[mmHg]16.407.4DispensingTime[s]10.3VariableTip[gauge]252520Cure150 Cfor1h24hunderUVlampRoomtemp.for2d Table7-6.Wirebondsettings. BallBondWedgeBond Force77Time55Power34 Thecircuitryassociatedwiththemicrophonepackage|abu eramplierwith powersupplyltercapacitors(0 : 1 Fceramicand10 Ftantalum)|isshownschematically 137

PAGE 138

X X X X X X X X X X Xz MicDie H H H H H H H H Hj Endcap Wirebonds toVias Figure7-7.CloseupphotographofapackagedMEMSpiezoelec tricmicrophone. inFigure 7-8 .TheamplierusedwastheLinearTechnologiesLTC6240CS8, whichwas chosenforavarietyofpositivecharacteristicsincluding lowoperatingcurrentandvoltage, lownoise(voltagenoise < 10nV = p Hz),andhighinputresistance(1Tn)[ 44 ].Inorderto reduceparasiticcapacitanceandtheassociateddetriment aleectsondevicesensitivity (refertoSection 5.3.1 ),theamplierwassituatedasphysicallyclosetothemicro phone dieaspossible.Figure 7-9 showsthecircuitboardlayout.Theboardsweremilled in-houseonthin0.028"FR4andcomponentswerehand-solder ed.Wiringterminatedin bananaconnectorsfor v + v ,andground,inadditiontoaBNCconnectorfortheoutput signal.TheBNCgroundwastiedtothepowersupplygroundonth eboard. Onedevice,116-1-J7-A,waspackagedfordierentmeasureme ntswithbothavoltage andchargeamplier.Thecircuitdiagramforthechargeampl ierisfoundinFigure 7-10 TheselectedoperationalamplierwasaTexasInstrumentsO PA129.Thefeedbackloop wascomposedofa1Gnfeedbackresistorandtwo4pFfeedbackc apacitorsinparallelfor atotalcapacitanceof C fb =8pF.Thesevalueswerechosentoyieldclosetounitygain andtomaintainalowcut-onfrequency.Theboardlayoutisno tshownbutwassimilar tothatforthevoltageamplier,exceptwithadditionallen gthfortheinclusionofthe feedbackresistorandcapacitors. 138

PAGE 139

+ v i 0 : 1 F 10 F v + v 0 : 1 F 10 F v o LTC6240 Figure7-8.Voltageampliercircuitryincludedinthemicr ophonepackage. Output Pads 6 Tantalum Capacitors J J J] Electrolytic Capacitors @ @I LTC6240 6 Input Pads Figure7-9.Voltageampliercircuitboardlayout[ 162 ]. Electromagneticinterference(EMI)isamajorproblemforh igh-impedancedevices [ 46 ]suchasthepiezoelectricmicrophone,andstepsweretaken tomitigateitsimpact. Thebrasstubingwasconnectedtogroundtoprovideashieldf ortheampliercircuitry [ 46 ].Inaddition,theamplierboardfeaturedaguardringtohe lplimitleakage currentsintothepositiveamplierterminal[ 44 ].Shieldedcoaxialcablewasusedfor themicrophoneoutputsignal. Thecompletedpiezoelectricmicrophonelaboratorytestpa ckageisshownin Figure 7-11 .Thepackagetsa3/8"holewith1/2"depth.Flushnesswasno tcharacterized butwasestimatedtobelessthan500 m. 139

PAGE 140

+ 4pF 1Gn 4pF v i 0 : 1 F 10 F v + v 0 : 1 F 10 F v o OPA129 Figure7-10.Chargeampliercircuitdiagram. Figure7-11.CompletepackagedMEMSpiezoelectricmicroph one. 7.4Summary Thischapterdiscussedmicrophonerealizationandpackagi ng.Thelaboratorytest packagedwasdevelopedtoenabledevicecharacterizationi nmeasurementsetupsatboth theInterdisciplinaryMicrosystemsGroupandBoeingCorpo ration.Thesubjectofthe nextchapterisexperimentalcharacterizationofthepacka gedmicrophones. 140

PAGE 141

CHAPTER8 EXPERIMENTALCHARACTERIZATION Thischapterdescribesthethoroughexperimentalcharacte rizationofseveralMEMS piezoelectricmicrophones.First,experimentalmethodsa reintroduced,startingwith dieselectionanddiaphragmtopographymeasurements,then proceedingtoacoustic andelectricalcharacterization.Anovelsetofparametere xtractionexperimentsare alsodescribed.Experimentalsetupsanddataprocessingte chniquesarecovered.The experimentalresultspresentedthereafterquantifymicro phoneperformanceinterms ofcommonmetrics,thengivewaytotheresultsofparametere xtractionexperiments. ComparisonstothelumpedelementmodelpresentedinChapte r 5 arealsomade. 8.1ExperimentalSetup Thissectionprovidesanoverviewoftheexperimentalsetup susedinmicrophone selection,characterization,andparameterextraction.T hemicrophonecharacterizationis dividedintoacousticcharacterization(directmeasureme ntsofthemicrophoneresponse inapressureeld)andelectricalcharacterization(deter minationofmicrophoneelectrical traits).8.1.1DieSelectionSetup AvagoTechnologiessuppliedeight6"waferswiththousandso fdieperwafer. ThedicingprocessdiscussedinSection 7.2 wascarriedoutonsmallportionsoftwo wafers,withayieldof439unbrokenmicrophonedie.Withthi smanydieavailablefor characterization,anecientdieselectionmethodwasneed edpriortoinvestingsignicant timeinthepackagingofindividualdie(asdescribedinSect ion 7.3 ). Electricalmeasurementsareadesirablemeansofdieinterr ogationbecausethey canoftenbedoneeasilyatthedielevelviaprobing.Electri calimpedanceisan obviousquantitytousefordiscriminatingbetweendie.Howe ver,theexpectationof mechanicalpropertyvariations(i.e.stress)beingthepri maryfactorseparatinggood diefrombadsuggestedtheneedforamoremechanical-orient edselectionmethod. 141

PAGE 142

Forexample,itisshowninChapter 5 thathighvaluesoftheeectivepiezoelectric coecient d a areassociatedwithbothhighsensitivityandlownoise.Byt hepiezoelectric constitutiverelations(Equation 5{2 ), d a isequivalenttovolumedisplacementpervolt ( 8 =V ),calculablefromanopticalscanofthemicrophonediaphra gmunderelectrical excitation.Unfortunately,opticalscanningofthediaphra gmisatime-consumingand equipment-intensiveprocedurethatisnotsuitedtobeperf ormedonalargenumberof die.However, d a = 8 =V mayberewrittenas d a = A eff w=V ,where w isthedisplacement atanarbitrarydiaphragmlocationand A eff isaneectivearea.Thissuggestedthata quicksinglepointinterrogationcouldstillprovideusefu lcomparativeinformation. Anotherusefulmetriceasilyobtainableviaopticalinterro gationofthediaphragm underelectricalexcitationistheopen-circuitresonantf requency, f r .Trackingshifts inresonantfrequencybeforeandafterthepackagingproces scanprovideinformation aboutchangesindiaphragmstinessduetounintendedpacka gingstress.Inaddition, resonantfrequencyprovidesasecondcomparativemeasure, andisparticularlyuseful whenselectionoflikedevicesisnecessary. TheexperimentalsetupispicturedinFigure 8-1 .Agriddedgelpackwithmicrophone dieinsituwasplaceddirectlyonthemicroscopestageofthe Polytecscanninglaser vibrometer(LV)system.Eachdiewasinterrogatedonlyatasi nglepoint,chosenasthe centerofthediaphragmformeasurementrepeatabilityandt omaximizetheLVsignal. ProbesdeliveredaperiodicchirpsignalfromtheLVfunctio ngenerator(50noutput impedance)totheindividualdieoverawidefrequencyrange ,andtheresonantfrequency wasselectedfromthedisplacementpervoltagefrequencyre sponsefunction, H vw ( f ). Next,asingletoneexcitationat1kHzwasusedtondtheapprox imateratbandvalueof H vw ,denoted S a; 0 .Ateachstage,thesignalpowerwasne-tunedtoobtaingrea terthan 0.98coherencebetweenexcitationsignalandLVoutput.The importantmeasurement settingsarefoundinTable 8-1 142

PAGE 143

n r r !n Figure8-1.Experimentalsetupfordieselection.Table8-1.Dieselectionlaservibrometersettings. Settings ParameterMeasurementof f r Measurementof S a; 0 Bandwidth0kHzto200kHz0kHzto20kHzFFTlines64006400FrequencyResolution31 : 25Hz3 : 125Hz Averages100(Complex)ExcitationPeriodicChirp1kHzSineWindowRectangular Earlyintheprocess,measurementswererepeatedforsevera ldiethatwereremoved fromthegelpackandplaceddirectlyonthemicroscopestage inordertocharacterize theimpactofthesoftacousticboundaryconditionpresente dbythegel.Nodierence betweenmeasurementswasobserved,anddatainSection 8.2.1 areonlygivenfor microphonestesteddirectlyongelpacks. Outlierrejectionwasemployedbeforedeterminingthemean sandstandarddeviations associatedwith S a; 0 and f r foreachmicrophonedesign.Withvaluesofboth f r and 143

PAGE 144

S a; 0 knownforeachdie,thedatawerebivariate[ 163 ].Asaresult,straightforward univariateoutlierdetection,suchastheModiedThompson -TauTechnique[ 164 ],wasnot appropriate.Instead,multivariateoutlierdetection,wh ichpresumesmultivariateoutliers areunivariateoutliersinaparticular1-Dprojection,was needed[ 165 ].Theadjusted outlyingness(AO)algorithm[ 166 ],partoftheLIBRAMATLABtoolbox[ 167 168 ] developedbytheRobustStatisticsResearchGroupattheKat holiekeUniversiteit Leuven,wasused.Inthealgorithm,ateststatisticknownas theAOisgeneratedfor eachobservationovermanyrandom1-Dprojections,withthe maximumAOestimatefor eachobservationretained.Anadjustedboxplot[ 169 ]isgeneratedfortheAOestimates andobservationswhoseAOestimatesexceedtheboxplotuppe rwhiskerareregardedas outliers.TheprimaryassumptionoftheAOalgorithmisunim odalityofthedata[ 165 ]. Grinetal.provideanaccessibleintroductiontotheAOalg orithm[ 165 ]. Afterdieselection,thesameLVmeasurementsusedfordiesel ectionwererepeated fortheendcap-packagedmicrophonedie(recallSection 7.3 ).Forthismeasurement,the endcapwassimplyplacedonthemicroscopestageandprobedi nasimilarmannertothe individualdie.Thesamemeasurementwasalsoperformedpri ortopackagingofdevices forparameterextraction(detailsinSection 8.1.5 ). 8.1.2DiaphragmTopographyMeasurementSetup Microphonediewerepackagedinmultiplerounds,withther stroundsubjectedto bothpre-andpost-packagingtopographicalmeasurements. Thelevelofstaticderection ofthemicrophonediaphragm,aby-productoflmstress,isi nformativeofthediaphragm stressstate.AZYGONewView7200scanningwhitelightinterfer ometer(SWLI)wasused toperformthemeasurements.ASWLIworksbyilluminatingas amplewithwhitelight, whichrerectsothesurfaceofthesampleandrecombineswit hareferencebeam,creating interferencefringes.Acharge-coupleddevice(CCD)camer acapturesthefringesasthe SWLIobjectiveisscannedvertically;thesurfacetopograp hyisdeducedfromthecaptured imagesviaasoftwarealgorithm[ 170 ].TheNewView7200featuredaverticalresolution < 144

PAGE 145

0 : 1nm.Measurementsweremadewitha5XMichelsonobjectivean d1Xeldzoomlens, yieldingameasurementareaof1 : 41mm 1 : 05mm.Thestandard640px 480pxhigh speedcameraprovidedalateralresolutionofapproximatel y2 : 2 m.Threeaverageswere usedinallmeasurements,whichwerereferencedtothesurro undingwafersurface.Other notablesoftwaresettingsarefoundinTable 8-2 Table8-2.Scanningwhitelightinterferometersoftwarese ttings. ControlParameterSetting MeasurementFDAResHigh ACGOnPhaseResSuper SurfaceMapRemovePlane RemoveSpikesODatallOFilterOTrim0 8.1.3AcousticCharacterizationSetup Acousticcharacterizationreferstoexperimentalquantic ationofthemicrophone responsetoacousticpressureexcitation.Thegoaloftheac ousticcharacterizationwas toquantifythepiezoelectricmicrophoneperformanceinte rmsoffrequencyresponse (sensitivity,bandwidth)andlinearity.8.1.3.1Frequencyresponsemeasurementsetup Thefrequencyresponseofthepiezoelectricmicrophones, H m ( f )[V = Pa],was determinedovertheaudiorangeviaasecondarycalibration ,andtheproceduresused hailedfromthefamilyofcomparisonmethods[ 171 ].Specically,theperformanceofthe DUTwasdeterminedviacomparisonwithameasurement-grader eferencemicrophone. Theacousticcharacterizationwasperformedinanapproxim ately1m-long, 2 : 2cm-thick,aluminumplanewavetube(PWT)witha1in 1induct.APWTisa rigidwaveguidedesignedsuchthatonlyplanarwavespropag atebelowacertainfrequency, calledthecut-ofrequencyofthetube, f c .Belowthisfrequency,higher-orderacoustic 145

PAGE 146

modesintroducedtothePWTareevanescent,meaningtheydec ayexponentiallyalong itslength.Twomicrophonesmountedatthesamelengthwisel ocationaretherefore simultaneouslyexposedtothesamepressurefordrivefrequ encieslessthan f c .Fora squarewaveguidewithcross-sectionaldimension a ,thecut-ofrequencyis[ 28 ] f c = c 0 2 a : (8{1) Equation 8{1 revealsthecut-ofrequencymaybetunedbythePWTcross-se ctional dimension a orchoiceofgas.Forthepurposesofdeterminingthefrequen cyresponseof anaudiomicrophone, f c 20kHzisdesirablebut f c inairfor a =1inisapproximately 6 : 7kHz.Helium'sfasterisentropicspeedofsoundmakesitpossi bletoincrease f c to approximately19 : 8kHz,allowingforamorecomprehensiveviewoftheaudioband responseofamicrophone. Asaresult,complementaryfrequencyresponsemeasurements wereperformedusing bothairandheliuminthePWT.Themeasurementinairwasinte ndedtoyieldaccurate sensitivityinformationundernormaloperatingcondition s.Theexpandedfrequencyrange oftheheliummeasurementenabledassessmentoftheratness ofthefrequencyresponse overnearlythefullaudiorange.Theuseofheliuminsteadof airhasaslighteectonthe performancesofboththeDUTandreferencemicrophones;fore xample,ahelium-lled cavityislesscompliantthananair-lledone,since C ac / 1 = 0 c 20 andthe 0 c 20 product ishigherinhelium.Lumpedelementmodelpredictionsforth emicrophonefrequency responseinairandinheliumareshowninFigure 8-2 fordesignD.Dependingonthe microphonedesign,thereductioninsensitivityinheliumc omparedtoairwaspredictedto be0 : 04dBto0 : 4dB. Theexperimentalsetupforthefrequencyresponsemeasurem entisshownin Figure 8-3 ,withboththereferencemicrophoneandDUTmountedattheend ofthe PWT.ThereferencemicrophoneusedwasaBruelandKjr4138 1/8"pressureeld microphone[ 50 ]mountedonaBruelandKjrUA0160adapterandconnectedtoa 146

PAGE 147

10 2 10 3 10 4 10 5 94 92 90 88 86 Frequency[Hz]j H m ( f ) j [dB] Air Helium Figure8-2.Predictedfrequencyresponsemagnitudeinaira ndheliumfordesignD. BruelandKjr2670preamplier.ABruelandKjrType3560 DMultichannelPortable PULSEsystemwithaType3032A6/1Ch.Input/OutputModuleand Type31094/2 Ch.Input/OutputModulewasusedtogeneratethetestsignal andacquiredata.Thetwo microphoneswereconnectedtoseparateinput/outputmodul estominimizecross-talk.A Techron7540PowerAmplierampliedthepseudorandomtests ignalbeforeitreached aBMS4590compressiondriver.ThepoorresponseoftheBMS45 90compressiondriver below300Hzrequiredallmeasurementstobeconductedstarti ngatthatfrequency. MeasurementsettingsarecollectedinTable 8-3 .Finally,fortheheliummeasurement,the PWTwasroodedwithheliumviaapressurizedcanisterregula tedat10psi.Thehelium exitedthePWTintoacupofwater. Inair,thefrequencyresponseoftheDUT, H m ( f ),wasdeterminedsimplyasthe frequencyresponsefunctionrelatingtheoutputoftheDUT[V ]andtheoutputofthe calibratedreferencemicrophone[Pa],acalculationperfo rmednativelyinthePULSE software.TheBruelandKjr4138frequencyresponsemagni tudewasregardedasrat inthiscalculationandonlyrelativephasewasdetermined. Inhelium,concernsabout straticationofthegasmediumandresultingwavefrontdis tortionledtotheuseofthe substitutionmethod[ 19 172 173 ]toimprovemeasurementqualityinhelium. Thesubstitutionmethodrequiredtwomeasurementswiththe microphonesinoriginal andswappedpositionsasindicatedinFigure 8-4 .Let H o 12 and H s 12 representthemeasured 147

PAGE 148

nn r n r n n Figure8-3.Planewavetubesetupforacousticcharacteriza tion. Table8-3.Settingsformicrophonefrequencyresponsemeas urementsinPULSE. GroupingParameterSettinginAirSettinginHelium AcquisitionFFTTypeZoomBaseband CenterFrequency[kHz]3.5N/ABandwidth[kHz]6.425.6FrequencyRange300Hz{6 : 7kHz0Hz{25 : 6kHz #ofFFTLines6400FrequencyResolution[Hz]14WindowRectangularOverlap0%#ofAverages100 GeneratorSignalPseudorandomnoise FrequencyRange300Hz{6 : 7kHz300Hz{25 : 9kHz SpectralLines6400 frequencyresponsefunctionsintheoriginalandswappedpo sitions,respectively,relating theoutputofmicrophone2(theDUT)tothatofmicrophone1(th ereference)inunitsof V = V.Alsoletthefrequencyresponsefunctionsofthetwomicrop honesbedenoted H 1 and H 2 [V = Pa].IntheoriginalandswappedpositionsdepictedinFigur e 8-4 148

PAGE 149

A B Figure8-4.Microphoneswitchingprocedure.A)Originalpos itions.B)Swappedpositions. H o 12 = G o12 G o11 = H ab H 1 H 2 H 1 H 1 (8{2) and H s 12 = G s12 G s11 = H ba H 1 H 2 H 1 H 1 ; (8{3) where denotescomplexconjugate, G 12 cross-spectraldensity, G 11 isautospectraldensity, and H ab and H ba [Pa = Pa]arefrequencyresponsefunctionsrelatingtheactualpr essuresat thetwomeasurementlocations.AkeyassumptionofEquation s 8{2 and 8{3 isthatthere isnochangeinthepressureeldbetweenmeasurements,i.e. thepressuresatlocation aandlocationbremainunchanged.Tohelpadheretothisassu mption,noalterations tothestateofthemeasurementsetup,particularlytheacou sticsource,weremade betweenmeasurementssaveforswappingofthemicrophones, whichwasaccomplishedvia removingandrotatingthePWTendplate.MultiplyingEquati ons 8{2 and 8{3 together, notingthat H ab H ba =1,takingthesquareroot,andrearranging, H 2 = H 1 p H o 12 H s 12 : (8{4) Therefore,withthefrequencyresponsefunctionoftherefe rencemicrophone, H 1 well-known,thefrequencyresponsefunctionoftheDUT, H m ( f )= H 2 ( f ),canbe 149

PAGE 150

deducedfromthegeometricmeanofmeasurementsfor H o 12 and H s 12 evenwhenthe twomicrophonesarenotexposedtopreciselythesamepressu re.Overtherangeof measurementfrequencies,itissucienttoregard H 1 ashavingconstantmagnitude[ 50 ] andnon-constantphase.Thephaseroll-oisapproximately 7 : 5 by20kHz[ 174 ]. Duetothelow-frequencylimitationsoftheBMS4590compres siondriverusedin thePWTsetup,additionalmeasurementstocharacterizethe low-frequencyroll-oof thepiezoelectricmicrophonewereperformedatBoeingCorp oration.Twopiezoelectric microphones,138-1-I2-Dand138-1-J3-F,weretransferred toBoeingforthismeasurement andothers.ThemeasurementsetupispicturedinFigure 8-5 andconsistedoftheDUT andBruelandKjr4136referencemicrophonemountedinasm allacousticcavity (thoughFigure 8-5 Bshows2BruelandKjr4136microphonesmountedthere)tha twas drivenbyaspeakerandterminatedintoan\innite"(100ft) coppertube.Theinnite tubeterminationwasactuallydesignedtosuppresstheform ationofstandingwavesand enablehighfrequencymeasurements,butthisexistingsetu pwasstillattractiveforthe lowfrequencymeasurement. AnHP35670spectrumanalyzerprovidedabroadbandwhitenoise signaland acquiredtheDUTandreferencemicrophonesignals.Measurem entsettingsarefound inTable 8-4 .Usingthespectrumanalyzer,thefrequencyresponsefuncti onrelatingthe DUToutputtothecalibratedreferencemicrophoneoutput[V = Pa]wascalculated.The frequencyresponsefunctionwasthenpost-processedtocor rectforthelow-frequency roll-ointhereferencemicrophone.Thelow-frequencycal ibrationofthereference microphonewasobtainedat1/3octavebandsdownto10Hzusing aBruelandKjr UA0033electrostaticactuatorwithaG.R.A.S.actuatorsupply Type14AA.Thetypical 3dBlowerlimitingfrequencyforaBruelandKjr4136is0 : 3Hzto3Hz. 8.1.3.2Linearitymeasurementsetup Characterizationofmicrophonelinearityreferstothequa nticationofhowthe voltageoutputoftheDUTchangeswithsoundpressurelevel.M easurementswere 150

PAGE 151

nr rr nr n nnr n A B C Figure8-5.Innitetubemeasurementsetup.A)Measurements chematic.B)TwoBruel andKjr4136microphonesmountedintheacousticcavity.C) Insidethe acousticcavity. Table8-4.Frequencyresponsemeasurementsettingsusedat Boeing. ParameterSetting Bandwidth1 : 6kHz FFTLines1600FrequencyResolution1HzTestSignalBroadbandwhitenoise performedatbothUniversityofFloridaandatBoeingCorpora tion.Fromthecollected data,totalharmonicdistortionwascalculated,rewritten herefromEquation 2{13 in termsofpowerspectraldensityas[ 47 ] THD= vuuut 1 P n =2 G xx ( f n ) G xx ( f 1 ) 100% ; (8{5) where f 1 isknownasthefundamentalfrequency,excitationfrequenc y,orrstharmonic, and f n isthe n thharmonic.Assuminguniformmicrophonesensitivityateac h f n G xx can beregardedinunitsofPa 2 = HzorV 2 = Hz. 151

PAGE 152

AtUniversityofFlorida,thesamesetupusedtondthefreque ncyresponse(in air),picturedinFigure 8-3 ,wasusedtoobtaindataforthetotalharmonicdistortion calculation.Asingletonesignalat1kHzdrovetheBMS4590co mpressiondriver,which couldreachaSPLofapproximately160dBwithoutexceedingi tspowerrating.APCB PiezotronicsModel377A51precisioncondensermicrophone, withamaximumSPLof 192dB(3%distortion),wasconnectedtoaBruelandKjr267 0preamplierandserved asthereference.TheDUTandreferencemicrophoneoutputsig nalswerecollectedusing thesamesettingsfoundinTable 8-3 atmultiplepressurelevels.Startingfromthelowest SPLwithadetectable2ndharmonic,theSPLwasincreasedins tepsof3{4dBSPLup to160dB.The6 : 4kHzbandwidthenabledtherstsixharmonicstobecaptured. An importantconsiderationingettingareliablepressureref erenceusingthePWTforthis measurementwasthatharmonicshigherthanthe6thpropagat eashigher-ordermodes andthusdonotcontributeequallytotheresponseoftheDUTan dreferencemicrophone. Therefore,powerdistributedtofrequencies f n for n> 6mustbenegligibleinorderforthe calculationtobevalid. ExperimentalresultstobediscussedinSection 8.2.3.2 showthatthesetupof Figure 8-3 ,apartfromthemicrophones,suersfromsignicantharmon iccontamination. Speakerdistortionisonecontributor,togetherwithharmo nicgenerationduringnonlinear acousticpropagationathighsoundpressurelevels[ 28 ];thelattersourceofdistortion worsenswithpropagationdistance. AmeasurementsetupatBoeingCorporationwasdesignedspec icallytominimize harmoniccontaminationathighsoundpressurelevels.Phot ographsofthesetupare foundinFigure 8-6 .Themeasurementapparatus,anacousticalcoupler[ 35 ],wasbetter knownas\thewedge,"insideofwhichwasalow-volumecavity drivenbyfourmanifolded speakers.Areferencemicrophone(inthiscasetheBruelan dKjr49381/4"pressure eldmicrophonewithBruelandKjr2670-W-001preamplie r)andtheDUTwere mountedfacingeachother,asshowninFigure 8-6B ,atcloseproximity(0.231").Asingle 152

PAGE 153

P P P Pq Wedge DUT > Speakers A @ @ @R Referencemic DUT Wedge B Figure8-6.LinearitymeasurementsetupatBoeingCorporat ion.A)Viewoftheentire wedgexture,withspeakers.B)ReferencemicrophoneandDUT mountedin thewedge.PhotographscourtesyofBoeingCorporation. tonesignalat2 : 5kHzwaschosenbasedonthespeaker'sfrequencyresponsecha racteristics toprovidethehighestsoundpressurelevels.ThemaximumSP Lachievableinthewedge wasapproximately172dBandwaslimitedbythespeakers'pow errating.Measurement settingsforanHP35670Aspectrumanalyzer,usedtocollectt hedataandperformthe THDcalculationwithtenharmonicsincluded,arefoundinTab le 8-5 Table8-5.Totalharmonicdistortionmeasurementsettings usedatBoeing. ParameterSetting Bandwidth25 : 6kHz FFTLines400FrequencyResolution64HzWindowFlattopFundamentalFrequency2 : 5kHz #ofHarmonics10 8.1.4ElectricalCharacterizationSetup Thereweretwomajorgoalsintheelectricalcharacterizati onofthepiezoelectric microphones.First,themicrophone'snoiseroorwasmeasur edtoenablecalculationof theimportantminimumdetectablepressuremetric.Inaddit ion,electricalelementsfound inthelumpedelementmodelofChapter 5 ,including C eb (or C ef ), C eo R ep ,and R es 153

PAGE 154

wereextractedfromimpedancemeasurements.Finally,thet otalparasiticcapacitances thatservedtoattenuatethemicrophonesensitivitiesfrom opencircuitvalues, C ep + C ea wereestimatedforasingledeviceviadatafromacombinatio nofelectricalandacoustic measurements.8.1.4.1Noiseroormeasurementsetup Thissectiondetailsthemeasurementstrategyforthemicro phone'sintrinsicnoise roor.Section 2.3.2 addressedthepresenceofbothintrinsicandextrinsicnois einsensors. Theintrinsicnoiseroorisofprimaryimportancebecauseit indicatesthebest-achievable noisecharacteristicsoftheMEMSmicrophoneswheneectiv elyshieldedfromextrinsic noisesources.Referringtheintrinsicelectricaloutputn oisetothemicrophoneinputyields theminimumdetectablepressureofthemicrophone. Themeasurementsetup[ 38 ]ispicturedinFigure 8-7 .TheDUTisplacedinside atripleFaradaycage,whichservestoattenuateelectromag neticinterferencefromthe labenvironment.TwosetsofAAbatteriespoweredtheDUTbuer amplierat 3V. TheDUToutputsignalwasfedthroughtheinnermostFaradayca getothemiddle Faradaycage,whereitwasconnectedtoaStanfordResearchS ystems(SRS)ModelSR560 Low-NoisePreamplier(itselfbattery-powered)andampli edbyafactorof1000.The amplieroutputwasthenfedthroughtheoutertwoFaradayca gestoaSRSModelSR785 2ChannelDynamicSignalAnalyzer.Acustom-programmedLabv iewVIperformedthe datacollectionviacomputercontroloftheSR785andsavedt hemeasuredoutputpower spectraldensity[V 2 = Hz]. MeasurementsettingsarefoundinTable 8-6 .Thenoisepowerspectraldensityof theDUTwascollectedoveratotalbandwidthfrom0Hztothemaxi mumfrequencyof 102 : 4kHzusingmultipleseparatemeasurements,eachwiththeins trumentmaximum 800FFTlines.Employingmultiplefrequencyspansenabledm easurementswithbetter frequencyresolutionatlowfrequenciesandmoreblocksath ighfrequencies,where measurementtimewasdramaticallyreduced.Thestartanden dfrequencies,frequency 154

PAGE 155

nr n Figure8-7.TripleFaradaycagesetupfornoiseroorcharact erization. resolution,andnumberofblocksforeachspanareshowngrap hicallyinFigure 8-8 TheSR560noiseroorwasalsomeasuredindependentlyviasho rtingoftheinput andwassubtracted,intermsofPSD,fromtheDUToutputinallr esultspresentedin Section 8.2.4.1 beforethenoisewasinput-referred. Table8-6.Noiseroormeasurementsettings. InstrumentParameterSetting SpectrumAnalyzerFFTlinesperspan800 FrequencyResolution SeeFigure 8-8 #BlocksWindowHanningOverlap75% AmplierGain1000 FilterBandpass0 : 03Hz{300kHz ModeLowNoiseCouplingAC 155

PAGE 156

06.412.825.638.451.276.8102.4 881616163232 1k1k5k5k10k10k10k f [Hz] f [kHz] #Blocks Figure8-8.Noiseroormeasurementsspans,frequencyresolu tion,andaverages. 8.1.4.2Impedancemeasurementsetup Thegoaloftheelectricalimpedancemeasurementwastoobta inimpedancedatafrom whichelectricalparameterscouldbeextracted.InSection 5.2.4 ,anexpressionwasderived fortheelectricalimpedanceofapiezoelectricmicrophone Z eq = R es + R ep 1+ j!R ep ( C ef + C eo ) ; (8{6) Usingthisequation,theelements R es R ep ,and C ef + C eo wereextractedfromimpedance measurementsperformedon2ofeachdesignfromwafer116sec tion3(14measureddiein total).AnHP4294Aimpedanceanalyzer[ 175 ]togetherwithaCascadeMicrotechM150 probestationwereusedtoperformthemeasurement.TheHP429 4Autilizestheaccurate low-frequencyauto-balancing-bridgemethod[ 176 177 ]andafourterminalconguration thatreducestheeectsofleadimpedancesonthemeasuremen t[ 176 ].Themeasurement setupisshowninFigure 8-9 .Thetwoterminalsofeachpair(Lc/LpandHc/Hp)come togetherattheverytipoftheprobeneedleandallfourtermi nalgroundswereconnected attheprobeinput.CalibrationwasperformedusingaGGBInd ustriesCS-8impedance standardsubstrateoftheground-short(GS)conguration. Acustom-writtenprogramin HPInstrumentBASICcollectedandstored31completeimpedanc emeasurementsweeps foreachdevice(noon-boardaveraging),enablingpost-pro cessingtoestablishcondence bounds.MeasurementsettingsarefoundinTable 8-7 Foracapacitance-dominateddeviceapproximatelyinthera ngeof1{10pF, themaximumbiaserrorwasnotguaranteedintheoperationma nual[ 175 ]tobe below10%untilthemeasurementfrequencyexceededbetween 0.4and4kHz.The providedbiaserrorpredictionequationswereinfactinval idforimpedancesexceeding 156

