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Development of an Ultrasonic Piezoelectric MEMS-Based Radiator for Nonlinear Acoustic Applications

Permanent Link: http://ufdc.ufl.edu/UFE0023480/00001

Material Information

Title: Development of an Ultrasonic Piezoelectric MEMS-Based Radiator for Nonlinear Acoustic Applications
Physical Description: 1 online resource (227 p.)
Language: english
Creator: Griffin, Benjamin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

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

Notes

Abstract: The development of a piezoelectric micromachined ultrasonic transmitter is presented. The transducer is evaluated as an ultrasonic source for parametric arrays. A parametric array is an acoustic technology that leverages the nonlinear demodulation of ultrasound to create a highly directional beam of audible sound analogous to a flashlight. The transmitter was formed by a circular composite diaphragm that was radially non-uniform. The diaphragm consisted of molybdenum annular electrodes and an aluminum nitride diaphragm. The composite diaphragm was released using a combination of deep reactive ion and oxide etching. Transduction of the diaphragm occurred when an electric field was induced across the annular piezoelectric layer of aluminum nitride. The electric field caused mechanical strain within the piezoelectric layer through the piezoelectric effect. The strain coupled into force and moment resultants that generated diaphragm deflection. By supplying an ac voltage between the electrodes, an oscillating electric field caused diaphragm vibration. The vibrating diaphragm in turn generated acoustic waves. The overall device model was formed using composite plate mechanics, lumped element modeling (LEM), and acoustic theory. Through LEM an equivalent circuit of the device was formed that incorporated electrical, mechanical, and acoustic components. Nonlinear acoustic theory was used to predict ultrasonic demodulation. The device performance model was used in a geometric constrained optimization scheme. The optimal design based on the LEM was used to form the primary design. A series of secondary designs were formed by constraining the device layer thicknesses and performing optimization using radial geometry as design variables and considering deviations in stress. The device was fabricated at Avago Technologies Limited's foundry. A unique package with back cavity depth control was designed and fabricated. This was followed by experimental characterization. Electrical characterization consisted of measurement of device impedance. Mechanical characterization included mode shape and resonant frequency measurements. Acoustic characterization encompassed farfield acoustic measurements of a single device. The characterization results showed significant mismatch between devices as well as the equivalent circuit model. Four out of the six devices tested had resonant frequencies near 60 kHz. The remaining two devices had resonant frequencies of 31 kHz and 44 kHz. The equivalent circuit predicted a resonance of 39 kHz. The variation between the results was attributed to stress variations across the wafer that occurred during fabrication. A variable back cavity was used to tune the devices and maximize sensitivity at resonance. A 220% improvement in the resonant deflection sensitivity of the 31 kHz resonant device was found by tuning the back cavity. A nonlinear acoustic calculation was used to project the performance of a 150 mm diameter device array as an acoustic source for a parametric array. The sound pressure level of a 5 kHz audible tone was 42 dB at 1 m. The low projected audible output does not show good promise of the application of this design to a conventional audible parametric array. Recommendations for future work focus on a more robust device design and packaging improvements for other ultrasonic applications.
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 Benjamin Griffin.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Sheplak, Mark.
Local: Co-adviser: Cattafesta III, Louis N.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0023480:00001

Permanent Link: http://ufdc.ufl.edu/UFE0023480/00001

Material Information

Title: Development of an Ultrasonic Piezoelectric MEMS-Based Radiator for Nonlinear Acoustic Applications
Physical Description: 1 online resource (227 p.)
Language: english
Creator: Griffin, Benjamin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

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

Notes

Abstract: The development of a piezoelectric micromachined ultrasonic transmitter is presented. The transducer is evaluated as an ultrasonic source for parametric arrays. A parametric array is an acoustic technology that leverages the nonlinear demodulation of ultrasound to create a highly directional beam of audible sound analogous to a flashlight. The transmitter was formed by a circular composite diaphragm that was radially non-uniform. The diaphragm consisted of molybdenum annular electrodes and an aluminum nitride diaphragm. The composite diaphragm was released using a combination of deep reactive ion and oxide etching. Transduction of the diaphragm occurred when an electric field was induced across the annular piezoelectric layer of aluminum nitride. The electric field caused mechanical strain within the piezoelectric layer through the piezoelectric effect. The strain coupled into force and moment resultants that generated diaphragm deflection. By supplying an ac voltage between the electrodes, an oscillating electric field caused diaphragm vibration. The vibrating diaphragm in turn generated acoustic waves. The overall device model was formed using composite plate mechanics, lumped element modeling (LEM), and acoustic theory. Through LEM an equivalent circuit of the device was formed that incorporated electrical, mechanical, and acoustic components. Nonlinear acoustic theory was used to predict ultrasonic demodulation. The device performance model was used in a geometric constrained optimization scheme. The optimal design based on the LEM was used to form the primary design. A series of secondary designs were formed by constraining the device layer thicknesses and performing optimization using radial geometry as design variables and considering deviations in stress. The device was fabricated at Avago Technologies Limited's foundry. A unique package with back cavity depth control was designed and fabricated. This was followed by experimental characterization. Electrical characterization consisted of measurement of device impedance. Mechanical characterization included mode shape and resonant frequency measurements. Acoustic characterization encompassed farfield acoustic measurements of a single device. The characterization results showed significant mismatch between devices as well as the equivalent circuit model. Four out of the six devices tested had resonant frequencies near 60 kHz. The remaining two devices had resonant frequencies of 31 kHz and 44 kHz. The equivalent circuit predicted a resonance of 39 kHz. The variation between the results was attributed to stress variations across the wafer that occurred during fabrication. A variable back cavity was used to tune the devices and maximize sensitivity at resonance. A 220% improvement in the resonant deflection sensitivity of the 31 kHz resonant device was found by tuning the back cavity. A nonlinear acoustic calculation was used to project the performance of a 150 mm diameter device array as an acoustic source for a parametric array. The sound pressure level of a 5 kHz audible tone was 42 dB at 1 m. The low projected audible output does not show good promise of the application of this design to a conventional audible parametric array. Recommendations for future work focus on a more robust device design and packaging improvements for other ultrasonic applications.
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 Benjamin Griffin.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Sheplak, Mark.
Local: Co-adviser: Cattafesta III, Louis N.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0023480:00001


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Isaiah40:31 ...butthosewhohopeintheLord willrenewtheirstrength. Theywillsoaronwingslikeeagles; theywillrunandnotgrowweary, theywillwalkandnotbefaint. 3

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FinancialsupportforthisworkhasbeenprovidedbygraduatefellowshipsfromtheNationalScienceFoundationandtheUniversityofFlorida.Ithankmyadvisors,MarkSheplakandLouisN.CattafestaIII,fortheirmanyhelpfultechnicaldiscussions,aswellastheircareerandpersonaladvice.Iamalsogratefultomycommitteemembers,HavanaV.Sanka,DavidArnold,andNabHoKim,fortheirexpertiseandassistanceinthesuccessofthisproject.IamespeciallygratefultomymanycolleaguesintheInterdisciplinaryMicrosystemsGroup.Iwouldliketothankmypredecessors,VenkataramanChandrasekaranandGuiqinWang,forestablishingarmfoundationuponwhichthisworkwasbuilt.FormercolleaguesDavidMartinandStephenHorowitzaregreatlyappreciatedfortheirmentorshipandtrain-ingduringourconcurrentassociationwiththeInterdisciplinaryMicrosystemsGroup.IhavemuchgratitudeforcontemporariesBrianHomeijerandVijayChandrasekharanaswehave\comeofage"asgraduatestudentstogether.Theirengagingtechnicaldiscussions,friend-ship,andcomraderyhavebeenasustainingforceinmygraduatecareer.IamalsoindebtedtoMatthewWilliams,whoIhaveworkedcloselywithonthisproject.WithouthisadditiontotheInterdisciplinaryMicrosystemsGroup,thisundertakingwouldnothavebeenassuc-cessful.Inaddition,IamgratefultoChaseComan,DylanAlexander,andJohnGrinfortheirassistancewithexperimentalsetups,packagefabrication,anddataacquisition.IwouldalsoliketoacknowledgealloftheInterdisciplinaryMicrosystemsgroupwhosecontributionsaretoonumeroustolist.IamparticularlythankfultoAvagoTechnologiesLimitedfortheaccesstotheirfab-ricationfacilities.SpecialthanksgoestoDavidMartinandOsvaldoBuccafuscaatAvagofortheirspecialattentionandpersonaltimedevotedtothisproject.IalsoacknowledgeDynatexInternationalfortheirskillinwaferseparation.IamespeciallygratefulfortheexcellentmachiningworkperformedbyKenReedatTMRengineering.TheMechanicalandAerospaceEngineeringdepartmentalstaisthankedfortheirkindassistance. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 11 ABSTRACT ........................................ 17 CHAPTER 1Introduction ...................................... 19 1.1ParametricArrays ............................... 19 1.2TransducerIssues ................................ 22 1.2.1CurrentLimitations ........................... 23 1.2.2PotentialTransducerSolution ..................... 24 1.3ResearchObjectives ............................... 24 1.4DissertationOverview ............................. 24 2NonlinearAcoustics .................................. 25 2.1FinitePerturbationAcousticTheory ..................... 25 2.2ParametricArray ................................ 30 2.2.1ModelEquationsofNonlinearAcousticTheory ............ 31 2.2.2SoundBeamSolutions ......................... 34 2.2.3ExistingImplementations ........................ 38 2.2.4MEMSParametricArrays ....................... 47 2.3Conclusion .................................... 50 3Air-CoupledMEMSUltrasonicTransducers .................... 51 3.1PrinciplesofTransmitterOperation ...................... 51 3.2AcousticSources ................................ 55 3.2.1PlanarRadiation ............................ 55 3.2.2ArrayofSources ............................. 61 3.2.3AcousticAttenuation .......................... 63 3.2.4Summary ................................. 66 3.3MEMSActuators ................................ 66 3.3.1ElectrostaticTransduction ....................... 67 3.3.2PiezoelectricTransduction ....................... 80 3.3.3ThermoelasticActuation ........................ 95 3.4Conclusion .................................... 100 6

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............................. 102 4.1Avago'sFBARProcess ............................. 102 4.2Fabrication ................................... 104 4.3Package ..................................... 105 4.4Conclusion .................................... 108 5Modeling ....................................... 109 5.1EquivalentCircuit ............................... 109 5.1.1AcousticalDomain ........................... 112 5.1.2ElectricalDomain ............................ 118 5.1.3Transduction ............................... 118 5.1.4EquivalentCircuit ............................ 119 5.1.5ApproximatePerformance ....................... 121 5.1.6ExampleDevice ............................. 122 5.2NonlinearAcousticModeling .......................... 130 5.3Conclusion .................................... 132 6DesignOptimization ................................. 133 6.1Methodology .................................. 133 6.2RadiatorOptimization ............................. 134 6.2.1Limitations-Constraints ......................... 135 6.2.2ProblemFormulation .......................... 137 6.3Results ...................................... 138 6.4AlternateDesigns ................................ 142 6.5Conclusions ................................... 143 7ExperimentalSetupandResults ........................... 145 7.1FabricationResults ............................... 145 7.2ElectricalCharacterization ........................... 148 7.2.1Setup ................................... 148 7.2.2Results .................................. 149 7.3DeviceTopography ............................... 151 7.4ElectromechanicalCharacterization ...................... 153 7.4.1Setup ................................... 155 7.4.2FrequencyResponseFunction ..................... 157 7.4.3DiaphragmResonance ......................... 159 7.4.4Linearity ................................. 160 7.4.5VariableBackCavity .......................... 163 7.4.6VacuumExperiments .......................... 167 7.5ElectroacousticCharacterization ........................ 171 7.5.1Setup ................................... 171 7.5.2Results .................................. 174 7.6PerformanceasaParametricArray ...................... 176 7

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.................................... 178 8ConclusionandFutureWork ............................. 180 8.1Conclusions ................................... 180 8.2RecommendationsforFutureWork ...................... 182 8.3RecommendationsforFutureDesign ..................... 183 APPENDIX ANONLINEARACOUSTICMODELING ...................... 186 A.1WesterveltParametricArraySolution ..................... 186 A.2BerktaySolution ................................ 189 BPLATEMODEL ................................... 191 B.1BasicAssumptions ............................... 192 B.2StaticEquilibrium ............................... 192 B.3ConstitutiveEquations ............................. 194 B.4GoverningDierentialEquations ....................... 196 B.5GeneralSolution ................................ 197 B.6BoundaryandMatchingConditions ...................... 198 B.7IncrementalPlateDeection .......................... 200 CUNCERTAINTYANALYSIS ............................ 201 C.1ElectricalCharacterizationUncertainties ................... 201 C.1.1ElectricalImpedance .......................... 201 C.1.2ElementExtraction ........................... 202 C.2ElectromechanicalCharacterizationUncertainties .............. 206 C.2.1Velocity ................................. 206 C.2.2VolumeVelocity ............................. 207 C.2.3ResonantFrequency ........................... 208 C.2.4DampingCoecientEstimation .................... 208 C.2.5VariableBackCavity .......................... 210 REFERENCES ....................................... 213 BIOGRAPHICALSKETCH ................................ 227 8

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Table page 2-1CMUTtransducerarrayspecicationsandperformancepresentedbyWygantetal.[15]. ........................................ 48 2-2Transducersusedinparametricarrayexperiments. ................ 49 3-1Air-coupledcMUTcharacteristics. ......................... 81 3-2Piezoelectriclmproperties. ............................. 87 3-3Air-coupledpMUTcharacteristics. ......................... 96 3-4ThermoelasticMEMScharacteristics. ........................ 100 4-1Typicalepoxydispenseandcureparametersusedfordieattachment.PCBboardswerepre-heatedat145Cfor5min.beforeepoxyapplication. .......... 106 4-2Wirebondersettings. ................................ 106 5-1Geometryofexampledevice. ............................. 123 5-2MaterialpropertiesofAlNandMo. ......................... 123 5-3Comparisonofelectricalelementttofullmodel. ................. 129 6-1AvagoTechnologiesLimitedprocessoptionswherej=1;2referstotheinnerandannularplatesections,respectively. ....................... 135 6-2Designoptimizationresults. ............................. 140 6-3Alternateoptimizationresults. ........................... 143 7-1FilmstresstargetandrealizedvaluesforthewaferprovidedbyAvago. ..... 146 7-2ObjectivefunctionofdevicedesignsindBreferencedtothetargetdesign,UFPA08.Blanksrefertodevicesthatviolatemodelingconstraints. ............. 147 7-3Theextractimpedanceparametersfromtheexperimentalimpedancemeasurement. 151 7-4DampingandqualityfactorvaluesforthedierentdevicesatvacuumandSTP. 169 7-5Performanceofthedie.Thecenterdeectionafterbackcavitytuningisshowninparentheses. .................................... 179 8-1Comparisonofair-coupledMEMSradiatorperformances. ............. 182 C-1Frequencyboundsonsystemt. ........................... 210 C-2Uncertaintiesintheresonantfrequencyandresponseasafunctionofmeasure-mentuncertainties. .................................. 211 9

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Figure page 1-1Observerofamuseumdisplayislisteningtocommentarywhiledisinterestedpedestrianspassbyinquiet(adaptedfromHolosonics[3]). ........... 20 1-2Soundeldsofaconventionallysizedradiatorataudioandultrasonicfrequencies(adaptedfromHolosonics[3]). ........................... 21 2-1Wavesteepeningduetononlinearwavepropagation.AdaptedfromBlackstock[8]. .......................................... 29 2-2Thefrequencydistributionofanonlinearlypropagatingwave. .......... 30 2-3Parametricarrayradiationofsound(adaptedfromHolosonics)[3]. ....... 31 2-4Thesoundeldandlengthscalesassociatedwiththesoundbeamsolutions.Notethatthisgureishighlyschematic. ........................ 35 2-5ATCbimorphpiezoelectricactuatorwithattachedconeforenhancedvolumevelocity[11]. ...................................... 44 3-1Genericacoustictransmitter[49]. ......................... 51 3-2Bulkacoustictransmitter. ............................. 52 3-3Genericbendingmodeacoustictransmitter[49]. ................. 52 3-4Frequencyresponseofanultrasonictransmitter. ................. 53 3-5Comparisonoftheidealandphysicaltransducersensitivitiesversustheforcingvoltageforaxedfrequency. ............................ 54 3-6Speakermodeledasabaedpiston. ........................ 56 3-7Arbitrarilyshapedbaedpiston(adaptedfromBlackstock[8]). ......... 57 3-8Circularpistoninaninnitebae(adaptedfromBlackstock[8]). ....... 57 3-9Comparisonofthedirectivityofacompactandnon-compactsource. ...... 59 3-10Radiationimpedanceofabaed,circularpiston. ................. 60 3-11LinearrayofNidealmonopoles(adaptedfromBlackstock[8]). ......... 62 3-12Arraydirectivityfor3and7monopolearraysatkd=1. ............. 63 3-13Soundabsorptioncoecientofairat70%relativehumidity(afterZuckerwar[54]). 65 3-14One-dimensionalelectrostatictransducer. ..................... 68 11

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................................ 70 3-16Plotofthemechanicalforce,FM,andtheelectrostaticforce,FE,atdierentvoltagesversusthegapdistancex(Notethatallvaluesarenon-dimensional). 71 3-17Deviceformedbycombiningmacro-andmicromachining(adaptedfromHiguchi[59]). ......................................... 72 3-18CapacitivecMUToftheE.L.GinztonLaboratory(adaptedfromHaller[66]). 73 3-19NitridediaphragmcMUTwithvacuumsealedbackcavity(adaptedfromJinetal.[71]). ....................................... 74 3-20Capacitivetransducerwithapolysilicondiaphragmanddopedbottomelectrode(adaptedfromEccardtetal.[78]). ......................... 75 3-21MicromachinedcapacitivedevicewithcoupledHelmholtzresonatorformedbytheresonantcavityandthroat(adaptedfromParvizetal.[81]). ........ 76 3-22AcMUTwhosecavitiesareformedbyananisotropicetch(adaptedfromTorndahletal.[82]). ...................................... 77 3-23Capacitivetransducerwithpolysiliconmoveablemembranes(adaptedfromBuhrdorfetal.[83]). ...................................... 77 3-24AcMUTfabricatedusingMUMPS(adaptedfromOppenheimetal.[84]). ... 77 3-25AcMUTfabricatedusinganSOIwaferbondedtoapatternedsubstrate(adaptedfromHuangetal.[85,86]). ............................. 78 3-26AcMUTwithnitridediaphragm(adaptedfromKimetal.[90]). ........ 79 3-27Isometricviewofanidealperovskitestructure.[91]. ................ 82 3-28Isometricviewsofidealwurtziteandperovskitestructure.[91]. ......... 83 3-29One-dimensionalpiezoelectrictransducer(adaptedfrom[16]). .......... 84 3-30Eectofthed31coecientinpiezoelectricthinlms. .............. 85 3-31Hysteresisloopsofpiezoelectricmaterials(adaptedfrom[91]) .......... 87 3-32Twoside-by-sidepMUTsformedbyanitridediaphragmandZnOannularring(adaptedfromPercinetal.[115]). ......................... 88 3-33DomeshapedpMUTwithnitridediaphragmandZnOpiezoelectriclayer(adaptedfromCheol-Hyunetal.[100]). ........................... 88 3-34SquarediaphragmpMUTusingPZT(adaptedfromMohamedetal.[102,120]). 89 12

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...... 90 3-36PZTmicrospeaker(adaptedfromZhuetal.[124]). ................ 90 3-37PZTdiaphragm(adaptedfromZhuetal.[131]). ................. 91 3-38PZTmicrospeaker(adaptedfromZhuetal.[53]). ................ 92 3-39SquarediaphragmpMUTutilizingaP(VDF-TrFE)lmforactuation(adaptedfromLam[101]). ................................... 93 3-40OxidediaphragmformedbytheBOXlayerofanSOIwaferandactuatingbyaPZTlm(adaptedfrom[103]). ........................... 93 3-41ApMUTusedinanenergyharvester(adaptedfrom[46]). ............ 94 3-42AnAlNaudiomicrospeakerpMUT(adaptedfrom[135]). ............ 94 3-43Proximitysensorthatutilizesthermoelasticactuationandpiezoresistivesensing(adaptedfromBrand[138]). ............................ 98 3-44Thermoelasticactuatorwithabuckleddiaphragm(adaptedfromPopescuetal.[147]. ....................................... 98 3-45Thermoelastic/piezoresistiveproximitysensorthatusesthedevicelayerofanSOIwafertoformacirculardiaphragm(adaptedfromChandrasekaranetal.[148]). 99 3-46Thermoelasticproximitysensorusingpolysiliconfortheheaterandpiezoresistors(adaptedfromRuferetal.[149]). ......................... 99 4-1Cross-sectionofthestructurallayersandbackcavitypossibleusingtheAvagoFBARprocess. .................................... 103 4-2MEMS-basedultrasonicradiatorforparametricarrayapplications. ....... 104 4-3PCBboardfordevicepackage. ........................... 105 4-4Thealuminumblockcontainingvariablebackcavity,dowels,screwholesforalign-menttothePCB,andananchorscrewhole. .................... 107 4-5Packageddevicemountedtovariablebackcavity. ................. 107 5-1Diagramofincrementaldiaphragmdeection. ................... 110 5-2Equivalentcircuitmodel. .............................. 111 5-3Rigidductmodel. .................................. 115 5-4Thebackcavityofthetransducerwiththedie,pcb,andaluminumsections. 116 5-5Circuitrepresentationofthebackcavityusingtransfermatrices. ........ 116 13

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.. 118 5-7Electricalelementsoftheequivalentcircuit. .................... 118 5-8Two-portrepresentationofelectro-acoustictransduction. ............ 120 5-9Equivalentacousticcircuitmodel. ......................... 120 5-10Volumevelocityfrequencyresponsecalculatedusingthefullsensitivityequivalentcircuitandthehighfrequencyapproximation. .................. 124 5-11Totalimpedanceoftheventandcavityinparallelversusjustthecavityimpedance. ............................................. 124 5-12Diaphragmimpedancecomparisontotheradiationimpedance. ......... 125 5-13Contributionstotheoverallacousticimpedance. ................. 125 5-14Inputandtheelectricalimpedancecomparison. .................. 126 5-15factoroftheinputimpedance. .......................... 126 5-16RadiationimpedancecomparisonofthefullmodelandLEM. .......... 127 5-17BackcavityimpedancecomparisonofthefullmodelandLEM. ......... 127 5-18InputimpedancecomparisonofthefullmodelandLEM. ............ 128 5-19FullmodelandcurvetcomparisonassumingaLEMoftheinputimpedance. 129 5-20Diagramoftransducerarrayshowingradiusdenitions(Notdrawntoscale). 130 5-21Outputoftheexampleparametricarrayat1m. ................. 132 6-1Outputoftheparametricarrayat1m. ...................... 138 6-2Sensitivityofthenormalizedobjectivefunctiontothenormalizeddesignvari-ables. ......................................... 140 6-3Sensitivityoftheobjectivefunctionduetochangesintheindividualdesignvari-ablesshowactiveconstraintsandbounds. ..................... 141 6-4Sensitivityoftheobjectivefunctiontothenonlineardeectionconstraint. ... 142 6-5Sensitivityoftheobjectivefunctiontotheminimumresonantfrequencycon-straint. ........................................ 142 7-1Reticlelabelingconvention. ............................. 148 7-2ThefrontpanelconnectionsontheHP4294AImpedanceAnalyzerwhereredisthehighconnection,blackisthelowconnection,andgreenisground. ...... 149 14

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........ 150 7-4Initialdiaphragmdeections. ............................ 152 7-5Scanninglaservibrometersystemusedforelectromechanicalcharacterization. 154 7-6Thescangridoverlayedwiththemicroscopepictureofthedevice. ....... 157 7-7Velocityfrequencyresponsefunctionforthecentergridpoint,wheretheuncer-taintyisthe95%condenceinthemeanestimate. ................ 158 7-8Displacementfrequencyresponsefunctionforthecentergridpoint,wheretheuncertaintyisthe95%condenceinthemeanestimate. ............. 158 7-9Volumevelocitysensitivity,wheretheuncertaintyisthe95%condenceinthemeanestimate. .................................... 159 7-10Displacementsensitivitycross-sections. ....................... 160 7-11Resonantmodeshapes. ................................ 161 7-12Thevelocitysensitivityversusexcitationvoltage. ................. 162 7-13Velocityanddisplacementresponseversusresonanttoneexcitationamplitude. 163 7-14Measurementsofthebackcavity. .......................... 166 7-15Idealpistonandbackcavity. ............................. 166 7-16Imaginarypartsofthediaphragmimpedanceandcavityimpedancesofvaryingbackcavitydepth. .................................. 167 7-17Vacuumchamberdiagram. .............................. 168 7-18DevicecomparisonofthefrequencyresponsefunctionbetweenroughvacuumandSTPconditions. .................................... 171 7-19ResonantperformancereferencetoSTPversusincreasevacuumchamberpressure. 172 7-20Setupfortheelectroacousticcharacterization. ................... 172 7-21Directivitymeasurementsalongwiththepredicteddirectivitiesfoundbyextrap-olatingtheLVmeasurementsusingRayleigh'sintegral. .............. 175 7-22Onaxispressureresponse. .............................. 177 7-23Parametricarrayoutputat1mofaanarrayof4,500radiators. ......... 178 8-1Stresseldsinsideadeectedclampedplate. .................... 184 8-2Alternatedesignforimprovedperformance. .................... 184 15

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......................................... 191 B-2Isometricviewofaninnitesimalplateelement.Threelayersareshownforillustrationbuttheplateelementisconsideredtohaveanarbitrarynumberoflayersinthederivationofthegoverningequations[161]. ............. 193 B-3Diagramofincrementaldiaphragmdeection. ................... 200 C-1Randomandbiaserrorsintherealandimaginarypartsoftheimpedance. ... 202 C-2Impedancetincomparisontotheexperimentaldata. .............. 205 C-3Schematicoftheuncertaintyintheresonantfrequencycalculation. ....... 208 C-4Dampingcoecientcurvet. ............................ 209 16