PAGE 157

nnr n Figure8-9.Impedancemeasurementsetupusingaprobestati on. Table8-7.Impedancemeasurementsettings. ParameterSetting SweepTypeLogarithmicSweepRange1kHzto200kHzNumberofPoints801PointDelayTime0sSweepDelayTime0sOscillatorLevel500mVDCBiasOBandwidth3SweepAveragingOPointAveragingO approximately100Mn,wheremaximumbiaserrorpredictions easilysurpassed100% [ 178 ].Asaresult,themeasurementwasconductedstartingfrom1k Hz,atwhicha4pF capacitancemeasurementwasguaranteedtohave < 10%biaserror(or < 3%fora16pF measurement).Theinstrument-minimumfrequencywas40Hz. Althoughimpedancewasthemeasurand,admittanceisoftenam oreconvenient representationforpiezoelectrics.Impedancedatapost-p rocessedintoadmittanceform ( Y eq =1 =Z eq )wasusedforthemodelt, Y eq = j!R ep ( C ef + C eo )+1 ( j!R ep ( C ef + C eo )+1) R es + R ep ; (8{7) Thebenetoftheadmittanceformisthatwhen R es issmall,theadmittancereducesto theverysimpleexpression Y eq 1 =R p + j! ( C ef + C eo ).ThettoEquation 8{7 was performedusingtheMATLABfunction invfreqs ,whichlikemostcurve-ttingtools 157

PAGE 158

attemptstominimizetheweightedsumofthesquaredresidua lsbetweenthedataand tateachmeasurementpoint.Theparticularbenetofinvfr eqsisthatitisspecically formulatedtottransferfunctionstocomplexfrequencyre sponsedata.Theformof Equation 8{7 thatMATLABusesforttingis Y eq = B 1 s + B 2 A 1 s + A 2 ; (8{8) where A 1 =1alwaysbyconvention.ComparingtoEquation 8{7 ,theelectricalparameters wereextractedas C ef + C eo = B 1 A 2 B 2 =B 1 ; (8{9) R ep = A 2 B 2 1 B 1 ; (8{10) and R es = 1 B 1 : (8{11) Astatisticaldistributionfortheseparameterswasobtain edviarepeatedttingto perturbedmeanmeasurementsinMonteCarlosimulations.Fr omthesedistributions,the meanand95%condenceintervalwerecalculated.Furtherde tailsontheMonteCarlo simulationsandaccompanyinguncertaintyanalysisarefou ndinSection C.4 8.1.4.3Parasiticcapacitanceextractionsetup Theexpressionsforthefrequencyresponseofamicrophonep ackagedwithacharge orvoltageamplierweredevelopedinSection 5.2.3 ,includingapproximateexpressionsfor ratbandsensitivity.Fromthoseexpressions,itispossibl etoestimateparasiticcapacitance andopen-circuitsensitivitywithappropriatemeasuremen ts.Equations 5{47 and 5{53 predicttheratbandsensitivityforamicrophonepackagedw ithavoltageamplierand chargeamplier,respectively.Equatingtheopencircuits ensitivity, S oc ,thatappearsin Equations 5{47 and 5{53 andrearrangingyieldsanestimateforparasiticcapacitan ce, C ep + C ea = S ca S va C fb ( C ef + C eo ) : (8{12) 158

PAGE 159

Tomakeuseofthisexpression,frequencyresponsemeasurem entsreplacethesingle-valued sensitivitiesinEquation 8{12 toyield C ep + C ea = H m;ca ( f ) H m;va ( f ) C fb ( C ef + C eo ) ; (8{13) where H m;ca and H m;va representthefrequencyresponsefunctions[V = Pa]associated withasinglemicrophonepackagedconsecutivelywithachar geandvoltageamplier. Microphone116-1-J7-Awaspackagedsolelyforthispurpose .Packagedwiththevoltage amplierarchitecture,microphone116-1-J7-Asharedcomm onelectronicsarchitecture, includingconsistenttracelengths,amplier,etc.withth eotherpiezoelectricmicrophones; thissuggestedconsistentparasiticcapacitancecouldbee xpected.Withtheparasitic capacitanceknownfor116-1-J7-Aandassumingthatitremai nedessentiallyunchanged fromdevice-to-device,theopencircuitsensitivityofall microphoneswasestimatedfrom therearrangedEquation 5{47 S oc = S va C ef + C eo + C ep + C ea C ef + C eo : (8{14) Estimatingtheopen-circuitsensitivitiesofthemicropho nesinthiswayalsoenabled avoidanceofthesubstantialriskofdamageassociatedwith packagingandre-packagingall ofthemicrophoneswithbothvoltageandchargeamplierarc hitectures. Measurementsof H m;ca and H m;va formicrophone116-1-J7-Awereperformedinair usingthesamePWTsetupdescribedinSection 8.1.3.1 .Valuesfor C ef + C eo valueswere obtainedfromimpedancemeasurementspresentedinSection 8.2.4.2 undertheassumption thatelectricalpropertieswereconsistentdevice-to-dev ice. 8.1.5ElectroacousticParameterExtraction Extractionofelectroacousticparametersenablesvalidat ionofindividuallumped elementpredictions.Relativelysimpleelementsrepresen ting,forexample,theacoustic backcavityarewell-known[ 28 ].However,elementswhosevaluesarepredictedfromthe diaphragmmodel,includingthediaphragmcompliance C ad andmass M ad ,inadditionto 159

PAGE 160

theeectivepiezoelectriccoecient d a ,requirevalidation.Inthissection,experiments fortheirextractionaredescribed,withtheapproachdrive nbythemoredemanding needsforextractionofcomplianceandmass.Ameasurementp rocedureformicrophone sensitivitycompatiblewiththerequirementsoftheparame terextractionexperimentis alsoaddressed.8.1.5.1Complianceandmassmeasurementsetup Thediaphragmcomplianceandmass,asdenedinSection 5.2.1.2 ,arecalculated fromthediaphragmdisplacementduetopressureloading.In nomenclatureappropriate forthemeasurementsetting,theymayberedenedas C ad = Z 2 0 Z a 2 0 H pw j V =0 ( r; ) rdrd (8{15) and M ad = R 2 0 R a 2 0 a H pw j 2V =0 ( r; ) rdrd h R 2 0 R a 2 0 H pw j V =0 ( r; ) rdrd i 2 ; (8{16) where H pw j V =0 [m = Pa]isthefrequencyresponsefunctionobtainedundershort -circuit conditionsrelatingthelocation-dependentdisplacement w ( r; )topressureactingonthe diaphragm, a 2 istheouterdiaphragmdiameter,and a istheaerialdensity,denedin Equation 5{13 .Notethatbecause a changesabruptlyat r = a 1 ,theintegralover r inthe numeratorofEquation 8{16 mustbeevaluatedpiece-wise. FromEquations 8{18 to 8{16 ,extractionof C ad and M ad requirestheabilitytoapply aknownpressuretothediaphragmwhileopticallyscanningi tsdisplacement.Although ameasurementsetupcouldbedevisedthatwouldallowsimult aneousexcitationand opticalmeasurementofthepackagedmicrophones,thedesig nofmeasurementxtures providingopticalaccessforalaservibrometersystemwith initsdepth-of-eldwouldbe asignicantchallengeandexpense.Instead,asimplermeas urementsetupwasusedin whichspecially-packagedmicrophoneswereexcitedwithak nownpressureviatheirback cavitiesandtheaccompanyingdiaphragmdisplacementwasm easuredfromthefrontside. 160

PAGE 161

Thepackagingrequirementsforthismeasurementweredicta tedlargelybythedesire touseanexistingpressurecoupler[ 149 ].Togetherwiththeneedforcompatibilitywith thepressurecoupler,theneedtoenablemeasurementsofmic rophonefrequencyresponse functionsviainclusionofintegratedinterfaceelectroni csledtothechoiceofacircuit boardtohousethemicrophonedie.A0 : 059inthickboardmilledin-housefromFR-4, withthemicrophonedieepoxiedintoarecessatoneend,wasu sed.Anexplodedview ofthepressurecouplerassemblyandpackagingsolutionare showninFigure 8-10 .The circuitboardwasclampedintopositionovertheopentopsid eofthepressurecoupler's acousticcavitywithaLuciteendplate.A0 : 03in(762 m)diameterholecenteredwithin thedierecessinthecircuitboardcoupledthemicrophoneba ckcavitywiththepressure couplercavity,whileanopticalwindowonthefrontsideena bledlaseraccesstothe diaphragm,asshowninFigure 8-11 Thepressurecouplerprovidedtwoaccesspointsforacousti cpressuremeasurements withinthecavity.Areferencemicrophonewasmountedatnor malincidenceinaplugthat insertedintotheendofthecavity,aslabeledinFigure 8-10 .Meanwhile,theDUTwas mountedatgrazingincidenceapproximately9mmupthecavit y.Forlowfrequencysound withwavelengthmuchgreaterthanthisdimension,thepress ureswereapproximately equal.ThepressuresatthereferencemicrophoneandDUTloca tionswere90 outof phaseatquarterwavelengthseparation(9 : 5kHzdrivingfrequencyinair). InChapter 5 C ad and M ad wereextractedfromthetheoreticalpredictionofstatic diaphragmderection.Theyareequivalentlycalculablefro mdynamicmeasurementsat sucientlylowfrequencies(i.e.frequenciesmuchlowerth antheresonantfrequencyof thediaphragm).Withresonantfrequenciesupwardsof100kHz foralldevicesmeasured, excitationat1kHzwassucientlylowtobeconsideredquasistatic.Thewavelengthat 1kHz(34cm)wasalso38timesthetestandreferencemicrophon eseparationandthus morethansucienttoregardthepressuresatthetwolocatio nsasnearlyequal.Thiswas conrmedexperimentally. 161

PAGE 162

@ @ @ @ @ @ @ @ @R Optical window Reference micport H H H H H Hj Micdie J J J J] Circuitboard Z Z Z Z Z Z~ Acoustic cavity 1 Pressure coupler Endplate 9 Interfaceelectronics n n n n Speakerconnection Figure8-10.Pressurecouplerassembly(fastenersnotshow n). Themeasurementsetupforextractionofacousticmassandco mplianceisshownin Figure 8-12 .A1kHzsinusoidgeneratedbyanAgilent33120Afunctiongener ator 1 and 1 Althoughthelaservibrometersystempossessesitsownfunc tiongeneratoraspartofthescanner controller,intermittentproblemswithprolongedusageof sinusoidsledtotheuseoftheexternalfunction generator. 162

PAGE 163

A A AK Backside Pressure Excitation ScanningLaser Figure8-11.Closeupdepictionofamicrophonedieinthepre ssurecouplersetup. ampliedviaaStewartElectronicsPA-1008200WattPowerAmpl ierdrovetheBMS 4590Pcompressiondriver.Thereferencemicrophonewasamp liedusingaSRSModel SR560Low-NoisePreamplierandtheampliedsignalserveda sthereferenceinthelaser vibrometersystem'snativedataacquisitionsystem.There ferencemicrophonecalibration wasentereddirectlyintothelaservibrometersoftwaretoa voidtheneedtoadjustdatain apost-processingstep.Thevelocitysignalfromthelaserv ibrometeritselfwastheother inputforthetwo-channelsystem.Specicmeasurementsett ingsarecollectedinTable 8-8 Table8-8.Pressurecouplermeasurementsettings. ParameterSetting Span5kHzFFTLines400Resolution12 : 5Hz #ofaverages100(Complex)WindowRectangularSignal1kHzSineLVsensitivityDC1mm = s = V Typicalpressurelevel95dBto105dBSR560gain100 Diaphragmscansweretakenoverpolargridsof20azimuthalp ointsand13-15 radialpoints,dependingondiaphragmsize;Figure 8-13 showsonesuchgrid.Priorto integration,theactualmeasureddata|whichwasreturnedf romthelaservibrometer systemasascattereddataset|wasinterpolatedtoformasur faceviaMATLAB's 163

PAGE 164

nrrn rrr rr r r r rrr rr rr r rr r r "rr # $rrr! r Figure8-12.Experimentalsetupforextractionofacoustic massandcompliance. TriScatteredInterp [ 138 ].Independentsurfaceswerecreatedfortherealandimagin ary partofthefrequencyresponsefunctionandthenrecombined forintegrationinMATLAB's numericalroutine dblquad [ 138 ],whichemploysGaussquadratureoverarectangular domainintwodimensions.Theintegrationwasperformedin r spaceusingsurfaces originallyinterpolatedinCartesianspace. 500 m Figure8-13.Laservibrometerscangridoverlayedondesign Emicrograph(diaphragm outerdiameterof756 m). 164

PAGE 165

Inordertopredictthequalityoftheinterpolationandinte grationroutineapriori, atestnumericalintegrationwasperformedusingananalyti calexpressionforthetypical staticderectionshapeofaclampedplatesubjectedtoaunif ormpressureload[ 121 ], w ( r )= w 0 1 r a 2 2 ; (8{17) interpolatedattheactualmeasurementscanpoints.Errorw asfoundtobeapproximately 1%relativetotheassociatedanalyticalvolumedisplaceme nt, 8 = w 0 a 2 = 3.The integrationprocedurewasalsocomparedtotrapezoidalint egrationofthesametest problemanditwasconrmedthattheGaussquadratureroutin ewasmoreaccurateby severaltenthsofapercent.8.1.5.2Frequencyresponsemeasurementsetup Therequirementofshortcircuitconditions,inadditionto inputchannellimitations ofthelaservibrometersystem,didnotenablesimultaneous acquisitionofmicrophone electricaloutputduringtheactualparameterextractione xperiment.Instead,the electricalacquisitionwasdoneinaseparatemeasurement, alsointhepressurecoupler,to determinethemicrophonesensitivities. n r n Figure8-14.Experimentalsetupforpressurecouplercalib ration. First,therelationshipbetweenthepressuresatthetwomea surementlocationswas conrmedviatheexperimentalsetuppicturedinFigure 8-14 ,inwhichtwoBrueland Kjr4138microphonesweremountedatthereferenceandDUTpo sitions.Thefrequency 165

PAGE 166

responsefunctionbetweenthetwomicrophones[Pa = Pa]wasthencomputedusingthe BruelandKjrPULSEsystemandsoftware. Fortheactualsensitivitymeasurement,theexperimentals etupofFigure 8-15 wasused,withtheDUTandreferencemicrophoneinstalledass hown.Againutilizing thePULSEsystem,thefrequencyresponsefunctionbetweenth eDUTandreference microphonewascomputed.Measurementsettingsforbothset sofmeasurementsarefound inTable 8-9 nr r Figure8-15.Experimentalsetupformicrophonecalibratio ninthepressurecoupler. Table8-9.Settingsforsensitivitymeasurementofpressur ecouplermicrophones. GroupingParameterSetting AcquisitionFFTTypeZoom CenterFrequency1 : 9kHz Bandwidth3 : 2kHz FrequencyRange300Hz{3 : 5kHz #ofFFTLines3200FrequencyResolution1HzWindowRectangularOverlap0%#ofAverages100 GeneratorSignalPseudorandomnoise FrequencyRange300Hz{3 : 5kHz SpectralLines3200 166

PAGE 167

8.1.5.3Eectivepiezoelectriccoecientmeasurementset up Theexpressionfortheeectivepiezoelectriccoecient,E quation 5{8 ,maybewritten forthemeasurementsettingas d a = Z 2 0 Z a 2 0 H vw j p =0 ( r; ) rdrd; (8{18) where H vw isthefrequencyresponsefunctionrelatingthedisplaceme nt w ( r; )toan excitationvoltage.Thesubscript p =0 followsfromthetheoreticaldenitionof d a and denotesanacousticshortcircuitcondition.Suchaconditi onisonlyachievablefor excitationwellbelowthevent/cavitybreakfrequency,whi chbasedonothersensitivity measurementsmustbeinthevicinityof50Hz.Becausethemeas urandofthelaser vibrometer,velocity,is / f foraharmonicinput[ 34 ],thesignal-to-noiseratioofthe measurementdegradesconsiderablyatlowfrequencies.Ins tead, H vw j p =0 wasobtained viaexcitationat1kHz.Thediaphragmdisplacementduetovol tageexcitationforthe n r n r Figure8-16.Experimentalsetupforextractionofeective piezoelectriccoecient. 167

PAGE 168

devicespackagedasdescribedinSection 8.1.5.1 wasmeasuredviatheexperimentalsetup showninFigure 8-16 .Inthissetup,thecircuitboardshousingthemicrophonesw ere axeddirectlytothemicroscopestageunderthelaservibro meterandelectricallydriven witha1kHzsinusoidalwaveformdeliveredviaprobeneedles. Theinterfaceelectronics presentontheboardweredisconnectedfromthemicrophonef orthismeasurement.The measurementsettingsforthelaservibrometerscanwerethe sameasthoseinTable 8-8 andthesameintegrationstrategydescribedinSection 8.1.5.1 wasalsoused. 8.2ExperimentalResults Experimentalresultsforeachofthemeasurementsdiscusse dinSection 8.1 arefound inthissection.Calculationdetailsfor95%condenceinte rval( U 95% )estimatespresented withmanyexperimentalresultsarefoundinAppendix C 8.2.1DieSelection Aseriesofwafermapscapturingthevariationof f r and S a; 0 overportionsofwafers 116and138arepresentedinFigures 8-17 { 8-22 .Outliersweredetectedandremovedfrom thedatasetsviathemethoddiscussedinSection 8.1.1 priortomapping.Inall,14/249die (5.6%)fromwafer116and7/190die(3.7%)fromwafer138were identiedasoutliers. -2 -1 0 1 2 A -2 -1 0 1 2 B Figure8-17.Mapsofdicedsectionofwafer116(alldesigns) withcolorcorrespondingto thenumberofstandarddeviationsfromindividualmeanofea chdesign.A) f r .B) S a; 0 Figure 8-17 and 8-18 respectivelyshowmapsofthewafer116and138subregionsin termsofthenumberofsamplestandarddeviationseachdiewa sfromthesamplemeanfor itsparticulardesign.Atrendclearlyexistedacrossbothw afers,with f r and S a; 0 trending 168

PAGE 169

-2 0 2 A -2 -1 0 1 B Figure8-18.Mapsofdicedsectionofwafer138(alldesigns) withcolorcorrespondingto thenumberofstandarddeviationsfromindividualmeanofea chdesign.A) f r .B) S a; 0 oppositelywithrespecttoeachotheracrosswafer116butla rgelythesamewaywith respecttoeachotheracrosswafer138. 140 145 150 155 A 135 140 145 B 120 125 130 135 C 110 112 114 116 118 120 122 D 95 100 105 110 115 E 94 96 98 100 102 F 90 95 100 G Figure8-19.Resonantfrequencymapsforwafer116[kHz].A)De signA.B)DesignB.C) DesignC.D)DesignD.E)DesignE.F)DesignF.G)DesignG. Figure 8-19 andFigure 8-20 show f r and S a; 0 ,respectively,forwafer116inindividual mapsforeachdesign.TrendsareclearfordesignsA{D,withre sonantfrequency increasingawayfromthewafercenteranddisplacementsens itivitydecreasing.In Figure 8-19 (E{G),thelackofacorrespondingcross-wafertrendin f r fordesignsE{G, withlargerdiaphragmsthataremoresusceptibletobucklin g,mayindicatethediaphragm responsetostressisnotstabledie-to-dieforthesedesign s. 169

PAGE 170

2 2.2 2.4 2.6 2.8 3 3.2 A 1.8 2 2.2 2.4 2.6 2.8 B 1.4 1.6 1.8 2 2.2 C 1.2 1.3 1.4 1.5 1.6 1.7 1.8 D 0.8 1 1.2 1.4 1.6 1.8 E 0.6 0.8 1 1.2 F 0.5 1 1.5 G Figure8-20.Centerdisplacementsensitivitymapsforwafe r116[nm = V].A)DesignA.B) DesignB.C)DesignC.D)DesignD.E)DesignE.F)DesignF.G)D esign G. Mapsfor f r and S a; 0 onwafer138,Figure 8-21 and 8-22 ,respectively,showan entirelydierenttrendthanwafer116.Onwafer138,theres onantfrequencyand sensitivitybothdecreasetogethertowardtheoutsideofth ewaferandthetrendis consistentforalldesigns. Table 8-10 collectsthesamplemeans(denotedwithoverbars)andsampl estandard deviations( s )of f r and S a; 0 foreachdesign.Asexpected,theresonantfrequency decreaseswithdesignletter,rerectingtheexpectedincre aseincompliancewithdiaphragm size.Perhapsunexpectedly, S a; 0 actuallydecreaseswithdiaphragmsizeforwafer116,but againthebucklednatureofthosediaphragmsmakesdrawingc onclusionsdicult.For wafer138, S a; 0 remainsessentiallyconstantforalldesigns,whichisalso unexpected.Itis showninSection 8.2.5 thatthenearlyconstant S a; 0 acrossalldesignsisonlyindicativeof aconsistentcenterderectionandthatcontrarytoexpectat ions,thepiezoelectriccoupling coecientdoesnotnecessarilytrendstronglywithcenterd erection. 170

PAGE 171

180 200 A 160 170 180 B 130 140 150 C 110 120 130 D 100 105 110 115 E 85 90 95 100 F 80 90 G Figure8-21.Resonantfrequencymapsforwafer138[kHz].A)De signA.B)DesignB.C) DesignC.D)DesignD.E)DesignE.F)DesignF.G)DesignG. 1.2 1.4 1.6 1.8 A 1 1.2 1.4 1.6 1.8 B 1.2 1.4 1.6 1.8 C 0.8 1 1.2 1.4 1.6 1.8 D 0.8 1 1.2 1.4 1.6 1.8 E 1 1.5 F 1 1.5 G Figure8-22.Centerdisplacementsensitivitymapsforwafe r138[nm = V].A)DesignA.B) DesignB.C)DesignC.D)DesignD.E)DesignE.F)DesignF.G)D esign G. Oneormoremicrophonesofseveraldesignswereselectedtob epackagedforrigorous characterizationandparameterextraction.Foreachdesig n,thediewiththehighest valuesof S a; 0 weregenerallyselectedforpackagingasmicrophoneswitht heexpectation thatthepiezoelectriccouplingcoecient d a ,andthussensitivity,wouldtrendsimilarly. Section 8.2.5 addressesthevalidityofthisassumption.Astheonlyotherc omparative measureavailable, f r wasusedasasecondmetricforchoosinglikemicrophonesoft he 171

PAGE 172

Table8-10.Waferstatistics. Wafer116Wafer138 f r [kHz] S a; 0 [nm = V] f r [kHz] S a; 0 [nm = V] Design f r s f r S a; 0 s S a; 0 f r s f r S a; 0 s S a; 0 A149.47.22.70.3195.010.01.60.3B138.55.72.20.2166.08.81.60.4C129.34.41.80.3142.98.91.60.3D118.14.41.50.2122.47.91.50.4E109.05.31.30.3107.55.81.50.4F100.13.41.10.295.86.11.50.3G92.63.90.90.383.66.61.50.3 Table8-11.Pre-andpost-packagingLVmeasurements. Pre-PackagingPost-Packaging DUT f r [kHz] S a; 0 [nm = V] f r [kHz] S a; 0 [nm = V] 116-1-I6-A147.33.18127.03.79116-1-C4-B133.02.51127.52.17116-3-F7-B144.02.08160.31.88116-1-E2-C124.92.11123.52.10138-1-E4-D132.61.80126.91.89138-1-I2-D133.51.91116.51.68138-1-I8-E114.41.85106.41.63138-1-H3-F103.91.8299.91.45138-1-J3-F103.91.87129.51.46 samedesign.Inall,10dieweresuccessfullyshepherdedthr oughthepackagingprocess describedinSection 7.3 ,withoneusedexclusivelyfortheparasiticcapacitanceex traction. Pre-andpost-packaginglaservibrometermeasurementsoft heninepackaged microphonesarecollectedinTable 8-11 .Changesinresonantfrequencywerelikelydue toinadvertentintroductionofpackagingstresstothemicr ophonediaphragmduringthe dieepoxystep,whichmodiestheeectivecomplianceofthe diaphragm.Acorresponding changein S a; 0 duetopackagingstresswasalsoobserved.Figure 8-23 showsthattheshifts in f r and S a; 0 followingthepackagingprocesswerenotatallconsistent, particularlythe directionoftheshifts.Thissuggeststheepoxyisnotentir elyconsistent;somenotable 172

PAGE 173

I6-A C4-B F7-B E2-C E4-D I2-D I8-E H3-F J3-F 20 10 0 10 20 DUT%Change f r S Figure8-23.Changesinresonantfrequencyanddisplacemen tsensitivityduetopackaging. possibilitiesthatwerenotinvestigatedfurtherareuneve nsealingofthemicrophonedieto theboardorepoxypenetratingslightlyintothebackcavity 8.2.2DiaphragmTopography Thediaphragmstaticderectionprolesforpre-andpost-pa ckagedmicrophones areshowninFigures 8-24A { 8-24B ,respectively.Sixdevicesrepresentingonebatchof packagedsensorsareincluded,andjusttheinnerregionsof thediaphragmsareshownfor clarity.Displacementsarereferencedtothesurroundings ubstratesandweretakenalonga linebisectingthediaphragmthroughtheventhole. MicrophoneC4-B,theonlydeviceincludedinFigure 8-24 thathailedfromwafer 116,isshowntobesignicantlybuckled.Thiswasexpectedg iventhevisiblebuckling ofalldevicesonwafer116.Thetotalderectionovertheinne rportionofthediaphragm beforepackagingwasapproximately2 : 7 mcomparedtoatotalthicknessinthatregionof 2 : 14 m.Thestaticderectionprolesoftheunpackagedmicrophon eshailingfromwafer 138wereconsistentandmuchlowerthanmicrophoneC4-B,typ icallyabout300nmfrom edgetocenter. Afterpackaging,thebuckledamplitudeofC4-Bwasreducedto approximately 2 : 3 mandthestaticderectionprolesofthewafer138deviceswe renolongerastightly grouped.Figure 8-25 showsthedierencesinstaticderectionafterthepackagin gprocess, 173

PAGE 174

400 2000200400 1 0 1 2 Substrate Radius[ m]StaticDerection[ m] 116-1-C4-B 138-1-E4-D 138-1-I2-D 138-1-I8-E 138-1-H3-F 138-1-J3-F A 400 2000200400 1 0 1 2 Substrate Radius[ m]StaticDerection[ m] 116-1-C4-B 138-1-E4-D 138-1-I2-D 138-1-I8-E 138-1-H3-F 138-1-J3-F B Figure8-24.Staticderectionprolesofseveralmicrophon ediaphragms(innerportions). A)Beforepackaging.B)Afterpackaging. 400 2000200400 0 : 6 0 : 4 0 : 2 0 0 : 2 Radius[ m]StaticDerectionDi.[ m] 116-1-C4-B 138-1-E4-D 138-1-I2-D 138-1-I8-E 138-1-H3-F 138-1-J3-F Figure8-25.Staticderectiondierencesforpre-andpostpackagedmicrophones. whichweretypicallyaround10nmintotalforwafer138micro phonesandclearlynotof anaxiallysymmetricnature. 174

PAGE 175

8.2.3AcousticCharacterization8.2.3.1Frequencyresponse Thefrequencyresponsefunctionmeasurementsmadeinheliu marecollectedin Figure 8-26 ,shownintermsofmagnitudeandrelativephasetotherefere ncemicrophone. Thefrequencyresponsemagnitudeisrattowellwithinthest atedgoalof 2dBover theportionoftheaudiobandmeasured(300Hz{20kHz).Deviati onsinthemagnitude andphasecloseto20kHzaretheresultofhigher-orderacoust icmodesbeginningto propagate.NotethatthephaseisrelativetotheBruelandKj r4138.Themicrophones werephase-matchedto < 2 outto15kHz. 05101520 100 95 90 85 80 Magnitude[dBre1V = Pa] 116-1-I6-A 116-1-C4-B 116-3-F7-B 116-1-E2-C 138-1-E4-D 138-1-I2-D 138-1-I8-E 138-1-H3-F 138-1-J3-F 05101520 0 45 90 135 180 Frequency[kHz]RelativePhase[ ] 116-1-I6-A 116-1-C4-B 116-3-F7-B 116-1-E2-C 138-1-E4-D 138-1-I2-D 138-1-I8-E 138-1-H3-F 138-1-J3-F Figure8-26.Microphonefrequencyresponsesinhelium. ThesensitivitiesoftheMEMSmicrophonesarecollectedinT able 8-12 inbothdB and V = Paformeasurementsperformedinair.Thesensitivitiesinh eliumwerealllower thaninairasexpected,byupto0 : 2dB(2 : 3%).Thephaseroll-oinheliumwaslessthan inairbyapproximately5 at6kHz. 175

PAGE 176

Table8-12.Microphonefrequencyresponsecharacteristic s y at1kHzinair. Magnitude DUTdBre1V = Pa V/PaRelativePhase[ ] 116-1-I6-A 90 : 68 0.0629.2 0.2176.8 0.1 116-1-C4-B 89 : 24 0.0634.5 0.2177.6 0.1 116-3-F7-B 90 : 87 0.0628.6 0.2177.0 0.1 116-1-E2-C 88 : 52 0.0637.5 0.2177.6 0.1 138-1-E4-D 89 : 86 0.0632.1 0.2177.3 0.1 138-1-I2-D 89 : 77 0.0632.5 0.2177.9 0.1 138-1-I8-E 88 : 71 0.0636.7 0.2178.1 0.1 138-1-H3-F 87 : 19 0.0643.7 0.3178.3 0.2 138-1-J3-F 88 : 25 0.0638.7 0.3178.0 0.2 y Uncertaintiesarefor95%condenceinterval(seeSection C.2 ). Thenormalizedfrequencyresponsemeasurementsformicrop hones138-1-I2-Dand 138-1-J3-FobtainedatlowfrequenciesinBoeingCorporati on's\innite"tubesetupare capturedinFigure 8-27 .The 2dBfrequenciesfor138-1-I2-Dand138-1-J3-Fwere85Hz and69Hz,respectively,whichcomparedwellwiththeoretica lpredictionsof70Hzand 75Hz.8.2.3.2Linearity Figure 8-28 showsplotsofDUTvoltageversusthereferencemicrophonepr essure level(bothtakenatthefundamentalfrequencyof1kHz)inbot hlinearunitsanddecibels for7ofthe9microphones.Somevariationfromlinearitycan bedetectedforseveral devicesinFigure 8-28A ,mostnotably116-1-I6-Aand116-1-C4-B.Theresponseofth e remainingtwodevices,116-3-F7-Band116-1-E2-C,areshow ninFigure 8-29 ,withabrupt deviationsfromlinearityhappeningnear1000Paand1500Pa ,respectively.Thisbehavior canlikelybeattributedtosuddensnap-throughofthebuckl eddiaphragms,anonlinear dynamicevent.Furtherinvestigationofthisunwantedphen omenonisbeyondthescopeof thisstudy,buttheinterestedreaderisreferredtovarious textsonnonlineardynamicsof structures[ 179 180 ]. 176

PAGE 177

10 1 10 2 10 3 4 2 0 Norm.Magnitude[dB] 138-1-I2-D 138-1-J3-F 10 1 10 2 10 3 5 0 5 Frequency[Hz]Norm.Phase[ ] 138-1-I2-D 138-1-J3-F Figure8-27.Piezoelectricmicrophonefrequencyresponse functionsatlowfrequencies normalizedtovaluesat1kHz TheTHDcalculationsforall9microphonesareshowninFigure 8-30 .Thelarge levelsofdistortion(30{40%)forthereferencemicrophone ,whichbyspecicationshould notexceed3%until190dB,indicatethemeasurementenviron mentisharmonic-rich. Nonlinearitiesintheamplier,speaker,andacousticpropa gationpathareallpossible contributors.Asaresult,thecalculatedTHDofFigure 8-30 arenotvalidinan absolutesense,thoughcomparisontothereferencemicroph one\THD"providesvaluable qualitativeinformation.ForDUTTHDthatalignscloselyenou ghtothatofthereference (asisthecasewithallwafer138microphones),onecanberea sonablycondentthatthe distortionlimitiswellabove160dB.Thesamecannotbesaid denitivelyforthewafer 116microphones,whichexhibitvaryinglevelsofdeviation fromthereferencemicrophone \THD." TheresultsoftheBoeinglinearitymeasurementsarecollec tedinTable 8-13 .The calculatedtotalharmonicdistortioninboththeDUTandrefe rencemicrophonearegiven foreachtest,andSPLsarereportedasthepressuremeasured at2 : 5kHz(thefundamental 177

PAGE 178

05001 ; 0001 ; 5002 ; 000 0 25 50 75 Pressure[Pa]Voltage[mV] 116-1-I6-A 116-1-C4-B 138-1-E4-D 138-1-I2-D 138-1-I8-E 138-1-H3-F 138-1-J3-F A 6080100120140160 120 100 80 60 40 20 SPL[dBre20 Pa]Voltage[dBre1V] 116-1-I6-A 116-1-C4-B 138-1-E4-D 138-1-I2-D 138-1-I8-E 138-1-H3-F 138-1-J3-F B Figure8-28.Linearitymeasurements.A)Linearscale.B)Ind B. 05001 ; 0001 ; 5002 ; 000 0 25 50 75 100 125 Pressure[Pa]Voltage[mV] 116-3-F7-B 116-1-E2-C Figure8-29.Linearitymeasurementsshowingunusualnonli nearbehavior. frequency)usingeachmicrophone.BecauseboththeBruela ndKjr4938andDUT distortwhilealsoservingassourcesoftheSPLmeasurement ,thereportedSPLsmustbe regardedaslowerboundsonthedistortionlimits.Thatis,r eporteddistortionoccursata SPLgreaterthanthatgivenhere.Therefore,thedevice1381-J3-Falmostcertainlymeets the172dBspecicationforPMAXgiveninSection 1.2 178

PAGE 179

100120140160 0 10 20 30 SPL[dB]THD[%] 116-1-I6-A PCB377A51 A 100120140160 0 10 20 SPL[dB]THD[%] 116-1-C4-B PCB377A51 B 100120140160 0 20 40 SPL[dB]THD[%] 116-3-F7-B PCB377A51 C 100120140160 0 20 40 SPL[dB]THD[%] 116-1-E2-C PCB377A51 D 100120140160 0 10 20 30 SPL[dB]THD[%] 138-1-E4-D PCB377A51 E 100120140160 0 10 20 30 SPL[dB]THD[%] 138-1-I2-D PCB377A51 F 100120140160 0 10 20 30 SPL[dB]THD[%] 138-1-I8-E PCB377A51 G 100120140160 0 10 20 30 SPL[dB]THD[%] 138-1-H3-F PCB377A51 H 100120140160 0 10 20 30 SPL[dB]THD[%] 138-1-J3-F PCB377A51 I Figure8-30.THDmeasurements.A)116-1-I6-A.B)116-1-C4-B.C )116-1-F7-B.D) 116-1-E2-C.E)138-1-E4-D.F)138-1-I2-D.G)138-1-I8-E.H) 138-1-H3-F.I) 138-1-J3-F. 179

PAGE 180

Table8-13.THDmeasurementsperformedatBoeingCorporatio n. MeasurementMicrophoneSPL[dB]THD[%] 1138-1-I2-D166.03.0 BruelandKjr4938167.62.4 2138-1-J3-F171.62.9 BruelandKjr4938171.311.5 8.2.4ElectricalCharacterization8.2.4.1Noiseroor Figure 8-31 showsthemeasuredoutput-referrednoiseroor.Eightofthe nine microphonesshowverysimilarbehavior,withone(138-1-I8 -E)servingastheoutlier.As predicted,thenoiseassociatedwith R ep dominatesatlowfrequenciesbeforeapproaching thethermalnoiseroor,wherethedominantnoisecontributo rtransitionstothebuer amplier.Theamplier'scurrentnoiseclearlydominateso veritsvoltagenoise,as predicted,sincethenoiselevelat100kHziswellabovethevo ltagenoiselevelof 8nV = p Hzto10nV = p Hz( 162dBto 160dB). AlthoughdierencesbetweenthenoisecurvesofFigure 8-31 aresmall,theyare greaterthanthemeasurementuncertainty(refertoSection C.3.1 ),andthemicrophones dohavesuccessivelylowernoiseroorsasthediaphragmdiam eterincreases(A F). ThisbehaviorisconsistentwithpredictionsinSection 5.3.3.1 ,whichshowedthatoutput noisePSDassociatedwith R ep intheroll-oregionwas / 1 =R ep C 2 et ;thisimpliesthatas predicted,theincreasein C 2 et acrossdesignswasdominantcomparedtothecorresponding decreasein R ep .Inaddition,thelowerampliercurrentnoisecontributio nbeyondthe cornerfrequencyfordesignswithlargediaphragmdiameter swasattributedtothereduced electricalimpedance(highercapacitance)ofthedevicesp erEquation 5{65 180

PAGE 181

10 1 10 2 10 3 10 4 10 5 160 140 120 Frequency[Hz]NoisePSD[dBreV/Hz 1 2 ] 116-1-I6-A 116-1-C4-B 116-3-F7-B 116-1-E2-C 138-1-E4-D 138-1-I2-D 138-1-I8-E 138-1-H3-F 138-1-J3-F Figure8-31.Output-referrednoiseroors. 10 1 10 2 10 3 10 4 10 5 20 40 60 80 Frequency[Hz]MDP[dBre20 Pa/Hz 1/2 ] 116-1-I6-A 116-1-C4-B 116-3-F7-B 116-1-E2-C 138-1-E4-D 138-1-I2-D 138-1-I8-E 138-1-H3-F 138-1-J3-F Figure8-32.Minimumdetectablepressurespectra. Figure 8-32 showstheminimumdetectablepressurespectraofthemicrop hones, calculatedfromthemeasuredoutputnoisePSD S v o as MDP=20log 10 0@ q S v o = j S va j 2 20 Pa = Hz 1 = 2 1A ; (8{19) with j S va j takenfromTable 8-12 .Because j S va j isaratbandsensitivityvalue,this calculationisnotvalidinthevicinityofandbeyond f 2dB Duetodierencesinsensitivity,theminimumdetectablepr essurecurvesareless tightlygroupedcomparedtotheoutput-referredequivalen ts.Figure 8-32 showsthatthe noiseroorintheaudiobandisbelow70dBforallmicrophones ,andbelowthetarget 1kHznarrowbinMDPof48 : 5dB,savefortheoutlier,138-1-I8-E.By20kHz,minimum 181

PAGE 182

detectablepressurelevelsdeclineto25{30dBSPL.Theunce rtaintiesintheMDPspectra rangefrom 0 : 10dBinthersttwofrequencyspans( < 12 : 8kHz)downto 0 : 04dB above38 : 4kHz;refertoSection C.3.1 fordetails. 10 1 10 2 10 3 10 4 10 5 160 140 120 Frequency[Hz]NoisePSD[dBre1V = Hz 1 = 2 ] VoltageAmp ChargeAmp A 10 1 10 2 10 3 10 4 10 5 40 60 80 Frequency[Hz]MDP[dBre20 Pa = Hz 1 = 2 ] VoltageAmp ChargeAmp B Figure8-33.Noiseroorspectrafor116-1-J7-Awithavoltage ampandchargeamp.A) NoisePSD.B)MDP. Thenoiseroorspectrafordevice116-1-J7-Apackagedwitha voltageandcharge amplieraregiveninFigure 8-33 .Figure 8-33A showsthatthenoisespectraofthe systemisnearly10dBhigherwhenthedeviceispackagedwith achargeamplier.Forthe chargeampliercaseatlowfrequencies,theequivalentres istor R ep k R efb isasourceof morenoisethanjust R ep inthevoltageampliercase.Athigherfrequencies,itisli kely thattheapproximately2 greatervoltagenoiseoftheOPA129amplierusedinthe chargeampliercircuitcomparedtotheLTC6240amplierus edinthevoltageamplier circuitdominates,especiallygiventheextragainfactorf orthisnoisesourceassociated withthechargeampliercircuitry,(1+ C et =C efb ) 2 (perEquation 5{75 ).IntermsofMDP, thechargeampliercongurationyieldsaminimumdetectab lepressureapproximately 6 : 5dBgreaterthanthatofthevoltageamplierconguration, evendespitethehigher sensitivityofthemicrophonefortheformer(discussedinS ection 8.2.4.3 ). FromthedatapresentedinFigure 8-32 ,severalvariantsonminimumdetectable pressurewerecomputedandarepresentedinTable 8-14 withestimated95%condence 182

PAGE 183

Table8-14.Minimumdetectablepressuremetrics. DUTdB y dBOASPL z dB(A)OASPL x 116-1-J7-A # 54.395.188.6 116-1-J7-A47.688.182.0116-1-I6-A45.787.280.4116-1-C4-B43.785.678.1116-3-F7-B45.587.280.4116-1-E2-C42.885.077.3138-1-E4-D43.585.378.3138-1-I2-D43.985.378.2138-1-I8-E50.689.286.3138-1-H3-F40.282.775.0138-1-J3-F40.483.275.4 y Narrowbin( f =1kHz, f =1Hz); j U 95% j < 0 : 10dB z j U 95% j < 0 : 01dB x j U 95% j < 0 : 007dB(A) # Packagedwithchargeamplier intervals(refertoSection C.3 fordetails).First,thealready-discussednarrowbinMDP isgiven.ThesecondandthirdcolumnsofTable 8-14 containintegratedmeasuresof MDP,theoverallsoundpressurelevel(OASPL)andA-weightedo verallsoundpressure level(AOASPL).Inbothcases,integrationofthenoiseroorw ascompletedoverthe individual1/3octavebandsfrom20Hz{20kHz,withA-weighting [ 33 ]alsoemployedin thelattercasebeforenalsummation.Everymicrophonehad anMDPmorethan3dB belowthespecicationof93dBOASPLexcept116-1-J7-Apacka gedwiththecharge amplier.TheA-weightedMDPislowerinallcasesbecauseA-we ightingde-emphasizes noisecontributionsatfrequenciesbelow1kHz,wheretheMDP spectrumsarethehighest. 8.2.4.2Impedance Atypicalimpedancemeasurement,presentedintermsofther ealandimaginary componentsofadmittance(conductance, G ,andsusceptance, B )arefoundinFigure 8-34 Thestandardandoveralluncertaintiesassociatedwith G and B arealsoincluded.The curvetdoesanexcellentjobmatchingsusceptance,whichi scapacitance-dominated; typicalR-squaredvaluesforthattwereunity.Thegeneral characteroftheconductance isalsocaptured,thoughclearlythereisroomforimproveme ntinthemodel;typical 183

PAGE 184

10 3 10 4 10 5 10 8 10 7 Conductance, G [S] Measured Fit 10 3 10 4 10 5 10 11 10 10 10 9 10 8 UncertaintyBounds[S] s G b G U 95% 10 3 10 4 10 5 10 7 10 6 10 5 Frequency[Hz]Susceptance, B [S] Measured Fit 10 3 10 4 10 5 10 10 10 8 Frequency[Hz]UncertaintyBounds[S] s B b B U 95% Figure8-34.Admittancemeasurementsandtsformicrophone B5-E. R-squaredvaluesforthistwere0.84.Notethattheresonanc eatapproximately118kHz isnotcapturedduetosimplicationsmadeintheequationfo relectricalimpedance Extractedparameters,togetherwiththeir95%condencebo unds,arecollected andcomparedwiththetheoryinTable 8-15 ,inwhichthenalletterintheDUTlabel continuestostandforthedesign.Capacitancepredictions arewithin7%to15%of extractedvalues,withthepredictionsimprovingwithdiap hragmsize.Thedierence betweenmeasuredandtheoreticalvaluesisessentiallycon stantatapproximately1pF foralldesigns,suggestingthepresenceofadditionalpara siticsand/orinherentbias intheparallelplatecapacitanceprediction.Itiswell-kn ownthattheparallel-plate approximationtendstounderpredictcapacitance,withthe underpredictionbecoming moresevereaslateraldimensionsapproachtheelectrodese parationdistance[ 181 ]. 8.2.4.3Parasiticcapacitanceextraction Figure 8-35 showsthefrequencyresponsefunctionsofmicrophone116-1 -J7-A packagedwithavoltageandchargeamplier.Themicrophone 'ssensitivitywhenpackaged 184

PAGE 185

Table8-15.Extractedelectricalparameters. Measurement y Theory z DUT C ef + C eo [pF] R ep [Mn] R es [kn] C ef + C eo [pF] R ep [Mn] 116-3-B9-A6.9 0 : 03%96 4%2.6 2%5.9401 116-3-C5-A6.9 0 : 03%98 4%2.5 2%5.9401 116-3-B8-B7.5 0 : 03%131 7%2.5 2%6.5362 116-3-C6-B7.5 0 : 03%127 7%2.4 2%6.5362 116-3-B7-C8.2 0 : 03%122 6%2.0 2%7.2326 116-3-C7-C8.2 0 : 02%120 7%2.0 2%7.2326 116-3-C8-D9.1 0 : 02%116 6%1.6 2%8.2286 116-3-E9-D9.2 0 : 02%115 6%1.6 2%8.2286 116-3-B5-E10.2 0 : 02%116 6%1.3 2%9.2253 116-3-C9-E10.1 0 : 02%113 6%1.3 2%9.2253 116-3-B4-F11.3 0 : 02%115 6%1.0 2%10.4225 116-3-C10-F11.3 0 : 02%116 6%1.0 2%10.4225 116-3-B1-G12.8 0 : 02%112 6%0.9 2%12.0195 116-3-D5-G12.8 0 : 02%112 6%0.9 2%12.0195 y Uncertaintiesare95%condenceintervalsinthecurvet z R es =4 : 1kn withthechargeamplierwasapproximately3dB(1 : 4 )higher,indicatingtheparasitic capacitancedidindeedplayaprominentroleinlimitingthe sensitivityofthedevice.The phasecomponentofFigure 8-35 showstheoutputofthemicrophonewithchargeampwas 180 outofphasewiththevoltageamplierconguration,aspred ictedinEquations 5{53 and 5{53 Theextractedparasiticcapacitance C ep + C ea isplottedagainstfrequencyin Figure 8-36 togetherwithits95%condenceinterval(seeSection C.5 ).Theparasitic capacitancewasrelativelyconstantoverthemeasurementb andwidth,indicatingthe characterofitsimpactwascapturedaccuratelybythemodel s.Takingthemeanofeachof thecurves,asingleextractedvaluefor C ep isapproximately4pF 1pF.Theuncertainty analysis,foundinSection C.5 ,suggeststhat C fb isadominanterrorsource.Alsonote thatthemeasuredcapacitanceapproximating C ef + C eo wereselectedfromtheB9-Aand C5-Adie. 185

PAGE 186

01234567 100 95 90 85 80 ChargeAmp VoltageAmp Mag.[dBre1V = Pa] 01234567 0 90 180 ChargeAmp VoltageAmp Frequency[kHz]RelativePhase[ ]Figure8-35.Frequencyresponsefunctionofmicrophone116 -1-J7-Atestedwithvoltage andchargeampliercircuitry. 01234567 0 2 4 6 8 10 Frequency[kHz]ParasiticCapacitance[pF] C ep + C ea C ep + C ea U 95% Figure8-36.Parasiticcapacitanceextractionformicroph one116-1-J7-A. Theestimatedopencircuitsensitivitiesofallmicrophone s,foundusingEquation 8{14 undertheassumptionofnegligiblechangein C p fromdevice-to-device,arefoundin Table 8-16 .DetailsoftheuncertaintyanalysisarepresentedinSecti on C.5 .Ingeneral, theextractedopencircuitsensitivitiesofthemicrophone swere3{4dB(40{60%)higher thanthesensitivitymeasuredwiththevoltageamplierpac kage(Section 8.1.3.1 ). Table 8-16 alsocomparestheextractedopencircuitsensitivitestoth osepredictedusing 186

PAGE 187

Table8-16.Open-circuitsensitivityestimates. S oc y [dBre1V = Pa] DUTMeasured z Predicted S oc S va z [dBre1V = Pa] 116-1-J7-A-87.2 0.8-89.84.2 0.8 116-1-I6-A-86.4 0.8-89.84.3 0.8 116-1-C4-B-85.2 0.8-88.74.0 0.8 116-3-F7-B-86.8 0.8-88.74.0 0.8 116-1-E2-C-84.8 0.7-87.73.7 0.7 138-1-E4-D-86.4 0.7-86.63.4 0.7 138-1-I2-D-86.4 0.7-86.63.4 0.7 138-1-I8-E-85.6 0.6-85.63.1 0.6 138-1-H3-F-84.3 0.6-84.62.9 0.6 138-1-J3-F-85.4 0.6-84.62.9 0.6 y Takenat1kHz z With95%condencebounds(Section C.5 ) thelumpedelementmodel.Theagreementisexcellentforwaf er138,butnotsoforthe buckleddevicesofwafer116.8.2.5ElectroacousticParameterExtraction Priortoperformingmeasurementsdirectlyforelectroacou sticparameterextraction, thepressurecouplerassemblywascharacterized.Figure 8-37 showsthe\gain"(really attenuation)betweentheDUTpositionandreferencemicposi tion.At1kHzthepressure wasshowntobeapproximately1 : 5%lessattheDUTlocation.Extractedparameters werethereforecorrectedforthisdierenceinpressure. Uncorrectedfrequencyresponsemeasurementsforeachofthe specially-packaged parameterextractionmicrophonesarepresentedinFigure 8-38 .Thesensitivitiestrend lowerthanlikedesignsmeasuredinthePWT,suggestingincr easedparasiticcapacitance wasassociatedwiththepackageforthisexperiment.Thiswa snotunexpected,asthe FR4boardsinwhichmicrophoneswerepackagedfeaturedlong ertracelengths,dierent routingoftraces,extrasolderconnections,etc.Measurem entuncertainty(95%condence) was < 1%anddominatedbybiaserrorinthereferencemicrophoneca libration(see Section C.2 ). 187

PAGE 188

0123 0 0 : 5 1 1 : 5 Gain[Pa = Pa] 0123 90 45 0 45 90 Frequency[kHz]Phase[ ]Figure8-37.Comparisonofpressureattestandreferencelo cationsinpressurecoupler. Modeshapesassociatedwiththediaphragmdisplacementres ponseunderpressure loading, H pw ,arecollectedinFigure 8-39 .Inadditiontosurfacemaps,prolestaken through x =0and y =0areprojectedontheplotbackplanestoenablecomparison Acleartrendofincreasingcenterdisplacementandvolumed isplacementfromdesign DtodesignFisobserved,providinganimmediatevisualindi cationthatthemeasured complianceincreaseswithdiaphragmsize.Thecorrespondi ngmodeshapepredictions (incrementalderection)obtainedfromthestaticdiaphrag mmodelrangedfrom approximately0 : 2nm = Pato0 : 3nm = Pa,lowerthantheobserved0 : 4nm = Pato0 : 65nm = Pa inFigure 8-39 Theacousticcomplianceandmassextractedfromthemodesha pesofFigure 8-39 arepresentedinTable 8-17 ,withbothcomplianceandmasstrendingasthemodels.The measuredandpredictedvaluesof M ad agreedtowithin10%,thoughitisimportant toacknowledgethatbothvaluessharedacommoninput|theae rialdensity a associatedwiththediaphragmmaterials(Equation 5{13 ).Ingeneral,theresolution 188

PAGE 189

0123 94 92 90 88 86 Sensitivity[dBre1V = Pa] 138-1-B6-D 138-1-F5-D 138-1-C9-E 138-1-D9-E 138-1-F7-F 0123 90 45 0 45 90 Frequency[kHz]Phase[ ] 138-1-B6-D 138-1-F5-D 138-1-C9-E 138-1-D9-E 138-1-F7-F Figure8-38.Frequencyresponseofpiezoelectricmicropho nesinpressurecoupler. ofalumpedelementmodelisexpectedtobeontheorderof10%. Measurementof C ad whichpossessednosuchsharedinput,yieldednearlydouble thepredictedvalue,though extractedvaluesforlikedesignswereconsistenttowithin < 7%.Measurementuncertainty estimateswerecalculatedviaMonteCarloasimulationasad dressedinSection C.6 Theunder-predictionof C ad viathediaphragmmodelcouldstemfromoneofseveral sources.Finiteelementanalysisvalidationofanexamplem icrophoneinSection 5.2.5 wasincloseagreementwiththeanalyticalpredictions,but thisanalysissharedthesame modelinputsforresidualstresses,materialproperties,e tc.Errorinresidualstressinputs couldhaveasignicantimpactonthemodelpredictions.Ina ddition,thegeometryin boththeanalyticaldiaphragmmodelandniteelementmodel wassimpliedfromthe truegeometry,whichdoesnottrulypossessthesharpstepdi scontinuityat r = a 1 .A nalpossibilityiscomplianceintheboundaryconditionso fthediaphragm,whichwould 189

PAGE 190

-500 0 500 -500 0 500 0 0.4 0.8 m mH pw [nm = Pa]A -500 0 500 -500 0 500 0 0.4 0.8 m mH pw [nm = Pa]B -500 0 500 -500 0 500 0 0.4 0.8 m mH pw [nm = Pa]C -500 0 500 -500 0 500 0 0.4 0.8 m mH pw [nm = Pa]D -500 0 500 -500 0 500 0 0.4 0.8 m mH pw [nm = Pa]E -500 0 500 -500 0 500 0 0.4 0.8 m mH pw [nm = Pa]F Figure8-39.Displacementperpressureplots.A)138-1-B6-D .B)138-1-F5-D.C) 138-1-C9-E.D)138-1-D9-E.E)138-1-F7-F.F)138-1-H7-F. 190

PAGE 191

leadtolargerderectioninrealitythanpredictedbythemod el,whichutilizedanidealized clampedboundarycondition.Table8-17.Extractedmechanoacousticparameters. C ad [10 17 m 3 = Pa] y M ad [10 4 kg = m 4 ] z DUTMeasuredPredictedMeasuredPredicted 138-1-F5-D5.32.72.42.2138-1-B6-D5.62.72.32.2138-1-C9-E8.74.21.81.8138-1-D9-E8.64.21.91.8138-1-F7-F12.76.31.51.5138-1-H7-F13.46.31.61.5 y j U 95% j < 1 : 4% z j U 95% j < 0 : 03%(notaccountingfordensity) Themodeshapesassociatedwithdiaphragmdisplacementres ponseundervoltage loading, H vw ,areshowninFigure 8-40 .Unlikeinthecounterpartmeasurementfor H pw thereisverylittlechangeincenterdisplacementfromdesi gntodesigninFigure 8-40 However,theincreasingdiaphragmareafromdesignDtodesig nFnaturallyleadsto substantialincreasesinvolumedisplacementgiventhesto ckynatureofthemodeshapes. Thus,thecenterdisplacementpervoltage H vw ( r =0)alone,asmeasuredinthedie selectionmethodologyofSections 8.1.1 and 8.2.1 ,doesnotprovideagoodmeasureofthe relativelylargerdierencesin d a amongdierentdevicedesigns.However,itprobably remainsagoodscreeningmetricamonglikedesigns.Alternat ively,usingthesinglepoint measurementtoscaleanexperimentaloranalyticalestimat eofthediaphragmmodeshape couldprovetobeasuperiorscreeningmethodamongalldesig ns. ElectroacousticparametersarecollectedinTable 8-18 ,withmeasurement/theory agreementontheorderof20%for d a .Theextractedvaluewasconsistentbetweenlike designs,with < 3%variation.Foranexampleofhow d a and H vw ( r =0)donottrack consistently,considerthevaluesassociatedwitheachqua ntityformicrophones138-1-H7-F and138-1-B6-DinFigure 8-40 andTable 8-18 .Itisseenthat d a for138-1-H7-Fwas approximately65%greaterthanfor138-1-B6-D,but H vw ( r =0)wasonly8%greater. 191

PAGE 192

-500 0 500 -500 0 500 0 1 2 3 m mH vw [nm = V]A -500 0 500 -500 0 500 0 1 2 3 m mH vw [nm = V]B -500 0 500 -500 0 500 0 1 2 3 m mH vw [nm = V]C -500 0 500 -500 0 500 0 1 2 3 m mH vw [nm = V]D -500 0 500 -500 0 500 0 1 2 3 m mH vw [nm = V]E -500 0 500 -500 0 500 0 1 2 3 m mH vw [nm = V]F Figure8-40.Displacementpervoltageplots.A)138-1-B6-D. B)138-1-F5-D.C) 138-1-C9-E.D)138-1-D9-E.E)138-1-F7-F.F)138-1-H7-F. 192

PAGE 193

Measurementuncertaintyestimatesfor d a werecalculatedviaMonteCarlosimulationand estimatesasdescribedinSection C.6 Table8-18.Extractedelectroacousticparameters. d a [10 18 m 3 = V] y a [Pa = V] z k 2 [10 3 ] x DUTMeasuredPredictedMeasuredPredictedEstimated Predicted 138-1-F5-D489396-9.4-14.50.490.76138-1-B6-D474396-8.7-14.50.440.76138-1-C9-E585502-6.8-12.10.380.71138-1-D9-E600502-7.1-12.10.410.71138-1-F7-F772634-6.1-10.10.410.67138-1-H7-F784634-6.0-10.10.400.67 y j U 95% j < 1 : 1% z j U 95% j < 1 : 8% x j U 95% j < 2 : 6% Estimatedusingnominalmeasuredvalues, C ef C ef + C eo ,foreachdesign Thebetteragreementbetweenmeasurementandtheoryfor d a ascomparedto C ad | bothofwhichhavesimilarsensitivitytoresidualstress|s uggestedthatuncertaintyin stressvaluesisnotthedominantcauseforthedisagreement .Alternatively,uncertainties inotherparametersusedtocalculate d a (forexample,in d 31 )couldhaveacompensatory eectthatisnotpresentfor C ad .Compliantboundaryconditionswouldalsohavea similarimpactonboth d a and C ad Boththemeasuredvalueofthetransductionfactor a (= d a =C ad )andthe estimatedvalueofelectromechanicalcouplingfactor k 2 (= d 2a =C ad C ef )arealsoincluded inTable 8-18 .Botharecalculatedfrommeasurementsof d a aswellas C ad (referto Section 5.2.1.1 )andthustheiragreementwiththemodelisdegradedduecalc ulation withthelatter.Notealsothatbecause C ef couldnotbeisolatedintheimpedance measurements, k 2 isestimatedfrommeasurementsusing C ef + C eo inplaceof C ef ;thebias inthecalculationisthustowardsunder-estimationof k 2 ontheorderof10%. Finally,Figure 8-41 collectsthetabulatedmechanoacousticandelectroacoust ic dataintoindividualplots,witheachplotcontainingthesi xdatapointstogetherwith thetheoreticaltrend.Thetrendsarewell-predicted,with visuallyconsistenterrorinall 193

PAGE 194

340360380400420 0 5 10 a 2 [ m]C ad [10 17 m 3 = Pa]A 340360380400420 0 1 2 3 4 a 2 [ m]M ad [10 4 kg = m 4 ]B 340360380400420 0 200 400 600 800 a 2 [ m]d a [10 18 m 3 = V]C 340360380400420 0 5 10 15 a 2 [ m]j a j [Pa = V]D 340360380400420 0 0 : 2 0 : 4 0 : 6 0 : 8 a 2 [ m]k 2 10 3E Figure8-41.Comparisonofmeasuredandtheoreticaltrends forextractedparameters versusdiaphragmsize.Measuredvalues(dots)andtheoreti calpredictions (lines)areshown.A)Diaphragmcompliance, C ad .B)Diaphragmmass, M ad C)Eectivepiezoelectriccoecient, d a .D)Transductionfactor, 2a .E) Electromechanicalcouplingfactor, k 2 quantitiesexcept C ad ,forwhichdisagreementbetweentheoryandmeasurementinc reases withdiaphragmradius. Theextractedparameters C ad M ad d a ,inadditiontotheelectricalimpedance C ef + C eo ,weresubstitutedintothelumpedelementmodel,whichwast henusedto predictthefrequencyresponsefunctionofthemicrophones .Dependentparameters suchas a werealsocalculatedfromtheextractedparameters.Thepre dictedfrequency responsefunctionsareplottedtogetherwiththemeasuredf requencyresponseofeach 194