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Thedevelopmentofapiezoelectricmicromachinedultrasonictransmitterispresented.Thetransducerisevaluatedasanultrasonicsourceforparametricarrays.Aparametricarrayisanacoustictechnologythatleveragesthenonlineardemodulationofultrasoundtocreateahighlydirectionalbeamofaudiblesoundanalogoustoaashlight. Thetransmitterwasformedbyacircularcompositediaphragmthatwasradiallynon-uniform.Thediaphragmconsistedofmolybdenumannularelectrodesandanaluminumnitridediaphragm.Thecompositediaphragmwasreleasedusingacombinationofdeepreactiveionandoxideetching. Transductionofthediaphragmoccurredwhenanelectriceldwasinducedacrosstheannularpiezoelectriclayerofaluminumnitride.Theelectriceldcausedmechanicalstrainwithinthepiezoelectriclayerthroughthepiezoelectriceect.Thestraincoupledintoforceandmomentresultantsthatgenerateddiaphragmdeection.Bysupplyinganacvoltagebe-tweentheelectrodes,anoscillatingelectriceldcauseddiaphragmvibration.Thevibratingdiaphragminturngeneratedacousticwaves. Theoveralldevicemodelwasformedusingcompositeplatemechanics,lumpedelementmodeling(LEM),andacoustictheory.ThroughLEManequivalentcircuitofthedevicewasformedthatincorporatedelectrical,mechanical,andacousticcomponents.Nonlinear 17

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TheoptimaldesignbasedontheLEMwasusedtoformtheprimarydesign.Aseriesofsecondarydesignswereformedbyconstrainingthedevicelayerthicknessesandperformingoptimizationusingradialgeometryasdesignvariablesandconsideringdeviationsinstress.ThedevicewasfabricatedatAvagoTechnologiesLimited'sfoundry.Auniquepackagewithbackcavitydepthcontrolwasdesignedandfabricated.Thiswasfollowedbyexperimentalcharacterization.Electricalcharacterizationconsistedofmeasurementofdeviceimpedance.Mechanicalcharacterizationincludedmodeshapeandresonantfrequencymeasurements.Acousticcharacterizationencompassedfareldacousticmeasurementsofasingledevice. Thecharacterizationresultsshowedsignicantmismatchbetweendevicesaswellastheequivalentcircuitmodel.Fouroutofthesixdevicestestedhadresonantfrequenciesnear60kHz.Theremainingtwodeviceshadresonantfrequenciesof31kHzand44kHz.Theequivalentcircuitpredictedaresonanceof39kHz.Thevariationbetweentheresultswasattributedtostressvariationsacrossthewaferthatoccurredduringfabrication.Avariablebackcavitywasusedtotunethedevicesandmaximizesensitivityatresonance.A220%improvementintheresonantdeectionsensitivityofthe31kHzresonantdevicewasfoundbytuningthebackcavity.Anonlinearacousticcalculationwasusedtoprojecttheperformanceofa150mmdiameterdevicearrayasanacousticsourceforaparametricarray.Thesoundpressurelevelofa5kHzaudibletonewas42dBat1m.Thelowprojectedaudibleoutputdoesnotshowgoodpromiseoftheapplicationofthisdesigntoaconventionalaudibleparametricarray.Recommendationsforfutureworkfocusonamorerobustdevicedesignandpackagingimprovementsforotherultrasonicapplications. 18

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Thegoalofthisresearcheortwastodevelopamicroelectromechanicalsystems(MEMS)basedultrasonictransmitterthatwasdesignedtoserveasanelementofaparametricacous-ticarraysoundsourcetransducer.Theparametricarrayisanacoustictechnologyanalogoustoaashlight.Inaparametricarray,thedistortionofhighamplitudeultrasoundcreatesahighlydirectionalbeamofaudiblesound,oran\AudioSpotlight"[1].Thesoundbeamissimilartoaashlightinthatobjectsareonly\illuminated"withsoundiftheyareinthebeampath. Thischapterbeginswithanintroductiontoparametricarrays.Next,generalissuesandlimitationsassociatedwithexistingparametricarraysoundsourcetransducersarediscussed,leadingtothemotivationforthedevelopmentofaMEMS-basedtransducer.Finally,theresearchobjectivesandoveralldissertationorganizationaregiven. Thecontrolofsounddistributionisusefulinenvironmentswhereaudiblecommuni-cationisneededbutdisturbanceofbystandersisundesirable.AnexampleisamuseumdisplaysimilartoFigure 1-1 whereanobserverofanexhibitlistenstoarecordingonthenerpointsofthedisplaywhiledisinterestedpedestrianspassbyundisturbed[3].Thefol-lowingareotherpossibleapplicationsofparametricarrays:longdistancecommunicationsimilartobullhorns[4],human-humanoidcommunication[5],drivethroughcommunica-tionsperceptibleonlytothecustomer,nondestructiveevaluation[6],activenoisecontrolsystems[7],virtualheadphonesthatprovideaprivateaudioexperiencewithoutphysicalear 19

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Figure1-1. Observerofamuseumdisplayislisteningtocommentarywhiledisinterestedpedestrianspassbyinquiet(adaptedfromHolosonics[3]). Theabilityofasinglesourcetocontrolthedistributionofsounddependsonthesizeofthesourceincomparisontothewavelengthoftheprojectedsound[8].AsourcewhosesizeissmallrelativetotheprojectedwavelengthwillcreateanomnidirectionalsoundeldthatcanbeheardequallyinalldirectionsasshowninFigure 1-2 .Forexample,aconventionalaudiosystemusesasinglesub-woofertogeneratebassfrequenciesontheorderof100Hzorlowerwherethecorrespondingwavelengthsareontheorderofmetersorlarger[9].Sincethewavelengthsaremuchlargerthanthesub-woofer,anomnidirectionalsoundeldisgenerated. Incontrast,aconventionalaudiosystemusesmultipletweeterstogeneratefrequenciesontheorderof1kHzorhigherwherethewavelengthsareontheorderof10centimetersorless[9].Atthesefrequenciesthetweeterisnolongermuchsmallerthanthewavelengths,resultinginasoundeldthatisnolongeromnidirectional.Thus,conventionalaudiosystemsusemultipletweetersthatcombinetocreateamoreuniformsounddistribution[9]. 20

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3 ).Ontheotherhand,theprojectionofultrasoundinairwithwavelengthsmeasuredinmillimetersbyaconventionallysizedsourcecanproduceabeam.Ultrasoundisdenedassoundwhosefrequencyrangeisgreaterthantheupperlimitofthehumanhearingrange,20kHz[9].Forinstance,aspeaker70mmindiameterthatgeneratesultrasoundatapproximately40kHzhasa-6dBbeamwidthof10degrees. Figure1-2. Soundeldsofaconventionallysizedradiatorataudioandultrasonicfrequencies(adaptedfromHolosonics[3]). Ultrasound,though,isnotperceptibletothehumanear,butitispossibletocreateaudiblesoundfromthedistortionofultrasound.Highintensityultrasounddistortsasitpropagatesduetoconvectiveandlocalheatingeects[8].Thedistortedwavecanbede-scribedbyaFourierseriesthatconsistsofthesummationofmultipleharmoniccomponents. 21

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2 Parametricarraytransducersusuallyoperateinthelowerultrasonicrangebecausesoundabsorptioninairincreaseswithfrequency[1,10{12].Operatingatlowerultrasonicfrequenciesmaximizestheamountofpowerthatisconvertedfromthesourcetotheaudiofrequencies.However,itisimportantforparametricarraystostayawayfromtheupperlimitofhumanhearingfrequencyrangeasasafetyprecaution.Mostparametricarraysoperatearound40kHzasatradeobetweenperformanceandsafety[1,10]. Therearespecicdesirableattributesofair-coupledultrasonictransducersforpara-metricarraysources.Themostimportantattributeistohavealargesoundoutputatthefundamentaltones.Alargesoundoutputincreasestheamountofenergyavailableforcon-versiontotheaudibledierencetone.Inaddition,alargenumberofradiatorsincreasestheoverallsoundoutputofthearray.Moretransducersperarea,knownaspackingdensity, 22

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2 [13].Thus,itisimportanttohavehighemissioneciencywithrespecttotheinputelectricalenergytoensurethatthemajorityoftheenergylostisintheultrasonictosonicconversion.Atransducerwithanadjustableresonantfrequencyisdesirablesothatthesametransducercanachievemaximumsoundoutputatmultipleultrasonictones.Forinstance,halfofthetransducersinanarraycouldemitatoneresonanttonewhiletheotherhalfemitatanotherresonanttone.Also,itisdiculttofabricateandpackageadevicewitharesonantfrequencythatexactlymatchesthemodelprediction.Thus,itisbenecialtohavetheabilitytoadjustthetransducer'sresonantfrequencytomatchoriginallydesignedperformance.Itisidealforthetransducertohavethefullhumanhearingrangebandwidthaboutitsresonantpeak. 2 atthisjunction.Suchadesignwouldhavetomeetuniquechallenges: 23

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1 presentsthebackground,motivation,andgoalsoftheproject.Chapter 2 discussesnonlinearacousticsincludingtheoryandimplementationsofparametricarrays.Chapter 3 reviewsdierenttypesofelectroacoustictransducerswithspecicattentionpaidtoultrasonic,air-coupledMEMS.Chapter 4 givesanoverviewofthedevicedesignandfabrication.Chapter 5 presentsthedevicemodeling.Chapter 6 summarizestheoptimizationmethodology.TheexperimentalsetupandresultsarepresentedinChapter 7 .Chapter 8 discussestheconclusionsandfuturework. 24

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Theeldofuiddynamicsdescribesuidparticlemotionanditsinteractionwithsolidbodies[17].Acompleteuiddynamicelddescriptionincludesthree-dimensional,thermo-viscous,andunsteadyeects.Alinearacousticdescriptionisderivedbyperturbingtheuideldvariablesbyaninnitesimalamountandlinearizingthegoverningequationsofcompressibleuiddynamics.Linearacousticsissuccessfulindescribingcommonacousticphenomenawherethesmall-signalassumptionisvalid[8].Thescienceofnonlinearacousticsformsabridgebetweenthefullviscouscompressibleowmechanicsdescriptionandlinearacoustics.Althoughthestudyofnonlinearacousticsistheanalysisandmeasurementofniteacousticperturbations,nonlinearacoustictheoryneglectsinsignicanteectsofafulluidmechanicsmodelwhenappropriate. Thischapterpresentsfundamentalsofnonlinearacoustictheory.Anexampleofnitewavepropagationisdemonstratedusingplaneacousticwaves.Thenonlinearacousticequa-tionsmostapplicabletotheanalysisofparametricacousticarraysareintroduced.Solutionstotheseequationsforsoundbeamsarepresented.Finally,parametricarrayimplementationsfoundintheliteraturearesummarized. Thecontinuityequationisgivenby[17] Dt+~r~V=0;(2{1) whereisthemassdensity,~Vistheuidvelocityvector,andD()/DtisthematerialderivativegivenbyD()/Dt=@()/@t+~V~r(). 25

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Dt+~rP=r2~V+B+1 3~r~r~V;(2{2) wherePisthethermodynamicpressure,r2istheLaplacianoperator,istheshearviscos-ity,andBisthebulkviscosity.Theshearviscosityrelatesthevelocitygradienttotheshearstressandisameasureofauid'sabilitytotransfermomentumtoparticlesperpendiculartothemomentumdirection.Thebulkviscosityaccountsfordierencesbetweenthethermo-dynamicpressureandthesumofallnormalstresses[18].Themomentumequationusingthebulkviscosityformulationisanapproximationthatisvalidatlowfrequencieswherethetimerequiredtoreachthermodynamicequilibriumismuchsmallerthantheperiodofoscillation.Athigherfrequencies,thebulkviscositymodelbreaksdownduetonon-equilibriumeects.Thechangesinthethermodynamicstateoccursorapidlythatthermodynamicequilibriumisnotreachedduringeachcycle.Theresultisanenergylosscalledmolecularrelaxationingases[17].Inair,molecularrelaxationisdominatedbythetimeittakesthevibrationenergymodeofnitrogen,100ms,andoxygen,1ms,toreachthermodynamicequilibrium[8].AswillbeshowninSection 3.2.3 ofChapter 3 ,theacousticattenuationofa40kHztoneis-1.29dBpermeter,ofwhichthermoviscouseectsaccountfor-0.26dBpermeter,relaxationfromnitrogenvibrationaccountsfor-0.005dBpermeter,andrelaxationfromoxygenvibrationaccountsfor-1.03dBpermeter. Ifthebulkviscositymodelholds,thentheappropriateentropyformulationoftheenergyequationis[18] Dt=r2T+B~r~V2+$"1 3$~r~V;(2{3) wheresisspecicentropy,$"isthestrain-ratetensorgiveninindicialnotationas"ij=1 2@ui Theequationofstateforasingle-phaseuidinthermodynamicequilibriumstatesthatanythermodynamicvariablecanbedescribedasafunctionoftwootherthermodynamic 26

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Thelinear,losslessacousticwaveequationsforhomogeneous,initiallyquiescentmediaarederivedwhentheperturbationsoftheuidareassumedtobeverysmallwithrespecttotheambientconditions.Sincetheambientpressureismanyordersofmagnitudegreaterthancommonacousticlevels,thechangeinthethermodynamicpressureisassumedtobesmallincomparisontotheisentropicbulkmodulus,0c20,suchthat wherec0isthesmall-signal,isentropicsoundspeeddenedby Themediumisassumedtobeinitiallyquiescent.Thus,theparticlevelocitymagnitudeisassumedsmallincomparisontotheisentropicspeedofsound, Theratioofthevelocityamplitudetotheisentropicspeedofsound,~V0.c0,isalsoknownastheacousticMachnumber.Inadditiontothesmallperturbationsassumptions,thespeedofsoundisassumedtobethesmall-signal,isentropicspeedofsound, Forlinear,losslesspropagationofacousticwavesinhomogeneousmedia,thegoverningequationintermsofvelocityperturbationis[8] Furtherdetailsonlinear,lossless,acousticmotioncanbefoundinthetextbyBlackstock[8]. 27

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2{5 2{7 ,and 2{8 areviolatedandthelinearacousticdescriptionbreaksdown.Therearetwophysicaleectsthatsimultane-ouslycontributetononlinearacousticpropagation:convectionandlocalheating[8]. Theconvectiveeectoccursathighsoundpressurelevels(SPL)whentheparticleveloc-itybecomessignicantincomparisontothespeedofsound.Forexample,thepropagationspeedofconstantphase,cph,ofaonedimensionalwavebecomesthesoundspeed,c(T),plustheone-dimensionalparticlevelocity,u0, wherec(T)=p Thelocalheatingeectistheresultofthespeedofsound'sdependenceupontem-perature.Largeamplitudewavescompressandexpandthegas,creatinganon-negligibletemperaturechangeandthusachangeinthespeedofsound.Itcanbeshownthattherstorderapproximationofthespeedofsoundforlossless,one-dimensionalniteamplitudeplanewavesis[18] whereisthecoecientofnonlinearityofagasgivenby[18], 2;(2{12) and=cp AsobservedinEquation 2{10 ,thepropagationspeedofaconstantphaseisdependentuponthemagnitudeoftheparticlevelocity.Givenu0astheamplitudeofthevelocity,thephasecorrespondingtothepeakofthesinewaveshowninFigure 2-1 willpropagateatcjpeak=c0+u0,whilethephasecorrespondingtothetroughpropagatesatcjtrough=c0u0.ThiswillleadtosteepeningofthewaveasshowninFigure 2-1 .Asevidentinthegure,eventuallythepeakofthewavewillcatchuptothetroughcausingthewavetoformashockasshownincase(d).Thermoviscousdissipationacrosstheshockpreventsthepeak 28

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Figure2-1. Wavesteepeningduetononlinearwavepropagation.AdaptedfromBlackstock[8]. Thedistortedwavesincases(b),(c),and(d)ofFigure 2-1 canbedescribedmathemat-icallywithaFourierserieswhosefundamentalfrequencyistheoriginatingfrequencyshownincase(a), Thus,asthewavepropagates,powerredistributesfromthefundamentalfrequencytoitsharmonics.Figure 2-2 showsthepowerdistributionbetweenthefrequenciesatdierentdistancesalongthepropagationpath.Noticethatasthewavetravels,thepoweroftheharmonicshasincreasedand,correspondingly,thepowerofthefundamentalhasdecreased.Thus,ifasourceproducesahighamplitudetone,f0,thespectrummeasuredatanitedistancefromthesourcewillalsoincludeitsharmonics(2f0;3f0;4f0;etc:)[18]. Inpractice,nonlineareectsareperceptibleonalogscaleatSPLswhereassumptions 2{5 through 2{8 arestillrelativelyvalid.Forexample,waveformsinairofSPLsabove120dBdistortenoughtoproducenoticeableharmonics.At120dBinairtheacousticMachnumberisapproximately~V0.c0=0:06foraplanewave.AlthoughtheacousticMachnumbermayseemsmallwithrespecttounity,wavedistortionissignicantenoughtoproduceharmonicsonalogscale.Forinstance,anexperimentalarraythatprojected128dB(distancenotgiven)at25and30kHzproduced100dBoftheaudibledierencefrequency,5kHz,at4m[19]. 29

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Thefrequencydistributionofthepowerattheorigin. (b) Thepowerinthesinusoidalacousticwaveredistributestoitsharmonicsasitpropagates. Thefrequencydistributionofanonlinearlypropagatingwave. 2-3 comparesthe"spotlight"[1]producedbyaparametricarraywiththedirectivityofaconventionaltransducer.Also,thefrequencycontentoftheparametricarrayatthesourceandatanitedistancefromthesourceisshown.Noticethatalongthepropagationpath,harmonicsareproducedattheexpenseofthepowerofthefundamentalfrequencies.Thegenerationofthedierencetoneisthefundamentalmechanismbehindparametricarrays. 30

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Parametricarrayradiationofsound(adaptedfromHolosonics)[3]. 2.2.2 31

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@t2c20@2 @xi@xi=@2 ThisisLighthill'sexactequationofuidmotionwherethestresstensorinEquation 2{15 actsasasourcetermfortherestoftheotherwisequiescentmedia[23].TherstterminEquation 2{15 accountsfordeviationsfromisentropicbehavior,thesecondtermistheReynold'sturbulentstress,andthethirdtermistheviscousstresstensor.InWestervelt'soriginalderivation,theviscousstresstensorisignored. ManipulationofEquation 2{14 resultsinWestervelt'ssecondorderwaveequation.ToarriveattheWesterveltequation,eachdependentvariableisperturbedaboutanominalvalue,thesecond-orderTaylorseriesexpansionofpressurewithrespecttodensityissubsti-tuted,andthelinearacousticequationsareback-substitutedtoarriveatanequationwithasingledependentvariable.Theresultingequationis[20] 0c40@2p02 wheretheprimereferstotheperturbationaboutthenominalvalue,isthenonlinearcoecientequalto+1 2,and2=@2 2{16 areincludedtheresultingequationis[18], c40@3p0 0c40@2p02 whereisthediusivityofsoundgivenby 3+B+ 01 32

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2{18 TheWesterveltequationissecondorderinthesmallparameter~"thatrepresentsthemagnitudeofboththeacousticMachnumber,"=u/c0,and=!/0c20.Thesmallparametermeasuresthebalancebetweenviscousshearstressandpressureuctuations.Divisionofthenumeratoranddenominatorofby!2resultsin 0!k2(2{19) wherek=!/c0isthewavenumber.Equation 2{19 isthesquareoftheratioofanoscillatorydiusionlengthscaletothewavelength. TheWesterveltequationalsoignorestheLagrangiandensitydenedby[18] 20u2p02 TheLagrangiandensityisthedierencebetweenthekineticandpotentialenergydensitiesofthewaveandiszeroforprogressiveplanewavesinaquiescentmedium[24]. TheWesterveltEquation 2{17 continuestobevalidtosecondorderfornon-planewaveswhenthecumulativeeectsdominatethelocaleects.Cumulativeeectsaresignicantasthewavepropagates,whereaslocaleectsareignoredafterawavelength.Theconvectiveandlocalheatingeectsareexamplesofcumulativeeects.Examplesoflocaleectsincludelinearizingavibrationtoaxedsurfaceataboundaryconditionorthesubstitutionoflinearacousticrelationsintothenonlinearequations[18].Afterawavelength,theerrorsintheseapproximationsaresmallincomparisontotheeectofcumulativewavedistortion.Localeectsbecomeimportantincompoundwaveeldssuchasstandingwavesandintheradiationpressure[18]. FurtherassumptionsleadtoasubsetoftheWesterveltequation,theKZKequation,thatissuitedfordirectionalsoundbeams. 33

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2{17 ,bytransformingthespatialscalessothattheeectsofdirac-tion,absorption,andnonlinearityarealloforder~"2andallotherhigherordertermsaredisregarded.ThefollowingscalesareintroducedintotheWesterveltequation: wherexandyareintheplaneoftheacousticsource,zisalongthepropagationdirection,andistheretardedtime.TheLaplacianintheD'AlembertoperatorinEquation 2{17 becomes IntroducingthisandthenewscalesintotheWesterveltEquation 2{17 whiledroppingthehigherordertermsgives ~"@2 @z1@+ c40@3p @3= 0c40@2p2 Rewritingtheequationintermsofx,y,and,theresultingKZKequationis @z@c0 {z }Diraction @3| {z }Absorption= {z }Nonlinearity;(2{24) wherer2?=@2 2{24 accountsforchangesintheplaneperpendiculartothez-axisofpropagation.Thus,torstorderin~",thisequationmodelsdirectional,quasi-planarbeamsandaccountsfordiraction,absorption,andnonlinearityeects[18]. 34

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Figure2-4. Thesoundeldandlengthscalesassociatedwiththesoundbeamsolutions.Notethatthisgureishighlyschematic. Beforediscussingsoundbeamsolutionsforparametricarrays,itisimportanttodiscussthescalesusedintheseanalysistomakeassumptionsthatallowananalyticalsolutiontobeformed.First,asshowninFigure 2-4 theprimarysoundeldisassumedtobeacollimatedwaveofradiusathatexperiencesattenuationduetoacousticabsorptionintheabsenceofsphericalspreading.Theprimaryeldsinthenexttwosoundbeamsolutionstakeontheform wherep0isthepressuregivenbyp0=0c0u0,0istheabsorptionattheaverageprimaryfrequency,andtheHeavisidefunctionisdenedby 2x=01x>0:(2{26) 35

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2ka2(fordetailsonneareldandfareldseeSection 3.2.1 ),thatthereisnolongerenoughpowertocontributesignicantlytotheproductionofthedierencefrequency.Theeectivelengthoftheparametricarray,alsoknownastheabsorptionlength,isgivenby 20:(2{27) Theconsequenceofthisassumptionisthatsphericalspreadingcanbeignoredsincenonlinearproductioninthefareldisnegligibletothatintheneareld.Iftheprimarysoundeldisstillsignicantinthefareld,thisassumptionwillbeviolatedandthefollowingsolutionswilloverpredictthenonlineardemodulatedeldsincetheprimarysoundeldsinthefareldwillnotexperienceattenuationduetosphericalspreading.AnotherassumptionasshowninFigure 2-4 isthatthedistanceofthemeasurementpointfromthesourceoftheparametricarray,z0,ismuchlargerthantheregionofnonlinearproduction.Thisassumptionisvalidaslongasz0La.Ifthisassumptionisviolated,theamplitudeofthedierencefrequencyatthepointz0wouldbeover-predictedsincebothsolutionsconsidertheentireprimaryeldouttoinnityasthesourceofthedierencefrequency.WithoutgoingtonumericalsolutionsoftheKZKequationthatrequirelargeresourceallocation[18],theanalyticalsoundbeamsolutionsaregenerallyusedinthisdissertationforrstorderapproximationsofthedemodulatedsoundeld. 2{16 issolvedforthepressureatthedierencefrequencyoftwoadjacenttonesproducedbyaradiatingcircularpiston.Theassumedpressureeldoftheprimarytonesarep1a(r;z)=p0aH(ar)eaz 36

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3 ). Theresultingpressureofthedierencefrequencyasafunctionofthedistancealongtheaxisnormaltotheradiator,z,andtheanglewithrespecttothenormal,,is z 2k ztan2);(2{30) wheretheminussubscriptindicatespropertiesthatcorrespondtothedierencefrequency,p0aandp0baretheoriginatingpressuresofthetwofundamentalfrequenciesgivenbyp0=0c0u0,u0isthepistonvelocity,aisthepistonradius,0andaretheabsorptionco-ecientsoftheoriginaltonesandthedierencetone[8],respectively,andtheWesterveltdirectivityandaperturefactoraregivenbyDW()=1 1+j(k atan) atan;respectively. (2{32) wherep0isthepressuregivenbyp0=0c0u0.Theamplitudeandphasemodulation(E(t)and'(t);respectively)areassumedtobeslowlyvaryingfunctionsoftimeincompar-isontothecarrierfrequency,!0.AderivationofBerktay'ssolutionisfoundinAppendix A .Theaudiblepressuresolutionis[18] 37

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Thus,Berktay'ssolutionpredictsa40dBperdecaderoll-oaslowermodulatingfrequenciesareapproached. 2.2.4 38

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AJapaneseresearchgroupassociatedwithNagoyaUniversityandtheUniversityofElectro-communications,Tokyo,headedbyYoneyamaandKamakurabeganthenextmajoreortin1983.Theirrstpaperwasalsotherstworkwherepracticalaudiooutputwasobtained[1].Thiseort,aptlynamedthe"AudioSpotlight,"employedanarrayof547PZTbimorphemitters(overallarrayandemittersizeomitted)arrangedinahexagonandoperatingwitharesonanceof40kHzandsecondarypeakjustbelow60kHz.Theywereabletoobtainanarrowbeamwithadierencefrequencyat1kHzofabout80dBat4musingamplitudemodulationofthesourcesignal.TheyusedBerktay'ssolutiontorecognizethecorrelationbetweentheaudiosignalandthemodulatingfunction.AspredictedbyBerktay'ssolution,experimentsshowedthedependenceoftheaudiosignalamplitudeontheaudiofrequencysquaredfromabout200Hzto2kHz.Forparametricgenerationofaudiofrequenciesabove2kHz,theresponseofthetransducerabove40kHzbegantodeclinesuchthattheSPLoutputoftheprimarysignalswasmuchlower.Theyalsorecognizedthatmodulatingthecarriersignalwithadcosetsinusoidgeneratedharmonicdistortionoftheaudiosignal.ThiscaneasilybeseenbyemployingamodulatingfunctionE(t)=1+mg(t)intheBertkaysolution,wheremisthemodulationindexandg(t)istheaudiosignal.Theresultingdemodulatedsignalis {z }ps+p20a2m2 {z }pd;(2{36) wherepsisthedesiredpressuresignalandpdisharmonicdistortion.AspredictedbytheBerktaysolution,ifg(t)isaharmonicfunction,thentheaudiooutputhasa40dB/decade 39