PAGE 195

microphone,correctedhereforthesmallpressuredierenc ebetweenreferenceand DUTlocations(Figure 8-37 ),inFigure 8-42 .Becausemeasurementswereperformed withvoltageamplierarchitectures,parasiticcapacitan cewasalsoaccountedforinthe analyticalmodelandwasestimatedsuchthatthetheoretica landmeasuredmagnitude ofthefrequencyresponsefunctionsmatchedat1kHz.Asaresul t,estimatedparasitic capacitancevaluesforeachmicrophoneareincludedinthel egendofFigure 8-42 Extractedparasiticcapacitancevaluesrangedfrom5 : 3pFto6 : 4pF,somewhathigher thanthoseextractedfromthetubular-packagedmicrophone sinSection 8.2.4.3 (4 1pF) asexpected. 0123 94 92 90 88 86 FRFMagnitude[dBre1V = Pa] B6-D(Measured) B6-D(Theory, C ep + C ea =6 : 4pF) F5-D(Measured) F5-D(Theory, C ep + C ea =6 : 1pF) C9-E(Measured) C9-E(Theory, C ep + C ea =5 : 6pF) D9-E(Measured) D9-E(Theory, C ep + C ea =5 : 9pF) F7-F(Measured) F7-F(Theory, C ep + C ea =5 : 3pF) Figure8-42.Correctedfrequencyresponsemagnitudeofmic rophonesinpressurecoupler togetherwiththeoreticalpredictionscalculatedusingex tractedparameters. 8.3Summary Inthischapter,variouscharacterizationandparameterex tractionexperiments performedonthepiezoelectricmicrophonesweredescribed .Ninemicrophoneswere characterizedintermsofacousticperformance(bandwidth ,sensitivity,linearity)and electricalproperties(impedance,parasiticcapacitance ).Oneadditionalmicrophonewas usedtoestimateparasiticcapacitance.Electroacousticp arameterswereextractedfrom6 moremicrophonesasanadditionalassessmentofanalytical modelpredictions.Inthenext section,nalconclusionsaredrawnandthepiezoelectricm icrophonedevelopedinthis studyiscomparedtothepriorart. 195

PAGE 196

CHAPTER9 CONCLUSION Thisstudyfocusedonthedevelopmentofmicroelectromecha nicalsystems(MEMS) piezoelectricmicrophones(Figure 9-1 )withtheperformancecharacteristicsneeded toenablesuperiortechnicalmeasurementsinfull-scaleri ghttests.Theaudio-band microphonewasrequiredtobesmall( 1 : 9mm),thin( < 1 : 3mm),passive,andhave alargemaximumpressure( 172dB)withmoderatenoiseroor( 48 : 5dBSPL).In previouschapters,themodeling,optimization,fabricati on,packaging,andexperimental characterizationofjustsuchaMEMSpiezoelectricmicroph onewasdiscussed.The ultimategoalwasnotjusttodevelopareplacementforexist ingmicrophones,but toenablethetypesofmeasurementsaircraftmanufacturers envisionforthefuture, potentiallyinvolvingseveralarrayscomposedofhundreds ofmicrophonesblanketingan aircraftfuselage. Figure9-1.AMEMSpiezoelectricmicrophonedieonaplaying card. InChapter 8 ,theMEMSmicrophonesdevelopedinthisstudywerethorough ly characterized,andtheresultsgenerallymetorexceededta rgetspecications.The collectedmicrophoneperformancecharacteristics,ascom paredbothtotargetspecications andtheKulitemicrophonepresentlyin-useforfull-scaler ighttestsatBoeingCorporation, arefoundinTable 9-1 .Mostnotably,theMEMSpiezoelectricmicrophoneswerewel l 196

PAGE 197

underthe48 : 5dBSPL/93dBOASPLMDPspecication(savefortheoutlier138 -1-I8-E) andhadalowernoiseroorwithhighersensitivity(26{40 greater)thantheKulite microphones.Inaddition,measurementsshowedthat5ofthe testedmicrophones representing3dierentdesignshadPMAX > 160dB,andofthetwomicrophones testedatevenhigherSPLs,one(138-1-J3-F)demonstratedP MAX 171 : 6dBSPL. Duetodistortioninthereferencemicrophoneduringthisme asurement(discussed inSection 8.2.3.2 ),thedeviceperformancealmostcertainlyexceededthetar get PMAX 172dB.On-boardgainofslightlyover20dBissucienttorea chthesensitivity targetof500 V = Pa.Althoughthemeasured f 2dB pointof70Hzslightlyexceeded the20Hzminimumtarget, f +2dB 20kHzwasmet.Measuredmicrophoneresonant frequenciesexceeding100kHzsuggestedasurplusofusableb andwidththatcouldexpand therangeofapplicationsfortheMEMSpiezoelectricmicrop honetomodel-scaletests. Finally,eventhediaphragmofthelargestmicrophoneteste d,havingadiameterof828 m (designF),wassmallerthantheKulitediaphragm(864 monaside)andwaslessthan halfofthemaximumdiametertargetspecication(1 : 9mm). Table9-1.RealizedMEMSpiezoelectricmicrophoneperform ancecomparedto specicationsandbenchmarkKulitesensor. MetricObtainedTargetSpecicationKuliteLQ-1-750-25SG Sensingelementsize 514{910 m 1 : 9mm864 864 m 2 Sensitivity29{44 V = Pa500 V = Pa y 1 : 1 V = Pa MDP40{51dB z 48 : 5dB z 48 : 5dB z 83{89dBOASPL 93dBOASPL93dBOASPL PMAX x > 171 : 6dBSPL # 172dBSPL 168dBSPL Bandwidth 70Hz # {20kHz+20Hz{20kHz < 20Hz{20kHz+ y Withon-boardgain z 1Hzbincenteredat1kHz x 3%distortion 2dB # 138-1-J3-F Microphone138-1-J3-Fdevelopedinthisstudyiscomparedi nTable 9-2 tonotable microphonesfromtheacademicliteraturewithsimilarappl icationareaortechnology utilization.AmongthepassivesensorsincludedinTable 9-2 ,138-1-J3-Ffeaturedthe highestveriedPMAX( 171 : 6dB),withthemicrophoneofHorowitzetal.(2007)[ 20 ] 197

PAGE 198

havingthesecondhighest(169dB),thoughthatresultwasli mitedbythetestsetup. Themaximumpressureveriedin[ 20 ]waslimitedbythemeasurementsetupandmay wellhaveexceeded172dB.However,theMDPofmicrophone1381-J3-F(andothers characterizedinthisstudy)wasasignicantimprovemento ver[ 20 ]intermsofdB(A). Thesensitivityobtainedfor138-1-J3-Fwasalsoa52 improvementoverthatin[ 20 ]. Microphone138-1-J3-Fandothersdevelopedinthisstudyar ethustheclosestpassive microphonesinexistencetomeetingaircraftmanufacturer needsforfull-scalerighttests. Theprimarycontributionsofthisstudyarethusasfollows: 1.DevelopmentofaMEMSpiezoelectricmicrophoneexhibiti ngthehighestconrmed PMAXamongpassiveMEMSmicrophonesandperformancecharact eristicsmore closelymatchingthoseneededforaircraftfuselageinstru mentationthananyprior passivesensor 2.Generalizationoftheradiallynon-uniformpiezocompos itediaphragmmechanical modelofWangetal.(2002)[ 127 ]toincludearbitrarylayercompositionandresidual stressesoneithersideofthestep-discontinuity,develop mentofageometrically nonlinearversionofthemodel,anduseofthesemodelsinthe microphonedesign process 3.Solutionofaformally-deneddesignoptimizationprobl emforaMEMSpiezoelectric microphoneutilizinglumpedelementmodeling 4.Executionofanovelsuiteofparameterextractionexperi mentstoassessthe accuracyofindividuallumpedelementpredictions,mostno tablythoseobtained viathediaphragmmechanicalmodel Thescopeofthisstudywasthedesignandcharacterizationo fMEMSpiezoelectric microphonesinthelaboratorysetting.Therefore,researc hremainstobedonebefore thedevelopedmicrophonescanserveastruereplacementsfo rKulitemicrophonesin full-scalerighttests.Inthenextsections,recommendati onsaregivenforfuturedesign modicationsandalsoforfutureworkrelatedtocharacteri zation. 9.1RecommendationsforFuturePiezoelectricMicrophones Severalimprovementstothepiezoelectricmicrophonedesi gnanddesignprocesscan bemadeinfutureiterations.Themostcriticalunmetneedfo rdeploymentonanaircraft 198

PAGE 199

Table9-2.PerformancecharacteristicsofMEMSpiezoelect ricmicrophone138-1-J3-F comparedtonotablemicrophonesfromtheacademicliteratu re. AuthorTransduction Method Sensing Element Dimensions SensitivityDynamic Range Bandwidth (Predicted) Franz1988[ 60 ] Piezoelectric(AlN) 0 : 72mm 2 1 m # 25 V = Pa # 68dB(A) # { N/R N/R{45kHz # Sheplaketal.1998[ 16 17 ] Piezoresistive105 m 0 : 15 m 2 : 24 V = Pa = V92dB z {155dB200Hz{6kHz (100Hz{ 300kHz) Arnoldetal.2001[ 18 ] Piezoresistive500 m 1 m0 : 6 V = Pa = V52dB z {160dB1kHz{20kHz (10Hz{40kHz) Huangetal.2002[ 68 ] Piezoresistive710 m y 0 : 38 m 1 : 1mV = Pa = V53dB z {174dB100Hz{10kHz Scheeperetal.2003[ 79 ] Capacitive1 : 95mm 0 : 5 m 22 : 4mV = Pa23dB(A){ 141dB 251Hz{20kHz Hillenbrandetal.2004[ 81 ] Piezoelectric(VHD40) 0 : 3cm 2 55 m2 : 2mV = Pa37dB(A){ 164dB 20Hz{140kHz 0 : 3cm 2 275 m 10 : 5mV = Pa26dB(A){ 164dB 20Hz{28kHz Martinetal.2007[ 71 72 89 ] Capacitive230 m 2 : 25 m 390mV = Pa41dB z {164dB300Hz{20kHz Martinetal.2008[ 73 ] Capacitive230 m 2 : 25 m 166 V = Pa22 : 7dB z { 164dB 300Hz{20kHz Horowitzetal.2007[ 20 ] Piezoelectric(PZT) 900 m 3 : 0 m 1 : 66 V = Pa35 : 7dB z / 95 : 3dB(A){ 169dB 100Hz{6 : 7kHz (100Hz{ 50kHz) Littrell2010[ 85 ] Piezoelectric(AlN) 0 : 62mm 2 { 2 : 3 m 1 : 82mV = Pa37dB(A){ 128dB 50Hz{8kHz (18 : 4kHz) Thisstudy x Piezoelectric(AlN) 414 m 2 : 14 m 39 V = Pa40 : 4dB z / 75 : 4dB(A){ 171 : 6dB+ 69Hz{20kHz ( > 104kHz) # References[ 62 88 ] Radiusofcirculardiaphragm y Sidelengthofsquarediaphragm z 1Hzbinat1kHz { 2cantilevers x Microphone138-1-J3-F fuselageisintegrationofthrough-siliconvias(TSVs)inpl aceoffront-sidewirebonds. Wirebondsareacommoncontributortofailureinmicrosyste ms[ 182 ]andwiththeneed forprotectivewireencapsulant,limittheachievablesens orsurfaceroughness.Waferswith customTSVsareavailableforpurchaseandonlyrequirequali cationinafacilitywith AlNcapabilitiestobeimplementedinfuturedesigns. 199

PAGE 200

TheoptimizationofChapter 6 showedthatduetothestressstatesofthelms, themoderatelytensilestructurallayerthicknesstendedt oitsupperboundinorderto mitigatetheimpactofhighstresses|particularlycompres sivestresses|intheother lms.HighervaluesofPMAXwereshowntobeachievablewithath ickerstructurallayer inexchangeforrelativelysmallsacriceinMDP(recallSec tion 6.4 ).Toachieveathicker structurallayer,thefabricationprocesscouldbetransit ionedtosilicon-on-insulator (SOI)wafers,withtheapproximatelystress-freesilicond evice-layerservingasthe structurallayer.SOIwafersareavailableforpurchasewit havarietyofsilicondevice-layer thicknessesandintegratedTSVs.Thepiezoelectric/metal lmdepositioncouldremain virtuallyunchanged,withprocessdevelopmentlargelynee dedonlyforintegrationofa newventstructure. Thelowfrequencytargetof f 2dB 20Hzwasnotquitemetinthisstudy.Modeling suggestedthatthedielectriclossinthepiezoelectriclm wasthelimitingagentinthe lowfrequencyreseponse.Newvaluesof R ep wereobtainedviaparameterextractionfrom impedancemeasurements,fromwhichresistivityiscalcula ble.Futuredesignoptimization processesshouldrstfocusonactivereductionof f 2dB usingtheseextractedresistivities. Dielectriclossmayalsobereducedatthemateriallevelwit himprovedAlNlmquality [ 183 ].Improvedlmqualityandlowerdielectriclosshavebeenl inkedinsomestudiesto thickAlNlms[ 85 ]ofupto2 m[ 184 ]. Thediaphragmmodelpresentedinthisstudywasasignicant stepforwardfrom priorworks[ 20 113 127 128 ],butadditionalimprovementscouldbemade.Alinear modelwasusedtopredictdiaphragmperformancebothinterm sofinitialderection (duetoresidualstress)andincrementalderection(duetov oltage/pressureloading). Sincetheincrementalderectionisthequantitythatmustbe linearwithrespectto pressure,themodelcouldbeextendedsuchthattheinitiald erectionissolvedasa nonlinearproblemandthenalinearproblemissolvedforinc rementalderectionwiththe initially-derecteddiaphragmservingasthereferencecon guration.Thisapproachwould 200

PAGE 201

increasecomputationtimewhenimplementedinanoptimizat ionalgorithm,butitwould alsorelaxconstraintsonnonlineartransitionbehaviorth atwereperhapstooconservative inthisstudy. Modelingofnonlineartransitionbehavior|characterized inthisstudyviaTHD| alsodeservesrenewedattention.Mostnotably,thestatice stimationofTHDutilizedin Chapter 6 hasnotbeenveried.Afocusedstudyutilizingniteelemen t-basednonlinear dynamicssimulationsofthemicrophonediaphragmcouldrev ealtherelationshipbetween staticnonlinearityandTHDforthemicrophonegeometriesin thisstudy.However,amore generalandcomputationally-ecientapproachisneeded.F orexample,theclassicalmodel foraDungspringmightbeusedtoestimateTHDgiveninputsfr omstaticmechanical models.Suchanapproachwouldbehighlyvaluabletomicroph onedesignersandintegrate wellwithdesignoptimizationapproaches. Thiswastherststudyforwhichformaloptimizationwasemp loyedinthedesign ofapiezoelectricMEMSmicrophone.Anumberofmodication scouldbemadeto theoptimizationformulation.First,usinganoverallmeas ureofMDPratherthanthe narrowbanddenitionmightservetolowertheoverallnoise roor,sincetheoptimization algorithmwouldthenhaveadditionalincentivetosimultan eouslyreducenoisedue to R ep andtheamplierratherthanjustthedominantsourceat1kHz. Inaddition, withcondenceinlow-frequencycut-opredictionsexperi mentallyestablished,future optimizationscouldincludeaconstrainton f 2dB toensuresponsorspecicationsaremet. Finally,theconstraintonaspectratiocouldberemovedinf avorofverifyingprediction qualityafteroptimizationiscompletedratherthanunnece ssarilylimitingthefeasible designspace. Theoveralloptimizationapproachcouldalsobetransition edfromdeterministicto robustoptimization,adesignmethodologyinwhichthebest designisn'tdenedsimply bymeanperformance,butalsobyhowsensitivemeanperforma nceistovariableslike materialproperties,processvariations,etc.[ 185 ].Inthisstudy,thethin-lmresidual 201

PAGE 202

stressmodelinputswerenotwell-knownbuthadsignicanti mpactonmicrophone performance;microphoneshailingfromwafer116,forexamp le,hadvisibly-buckled diaphragmsandhadtheworstperformanceofthosecharacter izedinChapter 8 .Material propertieswerealsodrawnfromavarietyofsourcesthatmay nothavebeentruly representativeofthematerialpropertiesassociatedwith theFBAR-variantprocess(e.g. d 31 forAlN).Robustoptimizationformulationshavebeenapplied previouslytothedesign ofaMEMSgyroscope[ 186 ]andmultistablemechanism[ 187 ],amongothers.Thereare twomajorhurdlestoimplementationofrobustoptimization inMEMSpiezoelectric microphonedesign:robustoptimizationisoftenmorecompu tationallyintensivethanits deterministiccounterpartanditideallyutilizescompreh ensivestatisticalinformationfor propertyandprocessvariationsthatisrarelyavailable.M ethodshavebeendeveloped forrobustoptimizationwhenadearthofstatisticalinform ationisavailable,thoughat increasedcomplexityandcomputationalcost[ 188 ]. 9.2RecommendationsforFutureWork Superiorstabilityisonemajorcharacteristicthatsepara tesmeasurementmicrophones fromthoseusedinotherapplications.Tobedeployedonanai rcraftfuselage,theMEMS piezoelectricmicrophonemustdemonstraterobustnesstom oistureandfreezing,in additiontotemperaturestabilityfrom 60 Fto150 F.Thiskindofcharacterization wasbeyondtheequipmentcapabilitiesatInterdisciplinar yMicrosystemsGroupand thusfelloutsidethescopeofthisstudy.Abatteryofenviro nmentaltestsareneededto characterizestabilityanddriftinthepiezoelectricmicr ophones.Suchmeasurementscould leadtodesignimprovementsorcompensationschemesifnece ssary.Environmentaltesting ofthiskindisalreadyinprogressatBoeingCorporation. Thepackagingschemeutilizedinthisstudywasdesignedfor laboratorycharacterization. Movingtotheaircraftfuselageapplicationrequiresdevel opmentofalow-cost,robust, thinpackagewithadequateelectromagneticinterference( EMI)shieldingforthe high-impedancesensors.Thedesireforlowcomplexityandh ighlevelsofintegration 202

PAGE 203

whendeployingthousandsofsensorsdemandsintegrationof interfaceelectronicsin thesurface-mountpackageaswell.Allrequiredcircuitrymu stideallyoperateoofa standard4mAconstantcurrentsourcecommonlyintegratedw ithcurrent-generation dataacquisitionsystems.Packagecostisalsoasignicant concernmovingforward,as packagingisknowntooftendominatethecostofMEMSsensors [ 43 ]. Modicationscouldalsobemadetothelaboratorypackageto improvefuture characterizationexperiments.Althoughpre-andpost-pack agemeasurementswere takentoestablishtheimpactofpackagingondieperformanc e,noeortwasmadeto systematicallyidentifycausesforbehavioralchangesand remedythem.Astudyinvolving multiplesubstratematerialsanddie-attachmethodsisnec essaryfordevelopment ofapackagethatdoesnotimpactmicrophoneperformance.Ac hangeinsubstrate materialalsohasthepotentialtoreduceparasiticcapacit ance.Inaddition,EMIissues wereoccasionallyencounteredinthelaboratorytestingof thesemicrophones,bothat InterdisciplinaryMicrosystemsGroupandBoeingCorporat ion.Focusedeortshouldthus bemadetoreduceEMIinthelaboratorypackage. Theparameterextractionexperimentscouldalsobeimprove d.Thepressure couplerhardwareusedintheseexperimentssueredfrominc onsistentsealinganda tendencytodriftunderneaththemicroscopeobjective.Cle ardesignmodicationsthat wouldreducedriftincludebetterpositioningofthepressu recoupler,perhapsusing micro-positioners,andarexibleconnectionwiththespeak ertohelpvibration-isolatethe pressurecoupleritself.Modicationofthemicrophonepac kageform-factorusedinthe parameterextraction(recallSection 8.1.5 )toavoidcantileveringisalsosuggested.The mostimportantmodicationtothepressurecouplerexperim ent,however,istheuseof chargeamplierratherthanavoltageampliercircuitry.Us ingthevoltageamplier, parasiticcapacitanceservedassomethingofaconfounding variableandlimitedtheability toverifyparameterextractionsviameasuredmicrophonefr equencyresponsefunctions. Utilizingachargeampliereliminatestheimpactofparasit iccapacitance. 203

PAGE 204

APPENDIXA DIAPHRAGMMECHANICALMODEL Inthisappendix,amodelofanaxisymmetric,laminated,pre -stressed,and radially-discontinuouscircularpiezoelectricplateexp osedtopressureand/orvoltage loadingispresented.MotivatedinSection 5.2.1.2 ,thismechanicalplatemodelprovides crucialinputstotheoverallpiezoelectricmicrophonelum pedelementmodelintheformof displacementpredictionsforparticularloadingscenario s.Themodelispartofanatural evolutionfrompriorwork,including[ 113 128 189 190 ],butmostspecicallyWangetal. (2002)[ 127 ].Earlierformsofthemodelwereutilizedin[ 20 139 140 140 ]. Figure A-1 ,repeatedfromSection 5.2.1.2 ,showsthegeometryofthepiezoelectric microphone,whichfeaturesanannularpiezoelectricringa ndotherwisepassivematerials. Themodelderivedhereisgeneralizedtoinclude,butnotbel imitedto,thisspecic geometry.Ingeneral,boththeinner(0 r a 1 )andouter( a 1 r a 2 )regions (or\domains")maycontainanarbitrarylayupofpiezoelect ricand/ornon-piezoelectric materials,witheachpiezoelectriclayerindividuallyadd ressablewithanelectriceld. Uniformpressureloadingandtheeectsofin-planeresidual stressarealsoincluded. n r FigureA-1.Laminatedcompositeplaterepresentationofthe thin-lmdiaphragm. Thederivationisbrokenintoseveralparts.First,thestra indisplacementrelationsfor small,nitedeformationsarederivedfromtheGreenstrain tensor.Next,theequationsof motion,andtheassociatedgeneralizedboundarycondition s,arederivedfromHamilton's principle.Theelectromechanicalconstitutiverelations relatingforces/moments, displacements,andelectriceldarethengivenandarecomb inedwiththeequations ofmotiontoyieldthedisplacement-basedgoverningequati onsforthepiezoelectric compositeplate.Bothlinearandnonlinearformsofthegove rningequationsarepresented. Withthegoverningequationsderived,particularsolution sforthesingleradial-discontinuity casedepictedinFigure A-1 arepresented.Thelineargoverningequationsaresolved analyticallyuptothestepofapplyingboundaryconditions ,atwhichtimeintegration 204

PAGE 205

coecientsaredeterminedfromthenumericalsolutionofas ystemoflinearalgebraic equations.Thenonlineargoverningequations,meanwhile, aremanipulatedintoaform solvableviaacommonboundaryvalueproblemsolver, bvp4c inMATLAB. Thederivationcontainedhereinstrivesformaximumgenera litywhilemaintaining adelicatebalancewithreadability.Simplicationsspeci ctotheproblemofinterest areemployedonlywhentheyarenecessary,usuallyatatimew hencontinuingwithout simplicationisnolongerpossibleorwouldbetoounwieldl y.Inthisway,additional referencematerialisprovidedforfuturemodelingeorts. A.1Strain-DisplacementRelations Thestartingpointofthisderivationlieswiththenonlinea rtheoryofelasticityand theGreenstraintensor ij ,givenas[ 98 191 ] ij = 1 2 @u i @X j + @u j @X i + @u k @X i @u k @X j = 1 2 [ ~u r + r ~u +( ~u r ) ( r ~u )](A{1) where u i ( X j )isthedisplacementvectorand X j aretheCartesiancoordinatesofparticles inthereferencecongurationandindicialnotation[ 192 { 194 ]isusedheretoimply summationoverrepeatedindices.TheGibbsnotation[ 194 ]equivalent,whichdoesnot presupposeacoordinatesystem,isalsogiven.TheGreenstr aintensorisaLagrangian measureofstrainandisapplicableforcasesinwhichabodyu ndergoeslarge,nite deformations[ 191 ].AnotherforminwhichtowritetheGreenstraintensoris ij = e ij + 1 2 ( e ik + ik )( e kj kj ) = e + 1 2 ( e + ) ( e ) ; (A{2) wheretheinnitesimalstraintensor e ij androtationtensor ij aredenedas[ 193 195 ] e ij = 1 2 @u i @X j + @u j @X i e = 1 2 ( ~u r + r ~u )(A{3) and ij = 1 2 @u i @X j @u j @X i = 1 2 ( ~u rr ~u ) : (A{4) Itisimportanttonotethat e ij issymmetricwhile ij isanti-symmetric[ 193 ]. Thin,rexiblestructuressuchasbeams,plates,andshellsa recharacterizedbylarge rotationsoftheircrosssectionsbutonlyminimalchangein shapeofindividualelements [ 196 ].TheGreenstraintensormaythereforebesimpliedundert heassumptionthat [ 191 195 ] e ij ij ; (A{5) thatis,thestrainsaremuchlessthantherotations.Thisis incontrasttothelinear theory,inwhichboth e ij and ij aremuchlessthanunity.Applyingtheassumption A{5 requirestheremovalofanytermscontainingproductsof e ij fromtheGreenstraintensor, Equation A{2 .Performingthisoperationandmakinguseoftheanti-symme tryof ij 205

PAGE 206

(meaning ij = ji ),theGreenstraintensorissimpliedto[ 191 195 ] ij e ij + 1 2 ik jk e + 1 2 T : (A{6) Equation A{6 isdirectlyapplicabletotheanalysisofathinplateandiss ometimes referredtoasthecaseof small,nitedeformations [ 191 ]. Aplatewithsurfacenormalorientedalongthe x 3 axisintheundeformedstatedoes notundergolargerotationsaboutthataxiscomparedtoaxes intheplaneoftheplate.In mathematicalterms, 12 31 ;! 32 (A{7) and 12 maybeneglected.Inmanytexts[ 193 197 ],asinglesubscriptnotationisused thatclariestheaxisofrotation.Inthisnotation, 32 = 1 31 = 2 ,and 12 = 3 Thesealsocorrespondtocomponentsofarotationvector.Und ertheassumptionof Equation A{7 ,thesixcomponentsofthereducedGreenstrainincylindric alcoordinates are rr = e rr + 1 2 2 rz ; (A{8) = e + 1 2 2 z ; (A{9) zz = e zz + 1 2 2 rz + 2 z ; (A{10) r = e r + 1 2 rz z ; (A{11) z = e z ; (A{12) and rz = e rz : (A{13) Thelinearstrains e ij androtations ij aredenedincylindricalcoordinatesas[ 98 ] e rr = @u r @r ; (A{14) e = u r r + 1 r @u @ ; (A{15) e zz = @u z @z ; (A{16) 2 e r = 1 r @u r @ + @u @r u r ; (A{17) 2 e rz = @u r @z + @u z @r ; (A{18) 2 e z = @u @z + 1 r @u z @ ; (A{19) 206

PAGE 207

2 r = 1 r @u r @ @u @r u r ; (A{20) 2 rz = @u r @z @u z @r ; (A{21) and 2 z = @u @z 1 r @u z @ : (A{22) WiththeGreenstraintensorsimpliedsignicantly,thene xtpartofthederivation focusesontheindividualdisplacementcomponents. A.2KirchhoHypothesis In1850,theGermanphysicistGustavKirchhoproposedakin ematicassumption forthedeformationofthinplates.Theso-called Kirchhohypothesis focusesonthe deformationofcrosssectionswithintheplate.Itstatesth atduringdeformation,lines initiallynormaltothereferencesurface(1)remainstraig ht(in-planedisplacements arelinearfunctionsof z ),(2)remainnormal( rz = z =0),and(3)donotextend ( u z = u z ( r; )).Plateequationsderivedundertheseassumptionsaresai dtocomefrom theclassicaltheoryofplates[ 121 ]. Theassumeddisplacementforms u r ( r;;z ; t )= u ( r; ; t ) z @w @r ; (A{23) u ( r;;z ; t )= v ( r; ; t ) z 1 r @w @ ; (A{24) and u z ( r;;z ; t )= w ( r; ; t )(A{25) areconsistentwiththeKirchhohypothesis.Here, u v ,and w representthedisplacements ofaparticleonthesurface z =0[ 121 ],calledthe\referenceplane"or\referencesurface" andchosenforconvenienceatanarbitrarylocationwithint hethicknessoftheplate. SubstitutingthedisplacementsintoEquations A{8 A{9 ,and A{11 yields 8<: rr 2 r 9=; = 8<: 0r 0 0r 9=; + z 8<: r r 9=; ; (A{26) wherethein-planestrains 0 andcurvatures aredenedas 0r = @u @r + 1 2 @w @r 2 ; (A{27) 0 = u r + 1 r @v @ + 1 2 r 2 @w @ 2 ; (A{28) 0r = 1 r @u @ v r + @v @r + 1 r @w @r @w @ ; (A{29) 207

PAGE 208

r = @ 2 w @r 2 ; (A{30) = 1 r 1 r @ 2 w @ 2 + @w @r ; (A{31) and r = 2 r @ 2 w @r@ 1 r @w @ : (A{32) Equation A{26 canbewrittencompactlyas = 0 + z ; (A{33) withboldfaceindicatingarrayquantities.Theremainings hearstrains rz and z and zz vanishperKirchho'shypothesis.Equations A{27 to A{32 arecollectivelyknownasthe vonKarmanstrains,andtheplatetheorymakinguseofthem issometimescalledthevon Karmanplatetheory[ 121 ]. A.3EquationsofMotion Thederivationoftheequationsofmotionforthepiezoelect riccompositeplate makesuseofvariationalmethods,whoseprimaryadvantagei sthatconsistentboundary conditionsarealsoproduced.Thevariationalformulation makesimmediateuseofthe vonKarmanstrainsderivedinSections A.1 { A.2 .ThederivationbeginswithHamilton's principleforaconservativesystem[ 142 195 ], Z t 2 t 1 Ldt =0 ; (A{34) wheretheintegrandisthevariationoftheLagrangianfunct ion[ 191 ]foranelasticbody, L = T ( U + V ) : (A{35) Here, T isthekineticenergyofthebody, U isthestrainenergystoredinthebody, and V isthepotentialenergyassociatedwithexternalforcesapp liedtothebody[ 142 ]. Hamilton'sprincipleisthedynamicanalogoftheprincipleo fvirtualwork,andmayin factbederivedfromitviatheuseofD'Alembert'sprinciple[ 191 195 ].DymandShames [ 195 ]summarizeasfollows: \Hamilton'sprinciplestatesthatofallpathsofadmissible congurationsthat thebodycantakeasitgoesfromconguration1attime t 1 toconguration 2attime t 2 ,thepaththatsatisesNewton'slawateachinstantduring theinterval(andisthustheactuallocusofcongurations) isthepaththat extremizesthetimeintegraloftheLagrangianduringthein terval." Virtualdisplacements(innitesimalvariationsfromthetr ueequilibriumcongurationto anarbitraryadmissibleconguration[ 92 ])mustvanishat t 1 and t 2 andonanyregionof thebodywheredisplacementisprescribed[ 191 ]. 208