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ThepaperbyYoneyamaetal.[1]spurredonfurthereortsintheJapanesecommunitytoeliminatetheharmonicdistortionoftheaudiosignalproducedwhenusingthemodulatingfunction.Kamakura[28]presentedresultsfromarectangulararrayof581PZTbimorphtransducersoperatingat40kHzwithaneectivearrayradiusof15cm.Theywereabletoobtainapproximately68dBof1kHzdierencefrequencyat9.5m. Aokietal.[32]showedgoodagreementbetweennumericalsolutionsoftheKZKequationandexperiments.Thearrayconsistedof1410piezoelectrictransducersof1cmdiameterandaresonanceat28kHz.Thearrayhadaradiusof21cm,andtheprimarytoneswere27and30kHz.Givenasourcelevelof112dB,theyobtainedapproximately65dBof3kHzdierencefrequencyat3m.Atasourcelevelof133dB,theyobtainapproximately103dBofthe3kHzdierencefrequencyat3m. InthenextpaperbyKamakuraetal.[30],thesquarerootoftheentiremodulatingfunctionwastakentoeliminateharmonicdistortionoftheaudiosignal.Theyalsointroducedalowfrequencyenvelopefunction,e(t),inaneorttoreducethepoweroutputoftheparametricarray.ThemodulatingfunctioninrelationtotheBerktaysolution[25]forthiscaseis wheres(t)isthedesiredaudiosignal.TheFouriertransformofEquation 2{37 consistsofaninnitenumberoffrequenciesduetothesquarerootfunction.Thusthetransducerrequiredalargebandwidtharounditsresonantpeaktobeabletoaccuratelyreplicatefunction 2{37 .Theywereabletoshowadecreaseintheharmonicdistortionoftheaudiosignalaswellasa36%decreaseinthepowerrequirementwithrespecttodoublesidedmodulationwhilemaintainingtonequality.Theirexperimentalimplementationuseda44by50cm2arrayof2000PZTbimorphswitha28kHzresonance. 40

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Next,Kamakuraetal.[19]conductedexperimentsonarectangulararrayandcomparedtheirresultstonumericalsimulations.NumericalsimulationsbasedontheKZKequationwereconductedwithbothrectangularandcircularsourcefacesandbothshowedexcellentagreementwiththeexperimentaldata.Thearrayconsistedof1102piezoelectricbimorphsina24by44cm2rectangle.Theyradiatedat25and30kHzat116dBat0.7mfromthesource.Theyobtainedabout80dBof5kHzat4mwiththisarrangement.Radiatingat128dB(positionnotgiven)oftheprimaryfrequenciestheyobtainedapproximately103dBat4mofthedierencefrequency. Pompei[12]presentedthenextimplementationoftheparametricarrayinair.Hispaperdiscussedsolutionsforsignalprocessinganddierentmodulationtechniquesthatemphasizedthedierencefrequencyincomparisontoitsharmonicdistortion.Thesetupusedanarray(35cmindiameter)ofwidebanddevicesgeneratingwithacarrierfrequencyof60kHz.Pompeirecognizedthattogetaatfrequencyresponseoftheaudiosignal,theparametricarraymustcontendwiththe40db/decadeslopewithincreasingfrequencythattheBerktaysolutionpredicts.Heclaimedthatusingthesquarerootofoneplusthedoubleintegralofthedesiredaudiosignal, asthemodulatingfunctionwouldcounteractthe40dB/decadeslope.Themodulatingfunction 2{38 hadaproblemsimilartotheequalizerproposedbyYoneyamaetal.[1].Thetransducerrequiredalargebandwidthtoproduceenoughfrequencycomponentsto 41

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2{38 .Instead,Pompeiutilizedtheroll-oofhistransducerstocompensatefortheproblemofthe40dB/decadesloperesultinginafairlyataudiospectrum.Therefore,equalizationbydoubleintegrationwasunnecessaryinthepreprocessing.Thedesignofthesetransducersisnotpresented.Theresultsshowedsignicantreductionintotalharmonicdistortionwithjustthesquarerootpre-processing.Thestudyobtainedabout77dBof1kHzdierencefrequencyat3m. HavelockandBrammer[35]madeacomparisonoftwodirectionalsourcesusingthesamecompressiondrivers;aperforatedpipeandaparametricarray.Theyusedfour50mmdiameterpiezoelectricdisksinanarraywithanoverallarraydiameterof170mm.Thesourcelevelfortheparametricarraywas134dBat0.25mand28kHz.At4mtheywereabletomeasure50dBat1kHzand40dBat300Hz.Inthe2-5kHzrangeofthedierencefrequency,theywereabletoshowthefrequencysquareddependencepredictedbyBerktay'ssolution. Inadditiontosoundreproductionapplications,parametricarrayshavealsobeenusedfornondestructiveevaluation(NDE).In2000,Kaduchaketal.usedaparametricarraytoexciteresonancesofelastic,uidlledcontainerstodeterminetheuidtype[6].Theparametricarrayconsistedof48o-the-shelfair-coupledpiezoceramictransducers(AirMarmodelnumberAT200)15.8mmindiameterwitharesonancearound200kHz.Thetrans-ducerswerearrangedinarectangulararraywithcentertocenterspacingof19mm.Theauthorsmeasuredthecarrierfrequencyof217kHzat0.5mtobe115dB.Theyalsoreportedadierencefrequencymeasurementof85dBat3m,butdidnotreporttheactualdierencefrequency.Theauthorsshowedgoodagreementbetweenabeamwidthmeasurementat3.9kHzandtheWesterveltdirectivity(soundpressurelevelsarenormalized). ATCproducedawhitepaperthatcontainedaliteraturereviewofparametricarrayimplementationsinairaswellasabriefdescriptionoftheirowneortsatATC[11].Heretheydescribedsomeoftheirsignalprocessingsolutionsaswellasamonolithiclmultra-sonictransducer.Theyintroducedsinglesideband(SSB)amplitudemodulationashaving 42

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2-5 .Aconewasincorporatedwiththepiezoelectricbimorphtoimprovevolumevelocitybyessentiallycreatingapistonthatvibratesatthecentervelocity.Aquar-terwavelengthbackcavityprovidedzerocavityimpedanceatthecarrierfrequency.Theywereabletoget123dBfromaNiceraAT40-12Ptransducerat30cm(frequencynotlisted).Akeypointofthepaperwasthatanarraywithalargerareabutsmalleroutputperareacanoutperformasmallerarraywithlargeroutputperarea.TheBerktaysolutionisdirectlyproportionaltoarea.Togetthesameamplitudeasalargearray,aconsiderableamountofpowermustbesuppliedtoasmallerarray.Iftheamplitudesofthefundamentaltonesarelargeenough,thewaveswillformshocksbeforesucientparametricconversioncantakeplace.Oncethewavesshock,dissipationacrosstheshockdominatesandabsorbsthepower 43

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Figure2-5. ATCbimorphpiezoelectricactuatorwithattachedconeforenhancedvolumevelocity[11]. Moonetal.[36]constructedaexural-typetransducerutilizingpiezoelectricactuation.ThePZTlayerhadadiameterofabout9mm.Theprimaryfrequenciesusedintheexperi-mentswere42.24and43.25kHz.Theymountedtheirtransducersatanglestoeachotherinanattempttoavoidsidelobes.Allexperimentalmeasurementsweremadeat120mmfromthearray.Allmeasuredamplitudesgivenwerenormalizedsoanabsolutemeasurementoftheparametricconversionisnotavailable[36]. RohandMoon[14]gaveresultsfromanarrayof60thicknessmodepiezoceramictrans-ducersoperatingat650kHz.Theactuatorshadadiameterof30mmwithaPZT-4thickness 44

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Anotherpaperfocusedonpre-processingwaswrittenbyYoungandSung[37].Theymitigatedtheaudiosignaldistortionresultingfromthesquarerootoftheequalizedmodu-lationsignalbymanipulatingthephaseandamplitudeofthefrequencycomponentsofthesquarerootedsignals.TheymeasuredtheSPLoutputandthenchangedtheamplitudeandphaseuntiltheTHDwasminimized.Theyoperatedatacarrierfrequencyof40kHz.Theprimarywaveswereabout120dBat2mfromthearray.TheyclaimedsignicantreductionintheTHDwithrespecttodoublesidedmodulation.TheydidnotprovidequantitativeSPLlevelsofthedierencefrequency. ARussiangroupreportedexperimentalresultswiththegoalofusingtheparametricarrayinairforatmosphericsounding[38].Theygaveresultsfrommultiplearraytypes:asinglecapacitivemembraneradiator,anarrayofbimorphpiezoelectricradiators,andafocusedarrayofradiators.Thecapacitiveradiatorwasalargemetallizedpolymermembranestretchedoveraconductingsurfacethathasbeenroughened.Theroughnesscreatedairpocketsoverwhichthemembranevibrates.Theywereabletomeasureademodulatedsignal 45

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Parametricarrayshavealsobeenevaluatedasaninstrumentforanactivenoisecontrolsystem[7].Brooksetal.measuredultrasonicanddemodulatedsignalsfromtheATCparametricarray.At1m,theymeasureda1kHzdemodulatedsignalofabout90dBfroma48kHzcarrierfrequency. In2006,Kamakuraetal.presentedaconferencepaperthatfocusedonincreasingthepowereciencyofparametricarrays[39].Theyreporteda12.5by25cm2arrayof286monomorphceramictransducers.ThetransducersareNiponCeramicCo.,Ltd,TypeAT/R40-10andare10mmindiameter.Theyobtained120dBofthe39.3kHzcarrierfrequencyat3m. Otherresearchershavelookedatusingparametricarraysincellphonestodecreasediusivityofaudiblesound[40].In2006,Nakashimaetal.mounted16piezoelectrictrans-ducersonacellphonesizeddevice.Theywereabletoobtain134dBat1mofthe40kHzcarrierfrequency.Thisresultedina65dBdierencefrequencyof1kHzat1m. In2007,Chenetal.[41]presentedaconferencepaperthatidentiedtheproblemoftheinnitebandwidththatwasrequiredforthetransducertoproduceasquarerootmodulatingfunctionsimilartoEquation 2{37 ,wheree(t)=1ands(t)wasasimplesinusoid.Theyconductedexperimentsusingrst,second,andthirdorderMaclaurinseriesapproximationsofthesquarerootmodulatingfunction.Theexperimentsshowedadecreaseinthetotalharmonicdistortionforthesecondandthirdorderseriesapproximations.Thearraywas 46

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Recently,Peifengetal.[42]presentedexperimentalresultsofaparametricarraylookingatapplicationsinvirtualrealitysystemsandhometheaters.Theyconductedexperimentswheretwoultrasonicsources,oneemittingat40kHzandtheotherat41kHz,intersectedorthogonallyat1m.Eacharrayconsistedof91o-the-shelfpiezoelectrictransducersof8mmradiusand45degreeindividualbeamangles.Theoutputofasinglearraywas140dBofthe40kHzcarrierfrequencyat1m.Attheintersectionpoint,40dBofthe1kHzdierencefrequencywasmeasured. YingandChenetal.[43]presentedasimilarconferencepapertoChenetal.[41]thatinvestigatedtruncatedsquarerootmodulatingfunctionswithadierenttransducer.Inthiscase,atransducermadeofPVDF(PolyvinylideneFluoride)wasusedtoproduceacarrierfrequencyof42kHz.Theyreported62dBat1m. In2006,Haksueetal.presentedamicromachinedparametricarraythatfocusedonrang-ingapplicationssuchasrobotics[44,45].Theapplicationrequiredaverynarrowbeamwidthfromasmallactuator.Thus,theauthorschoseanultrasonicdierencefrequencyof40kHzandapiezoelectricmicromachinedultrasonictransducer(pMUT)astheactuator.PMUTsarecoveredinmoredetailinSection 3.3.2 .ThepMUTwasfabricatedbydepositinga2.5mleadzirconatetitanate(PZT)layeronasilicon-on-insulator(SOI)wafersimilartotheworkofHorowitzetal.[46]coveredinSection 3.3.2.1 .Theauthorsfabricatedtwodiaphragmswithradiiof1200mand1420m,bothwith15mthicksiliconlayers.Theresonantfrequenciesofthetwodeviceswere135and95kHz,respectively.Thetwodiaphragmswerehexagonallyarrangedtoformadevicearray35by30mm2.At0.18m,theSPLofthe40kHzdierencefrequencywas85dB.Theexcitationsignalwas8.5voltspeak-to-peak. 47

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CMUTtransducerarrayspecicationsandperformancepresentedbyWygantetal.[15]. Membranediameter(mm) 4 4 Membranethickness(m) 40 60 Cavitydepth(m) 36 16 ACexcitation(Vpeak-to-peak) 200 200 DCbias(V) 380 350 Centerfrequency(kHz) 46 55 -3dBbandwidth(kHz) 2.0 5.4 Pressureat3m(dBre20Pa) 115 107 3.3.1 .ThefabricationofthesedeviceswassimilartothosepresentedbyErgunetal.[47],whichiscoveredinSection 3.3.1.1 .Thedevicewasformedbyanelectricallyconductivediaphragmoverabackcavityatvacuum.Twodierenttransducerdesignswerefabricated.Bothhad4mmdiameters,butdesignsAandBhaddiaphragmthicknessesof40mand60m,respectively.TheAandBdesignpropertiesareoutlinedinTable 2-1 .Thediaphragmshadlargestaticdeectionssincethebackcavitieswereatvacuum.Thestaticdeectionwasaveragedovertheareaofthediaphragmandwasreportedasanaver-ageof2m.Thetransducerperformanceincludingthelargestaticdeectionwasmodeledusingniteelementanalysis(FEA).Itwasnotreportedifanonlinearanalysiswasused.Thestinessofthediaphragmwasdominatedbythestaticdeectionduetoatmosphericpressureacrossthediaphragm[15].TheresonantfrequencyofdesignsAandBwere46and55kHz,respectively.Anentirearrayofdeviceswas8cmindiameter.Thepressureat3mwas115and107dBfortheAandBarrays,respectively.Thediaphragmswereactuatedwitha200Vpeak-to-peakACsignalwitha350-380Vbiasvoltage.Todemonstratetheparametricarrayeect,thedesignBarraywasexcitedat52kHz(100dB)and57kHz(110dB)resultingina58dB,5kHzdierencefrequencyat3m. 48

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Bennettetal.1975[10] 1 oil-lledhydrophone 18.6&23.6 110 5 50 0.3 Yoneyamaetal.1983[1] 547 PZTbimorph 40 1 80 4 Kamakuraetal.1984[28] 581 PZTbimorph 40 1 68 9.5 Aokietal.1991[32] 1410 27&30 112133 3 65103 3 Kamakuraetal.1991[30] 2000 PZTbimporph;4450cm2array 28 Aokietal.1994[34] 91 38.5&41.5 134.5 3 88 1 Kamakuraetal.1994[19] 1102 PZTbimporph24by44cm2array 25&30 116128 5 79100 Pompeietal.1999[12] 60 1 77 3 Havelocketal.2000[35] 5 134 1 50 0.25 Kaduchaketal.2000[6] 48 217 115 3 Croftetal.2001[11] 85 37.23 136.5 Moonetal.2002[36] 42.24&43.25 Rohetal.2002[14] 60 650 17.5y Kimetal.2002[48] 40 120 2 Vinogradovetal.2005[38] PA:49;FA:90 40 85z 286 39.3 120 3 Nakashimaetal.2006[40] 16 piezoelectrictransducers 40 134 1 65 0.5 Haksueetal.2006[44] 16 pMUT 95&135 40 85 0.18 Chenetal.2007[41] 300by300mm2array 40 2 70.58{72.65 1 Peifengetal.2007[42] 91 40 140 1 40x Yingetal.2007[43] PVDFtransducer 42 62 1 Wygantetal.2007[15] 52&57 100&110 5 58 3

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2.2.4 .Thebimorphtransducerswereotheshelfcomponentsandwerenotdesignedspecicallyasparametricarraysources.Recentworkfocusedoninno-vativetransducerdesignsthatwerespecictoparametricarrayssuchasthemicromachinedtransducerarraysofHaksueetal.[45]andWygantetal.[15].Thusfar,thepublishedpMUTdesignsofHaksueetal.havefocusedonrangingapplicationswherethedemodulatedsignalswereultrasonicandnotapplicabletosoundreproduction.Wygantetal.werethersttoproduceanaudibletoneusingamicromachinedparametricarray.Theprimarysoundlevels,however,wereweakincomparisontopreviousworks.Thisledtoalowaudiooutputof58dBevenatadierencefrequencyof5kHz.Also,thecapacitivetransductionschemeusedbyWygantetal.requiredlargeacvoltagesinadditiontoalargedcbiaswherethepeakvoltagesignalcouldbealmost600V[15].ThevoltagerequirementprecludedthecMUTarrayfromgeneralcommercialapplications.Inallcases,anoptimaltransducerdesignforparametricarrayapplicationsisnotfoundinthecurrentliterature. 4 ,thetransducerdesignispresented.InChapters 5 and 6 ,thenonlinearacoustictheoryofthecurrentchapteriscombinedwiththetransducerbasicsofChapter 3 toformanelectroacousticmodelandanobjectivefunctionforoptimization.Finally,theexperimentalresultsofChapter 7 areevaluatedforapplicabilitytoparametricarraysusingthenonlinearacoustictheoryofthecurrentchapter. 50

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Thischapterpresentsanoverviewofair-coupledMEMSultrasonictransmitters.Thekeycharacteristicsofair-coupledtransmittersaredescribed.Next,themajorsensingandtransmittingmechanismsusedinmicromachinedultrasonictransducers(MUTs)arepre-sented.Finally,existingair-coupledMUTsarereviewed. 3-1 displaysarepresentationofagenericacoustictransmitter.Thetransmitteractsasanelectroacoustictransferfunctionthatshapesthefrequencyandphasecontentoftheradiatedacousticwave.Anelectricalsignalisappliedtotheacoustictransmitter.Thetransductionmechanismcausesamechanicalelementtodeect.Themechanicalelementimpartsmotiontoneighboringuidparticles.Thisuidparticlemotionconsistsoftwoparts:hydrodynamicuctuationandgascompression.Theenergyofthecompressedgaspropagatesintotheuidmediumintheformofanacousticwave. Figure3-1. Genericacoustictransmitter[49]. Acousticradiationinultrasonictransmittersisnormallyaccomplishedbyeitherlongi-tudinalbulkresonatorsorbendingmodedevices.ThevibrationofabulkmaterialshowninFigure 3-2 ispredominatelyusedinmacro-sizedtransmitters[50]. DiaphragmdeectionisusedpredominantlyinMUTs.ThetransductionmechanismcausesbendingofthediaphragmthatleadstooutofplanemotionasshowninFigure 3-3 51

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Bulkacoustictransmitter. Figure3-3. Genericbendingmodeacoustictransmitter[49]. Ifthetransmitterislinear,thenthepressureoutputatagivendistance,Pout(!),isrelatedtotheinputelectricalsignal,Vin(!),as whereHtrans(!)isafunctionoftheacousticfrequencyresponsefunctionofthetransmitterandtheacousticpropagationtothepointofinterest.Thefrequencyresponsefunctionisseparatedintoitsmagnitudeandphase, Themagnitudeofthefrequencyresponsefunction,jHtrans(!)j,isameasureofthetrans-missionsensitivity.Thephase,\Htrans(!),isthedelayofacousticpropagationplusthatofthetransmitterwithrespecttotheinputelectricalsignal. 52

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Frequencyresponseofanultrasonictransmitter. Transmittersaredesignedtohavelargesoundpressureleveloutputsatagivendistance.Inordertomaximizethesoundpressureleveloutput,ultrasonictransmitterstypicallyop-eratearoundtheirfundamentalresonantfrequency.Theresonantperformanceisdominatedbydamping.Arepresentativefrequencyresponsefunctionofanunder-dampedultrasonictransmitterisshowninFigure 3-4 .Thebandwidthisameasureofthewidthofthefunda-mentalresonantpeak.Itisoftendenedasthefrequencydierence-6dB,orquarterpower,belowthepeak[50].Thequalityfactor,Q,isameasureofthesharpnessoftheresonantpeak.Thequalityfactorisdenedas[51] where!risthedampedresonantfrequencyandH(0)isthefrequencyresponsefunctionatdc.Thequalityfactorofanunder-damped,secondordersystemwhere2<1/2isgiven 53

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2p whereisthedampingratio.Asthedampingdecreases,thequalityfactorincreasesandthepeakbecomessharper.Althoughthepeaksensitivityincreases,thereisatradeoinlossofbandwidth.Thehalf-powerbandwidth(-3dB),Br,ofanunder-dampedsecondordersystem,where0:1,isapproximately[51] Thedampingratioofmanymacroultrasonictransducersisdesignedtomediatethetradeobetweensensitivityandbandwidthtomeettheneedsoftheapplication. Figure3-5. Comparisonoftheidealandphysicaltransducersensitivitiesversustheforcingvoltageforaxedfrequency. Anidealtransducer'ssoundoutputsensitivityatagivendistanceremainsconstantre-gardlessoftheinputelectricalpower.However,aphysicaldevice'ssensitivitydecreasesatlargeforcingvoltagesduetononlinearitiesasshowninFigure 3-5 .Materialstieningatlargedeectionsandacousticharmonicgeneration(discussedinChapter 2 )areexamplesofsourcesofnonlinearities.Thenonlinearitiescausedistortioninthepressureoutputofthetransmitter.Thedistortionismeasuredintermsoftheharmonicsgeneratedinreferencetothefundamentalsignal.Inparametricarrays,acousticallygeneratedharmonicsaredesirable 54

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Transmittersaredesignedtohavelargeacousticoutput,butatthecostofincreasingexcitationpowerrequirements.Thus,thetransmittersaredesignedtohavealargetrans-missionsensitivitytominimizepowerrequirements.Thepowertransmissioneciencyfromthemechanicaltoacousticdomainshasalargeimpactontheoveralldeviceeciency.Sec-tion 3.2 onacousticsourcesgivesspecicdetailsonhowpowereciencyofacousticsourcesvariesdependingontheirsizeincomparisontoacousticwavelengths. Transmittersaredesignedtohavespecicsoundelds.Forinstance,conventionalspeak-ersideallyllanentireroomwithsound.Ultrasonictransmitters,however,areusuallydesignedtoproduceaconnedsoundeldsuchasabeam.Themeasureofanacoustictransducer'sabilitytodistributesoundisthedirectivitypattern.Examplesoffundamentalacousticsourcesandtheirdirectivitypatternsaregiveninthefollowingsectionasexamples. 3-6 canbeapproximatedasabaedpistonwithareasonabledegreeofprecision[8]. 55

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Speaker. (b) Baedpiston. Speakermodeledasabaedpiston. Abaedpistonisarigidplanewhereallpointsarexedexceptforanareathatvibratesnormaltotheplane.InFigure 3-7 ,thewhiterigidbaeremainsstationarywhilethegraypistondeectsinthezdirection.TheanalyticalapproximationoftheacousticfareldduetobaedpistonvibrationiscalledRalyeigh'sintegral[8]. ForanarbitrarilyshapedpistonasshowninFigure 3-7 ,Rayleigh'sintegralgivesthepressureinthefareldatpointL(x;y;z)as[8] 2RdS;(3{6) whereR=q Manyspeakersystemscanbeapproximatedbyanaxisymmetriccircularpistonwhereallpointswithinthecirclearevibratingharmonicallywiththesamevelocityamplitude,u0.Notethatthesignconventionofharmonicvibrationsisapositiveexponent,i.e.ej!t.Thepressureatthepoint(r;)showninFigure 3-8 ,whereristhedistancefromthepistoncenterandistheanglemadewiththez-axis,is[8] 56

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Arbitrarilyshapedbaedpiston(adaptedfromBlackstock[8]). whereJ1()istherstorderBesselfunctionoftherstkind[52],R0=1 2ka2istheRayleighdistance,andP0=0c0u0. Figure3-8. Circularpistoninaninnitebae(adaptedfromBlackstock[8]). TheRayleighdistanceisalengthscalethatestimatesthetransitionfromaradiator'sneareld,wherepressureandvelocityhaveaphasedierenceandwavesarecollimated,totheradiator'sfareld,wherepressureandvelocityareinphaseandthewavesspreadspherically[8].ThepressureP0shouldnotbeconfusedwiththepressureonthepistonface. 57

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Manytimes,radiatorsaredesignedtohaveacertainSPLatagivendistance.Sincethisdistancecanvarydependingontheapplication,thesourcelevel(SL)isintroducedtocomparetherelativestrengthofacousticsources[8].TheSListheSPLmeasuredontheradiationaxisofasoundsourcethatisextrapolatedtoa1mdistancebyassumingsphericalspreading.Forinstance,ifprmsisthermspressureontheradiationaxisofasoundsourceatadistancer,thesourcelevelisgivenas SL=20log10prmsr pref(1m):(3{8) Thedirectivity,D(),givesthepatternofthepressureinthefareld[8].Thedirectivityfunctionisthefareldpressuredividedbythemaximumpressureatthesamedistance, wheremaxistheangleofmaximumfareldpressure.Thedirectivityofthebaedcircularpistonis TheMaclaurinseriesexpansionofEquation 3{10 forsmallkais[8] 8(kasin)2+1 192(kasin)4+:(3{11) Foracompactsourcewhereka1,thedirectivityfunctionapproaches1.Thus,acompactsourcehasauniformpressureintermsofinthefareld. Foranon-compactsourcewhereka1,J1(kasin)willbezeroatmultipleanglesasshowninFigure 3-9 .Thesearetermednulls.Inbetweenthenullsareminorlobescalledsidelobes.Notethatthesidelobesdecreaseinamplitudefurtherfromthecentrallobe.Also,eachsidelobeis180outofphasewithitsneighboringsidelobes.Figure 3-9 isthedirectivitypatternofapistoninaninnitebae.Amorerealisticacousticsourcewillhave 58

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Itisgeneralpracticetoreportthe-6dBbeamwidthofthemainlobeasameasureofthesource'sdirectivity.The-6dBbeamwidthdenotestheangleatwhichthesignalisaquarterofitsmaximumpower.InFigure 3-9 ,thebeamwidthofthecompactsourceismuchlargerthanthatofthenon-compactsource. Figure3-9. Comparisonofthedirectivityofacompactandnon-compactsource. Equation 3{7 isthefareldpressure.Theradiationimpedanceseenbythepistonisdenedas[8] wherePavistheaveragepressureatthepistonface.Theresultingradiationimpedance,showninFigure 3-10 ,is[8] 59

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Radiationimpedanceofabaed,circularpiston. TheMaclaurinseriesexpansionofthepistonradiationimpedanceforsmallkais Thus,foracompactsourcewhereka1,theradiationimpedancebecomes Clearly,forka1,theimaginarypartoftheradiationimpedancewilldominate.Equa-tion 3{13 isrewrittenas ThesecondterminEquation 3{16 isequivalenttoalumpedmass.Thus,thedominantloadingonacompactradiatorisacylindricalslugofuidwiththeradiusofthepistonandaheightof8a Ontheotherhand,theradiationimpedanceofanon-compacttransducerapproachesthecharacteristicspecicacousticimpedance,0c0,askabecomeslargeasshowninFigure 3-10 60

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Theeciencyofcompactversusnon-compacttransducersiscontrastedbylookingatthepoweremittedbythebaedpiston, 2ka:(3{17) Thepoweremittedbycompactandnon-compactradiatorsreducestoWc=a2u200c0 (3{18)andWnc=a2u200c0 respectively.Thepoweremittedbythecompactradiatorwhereka1isdependentupon(ka)2whilethenon-compactradiatorisindependentoffrequency. 2 ,aparametricarraycommonlyusesmanysmallersourcestoformitsultrasonicspeaker.Thusfar,baedcircularpistonshavebeenintroducedasacoustictransmitters.Theacousticeldofmanyofthesesources,calledanarray,isdiscussedhere.Asanexample,alinearrayofidealpointsources,ormonopoles[8],isdiscussed.Acorrectionfornon-idealsourcesisaddressedattheendofthesection. Thepressureeldofasinglemonopoleis[8] rej(!tkr);(3{20) whereAistheamplitude. ConsideralineararrayofNideal,equallyspacedmonopolesseparatedbyadistancedasshowninFigure 3-11 .Thepressureofthesumofthemonopolesis[8] 61

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LinearrayofNidealmonopoles(adaptedfromBlackstock[8]). wheredisthespacingbetweenmonopoles.FromEquation 3{21 ,thepressureonaxisisp(=0)=Np0.Thus,thearraydirectivityis ThedirectivityfunctionisplottedinFigure 3-12 forarrayswithdierentnumberofsourcesatanequalspacingofkd=1.Asshown,increasingthenumberofelementsinthearraydecreasesthebeamwidthofthesoundeld.NotethatinEquation 3{22 ,askdapproacheszero,thedirectivityfunctionapproachesone.Askdapproacheszerothemonopoleslineupontopofeachotherformingasinglesource. Theacousticeldofalinearrayofdirectionalsourcesissimplythemonopoleeldmultipliedbythedirectivityofthedirectionalsources,De()[8].Thus,thedirectivityofanarrayofNnon-idealsourcesis[8] Thebasicprinciplesoftheanalysisofalinearraycanbeextendedtoatwo-dimensionalarraysuchasaparametricarraysource.AswillbeshowninSection 3.3 ,thesizeofaMEMSbendingmodeultrasonictransmitterscalesontheorderof1mmforresonantfrequenciesinthelowultrasonicrangearound40kHz[53].Thedirectivityofasingleofthesetransduceris 62

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Arraydirectivityfor3and7monopolearraysatkd=1. fairlyomnidirectionalwithaka0:7.Ontheorderof103devices[1]wouldformthesourceoftheparametricarray.Similartotheexampleofthelineararray,theincreaseinnumberoftransducersandsubsequentincreaseinarraydiametercreatesanarrowarraydirectivity,resultinginthebeamgenerallyrequiredforparametricarrayoperation. Tondananalyticalsolutionofabsorptionthatincludesthecontributionsofattenu-ation,thephysicalmechanismsareindividuallyincludedinthegoverninguidmechanicsequations.Theequationsarecombinedtoformawaveequation[8].Planewavesarecon-sideredhereforthepurposeofdiscussion.Itcanbeshownthatthesolutiontothewave 63

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InEquation 3{24 ,kisallowedtobecomplexandassumestheform[8] Thus,adispersionrelationshipwherethephasevelocityisdependentuponfrequencyisintroducedintoEquation 3{24 [8]resultingin whereiscalledtheabsorptioncoecient.Thenewphasespeed,cph,isrelatedtoby :(3{27) Theabsorptioncoecientforeachlossmechanismisfoundfromthecorrespondingsolutiontothewaveequation.Theabsorptioncoecientsarethensummedtoobtainthetotalabsorption.Forexample,tondtheabsorptioncoecientthatincludesthermoviscousandrelaxationeects,thesumoftheindividualabsorptioncoecientsistaken, whereimristherelaxationabsorptioncoecientforeachconstituentofthegas.AlthoughtheabsorptioncoecientsfromEquation 3{28 aresolvedusingplanewavesolutions,thesamecoecientsareusedintheanalysisofsphericalwaves[8]. Thethermoviscousabsorptioncoecientisproportionaltofrequencysquared[8], 64

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wherefr;iistherelaxationfrequencyofeachofthemolecularconstituentsofthegas. Acousticabsorptioninairisdominatedbythermoviscousabsorptionandrelaxationassociatedwiththevibrationalmodesofnitrogenandoxygen[8]: Figure3-13. Soundabsorptioncoecientofairat70%relativehumidity(afterZuckerwar[54]). Therelaxationfrequenciesaredependentuponthewatervaporcontent.Thus,theabsorptioncoecientissubstantiallydependentuponhumidity.Absorptionat70%relativehumidityisgiveninFigure 3-13 .At40,80,and120kHz,theabsorptionat70%relative 65

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FromtheBertkaysolutionfromSection 2.2.2.2 ,thedemodulatedsoundscalesastheinverseoftheabsorptioncoecientofthefundamentals.Oncethedemodulatedsoundiscreated,ittravelsasanormalacousticwaveandexperiencesacousticabsorption.How-ever,sincethefrequenciesofthedemodulationsoundareusuallyintheaudiblerange,itsattenuationiscommonlyneglected. 5 whereVisvoltage,Fisforce,Iiscurrent,Uisvelocity,ZEB=V IU=0istheblockedelectricalimpedance,ZMO=F UI=0istheopen-circuitmechanicalimpedance,TEM=V UI=0

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IU=0istheblockedelectromechanicaltransductionfactor.Notethattheproductofpowerconjugatevariables,suchasforceandvelocityorcurrentandvoltage,resultsinpower. ThemajorsensingandtransmittingmechanismsusedinMUTsdescribedinthelit-eraturearepresented.Themajormechanismscanbesplitintotwocategories:reciprocalandnon-reciprocal.Reciprocalmechanismsareusedtobothactuateandsensechangesinaphysicalsystem.Bydenition,reciprocityindicatesthattheopen-circuitandblockedelec-tromechanicaltransductionfactorsinEquation 3{32 areequal[55].ReciprocalmechanismsprevalentintheMUTliteratureareelectrostaticandpiezoelectric.Electrostatictransduc-tionutilizesthecapacitance-displacementrelationshipbetweentwoormoreelectrodes[56].Piezoelectrictransductionmakesuseofthepiezoelectriceect,wheredirectpolarizationofapiezoelectricmaterialgivesrisetoanelasticstrain.Conversely,elasticstressofthematerialcreatesachangeinelectricpotential[56]. Thermoelasticactuationisthemajornon-reciprocalactuatingmechanismobservedintheliterature.Temperaturegradientscreatedwithinastructurecausenon-uniformthermalexpansionandthusaniteamountofdisplacement[57]. Theliteraturereviewisorganizedaccordingtothetransductionmethod.Abriefreviewispresentedofelectrostatic,piezoelectric,andthermoelastictransductionwitheachfollowedbyareviewofMEMS-basedactuators.Theperformanceandfabricationofthedevicesispresented.Finally,thetradeosamongthedierentmechanismsarediscussedintheconclusion. 3-14 .Thebottomplateisxed.Thebalancebetweenthemechanicalrestoringforce(representedbyaspring),theelectrostaticforcebetweenthetwoplates,andthegravitationalforceholdsthetopplateinplace. 67

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One-dimensionalelectrostatictransducer. Displacementofthetopplateisdenotedx0(t)suchthatthedistancebetweenelectrodesisx0x0(t).Thecapacitancebetweentheplatesisgivenby whereCEB="0A/x0istheblockedcapacitance[56].Theblockedimpedanceofanelectro-statictransduceristheratioofthevoltagetocurrentwhilemaintainingthegap.Ifavoltageissuppliedacrosstheplates,achargeisstoredonthecapacitor.Thevoltageisrelatedtothechargeby[56] Thetotalforceactingontheplateisthesumoftheelectrostaticandmechanicalforces(ignoringgravitationaleects).Tocalculatetheelectrostaticforce,thedisplacementderiva-tiveoftheelectricalpotentialenergyistaken[56].Theelectricpotentialenergyisgivenby 2CE(t):(3{35) Thus,theelectrostaticforceis 2Q2(t) 68

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3-14 .Thetotalforceactingonthetopplateis 2Q2(t) Equations 3{33 3{34 ,and 3{37 formthreecoupled,nonlinearequations. Itiscommonpracticetosupplyadcbiasaswellasanacvoltageacrosstheplatesforlinearizationpurposes,V(t)=V0+V0(t).Themagnitudeofthebiasisadjustedtoimprovethelinearitybetweentheacsignalandtheplatedeection.ThepotentialacrosstheplatesinducesachargethatalsoconsistsofdcandaccomponentsQ(t)=Q0+Q0(t),whereQ0=V0CEB.Tondthelineartransductionequations,itisassumedthattheaccomponentsofcharge,voltage,anddisplacementaremuchsmallerthatthedccomponents.Thus,thelinearizedequationsareCE(t)=CEB1x0(t) Assumingtimeharmonicsignals,thelinearizedtransductionEquations 3{39 and 3{40 arewritteninmatrixformas whereI0andU0areaccurrentandvelocity.TheupperrightandlowerlefttermsinEqua-tion 3{41 areknownastheopenandblockedelectromechanicaltransductionfactors,re-spectively[55].Linearelectrostatictransductionisreciprocalbecausethesetwotermsareequal. 69

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3{41 isalinearizedtransductionequationwhosedegreeofrobustnessdependsuponsmalldeectionsandsmallacsignals.Whendeectionsbecomelarge,thelinearassumptionisviolatedandsubstantialharmonicsareproduced.ThechargedependenceofEquation 3{36 isconvertedtoavoltagedependencebysubstitutingEquation 3{34 : 2"0AV2(t) (x0x0(t))2:(3{42) ThevoltagesquareddependenceofEquation 3{42 isnulliedbysupplyingadcbiasvoltageacrosstheplatesasdescribedabove.However,thedependenceofEquation 3{42 ontheinverseofthegapsquaredremains.Forsmalldeections,x0(t),withrespecttothenominalgap,x0,thedenominatorofEquation 3{42 canbelinearized, 1 (1x0(t)/x0)2'1+2x0(t) Ontheotherhand,ifthedynamicdeectionsaresubstantialwithrespecttothegap,oddharmonicsareproducedbythedisplacementdependenceofthesecondterminEquation 3{42 [58]. Figure3-15. Pull-inpointwherethechangeintheelectrostaticforceisequaltothechangeinthemechanicalforce. Forthetopplatetobeinstaticequilibrium,themechanicalandelectricalforcesmustbalanceeachother.Forasmalldcvoltage,thereexistsasmalldisplacementwheretheforcesbalanceandthenetforce,givenbythestaticversionofEquation 3{42 ,equalszero.Atthis 70

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3-15 .Asthevoltageincreases,thegapdecreasesandeventuallyreachesanunstablepointatwhichthegradientofthenetforceequalszero.Beyondthispoint,theelectrostaticforcegrowsfasterthanthemechanicalrestoringforceasshowninFigure 3-15 .Thiscombinationofgapandvoltageisknownasthestaticpull-inpoint.AplotofthemechanicalrestoringandelectricalforcesforincreasingvoltageareshowninFigure 3-16 .Tondthestaticpull-involtageandgap,thenetforceandnetforcegradientarebothsetequaltozeroandsolved,resultinginxPI=2 3x0 Figure3-16. Plotofthemechanicalforce,FM,andtheelectrostaticforce,FE,atdierentvoltagesversusthegapdistancex(Notethatallvaluesarenon-dimensional). intothebackplate.Evenifthetopplateisundamaged,thevoltagemustbeloweredbelowthepull-involtagebeforeitcanbereleasedsinceintermolecularcohesiveforcesacttokeep 71

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ThefollowingsectionisareviewofMEMSelectrostaticactuatorsforairapplications.Acomparisonoftheresonantfrequenciesandoutputsofthedevicestothoseofparametricarrayimplementationstodate(seeSection 2.2.3 )provideinsightintotheapplicabilityofthistechnologytoparametricarrays.Thefollowingdevicereviewalsogivesinsightintothefabricationtechnologiesusedtocreatethesedevices. Figure3-17. Deviceformedbycombiningmacro-andmicromachining(adaptedfromHiguchi[59]). EortsbeganwithHiguchietal.'s[59]presentationofahybridultrasonicsensor/actuatorshowninFigure 3-17 thatutilizesbothmicro-andmacro-machining.Thecavityandbackelectrodeswereformedbyanisotropicetchingofthesiliconandsubsequentaluminumdepo-sition.Thewaferwasdicedinto20mmby30mmchipscontaining32elements.A12mthickpolyesterlmwitha50nmAllayerwasstretchedoverthecavitiesandanchoredwithmachinescrews.Threesquarecavities80m,40m,and10monasidewerefabricated. 72

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Suzukietal.[60]continuedtheworkofHiguchietal.andpresentedasimilarhybridultrasonicsensor/actuator.Micromachiningwasusedtodenepyramidshapedbackcavi-ties10-40monasideandanaluminumbackelectrode.Then,thewaferwasdicedandpackagedandastretched,metallizedpolyesterdiaphragmwasattachedoverthecavities.Theyachievedatransmitsensitivityof119.1dBre1Pa/Vat50cmfromthedevicesat150kHzandareceivingsensitivityof0.47mV/Pafrom10-130kHzatabiasvoltageof30V. ThetransducerspresentedinbySchindelandHutchinsetal.[61{65]aresimilartothoseofSuzukietal.[60].AKOHetchwasusedtodenecavitiesonthesiliconwafer40monasidewith80mcenter-to-centerspacing.Thecavitieswerearrangedinarectangulararray25mm2.Ametallized,5mthickKaptonlmwasstretchedandmechanicallyattachedoverthecavities.Thereportedexperimentalresultswerenormalized. Cross-section. (b) Top-view. CapacitivecMUToftheE.L.GinztonLaboratory(adaptedfromHaller[66]). Therstair-coupledultrasonicMEMSoftheE.L.GinztonLaboratoryatStanfordUniversityisshowninFigure 3-18 [66{69].Thedeviceconsistedofanarrayona1cmsquaredieofnitridemembranes750nmthickwith50nmthickgoldtopelectrodeandahighlydoped(100)siliconsubstratebackelectrode(substratealsohasalayerofgoldonthebacksideforelectricalcontact).Thenitrideandgoldelectrodelayersweredepositedovera 73

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AsimilardevicewasfabricatedbyLadabaumetal.[70].Inthisdevicetheholesusedforthepreviouslydescribedtimedetchedwerelledwithalowpressurechemicalvapordepositionnitride.Theresultingnitridediaphragmthicknesswas600nm.Theresonantfrequencieswere1.8MHzfora100mdiametermembraneand12MHzfora12mdiametermembrane. Figure3-19. NitridediaphragmcMUTwithvacuumsealedbackcavity(adaptedfromJinetal.[71]). Jinetal.presentedanarrayofcMUTsformedbyanitridediaphragmwithanalu-minumtopelectrodeasshowninFigure 3-19 [47,70{77].Theheavilydopedsiliconsubstrateformedthebottomelectrode.A200-1000nmamorphoussiliconlayerwasusedasasacri-ciallayerbecauseithadexcellentselectivitywithrespecttothenitridediaphragm.Theamorphoussiliconwasetchedintohexagonallyshapedislandsthatdenedthecavitiesafterdiaphragmrelease.Nitridewasdepositedovertheislandsandpatternedforaccesstothe 74

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Figure3-20. Capacitivetransducerwithapolysilicondiaphragmanddopedbottomelec-trode(adaptedfromEccardtetal.[78]). Eccardtetal.[79,80]presentedthecapacitivetransducerillustratedinFigure 3-20 .Hexagonal,polysiliconmembraneswithsidelengthsof40mandthicknessof400nmandcavitiesof450nmdepthformedthetransducer.Theyachievedapproximately10nmofdeectionaroundaresonanceof10MHz.Anewfabricationstepwasaddedinhopesofincreasingtheelectrostaticforce[78].TheprocesswasamodiedCMOSprocessthatxedthegapthickness.Bumpswereintroducedonthebottomofthediaphragm.Whenthediaphragmwascollapsedduetopull-in,thesebumpsformedtheedgeofasmallerdiaphragmwithagapthicknessthesizeofthebumps.Ofcourse,eventhoughtheelectrostaticforcewasincreasedbyreducingthegapthickness,pull-inrestrictedthedeectionandthusthepressureamplitude.Thus,themaximumcenterdeectionofthesmallermembranewasthenewgapthickness.Testingofthesetransducerswasconductedinwater. 75

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MicromachinedcapacitivedevicewithcoupledHelmholtzresonatorformedbytheresonantcavityandthroat(adaptedfromParvizetal.[81]). Parvizetal.[81]describedtheuseofanelectrostaticactuatorcoupledtoaHelmholtzresonatorinaneorttoproduceacousticstreamingathighfrequencies(100kHz).Acous-ticstreamingisthecreationofameanowbyahighfrequencyacousticeld[18].Appli-cationsofthedeviceincludedpropulsion,micro-cooling,andmicro-pumping.Asquare,polysiliconmembrane1.36mthickand1.2mmonasidewassuspended3mbelowaperforated,borondopedsinglecrystalsiliconlayer.Oxideandnitridelayersservedaspas-sivationofthemembrane.Ontheoppositesideofthemembranewasaresonantchamberthatledtoathroat.Bysupplyinganalternatingvoltageacrossthediaphragm,anacousticeldwasestablishedwithintheresonantchamber.Iftheacousticsignalwashighenough,itgeneratedacousticstreamingthroughthethroat.Actuatorscoveredanentire4inchwaferin4quadrants(992totaldevices).Thediaphragmwasclaimedtocollapsewhenavoltagewasapplied.Themeasurementmicrophonesusedtocharacterizethedeviceswereaudiogradeandthereforeonlyhadaatbandwidthoutto20kHz.Thus,themeasurementsathigherfrequenciesweresubjecttotheroll-oresponseofthemicrophone.Theyreportedapeakat96kHz.Theyalsonotedtheharmonicsgeneratedbythecollapsemodeactuation. Torndahletal.[82]presentedacapacitivetransducerwithapyramidshapedbackcavityandsidesoflength40or60manddiaphragmthicknessesof8or12m.ThediaphragmofthecMUTwasformedbyametalontopofapolymericmembrane.Thedopedsiliconsubstrateservedasthebackplate.TheresonancesofthecMUTsvariedfrom400kHzto1MHz.Lightdiractiontomographywasusedtomeasuretheacousticeldofthetransducer. 76

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AcMUTwhosecavitiesareformedbyananisotropicetch(adaptedfromTorn-dahletal.[82]). Experimentsonanarrayofthe40msizedcavitieswiththe8mthickdiaphragmproduced137.5dBat450kHzat35mmfromthedevicesurfaces. Figure3-23. Capacitivetransducerwithpolysiliconmoveablemembranes(adaptedfromBuhrdorfetal.[83]). Buhrdorfetal.[83]detailedacMUTthatutilizedapolysilicondiaphragm.Adopedpolysiliconregiondenedtheupperelectrode.Thediaphragmwasabout1mthickwithagapheightbetween300-400nm.Linearraysofthedeviceshadawidthof100m.Thehexagonaldiaphragmswerepackedinthisregioninrowsoftwoorthreedevices.Thedevicesobtainedover100nm/Vdrivesensitivityataresonanceofapproximately5.7MHzwhenadcbiasof82Vwassuppliedacrossthediaphragm. Figure3-24. AcMUTfabricatedusingMUMPS(adaptedfromOppenheimetal.[84]). 77

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3-24 .Thediaphragmthicknessandbackcavitywereconstrainedbytheprocesstobe2m.Thedevicehadahexagontopplatethatwas45moneachside.Etchholes5mindiameterallowedtheoxidelayertobesacriced.Theetchholesalsoservedasventssothatthedevicedidnotrespondtochangesinambientpressure.Theresonantfrequencywas3.47MHz. Figure3-25. AcMUTfabricatedusinganSOIwaferbondedtoapatternedsubstrate(adaptedfromHuangetal.[85,86]). AnotherpaperbytheStanfordgroupdescribedcMUTssimilartothosealreadypre-sentedcreatedusinganewfabricationmethod[85,86].Theydenedthecavityononewaferusingeitherasimpleisotropicoxideetchoracombinationofoxideetchandsiliconetchdependingonthecavitydepthdesired.AnSOIwaferwasbondedtotheoriginalwafer.ThediaphragmwasformedbythesinglecrystalsiliconlayeroftheSOIwaferbygrindingthebulksiliconoftheSOIandsacricingtheBOXlayer.Squaremembranesupto750minsizewerefabricated.Fora650m,4.2mthickdevicewithagapof11.5m,theymeasuredaresonantfrequencyof310kHz.Thisfabricationmethodavoidedsomeofthedicultiesofthepreviousmethodssuchasthenitridell-intovacuumsealthebackcavity.Theyconductedpulse/receiveexperimentsinairwherethetransducerhasanarrowbandofabout278kHz. TheStanfordresearchgroupalsoexploredthreedierentoperatingmodesfortheircMUTtransducers[76,87{89]:conventionalmodewherethetotalofdcbiasandacvoltageneverexceedthepull-involtage,collapsemodewherethedcbiasexceedsthepull-involtage 78

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Figure3-26. AcMUTwithnitridediaphragm(adaptedfromKimetal.[90]). Kimetal.[90]designedandfabricatedacMUTwitha0.4mthicknitridemembrane45mindiameterand15mspacing.Thetopelectrodewasformedby0.2mofaluminumasshowninFigure 3-26 .Thebottomelectrodewasformedbythesiliconsubstratewithametalback-plate.Theairgapwas0.3m.Finiteelementanalysispredictedaresonanceof7.4MHz.Witha40Vdcbiasand5Vacsignal,0.2mofdeectionwasobtained. 3-1 .Allthedeviceshaveresonantfrequenciesfromthe100'sofkHzuptotheMHzrange.Thesefrequenciesaremostlikelynotsuitableforparametricarraytransducersduetolargeattenuationathighfrequenciesandthelongerworkingdistancerequiredoftheparametricarray.Itisalsoextremelydiculttomakequantitativemeasurementsofthesoundeldathighfrequenciesinair.Generally,B&Kserveasmeasurementstandardsformicrophones.TheB&K41381/4-inchfreeeld 79

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Inaddition,thebacksidesofmanycMUTs'diaphragmsarevacuumsealedsuchthattheatmosphericpressurewillcauseastaticdeection.Astaticdeectionwillcauseachangeinstinessofthediaphragm.Thus,achangeinatmosphericconditionswouldcauseachangeintheresonantperformanceofthedevice. ThetransductionmechanismofcMUTsisinherentlynonlinearresultinginharmonicdistortionproducedbythediaphragmvibrations.Also,thenonlineartransductionmecha-nismresultsinapull-involtagethatfurtherconstrainsdeviceperformance. Incontrast,piezoelectrictransductioninvolvestherelationshipbetweentheelectriceldandthestress/straininthedielectric.Thedielectricinthiscaseisapiezoelectricmaterial,suchasquartz(SiO2)orleadzirconatetitanate(Pb[ZrxTi1x]O30
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Air-coupledcMUTcharacteristics. Higuchietal.1986[59] 80mby12m NA NA 105dBat50cm 140kHz Suzukietal.1989[60] 40mby12m NA 19.1dBre1bar/Vat50cm NA 150kHz Schindeletal.1995[64] 40mby5m NA NA NA NA Eccardtetal.1996[79] 40mby400nm 450nm NA 10nm 10MHz Halleretal.1996[66] 1m 23nm/V 300nm 1.8MHz Ladabaumetal.1998[70] 1m NA NA 1.8MHz Jinetal.1998[72] 1m NA NA 2.3MHz Parvizetal.2000[81] 1.2mmby1.36m 3m NA NA 96kHz Torndahletal.2002[82] 40mby8m NA NA 137.5dBat35mm 450kHz Huangetal.2003[85] 650mby4.2m 11.5m NA NA 310kHz Buhrdorfetal.2003[83] 50mby1m 300-400nm 100nm/V NA 5.7MHz Oppenheimetal.2003[84] 45mby2m 2m NA NA 3.47MHz Kimetal.2005[90] 0.3m NA 0.2m 7.4MHz Apiezoelectricmaterialismostgenerallydenedasacrystallinematerialwithnon-centrosymmetricstructure,withtheexceptionbeingthepointgroup43piezoelectricmaterial[91].Examplesofcentrosymmetricandnon-centrosymmetriccrystallinestructuresaregiveninFigure 3-27 .Piezoelectricmaterialsexhibitacouplingbetweenelectricalandmechanicalenergydomains.Mechanicalstressappliedtoapiezoelectricmaterialinducesanelectriceld.Thisisknownasthedirectpiezoelectriceect.Ontheotherhand,theconverse 81

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Centrosymmetriccrystalunitcell. (b) Non-centrosymmetriccrystalunitcell. Isometricviewofanidealperovskitestructure.[91]. Asubsetofpiezoelectricmaterialsareclassiedaspyroelectric.Thesematerialsexhibitachangeinpolarizationduetoatemperaturechange[91].Polarmaterialshaveanetchargedistributionintheabsenceofanelectriceld[92].Onesuchdistributionisadipolewhichisthecreatedbytheseparationofpositiveandnegativecharges[93].Onlypiezoelectricmaterialsthatarepolararealsopyroelectric.Anexampleofanon-polarpiezoelectricmaterialisSiO2.Aluminumnitride(AlN)andzincoxide(ZnO)areexamplesofpyroelectric(i.e.,piezoelectricandpolar)materials[91].BothmaterialshavewurtzitestructuresasshowninFigure 3-28 a.Thestructureconsistsoftwohexagonalcrystallatticesthataretetrahedrallyinterconnected[91]. Somepyroelectricmaterialsexhibitapropertywherethedirectionoftheirdipolecanbechangedbythetemporaryapplicationofanelectriceld.Thesematerialsaretermedferroelectric[91].AnexampleofapyroelectricmaterialthatisnotferroelectricisAlN.PZTisanexampleofaferroelectricmaterial.ThecrystallinestructureofPZTisperovskiteasshowninFigure 3-28 b.Ferroelectricmaterialslosetheirpiezoelectricpropertywhen 82