PAGE 209

Therstvariationsofkineticenergy T andstrainenergy U are[ 122 195 ] T = Z 8 u i u i d 8 T = Z 8 ~ u ~ ud 8 (A{36) and U = Z 8 ij ij d 8 U = Z 8 : "d 8 ; (A{37) whereanoverdotdenotespartialdierentiationwithrespe cttotime, @=@t ,andthe integralsareovertheplatevolume, 8 .Restrictingtheexternalloadingtoanarbitrary distributedloaddirectedinthe z direction, q z ,therstvariationofthepotentialenergyof thisappliedloadis V = Z 8 q z u z d 8 : (A{38) Thekeytoderivingtheequationsofmotionusingvariationa lmethodsisto manipulatetheintegrandofEquation A{34 viaintegrationbypartsuntilthegoverning equationsandboundaryconditionscanbeextracted.Thevir tualdisplacementsfor thisproblemarethereferenceplanedisplacements, u v ,and w .Forconvenience,the individualterms U T ,and V aremanipulatedindependentlyandthencombinedinto Equation A{34 attheendofthederivation.Thesimplestexpression, V ,requiresonly substitutionofEquation A{25 ,yielding V = Z 8 q z wd 8 : (A{39) Next,Equation A{36 ( T ),maybetreated.Performingthevectordotproduct, T = Z 8 (_ u r u r +_ u u +_ u z u z ) d 8 : (A{40) Integratingbypartsovertimeyields Z t 2 t 1 Tdt = Z t 2 t 1 Z 8 ( u r u r + u u + u z u z ) d 8 dt + Z 8 [_ u r u r +_ u u +_ u z u z ] t 2 t 1 d 8 ; (A{41) wherethesecondtermontheright-handsideofEquation A{41 mustvanishbecause admissiblevirtualdisplacements u r u ,and u z arerequiredtobezeroat t = t 1 and t = t 2 .Thus, T = Z 8 ( u r u r + u u + u z u z ) d 8 : (A{42) 209

PAGE 210

SubstitutingEquations A{23 to A{25 intotheabove,noting d 8 = rdrddz ,and integratingwithrespectto z yields T = Z I 0 u I 1 @ w @r u + I 0 v I 1 1 r @ w @ v + I 0 ww + I 2 @ w @r I 1 u @w @r + 1 r I 2 1 r @ w @ I 1 v @w @ rdrd; (A{43) wherethemomentsofinertia I 0 { I 2 are f I 0 ;I 1 ;I 2 g = Z z t z b 1 ;z;z 2 dz (A{44) andtheintegrationlimitsarefromthebottomsurfaceofthe plate( z = z b )tothe topsurface( z = z t ).Notethat I 0 and I 2 maybereferredtoastheaerialdensityand rotaryinertia,respectively.Theterm I 1 isonlynonzeroifthedensityoftheplateisnot symmetricaboutthereferenceplane( z =0).Performingintegrationbypartsonthenal twotermsintheintegrandnallyyields T = Z A I 0 u I 1 @ w @r u + I 0 v I 1 1 r @ w @ v + I 0 ww 1 r @ @r rI 2 @ w @r rI 1 u + @ @ I 2 1 r @ w @ I 1 v w rdrd Z r I 2 @ w @r I 1 u w r = r 2 r = r 1 d Z r I 2 1 r @ w @ I 1 v w =2 =0 dr; (A{45) wheretherstintegralwillcontributetotheequationsofm otionandtheremaining integralswillcontributetotheboundaryconditionsforth eequationsofmotion.The integrationisperformedhereoverageneraldomain[ r 1 ;r 2 ]whichcouldpresent[0 ;a 1 ]or [ a 1 ;a 2 ],forexample. Attentionisnowturnedtotheexpressionfor U ,Equation A{37 .Giventhat rz = z =0andtheplateisinastateofplanestress( zz 0),Equation A{37 canbe writtensimplyas U = Z 8 ( rr rr + +2 r r ) d 8 : (A{46) SubstitutingEquation A{26 andintegratingwithrespectto z gives U = Z A N r 0r + M r r + N 0 + M + N r 0r + M r r rdrd; (A{47) withtheforceandmomentresultants[ 121 ]denedas f N r ;N ;N r g = Z z t z b f rr ; ; r g dz (A{48) 210

PAGE 211

and f M r ;M ;M r g = Z z t z b f rr ; ; r g zdz; (A{49) respectively.Next,substitutingEquations A{23 to A{25 intoEquation A{47 U = Z A N r u N r r v + N r @u @r + N r @v @r + N r r @u @ + N r @v @ + rN r @w @r + N r @w @ M 1 r @w @r + rN r @w @r + N @w @ +2 M r 1 r 2 @w @ M r @ 2 w @r 2 + M 1 r 2 @ 2 w @ 2 + M r 1 r @ 2 w @r@ + M r 1 r @ 2 w @@r rdrd: (A{50) Integratingbypartsonce(andpayingspecialattentiontot he M r termsper[ 142 ]), U = Z A N r u N r r v + N r @u @r + N r @v @r + N r r @u @ + N r @v @ + rN r @w @r + N r @w @ + @ @r ( rM r ) M + @M r @ 1 r @w @r + rN r @w @r + N @w @ +2 M r + r @M r @r + @M @ 1 r 2 @w @ rdrd Z rM r @w @r + M r @w @ r = r 2 r = r 1 d Z r M 1 r @w @ + M r @w @r = 0 =0 dr: (A{51) Integratingbypartsasecondtimecompletestheprocess: U = Z @ ( rN r ) @r + @N r @ N 1 r u + N r + @ ( rN r ) @r + @N @ 1 r v + 1 r @ @r rN r @w @r + N r @w @ + @ ( rM r ) @r M + 1 r 2 @ @ rN r @w @r + N @w @ +2 @ @r ( rM r )+ @M @ w rdrd + Z N r u + N r v + N r @w @r + N r 1 r @w @ + 1 r @ ( rM r ) @r 1 r M + 2 r @M r @ w M r @w @r r = r 2 r = r 1 rd + Z N r u + N v + 1 r rN r @w @r + N @w @ +2 @ ( rM r ) @r + @M @ w M 1 r @w @ = 0 =0 dr 2 M r w j ( r; )=( r 1 ; 0) ; ( r 2 ; 0 ) ( r; )=( r 2 ; 0) ; ( r 1 ; 0 ) : (A{52) Theterms T U ,and V inequationsEquation A{45 ,Equation A{52 ,and Equation A{39 ,respectively,maynowbecombinedintothesingleexpressi onof Equation A{34 ;thecompleteexpressionisnotgivenhereforbrevity.Beca usethevirtual displacementsarearbitrary,the\coecients"foreachmus tbezerotosatisfyHamilton's 211

PAGE 212

principle.Theextractedequationsofmotionarethen,afte rmovinginertialtermstothe right-handside, @N r @r + 1 r @N r @ + N r N r = I 0 u I 1 @ w @r (A{53) @N r @r + 1 r @N @ + 2 N r r = I 0 v I 1 1 r @ w @ ; (A{54) and @ 2 M r @r 2 + 2 r @M r @r + 1 r 2 @ 2 M @ 2 1 r @M @r + 2 r @ 2 M r @r@ + 2 r 2 @M r @ + 1 r @ @r rN r @w @r + N r @w @ + 1 r 2 @ @ rN r @w @r + N @w @ + q z = I 0 w + 1 r @ @r rI 1 u rI 2 @ w @r + 1 r @ @ I 1 v I 2 1 r @ w @ : (A{55) Althoughtheyhavebeencarriedthroughtothispointforcomp leteness,terms containingin-planeaccelerations u and v arenegligiblebecausethemotionoftheplateis primarilyinthe z -direction.Rotaryinertiatermscontaining I 2 canalsobeneglected,as theyprimarilycontributetohigher-ordervibrationmodes [ 121 198 ].Therstvibration modeistheprimaryoneofinterestforthisinvestigation.T heequationsofequilibrium thenbecome @N r @r + 1 r @N r @ + N r N r =0 ; (A{56) @N r @r + 1 r @N @ + 2 N r r =0 ; (A{57) and @ 2 M r @r 2 + 2 r @M r @r + 1 r 2 @ 2 M @ 2 1 r @M @r + 2 r @ 2 M r @r@ + 2 r 2 @M r @ + 1 r @ @r rN r @w @r + N r @w @ + 1 r 2 @ @ rN r @w @r + N @w @ + q z = I 0 w: (A{58) Theseequationsaresubjecttoboundaryconditionsthatare alsoextractedfromthe combinedequationforHamilton'sprinciple.Oneachboundar y,thereisanessential(or geometric)boundaryconditionandanaturalboundarycondi tion,fromwhichonemustbe specied[ 121 ].On r = r 1 and r = r 2 ,specify[ 122 ]: u or N r (A{59) v or N r (A{60) w or Q r + N r @w @r + N r 1 r @w @ + 1 r @M r @ (A{61) @w @r or M r : (A{62) 212

PAGE 213

Similarly,on =0 ; 0 specify[ 122 ]: u or N r (A{63) v or N (A{64) w or Q + N r @w @r + N 1 r @w @ + @M r @r (A{65) @w @ or M : (A{66) Finally,at( r; )=( r 1 ; 0) ; ( r 2 ; 0 ) ; ( r 2 ; 0) ; ( r 1 ; 0 ),specify[ 122 ]: w or M r : (A{67) Theshearintensities[ 122 ]appearinginEquation A{61 andEquation A{65 aredenedas Q r = 1 r @ @r ( rM r )+ @M r @ M (A{68) and Q = 1 r @ @r ( rM r )+ M r + @M @ : (A{69) NotethatEquations A{56 to A{58 arecompletelygeneralwithintheconnesofthevon Karmanplatetheory,i.e.theyarevalidforacircularpla tewitharbitrarycompositelayup andarbitrarydistributedload q z Restrictingtheproblemtooneexhibitingaxialsymmetry,a llquantitiesarenolonger regardedasfunctionsof ( @=@ =0).Inaddition,the -directeddisplacement, v ,is necessarilyzero.Fortheaxisymmetriccase,theequations ofmotionsimplifyto @N r @r + N r N r =0(A{70) and @ 2 M r @r 2 + 2 r @M r @r 1 r @M @r + 1 r @ @r rN r @w @r + q z = I 0 w: (A{71) Theboundaryconditionsaresimpliedaswell.On r = a and r = b ,specify: u or N r (A{72) w or Q r + N r w r (A{73) @w @r or M r : (A{74) Theremainingderivationwillfocusontheaxisymmetriccas e,ascarryingthemathematics throughforanon-axisymmetric,nonlinearcircularcompos iteplatewithunsymmetric layupisanunnecessarilylaborioustask.Theaxisymmetric restrictionalsoimpliesthat thematerialscomposingthecompositelaminatemustbetran sverselyisotropicandthat bothexternalloadingsandboundaryconditionsmustnotvar yin .Notealsothatwith 213

PAGE 214

theaxisymmetricrestrictioninplace,non-axisymmetricb ucklingorvibrationmodes| eventhoseresultingfromsymmetricloadings|cannotbepre dictedinabucklingor dynamicanalysis,respectively.Thenonlineartreatmento fanon-axisymmetricisotropic circularplatecanbefoundin[ 122 ]anditisrelativelystraightforwardtoextendittothe symmetriclaminatecasestartingfromEquations A{56 to A{58 A.4ConstitutiveEquation Tosolveforthereferenceplanedisplacements,theequatio nsofmotionmustbe writtenintermsofthesequantities.Thus,theforcesandmo mentsmustberelatedtothe displacements;thisisaccomplishedthroughincorporatio noftheconstitutivebehaviorof thematerial(s)ofwhichtheplateiscomposed.Ingeneral,t heplateconsideredhereis anasymmetricallylaminatedcompositewithintegratedpie zoelectriclayers.Thegeneral constitutiverelationshipforapiezoelectricmaterialis = S E f + d T E f ; (A{75) where S E f istheelasticcompliancematrix(measuredunderconstante lectriceld), d is thematrixofpiezoelectricconstants,and E f istheelectriceldvector.Forapiezoelectric materialoftheTetragonal4mmorHexagonal6mmcrystalclass (e.g.PZTandaluminum nitride,respectively),thespecicformoftheconstituti verelationis 8>>>>>><>>>>>>: r z 2 z 2 rz 2 r 9>>>>>>=>>>>>>; = 266666664 1 E p p E p zp E z 000 p E p 1 E p zp E z 000 zp E z zp E z 1 E z 000 000 1 G zp 00 0000 1 G zp 0 00000 1 G p 377777775 8>>>>>><>>>>>>: r z z rz r 9>>>>>>=>>>>>>; + 26666664 00 d 31 00 d 31 00 d 33 0 d 15 0 d 15 00 000 37777775 8<: E fr E f E fz 9=; ; (A{76) wherethesubscript p referstopropertiesintheplaneoftheplateand G p =2(1+ p ) =E p Notealsothatbasedonthegivendenitionfor S E f representengineering|not tensoral|strains[ 98 ]. Recognizingthatathinplateexistsinastateofplanestres sandthatelectrodelayers promotepotentialgradientsonlyinthe z -direction,theconstitutiveequationmaybe reducedto[ 198 ] 8<: r 2 r 9=; = 24 1 =E =E 0 =E 1 =E 0 002(1+ ) =E 35 8<: r z r 9=; + 8<: d 31 d 31 0 9=; E f : (A{77) Equation A{77 isconsistentwithamaterialexhibitingtransverseisotro py,anecessityfor thisderivationgiventheassumptionofaxisymmetry.The p subscripthasbeendropped hereforconvenience,anditwillbeunderstoodhencefortht hattheYoung'smodulus, E ,andPoisson'sratio, ,correspondtothepropertiesintheplaneoftheplate.The subscript z hasalsobeendroppedfromtheelectriceldterm,whichisno wunderstoodto 214

PAGE 215

beorientedinthe z -direction.Next,letting = r 2 r T ; (A{78) = r r T ; (A{79) and d = d 31 d 31 0 T ; (A{80) Equation A{77 issolvedforthestresses, = Q ( d E f ) ; (A{81) where Q = 24 Q 11 Q 12 0 Q 12 Q 11 0 00 Q 66 35 (A{82) = E 1 2 24 1 0 10 00(1 ) = 2 35 (A{83) aretheplanestress-reducedstinesses. In-planeresidualstresses|analogoustothermalstresses |areintroducedherevia addinganextratermtotheconstitutiverelation,Equation A{81 .Theresultisthen = 0 + Q ( E f d ) ; (A{84) where 0 = 0 0 0 T : (A{85) Noassumptionsaremadeatthistimeaboutthespatialdistrib utinofthein-plane stresses.Next,substitutingEquation A{26 intoEquation A{84 givesthestressesinterms ofthereferencesurfacestrainsandcurvaturesas = 0 + Q 0 + z E f d (A{86) IntegratingEquation A{86 throughthethickness(i.e.from z = z b to z t )subjectto thedenitionsofthein-planeforcesandmomentsfoundinEq uations A{48 to A{49 yields N = N 0 + A" 0 + B N p (A{87) and M = M 0 + B" 0 + D M p ; (A{88) where N = N r N N r T ; (A{89) M = M r M M r T ; (A{90) 215

PAGE 216

N 0 = Z z t z b 0 dz; (A{91) M 0 = Z z t z b 0 zdz; (A{92) N p = Z z t z b E f Qd dz; (A{93) and M p = Z z t z b E f Qd zdz: (A{94) Theextensionalstinesses A ,bending-extensionalcouplingstinesses B ,andbending stinesses D aregivenas A = Z z t z b Q dz; (A{95) B = Z z t z b Q zdz; (A{96) and D = Z z t z b Q z 2 dz; (A{97) respectively.Forasymmetriclaminate,i.e.oneinwhichth elayersabovethereference surfaceareexactmirrorimagesofthosebelowthereference surface(intermsofmaterial properties,orientation,andthickness), B = 0 .Inthetypicalcaseofconstantproperties withineachindividuallayerofthecomposite,theintegral susedinEquations A{91 to A{97 canberewrittenassummationsintermsofindividuallayerc oordinates (Figure A-2 )as n FigureA-2.Layercoordinatesforanarbitrarycompositelay up. Z z t z b () dz = L X i =1 () i ( z i +1 z i )= L X i =1 () i H i ; (A{98) Z z t z b () zdz = 1 2 L X i =1 () i z 2 i +1 z 2 i = L X i =1 () i z i H i ; (A{99) 216

PAGE 217

and Z z t z b () z 2 dz = 1 3 L X i =1 () i z 3 i +1 z 3 i = L X i =1 () i H 3 i 12 + H i z 2i ; (A{100) where() i referstothevalueoftheintegrandinthe i thlayer, H i isthe i thlayerthickness, and z i isthecoordinateofthecenterofthe i thlayer.Equations A{87 to A{88 areoften writteninamorecompactformas N M = N 0 M 0 + AB BD 0 M p N p : (A{101) Also,forconvenience,let ~ N ~ M = AB BD 0 : (A{102) suchthattheoverallconstitutiveequationforthelaminat edcompositebecomes N M = N 0 M 0 + ~ N ~ M M p N p : (A{103) Observingtheaxisymmetricassumption, N r = M r =0andthethirdcomponentcan bedroppedfrom N and M because r = r =0.Thevariousstinessmatricesthen onlyneedberegardedas2 2. A.5DisplacementDierentialEquationsofMotion Theaxisymmetricformoftheconstitutiverelationsdevelo pedinSection A.4 may nowbecombinedwiththeaxisymmetricequationsofmotion,E quations A{70 to A{71 toyieldgoverningdierentialequationsforthereference planedisplacements u ( r ; t ) and w ( r ; t ).First,Equation A{102 isexpandedtoexplicitlydeneeachoftheforceand momentterms, ~ N r = A 11 du dr + 1 2 dw dr 2 # + A 12 u r B 11 d 2 w dr 2 B 12 1 r dw dr ; (A{104) ~ N = A 11 u r + A 12 du dr + 1 2 dw dr 2 # B 11 1 r dw dr B 12 d 2 w dr 2 ; (A{105) ~ M r = B 11 du dr + 1 2 dw dr 2 # + B 12 u r D 11 d 2 w dr 2 D 12 1 r dw dr ; (A{106) and ~ M = B 11 u r + B 12 du dr + 1 2 dw dr 2 # D 11 1 r dw dr D 12 d 2 w dr 2 : (A{107) Continuing,Equation A{70 isrstsolvedfor N andthensubstitutedintoEquation A{71 toyield @ 2 M r @r 2 + 2 r @M r @r 1 r @M @r + 1 r @ @r rN r @w @r + q z = I 0 w: (A{108) 217

PAGE 218

SubstitutingEquations A{106 to A{107 intoEquation A{108 yields D 11 r 4 w + B 11 @ 3 u @r 3 + 2 r @ 2 u @r 2 1 r 2 @u @r + 1 r 3 u + @ 2 w @r 2 2 + @w @r @ 3 w @r 3 + 2 r @w @r @ 2 w @r 2 # B 12 r @w @r @ 2 w @r 2 + r 2 ( M 0 M p )+ 1 r @ @r rN r @w @r + q z = I 0 w; (A{109) wherethefamiliarbiharmonicandLaplacianoperatorsared enedfortheaxisymmetric problemas r 4 ()= 1 r d dr r d dr 1 r d dr r d () dr = d 4 () dr 4 + 2 r d 3 () dr 3 1 r 2 d 2 () dr 2 + 1 r 3 d () dr (A{110) and r 2 ()= 1 r d dr r d () dr = d 2 () dr 2 + 1 r d () dr ; (A{111) respectively.Similarly,taking 1 r @ [ r ()] @r ofEquation A{70 andthensubstitutinginfor N r and N usingEquations A{104 to A{105 yields B 11 r 4 w + A 11 @ 3 u @r 3 + 2 r @ 2 u @r 2 1 r 2 @u @r + 1 r 3 u + @ 2 w @r 2 2 + @w @r @ 3 w @r 3 + 2 r @w @r @ 2 w @r 2 # A 12 r @w @r @ 2 w @r 2 + r 2 ( N 0 N p )=0 : (A{112) ClearlybothEquations A{112 and A{109 haveverysimilarforms.MultiplyingEquation A{112 by B 11 =A 11 andsubtractingEquation A{109 fromtheresultgivesthegoverningequation for w I 0 w + D 11 r 4 w = q z B 12 1 r @w @r @ 2 w @r 2 + 1 r @ @r rN r @w @r + r 2 ( M 0 M p ) B 11 A 11 ( N 0 N p ) : (A{113) where D 11 = D 11 B 2 11 A 11 (A{114) and B 12 = B 12 B 11 A 12 A 11 : (A{115) Equation A{113 containstwounknowns, w and N r .Asecondequationfor N r is thereforerequired.Tondthisequation,Equation A{28 issolvedfor u andsubstituted 218

PAGE 219

intoEquation A{27 toyield @" 0 @r + 0 0r r + 1 2 r @w @r 2 =0 : (A{116) Thisisknownasacompatibilitycondition.InvertingEquat ion A{102 tond 0r and 0 in termsof ~ N r ~ N ,and w andthensubstitutingtheresultintoEquation A{116 anddividing by r 2 yields @ 2 ~ N r @r 2 + 3 r @ ~ N r @r = B 12 1 r @ 3 w @r 3 + 1 r 2 @ 2 w @r 2 1 r 3 @w @r A 211 A 212 A 11 1 2 r 2 @w @r 2 : (A{117) Together,Equations A{113 and A{117 arethemixed-formdierentialequationsfor themotionofthepiezoelectriccompositeplate.Alternativ ely,Equation A{112 maybe rearrangedintoadierentialequationfor u intermsof w @ 3 u @r 3 + 2 r @ 2 u @r 2 1 r 2 @u @r + 1 r 3 u = B 11 A 11 r 4 w @ 2 w @r 2 2 @w @r @ 3 w @r 3 2 r @w @r @ 2 w @r 2 + A 12 A 11 1 r @w @r @ 2 w @r 2 1 A 11 r 2 ( N 0 N p )(A{118) and N r maybesubstitutedintoEquation A{113 togiveasetofgoverningdierential equationspurelyintermsofdisplacement. A.6EquationsofEquilibrium Atthisjuncture,thefocusshiftstoparticularsoftheprob lembeingpursued,and severalnewassumptionsaremade.First,theproblemisrest rictedtothestaticcase forwhich w =0;thepartialdierentialequations(PDEs)thereforebec omeordinary dierentialequations(ODEs).Next, 0 and E f arerestrictedtobeconstantinanygiven layerofthecompositeplate,whichresultsin N 0 M 0 N p ,and M p notbeingfunctionsof r Finally,theloadingisrestrictedtoauniformpressureact inginthe z -direction,i.e. q z = p Inthefollowingtwosections,theseassumptionsareapplie dtothenonlinearequationsof motion,whicharethenlinearized.A.6.1Nonlinear Undertheassumptionspresentedintheintroduction,thegov erningequations become,withsomemanipulation, D 11 r 4 w = p B 12 1 2 r d dr dw dr 2 # + 1 r d dr rN r dw dr ; (A{119) with r 4 ()= 1 r d dr r d dr 1 r d dr r d () dr : (A{120) 219

PAGE 220

Equation A{119 ismultipledby r ,integratedwithrespectto r ,andthendividedby D 11 r toyield d 3 w dr 3 + 1 r d 2 w dr 2 1 r 2 dw dr = pr 2 D 11 + N r D 11 dw dr B 12 2 r dw dr 2 (A{121) Writingthisequationintermsoftransverserotation, = dw dr ; (A{122) yields d 2 dr 2 + 1 r d dr r 2 = pr 2 D 11 + N r D 11 + B 12 2 D 11 2 r (A{123) EitherofEquations A{121 or A{123 maybetakenasthegoverningequationfor transversereferencesurfacedisplacements. Thegoverningequationforin-planereferencesurfacedisp lacementsisfoundfrom Equation A{118 ,whichrstmaybeequivalentlyrewrittenas 1 r d dr r d dr 1 r d ( ru ) dr = B 11 A 11 r 4 w 1 r d dr r dw dr d 2 w dr 2 + A 12 A 11 1 1 2 r d dr dw dr 2 # (A{124) SubstitutingEquation A{122 intoEquation A{124 ,multiplyingEquation A{124 by r integratingwithrespectto r ,anddividingby r gives d 2 u dr 2 + 1 r du dr u r 2 = B 11 A 11 d 2 dr 2 + 1 r d dr r 2 1 A 12 A 11 2 2 r d dr : (A{125) Equations A{123 and A{125 areadditionallylinkedby N r .SubstitutingEquation A{122 intoEquation A{117 andEquation A{102 intoEquation A{123 gives d 2 ~ N r dr 2 + 3 r d ~ N r dr = B 12 1 r d 2 dr 2 + 1 r 2 d dr r 3 A 211 A 212 A 11 2 2 r 2 : (A{126) and d 2 dr 2 + 1 r d dr N 0 N p D 11 + 1 r 2 = pr 2 D 11 + ~ N r D 11 + 2 B 12 2 rD 11 ; (A{127) whichtogetherarethemixed-formdierentialequationsof equilibrium.Alternatively, Equation A{125 andEquation A{127 ,with ~ N r substitutedinfromEquation A{102 togethercomposethedisplacement-baseddierentialequa tionsofequilibrium. Forconvenienceinfuturesteps,letthein-planestresspar ameterbedenedas k 2 = j N 0 N p j a 2 D 11 ; (A{128) 220

PAGE 221

where a isacharacteristicdimensionoftheplate(suchasouterrad ius).Substitutinginto Equation A{127 d 2 dr 2 + 1 r d dr x k 2 a 2 + 1 r 2 = pr 2 D 11 + ~ N r D 11 + 2 B 12 2 rD 11 ; (A{129) where x isaragdenotingthenetsenseofthein-planeforceterms,i. e. x =sgn N 0 N P r : (A{130) A.6.2Linear InordertolinearizeEquations A{125 to A{127 ,secondorderproductsofdisplacements areneglected,includingtheproduct ~ N r since ~ N r isafunctionofthedisplacements.In addition,since N p isproportionaltovoltage,therealwaysexistsasucientl ysmall voltageinputforwhich N p N 0 ,andthereforeitmaybeneglectedfromthegoverning dierentialequation.Alternatively,forthecaseofsmall N 0 ,bothmaybenegligible comparedtotheremainingtermsinthelinearizedgoverning dierentialequation.Thus, neglectingsecondorderproductsofdisplacementsandthe N p termfromthegoverning dierentialequationsyields d 2 u dr 2 + 1 r du dr u r 2 = B 11 A 11 d 2 dr 2 + 1 r d dr r 2 : (A{131) d 2 ~ N r dr 2 + 3 r d ~ N r dr = B 12 r d 2 dr 2 + 1 r d dr r 2 ; (A{132) and d 2 dr 2 + 1 r d dr x k 2 a 2 + 1 r 2 = pr 2 D 11 ; (A{133) where k 2 isnowredenedas k 2 = j N 0 j a 2 D 11 ; (A{134) ThesolutionofEquation A{133 takesonthreeformsthataredependentonitsclassication Itisarst-order,non-homogeneousmodiedBesselequatio nfor x> 0,arst-order, non-homogeneousBesselequationfor x< 0,andanon-homogeneousCauchy-Euler equationwhen x =0[ 199 ].Equation A{133 isnextsubstitutedintoEquations A{131 to A{132 ,givingthesimpliedforms d 2 u dr 2 + 1 r du dr u r 2 = B 11 A 11 pr 2 D 11 + x k 2 a 2 : (A{135) and d 2 ~ N r dr 2 + 3 r d ~ N r dr = B 12 r pr 2 D 11 + x k 2 a 2 : (A{136) Equation A{133 andEquations A{135 to A{136 arenowsequentiallycoupled,inthatthe solutionfor mustrstbeobtainedbeforethesolutionto ~ N r or u maybe.Inaddition, 221

PAGE 222

itisalsonowclearthatbothEquation A{135 andEquation A{136 areformsofthe non-homogeneousCauchy-Eulerequation[ 199 ]. A.7ProblemSolutions Inthissection,solutionsarepresentedforthelinearandn onlinearformsofthe equilibriumequationsofthepiezoelectriccompositeplat e.Thelinearsolutionisfully analytical,thoughtheequationsbecomesucientlycumber somethatamatrixinversion issuggestedforuseindeterminingintegrationcoecients .Meanwhile,thenonlinear equationsarewritteninaconvenientformfornumericalsol utionviareadily-available multi-pointboundaryvalueproblemsolvers. Thesolutiondomainisdividedintoaninnerandouterregion andthesolutions, materialproperties,geometricproperties,etc.withinap articulardomainwillbedenoted byasuperscript (1) fortheinnerregionand (2) fortheouterregion.Letthearbitrary lengthscale a foundinthegoverningequationscorrespondtotheouterrad iusofthe regionofinterest. TheboundaryconditionsfollowfromthechoicesgiveninEqu ations A{72 to A{74 .Usingthetwo-domainnotation,theboundaryconditionsfor theprobleminclude symmetryconditionsattheplatecenter( r =0), (1) (0)=0 ; (A{137) u (1) (0)=0 ; (A{138) matchingconditionsattheinterfacebetweentheinnerando uterregion( r = a (1) ), (1) a (1) = (2) a (1) ; (A{139) u (1) a (1) = u (2) a (1) ; (A{140) w (1) a (1) = w (2) a (1) ; (A{141) M (1) r a (1) = M (2) r a (1) ; (A{142) N (1) r a (1) = N (2) r a (1) ; (A{143) andboundaryconditionsontheouterradius( r = a (2) ) M (1) r a (2) = k (2) a (2) (A{144) u (2) a (2) =0(A{145) w (2) a (2) =0(A{146) ThecompliantboundaryconditionofEquation A{144 eectivelymeansboththe simply-supported( k =0)andclamped( k = 1 )casesareavailablefromthenal solution. ThesolutionsinthecomingsectionsmakeuseoftheBesselfu nctionsoftherstand secondkind, J n and Y n ,respectivelyandthemodied-Besselfunctionsoftherst and secondkind, I n and K n ,respectively[ 200 ]. 222