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WurtzitecrystalstructureofAlNandZnO. (b) PerovskitecrystalstructureofPZT. Isometricviewsofidealwurtziteandperovskitestructure.[91]. thetemperatureisraisedbeyondtheCurietemperature.Atthispoint,theatomsrelaxintotheircentrosymmetricpositions[93].Withinasingleferroelectriccrystal,domainsofoppositebutequalpolarizationwillformsuchthatthenetpolarizationiszero.Onamacro-scale,thesymmetryofthepolarizationisaccentuatedbythemulti-crystallinestructureofmanypiezoelectricmaterials[91].Thus,onamacro-scale,thepiezoelectricmaterialisnotpiezoelectricallyactive.Aferroelectricmaterialismadepiezoelectricallyactivebyapplyinganelectriceldthatisstrongenoughtoreversethedirectionofthepolarizationofthecrystaldomainssothattheyaremostcloselyalignedtotheelectriceld.Theeldstrengthatwhichreversalbeginsisknownasthecoerciveeld[94].Thisprocessistermedpoling.Manytimes,polingisachievedbyheatingbeyondtheCurrietemperatureandcoolingunderaweakerelectriceld[91]. 83

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whereDiistheelectricdisplacementvector[C/m2],"pistheengineeringstrainvector,pisthemechanicalstressvector[Pa],andEjistheelectriceldvector[V/m],diqisthepiezoelectricconstanttensor[C/N],ijisthepermittivitytensoratconstantstress[F/m],andSEpqistheelasticcompliancetensoratconstantelectriceld[1/Pa].Thethreetensorsarematerialconstants.ThetermsofthepiezoelectricconstanttensorinEquations 3{46 and 3{47 determinethelevelofcouplingthebetweentheelectricalandmechanicaldomain. Figure3-29. One-dimensionalpiezoelectrictransducer(adaptedfrom[16]). Considertheone-dimensionalsimpliedcaseofabulkvibratorshowninFigure 3-29 [56].Todescribethebehavior,theconstitutiveequations, 3{46 and 3{47 ,arerewrittentoexpresstheelectricdisplacementintermsofchargeq,thestrainintermsofdisplacementx,thestressintermsofmechanicalforceF,andtheelectriceldintermsofvoltageV,q=CEFV+dF respectively,wheredisthepiezoelectricmodulus,CMSistheshort-circuitmechanicalcom-pliance,andCEFisthefreeelectricalcompliance.FromEquation 3{48 ,thepiezoelectric 84

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3{49 ,thepiezoelectricmaterialbehaveslikeanormaldielectricmaterialbetweencapacitorplateswhentheplateis"free"suchthatF=0. Foratimeharmonicsystem,thetransductionequationsarewritteninconjugatepowervariableformas[56] NotethatEquation 3{50 iswritteninadmittance(i.e.,inverseofimpedance)formsincetheeortvariablesaregivenasoutput.Theo-diagonaltermsareequalindicatingthatthelinearmodelofthepiezoelectrictransducerisreciprocal[56]. Thepiezoelectricmodulus,d,forthethicknessmodetransducershowninFigure 3-29 isdirectlyrelatedtothed33termofthepiezoelectricconstanttensorthatrelatesanelectriceldinthe3directiontoelasticstraininthesamedirectionthroughtheconversepiezoelectriceect.Incontrast,manyMEMSdevicesthatutilizepiezoelectrictransductionarebendingmodestructureswithapiezoelectricthinlm.Thesedevicesrelyonthed31coecientfortransduction.Thiscoecientrelatesanelectriceldinthe3direction(transversetotheplaneofthelm)totheelasticstraininthe1direction(alongtheplaneofthethinlm).AsillustratedinFigure 3-30 ,strainwithinthethinlmdepositedonthemembraneorbeamcancauseoutofplanedeections[16]. Figure3-30. Eectofthed31coecientinpiezoelectricthinlms. 85

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3{50 iswrittenintheelectricalandmechanicaldomains.Often,itisconve-nienttomodelthetransductionintheelectricalandacousticdomains.Inthiscase,velocityandforcebecomevolumevelocity,Q,andpressure,P,andthetwo-portelectro-acousticmodelisexpressedas wheredAistheeectiveacousticpiezoelectriccoecientandCADistheshortcircuitacousticcomplianceofthediaphragm[95]. AlthoughEquations 3{46 and 3{47 arefrequentlyusedtodescribepiezoelectrictrans-duction,acommoncharacteristicofpiezoelectricmaterialsisanonlinearmaterialsresponseknownashysteresis.Thisphenomenonischaracterizedbyaphaselaginresponsetoex-ternalforcing,whethermechanicalorelectrical.Becauseofthelaginresponse,materialsthatdemonstratehysteresisarememorydependentwheretheircurrentreactionisdependentupontheirpastresponsetoforcing[96].Hysteresisiscommonlycharacterizedbythe\hys-teresisloops"showninFigure 3-31 thatshowsthemulti-valueddependenceofamaterial'sresponsetoexternalforcing.Thecauseofhysteresisiscomplexandinsomedenitionsmayincludeothernonlinearities.Hystereticsourcesareclassiedaseitherintrinsicorextrinsic.Intrinsicpropertiesareinherenttotheunitcellofthematerial.Extrinsicpropertiesareduetoeectsofnon-uniformitysuchasdefects,grainboundaries,anddomainboundaries.Forexample,hysteresisinPZTisdominatedbythemovementofdomainboundariesundertheapplicationofanelectriceld[91,97]. PiezoelectricmaterialscommonlyfoundinmicrosystemsareAlN[98],zincoxide(ZnO)[99,100],copolymerofvinylideneuoridewithvinyltriuorethylene(P(VDF/TrFE))[101],andPZT[46,102,103].P(VDF/TrFE)isaferroelectricpolymerthatcanbepoled[91].PZTisnotCMOScompatibleduetoleadcontamination.Inaddition,highprecisionchemicaletchesofPZTdonotexist[91].SincePZTisferroelectric,itmustbepoledintheproper 86

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Hysteresisloopsofpiezoelectricmaterials(adaptedfrom[91]) Table3-2. Piezoelectriclmproperties. MaterialPropertyAlN[111]PZT[112,113]ZnO[114] orientation[104].AlNandZnOarenotferroelectricmaterials,andthereforecannotbepoledbytheapplicationofanexternaleld.Asaresult,theirfabricationmustspecicallyorientthecrystalsinthesamedirectiontomakethematerialpiezoelectricallyactive.Althoughthisisdicultforbulkpiezoelectrics,itispossibleforthinlmdeposition,suchasRF-sputtering[105{108]orpulsedlaserdeposition[109].AlthoughAlNandZnOhavelowerpiezoelectricconstantsincomparisontoPZT(seeTable 3-2 foracomparisonofmaterialproperties),theydonotrequirepolingandhavethebenetofbeingleadfree.InthecaseofAlN,stablelmshavebeenobtained[98,110]anditiscompatiblewithintegratedcircuit(IC)fabrication. Thefollowingsectiongivesanoverviewofpiezoelectricmicromachinedultrasonictrans-ducers(pMUTs).TheresonantfrequenciesandoutputsofthedevicesaresimilartotherangesobservedinthereviewofaparametricarrayimplementationsinSection 2.2.3 .These 87

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3-32 .Thediaphragmsarereleasedusingtwodierentmethods.Intherstmethodthesacricialoxideisetchedviaaccessholesinthesiliconmembrane.Intheothermethod,DRIEisusedtoetchthroughthebulksilicontoaccessthesacricialoxide.Thediaphragmsare100mindiameterandarearrangedwith150mspacing.Atransmitsensitivityof0.15m/Visobservedataresonantfrequencyof2.85MHz. Figure3-32. Twoside-by-sidepMUTsformedbyanitridediaphragmandZnOannularring(adaptedfromPercinetal.[115]). Figure3-33. DomeshapedpMUTwithnitridediaphragmandZnOpiezoelectriclayer(adaptedfromCheol-Hyunetal.[100]). 88

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3-33 .Thetransducerfeatureda1.5mthicknitridediaphragm2mminradiuswitha0.5mthicklayerofdepositedpiezoelectricmaterial(ZnO)toprovidetransduction.Theuniquefabricationprocessincludedphotolithographymethodsusedtopatternelectrodesontothethree-dimensionaldome.Thesoundoutputfromthedomedtransducerwasabout113dBat140kHz(measured2mmfromthetransducerswithaBruel&Kjaer41351/4-inchfree-eldmicrophonewithanupperfrequencyresponseof100kHz).Cheol-HyunandSokclaimedthatthemajorbenetsofusingadomeareawrinkleandcrackfreediaphragmandanincreasedtransductionofinplanestrainsintoverticaldeections. Figure3-34. SquarediaphragmpMUTusingPZT(adaptedfromMohamedetal.[102,120]). ApiezoelectrictransducerwasdevelopedbyMohamedetal.forechorangingef-fects[102,120].ThetransducerconsistedofasquarediaphragmofPZTonnitride.Thediaphragms,rangingfrom100mto1500monasideand2.5mthick,werereleasedusingDRIE.Echorangingtestswereconductedwithadegreeofsuccess,althoughtheauthorspointedoutthatthetransducerimpedancewasnotmatchedtothatofthepulse-receivecontrollerusedintheexperiments. Beginningin2004,Leeetal.presentedaseriesofpapersonasquarepMUTformedbydepositingPZTonanSOIsubstrate[121{123].ThreevariationsofthesquarediaphragmshowninFigure 3-35 werefabricated,each700monaside.Theburiedoxide(BOX)layer 89

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SquarediaphragmpMUTusingPZT(adaptedfromLeeetal.[121]). oftheSOIsubstratewasremovedfortwodevices,resultingintotalthicknessesof4.65and3.86mandresonantfrequenciesof90.8and87.6kHz,respectively.TheBOXlayerwasnotetchedforthethirddeviceresultinginatotalthicknessof5.60mandaresonantfrequencyof111kHz.Theauthorsnotedsubstantialinitialdeectionof4-8minthediaphragmduetoresidualstresses.Actuationsensitivitieswereabsentfromthestudies. Figure3-36. PZTmicrospeaker(adaptedfromZhuetal.[124]). Zhuetal.introducedasequenceofpapersfrom2004through2007coveringtheirPZTbasedmicrospeakershowninFigure 3-36 [124{130].ThesquarediaphragmwascomposedofaSiO2/Pt/PZT/Pt/Ti/SiO2compositeformedbyananisotropicetchfromthebacksideofthesubstrate.Thediaphragmwas600by600m2and2.2mthick.Usinga4Vppexcitation,8.36mofdisplacementwasobtainedataresonantfrequencyof67.2kHz.Twosmallerpeaksintheirdisplacementspectrumat26.9and44.7kHzwerenotedandattributed 90

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Figure3-37. PZTdiaphragm(adaptedfromZhuetal.[131]). Additionallyworkin2007fromZhuetal.focusedonanin-planepolarizedPZTdia-phragmwithinterdigitatedelectrodesasshowninFigure 3-37 formicrospeakersandsen-sors[131,132].Themembranewas1000by1000m2and3mthick.ApotentialwasestablishedbetweenalternatingPtelectrodessothatthefringeoftheelectriceldtraveledthroughthePZTdiaphragm.ThePZTlmwaspolarizedin-planeat100kV/cmsothattheelectriceldgeneratesstrainviathed33piezoelectriccoecient,whosemagnitudewasovertwotimesthed31piezoelectriccoecientutilizedinpreviouslydescribedpMUTs.Givenan 91

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Figure3-38. PZTmicrospeaker(adaptedfromZhuetal.[53]). In2005,Wangetal.presentedthepMUTdesignshowninFigure 3-38 thatutilizedathickPZTlayerforactuation[53].ThediaphragmwasformedbythickPZTandsiliconlayersaswellassilicondioxide,siliconnitride,andplatinum/titaniumlayers.Deviceswithseparatedimensionswerefabricatedandcharacterized.Therstdiaphragmwas2mmby2mmandutilizeda7mthickPZTlayerwitha10msiliconlayer.Foranexcitationsignalconsistingof20Vdcand30Vpp,107dBwasmeasuredat12mmforaresonantfrequencyof41.2kHz.Theseconddiaphragmwas1.5mmby1.5mmwitha3.5mthickPZTlayer.Thesiliconlayerofthemembranewascompletelyetched.Fora20Vdcplusa25Vppexcitationsignal,a120dBacousticsignalwasmeasuredataresonantfrequencyof76kHz. Lametal.[101]describeda2.3mmsquaresilicon/oxidecompositediaphragmactuatedusingaP(VDF-TrFE)lm.Thelmwassandwichedbetweentwoaluminumelectrodes.Theelectrodesandlmweredepositedafterthediaphragmwasreleasedandthenpoled.Themaximumdisplacementataresonanceof40.8kHzwas0.9m(driveof7V). Muraltetal.[103,133]presentedapiezoelectrictransducerdesignthatwasadaptedforuseinair.Bridgesconnectedthediaphragmtothesubstratetoobtainalargervolume 92

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SquarediaphragmpMUTutilizingaP(VDF-TrFE)lmforactuation(adaptedfromLam[101]). Figure3-40. OxidediaphragmformedbytheBOXlayerofanSOIwaferandactuatingbyaPZTlm(adaptedfrom[103]). velocityperappliedvoltage.Thediaphragmwasformedfromthesingle-crystalsiliconofanSOIwaferwiththeBOXformingthefourbridgesthatanchorthediaphragmtothesubstrate.Theyformeda1.1mmdiameter,8.2mthickdiaphragmthatresonatedat49kHzwithanapproximateamplitudeof4m. AnacousticenergyharvesterwasdesignedbyHorowitzetal.usingaMEMSpiezo-electrictransducer[46].Measurementsofthetransduceroutputcharacteristicswerealsoconductedintheauthor'sdissertation[134].The3mthick,1.2mmradius,circulardia-phragmwasformedfromthedevicesiliconlayerofanSOIwaver.Anannularring(1.1mm 93

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ApMUTusedinanenergyharvester(adaptedfrom[46]). innerradius)ofPZTwasformedattheedgeofthediaphragm.Foraresonantfrequencyof34.25kHz,thedrivesensitivitywas450nm/V. Figure3-42. AnAlNaudiomicrospeakerpMUT(adaptedfrom[135]). Recently,Seoetal.designedandfabricatedAlNmicrospeakersasshowninFigure 3-42 foraudioapplications[135].Asquarediaphragm4mmby4mmandacirculardiaphragm4mmindiameterwerefabricated.Thediaphragmconsistsofa0.5mthickAlNlayerbetweenMolayersona1mthicksiliconnitridelayer.ThetopMolayerissplitinto2electrodes.Thebottomelectrodewasgrounded.Theexcitationsignalsappliedtothetoptwoelectrodeswere180degreesoutofphase.Fora20Vppexcitationsignal,acousticmeasurementsat3mmwithaB&K41921/2-inchpressure-eldmicrophonewere100dBataresonantfrequencyof10kHzforthecirculardiaphragmand76dBataresonantfrequencyof10.5kHzforthesquarediaphragm. 3-3 .Thefrequencyrangesofthedevicesarecomparabletothoseoftheparametricarrayimplementationstodate 94

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2.2.3 .IncontrasttothecMUTs,adcvoltageisnotnecessarytolinearizethepiezoelectrictransductionmechanism.HarmonicdistortionofdiaphragmdeectionforpMUTsislimitedtogeometricandhysteresiseects.Piezoelectrictransducersdo,however,haveanupperlimittothedrivingvoltage.Largevoltagescancausedepolingofferroelectricmaterialsorevendielectricbreakdownofgeneralpiezoelectricmaterials. SimilartothesoundmeasurementsoftheelectrostatictransducersinSection 3.3.2.2 ,thesoundoutputmeasurementsoutlinedinTable 3-3 suerfrommeasurementdiculties.TheB&K4135microphoneusedbyCheol-HyunandSok[100]tomeasurea140kHzsignalhasanupperbandwidthlimitof100kHz.Unlessthemeasurementiscompensated,themicrophonesensitivitywillnotbethesameastheat-bandsensitivityduetotheroll-oofthemicrophoneresponsebeyondresonance.Also,thewavelengthat140kHzinairis2.5mm.Sincethetransducerislessthanawavelengthfromthemicrophone,thereectedwavesfromthemicrophonewillloadthediaphragmandchangeitsresponse.The10and10.5kHzmeasurementsbySeoetal.[135]at3mmsuerfromasimilarproximityissuesbetweenthetransducerandmicrophone. Jouleheatingisquadraticallydependentupontheappliedvoltage.Tocreateoscillationsofthesameinputfrequency,theinputsignalisosetbyadcbias.Thus,theresultant 95

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Air-coupledpMUTcharacteristics. Percinetal.1998[115] ZnO 0.15m/V NA 2.85MHz Cheol-Hyunetal.1999[100] ZnO NA 113dBat2mm 140kHz Mohamedetal.2001[102,120] 0.1-1.5mmby2.5m PZT NA NA NA Leeetal.2004[121] 700by700by4.65m3 NA NA 90.8kHz 700by700by3.86m3 700by700by5.60m3 Zhuetal.2004[124] 600by600by2.2m3 4.18m/V 8.36m 67.2kHz 1000by1000by2.15m3 54kHz Wangetal.2005[53] 2by2mm2by20m PZT NA 107dBat12mm 41kHz 1.5by1.5mm2by13m 120dB 76kHz Lametal.2005[101] 2.3mmbyNA P(VDF-TrFE) NA 0.9m 40.8kHz Muraltetal.2005[103] PZT NA 4m 49kHz Horowitzetal.2006[46] PZT 450nm/V NA 34.25kHz Zhuetal.2007[131] 1000by1000by4.65m3 NA 73.8dBat100mm 43.8kHz Seoetal.2007[135] 4by4mm AlN NA 100dBat3mm 10kHz 76dBat3mm 10.5kHz 2V2ac+VdcVacsin(!t)1 2V2accos(2!t):(3{52) Themagnitudesoftheinputvoltagescanbevariedtoemphasizetheoriginalfrequencycomponent.Thestaticcomponent,however,createsastaticdeectionofthestructure,changingthecomplianceandresonantfrequencyofthedevice.Thermoelasticdeviceshave 96

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MEMSthermoelasticactuatorsforairapplicationsarereviewedinthefollowingsection. 3-43 .Acompressiveresidualstressintheoxidelayerwasaresultoffabricationandcausedbucklingofsucientlythindiaphragms.Reactiveionetchvariedthediaphragmthicknessfrom8-16mtostudytheeectsofbucklingincludingstaticdeectionheight,resonantfrequency,andvibrationamplitude.Theresonantfrequencyreachedaminimumwhenthethicknessreachedthecriticalvalueatwhichbucklingoccurs.Atsomethicknessjustbelowthecriticalthickness,Brandetal.achievedbenecialkinematicamplicationwhilereducingtheresonantfrequency.Furtherreductionofthethicknessresultedinbucklingoftheplateandanincreaseintheresonantfrequency,aswellasadecreaseinthevibrationamplitudeduetostiening.Thecriticalbucklingthicknessofthedeviceoccurredatabout6.7mwhentheresonancewasapproximately50kHzwithadeectionamplitudeontheorderof900nm.Theauthorswereabletoshowgoodagreementbetweenexperiments,FEA,andanalyticalsolutionsuptothebucklingpoint;afterwardstheslopeofthedatashowsagreement.AsummaryoftheworkonthisproximitysensorwasfoundinthepaperbyBrandetal.[138].Also,HornungandBrand[57]authoredabookonthedesignofthermoelasticproximitysensors. Anothermicromachinedthermoelastictransducerwithpiezoresistivesensingwaspre-sentedbyPopescu[147].Asilicon/oxidecompositeformedthediaphragm.Thediaphragm 97

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Proximitysensorthatutilizesthermoelasticactuationandpiezoresistivesensing(adaptedfromBrand[138]). Figure3-44. Thermoelasticactuatorwithabuckleddiaphragm(adaptedfromPopescuetal.[147]. alsocontainedapolysiliconheaterandanaluminumringlayerthatservedasathermalcon-duit.Themembranewas4mmby4mmandhadaninitialbuckledheightof20m.Staticdeectionsofupto50mwereobtainedwithdissipatedpowerofmorethan1.3Watts. Chandrasekaranetal.[148]presentedathermoelasticproximitysensorsimilartothatofBrandetal.ThediaphragminthiscasewascircularandwasreleasedusingDRIE.Thediameterwas1mmandthethicknessvariedfrom6-10m.Themajorinnovationintroducedinthisworkwastheuseofelectronic-through-waferinterconnects(ETWIs)forelectricalconnections.ETWIsallowedelectricalconnectionstothebackofthewafercreatingamorerobustpackage.Chandrasekaranetal.fabricatedthreedevicedesignsandobservedbucklingphenomenonsimilartothatreportedbyBrandetal.A9mthickdeviceachievedalmost200nmofdeectionataresonanceof55kHz. 98

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Thermoelastic/piezoresistiveproximitysensorthatusesthedevicelayerofanSOIwafertoformacirculardiaphragm(adaptedfromChandrasekaranetal.[148]). Figure3-46. Thermoelasticproximitysensorusingpolysiliconfortheheaterandpiezoresis-tors(adaptedfromRuferetal.[149]). Anotherthermoelasticproximitysensor[149],showninFigure 3-46 ,wasformedusinga0.8mcomplementarymetal-oxide-semiconductor(CMOS)process.Thediaphragmwascomposedofoxideandnitridelayersembeddedwithpolysiliconresistorsthatservedasfourpiezoresistorsandaheater.Fourthermopileswerealsocontainedwithinthediaphragmtomeasureitstemperature.Thediaphragmthicknessvariedfrom4.2to5.2manditssidelengthwas1.3mm.Forasingledevice,a1/4"Bruel&Kjaer4135measured48dBataresonantfrequencyof41.3kHzatadistanceof10mm. 3-4 .ThefrequencyrangesofthethermoelasticdevicesarealsocomparabletothoseoftheparametricarrayimplementationstodatereviewedinSection 2.2.3 .However,thetransductionmechanismof 99

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3{52 ,theinputelec-tricalsignalisconvertedintodcheatingandharmonicdeectionaswellasthefundamentalcomponent.Itwouldbeproblematictousethermoelasticactuationtosupplythemultiplefrequencycomponentsneededtoproduceanaudiosignal. FromSection 3.3.1 ,theelectrostaticforceisdependentupontheinverseofthegaptothesecondpower.Itwouldseemadvantageoustominimizethegap.Asdiscussedintheelectrostaticsection,however,reducingthegapreducesthedeectionthattheelectrostatictransducercanachievebeforepull-inoccurs.Therefore,enhancingthetransmitsensitivityreducesthedeectioncapabilitiesofanelectrostaticspeaker[55].Theelectrostatictrans-ductionmechanismisinherentlyquadraticandthusrequiresabiasvoltageto\linearize"thetransducer.Generally,thecMUTsintheliteraturedonotreectthefrequencyrangeoroutputsestablishedintheliteratureforparametricarraytransducers.Thus,electro-statictransductionisnotchosenasthetransductionmechanismfortheparametricarraytransducer. Table3-4. ThermoelasticMEMScharacteristics. Brandetal.1997[138] 1mmby6.7m P-typedoped 900nm 50kHz Popescuetal.1996[147] 4mmby15m aluminumring 50m(static) NA Chandrasekaranetal.2002[148] 1mmby9m P-typedoped 200nm 55kHz Ruferetal.2006[149] 1.3mmby4.2-5.2m polysilicon 5mPaat10mm 41.3kHz

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Piezoelectrictransductiondoesnotrequireabiasvoltageforalinearinput-outputrelationship.Ontheotherhand,piezoelectricdevicesdoexhibithysteresiseectswhichcangenerateharmonicsanddegradedeviceperformance.TheliteratureonpMUTs,though,comparesfavorablywiththeperformancerequiredofaparametricarraytransducer.Thus,piezoelectrictransductionisselectedforthedevicedesign. 101

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Thischapterdescribesthepiezoelectricultrasonicradiatordesignedfornonlinearacous-ticapplications.ThedevicesweredesignedaroundAvagoTechnologiesLimited'shighvolume,lmbulkacousticresonator(FBAR)process.ThedesignofstructuresusinganestablishedfabricationprocessisacommongoalofMEMSdesignsinceitavoidsprocessdevelopmentandtheassociatednon-recoverableengineeringcosts[16].Thetradeoofthismethodistheconstraintofthedevicedesignbythelimitationsanduncertaintiesassociatedwiththeprocess.Also,uncertaintieswithintheprocessmayarisethatwerenotcrucialtotheoriginalapplicationofthefabricationprocess.Consideringthesetradeos,theopportu-nitytoleverageanexistinghighvolumecommercialfabricationfacilitysuchasAvago'swasfavorableforthedesignofanultrasonicradiatorfornonlinearacousticapplications. ThestructuresavailableusingAvago'sFBARprocessiscoveredinSection 4.1 includinganoverviewoffabricationsteps.Afterwards,theimplementationoftheFBARprocesstoformtheultrasonicradiatorstructureisoutlinedinSection 4.2 .Finally,thepackagingdesignandmethodsaresummarizedinSection 4.3 TheFBARfabricationwasleveragedtoformanAlN/Mo/AlN/Mo/AlN/AlNcompositediaphragmonasiliconsubstratewithabackcavityasshowninFigure 4-1 [151].Thefabricationbeginswitha150mmdiameter,675mthicksiliconsubstrate.First,ashallowcavitythatdenesthediaphragmboundaryisetchedinthefrontofthesilicon[98,151]. 102