PAGE 223

A.7.1LinearA.7.1.1Generalsolutions Thegeneralsolutionstothegoverninglinearequationsofe quilibrium,Equations A{133 A{135 ,and A{136 ,are ( r )= 8>>>>>><>>>>>>: c 1 I 1 k r a + c 2 K 1 k r a + 1 2 pa 2 r D 11 k 2 ;x> 0 c 1 r + c 2 r 1 16 pr 3 D 11 ;x =0 c 1 J 1 k r a + c 2 Y 1 k r a 1 2 pa 2 r D 11 k 2 ;x< 0 ; (A{147) u ( r )= 8>>>>><>>>>>: c 3 r + c 4 r B 11 A 11 h c 1 I 1 k r a + c 2 K 1 k r a i ;x> 0 c 3 r + c 4 r + 1 16 B 11 A 11 pr 3 D 11 ;x =0 c 3 r + c 4 r B 11 A 11 h c 1 J 1 k r a + c 2 Y 1 k r a i ;x< 0 ; (A{148) and w ( r )= 8>>>>>><>>>>>>: c 1 a k I 0 k r a + c 2 a k K 0 k r a 1 4 pa 2 r 2 D 11 k 2 + c 5 ;x> 0 c 1 1 2 r 2 c 2 ln( r )+ 1 64 pr 4 D 11 + c 5 ;x =0 c 1 a k J 0 k r a + c 2 a k Y 0 k r a + 1 4 pa 2 r 2 D 11 k 2 + c 5 ;x< 0 : (A{149) Followingfromthesesolutionsaretheforceandmomentresu ltants,Equation A{102 whichwithnonlineartermsneglectedare N r ( r )= 8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>: N 0 N p + B 12 I 1 k r a 1 r c 1 + B 12 K 1 k r a 1 r c 2 +( A 11 + A 12 ) c 3 ( A 11 A 12 ) 1 r 2 c 4 + 1 2 ( B 11 + B 12 ) pa 2 D 11 k 2 ; x> 0 N p +( B 11 + B 12 ) c 1 ( B 11 B 12 ) 1 r 2 c 2 +( A 11 + A 12 ) c 3 ( A 11 A 12 ) 1 r 2 c 4 B 12 1 16 pr 2 D 11 ; x =0 N 0 N p + B 12 J 1 k r a 1 r c 1 + B 12 Y 1 k r a 1 r c 2 +( A 11 + A 12 ) c 3 ( A 11 A 12 ) 1 r 2 c 4 1 2 ( B 11 + B 12 ) pa 2 D 11 k 2 ; x< 0 ; (A{150) 223

PAGE 224

and M r ( r )= 8>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: M 0 M p + D 11 k a I 0 k r a ( D 11 D 12 ) I 1 k r a 1 r c 1 D 11 k a K 0 k r a +( D 11 D 12 ) K 1 k r a 1 r c 2 +( B 11 + B 12 ) c 3 ( B 11 B 12 ) 1 r 2 c 4 + 1 2 ( D 11 + D 12 ) pa 2 D 11 k 2 ; x> 0 M 0 M p +( D 11 + D 12 ) c 1 ( D 11 D 12 ) 1 r 2 c 2 +( B 11 + B 12 ) c 3 ( B 11 B 12 ) 1 r 2 c 4 (3 D 11 + D 12 ) 1 16 pr 2 D 11 ; x =0 M 0 M p + D 11 k a J 0 k r a ( D 11 D 12 ) J 1 k r a 1 r c 1 + D 11 k a Y 0 k r a ( D 11 D 12 ) Y 1 k r a 1 r c 2 +( B 11 + B 12 ) c 3 ( B 11 B 12 ) 1 r 2 c 4 1 2 ( D 11 + D 12 ) pa 2 D 11 k 2 ; x< 0 : (A{151) A.7.1.2Particularsolutions Intotal,thereare5unknownintegrationconstantsintrodu cedviaEquations A{147 to A{149 : c 1 c 2 c 3 c 4 ,and c 5 .Asthesolutiontoeachoftheseequationsissoughtin eitherdomain,thereareactuallyatotalof10unknowninteg rationconstantsinthe two-domainproblem: c (1)i and c (2)i with i =1 ::: 5.Withthetenboundaryconditionsof Equations A{137 to A{146 ,theproblemisthuswell-posed. GiventhelengthoftheequationsintroducedinSection A.7.1.1 ,solvingforthe integrationconstantsexplicitlyisaburdensometask.Ins tead,theyarefoundviawriting theboundaryconditionsasasystemoflinearequationssolv edviamatrixinversion. Beforethatprocess,though,thesymmetryconditionsofEqu ations A{137 to A{138 immediatelyrevealthat c (1)2 = c (1)4 =0becauseeachofthetermstheyareassociated withinEquations A{147 to A{148 areunboundedat r =0.Thisreducestheunknown integrationconstantsto8intotal.Aftersubstitutingthea ppropriategeneralsolutionsof Section A.7.1.1 intoanyoftheremainingboundaryconditions,theresultma yalwaysbe writtenintheform C (1) i T c (1) + f (1) i = C (2) i T c (2) + f (2) i ; (A{152) whereeach C ( j ) i isanarraywhichcontainsthecoecientsoftheintegration constants, c ( j ) containstheintegrationconstantsthemselves,and f ( j ) i representsthecollected freetermsforthe i thboundarycondition( i =1 ; 2 ::: 8)inthe j thdomain( j =1 ; 2). CollectingtheeightequationsrepresentedbyEquation A{152 intoasinglematrix equationgives h C (1) 8 x 3 C (2) 8 x 5 i 8<: 3 x 1 c (1) c (2) 5 x 1 9=; = f (1) 8 x 1 f (2) 8 x 1 : (A{153) 224

PAGE 225

TheutilityofEquation A{153 isthattheintegrationconstantsarefoundinamodular mannerthatallowsforthecoupledsolutionofanyinnerregi oncase( x (1) = 1 ; 0 ; 1)and anyouterregioncase( x (2) = 1 ; 0 ; 1)simplyviamatrixinversionofthecombined[ C ] matrix. Derectionofthediaphragmoccursduetoanyorallof3inputs :initialstress, pressure,orvoltage.Inthecaseofaninitiallystresseddi aphragm,thereisanexisting staticderectionbeforetheapplicationofvoltageorpress ure.Voltageorpressureloading leadstoanadditionalincrementalderectionwhichintheco ntextoflumpedelement modelingisthequantityofinterest.Itisthusconvenientt osolvefortheincremental derectiondirectly.Thisismadepossibleviadividingthea rrayofforcingterms, f ( j ) ,into itscomponentsparts, f ( j ) = f ( j ) 0 + f ( j ) p + f ( j ) v ; (A{154) whereeachof f ( j ) 0 f ( j ) p ,and f ( j ) v includeonlythosetermsrelatingtoin-planestress, pressure,andvoltage,respectively,withallotherszero. Tosolvefortheinitialderection alone,replace f ( j ) withonlythein-planecomponent, f ( j ) 0 (equivalenttoletting v = p = 0).Tosolvefortheincrementalderectiondirectly,replac ethetotal f ( j ) by f ( j ) p or f ( j ) v forincrementalderectionduetopressureandvoltage,resp ectively.Intheselattercases, thein-planestressstillaectsthestinessviaitsinclus ionin C Ineachofthefollowingsubsections,thespecicdenition softhe C ( j ) andeachof thecomponentsof f ( j ) aregivenforallcases.Thesolutionoftheproblemisobtain edby usingtheseexpressionstoassembleEquation A{153 ,solvingfortheintegrationconstants numericallyviamatrixinversion,andthenpluggingthenum ericalvaluesintoanyof Equations A{147 to A{151 dependingonthequantityofinterest.Thesubstitutionpar tof theprocessisperformedforeachdomain,sointheendtherea retwoequationsforeachof thedisplacementsandforcecomponents|correspondingtot heinnerandouterdomain |withvalidityover0 r a (1) and a (1) r a (2) ,respectively. 225

PAGE 226

A.7.1.3Innerregion:tension( x (1) > 0 ) C (1) = 266666666666664 I 1 k (1) 00 B (1) 11 A (1)11 I 1 k (1) a (1) 0 a (1) k (1) I 0 k (1) 01 h D (1) k (1) I 0 k (1) D (1) D (1) 12 I 1 k (1) i 1 a (1) B (1) 11 + B (1) 12 0 B (1) 12 I 1 k (1) 1 a (1) A (1)11 + A (1)12 0 000000000 377777777777775 (A{155) f (1) = 8>>>>>>>>>><>>>>>>>>>>: 000 M (1) 0 N (1) 0 000 9>>>>>>>>>>=>>>>>>>>>>; + 8>>>>>>>>>>>>><>>>>>>>>>>>>>: 1 2 pa (1)3 D (1) k (1)2 0 1 4 pa (1)4 D (1) k (1)2 1 2 D (1) 11 + D (1) 12 pa (1)2 D (1) k (1)2 1 2 B (1) 11 + B (1) 12 pa (1)2 D (1) k (1)2 000 9>>>>>>>>>>>>>=>>>>>>>>>>>>>; + 8>>>>>>>>>><>>>>>>>>>>: 000 M (1) p N (1) p 000 9>>>>>>>>>>=>>>>>>>>>>; (A{156) A.7.1.4Innerregion: x (1) =0 C (1) = 2666666666664 a (1) 00 0 a (1) 0 1 2 a (1) 2 01 D (1) 11 + D (1) 12 B (1) 11 + B (1) 12 0 B (1) 11 + B (1) 12 A (1)11 + A (1)12 0 000000000 3777777777775 (A{157)226

PAGE 227

f (1) = 8>>>>>>>>>><>>>>>>>>>>: 000 M (1) 0 0000 9>>>>>>>>>>=>>>>>>>>>>; + 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>: 1 16 pa (1) 3 D (1) 1 16 B (1) 11 A (1)11 pa (1)3 D (1) 1 64 pa (1)4 D (1) h 3 D (1) + D (1) 12 i 1 16 pa (1)2 D (1) B (1) 12 1 16 pa (1)2 D (1) 000 9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>; + 8>>>>>>>>>><>>>>>>>>>>: 000 M (1) p N (1) p 000 9>>>>>>>>>>=>>>>>>>>>>; (A{158) A.7.1.5Innerregion:compression( x (1) < 0 ) C (1) = 266666666666664 J 1 k (1) 00 B (1) 11 A (1)11 J 1 k (1) a (1) 0 a (1) k (1) J 0 k (1) 01 h D (1) k (1) J 0 k (1) D (1) D (1) 12 J 1 k (1) i 1 a (1) B (1) 11 + B (1) 12 0 B (1) 12 J 1 k (1) 1 a (1) A (1)11 + A (1)12 0 000000000 377777777777775 (A{159) f (1) = 8>>>>>>>>>><>>>>>>>>>>: 000 M (1) 0 N (1) 0 000 9>>>>>>>>>>=>>>>>>>>>>; + 8>>>>>>>>>>>>><>>>>>>>>>>>>>: 1 2 pa (1)3 D (1) k (1)2 0 1 4 pa (1)4 D (1) k (1)2 1 2 D (1) 11 + D (1) 12 pa (1)2 D (1) k (1)2 1 2 B (1) 11 + B (1) 12 pa (1)2 D (1) k (1)2 000 9>>>>>>>>>>>>>=>>>>>>>>>>>>>; + 8>>>>>>>>>><>>>>>>>>>>: 000 M (1) p N (1) p 000 9>>>>>>>>>>=>>>>>>>>>>; (A{160)227

PAGE 228

A.7.1.6Outerregion:tension( x (2) > 0 ) C (2) = 266666666666666664 I 1 k (2) K 1 k (2) 000 B (2) 11 A (2)11 I 1 k (2) B (2) 11 A (2)11 K 1 k (2) a (1) 1 a (1) 0 a (2) k (2) I 0 k (2) a (2) k (2) K 0 k (2) 001 D (2) k (2) a (2) I 0 k (2) D (2) D (2) 12 I 1 k (2) 1 a (1) D (2) k (2) a (2) K 0 k (2) D (2) D (2) 12 K 1 k (2) 1 a (1) B (2) 11 + B (2) 12 B (2) 11 B (2) 12 1 a (1)2 0 B (2) 12 I 1 k (2) 1 a (1) B (2) 12 K 1 k (2) 1 a (1) A (2)11 + A (2)12 A (2)11 A (2)12 1 a (1)2 0 D (2) k (2) I 0 k (2) D (2) D (2) 12 k a (2) I 1 k (2) # 1 a (2) D (2) k (2) K 0 k (2) + D (2) D (2) 12 k a (2) K 1 k (2) # 1 a (2) B (2) 11 + B (2) 12 B (2) 11 B (2) 12 1 a (2)2 0 B (2) 11 A (2)11 I 1 k (2) B (2) 11 A (2)11 K 1 k (2) a (2) 1 a (2) 0 a (2) k (2) I 0 k (2) a (2) k (2) K 0 k (2) 001 377777777777777775 (A{161) f (2) = 8>>>>>>>>>>><>>>>>>>>>>>: 000 M (2) 0 N (2) 0 M (2) 0 00 9>>>>>>>>>>>=>>>>>>>>>>>; + 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>: 1 2 pa (2)2 a (1) D (2) k (2)2 0 1 4 pa (2)2 a (1)2 D (2) k (2)2 1 2 D (2) 11 + D (2) 12 pa (2)2 D (2) k (2)2 1 2 B (2) 11 + B (2) 12 pa (2)2 D (2) k (2)2 D (2) 11 + D (2) 12 + k a (2) 1 2 pa (2)2 D (2) k (2)2 0 1 4 pa (2)4 D (2) k (2)2 9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>; + 8>>>>>>>>>>><>>>>>>>>>>>: 000 M (2) p N (2) p M (2) p 00 9>>>>>>>>>>>=>>>>>>>>>>>; (A{162)228

PAGE 229

A.7.1.7Outerregion: x (2) =0 C (2) = 266666666666664 a (1) 1 a (1) 000 00 a (1) 1 a (1) 0 1 2 a (1)2 ln a (1) 001 D (2) 11 + D (2) 12 D (2) 11 D (2) 12 1 a (1)2 B (2) 11 + B (2) 12 B (2) 11 B (2) 12 1 a (1)2 0 B (2) 11 + B (2) 12 B (2) 11 B (2) 12 1 a (1)2 A (2)11 + A (2)12 A (2)11 A (2)12 1 a (1)2 0 D (2) 11 + D (2) 12 + a (2) k D (2) 11 D (2) 12 a (2) k 1 a (2)2 B (2) 11 + B (2) 12 B (2) 11 B (2) 12 1 a (2)2 0 00 a (2) 1 a (2) 0 1 2 a (2)2 ln a (2) 001 377777777777775 (A{163) f (2) = 8>>>>>>>>>>><>>>>>>>>>>>: 000 M (2) 0 N (2) 0 M (2) 0 00 9>>>>>>>>>>>=>>>>>>>>>>>; + 8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>: 1 16 pa (1) 3 D (2) 1 16 B (2) 11 A (2)11 pa (1)3 D (2) 1 64 pa (1)4 D (2) 3 D (2) + D (2) 12 1 16 pa (1)2 D (2) B (2) 12 1 16 pa (1)2 D (2) 3 D (2) + D (2) 12 + k a (2) 1 16 pa (2)2 D (2) 1 16 B (2) 11 A (2)11 pa (2)3 D (2) 1 32 a (2)3 D (2) 1 2 pa (2) 9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>; + 8>>>>>>>>>>><>>>>>>>>>>>: 000 M (2) p N (2) p M (2) p 00 9>>>>>>>>>>>=>>>>>>>>>>>; (A{164)229

PAGE 230

A.7.1.8Outerregion:compression( x (2) =0 ) C (2) = 26666666666666666666664 J 1 k (2) Y 1 k (2) 000 B (2) 11 A (2)11 J 1 k (2) B (2) 11 A (2)11 Y 1 k (2) a (1) 1 a (1) 0 a (2) k (2) J 0 k (2) a (2) k (2) Y 0 k (2) 001 D (2) k (2) a (2) J 0 k (2) D (2) D (2) 12 J 1 k (2) 1 a (1) D (2) k (2) a (2) Y 0 k (2) D (2) D (2) 12 Y 1 k (2) 1 a (1) B (2) 11 + B (2) 12 B (2) 11 B (2) 12 1 a (1)2 0 B (2) 12 J 1 k (2) 1 a (1) B (2) 12 Y 1 k (2) 1 a (1) A (2)11 + A (2)12 A (2)11 A (2)12 1 a (1)2 0 D (2) k (2) J 0 k (2) D (2) D (2) 12 k a (2) J 1 k (2) # 1 a (2) D (2) k (2) Y 0 k (2) D (2) D (2) 12 k a (2) Y 1 k (2) # 1 a (2) B (2) 11 + B (2) 12 B (2) 11 B (2) 12 1 a (2)2 0 B (2) 11 A (2)11 J 1 k (2) B (2) 11 A (2)11 Y 1 k (2) a (2) 1 a (2) 0 a (2) k (2) J 0 k (2) a (2) k (2) Y 0 k (2) 001 37777777777777777777775 (A{165) f (2) = 8>>>>>>>>>>><>>>>>>>>>>>: 000 M (2) 0 N 0 (2) M (2) 0 00 9>>>>>>>>>>>=>>>>>>>>>>>; + 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>: 1 2 pa (2)2 a (1) D (2) k (2)2 0 1 4 pa (2)2 a (1)2 D (2) k (2)2 1 2 D (2) 11 + D (2) 12 pa (2)2 D (2) k (2)2 1 2 B (2) 11 + B (2) 12 pa (2)2 D (2) k (2)2 D (2) 11 + D (2) 12 + k a (2) 1 2 pa (2)2 D (2) k (2)2 0 1 4 pa (2)4 D (2) k (2)2 9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>; + 8>>>>>>>>>>><>>>>>>>>>>>: 000 M (2) p N (2) p M (2) p 00 9>>>>>>>>>>>=>>>>>>>>>>>; (A{166)230

PAGE 231

A.7.2Nonlinear Inthissection,thesolutionmethodologyforthenonlinear displacement-based governingequationsisaddressed.Theapproachistomanipu latethegoverningequations intoaformthatiseasilysolvedusingexistingboundaryval ueproblemsolvers,for example bvp4c or bvp5c inMATLAB[ 138 ].Aprospectivesolvermustbecapableof simultaneouslyhandlingtheclassesofsingularboundaryv alueproblems(duetothe1 =r termsthatappearsinthegoverningequations)andmultipoi ntboundaryvalueproblems (duetoboundaryconditionsappliedattheinterfacebetwee ntheinnerandouterregions). Both bvp4c and bvp5c meetthiscriteria.Theyspecicallysolvesystemsofrstorder ODEsoftheform[ 138 ] y 0 = 1 r Sy + f ( r; y ) ; (A{167) subjecttotheconditionthat Sy ( 0 )= 0 ; (A{168) where S isamatrixofconstants.Thegoalofthissection,then,isto manipulatethe nonlineargoverningequationsintosuchaform. First,the1 =r 2 singularityisremovedviachangingtheindependentvariab lesin Equations A{125 and A{129 from and u to =r and u=r ,respectively.Performingthe manipulation, d 2 dr 2 r + 3 r d dr r x k 2 a 2 r = p 2 D 11 + ~ N r D 11 r + B 12 2 D 11 r 2 (A{169) and d 2 dr 2 u r + 3 r d dr u r = B 11 A 11 x k 2 a 2 + ~ N r D 11 r p 2 D 11 + B 12 2 D 11 r 2 # 1 A 12 A 11 1 2 r 2 r d dr r + r r : (A{170) Thegoverningequationsarethustobesolveddirectlyfor =r and u=r ,whichareeasily post-processedbackto and u Next,Equations A{169 to A{170 arerewrittenasaseriesofrst-orderODEsviathe followingdenitions: y 1 = r ; (A{171) y 3 = u r ; (A{172) and y 5 = w: (A{173) Thesystemofrst-orderODEsisthen y 0 1 = y 2 = d dr r (A{174) 231

PAGE 232

y 2 0 = 3 r y 2 + x k 2 a 2 + ~ N r D 11 y 1 p 2 D 11 + B 12 2 D 11 y 1 2 (A{175) y 3 0 = y 4 = d dr u r (A{176) y 4 0 = 3 r y 4 B 11 A 11 x k 2 a 2 + ~ N r D 11 y 1 p 2 D 11 + B 12 2 D 11 y 1 2 # 1 A 12 A 11 1 2 y 1 2 ( ry 2 + y 1 ) y 1 (A{177) y 0 5 = dw dr = = ry 1 (A{178) WritingtheseequationsexplicitlyintheformofEquation A{167 gives 8>>>><>>>>: y 0 1 y 0 2 y 0 3 y 0 4 y 0 5 9>>>>=>>>>; = 1 r 266664 000000 3000 00000000 30 00000 377775 8>>>><>>>>: y 1 y 2 y 3 y 4 y 5 9>>>>=>>>>; + 8>>>>>><>>>>>>: y 2 x k 2 a 2 + ~ N r D 11 y 1 p 2 D 11 + B 12 2 D 11 y 1 2 y 4 B 11 A 11 h x k 2 a 2 + ~ N r D 11 y 1 p 2 D 11 + B 12 2 D 11 y 1 2 i 1 A 12 A 11 1 2 y 1 2 ( ry 2 + y 1 ) y 1 ry 1 9>>>>>>=>>>>>>; : (A{179) TosatisfyEquation A{168 ,itmustbetruethat y 2 (0)= d dr r 0 =0(A{180) and y 4 (0)= d dr u r 0 =0 : (A{181) Forproof,startbyexpanding y 2 y 2 = 1 r d dr 1 r 2 : (A{182) 232

PAGE 233

Thesecondtermisindeterminateat r =0byboundarycondition A{137 .Applying L'Hospital'sruletothistermandcombiningtheresultwitht hersttermgives y 2 (0)= 1 2 r d dr r =0 (A{183) ProofthatEquation A{183 istrueisfoundviaapplicationofL'Hospital'sruleto Equation A{129 ,whichyieldsEquation A{183 exactly.Thesameanalysisholdsfor y 4 (usingEquation A{125 forthenalstep),provingthatcondition A{168 issatised. Tocompletethenonlinearsolutionstrategy,theboundaryc onditionsmustalsobe considered.Itisthereforerstconvenienttoreformulate equationsfor N r and M r interms ofthenewindependentvariables.First,theybecomeN r = xk 2 D 11 a 2 + A 11 ( r d dr u r + u r + 1 2 r 2 r 2 ) + A 12 u r + B 11 r d dr r + r + B 12 r (A{184) andM r = M 0 M p + B 11 ( r d dr u r + u r + 1 2 r 2 r 2 ) + B 12 u r + D 11 r d dr r + r + D 12 r ; (A{185) whichareequivalenttothesolver-friendlyforms N r = xk 2 D 11 a 2 + A 11 ry 4 + y 3 + 1 2 ( ry 1 ) 2 + A 12 y 3 + B 11 ( ry 2 + y 1 )+ B 12 y 1 (A{186) and M r = M 0 M P r + B 11 ry 4 + y 3 + 1 2 ( ry 1 ) 2 + B 12 y 3 + D 11 ( ry 2 + y 1 )+ D 12 y 1 : (A{187) 233

PAGE 234

Theboundaryconditions,transformedfromEquations A{137 to A{146 arethen y (1) 2 (0)=0 ; (A{188) y (1) 4 (0)=0 ; (A{189) y (1) 1 a (1) = y (2) 1 a (1) ; (A{190) y (1) 3 a (1) = y (2) 3 a (1) ; (A{191) y (1) 5 a (1) = y (2) 5 a (1) ; (A{192) M (1) r a (1) = M (2) r a (1) ; (A{193) N (1) r a (1) = M (2) r a (1) ; (A{194) M (2) r a (2) = k ry (2) 1 a (2) ; (A{195) y (2) 3 a (2) =0 ; and(A{196) y (2) 5 a (2) =0 : (A{197) Thiscompletesthesolutionstrategyforthenonlineargove rningdierentialequations. Notethatboundaryvalueproblemsolversareofcoursealsoca pableofsolvinglinear dierentialequations,andinfactastraightforwardwayof solvingthelinearproblemisto programavariantofthesolutionpresentedinthissectionw ithnonlineartermsremoved. A.8Closing Inthisappendix,solutionproceduresforboththelinearan dnonlinearproblemofa anaxisymmetric,laminated,pre-stressed,andradially-d iscontinuouscircularpiezoelectric plateexposedtopressureand/orvoltageloadingwereprese nted.Theselinearsolutionis utilizedtoprovideinputstothelumpedelementmodelinCha pter 5 ,whilethenonlinear solutionisusedtoformaconstraintintheoptimizationofC hapter 6 .Themodelsare validatedagainstniteelementanalysisinSection 5.2.5 234

PAGE 235

APPENDIXB BOUNDARYCONDITIONINVESTIGATION Thisappendixbrierydescribesaninvestigationoftheoute rboundarycondition(at r = a 2 )usedinthediaphragmmodel.Theclampedboundaryconditio nutilizedinthe modeldevelopmentofAppendix A andalsointhevalidationexerciseofSection 5.2.5.1 isanidealizationthatdoesnottakeintoaccountthecompli anceofthesubstrate.A niteelementmodelthatincludesthesubstrateisdevelope dheretocomparetothe niteelementsimulationcompletedinSection 5.2.5.1 ,whichutilizedaclampedboundary condition. @ @R FixedBC @ @R RollerBC Piezoelectriclmstack Siliconsubstrate R Pressureloading A B FigureB-1.Finiteelementmodelforinvestigationofbound arycompliancy.A)Boundary geometrywithboundaryconditions.B)Mesh. Aniteelementmodelthatincludesasectionofthesilicons ubstrate|withthe substratereducedhereto60 m 30 mbecausestressisconcentratedinthesurface region|ispicturedinFigure B-1 .Boundaryconditions,whichincludeaxedboundary conditiononthebottomofthesubstrateandarollerboundar yconditiononthefarright side,areshowninFigure B-1A .Thedisplacementproleswerefoundforapressure loadingalongtheentiretopsurfaceofthemodel.Allotherco nditions,includingthose involvingthepiezoelectriclmstackandstress,areretai nedfromSection 5.2.5.1 Thelinearmodeshapespredictedfromtheniteelementmode lswithbothclamped andcompliantboundariesarefoundinFigure B-2 foranappliedpressureof111dB. 235

PAGE 236

Agreementisverygood;whenintegrated,thetotaldierence involumedisplacement 8 islessthan1%.Thisindicatesthatatleastforthegeometry usedinthismodel(design D),compliantboundaryconditionsaren'tlikelytobealarg econtributortoerrorinlinear modelpredictions. 050100150200250300 0 0 : 5 1 1 : 5 Radialcoordinate[ m]w inc (0)[nm] ClampedBC CompliantBC FigureB-2.DerectionprolesfromFEAwithclampedandcomp liantboundary conditions(P=111dB). Figure B-3A showsacomparisonbetweentheincrementalderectionat r =0found fromtheniteelementmodelwithaclampedouterboundaryco ndition(Section 5.2.5.1 ) andthatfoundfromtheniteelementmodelwithsubstrate.T hetwoagreeextremely closelyovermuchoftheinterval.Aplotoftherelativeerro rbetweenthetwosetsof simulationresultsisfoundinFigure B-3B ,whichshowsthatthecomplianceinthe boundaryconditionsbecomesmoreimportantathighpressur elevels.Therelativeerroris below5.5%upto172dB,withamaximumof11.4%at190dB. 100120140160180 10 3 10 1 10 1 Pressure[dBre20 Pa]w inc (0)[ m] ClampedBC CompliantBC A 100120140160180 0 5 10 15 Pressure[dBre20 Pa]Rel.Errorin w inc (0)[%]B FigureB-3.FEAresultsformodelswithclampedandcomplian tboundaryconditions (versuspressure).A)Incrementalcenterderection.B)Rela tiveerror. 236

PAGE 237

APPENDIXC UNCERTAINTYANALYSIS Thisappendixaddressesthecalculationofuncertaintyest imatesformeasured quantitiesinChapter 8 C.1Approach Thegeneralapproachtouncertaintyestimationtobeusedhe reisconsistentwiththe methodologypresentedinColeman&Steele[ 201 ],whichisdrawnfromtheISOstandard [ 202 ].Ineachcase,thecombinedstandarduncertaintyforarand omvariableisrstfound viatherootsumsquaremethod u = vuut s 2 + M X k =1 b 2k ; (C{1) where s isthestandarddeviationestimatefortherandomuncertain tyand b k isthe standarddeviationestimateforthe k thsystematicuncertainty(bias).Notethattheuse ofEquation C{1 requiresthaterrorsourcesareuncorrelated.Thecondenc eboundsare thendeterminedusingacoveragefactorvia U % = t % u; (C{2) where t % isthe t -statisticwith degreesoffreedomassociatedwithaselectedcondence levelinpercent.The t -statisticmaybedrawnfromstandardtables[ 34 163 201 ]orfrom MATLABastinv(1/2, )forthe(1 ) 100%condencebound.Thedegreesof freedom inthepresenceofbiaserrorscanbeestimatedfromtheWelch -Satterthwaite formula[ 201 ].Thestatistic t 95% istypicallytakenas2for > 30,whichisoftenthecase whenbiaserrorsarewell-denedandalargenumberofmeasur ements( N = +1 > 31) weretaken.Thisassumptionwillbemadethroughouttheunce rtaintyanalysispresented inthefollowingsections.Inthecaseofuncertaintyinares ultcalculatedfromseveral variables,theTaylorSeriesMethod(TSM)isemployed[ 201 ].Thecombinedstandard uncertaintyforafunction r = r ( x 1 ;x 2 ;:::;x n )isgivenas[ 201 ] u r 2 = J X i =1 @r @x i 2 b 2x i + J X i =1 @r @x i 2 s 2x i ; (C{3) where b x i and s x i areassociatedwiththe i thvariable. Theuncertaintyestimationapproachdiscussedhererequir essomeknowledgeof thestatisticalpropertiesofbiaserrors.Intheabsenceof data,adistributionmustbe assumed.Inthesectionstofollow,abiaserrorof a istypicallydrawnfromauniform distribution,yieldingastandarddeviationestimateof a= p 3. Errorestimatesforspectralquantities[ 34 ]usedrepeatedlyincludethenormalized standarderrorsofautospectraldensity, s G xx G xx = 1 p n d eff ; (C{4) 237

PAGE 238

frequencyresponsefunctionmagnitude, s j H xy j j H xy ( f ) j = q 1 r 2 xy ( f ) j r xy ( f ) j p 2 n d eff ; (C{5) andphase(inradians), s xy = q 1 r 2 xy ( f ) j r xy ( f ) j p 2 n d eff ; (C{6) where n d eff istheeectivenumberofaveragesand r 2 xy istheordinarycoherencefunction. Forarectangularwindowwith0%overlap, n d eff = n d ,theactualnumberofaverages.For anarbitrarilywindowedmeasurementwithoverlap, n d = b 1+( b blocks 1) = (1 r ) c ; (C{7) where b blocks isthenumberofblocksand r isthefractionofoverlap.Theeectivenumber ofaveragesdependsonboththewindowandtheoverlapas[ 203 ] n d eff = n d ; (C{8) wherefortheHanningwindowwith75%overlap( r =0 : 75), =0 : 52.Theconceptofthe eectivenumberofaverages|withdierentnomenclature|w asaddressedbyWelch [ 204 ]. C.2FrequencyResponseFunction Thissectionaddressestheuncertaintyestimatesforfrequ encyresponsefunctionsof thedevelopedmicrophones,forinstanceaspresentedinSec tion 8.2.3.1 ThePULSEsoftwareonlyenablesspecicationofthepistonph onepressurelevel (whichiscorrectedforatmosphericpressure)uptothenear est0 : 1dBSPL;thisimplies abiaserrorofupto 0 : 05dBpropagatesintotheDUTfrequencyresponsefunction. Drawingthebiaserrorfromauniformdistributionover 0 : 05dB,thestandardbiaserror estimateis b j H xy j = 10 0 : 05 = 20 1 p 3 j H xy ( f ) j (C{9) 0 : 0033 j H xy ( f ) j : Thestandarddeviationestimatefortherandomuncertainty isfoundfromEquation C{5 and U 95% iscalculatedfromEquations C{1 to C{2 with t 95% =2since100averageswere takeninallfrequencymicrophonefrequencyresponsemeasu rements. C.3NoiseFloor Thissectionaddressesuncertaintyfornoiseroormeasurem entspresentedin Section 8.2.4.1 .Uncertaintyforminimumdetectablepressuremetricscalcu latedfrom noiseroormeasurementsarealsogiven. 238