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Figure4-1. Cross-sectionofthestructurallayersandbackcavitypossibleusingtheAvagoFBARprocess. Oncethesubstratehasbeenplanarized,AlNandMolayerdepositionsbegin.Althoughtheexactdepositionprocessfortheselayersisproprietary,sputteringofAlNandMolmsiscommonintheliterature[105{108,110].First,thestructurallayer(referredtointhisdisser-tationas\scaolding")ofAlNisdeposited.Next,athinAlNlayerisdeposited(notshowninFigure 4-1 ).ThinAlNlayersarecommonlyusedinAlN/MocompositesasseedlayerstoorientthesubsequentlydepositedMoandAlNlms[106].ThebottomMoelectrodeisdepositedovertheAlNseedlayer.Next,theactivepiezoelectricAlNlayerisdepositedfollowedbythetopMoelectrode.Finally,anAlNpassivationlayerisdeposited[98].SinceAlNisadielectric,thislayerelectricallyisolatesthetopMoelectrodefromtheenvironment.AllMoandAlNlayersareresiduallystressedineithertensionorcompression.Moredis-cussiononlayerstressesiscontainedinthefollowingparagraph.AlllayersotherthantheAlNscaoldingandseedlayerscanbepatternedaccordingtodesignerdenedgeometry.Oncethelayerdepositionsarecomplete,thebackcavityisformedusingadeepreactiveionetch(DRIE)[152]fromthebackofthesubstratetothesacriciallayer.Theresultingcavityisstraightwalled.ThesacriciallayeristhenetchedfromthebacksidetoreleasetheAlN/Mo/AlN/Mo/AlN/AlNcompositeasshowninFigure 4-1 Asnotedpreviously,eachoftheAlNandMolayersemergefromthefabricationpro-cesswithresidualstresses.Avagoattemptsmanipulationofthestressineachlayerby 103

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6 .MeasureofstressesarereportedinChapter 7 4-2 wasproduced.ThestructureinFigure 4-2(a) possessestopandbottomMoannularelectrodes.TheAlNpassivationlayerisalsoannular.TheactivepiezoelectricAlNandAlNpassivationlmsarecontinuousacrossthediaphragm. Gold(Au)bondpadsandafrontsideventarealsooptionsinAvago'sprocessasseeninthetopviewimageofthedeviceinFigure 4-2(b) .Theventallowsstaticpressureequalizationbetweenthefrontofthediaphragmpreventingastaticdeectionofthediaphragmduetoatmosphericpressurechanges. Devicecross-section. (b) TopviewofthedeviceshowingtheAubondpadsandfrontsidevent. MEMS-basedultrasonicradiatorforparametricarrayapplications. Oncebatchfabricationwascomplete,thewafersweresingulatedintodiebyDynatexInternationalusingascribeand"smartbreak"process[153].Theseparateddieareapprox-imately3mmby3mm. 104

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Figure4-3. PCBboardfordevicepackage. Thedevicewasmountedintheprintedcircuitboard(PCB)boardasshowninFigure 4-3 .ThePCBboardwasformedofFR4materialwithgoldplatedbondpads.ArecesswasmachinedinthePCBboardsothatthediecouldbeushmounted.Thecircularholesatthefourcornersoftherecesswerewellsforepoxytoholdthedieinplace.Thewellsgavetheepoxyroomtoexpandwhilecuringtopreventimpartingstresstothedie.ThebackcavityofthedevicewasextendedviaaholemachinedthroughthePCBboardinthemiddleoftherecess.DowelandbeveledmachinescrewholeswereaddedtoalignandmountthePCBboardtothealuminumblockshowninFigure 4-4 ,whichcontainsthevariablebackcavity.MachiningofthePCBboardswasaccomplishedateitherTMREngineeringorinhouseusinga Sherline2000-series CNCmill. Theattachmentofdietoasupportingstructurehasthepotentialtoaectdeviceper-formance.Itispossibleforanepoxytoconductheatand/orelectricityoractasaninsulator. 105

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Typicalepoxydispenseandcureparametersusedfordieattachment.PCBboardswerepre-heatedat145Cfor5min.beforeepoxyapplication. 45 45 Vacuum(mmHg) 29.5 29.5 DispenseTime(s) 2 2 Tip(gauge) 30 30 CureType Heat/Optical Optical Oven 135C,3min. NA Lamp(405nm)0:5mW cm2at0.3m 10min. 10min. Cyberbond 'sDualbond707andKatiobond45952.Bothareone-part,modiedresinepoxiesthatcureopticallyusingvisiblewavelengthsfrom400to550nm.Dualbond707hastheaddedbenetofcuringther-mally.Theepoxiesweredispensedusing EFD 'sUltra2400SeriesDispensingWorkstation,apneumaticepoxydispenser.DispenseandcureparametersarecontainedinTable 4-1 Table4-2. Wirebondersettings. BallBond* 3 5 3 WedgeBond* 7 5 4.5 OncethediewasattachedtothePCB,electricalconnectionsweremadeviaAuwirebondsusinga Kulicke&Soa 4124SeriesManualBallBondingSystem.Therstbondwasaball-bondandthesecondawedgebond.Toaugmentsuccessfulwirebonding,thePCBwasplacedonaheatedworkholder.WirebondingparametersandsettingsarecontainedinTable 4-2 106

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IsometricviewoftheAlblockonwhichthePCBismounted. (b) Cross-sectionofAlblockshowingthevariablebackcavity. Thealuminumblockcontainsvariablebackcavity,dowelsandscrewholesforalignmenttothePCB,andananchorscrewhole. 3-Drenderedimage. (b) Photograph. Packageddevicemountedtovariablebackcavity. OncetheconnectionsbetweenthedieandthePCBboardwerecomplete,thePCBwasalignedandmountedtothealuminumblockshowninFigure 4-4 viatwodowelsandfourbeveledmachinescrews.ThePCBboardwasalignedwiththealuminumblocksuchthatthethroughholeinthePCBwascenteredonthevariablebackcavityholeinFigure 4-4(a) .AcrosssectionofthealuminumblockisshowninFigure 4-4(b) .A#0machinescrewwasusedtochangethedepthofthebackcavity.Byturningthescrew,thedepthofthebackcavitywasadjusted.Asingleturnconstitutesachangeindepthof320m.Quarterturn 107

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7 .Thefullyassembledpackage,includingBNCelectricalconnection,isshowinFigure 4-5 5 presentsthemodelofthepackageddevice. 108

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Thedesignofanultrasonicradiatorformaximumpressureoutputrequireddetailedun-derstandingoftheentiresystem.Thedevicebehaviorwasdeterminedbygeometry,materialproperties,fabricationinducedeectssuchasresidualstress,andacousticinteractionsofthepackage.Acomprehensivedesignmodelwasnecessarytoelicittherelationshipsbetweendesignparametersanddeviceperformancesooptimizationcouldbeperformed. Thischapterpresentsadetailedmodelofthelinearperformanceoftheultrasonicra-diator.Themodelincorporatestheinteractionbetweentheelectrical,mechanical,andacousticaldomains.Theinteractionbetweenthedomainsiscapturedusinganequivalentcircuit.Intheequivalentcircuit,thedomainsarerepresentedusingcombinationsofcircuitelements.ThecircuitelementsareformedusingmodelsdiscussedinthischapterandinAppendix B Section 5.1 presentstheequivalentcircuitmodeloftheultrasonicdevice.Derivationoftheacoustical,mechanical,andelectricalcircuitelementsareincluded.Section 5.2 outlinesanonlinearacousticmodelforanarrayofMEMStransmitters.Thesourceconditionforthenumericalsolutionissolvedfromthelumpedelementmodel.Thenalsectionpresentsmodelingresultsfromanon-optimalexampleofadesignthatdoesnotincorporatein-planestress. 5-1 .Thedevicecomponentsarerepresentedbylumpedelementswhenthedevicedimensionsaresmallerthanthewavelengthsofinterest.Forexample,thediaphragmismodeledasasecondordersystemwithmass,compliance,andresistance.Thecircuitrepresentationofasecondordersystemisainductor,capacitor,andresistorinseries.Resistorsrepresentdissipativeelements. 109

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Figure5-1. Diagramofincrementaldiaphragmdeection. TheequivalentcircuitthatmodelsthedeviceperformanceisformedusingmodelsfromboththeacousticandelectricaldomainsasshowninFigure 5-2 .Atransformerisusedtoconnectthetwoenergydomains.Thediaphragmismodeledasapistonwithanacousticmass,MAD,compliance,CAD,andresistance,RAD.Theequivalentresistancecombinesthedissipationeectsinthediaphragmduetothermoelasticdissipation,acousticradiationintothesupports,andothereects.Thediaphragmalsoisloadedbyaradiationimpedance,Zrad,duetoloadingoftheair.Theradiationimpedanceiscomplex.Thereactancerepresents 110

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Figure5-2. Equivalentcircuitmodel. Theelectricaldomainismodeledasaresistanceinserieswithacapacitanceandresis-tanceinparallel.ThecapacitanceisformedbetweenthetopandbottomelectrodesacrossthepiezoelectricAlNlayer.TheparallelresistancemodelsdissipationduetoleakagethroughthedielectricformedbytheAlNpiezoelectriclayer.Theseriesresistancemodelsdissipationduetotheelectricalleads,wirebondcontacts,andinterconnectsbetweenthedevicepadsandthemolybdenumelectrodes. 111

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f:(5{1) Whentheacousticwavelengthismuchlargerthanthediaphragmradius,thepressureisassumeduniformoverthediaphragm. Thedenitionofthebendingwavelengthofthisspecicstructureiscomplexduetotheradialnon-uniformityofthediaphragm.Thescalingofthebendingwavelengthismoreeasilyillustratedbyconsideringauniform,isotropicplatewithclampedboundaryconditions.Thebendingwavelengthscalesasfollows, A!2:(5{2) Whenthebendingwavelengthislargerthanthediaphragmradius,thediaphragmisassumedtovibrateinthefundamentalmodesuchthattheentiredeectionmodeshapeisinphase.Thus,thediaphragmcanbeapproximatedasalumpedmassandcompliance.Dissipationintheformofaresistanceisintroducedsothatthediaphragmresponseatresonanceisnite. 112

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PV=0:(5{3) Thus,CAD,isknownastheshortcircuitacousticcompliance.Thevolumedeectionisfoundfromtheincrementaldeectionofthediaphragm,winc,as {V=Zwinc(r)2rdr:(5{4) TheincrementaldeectionoftheplateisfoundfromthemechanicalmodelinAppendix B Theacousticmassofthediaphragm,MAD,is[56] {VV=02rdr;(5{5) whereAisthearealdensitygivenby Notethatthearealdensityisafunctionofradiusduetothenon-radiallyuniformcompositediaphragm. ByinsertingEquation 5{3 intoEquation 5{5 ,theacousticmassisrewrittenintermsoftheacousticcomplianceas C2ADZA(r)winc(r) Aresistance,RAD,isaddedtothediaphragmtoaccountfordissipation.Anaccuratemodelthatincorporatesallofthedissipationmechanismswithinthediaphragmisbeyondthescopeofthiswork.Theresistancevalueisbasedonpreviousexperimentalresultsofpiezoelectricdiaphragmtransducers[134].Theresistanceisfoundfromthedampingcoecientby[51] 113

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3 .Theexpression,repeatedhereforcompleteness,is Notethatthepistonradiusisnottheouterradiusofthediaphragmbuttheeectivepistonradiusfoundbyequatingthevolumedisplacementofthediaphragmtoapistonwithaneectiveradius,aeff.Theeectivearea,Aeff,isgivenby[56] winc(r=0)V=0:(5{10) Thus,thepistonradiusisaeff=p 5-1 ,thebackcavityconsistsofthreesectionsofdieringcross-sectionsandmaterials.Thebackcavityismodeledasthreesoundhardrigidductsofvaryingcross-sectionalarea.Dissipationinthebackcavityisaccountedforintheformofacousticabsorp-tionduetothermoviscousdissipationandmolecularrelaxationaswellasboundarylayerabsorption.Acousticabsorptionduetomolecularrelaxationandthermoviscousdissipationinair,air,iscoveredinSection 3.2.3 .Dissipationduetoboundarylayerabsorptionoccursduetotwophenomenon.First,thereisano-slipboundaryconditionatthebackcavitywallsduetotheuidviscosity.Thetransitionregionbetweenthewallandthebulkacousticos-cillationisknownastheacousticboundarylayer[8].Thesecondboundarylayerabsorptionphenomenonisthethermalboundarylayer.Atthewall'sedge,theexpansion/contractionoftheuidthatisnormallyassumedadiabaticinfree-spaceisnowisothermalsincetheboundaryideallyactsasaninniteheatsink.Theabsorptionofbothboundarylayerscanbeaccountedforinacombinedboundarylayerabsorptioncoecient[8], 114

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Figure5-3. Rigidductmodel. Toderivethebackcavityimpedance,rstconsiderplanewavestravelingalongastraightwalledductofcircularcross-section,asinFigure 5-3 .Thepressureandvolumevelocityintheductaregivenby[8]P=Aej^kx+Bej^kx Z0ej^kxBS Z0ej^kx; respectively,whereSisthecross-sectionalareaoftheductand^kisthecomplexwavenumber,[8] ^k=kj:(5{15) Nowconsideraniteductoflength`.Thepressureandvolumevelocityattheductentrance(x=0)areP0=A+B Z0BS Z0; respectively. 115

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Z0ej^k`BS Z0ej^k`: Equations 5{16 and 5{18 arecombinedtoyield where[T]isthetransfermatrixthatrelatesthepressureandvolumevelocityattheendoftheducttothatatthebeginningoftheduct.Thetransfermatrixisgivenby[8] [T]=264cos^k`jZ0 Z0sin^k`cos^k`375:(5{21) Figure5-4. Thebackcavityofthetransducerwiththedie,pcb,andaluminumsections. Figure5-5. Circuitrepresentationofthebackcavityusingtransfermatrices. 116

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5-4 .Thesectionsareestimatedasrigidductsofdieringcrosssectionandlength,eachwiththeirowntransfermatrix.Theroughnessofthreadsinthealuminumsectionareassumednegligiblesincetheirpitchismuchsmallerthantheacousticwavelengthsofinterest.Theback-cavityterminatesatthealuminumscrew,whichisassumedtobeasoundhardboundarycondition.Thetotalbackcavitytransfermatrixisgivenbythemultiplicationofthetransfermatricesofeachsection, [TBC]=[TSi][Tpcb][TAl]:(5{22) ThecircuitrepresentationofthebackcavityisshowninFigure 5-5 .Notethatthesoundhardboundaryterminationdictatesanopencircuitafterthelasttransfermatrix.Theimpedanceofthebackcavityisfoundfromtheratioofthepressureandthevolumevelocityatthebackcavityentrance,givenas wherethe'0'and'`'subscriptsrefertothecavityentranceandtermination,respectively.Thus,thecavityimpedanceisgivenby 5-6 .Theventisa2mthickchannelthatis50minlengthand25minwidth.Theventismodeledassumingfully-developedpressuredrivenow[17].Theresistanceofthechannelis[16] Wh3;(5{25) wherehistheheight,Wisthewidth,andListhelength. 117

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Edgeofthediaphragmshowingthefronttobackcavityvent(nottoscale). 5-7 .TheresistancesarefoundfromexperimentalcurvetsoftheelectricalimpedanceinChapter 7 .Thefreecapacitance,CEF,isgivenas whereistheabsolutepermittivity,Apistheplanformareaoftheelectrodes,andhpisthethicknessofthepiezoelectricaluminumnitridelayer. Figure5-7. Electricalelementsoftheequivalentcircuit. 118

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3 ,repeatedhereforconvenience[95],are264IU375=264j!CEFj!dj!dj!CMS375264VF375: whereQisthevolumevelocityanddAistheeectiveacousticpiezoelectriccoecientfoundas[56], VP=0:(5{28) Thetwoportmodel,Equation 5{27 ,canberepresentedusingatransformerwithaparallelshuntcapacitanceontheelectricalsideandaseriescomplianceonthemechanicalsideasshownbelowinFigure 5-8 .ThetransformerturnsratioAistheelectro-acoustictransduc-tioncoecient, Theblockedelectricalcomplianceis wherek2istheimpedance-couplingfactorgivenas 5.1.1 throughSection 5.1.3 arecombinedtoformequiv-alentcircuitasshownpreviouslyinFigure 5-2 .Theelectricalsideofthetransformeris 119

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Two-portrepresentationofelectro-acoustictransduction. convertedintotheacousticdomainusingtheelectroacoustictransductioncoecient.TheelectricalelementsintheacousticdomainaredenedasCEBA=CEB TheresultingequivalentacousticcircuitisgiveninFigure 5-9 ,whereacousticequivalentinputvoltageisp2=AV.TheimpedancesinFigure 5-9 are Figure5-9. Equivalentacousticcircuitmodel. 120

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V=Q p2A=A AsissubsequentlydiscussedinSection 5.2 ,thevolumevelocitysensitivityisausefulmetricwhenforminganarrayofradiators. Anotherimportantperformanceparameteristheinputelectricalimpedancethatisdrivenbythevoltagesource, 1+;(5{40) where =ZEBAZAC 5{39 and 5{40 aresimpliedandinsightintotheimportanceofparticulardeviceelementsisgained.First,theserieselectricalresis-tanceisexpectedtobemuchsmallerthanthetotalimpedanceoftheblockedcapacitanceandparallelelectricalresistance, ThisisconrmedexperimentallyinChapter 7 .TheresultofthisassumptionisthattheacousticequivalentinputvoltageisapplieddirectlyacrosstheacousticelementsonthelefthandsideofFigure 5-9 Also,theventimpedanceisonlyimportantatlowfrequencieswherethebackcav-ityimpedanceislargeenoughtocauseacurrentdivider.Sincethedeviceoperatesnearresonance,theeectoftheventinparalleltothebackcavityimpedanceisnegligiblysmall, 121

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5.1.6 Applyingtheseassumptions,thesensitivitysimpliesto Thus,thehighfrequencyperformanceofthedeviceismainlydependentonthediaphragmandbackcavityimpedances.Theresonantperformanceisdominatedbytheresistanceofthediaphragm,radiation,andcavityimpedances.GivenEquation 5{43 ,theinputelectricalimpedance,Zinput,simpliesto 1+;(5{45) where =ZEBA Thefollowingsectiongivesthepredictedperformanceofanexampledevice. 5-1 .Forthisexample,thestressinthediaphragmlayersisassumedtobeneutral.Also,thebackcavityisassumedtobeexactlyaquarterofthediaphragm'sresonantwavelengthsuchthatatresonancethebackcavityimpedance,ZAC,iszero.Dissipationinthebackcavityisalsoneglectedintheexample.Thedampingratioisassumedtobe0.03basedonpreviousexperience[46].TheparallelelectricalresistanceisbasedonaresistivityextractedfrominitialexperimentalimpedanceresultsandisgiveninTable 5-2 alongwithothermaterialproperties.Theserieselectricalresistanceisassumedtobe40basedoninitialexperimentalimpedanceresults. TheequivalentcircuitresultsinFigure 5-10 showaresonantfrequencyof45:2kHzandavolumevelocityradiationsensitivityof34:2mm3/s/Vatresonance.Furtherinsightintothedevicemodelisgainedbymakingrelevantcomparisonsofdeviceperformanceandimpedance. 122

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Geometryofexampledevice. MaterialpropertiesofAlNandMo. InFigure 5-10 ,thefullsensitivityresult,Equation 5{39 ,anditsapproximation,Equation 5{44 ,areplotted.AsobservedinFigure 5-10 ,theapproximatesensitivityestimationalmostexactlyreproducesthefullsensitivitycalculationoverthefrequencyrangeofinterest.ThisjustiestheuseofthehighfrequencyestimationofthesensitivityintheoptimizationschemeinChapter 6 AplotofthetotalimpedanceofthebackcavityinparallelwiththeventresistanceisshowninFigure 5-11 .Clearly,theventimpedanceislargeenoughincomparisontothecavityimpedancethatitcanbeneglected.Thus,thehighfrequencyassumptionthatZAC=ZAV1isfurtherjustied. ItisalsoinformativetonotetherelativecontributionoftheradiationimpedanceincomparisontothediaphragmimpedanceinFigure 5-12 .Theradiationimpedanceaddsasignicantcontributiontotherealcomponent.Italsocontributestothereactivecompo-nent,leadingtoaslightlydierentresonantfrequency.Theshiftinresonanceleadstoa 123

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Volumevelocityfrequencyresponsecalculatedusingthefullsensitivityequiv-alentcircuitandthehighfrequencyapproximation. Figure5-11. Totalimpedanceoftheventandcavityinparallelversusjustthecavityimpedance. largepercentagerelativeerrornearresonancebetweentheimaginarypartofthediaphragmimpedance,ZAD,andthetotalimaginaryimpedance,ZAD+Zrad.Basedontheseresults,theradiationimpedanceisincludedinthemodelusedintheoptimization. Next,thecontributionsofthebackcavityimpedanceandthediaphragmimpedancetotheoverallacousticimpedancearecomparedinFigure 5-13 whereZAMisthetotalimpedanceoftheacousticelementsgivenby 124

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Diaphragmimpedancecomparisontotheradiationimpedance. Figure5-13. Contributionstotheoverallacousticimpedance. Itisimportanttonotethatinthisanalysis,thebackcavitydepthistunedtomatchthequarterwavelengthoftheresonantfrequencyofthediaphragm.Thus,althoughthecon-tributionofthecavityimpedanceseemsinsignicantincomparisontothediaphragm,itshouldbeincludedinthedevicemodelsincethephysicallyrealizedcavitydepthdoesnotexactlymatchthequarterwavelength. Theinputelectricalimpedance,Equation 5{40 ,isplottedinFigure 5-14 togetherwiththeuncoupledelectricalimpedance,RES+ZEB,whereZEB=ZEBA=2A.Asobservedin 125

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5-15 forbothEquation 5{41 andthehighfrequencysimplication,Equation 5{46 .Clearly,thehighfrequencyestimateofisaccurateoverthefrequencyrangeofinterest.Thepeakinisalsoobservedtooccurattheresonantfrequency.ThisiseasilyconrmedsincethedenominatorofisthesameasthatofthevolumevelocitysensitivityinEquation 5{44 Figure5-14. Inputandtheelectricalimpedancecomparison. Figure5-15. factoroftheinputimpedance. InChapter 7 ,thecapacitance,parallelresistance,andseriesresistanceareextractedfromexperimentalmeasurementsoftheinputimpedance.AsisshowninFigure 5-14 ,theacousticandelectricaldomainsarecoupled,resultinginthepeakintheresistivecomponentoftheinputimpedance.Equation 5{45 containsmodelsoftheradiationandbackcavityimpedancesthatarenotlumpedelements.Thus,afrequencyresponsefunctionestimationoftheexperimentallymeasuredinputimpedancetoEquation 5{45 usingbasiccircuitsanalysisisnotpossible.AloworderTaylorseriesapproximation,orLEM,oftheradiationandcavityimpedancesresultsinaZinputthatcanbettoafrequencyresponsefunction.First,the 126

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3{16 andisrepeatedhereforconvenience,ZradRrad+j!Mrad=0c0(kaeff)2 5{9 tothelumpedelementmodelisgiveninFigure 5-16 .Theerrorisshowntobelessthan5%intherealandimaginarypartsoverthefrequencyrangeofinterest. Figure5-16. RadiationimpedancecomparisonofthefullmodelandLEM. Figure5-17. BackcavityimpedancecomparisonofthefullmodelandLEM. 127

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where{VandSarethecavityvolumeandcross-sectionalarea,respectively.AcomparisonofthefullcavityimpedancemodeltotheLEMisshowninFigure 5-17 .TheresultsshowasignicanterroraroundtheresonantfrequencywhereLEMbreaksdownsincetheacousticwavelengthisnolongermuchlargerthanthecavitydepth. Figure5-18. InputimpedancecomparisonofthefullmodelandLEM. AcomparisonofthetotalinputimpedancebasedonthefullmodelversusthelumpedelementmodelisgiveninFigure 5-18 .Theerrorintheresistivecomponentrisesto20%.TheerrorintheextractionoftheelectricalcomponentsfromexperimentaldataissimulatedbyttingtheLEMinputimpedancemodeltothedistributedimpedancemodel.Thetransferfunctionusedtottheimpedanceis 128

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5{49 isshowninFigure 5-19 .TheelectricalimpedancesarefoundfromthefollowingequationsRES=a0 TheextractedelectricalimpedancesaregiveninTable 5-3 .Clearly,theuseoftheLEMtottheelectricalelementsinChapter 7 isjustied. Figure5-19. FullmodelandcurvetcomparisonassumingaLEMoftheinputimpedance. Table5-3. Comparisonofelectricalelementttofullmodel. 129

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2.2.2.2 isusedtoestimatethenonlinearacousticoutputofanarrayoftransducers[25].ThesourceconditionoftheBerktaysolutionassumesasinglepiston.TheimplementationoftheMEMS-basedpara-metricarrayutilizesanarrayofmanysources,eachwithavolumevelocitypredictedbytheequivalentcircuitoftheSection 5.1.4 .Thus,thesinglepistonvolumevelocityistakentobethesummationofthevolumevelocityofanarrayofNradiators.Whenformingthearraypattern,hexagonalpackingofthedevicesisassumedwithacentertocenterspacingof(1+)dtrans,wheredtransisthediameterofthetransducerandisapositivenumberthataccountsforthespacebetweenthediaphragmedges.Forhexagonallypackedcircles,thetotalsurfaceareaofthetransducersplustheirspacingaccountsfor90%[156]ofthetotalarrayareaasillustratedinFigure 5-20 .Thus,thearraydiameter,d,isrelatedtothetransducerdiameterby, Figure5-20. Diagramoftransducerarrayshowingradiusdenitions(Notdrawntoscale). 130

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wherethemodulatingfunctionisgivenby where!a=2faandfaistheaudibletoneofinterest.Thus,thesourceconditionisrewrittenas NotethattheBertkaysolutionassumesthatthesourceamplitudesatbothoftheorigi-natingfrequenciesarethesame.Althoughtheamplitudeofthevolumevelocityfrequencyresponseofthetransducervariesaroundresonance,thesamevolumevelocityamplitudecanbeachievedusingtwodierentvoltageexcitationsignals.Theupperlimitofthevolumeve-locityamplitudeisdeterminedbyitslinearlimit.AsisshownexperimentallyinSection 7.4.4 thatupperlimitoflinearityisdictatedbythevelocityamplitude.Thus,bycompensatingthetwovoltageexcitationsignals,thesamevelocityisachievedateachfrequency.Thus,p0isfoundas whereQ0isthevolumevelocityofhalfofthetransducers. TheequivalentcircuitresultsoftheexampleinSection 5.1.6 showaresonantfrequencyof45:2kHzandavolumevelocityradiationsensitivityof34:2mm3/s/Vatresonance.Con-sideranexampleofaparametricarray141mm(5.6in)indiameter.Eight-thousandtrans-ducerstwithinthearraydiameterwithacentertocenterspacingof1.5mm.Aconserva-tiveestimatebasedonexperimentalresultsinSection 7.4.4 isa1Vexcitationatresonance.GiventheparametricarrayanalysisbasedonBerktay'ssolution[25]outlinedinSection 5.2 ,theaudiblefrequencyresponseat1misgiveninFigure 5-21 .Notethatthereis20dbper 131

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Figure5-21. Outputoftheexampleparametricarrayat1m. 6 inanoptimizationscheme. 132

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Itisdiculttointuitivelychoosedevicegeometrytocreateanultrasonicradiatorde-signthatmeetsorexceedstheapplicationrequirements.Therefore,anoptimizationschemeisusedintelligentlydeterminedevicedimensions.Inthischapter,theobjectiveoftheopti-mizationandtheconstraintsonthedesignaregiven.Optimizationresultsincludingdesignsensitivityaregiven.Theoptimizationwillalsoprovideinsightintothesensitivityofthedevicebehaviorwithrespecttoindividualdesignvariables. 5 ,thearrayoftransducerscanbemodeledbyasinglepistonbyequatingvolumevelocities.AsshowninChapter 3 ,theon-axisoutputofanarrayoftransducersisrelatedtotheoutputofasingletransducerbyaconstant.Thus,theoptimizationoftheoutputofanarrayofmanytransducerscanbereducedtotheoptimizationoftheoutputofasingletransducer. AsdiscussedintheintroductionofparametricarraysinChapter 1 andChapter 2 ,thepressuresofthefundamentaltonesactassourcesforthedierencefrequency.ThepressuresofthefundamentaltonesareassumedtobefoundfromRayleigh'sintegralforabaedpiston,discussedinChapter 3 ,becausethereisweaknonlinearconversion.ThepressureisrelatedtothediaphragmvelocityasgiveninChapter 3 .Maximizingthevelocityorvolumevelocityofasingletransducer,however,doesnottakeintoaccountacousticattenuationduetomolecularrelaxationastheacousticwavespropagateawayfromthetransducer. AmoreappropriateoptimizationschemeisbasedonthedierencepressureresultoftheBerktaysolutionoutlinedinChapter 2 .TheBerktaysolutionaccountsforacoustic 133

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MATLAB'sfminconfunctionwillserveastheoptimizationtoolinthismethodology.Thisfunctionusesageneralclassofgradient-basedlocaloptimizationcalledsequentialquadraticprogrammingwithnonlinearconstraints[157].DetailsonfminconcanbefoundinMATLAB'shelple. p2(r=0;z=1m)'0u20(fc)Af2a where,c0,and0arepropertiesofthesurroundingair.Thearrayarea,A,andtheaudiofrequency,fa=5kHz,arexedfortheproofofconceptapplication.Thevelocityamplitude,u0,isdirectlyrelatedtotheaveragevelocityofthediaphragmatresonancebyaconstant.Theattenuationisdependentuponthecarrierfrequency,0(fc).Thecarrierfrequency,fc,isalsotheresonantfrequencyofthedevice.Thus,theoptimizationwillmaximizethefollowingratiofromEquation 6{1 : Notethattheresonantfrequency,fc,andtheaveragevelocityatresonance,u0,arearesultofthemechanicsofthediaphragmcoupledwithitsradiationimpedance. Thevolumevelocitysensitivitytoappliedvoltage,Equation 5{39 ,derivedfromtheequivalentcircuitisconvertedtovelocitysensitivitybydivisionoftheeectivearea,Aeff.Theappliedvoltageislimitedbythebreakdownvoltage,VBD,oftheAlNpiezoelectriclayer.ThebreakdownvoltageisfoundbymultiplyingthedielectriceldstrengthofAlN,Em[159],bythethicknessoftheAlNpiezoelectricthickness,hp.Notethatthebreakdownvoltageismostlikelyoverlygenerousasnonlineareectswilllikelybecomeappreciableatlower 134

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6{2 areintermsofvelocity.Thus,theobjectivefunctioninEquation 6{2 becomes SincethedielectricstrengthisamaterialpropertyxedbyusingtheAvagoFBARprocess,itcanberemoved.Thenalobjectivefunctionis 6-1 Table6-1. AvagoTechnologiesLimitedprocessoptionswherej=1;2referstotheinnerandannularplatesections,respectively. ScaoldingAlNH(j)1500nm2,000nmBottomMoElectrodeH(j)3200nm400nmPiezoelectricAlNH(j)4300nm800nmTopMoElectrodeH(j)5200nm700nmTopAlNPassivationH(j)650nm300nm 250mR2750m:(6{5) 135

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Severalmodelingconstraintsarealsonecessary.First,itiscrucialtoensurethatthediaphragmsectionsareaccuratelymodeledasplates.Theaspectratioisthusconstrainedaccordingto[161]R1 ThisconstraintcouldberelaxedbyimplementingMidlinplatetheorythataccountsforsheardeformationstogiveaccurateresultsforthickerplates. Theeectsofresidualstressesandradialnon-uniformityofthediaphragmleadtoinitialdeectionthatisdiculttopredictusingthelinearplatetheorypresentedinAppendix B .ForthelinearplatemodeltoaccuratelyrepresentthediaphragmmassandcomplianceusedintheequivalentcircuitofChapter 5 ,theinitialdeectionmustremainlinear.Inaddition,constrainingtheinitialdeectionswillalsoensurethattheplateispre-buckled.Buckleddiaphragmsofsimilarresonantfrequencyandsensitivitytopre-buckleddiaphragmshaveshownadecreaseinlinearrange[162].Also,higherorderbucklingmodescanhaveareasofthediaphragmthatvibrateoutofphase,reducingtheeciencyofthedeviceasaradiator.Thus,itisadvantageoustoconstrainthedevicetobepre-buckled. Toformulatetheconstraintontheinitialdeection,thedeectionmodeshapesfoundfromthelinearandnonlinearplatemodelsinAppendix B mustbecompared.Insteadofcomparingthemodeshapespointbypoint,thetotalstaticvolumedeections,{V,arecomparedbetweenthelinearandnonlinearsolutions.Theconstraintontheinitialdevicedeectioniswrittenas 136

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Anoperationalconstraintisappliedtothecarrierfrequencytoensurethatthedevicesoperatesafely.Thecarrierfrequencymustbehighenoughtoguaranteethattheamplitudemodulationwillnotshiftoneoftheoriginatingtonesintothehumanhearingrangewheretheamplitudewouldbeabovethehumanthresholdforpain.Althoughthisconstraintcouldbelowereddependingonthemodulationtype,intheliterature[11]thecarrierfrequencyisusuallyconstrainedtobe35kHzorhigher, Asensitivitystudyofeachoftheconstraintsisprovidedintheoptimizationresults. Minimize[x]=hR1;R2R1;H(j)1;H(j)3;H(j)4;H(j)5;H(j)6ifobj([x])suchthatLB[x]UB1R1 137

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6{4 .Thedesignvariablesaretheinnerradius,thedierencebetweentheinnerandouterradius,andthethicknessesofthebottomMoelectrode,piezoelectricAlN,topMoelectrode,andtopAlNpassivationlayers.Thestressinallofthelayersisassumedtobe20MPacompressivestresswiththeexceptionoftheAlNscaoldinglayerwhichisassumedtobestressneutral. 6-2 .Thevolumevelocitysensitivityis68.2mm3/sataresonantfrequencyof35kHz.Usingthesamenonlinearacous-ticcalculationoutlinedinSections 5.2 and 5.1.6 ,theparametricarrayfrequencyresponseat1misgiveninFigure 6-1 .Notethatthearraydiameterisslightlylargerat154mm(6in)diameterduetothelargertransducerdiameter.TheSPLis54.5dBre20Paat5kHz.Thereisa10.5dBimprovementincomparisontothenon-optimalexamplewithzeroresidualstressesofSection 5.1.6 Figure6-1. Outputoftheparametricarrayat1m. Insightintotheimportanceofthedesignparametersisgainedbynotingthesensitivityoftheobjectivefunctiontochangesinthedesignvariables.Thenormalizedchangeintheobjectivefunctionversusthenormalizeddesignvariables,representedbyX,isshowninFigure 6-2 .Thedesignvariablesandobjectivefunctionarenormalizedbytheiroptimalvalues,Xoptandfobj(Xopt),respectively.Therangeinthedesignvariablesis20%.From 138

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6-2 ,theobjectivefunctionismostsensitivetotheAlNscaoldinglayerthickness,H1,andtheinnerradiusoftheannularpiezoelectricring,R1.Next,theannularringthickness,R2R1,andpiezoelectricAlNthickness,H4,alsoaectthedesignperformance.Theobjectivefunctionisleastsensitivetothemolybdenumelectrodethicknesses,H3andH5,andtheupperpassivationlayerthicknessofAlN,H6.ItwouldseemthatadecreaseintheAlNscaoldinglayerthicknessoranincreaseintheinnerradiusoftheannularpiezoelectricringwouldleadtoenhancedperformanceofthedevice.AsshowninFigure 6-3 ,however,thiswouldleadtoconstraintviolations. TheobjectivefunctionandactiveconstraintsensitivitiesfortheAlNscaoldinglayerthickness,H1,theinnerradiusoftheannularpiezoelectricring,R1,theannularringthick-ness,R2R1,andpiezoelectricAlNthickness,H4,aregiveninFigure 6-3 .Thenormalizedconstraintsareplottedsuchthattheyareactivewhentheyexceedavalueofone.Theresonantfrequencyandnonlinearinitialdeectionconstraintsareactive.Thedesignspaceregionswheretheconstraintsareactiveareshadedincolor.Theupperorlowerboundofeachdesignvariableisviolatedinthehatchedregion.Forexample,inFigure 6-3(b) ,theAlNpiezoelectricthicknessisatitsupperbound.AnincreaseintheAlNpiezoelectricthicknesswouldresultinahigherobjectivefunctionbutitisconstrainedbyitsupperbound.Ade-creaseintheAlNpiezoelectricthicknesswouldresultinadecreaseintheobjectivefunctionaswellasaviolationofbothnonlinearconstraints. Theoptimizationisalsosensitivitytotheheuristicnonlinearconstraints,specicallythepercentnonlinearityinthestaticdeectionandtheminimumresonantfrequency.Theoptimizationwasrepeatedusingdierentvaluesfortheheuristicconstraints.InFigure 6-4 ,therelativesensitivityoftheobjectivefunctiontothechoiceofthepercentdierencebetweenthelinearandnonlinearstaticdeectionisshown.Theobjectivefunctionisreducedbyalmost30%ifamoreconservativeconstraintof5%deviationischosen.Alessconservativechoiceofconstraintof15%resultsinanincreaseintheobjectivefunctionofalmost20%. 139

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Designoptimizationresults. Figure6-2. Sensitivityofthenormalizedobjectivefunctiontothenormalizeddesignvari-ables. Thus,theoptimizationismoresensitivetoareductionintheconstraintvalueratherthanarelaxation. TherelativechangeintheobjectivefunctionwithrespecttotheminimumresonantfrequencyconstraintisshowninFigure 6-5 .Anincreaseintheminimumresonantfrequencyconstrainttoamoreconservativevalueof40kHzresultsina10%decreaseintheobjective 140

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(b) (c) (d) Sensitivityoftheobjectivefunctionduetochangesintheindividualdesignvariablesshowactiveconstraintsandbounds. 141

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Anincreaseintheobjectivefunctionontheorderof10-30%couldbeachievedbyarelaxationofeitherofthenonlinearconstraints.Theaudibleoutputoftheparametricarray,however,isreportedindecibels.Therefore,theincreaseinSPLisnotsignicantenoughtowarrantarevisionofthenonlinearconstraints. Figure6-4. Sensitivityoftheobjectivefunctiontothenonlinearde-ectionconstraint. Figure6-5. Sensitivityoftheobjectivefunctiontotheminimumres-onantfrequencyconstraint. 6-2 wascreatedusingfminconbyassumingthatallofthelayershada20MPacompressivestresswiththeexceptionoftheAlNscaoldinglayerwhichisassumedtobestressneutral.ThiswasthestresstargetofAvagoTechnologiesLimitedfortheproductionrun.ThereisevidenceoflargestressvariationsfrompriorAlN/Mostructures,bothwithreferencetothestresstargetsandacrossthewafer.Takingthisintoaccount,asetofoptimizationswereperformedconsideringvariationinthestressof200%.Thedesignvariablesforthesecondarydesignsweretheinnerandouterradiioftheannularringsincethethicknessesarexedbytheprimaryoptimaldesign.Thisredundantdesignmethodprovidedadegreeofrobustnessthatensuredthatsomepercentageofthenaldevicesdidnotseverelyunder-performtheoriginaloptimaldesignorviolateconstraints.The 142

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6-3 alongwiththeobjectivefunctionreferencedtothatoftheoptimaldesign,UFPA08. Table6-3. Alternateoptimizationresults. UFPA0115512728-12.4UFPA0210483700-10.5UFPA035455672-8.5UFPA041398611-7.1UFPA05-5403618-4.3UFPA06-10379665-2.3UFPA07-15360572-0.3UFPA08-203995440.0UFPA09-25390507-1.3UFPA10-30356463-2.9UFPA11-35330429-4.2UFPA12-40308401-5.2UFPA13-45291378-6.1UFPA14-50276359-6.8UFPA15-55263342-7.5UFPA16-60252327-8.1 5 togetherwithBerktay'snonlinearacousticsolutionareusedtoformanobjectivefunction.Thelayerthicknessesandtopographicalgeometryofthediaphragmarethedesignvariables.AprimarydesignisfoundusingthefminconoptimizationtoolinMatlab.AsensitivityanalysisoftheobjectivefunctionshowsthattheAlNscaoldinglayerthicknessandtheinnerradiusoftheannularpiezoelectricringarethemostcrucialdesignvariables.Activeconstraintsincludeupperandlowerboundsonthelayerthicknessesaswellastheresonantfrequencyandnonlinearinitialdeectionconstraints.Asecondaryoptimizationisperformedonthetopographicalgeometrywhileconstrainingthelayerthicknessesandconsideringalternativein-planestress.Theresultis 143

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144

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Thischapterdescribestheexperimentalsetupsandresultsusedtoperformcharacteri-zation.First,initialresultsofthefabricationarediscussedalongwithamethodforpickingthebestcasedesign.Next,electricalexperimentalsetupandcharacterizationresultsareoutlined.Third,thediaphragminitialdeectionisgiven.Next,theelectromechanicalchar-acterizationresultsusingascanninglaservibrometerarepresented.Finally,ultrasoundmeasurementsofthedevicethatdeneitsdirectivityandonaxisresponsearegivenintheelectroacousticcharacterizationresultssection.Themeasurementresultsareusedtoestimatethedierencefrequencyproductionofanarrayofdevices. 6 ,sixteendierentdevicegeometrieswerefabricated.Fromthemeasurementofstressduringthefabricationprocess,theperformanceofeachdevicedesignwasprojectedusingtheequivalentcircuitfromChapter 5 .Themostpromisingdevicewaschosenbasedonthepredictedperformance. First,thestressmeasurementresultsaregiven.StressexistedineachoftheAlNandMolayersduetofabricationeectsasdiscussedinChapter 4 .Filmstresscauseswaferbowduetoamismatchinstressbetweenthelmandthesubstrate.Thewaferbowcausedbyalmdepositionwasfoundbymeasuringthewaferprolebeforeandafterlmdeposition.ThelmstresswasfoundfromthewaferbowmeasurementusingStoney'sformula[152].ThelmstressmeasurementswereconductedatAvagousingaTencorFlexusFLX5400.ThemeasurementresultsalongwiththestresstargetsaregiveninTable 7-1 .UncertaintiesinthestressmeasurementswereestimatedbyAvagoat20MPaforstresseswhosemagnitudewaslessthan60MPaand30%forlargerstresses.ItwasalsoimportanttonotethatAvagoclaimsthatthestressvariedacrossthewafer,withtheinnerportionofthewaferbeingmoretensilethantheouterportion.Thiswasconrmedbyavisualinspectionofthewafer.Thediaphragmsalongtheouterportionofthewaferappearedtohavelargeinitialdeections 145

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7-1 wastensilewhiletheouterpartofthewaferwasclearlyundercompressivestress,itwasinferredthattheinnerreticlesareunderahighertensilestressthanthatmeasuredbyAvago.TheramicationwasthatthedevicepredictedgiventhestressesinTable 7-1 wasmorecompliantthattheactualdevicestested.Thus,itwasprobablethatthetheoreticalmodelwouldunderpredicttheresonantfrequencyandoverpredictthedevicesensitivity.Thishypothesiswasconrmedfromthecomparisonoftheexperimentalandtheoreticalresultsofthefollowingsections. Table7-1. FilmstresstargetandrealizedvaluesforthewaferprovidedbyAvago. ScaoldingAlN(2m)017BottomMoElectrode(0.2m)-2090.1PiezoelectricAlN(0.8m)-20-20TopMoElectrode(0.2m)-20-31TopAlNPassivation(0.05m)-20-365.1 7-1 ,theequivalentcircuitfromChapter 5 wasusedtopredicttheperformanceofthefabricateddevices.ThepredictedobjectivefunctionofthefabricateddevicesingiveninTable 7-2 indBreferencedtothetargetdesign,UFPA08.Forexample,theperformanceofthefabricatedUFPA08deviceinthesecondcolumnshowsapredictedlossintheobjectivefunctionof-12.9dBwithrespecttothetargetUFPA08deviceintherstcolumn.ThepredictedperformanceofthefabricatedUFPA01device,ontheotherhand,showsalossintheobjectivefunctionof-11.8dBwithrespecttothetarget 146

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Table7-2. ObjectivefunctionofdevicedesignsindBreferencedtothetargetdesign,UFPA08.Blanksrefertodevicesthatviolatemodelingconstraints. UFPA01-11.8UFPA02-11.9UFPA03-12.0UFPA04-12.4UFPA05-12.3UFPA06-13.4UFPA07-12.8UFPA080.0-12.9UFPA09-3.6-14.0UFPA10-6.2-14.5UFPA11-8.0-15.0UFPA12-9.4-15.5UFPA13-10.7-16.1UFPA14-11.7-16.7UFPA15-12.7-17.3UFPA16-13.7-18.0 4.1 ,projectionlithographywasusedtoformtheplanformgeometryofthedevices.Projectionphotolithographystepsasinglemaskpatternacrossawaferinagridtoformmultiplecopiesofthesamemaskgeometryonasinglewafer[163].Eachcopyofthemaskisknownasareticle.TheresultofthefabricationwasagridofreticlesthateachcontainedacopyoftheUFPA01device.Thedevicesarelabeledaccordingtotheirreticlerow(RR)andreticlecolumn(RC)asshowninFigure 7-1 .ThedevicescharacterizedwereRR1RC3,RR2RC3,RR2RC4,RR3RC5,RR5RC5,andRR5RC2,whichareindicatedinFigure 7-1 .AllofthediewiththeexceptionofRR5RC2weremountedtoapcbboardthatallowedforavariablebackcavity,asdescribedinChapter 4 .TheexceptionwastheRR5RC2devicewhichwasmountedtoapcbboardwithnobackcavitythroughhole.Thus,thebackcavitydepthofRR5RC2waslimitedtothesiliconthickness. 147

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Reticlelabelingconvention. HP4294AImpedanceAnalyzer .Theblockedcapacitance,CEB,parallellossresistance,RP,andserieslossresistance,RS,wereextractedfromtheimpedancemeasurements. HP4294AImpedanceAnalyzer has4portsusedtoconnecttotheDUT.Tomakeanimpedancemeasurement,thelowcurrentandlowpotentialterminalsweretiedtogetherasarethehighcurrentandhighpotentialterminalsasshowninFigure 7-2 .TheresultinglowandhighterminalsareconnectedacrosstheDUTforimpedancetesting.TheHP4294AimpedanceanalyzerusesanI-Vmethodtomeasureimpedancewhere I=VHpLp Theimpedanceanalyzerwassettoperformalinearfrequencysweepfrom1to200kHzwith801pointswitha100mVsource.Thebandwidthsettingsrangefrom1to5,where5correspondstothenarrowestnotchltertheimpedanceanalyzerusedtomeasuretheimpedanceatafrequencybin.Thetradeoisanincreaseintimeofthemeasurement.Pointaveragingandaveragingofsweepsareturnedo.Themeasurementwasrepeated30 148

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ThefrontpanelconnectionsontheHP4294AImpedanceAnalyzerwhereredisthehighconnection,blackisthelowconnection,andgreenisground. timesandaveragingwasconductedduringpost-processingtoobtainestimatesoftherandomerrorinthemeasurement.Theestimationofthebiaserrorwastakenfromtheoperationmanual[164].Thecontributionoftherandomandbiaserrorsareindicatedinthefollowingresults.AcomparisonoftherandomtobiaserrorsispresentedinAppendix C Impedancemeasurementswereconductedpre-andpost-packagingtoidentifypackagingeectsontheimpedance.Thepre-packagedimpedancemeasurementswereconductedusingaWentworthLabsprobestationandSignatoneModelS-725probes.Non-repeatabilityoftheshortcalibrationanddriftoftheprobestationledtoerroneousimpedancemeasurements.Thus,thepre-packagedimpedancemeasurementsarenotpresentedinthiswork. 5.1.4 inFigure 7-3 .Theseriesresistance,RES,andparallelresistance,REP,usedinthetheoreticalpredictionofZinputinEquation 5{40 werebasedonanaverageofthevaluesextractedfromtheexperimentas39and31M,respectively.Asobserved,thereisagoodmatchbetweentheimaginarypartofthemeasuredinputimpedanceofallthedevicesaswellasthetheoreticalmodel.AswasshowninSection 5.1.6 ,theblockedcapacitancedominatestheimaginarycomponentoftheimpedance.Sincethedevicescomefromthesamewaferandwereformedusingprojectionlithography,whichindicatesthattheplanformgeometryshouldbewellmatched,thisresultisexpected. 149

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5.1.6 tocoincidewiththeresonantfrequencyoccurredatdierentfrequencies.ThiswasevidencethattheresonantfrequencyvariesbetweendevicesandisconrmedinSection 7.4 .Second,thetheoreticalmodelbeginstoasymptotetotheseriesresistancevalueof39afterthepeakintheresistance.Beforethepeak,thetheoreticalresistanceisdroppingatapproximately40dBperdecade.Theimplicationoftheapproximately20dBperdecadedierenceintheresistanceroll-obetweentheexperimentsandthetheoryisthatthereisatermthatmultipliestheresistancethatincreaseslinearlywithfrequencythathasnotbeenaccountedforinthetheory.Thedirectresultantoftheincreaseintheresistivecomponentoftheimpedanceisareductioninthevolumetricowratesensitivityofthedevice. Figure7-3. Therealandimaginarypartoftheelectricalimpedancedisplayingboththeexperimentaldataforalltransducersandthetheoreticalmodel. 150

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5 whichisrepeatedhereforconvenience,Zinput=RES+REP 1+;where=ZEBA 7-3 .Theerrorbetweenthecurvetandtheexperimentaldataoverthefrequencyrangeofthetislessthan10%fortheresistivecomponentotherthanatresonanceandlessthan5%forthereactivecomponentoverallfrequencies.AcomparisonofthecurvetstotheexperimentaldataisgiveninAppendix C .Asobserved,thereisawidevariationintheseriesandparallelresistances.Thisisprobablyduetoalowsensitivityofthecurvettothesecomponents. Table7-3. Theextractimpedanceparametersfromtheexperimentalimpedancemea-surement. RR1RC3938113RR5RC5943363RR2RC3941257RR2RC4941841RR3RC5942639RR5RC2931821Theory1013139 151

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4 .Anindicationoftheeectivenessoftheepoxywellswouldbeacomparisonofstatictopographybeforeandafterpackaging.Thepre-packagedtopographymeasurementswerecorruptedbymultiplelayerreections.Unfortunately,thedeviceswerealreadypackagedbeforethisaectwasdiscoveredsuchthatthesemeasurementsarenotrepresentedinthecurrentwork. Measurementsofthestaticdiaphragmdeectionsweremadewitha ZygoNewView7200 non-contactscanningwhitelightinterferometer.A5Xobjectivecombinedwitha0.5Xeldzoomlenstogivea2:832:12mm2eldofview.Theverticalresolutionofthesystemislessthan0.1nm.ThismeasurementpresentedachallengeduetothemultiplethinlmlayersoftheAlNandMo.Focusoftheopticalsystemresultedatmultiplelayerinterfacesresultinginambiguityastothezeroreferenceofeachmeasurement.Thus,themeasurementswereallreferencedtothetopMoelectrodesincefocusingonthislayerwasrepeatableinbetweenmeasurements. Figure7-4. Initialdiaphragmdeections. 152

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7-4 .Theverticalsteptowardstheouteredgeofthediaphragmsmarksthebeginningoftheannularregion.Notethattheverticalheightsarereferencedtothesurroundingsubstrate.Asisshowninthegure,allofthedeviceswiththeexceptionofRR5RC5hadsimilarinitialdeectiontopography.Therewasaslightwavinessontheorderof100nminthecentralregionofdevicesRR2RC3,RR2RC4,andRR3RC5.Clearly,theinitialdeectionoftheRR5RC5devicestoodoutfromtheothertopographies.Thecenterdeectionwas-1.3mdownfromthesurroundingsubstrate.Sincethecenterdeectionwasontheorderofthediaphragmthickness,thediaphragmwaslikelybuckled.ThiswasanindicationthattheperformanceofRR5RC5variessignicantlywithrespecttotheotherdevices,asconrmedinthefollowingsections. 3.3.2 foradiscussionofthedirectversusconversepiezoelectriceect).Measurementswereperformedusingopticalinterferometrictechniques.AnacvoltageexcitationsignalwasappliedtotheDUT.Thevibrationofthediaphragmwasmeasuredusinga Polytec MSV300scanninglaservibrometershowninFigure 7-5 .Thelaservibrometerisanon-contactmeasurementsystemthatutilizesopticalinterferometrytomeasuresurfacevibrationvelocities.Aheliumneonlaserisscatteredothediaphragmandintotheinterferometeralongwiththereferencelaserbeam.ThereectedbeamexperiencesaDopplershiftduetothevelocityofthediaphragm[167].Comparisonofthefrequencydif-ferencesmeasuredbetweenthereectedandreferencesignalsgivetheinstantaneousvelocityofthediaphragm. Bycouplingthevibrometertoamicroscopeandpiezoelectricmotors,thelaserwasscannedacrossthediaphragmsurfacetocapturephase-lockedmodeshapes.Theresonantfrequencywasfoundfromthefrequencyresponseofthediaphragm.Theresonantmodeshapewasextractedfromthephase-lockedscanofthediaphragmatresonance.Thelinearity 153