PAGE 239

C.3.1Spectra Therandomuncertaintyofthenoiseroormeasurementwasest imatedusing Equation C{4 forpowerspectraldensity,takingintoaccountthedieren tnumbersof blockscollectedovereachfrequencyspan,the75%overlap, andHanningwindowviathe procedureaddressedintheopening.The95%condenceinter valisthen U 95% =2 s S v o (C{10) =2 S v o p n d eff : (C{11) Letting ~ U 95% = U 95% =S v o ,thegenerallyunsymmetric errorboundsindBcanbe determinedas10log 10 1+ ~ U 95% and 10log 10 1 ~ U 95% ,respectively.Subtractingthe twoboundsgives10log 10 1 ~ U 95% andoneseesthattheasymmetryisnotimportant when ~ U 2 95% 1.TheuncertaintyforeachspanisshowngraphicallyinFigu re C-1 06.412.825.638.451.276.8102.4 0.190.190.080.080.060.060.06 f [kHz] U 95% [dB] FigureC-1.Noisespectra95%condenceintervals. Thestandardcombineduncertaintyoftheminimumdetectabl epressurespectra accountingfortheuncertaintyinboththenoiseroorandmic rophonesensitivitieswas estimatedusingtheTSMas u 2MDP = @MDP @S v o 2 s S v o + @MDP @ j S j 2 u 2j S j ; (C{12) whichafterperformingthepartialdierentiationsandnor malizinggives u 2MDP MDP 2 = 1 2 s S v o S v o 2 + u j S j j S j 2 : (C{13) Notetheuseofthecombinedstandarderrorfor j S j ,whichaccountsforbothrandomand biaserrorintheindividualmicrophonesensitivities.The uncertaintiesarethesameforall microphonesandareshownforeachfrequencyspaninFigure C-2 06.412.825.638.451.276.8102.4 0.100.100.050.050.040.040.04 f [kHz] U 95% [dB] FigureC-2.MDPspectra95%condenceintervals. 239

PAGE 240

C.3.2NarrowBand Theexpressionofthe95%condenceintervalofthenarrowba ndMDPisunchanged fromEquation C{13 .ItissimplytheuncertaintyvaluefortheMDPspectrumat1k Hz, U 95% = 0 : 19dB. C.3.3Integrated UncertaintyoftheintegratedMDPmeasures,inOASPLandAOASPL ,required complexintegrationprocessesandwerethereforeobtained viaMonteCarlosimulations. Randomperturbationsoftheminimumdetectablepressuresp ectraweretakenfroma normaldistributionwithastandarddeviationequaltothec ombinedstandarduncertainty u p min (=20 Pa10 u MDP 20 )denedinSection C.3.1 .Thetwointegrationprocesseswere performedon5000randomlyperturbedspectra,whichyielde dconvergedstatistics. Calculationof U 95% fromMonteCarloresultsvia t 95% u andalsofromanempirical cumulativedistributionfunctionagreedtoatleastthenum berofdecimalplacesreported inTable 8-14 foralldesigns. C.4Impedance ImpedancemeasurementresultsweregiveninSection 8.2.4.2 .The k thimpedance measurementfromtheHP4294Aimpedanceanalyzerpost-proce ssedintoadmittance formiswrittenas Y k ( f )= G k ( f )+ jB k ( f ) : (C{14) From n totalmeasurements,themeanvalues G and B werecomputed,togetherwith theirsamplestandarddeviations, s G = s G = p n and s B = s B = p n .FivethousandMonte CarlosimulationswereusedtotthedatatoEquation 8{7 ,withperturbationsto G and B andacurvetperformedateachiteration.Theperturbation srepresentingthe randomerrorweredrawnfromanormaldistribution(meanzer oandstandarddeviation s G and s B ,respectively),andthoserepresentingthebiaserrorwere drawnfromauniform distributionwithboundsequaltothebiaserror,calculabl efromexpressionsinthe equipmentmanual[ 175 ].Theextractedvalues C ef + C eo R ep ,and R es weresavedateach iterationtogetherwithR-squaredvaluesforthegoodnesso fttotheexperimentaldata, yieldingstatisticaldistributionsforallofthosequanti ties.Fromthose,the95%condence boundswereextracteddirectlyfromanexperimentalcumula tivedistributionfunction. C.5ParasiticCapacitanceExtraction TheTSMwasappliedtoEquation 8{12 toyieldthecombinedstandarduncertainty expression, u 2C ep + C ea =( C et ) 2 u 2S ca + b 2S ca S 2 ca + u 2S va + b 2S va S 2 va + s 2C fb C 2 fb # + u 2C ef ; (C{15) whichbecause S ca = S ca ( f )and S va = S va ( f ),isevaluatedateachfrequency.Theerrors usedinEquation C{15 arefoundinTable C-1 .Similarly,theuncertaintyintheopen circuitsensitivitywasestimatedfrom u 2S oc = C et C ef 2 s 2S va + b 2S va + C ep + C ea C 2 ef S va 2 u 2C ef + S va C ef 2 u 2C ep + C ea : (C{16) Inbothcases, U 95% =2 u 240

PAGE 241

TableC-1.Parasiticcapacitanceextractionuncertaintie s. UncertaintyValue s S ca Equation C{5 b S ca 0.0033 S ca (seeEquation C{9 ) s S va Equation C{5 b S va 0.0033 S va (seeEquation C{9 ) s C fb 0 : 25pF C.6ParameterExtraction Uncertaintyestimatesfortheprimaryparameterextraction quantitiesfoundin Section 8.2.5 C ad M ad ,and d a ,wereobtainedviaMonteCarlosimulations.Thespatial frequencyresponsefunctionsfromwhichtheywerecalculat ed, H pw ( r; ; f )and H vw ( r; ; f ) containedrandomerror(denedbyEquation C{5 )andbiaserrorfromthemicrophone calibrationinthecaseof H pw (denedviathegeneralformofEquation C{9 ).Thebias errorassociatedwiththeactuallaservibrometermeasurem entwasdeemednegligible. Theperturbationsrepresentingtherandomerrorweredrawn fromanormaldistribution, witheachindividualscanpointperturbedindividually.Pe rturbationsassociatedwiththe biaserrorweredrawnfromauniformdistributionandapplie duniformlytoeachscan point.Theintegrationroutinesassociatedwiththecalcul ationof C ad M ad and d a were performedforeachperturbationtobuildstatisticaldistr ibutionsand U 95% foreachwas calculateddirectlyfromtheexperimentalcumulativeprob abilitydistributionfunction. Secondaryparameters a and k werecalculatedviathebelowequationsusingthe standarduncertaintiesderivedfromtheMonteCarlosimula tions: u a a 2 = u d a d a 2 + u C ad C ad 2 (C{17) and u k 2 k 2 2 = 2 u d a d a 2 + u C ef C ef 2 + u C ad C ad 2 : (C{18) Uncertaintyin C ef intheaboveequationwasdrawnfromTable 8.2.4.2 241

PAGE 242

APPENDIXD MATERIALPROPERTIES Thisappendixcollectsthematerialpropertiesusedinsimu lationsthroughoutthis studyintotwotables:oneforpropertiesofmaterialsusedi nthemicrophonediaphragm andoneforpropertiesofgasesinwhichthemicrophonewaste sted. TableD-1.Propertiesofmicrophonediaphragmmaterials. PassivationMolybdenum(Mo)AluminumNitride(AlN)[ 141 ]Structural E [GPa]7332928373 0.170.310.270.17 [kg = m 3 ]22001028932502200 d 31 [m = V]-2 : 65 10 12 [F = m]--9 : 5 10 11 e [Mnm]--22.8TableD-2.Propertiesofgases. AirHelium c 0 [m = s]3431007 0 [kg = m 3 ]1.210.161 [mkg = s]1 : 81 10 5 1 : 9 10 5 242

PAGE 243

REFERENCES [1]B.Burnett,\Ssshh,we'reryingaplanearoundhere," BoeingFrontiers ,vol.4, no.8,Jan.2006. [2]P.A.SheikhandC.Uhl,\Airplanenoise:Apervasivedisturb anceinPennsylvania parks,USA," JournalofSoundandVibration ,vol.274,no.1-2,pp.411{420,Jul. 2004.[Online].Available: http://dx.doi.org/10.1016/j.jsv.2003.09.014 [3]\Aeronauticsandspace,noisestandards:Aircrafttypean dairworthiness certication," Title14USCodeofFederalRegulations,pt.36 ,2004. [4]V.G.Mengle,U.Ganz,E.J.Bultemeier,andF.T.Calkins,\C lockingeects ofchevronswithazimuthally-varyingimmersionsonshockc ell/cabinnoise,"in Proceedingsofthe29thAIAAAeroacousticsConference ,Vancouver,Canada,May 5{7,2008,AIAApaper08-3000,pp.1{14. [5] Occupationalsafetyandhealthprotectionsforcabincrewm embers ,International CivilAviationOrganizationWorkingPaper,Rev.A36-WP/208, Sep.2007. [6]W.H.Herkes,R.F.Olsen,andS.Uellenberg,\Thequiettechn ologydemonstrator program:Flightvalidationofairplanenoise-reductionco ncepts,"in Proceedings ofthe12thAIAA/CEASAeroacousticsConference(27thAeroa cousticsConference) Cambridge,MA,May8{10,2006,AIAApaper2006-2720,pp.1{9. [7]E.J.Bultemeier,U.Ganz,J.Premo,andE.Nesbitt,\Eecto funiformchevrons oncruiseshockcellnoise,"in Proceedingsofthe12thAIAA/CEASAeroacoustics Conference(27thAIAAAeroacousticsConference) ,Cambridge,MA,May8{10,2006, AIAApaper2006-2440,pp.1{15. [8]D.Reed,S.Uellenberg,andE.Davis,\Anin-rightstudyofc abinbuzz-sawnoise," TheJournaloftheAcousticalSocietyofAmerica ,vol.112,no.5,pp.2347{2347, 2002.[Online].Available: http://link.aip.org/link/?JAS/112/2347/2 [9]W.Herkes,E.Nesbitt,B.Callender,B.Janardan,J.Moe,an dJ.Yu,\Thequiet technologydemonstratorprogram:Statictestofairplanen oise-reductionconcepts," in Proceedingsofthe13thAIAA/CEASAeroacousticsConferenc e(28thAIAA AeroacousticsConference) ,Rome,Italy,May21{23,2007,AIAApaper2007-3670, pp.1{11. [10]S.Bailey,G.Kunkel,M.Hultmark,M.Vallikivi,J.Hill,K .Meyer,C.Tsay, C.Arnold,andA.Smits,\Turbulencemeasurementsusinganano scalethermal anemometryprobe," JournalofFluidMechanics ,vol.663,pp.160{179,Nov.2010. [Online].Available: http://dx.doi.org/10.1017/S0022112010003447 [11]J.W.NaughtonandM.Sheplak,\Moderndevelopmentsinsh ear-stressmeasurement," ProgressinAerospaceSciences ,vol.38,no.6{7,pp.515{570,Aug.{Oct.2002. [Online].Available: http://dx.doi.org/10.1016/S0376-0421(02)00031-3 243

PAGE 244

[12]H.TennekesandJ.Lumley, Arstcourseinturbulence .Cambridge,MA:MIT Press,1972. [13]L.LofdahlandM.Gad-el-Hak,\MEMS-basedpressureand shearstresssensorsfor turbulentrows," MeasurementScienceandTechnology ,vol.10,no.8,p.665,Aug. 1999.[Online].Available: http://stacks.iop.org/0957-0233/10/i=8/a=302 [14]T.J.Mueller,Ed., AeroacousticMeasurements .Berlin,Germany:Springer-Verlag, 2002,ch.3,pp.98{217. [15]D.P.Arnold,T.Nishida,L.N.Cattafesta,andM.Sheplak,\ Adirectional acousticarrayusingsiliconmicromachinedpiezoresistiv emicrophones," Journalof theAcousticalSocietyofAmerica ,vol.113,no.1,pp.289{298,Jan.2003.[Online]. Available: http://dx.doi.org/10.1121/1.1527960 [16]M.Sheplak,K.S.Breuer,andM.A.Schmidt,\Awafer-bond ed,silicon-nitride membranemicrophonewithdielectrically-isolatedsingle crystalsiliconpiezoresistors," in Proceedingsof1998Solid-StateSensorandActuatorWorksh op ,HiltonHead Island,SC,Jun.8{11,1998,pp.23{26. [17]M.Sheplak,M.Seiner,K.S.Breuer,andM.A.Schmidt,\AM EMSmicrophone foraeroacousticmeasurements,"in Proceedingsof37thAIAAAerospaceSciences Meeting ,Reno,NV,Jan.11{14,1999,AIAApaper99-0606. [18]D.P.Arnold,S.Gururaj,S.Bhardwaj,T.Nishida,andM.Sh eplak,\Apiezoresistive microphoneforaeroacousticmeasurements,"in ProceedingsofASMEIMECE2001 InternationalMechanicalEngineeringCongressandExposi tion ,NewYork,NY,Nov. 11{16,2001,paperMEMS-23841,pp.281{288. [19]P.J.KingandJ.R.Underbrink,\Characterizationofami croelectromechanical systems(MEMS)microphone,"in Proceedingsofthe14thAIAA/CEASAeroacoustics Conference ,Vancouver,Canada,May5{7,2008,AIAApaper2008-2912. [20]S.Horowitz,T.Nishida,L.Cattafesta,andM.Sheplak,\D evelopmentofa micromachinedpiezoelectricmicrophoneforaeroacoustic sapplications," Journalof theAcousticalSocietyofAmerica ,vol.122,no.6,pp.3428{3436,Dec.2007.[Online]. Available: http://dx.doi.org/10.1121/1.2785040 [21]D.Martin,\Design,fabrication,andcharacterizatio nofaMEMSdual-backplate capacitivemicrophone,"Ph.D.dissertation,Universityof Florida,Gainesville,FL, Aug.2007.[Online].Available: http://purl.fcla.edu/fcla/etd/UFE0017526 [22]V.G.Mengle,U.Ganz,E.Nesbitt,E.J.Bultemeier,andR.Th omas,\Flighttest resultsforuniquelytailoredpropulsion-airframeaeroac ousticchevrons:Shockcell noise,"in Proceedingsofthe27thAIAAAeroacousticsConference ,Cambridge,MA, May5{7,2008,AIAApaper08-3000. 244

PAGE 245

[23]P.SijtsmaandR.Stoker,\Determinationofabsoluteco ntributionsofaircraft noisecomponentsusingry-overarraymeasurements,"in Proceedingsofthe10th AIAA/CEASAeroacousticsConference ,Manchester,GreatBritain,May10{12,2004, AIAApaper2004-2958. [24]H.Siller,M.Drescher,G.Saueressig,andR.Lange,\Fly -oversourcelocalisationon aBoeing747-400,"in Proceedingsofthe3rdBerlinBeamformingConference ,Berlin, Germany,Feb.24{25,2010,pp.1{11. [25]K.Veggeberg,\Highchannel-countaircraftnoisemappi ngapplications," Soundand VibrationMagazine ,pp.14{16,May2009. [26]J.Lan,J.Premo,G.Zlavog,C.Breard,B.Callender,and M.Martinez,\Phasedarray measurementsoffull-scaleengineinletnoise,"in Proceedingsofthe13thAIAA/CEAS AeroacousticsConference(28thAIAAAeroacousticsConfer ence) ,Rome,Italy,May 21{23,2007,AIAApaper2007-3434,pp.1{14. [27]L.Brusniak,J.R.Underbrink,E.Nesbitt,D.Lynch,andM. Martinez,\Phased arraymeasurementsoffull-scaleengineexhaustnoise,"in Proceedingsofthe13th AIAA/CEASAeroacousticsConference(28thAIAAAeroacoust icsConference) Rome,Italy,May21{23,2007,AIAApaper2007-3612,pp.1{18. [28]D.T.Blackstock, FundamentalsofPhysicalAcoustics .NewYork,NY:JohnWiley &Sons,Inc,2000,ch.1,4,14. [29]J.E.F.Williams,\Hydrodynamicnoise," AnnualReviewofFluid Mechanics ,vol.1,no.1,pp.197{222,Jan.1969.[Online].Available: http://dx.doi.org/10.1146/annurev.r.01.010169.00121 3 [30]M.Lighthill,\Soundgeneratedaerodynamically," PhilosophicalTransactionsofthe RoyalSocietyA:Mathematical,PhysicalandEngineeringScie nces ,vol.267,no.1329, pp.147{182,Jun.1961.[Online].Available: http://www.jstor.org/stable/2414246 [31] MicrophoneHandbook ,BruelandKjr,Nrum,Denmark,Jul.1996.[Online]. Available: http://www.bksv.com/doc/be1447.pdf [32]H.H.Hubbard,\Aeroacousticsofrightvehicles:theoryand practice,"National AeronauticsandSpaceAdministration,Tech.Rep.90-3052,19 91. [33]C.M.Harris, HandbookofAcousticalMeasurementsandNoiseControl ,3rded. Woodbury,NY:AcousticalSocietyofAmerica,1998,ch.1. [34]J.S.BendatandA.G.Piersol, RandomDataAnalysisandMeasurementProcedures 3rded.NewYork,NY:JohnWileyandSons,2000,ch.2,6,9. [35]M.Rossi, AcousticsandElectroacoustics .Norwood,MA:ArtechHouse,Inc.,1988, ch.4{6,8. 245

PAGE 246

[36]L.L.Beranek, Acoustics .NewYork,NY:AcousticalSocietyofAmerica,1993,ch. 3,5,6. [37]S.S.Rao, VibrationofContinuousSystems .Hoboken,N.J.:Wiley,2007,ch.14. [38]R.Dieme,G.Bosman,M.Sheplak,andT.Nishida,\Sourceo fexcess noiseinsiliconpiezoresistivemicrophones," JournaloftheAcousticalSociety ofAmerica ,vol.119,no.5,pp.2710{2720,May2006.[Online].Availabl e: http://dx.doi.org/10.1121/1.2188367 [39]C.MotchenbacherandJ.Connelly, LowNoiseElectronicSystemDesign .NewYork, NY:JohnWiley&Sons,1993,ch.1,2. [40]T.B.Gabrielson,\Mechanical-thermalnoiseinmicrom achinedacousticandvibration sensors," IEEETransactionsonElectronDevices ,vol.40,no.5,pp.903{909,May 1993.[Online].Available: http://dx.doi.org/10.1109/16.210197 [41]H.B.CallenandT.A.Welton,\Irreversibilityandgenera lizednoise," PhysicalReview ,vol.83,no.1,pp.34{40,Jul.1951.[Online].Available: http://dx.doi.org/10.1103/PhysRev.83.34 [42]T.B.Gabrielson,\Fundamentalnoiselimitsforminiat ureacousticandvibration sensors," JournalofVibrationandAcoustics ,vol.117,no.4,pp.405{410,Oct.1995. [Online].Available: http://dx.doi.org/10.1115/1.2874471 [43]S.D.Senturia, MicrosystemDesign .Norwell,MA:KluwerAcademicPublishers, 2001,ch.1,2,5,13,16. [44] LTC6240/LTC6241/LTC6242Single/Dual/Quad18MHz,LowNoi se,Rail-to-Rail Output,CMOSOpAmps ,LinearTechnology,2011. [45]G.Vasilescu, ElectronicNoiseandInterferingSignals:PrinciplesandA pplications Berlin,Germany:Springer,2005,ch.1. [46]P.HorowitzandW.Hill, TheArtofElectronics ,2nded.Cambridge,UK:Cambridge UniversityPress,1989,ch.7. [47]B.Boulet, FundamentalsofSignalsandSystems .Hingham,MA:DaVinci EngineeringPress,2006,vol.1,ch.4. [48] AudioDistortionMeasurements ,BruelandKjr,Nrum,Denmark,1992.[Online]. Available: http://www.bksv.com/doc/BO0385.pdf [49] MeasurementMicrophones ,2nded.,BruelandKjr,Nrum,Denmark,Aug.1994. [Online].Available: http://www.bksv.com/doc/br0567.pdf [50] 1/8inPressure-FieldMicrophone-Type4138 ,BruelandKjr,Sep.2008. [51] 1/4inPressure-FieldMicrophone-Type4938 ,BruelandKjr,May2008. 246

PAGE 247

[52] KuliteHighIntensityMicrophones ,Kulite,Feb.2011.[Online].Available: http://www.kulite.com/pdfs/pdf Data Sheets/Mic.pdf [53]P.R.Scheeper,A.G.H.vanderDonk,W.Olthuis,andP.Berg veld,\Areviewof siliconmicrophones," SensorsandActuatorsA ,vol.44,no.1,pp.1{11,Jul.1994. [Online].Available: http://dx.doi.org/10.1016/0924-4247(94)00790-X [54]B.Homeijer,\Design,fabricationandcharacterizatio nofaMEMSpiezoresistive microphoneforuseinaeroacousticmeasurements,"Ph.D.di ssertation, UniversityofFlorida,Gainesville,FL,Dec.2008.[Online] .Available: http://purl.fcla.edu/fcla/etd/UFE0022840 [55]M.Royer,J.O.Holmen,M.A.Wurm,O.S.Aadland,andM.Glenn ,\ZnOonSi integratedacousticsensor," SensorsandActuators ,vol.4,no.3,pp.357{362,1983. [Online].Available: http://dx.doi.org/10.1016/0250-6874(83)85044-6 [56]E.S.KimandR.S.Muller,\IC-processedpiezoelectric microphone," IEEE ElectronDeviceLetters ,vol.8,no.10,pp.467{468,Oct.1987.[Online].Available: http://dx.doi.org/10.1109/EDL.1987.26696 [57]R.S.MullerandE.S.Kim,\ICprocessedpiezoelectricm icrophone,"U.S.Patent 4783821,Nov.8,1988. [58]E.S.Kim,R.S.Muller,andP.R.Gray,\Integratedmicro phonewithCMOS circuitsonasinglechip,"in Proceedingsof1989InternationalElectronDevices Meeting ,Washington,DC,Dec.3{6,1989,pp.880{883.[Online].Avai lable: http://dx.doi.org/10.1109/IEDM.1989.74193 [59]E.S.Kim,J.R.Kim,andR.S.Muller,\ImprovedIC-compa tiblepiezoelectric microphoneandCMOSprocess,"in Proceedingsof1991InternationalConference onSolid-StateSensorsandActuators ,SanFrancisco,CA,Jun.24{27,1991,pp. 270{273.[Online].Available: http://dx.doi.org/10.1109/SENSOR.1991.148858 [60]J.Franz,\Piezoelektrischesensorenaufsiliziumbas isfurakustischeanwendungen," Ph.D.dissertation,DarmstadtUniversityofTechnology,Da rmstadt,Germany,1988. [61]R.Schellin,G.Hess,W.Kuehnel,G.M.Sessler,andE.Fuk ada,\Silicon subminiaturemicrophoneswithorganicpiezoelectriclaye rs:Fabrication andacousticalbehaviour,"in Proceedingsof7thInternationalSymposiumon Electrets ,Berlin,Germany,Sep.25{27,1992,pp.929{934.[Online]. Available: http://dx.doi.org/10.1109/ISE.1991.167338 [62]R.Schellin,G.Hess,R.Kressmann,andP.Wassmuth,\Mic romachined siliconsubminiaturemicrophoneswithpiezoelectricP(VDF /TRFE)-layersand silicon-nitride-membranes,"in Proceedingsof8thInternationalSymposiumon Electrets ,Paris,France,Sep.7{9,1994,pp.1004{1009.[Online].Ava ilable: http://dx.doi.org/10.1109/ISE.1994.515262 247

PAGE 248

[63]R.P.Ried,E.S.Kim,D.M.Hong,andR.S.Muller,\Piezoel ectricmicrophonewith on-chipCMOScircuits," JournalofMicroelectromechanicalSystems ,vol.2,no.3, pp.111{120,Sep.1993.[Online].Available: http://dx.doi.org/10.1109/84.260255 [64]S.S.Lee,R.P.Reid,andR.M.White,\Piezoelectriccan tilevermicrophoneand microspeaker," JournalofMicroelectromechanicalSystems ,vol.5,no.4,pp.238{242, Dec.1996.[Online].Available: http://dx.doi.org/10.1109/84.546403 [65]S.S.LeeandR.M.White,\Piezoelectriccantileveraco ustictransducer," Journal ofMicromechanicsandMicroengineering ,vol.8,no.3,pp.230{238,1998.[Online]. Available: http://dx.doi.org/10.1088/0960-1317/8/3/009 [66]A.Naguib,E.Soupos,H.Nagib,C.Huang,andK.Naja,\Charact erizationofa MEMSacoustic/pressuresensor,"in Proceedingsof37thAIAAAerospaceSciences Meeting ,Reno,NV,Jan.11{14,1999,AIAApaper99-0520,pp.1{7. [67]||,\ApiezoresistiveMEMSsensorforacousticnoiseme asurements,"in Proceedings of5thAIAA/CEASAeroacousticsConference&Exhibit ,Bellevue,WA,May10{12, 1999,AIAApaper99-1992,pp.1{8. [68]C.Huang,A.Naguib,E.Soupos,andK.Naja,\Asiliconmicro machined microphoneforruidmechanicsresearch," JournalofMicromechanicsand Microengineering ,vol.12,no.6,pp.767{774,Nov.2002.[Online].Available: http://dx.doi.org/10.1088/0960-1317/12/6/307 [69]D.P.Arnold,\AMEMS-baseddirectionalacousticarrayf oraeroacoustic measurements,"Master'sthesis,UniversityofFlorida,Gai nesville,FL,Dec.2001. [70]K.Kadirvel,R.Taylor,S.Horowitz,M.Sheplak,andT.Nis hida,\Designand characterizationofMEMSopticalmicrophoneforaeroacous ticmeasurement,"in Proceedingsof42ndAerospaceSciencesMeetingandExhibit ,Reno,NV,Jan.5{8, 2004,AIAApaper2004-1030. [71]D.T.Martin,L.Jian,K.Kadirvel,R.M.Fox,M.Sheplak, andT.Nishida,\A micromachineddual-backplatecapacitivemicrophonefora eroacousticmeasurements," JournalofMicroelectromechanicalSystems ,vol.16,no.6,pp.1289{1302,Dec.2007. [Online].Available: http://dx.doi.org/10.1109/JMEMS.2007.909234 [72]D.T.Martin,K.Kadirvel,J.Liu,R.M.Fox,M.Sheplak,a ndT.Nishida,\Surface andbulkmicromachineddualback-platecondensermicropho ne,"in Proceedingsof 18thIEEEInternationalConferenceonMicroElectroMechan icalSystems(MEMS 2005) ,Miami,FL,Jan.30{Feb.3,2005,pp.319{322.[Online].Avai lable: http://dx.doi.org/10.1109/MEMSYS.2005.1453842 [73]D.T.Martin,K.Kadirvel,T.Nishida,andM.Sheplak,\As urfacemicromachined capacitivemicrophoneforaeroacousticapplications,"in Solid-statesensor,actuator, andmicrosystemsworkshop ,HiltonHead,SC,Jun.1{5,2008,pp.1{4. 248

PAGE 249

[74]L.-T.Zhang,T.-L.Ren,J.-S.Liu,L.-T.Liu,andZ.-J.L i,\Fabricationof acantileverstructureforpiezoelectricmicrophone," JapaneseJournalofApplied Physics ,vol.41,no.11B,pp.7158{7159,Nov.2002.[Online].Availab le: http://dx.doi.org/10.1143/JJAP.41.7158 [75]H.J.Zhao,T.L.Ren,J.S.Liu,L.T.Liu,andZ.J.Li,\Fabr ication ofhigh-qualityPZT-basedpiezoelectricmicrophone,"in Proceedingsof12th InternationalConferenceonSolidStateSensors,Actuator s,andMicrosystems vol.1,Boston,MA,Jun.8-12,2003,pp.234{237.[Online].Ava ilable: http://dx.doi.org/10.1109/SENSOR.2003.1215296 [76]H.Zhao,T.Ren,J.Liu,L.Liu,andZ.Li,\Anovelstructur eforPZT-based piezoelectricmicrophone,"in 2003NanotechnologyConferenceandTradeShow vol.1,SanFransisco,CA,Feb.23{27,2003,pp.356{359. [77]S.C.Ko,Y.C.Kim,S.S.Lee,S.H.Choi,andS.R.Kim, \Micromachinedpiezoelectricmembraneacousticdevice," SensorsandActuator A ,vol.103,no.1{2,pp.130{134,Jan.2003.[Online].Availab le: http://dx.doi.org/10.1016/S0924-4247(02)00310-2 [78]M.NiuandE.S.Kim,\Piezoelectricbimorphmicrophoneb uilt onmicromachinedparylenediaphragm," JournalofMicroelectromechanical Systems ,vol.12,no.6,pp.892{898,Dec.2003.[Online].Available: http://dx.doi.org/10.1109/JMEMS.2003.820288 [79]P.R.Scheeper,B.Nordstrand,B.L.J.O.Gullov,T.Claus en,L.Midjord,and T.Storgaard-Larsen,\Anewmeasurementmicrophonebasedo nMEMStechnology," JournalofMicroelectromechanicalSystems ,vol.12,no.6,pp.880{891,Dec.2003. [Online].Available: http://dx.doi.org/10.1109/JMEMS.2003.820260 [80]R.Polcawich,\ApiezoelectricMEMSmicrophonebasedo nlead zirconatetitanate(PZT)thinlms,"ArmyResearchLaborato ry, Adelphi,MD,Tech.Rep.ARL-TR-3387,Nov.2004.[Online].Avail able: http://handle.dtic.mil/100.2/ADA429041 [81]J.HillenbrandandG.M.Sessler,\High-sensitivitypiez oelectricmicrophones basedonstackedcellularpolymerlms," JournaloftheAcousticalSocietyof America ,vol.116,no.6,pp.3267{3270,Dec.2004.[Online].Availab le: http://dx.doi.org/10.1121/1.1810272 [82]R.S.Fazzio,T.Lamers,O.Buccafusca,A.Goel,andW.Dau ksher, \Designandperformanceofaluminumnitridepiezoelectric microphones," in 14thInternationalConferenceonMicro-Sensors,Actuator sandMicrosystems Lyon,France,Jun.10{14,2007,pp.1255{1258.[Online].Ava ilable: http://dx.doi.org/10.1109/SENSOR.2007.4300365 [83]T.L.LamersandR.S.Fazzio,\Acceleratingdevelopment ofamemspiezoelectric microphone,"in ProceedingsoftheASME2007InternationalDesignEngineer ing 249

PAGE 250

TechnicalConferences&ComputersInformationinEngineer ingConference ,Las Vegas,NV,Sep.4{7,2007,paperDETC2007-34958,pp.593{601. [Online].Available: http://dx.doi.org/10.1115/DETC2007-34958 [84]R.S.Fazzio,W.Dauksher,A.Goel,andT.Lamers,\Analysi sofmicromachined piezoelectrictransducersoperatinginarexuralmode,"in ProceedingsofIMECE 2007,ASMEInternationalMechanicalEngineeringCongress andExposition ,Nov. 11{15,2007,paperIMECE2007-41460,pp.1{9. [85]R.J.Littrell,\HighperformancepiezoelectricMEMSmi crophones,"Ph.D. dissertation,UniversityofMichigan,AnnArbor,MI,2010. [86]M.Pedersen,W.Olthius,andP.Bergveld,\Harmonicdist ortioninsiliconcondenser microphones," JournaloftheAcousticalSocietyofAmerica ,vol.102,no.3,pp. 1582{1587,Sep.1997.[Online].Available: http://dx.doi.org/10.1121/1.420070 [87]T.Hoevenaars,K.LeDoux,andM.Colosino,\Interpretin gIEEESTD519 andmeetingitsharmoniclimitsinVFDapplications,"in IEEEIndustry ApplicationsSociety50thAnnualPetroleumandChemicalIn dustryConference Houston,TX,Sep.15{17,2003,pp.145{150.[Online].Availabl e: http://dx.doi.org/10.1109/PCICON.2003.1242609 [88]G.M.Sessler,\Acousticsensors," SensorsandActuatorsA:Physical ,vol.26,no.1{3,pp.323{330,Mar.1991.[Online].Availabl e: http://dx.doi.org/10.1016/0924-4247(91)87011-Q [89]D.T.Martin,J.Liu,K.Kadirvel,R.M.Fox,M.Sheplak,a ndT.Nishida, \DevelopmentofaMEMSdualbackplatecapacitivemicrophon eforaeroacoustic measurements,"in Proceedingsof44thAIAAAerospaceSciencesMeetingand Exhibit ,Reno,NV,Jan.9{12,2006,AIAApaper2006-1246. [90]W.S.LeeandS.S.Lee,\Piezoelectricmicrophonebuilt oncirculardiaphragm," SensorsandActuatorsA:Physical ,vol.144,no.2,pp.367{373,Jun.2008.[Online]. Available: http://dx.doi.org/10.1016/j.sna.2008.02.001 [91]J.Raukola,N.Kuusinen,andM.Paajanen,\Cellularelec trets-frompolymer granulestoelectromechanicallyactivelms,"in 11thInternationalSymposiumon Electrets ,Victoria,Australia,Oct.1{3,2002,pp.195{198.[Online]. Available: http://dx.doi.org/10.1109/ISE.2002.1042977 [92]R.SoleckiandR.J.Conant, AdvancedMechanicsofMaterials .NewYork,NY: OxfordUniversityPress,2003,pp.167{182. [93]A.SafariandE.Akdo~gan,Eds., PiezoelectricandAcousticMaterialsforTransducer Applications .NewYork,NY:Springer,2008,ch.2,20. 250

PAGE 251

[94]N.Setter,Ed., PiezoelectricMaterialsinDevices:ExtendedReviewsonCu rrent andEmergingPiezoelectricMaterials,Technology,andAppl ications .Lausanne, Switzerland:N.Setter,2002,ch.1,12,13. [95] IEEEstandardonpiezoelectricity ,ANSI/IEEEStd.176-1987,1988.[Online]. Available: http://dx.doi.org/10.1109/IEEESTD.1988.79638 [96]B.A.Auld, AcousticFieldsandWavesinSolids ,2nded.Malabar,Fla.:R.E. Krieger,1990,ch.8. [97]C.A.Balanis, AdvancedEngineeringElectromagnetics .Hoboken,NJ:JohnWiley& Sons,1989,ch.2. [98]A.P.BoresiandR.J.Schmidt, AdvancedMechanicsofMaterials ,6thed.New York,NY:JohnWiley&Sons,2003,ch.3,13. [99]M.A.DuboisandP.Muralt,\Propertiesofaluminumnitri dethinlms forpiezoelectrictransducersandmicrowavelterapplica tions," Applied PhysicsLetters ,vol.74,no.20,pp.3032{3034,1999.[Online].Available: http://dx.doi.org/10.1063/1.124055 [100]S.Trolier-McKinstryandP.Muralt,\Thinlmpiezoel ectricsforMEMS," Journal ofElectroceramics ,vol.12,no.1/2,pp.7{17,Jan2004.[Online].Available: http://www.springerlink.com/content/t7415r71417825w 7 [101]R.Ruby,\FBAR|fromtechnologydevelopmenttoproduct ion,"in Proceedings ofthe2ndInternationalSymposiumonAcousticWaveDevicesf orFutureMobile CommunicationSystems ,Chiba,Japan,Mar.2{5,2004,pp.1{5. [102]Y.Oshmyansky,J.D.Larson,R.Ruby,andS.Mishin,\Spu tteringprocessesforbulk acousticwavelters," SemiconductorInternational ,Mar.2003. [103]S.TadigadapaandK.Mateti,\PiezoelectricMEMSsens ors:state-of-the-artand perspectives," MeasurementScienceandTechnology ,vol.20,no.9,p.092001,sep 2009.[Online].Available: http://dx.doi.org/10.1088/0957-0233/20/9/092001 [104]S.Wilson,R.Jourdain,Q.Zhang,R.Dorey,C.Bowen,M. Willander, Q.Wahab,S.Alhilli,andO.Nur,\Newmaterialsformicro-scale sensors andactuatorsanengineeringreview," MaterialsScienceandEngineering:R: Reports ,vol.56,no.1-6,pp.1{129,Jun.2007.[Online].Available: http://dx.doi.org/10.1016/j.mser.2007.03.001 [105]R.Ruby,P.Bradley,Y.Oshmyansky,A.Chien,andJ.Larso n,\Thinlmbulkwave acousticresonators(FBAR)forwirelessapplications,"in Proceedingsofthe2001 IEEEUltrasonicsSymposium ,vol.1,Oct.10,2001,pp.813{821.[Online].Available: http://dx.doi.org/10.1109/ULTSYM.2001.991846 [106]N.Maluf, Introductiontomicroelectromechanicalsystemsengineeri ng ,2nded. Boston,MA:ArtechHouse,2004,pp.208{211. 251

PAGE 252

[107]M.Madou, FundamentalsofMicrofabrication .BocaRaton,FL:CRCPress,1997, pp.261{272. [108]S.Franssila, IntroductiontoMicrofabrication .Boulder,CO:JohnWiley&Sons, 2004,p.401. [109]S.Beeby, MEMSMechanicalSensors .Boston,MA:ArtechHouse,2004,ch.6. [110]H.Lee,D.Kang,andW.Moon,\Amicro-machinedsourcetr ansducerfora parametricarrayinair,"vol.125,no.4,pp.1879{93,Apr.20 09.[Online].Available: http://dx.doi.org/10.1121/1.3081385 [111]S.Trolier-McKinstry,R.Smith,S.Krishnaswamy,and C.Freidho,\Vibration ofmicromachinedcircularpiezoelectricdiaphragms," IEEETransactionson Ultrasonics,FerroelectricsandFrequencyControl ,vol.53,no.4,pp.697{706,Apr. 2006.[Online].Available: http://dx.doi.org/10.1109/TUFFC.2006.1621496 [112]P.Blanchard,R.Devaney,andG.Hall, Dierentialequations .Belmont,CA: Brooks/ColeThomsonLearning,2006. [113]S.A.N.Prasad,Q.Gallas,S.Horowitz,B.Homeijer,B.V.San kar,L.N.Cattafesta, andM.Sheplak,\Analyticalelectroacousticmodelofapiezo electriccomposite circularplate," AIAAJournal ,vol.44,no.10,pp.2311{2318,Oct.2006.[Online]. Available: http://dx.doi.org/10.2514/1.19855 [114]J.Merhaut, TheoryofElectroacoustics .NewYork,NY:McGraw-Hill,1981,ch. 1,2,6. [115]F.A.Fischer, FundamentalsofElectroacoustics .NewYork,NY:Interscience Publishers,1955,ch.1. [116]S.T.ToddandH.Xie,\Anelectrothermomechanicallumped elementmodel ofanelectrothermalbimorphactuator," JournalofMicroelectromechanical Systems ,vol.17,no.1,pp.213{225,Feb.2008.[Online].Available: http://dx.doi.org/10.1109/JMEMS.2007.908754 [117]A.N.NorrisandD.M.Photiadis,\Thermoelasticrelaxati oninelasticstructures, withapplicationstothinplates," QuarterlyJournalofMechanicsandApplied Mathematics ,vol.58,no.1,pp.143{163,Feb.2005.[Online].Available: http://dx.doi.org/10.1093/qjmamj/hbi002 [118]S.G.DavidS.Bindel,\ElasticPMLsforresonatoranch orlosssimulation," InternationalJournalforNumericalMethodsinEngineerin g ,vol.64,no.6,pp. 789{818,Oct.2005.[Online].Available: http://dx.doi.org/10.1002/nme.1394 [119]S.B.Horowitz,\DevelopmentofaMEMS-basedacoustice nergyharvester,"Ph.D. dissertation,UniversityofFlorida,Gainesville,FL,Dec. 2005. 252

PAGE 253

[120]B.R.Munson,D.F.Young,andT.H.Okiishi, FundamentalsofFluidMechanics 4thed.NewYork,NY:JohnWiley&Sons,2002,pp.352{354. [121]J.N.Reddy, TheoryandAnalysisofElasticPlates .Philadelphia,PA:Taylor& Francis,1999,ch.1{3,5. [122]A.H.NayfehandP.F.Pai, LinearandNonlinearStructuralMechanics .Hoboken, N.J.:Wiley-Interscience,2004,ch.7. [123]I.Chopra,\Reviewofstateofartofsmartstructuresa ndintegratedsystems," AIAAJournal ,vol.40,no.11,pp.2145{2187,2002.[Online].Available: http://dx.doi.org/10.2514/2.1561 [124]Y.T.AntonyakandM.Vassergiser,\Calculationofthech aracteristicsofa membrane-typerexural-modepiezoelectrictransducer," SovietPhysicalAcoustics vol.28,no.3,pp.176{180,May{Jun.1982. [125]Y.B.Evseichik,S.I.Rudnitskii,V.M.Sharapov,andN.A.S hul'ga,\Sensitivityof ametal-piezoceramicbimorphtransducer," InternationalAppliedMechanics ,vol.26, no.12,pp.1174{1181,Dec.1990. [126]S.H.ChangandB.C.Du,\Optimizationofasymmetricbim orphicdisktransducers," TheJournaloftheAcousticalSocietyofAmerica ,vol.109,no.1,pp.194{202,Jan. 2001.[Online].Available: http://dx.doi.org/10.1121/1.1310669 [127]G.Wang,B.V.Sankar,L.N.Cattafesta,andM.Sheplak,\An alysisofacomposite piezoelectriccircularplatewithinitialstressesforMEM S,"in Proceedingsof2002 ASMEInternationalMechanicalEngineeringCongressandEx position ,NewOrleans, LA,Nov.17{22,2002,paper34337,pp.339{346. [128]S.Prasad,B.Sankar,L.Cattafesta,S.Horowitz,Q.Gal las,andM.Sheplak, \Two-portelectroacousticmodelofanaxisymmetricpiezoe lectriccompositeplate," in 43rdAIAAStructures,StructuralDynamics,andMaterialsCo nference ,Denver, CO,Apr.22{25,2002,AIAApaper2002-1365. [129]S.LiandS.Chen,\AnalyticalanalysisofacircularPZT actuatorforvalveless micropumps," SensorsandActuatorsA:Physical ,vol.104,no.2,pp.151{161,Apr. 2003.[Online].Available: http://dx.doi.org/10.1016/S0924-4247(03)00006-2 [130]S.Kim,W.W.Clark,andQ.M.Wang,\Piezoelectricener gyharvesting withaclampedcircularplate:analysis," JournalofIntelligentMaterialSystems andStructures ,vol.16,no.10,pp.847{854,Oct.2005.[Online].Available : http://dx.doi.org/10.1177/1045389X05054044 [131]C.Mo,R.Wright,W.S.Slaughter,andW.W.Clark,\Beha viour ofaunimorphcircularpiezoelectricactuator," SmartMaterialsand Structures ,vol.15,no.4,pp.1094{1102,Aug.2006.[Online].Available : http://dx.doi.org/10.1088/0964-1726/15/4/023 253

PAGE 254

[132]C.Mo,R.Wright,andW.W.Clark,\Theeectofelectrod epatternonthebehavior ofpiezoelectricactuatorsinacirculardiaphragmstructu re," JournalofIntelligent MaterialSystemsandStructures ,vol.18,no.5,pp.467{476,May2007.[Online]. Available: http://dx.doi.org/10.1177/1045389X06067112 [133]M.DeshpandeandL.Saggere,\Ananalyticalmodelandwo rkingequationsfor staticderectionsofacircularmulti-layereddiaphragm-t ypepiezoelectricactuator," SensorsandActuatorsA:Physical ,vol.136,no.2,pp.673{689,May2007.[Online]. Available: http://dx.doi.org/10.1016/j.sna.2006.12.022 [134]M.Papila,M.Sheplak,andL.N.C.III,\Optimizationof clamped circularpiezoelectriccompositeactuators," SensorsandActuatorsA:Physical ,vol.147,no.1,pp.310{323,Sep.2008.[Online].Available : http://dx.doi.org/10.1016/j.sna.2008.05.018 [135]C.-K.Lee,\Piezoelectriclaminatesfortorsionalan dbendingmodalcontrol:theory andexperiment,"Ph.D.dissertation,CornellUniversity,I thaca,NY,1987. [136]||,\Theoryoflaminatedpiezoelectricplatesforthe designofdistributed sensors/actuators.PartI:Governingequationsandrecipr ocalrelationships," The JournaloftheAcousticalSocietyofAmerica ,vol.87,p.1144,Mar.1990.[Online]. Available: http://dx.doi.org/10.1121/1.398788 [137]J.N.Reddy,\Onlaminatedcompositeplateswithintegr atedsensorsandactuators," EngineeringStructures ,vol.21,no.7,pp.568{593,Jul.1999.[Online].Available: http://dx.doi.org/10.1016/S0141-0296(97)00212-5 [138] MATLABHelpFile ,MathWorks,Natick,MA,2010,releaseR2010a. [139]B.A.Grin,M.D.Williams,C.S.Coman,andM.Sheplak, \Aluminum nitrideultrasonicair-coupledactuator," JournalofMicroelectromechanical Systems ,vol.20,no.2,pp.476{486,Apr.2011.[Online].Available: http://dx.doi.org/10.1109/JMEMS.2011.2111357 [140]B.A.Grin,\Developmentofanultrasonicpiezoelectr icMEMS-basedradiator fornonlinearacousticapplications,"Ph.D.dissertation ,UniversityofFlorida, Gainesville,FL,May2009. [141]K.TsubouchiandN.Mikoshiba,\Zero-temperature-coe cientSAWdevicesonAlN epitaxiallms," IEEETransactionsonSonicsandUltrasonics ,vol.32,no.5,pp. 634{644,Sep.1985.[Online].Available: http://dx.doi.org/10.1109/T-SU.1985.31647 [142]J.Reddy, EnergyPrinciplesandVariationalMethodsinAppliedMecha nics Hoboken,NJ:JohnWiley&Sons,Inc.,2002,ch.4{6,8. [143] ABAQUSHelpFile ,Simulia,Providence,RI,2008,version6.8. [144]A.S.SedraandK.C.Smith, MicroelectronicCircuits ,4thed.NewYork,NY: OxfordUniversityPress,1998,p.21. 254

PAGE 255

[145]D.JohnsandK.Martin, AnalogIntegratedCircuitDesign .NewYork,NY:John Wiley&Sons,2009. [146]M.SerridgeandT.R.Licht, PiezoelectricAccelerometerandVibrationPreamplier Handbook ,BruelandKjr,Nov.1987. [147]E.Bartolome,\Signalconditioningforpiezoelectri csensors," AnalogApplicationsJournal ,no.1Q,p.10,2010.[Online].Available: http://focus.ti.com/lit/an/slyt369/slyt369.pdf [148]G.Gautschi, PiezoelectricSensorics .Berlin,Germany:Springer-Verlag,2002,ch.11. [149]K.Kadirvel,\Developmentofclosed-loopinterfacec ircuitsforcapacitive transducerswithapplicationtoaMEMScapacitivemicropho ne,"Ph.D. dissertation,UniversityofFlorida,Gainesville,FL,Dec. 2007.[Online].Available: http://purl.fcla.edu/fcla/etd/UFE0020083 [150]R.T.HaftkaandZ.Gurdal, ElementsofStructuralOptimization ,3rded.Dordrecht, TheNetherlands:KluwerAcademicPublishers,1992,ch.1,2. [151]R.Spencer,B.Fleischer,P.Barth,andJ.Angell,\Athe oreticalstudyoftransducer noiseinpiezoresistiveandcapacitivesiliconpressurese nsors," IEEETransactions onElectronDevices ,vol.35,no.8,pp.1289{1298,Aug.1988.[Online].Available : http://dx.doi.org/10.1109/16.2550 [152]M.Papila,R.Haftka,T.Nishida,andM.Sheplak,\Piezor esistivemicrophone designparetooptimization:tradeobetweensensitivitya ndnoiseroor," Journalof MicroelectromechanicalSystems ,vol.15,no.6,pp.1632{1643,Dec.2006.[Online]. Available: http://dx.doi.org/10.1109/JMEMS.2006.883884 [153]J.Branke,K.Deb,K.Miettinen,andR.S lowinski,Ed s., MultiobjectiveOptimization :InteractiveandEvolutionaryApproaches .Berlin,Germany:Springer,2008,ch.1. [154]K.Deb,A.Pratap,S.Agarwal,andT.Meyarivan,\Afastan delitist multiobjectivegeneticalgorithm:NSGA-II," IEEETransactionsonEvolutionary Computation ,vol.6,no.2,pp.182{197,Apr.2002.[Online].Available: http://dx.doi.org/10.1109/4235.996017 [155]L.LibertiandN.Maculan, GlobalOptimization:FromTheorytoImplementation Berlin,Germany:Springer,2006,p.248. [156]G.G.Stoney,\Thetensionofmetalliclmsdepositedb yelectrolysisthetension ofmetalliclmsdepositedbyelectrolysis." ProceedingsoftheRoyalSocietyof LondonSeriesA ,vol.82,no.553,pp.172{175,May1909.[Online].Available : http://www.jstor.org/stable/92886 [157](2010)Services.DynatexInternational.[Online].Av ailable: http://www.dynatex.com/services.html 255

PAGE 256

[158]T.Lizotte,\Laserdicingofchipscaleandsiliconwaf erscalepackages," in IEEE/CPMT/SEMI28thInternationalElectronicsManufactu ringTechnology Symposium ,SanJose,CA,Jul.16{18,2003,pp.1{5.[Online].Available: http://dx.doi.org/10.1109/IEMT.2003.1225869 [159] Ablebond84-1LMI ,Ablestik,Jan.2005. [160] Dualbond707 ,CyberbondLLC,Feb.2011. [161] 3145RTVMIL-A-46146Adhesive/Sealant ,DowCorning,Feb.2011. [162] UltiboardHelpFile ,NationalInstruments,Jan.2007,version10.0.343. [163]W.Navidi, StatisticsforEngineersandScientists .McGraw-Hill,2006,ch.1,5. [164]W.R.Thompson,\Onacriterionfortherejectionofobs ervationsandthe distributionoftheratioofdeviationtosamplestandardde viation," TheAnnals ofMathematicalStatistics ,vol.6,no.4,pp.214{219,1935.[Online].Available: http://dx.doi.org/10.1214/aoms/1177732567 [165]J.Grin,T.Schultz,R.Holman,L.Ukeiley,andL.Cattaf esta,\Application ofmultivariateoutlierdetectiontoruidvelocitymeasure ments," Experiments inFluids ,vol.49,no.1,pp.305{317,Jul.2010.[Online].Available: http://dx.doi.org/10.1007/s00348-010-0875-3 [166]M.HubertandS.VanderVeeken,\Outlierdetectionfors keweddata," Journalof Chemometrics ,vol.22,no.3{4,pp.235{246,Mar.{Apr.2008.[Online].Avai lable: http://dx.doi.org/10.1002/cem.1123 [167]S.VerbovenandM.Hubert,\LIBRA:AMATLABlibraryforrob ustanalysis," ChemometricsandIntelligentLaboratorySystems ,vol.75,no.2,pp.127{136,Feb. 2005.[Online].Available: http://dx.doi.org/10.1016/j.chemolab.2004.06.003 [168]M.Hubert.(2009,Nov.)LIBRA:AMATLABlibraryforrobusta nalysis.Robust StatisticsResearchGroup,KatholiekeUniversiteitLeuven .Leuven,Belgium. [Online].Available: http://wis.kuleuven.be/stat/robust/LIBRA.html [169]M.HubertandE.Vandervieren,\Anadjustedboxplotfors keweddistributions," ComputationalStatistics&DataAnalysis ,vol.52,no.12,pp.5186{5201,Aug.2008. [Online].Available: http://dx.doi.org/10.1016/j.csda.2007.11.008 [170]S.Woods,\Understandingscanningwhitelightinterfe rometry," MICRO Manufacturing ,vol.2,no.4,pp.19{21,Winter2009.[Online].Available: http://www.micromanufacturing.com/past editions/pdf/Winter09.pdf [171]G.S.K.WongandT.F.W.Embleton, AIPHandbookofCondensorMicrophones NewYork,NY:AmericanInstituteofPhysics,1995,ch.13. 256

PAGE 257

[172] MeasurementMicrophones-Part5:MethodsforPressureCali brationofWorking StandardMicrophonesbyComparison ,InternationalElectrotechnicalCommission InternationalStandardCEI/IEC61094-5:2001,Oct.2001. [173]D.Alexander,C.Barnard,M.Sheplak,andB.Grin,\Cha racterizationofa high-frequencypressure-eldcalibrationmethod,"in Proceedingsofthe159th MeetingoftheAcousticalSocietyofAmerica ,Baltimore,MD,Apr.19{23,2010. [174] CondenserMicrophonesandMicrophonePreampliersforAco usticMeasurements BruelandKjr,Nrum,Denmark,1982. [175] Agilent4294APrecisionImpedanceAnalyzerOperationManua l ,7thed., AgilentTechnologies,Japan,Feb.2003.[Online].Available : http://cp.literature.agilent.com/litweb/pdf/04294-9 0060.pdf [176] ImpedanceMeasurementHandbook ,AgilentTechnologies,Dec.2003.[Online]. Available: http://cp.literature.agilent.com/litweb/pdf/5950-30 00.pdf [177] NewTechnologiesforAccurateImpedanceMeasurement(40Hz to110MHz) AgilentTechnologies,Jun.2009,productnote.[Online].Ava ilable: http://cp.literature.agilent.com/litweb/pdf/5968-45 06E.pdf [178]M.Robinson,technicalsupportcall,AgilentTechnolo gies,Mar.2010. [179]L.N.Virgin, IntroductiontoExperimentalNonlinearDynamics .Cambridge,UK: CambridgeUniversityPress,2000. [180]||, VibrationofAxiallyLoadedStructures .Cambridge,UK:CambridgeUniversity Press,2007. [181]A.NainiandM.Green,\Fringingeldsinaparallel-plat ecapacitor," American JournalofPhysics ,vol.45,no.9,pp.877{879,Sep.1977.[Online].Available: http://dx.doi.org/10.1119/1.11075 [182]G.ServaisandS.Brandenburg,\Wirebonding|acloser look,"in 17thInternational SymposiumforTesting&FailureAnalysis ,LosAngeles,CA,Nov.11{15,1991,pp. 525{529. [183]R.Vispute,H.Wu,andJ.Narayan,\Highqualityepitaxiala luminumnitridelayers onsapphirebypulsedlaserdeposition," AppliedPhysicsLetters ,vol.67,no.11,pp. 1549{1551,Sep.1995.[Online].Available: http://dx.doi.org/10.1063/1.114489 [184]F.Martin,P.Muralt,M.Dubois,andA.Pezous,\Thickne ssdependenceofthe propertiesofhighlyc-axistexturedAlNthinlms," JournalofVacuumScience& TechnologyA:Vacuum,Surfaces,andFilms ,vol.22,no.2,p.361,Mar./Apr.2004. [Online].Available: http://dx.doi.org/10.1116/1.1649343 257

PAGE 258

[185]S.Sanchez,\Robustdesign:seekingthebestofallpos sibleworlds,"in 2000Winter SimulationConferenceProceedings ,Orlando,FL,Dec.10{13,2000,pp.69{76. [Online].Available: http://dx.doi.org/10.1109/WSC.2000.899700 [186]J.S.HanandB.M.Kwak,\Robustoptimaldesignofavibra tory microgyroscopeconsideringfabricationerrors," JournalofMicromechanics andMicroengineering ,vol.11,pp.662{671,2001.[Online].Available: http://dx.doi.org/10.1088/0960-1317/11/6/307 [187]J.Wittwer,M.Baker,andL.Howell,\Robustdesignandm odelvalidation ofnonlinearcompliantmicromechanisms," JournalofMicroelectromechanical Systems ,vol.15,no.1,pp.33{41,Feb.2006.[Online].Available: http://dx.doi.org/10.1109/JMEMS.2005.859190 [188]S.Gurav,J.Goosen,andF.vanKeulen,\Bounded-but-u nknownuncertainty optimizationusingdesignsensitivitiesandparallelcomp uting:Applicationto mems," Computers&Structures ,vol.83,no.14,pp.1134{1149,May2005.[Online]. Available: http://dx.doi.org/10.1016/j.compstruc.2004.11.021 [189]M.SheplakandJ.Dugundji,\Largederectionsofclamp edcircularplates undertensionandtransitionstomembranebehavior," JournalofApplied Mechanics ,vol.65,no.1,pp.107{115,Mar.1998.[Online].Available: http://dx.doi.org/10.1115/1.2789012 [190]B.Grin,B.Homeijer,M.Williams,B.Sankar,andM.She plak,\Largederections ofclampedcompositecircularplateswithinitialin-plane tension,"in IMACXXVI: ConferenceandExpositiononStructuralDynamics ,Orlando,FL,Feb.4{7,2008. [191]Y.-Y.Yu, VibrationsofElasticPlates:LinearandNonlinearDynamica lModeling ofSandwiches,LaminatedComposites,andPiezoelectricLa yers .NewYork,NY: Springer,1996,ch.1,7. [192]W.M.Lai,D.Rubin,andE.Krempl, IntroductiontoContinuumMechanics ,1sted. NewYork,NY:PergamonPress,1974. [193]L.E.Malvern, IntroductiontotheMechanicsofaContinuousMedium .Englewood Clis,NJ:Prentice-Hall,1969,ch.4. [194]R.L.Panton, IncompressibleFlow ,3rded.Hoboken,NJ:JohnWiley&Sons,2005, ch.1. [195]C.L.DymandI.H.Shames, SolidMechanics:aVariationalApproach .NewYork, NY:McGraw-Hill,1973,ch.4,6{8. [196]A.J.M.Spencer, ContinuumMechanics .Mineola,NY:DoverPublications,2004, p.83. [197]C.-Y.Chia, NonlinearAnalysisofPlates .NewYork,NY:McGraw-Hill,1980,ch. 1,3. 258

PAGE 259

[198]J.N.Reddy, MechanicsofLaminatedCompositePlates:TheoryandAnalysi s BocaRaton:CRCPress,1997,ch.5. [199]K.R.Nagle,E.B.Sa,andA.David, FundamentalsofDierentialEquations andBoundaryValueProblems ,3rded.Reading,MA:Addison-Wesley,2000,pp. 187{189,510{514. [200]A.King,J.Billingham,andS.Otto, DierentialEquations .Cambridge,UK: CambridgeUniversityPress,2003,ch.3. [201]H.W.ColemanandW.G.Steele, Experimentation,Validation,andUncertainty AnalysisforEngineers ,3rded.Hoboken,NJ:JohnWiley&Sons,2009,ch.1{3. [202] GuidetotheExpressionofUncertaintyinMeasurement ,InternationalOrganization forStandardizationInternationalStandard,1993. [203]L.Cattafesta,\Spectralfunctions,"Spring2007,un published. [204]P.D.Welch,\Theuseoffastfouriertransformforthee stimationofpowerspectra," IEEETransactionsonAudioandElectroacoustics ,vol.15,no.2,pp.70{73,Jun. 1967.[Online].Available: http://dx.doi.org/10.1109/TAU.1967.1161901 259

PAGE 260

BIOGRAPHICALSKETCH MatthewDavidWilliamswasbornin1982inPlano,TXandsubse quentlylived inGarland,TX,Maryville,TN,andBatesburg-Leesville,SCbe foregraduatingfrom Batesburg-LeesvilleHighSchoolinJune2001.HeenrolledatC lemsonUniversity (Clemson,SC)inAugust2001andwasselectedarecipientofth eBarryM.Goldwater Scholarshipin2004beforegraduatingsummacumlaudewitha bachelor'sdegreein mechanicalengineeringinMay2005.InAugust2005,Mattenro lledatUniversityof Florida(Gainesville,FL)asaNationalScienceFoundationG raduateResearchFellow, joiningInterdisciplinaryMicrosystemsGroupinApril2006 .Mattreceivedhismasters degreeinmechanicalengineeringinMay2008beforeserving asavisitingresearchatthe DelftUniversityofTechnologyfromSeptember2008{Septemb er2009.Uponreturning toUniversityofFlorida,Mattcompletedhisdoctoraldegree inmechanicalengineering inMay2011.Matt'sresearchinterestsincludethedesignan doptimizationofmicroscale sensorsandactuators,inadditiontononlinearmechanics, particularpost-bucklingand snap-throughofmultistableelectromechanicalmicrostru ctures. 260