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Scanninglaservibrometersystemusedforelectromechanicalcharacterization. ofthedeviceatresonancewasfoundbymeasuringthevelocityamplitudeofthedeviceataxedfrequencyunderincreasingexcitation. Thelaservibrometerwasalsousedtodeterminetheimpedanceloadingeectofthevariablebackcavity.ThefrequencyresponseoftheDUTwasrecordedforaseriesofbackcavitydepths.Thedependenceoftheresonantfrequencyandsensitivityonthebackcavitydepthwasinvestigated. Inadditiontomeasuringthedeviceresponseinair,vibrometermeasurementswerealsomadeinavacuumchamber.Bymeasuringthedeviceresponsewhileundervacuum,theeectsoftheradiation,backcavity,andventimpedancesweremitigated.Bycomparisonofthemeasurementinairandundervacuum,theeectofdampingwithinthediaphragm,suchasthermoelasticdissipationoranchorloss,wasisolatedfromtheradiationdamping,backcavitydissipation,andventresistance.Aconvergencestudywasdonetoinvestigatewhetherthevacuumlevelreachedwaslowenoughtonegatetheeectsofairondeviceperformance. 154

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Thesignalsmeasuredbythevibrometerincludeboththereferenceexcitationsignal,Vref,andthesurfacevelocity,Uvib,atascanpointwhoselocationisdenoted(x;y).Thedataextractedfromthevibrometerwasinthefrequencydomain.Forfrequencydomaindata, 155

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ThephaseofthevibrometersignalwasreferredtothereferencesignalsuchthattheFFT'sextractedfromthevibrometerwereactuallyVref(f)=j Thevibrometersoftwarealsoreportedtheauto-spectrumsandcross-spectrums,GVV=2 Fromtheauto-andcross-spectrums,thevibrometersoftwarecalculatesthecoherenceas Theauto-andcross-spectrumsarealsousedtoestimatethefrequencyresponsefunction(alsoknownasvelocitysensitivity)atameasurementpointusingthe\optimal"WienerFiltertominimizedependenceontheoutputnoiseofthemeasurement, AdiscussionofthemeanestimationanduncertaintyinthevibrometerdataisdiscussedinAppendix C 156

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7-6 Figure7-6. Thescangridoverlayedwiththemicroscopepictureofthedevice. ThevelocityfrequencyresponsefunctionatthecenterofthediaphragmisgiveninFigure 7-7 withthe95%condenceoftheuncertaintyinthemeanestimateofthefrequencyresponse.Similarly,thecalculatedcenterdisplacementsensitivityisgiveninFigure 7-8 Figure 7-7 isonlythevelocitysensitivityatasinglegridpoint.AsisseeninRayleigh'sintegralinChapter 3 ,iftheentirediaphragmisnotdeectinginphase,thenthetotalsoundoutputisdiminished.Therefore,thevolumevelocitysensitivityisamoretellingperformancecharacteristic.Also,thevolumevelocitysensitivitycanbereadilycomparedtotheresultoftheequivalentcircuitinChapter 5 sincevolumevelocityistheowvariableoftheacousticdomaininLEM.Inaddition,thevolumevelocitycanbeusedtosolveforanequivalentpistonvelocityofanarrayofmanyradiatorsfornonlinearacousticcalculations. ThevolumevelocitysensitivityshowninFigure 7-9 wasfoundbynumericallyintegratingthemeancomplexfrequencyresponsefunction,bH,overthecartesiangridgiveninFigure 7-6 157

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Velocityfrequencyresponsefunctionforthecentergridpoint,wheretheuncer-taintyisthe95%condenceinthemeanestimate. Figure7-8. Displacementfrequencyresponsefunctionforthecentergridpoint,wheretheuncertaintyisthe95%condenceinthemeanestimate. 158

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Volumevelocitysensitivity,wheretheuncertaintyisthe95%condenceinthemeanestimate. foreachfrequencycomponent, Thecartesiancoordinatesystemofgridpointswasutilizedtosimplifythetwo-dimensionalnumericalintegration.Thetradeowastheadditionofnoisefrompointsnotonthedia-phragm.AsshowninFigure 7-9 ,the95%condenceintheamplitudeofthemeanestimateofthevolumevelocityisatleasttwoordersofmagnitudebelowthemeanestimate. 7-9 ,thefrequencyresponseofdevicesvaries,althoughfourarewellmatchedwithresonantfrequenciesaround60kHz.ThedeviceperformancecharacteristicsaresummarizedinTable 7-5 .TheRR5RC5devicegavethehighestsensitivityatresonanceandthelowestresonantfrequency.Itwasexpectedthatthisdevicewouldbehavedrasti-callydierentduetoitslargeinitialdeection.TheRR3RC5devicealsoshowedahighersensitivityandlowerresonantfrequencythanthe60kHzdevices.TheRR3RC5devicedid 159

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Displacementsensitivitycross-sectionsatresonanceareshowninFigure 7-10(a) witherrorbarsindicating95%condence.Also,thethree-dimensionalresonantmodeshapesarecontainedinFigure 7-11 .InFigure 7-10(b) ,thedisplacementsensitivitycross-sectionsarenormalizedbytheirpeakvaluetogiveresonantmodeshapecross-sections.Themodeshapesaresimilarindicatingthatthemodalmassofthediaphragmsarealsosimilar.Thisindicatesthattheshiftinresonantfrequencybetweendeviceswasduetoachangeinstinessbetweendevices.Thiswasanexpectedresultsince,aspreviouslynotedinSection 7.1 ,thestresswasknowntovaryacrossthewafer.Shiftsinstressbetweendeviceswiththesamegeometrywillcreateashiftinstinessresultinginvariationsinsensitivityandresonantfrequency. Resonantdisplacementsensitivitycross-sectionswith95%condenceintervals. (b) Modeshapecross-sections. Displacementsensitivitycross-sections. 160

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RR5RC5 (b) RR1RC3 (c) RR2RC3 (d) RR2RC4 (e) RR3RC5 (f) RR5RC2 Resonantmodeshapes. 161

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Thevelocitysensitivityversusexcitationvoltage. pointatthediaphragmcenterusingthelaservibrometer.Theonsetofnonlinearitywasdeterminedbymonitoringthevelocitysensitivity.AplotofthevelocitysensitivityasafunctionofincreasingvoltageexcitationisshowninFigure 7-12 .Theonsetofnon-linearityforeachdeviceismarkedbythesuddendropinsensitivity. Themaximumlinearrangeofthedevicecanalsobewitnessedthroughmeasurementsofthedisplacementandvelocityasfunctionsofincreasingexcitationvoltage.AsshowninFigure 7-13 ,theonsetofnonlinearitymarksasuddendecreaseintheoverallvelocityanddisplacement.ItisinterestingtonotethatthedisplacementattheonsetofnonlinearityisdierentforalldevicesasshowninFigure 7-13(a) .Thisindicatesthattheonsetofnonlin-earityisnotageometricaectasisnormallyconsideredinthenonlinearstaticdeectionofplates.Instaticdeection,theonsetofnonlinearitynormallyoccurswhenthedisplace-mentbecomesacertainpercentageofthediaphragmthickness.Incontrast,Figure 7-13(b) showsthatthevelocityamplitudesattheonsetofnonlinearityarenearlythesameforalldevices.Thisindicatesthatthelimitofthelinearrangeislikelyduetoavelocitythresholdphenomenon. Althoughthestudyofnonlinearitiesinactuatorsisanimportantsubjectmatter,itisbeyondthescopeofthiswork.Here,itissucienttorequirethattheexcitationwaskept 162

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Displacement (b) Velocity Velocityanddisplacementresponseversusresonanttoneexcitationamplitude. smallenoughinallotherteststhattheoverallvelocityresponsewasbelowthelinearvelocitythresholdindicatedinFigure 7-13(b) 4 ,thedeviceswerepackagedwithavariablebackcavity(withtheexceptionofRR5RC2).Testingwasconductedatvaryingcavitydepthstoinves-tigatetheeectofthebackcavityondeviceperformance.Qualitativecomparisonsweremadewiththeoreticalpredictions. C 163

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7-14(a) throughFigure 7-14(f) .Theexperimentalresultsshowthatthereisanoptimalbackcavitydepthforperformance.Forinstance,thecentervelocitysensitivityatresonanceoftheRR5RC5deviceincreasesbyafactorof2:3whenthescrewdepthistuned.NotethatinFigure 7-14(c) ,thenegativebackcavitydepthmeansthatthescrewwaspositionupintothepcbboard,decreasingtheoverallbackcavitydepth.Similarincreasesinthecentervelocityaftertuningareseenforalldevices.Ifthecavityisextendbeyondthedepthformaximumperformance,theperformancedecreasesmarkinganincreaseinthebackcavityimpedance.Ifthedepthisextendmuchbeyondtheoptimalrange,asinFigure 7-14(b) ,theperformancereachesaminimum.Atthispoint,thebackcavityimpedanceisamaximum. ThisissimilartotheidealcaseofavibratingdiaphragmattheendofaperfectlyrigidbackcavityasshowninFigure 7-15 .TheimpedanceofthediaphragmandbackcavityaregivenbyZAD=j!MAD+1 (7{13) Thevibrationwillreachamaximumwhenthebackcavityanddiaphragmhavethesameresonantfrequency.Thiscaseismarkedbythered\+"inFigure 7-16 wheretheimaginarypartofthebackcavityanddiaphragmimpedancebothgotozero.Thisoccurswhenthedepthisone-quarterthewavelengthoftheresonantfrequencyofthediaphragm.Fortheidealcasethatlacksdissipation,thebackcavityprovideszeroimpedanceatthediaphragm'sresonantfrequency.Ifthedepthisextendedbeyondthequarterwavelength,theimpedanceofthebackcavityincreasescorrespondingtoadecreaseintheperformanceandoverallreso-nantfrequency.Ifthebackcavitydepthisextendedtoahalf-wavelengthofthediaphragm'sresonantfrequency,anidealbackcavityactsasaninniteimpedance.Beyond,thehalf 164

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RR3RC5 (b) RR1RC3 (c) RR5RC5 (d) RR2RC4 165

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RR2RC3 (f) Theory Measurementsofthebackcavity. wavelength,thebackcavityimpedancebecomesperiodic.Thus,thereisadiscontinuityintheresonantfrequencyasthecavityimpedanceactsasifitislessthanaquarterwavelength.Thediaphragmvibrationwillagainreachamaximum,thistimeatthree-quartersthewave-length.Asthecavitydepthcontinuestodecrease,thebehaviordescribedwillrepeat. Figure7-15. Idealpistonandbackcavity. TheequivalentcircuitofChapter 5 isusedtomodeltheeectofthebackcavityforcomparisonwiththeexperimentalresults.TheresultofthetheoryinFigure 7-14(f) showsamatchingtrendtotheexperimentalresults.Asthescrewdepthisincreased,theresonantresponseincreasesslightly.Iftheresonantfrequencyofthediaphragmwaslower,thebackcavitywouldprovideevenmorestieningtothedeviceperformanceatsmallcavitydepthsandtherewouldbeamorepronouncedincreasedinresonantresponse.Thistypeoftrend 166

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Imaginarypartsofthediaphragmimpedanceandcavityimpedancesofvaryingbackcavitydepth. isviewedinFigure 7-14(c) wheretheresonantfrequenciesofdeviceRR5RC5arecloserto30kHzwithaquarterwavelengthnear2.9mmascomparedtothetheoreticalresonancepredictionwhichiscloserto40kHzandaquarterwavelengthnear2.2mm. ThetheoreticalresultbasedontheequivalentcircuitinFigure 7-14(f) alsoshowsadecreaseintheresonantfrequencywithanincreaseinscrewdepth.Thereisadiscontinuityintheresonantfrequencyandresonantresponse.Atthispoint,ananti-resonanceofthebackcavitymovesthroughthefrequencyresponse.Thefrequencyoftheanti-resonancecorrespondscloselytothehalfwavelengthequallingthecavitydepthwhichissimilartotheidealbackcavitydiscussedabove. 167

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7-17 isa150by150by40mm3rectangularsteelboxcapableofachievingaroughvacuumofaround100mTorr.Itisequippedwithseven1-1/3"CFangedportsthatsupportfoursingle-endedBNCfeedthroughs,avacuumport,apressuredetectionport,andanitrogensupplyport.AJBIndustries'PlatinumSeriesLAV-3vacuumpumpisattachedtothevacuumport.ThepressureinthechamberismonitoredbyaKJLCConvectronEquivalentGauge,ratedto0.1mTorr.LVmeasurementsareconductedthroughanopticalportttedwithaThorLabsBK7broadbandwindowthatis25.4mmdiameterand5mmthick.Theportprovidesa10mmdiameterviewingarea.Thefocalplaneusinga5xobjectivecanpenetrateupto10.5mmbelowtheinnerwallofthevacuumchamber. Figure7-17. Vacuumchamberdiagram. SinglepointLVmeasurementsatthecenterofthediaphragmweremadethroughtheviewingwindowofthevacuumchamber.Themeasurementsweremadeatdierentlevelsofvacuumtomakesuretheperformancehadasymptotedtothepointwhereacousticeectshadbeeneliminated.Aleakinthevacuumchamberpreventedthechamberfromachievingavacuumlevellowerthan100mTorr.Also,theleakpreventedphaselockeddiaphragm 168

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7-18(a) throughFig-ure 7-18 forbothstandardtemperatureandpressure(STP)androughvacuumconditions.ThedevicesshowasignicantdecreaseindampingratioandsubsequentincreaseinthequalityfactorasshowninTable 7-4 .Thedampingratio,,iscalculatedbyttingasecondordersystemtransferfunctiontothedeectionsensitivity.DetailsofthetarecontainedinAppendix C .Thequalityfactoriscalculatedas[51] 2p Notethatthedampingcoecientdecreasedbyatleastanorderofmagnitudeforallofthedevicestested.Thisindicatesthatthedominantdampingmechanismsforthedeviceareduetouidandacousticinteraction.Acomparisonoftheexperimentalresultswiththetheoreticalpredictionbasedontheequivalentcircuitmodelshowsanunder-predictionofthedampingratioatSTP.AscoveredinChapter 5 ,thefullequivalentmodelatSTPaccountsfordampingfromthediaphragm,backcavity,radiation,andventimpedances.Atvacuum,thedominantdampingmechanismfromthetheoryisthedampingratioofthediaphragm,whichisassumedtobe0.001basedontheresultsofRR2RC4whichhasasimilarresonantfrequencypredictedbythetheory. Table7-4. DampingandqualityfactorvaluesforthedierentdevicesatvacuumandSTP. RR5RC530.4729.020.0210.002224224RR1RC357.7760.330.0140.0003351945RR2RC359.4861.220.0150.0002342001RR2RC456.6761.200.0180.001027485RR3RC543.7344.880.0140.0004361397EquivalentCircuit38.8038.790.0080.001063499

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RR3RC5 (b) RR1RC3 (c) RR5RC5 (d) RR2RC4 170

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RR2RC3 (f) EquivalentCircuit DevicecomparisonofthefrequencyresponsefunctionbetweenroughvacuumandSTPconditions. Measurementsweremadeatdierentlevelsofvacuumtotestwhethertheuidand/oracousticeectshadbeencompletelymitigated.ThedampingratioversuspressureisplottedinFigure 7-19 .Asshowninthegure,completeconvergenceofthedampingratiowasnotobtainableduetolimitationsofthevacuumchamber.ThedrasticdropindampingfromtheSTPconditions,however,isagoodindicationthattheacoustic/uidiceectshavebeeneectivelyifnotcompletelyisolated. 7-20(b) .Themicrophonemountisattachedtotheopticaltableusingamicropositionerthatisusedforalignment. 171

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ResonantperformancereferencetoSTPversusincreasevacuumchamberpres-sure. TheultrasonicradiatorwasmountedtoabaeconstructedusingaSpectrumZ415rapidprototypingmachineasshowninFigure 7-20(a) .Thebaeis200mmindiameter.Thebaeismounteddirectlytoamanualradialtraverse.Thebaeandradialtraversearethenmountedtoa3axismicropositionerattachedtoauni-axismotorizedtraversecontrolledbyaVelmexVXMSteppingMotorControllerandVextaModelPK264-03A-P1.Across-hairedlaseralignmenttoolandthemicrophoneandradiatormicropositionerswereusedtoalignthemicrophonewiththeultrasonicradiator. Baedultrasonicradiator. (b) Microphonemount. Setupfortheelectroacousticcharacterization. 172

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TheexcitationsignalwassuppliedbyanAgilent33120Afunctiongenerator.TheB&K4939freeeldmicrophonewaspoweredbyaB&K2804batterymicrophonepowersupply.ThemicrophonesignalwasampliedbyaStanfordResearchSystemsSR560low-noiseam-plier.Thelevelofamplicationvarieddependingonthesignallevel.Abandpasslterwasappliedbytheamplierwithacut-onfrequencyof10kHzandacut-ofrequencyof100kHz.Theroll-ooftheamplieris-6dB/decade.TheexcitationandampliedmicrophonesignalsweremeasuredusingaNationalInstrumentsPXI-522100MHz,100MS/s,14-BitdigitizermountedinaNationalInstrumentsPXI-1045General-Purpose18-SlotChassisforPXI.Thesignalsweresampledat500kS/sandtriggeredothesyncsignalfromtheAgilentfunctiongenerator.One-hundredrecordsof5,000samplesweretakenforeachmeasurement.Com-plexaveragingofthefouriertransformofeachrecordwasperformed.LabVIEWwasusedtocontrolthedataacquisitionandmotorizedon-axistraverse. TheB&K4939freeeldmicrophonewascalibratedusingaB&K4228pistonphone.Forthecalibration,theamplicationofthelow-noiseamplierwassetto1andthelterwassettohigh-passanddc.Thepistonphoneemitsanominalvalueof124dBat250kHz.Themicrophonesignalwassampledat10kS/sand800samplesweretakenperrecordlength.TheFouriertransformsof1,000recordswereaveraged.Theresultofthecalibrationisa5:72mV/Pasensitivity. 173

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3.2.1 formoreinformationonRayleigh'sintegral).Theonaxisresponseisfoundfrom wherex0andy0refertocoordinatesonthetransducerface,_Histheaccelerationsensitivitymeasuredbythevibrometerateachpointx0andy0andfrequencyf,R=p 3.2.3 formoreinformationonacousticattenuation),andzisthedistancefromthetransducertothemi-crophone.NumericalintegrationoverthegridofscanpointswasusedtoevaluatethedoubleintegralinEquation 7{16 .TheresultsoftheonaxismeasurementsarecomparedtothecomputationinFigure 7-22 ThedirectivityresponsewasfoundusingRayleigh'sintegralbyconsideringthemea-surementpointtobeataxeddistancerandananglewithrespecttotheaxisofthetransducer.Forthesecalculations,thedirectivitywasfoundinthexzplane.First,thepressureresponseataxeddistancerforangleswasfound, wherenowR=q ThedirectivitymeasurementresultsandcomputationsbasedontheprojectionofLVdatausingRayleigh'sintegralareshowninFigure 7-21 .NotethattheresultsarereportedindBscale.TheprojectionoftheLVdatausingRayleigh'sintegralisfairlyomnidirectional.Thisistobeexpectedsincethekaofthedevicesrangefromapproximately0.4at30kHzto0.8at60kHz.Atthesevaluesofka,theradiatorsarefairlycompact.Inaddition,thedeection 174

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RR5RC5 (b) RR1RC3 (c) RR2RC3 (d) RR2RC4 (e) RR3RC5 (f) RR5RC2 Directivitymeasurementsalongwiththepredicteddirectivitiesfoundbyex-trapolatingtheLVmeasurementsusingRayleigh'sintegral. 175

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3.2.2 formoreinformationonarraydirectivitypatterns).TheexperimentalresultsshowedaqualitativematchtotheprojectionoftheLVdata.Theexperimentalresultsshowsomevariation,butallthedirectivitypatternsarewellwithin6dB.Sourcesoferrorsandnon-uniformitiesinthemeasurementcomefromthenitenessofthebae,alackofidealfree-spacesurroundingthemicrophone,andmisalignmentofthetransducertothemicrophone. AsseeninFigure 7-22 ,the1=rroll-ointhefareldpredictedbyRayleigh'sintegraliswellcapturedbytheexperimentalresults.Themagnitudeoftheresponse,however,isnotwellmatchedformostofthedevices.The95%condencerandomerrorsaresosmallthattheycouldnotberepresentedinthegures.Singlepointfrequencyresponsemeasurementsbeforeandaftertheon-axisexperimentshoweddriftinthedeviceresponse.Forexample,theexperimentallymeasuredon-axispressureresponseinFigure 7-22(a) hasapositiveosetfromtheextrapolatedresultsusingRayleigh'sintegral.Thecentervelocitysensitivityat27kHz,however,wasshowntodriftfrom26.2mm/s/Vto30.5mm/s/Voverthecourseofthemeasurement.Thisleadtodiscrepanciesbetweentheprojectedandmeasuredultrasoundeld. 7.4.2 wereusedtosimulateanequivalentsourceofanarrayoftransducersforaparametricar-ray.ThenonlinearacousticcalculationsoutlinedinSections 5.2 and 5.1.6 arerepeatedforarraysoftheDUTs.Consideranarraydiameterof154mm(6.1in)indiameter.Forty-vehundredtransducerstwithinthearraydiameterwithacentertocenterspacingof2.2mm.Thetransducersaredrivenat80%oftheirmaximumcentervelocityfoundinSection 7.4.4 andareassumedtovibrateinphasesuchthattheirvolumevelocitiesareperfectlysummed. 176

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RR5RC5 (b) RR1RC3 (c) RR2RC3 (d) RR2RC4 (e) RR3RC5 (f) RR5RC2 Onaxispressureresponse. 177

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5.2 ,theaudiblefrequencyresponseat1misgiveninFigure 7-23 foreachtransducer.ThemaximumoutputisbasedonanarrayoftheRR5RC5transducers,witha41.9dBsignalat5kHzand1m.Notethatthecalculationbasedonthenon-buckledRR5RC2devicewasnotfarbehindwitha40.6dBtoneat5kHzand1m.Loweringthedistancerequirementto10cmat5kHzresultsina61.9dBsignalfortheRR5RC5arrayand60.6dBfortheRR5RC2device. Figure7-23. Parametricarrayoutputat1mofaanarrayof4,500radiators. 7-5 .TherealizedlayerstressesdidnotmatchthetargetsfromChapter 6 .Also,thelargeuncertaintyofthestressmeasure-mentsplusthenon-uniformityacrossthewaferledtopoortheoreticalmatchbetweentheexperimentalandtheoreticalresults. ThedeectionsensitivitiesandresonantfrequenciesareverycomparabletothedevicessummarizedinChapter 3 .Itwasshownthatthedeectionsensitivitycouldbeimprovedbybackcavitytuning.Thecenterdeectionsensitivityof961.7m/VoftheRR5RC5deviceisremarkable.Thisdevice,however,wasbuckledanditisunlikelytobereproducedwithagooddegreeofcondence.Theperformanceofthedevicesarealsohamperedbytheearlyonsetofnon-linearityatrelativelylowexcitationvoltages. 178

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Performanceofthedie.Thecenterdeectionafterbackcavitytuningisshowninparentheses. RR5RC530.47+0:020:0938.5890.008442.00.2(961.7)RR1RC357.77+0:050:0537.5120.008181.00.3(211.6)RR2RC359.48+0:060:0927.4380.009113.30.5(147.4)RR2RC456.67+0:060:0527.5880.009144.81.0(150.4)RR3RC543.73+0:020:0343.3860.008285.30.5(359.7)RR5RC257.00+0:030:0546.4510.012224.50.5 2 andincreasetheoverallsoundoutput.Thiswouldleadtoanincreaseinthenumberofdevicesbeyondthe4,500usedinthecalculationandmaypushthelimitsofpracticality.Inaddition,itisproblematicthatthiscalculationismostlikelyagenerousestimategiventhelackofphasematchingoftherealizeddevicesofsimilarresonantfrequencies.Mismatchedphaseleadstoanoveralldecreaseinthetotalvolumevelocityoutputofthearrayandthusadecreaseinparametricconversion.Giventheresults,itisunlikelythatapracticalparametricarraycanberealizedwiththisdevicedesign.InChapter 8 ,asummaryisprovidedalongwithrecommendationsforfuturework. 179

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Thischapterpresentsasummaryofresearchgoals,objectives,andkeyresults.Recom-mendationsforfutureworkanddesignsarealsopresented. Atthebeginningofthiswork,therewasnoevidenceoftheapplicationofMEMStechnologytoparametricarrays.In2006,however,Haksueetal.[44,45]showedresultsforaninaudibleparametricarrayofpMUTs.Thedierencetonewas40kHzandthusnotappropriateforconventionalaudioapplicationsofparametricarrays.AnotherexampleofaMEMS-basedparametricarraythataroseduringthecourseofthisstudywasacMUTtransducerarraypresentedbyWygantetal.in2007[15].Theymeasured58dBofadierencefrequencyof5kHzat3mfromthearray.Thisrequiredanexcitationsignalcomposedofa200Vpeak-to-peakACcomponentinadditiontoa350-380Vbiasvoltage.Thehighvoltagerequirementsdegradetheappealofthistechnology.Neitherstudyutilizedformaloptimizationmethodswhendesigningthetransducers. Incontrasttopreviousparametricarrayworks,theaimofthisstudywastodesignanultrasonictransducerforaMEMS-basedparametricarraybydevelopingacomprehensivesystems-levelmodelofanAlN-basedultrasonicresonatorandoptimizationtoolsforparamet-ricarraytransducers.ThedevicemodelusedLEMtoformtheelectromechanicalcouplinganddiaphragmmodel.Includedinthemodelwerepackagingeectssuchasavariablebackcavitywhoseequivalentimpedancewasformedusingasetoftransfermatrices.Intermsof 180

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Thedevicecharacterizationshowedaquantitativedierencebetweenthepredictionoftheequivalentcircuitmodelandthemeasureddeviceperformance.Thedeviationwaslikelytheresultofnon-uniformlmstressacrossthewaferandlargeuncertaintiesinthemeanstressmeasurements.Thestressmeasurementsindicatedanaverageeectivetensilestressacrossthewaferwithestimateduncertaintiesofgreaterthan30%.Byvisualinspection,however,thediaphragmsalongtheouterportionofthewaferappearedtohavelargeinitialdeectionsindicatinganoverallcompressivestress.Alongtheinnerportionofthewafer,however,thediaphragmsappearedat.Theimplicationwasthattherewerelargestressgradientsacrossthewafer.Thiswasconrmedbythelargevariationinresonantfrequenciesbetweendevicesrangingfromapproximately30to60kHz.Thestressgradientsalongwiththelargelmstressuncertaintiesledtoquantitativediscrepanciesbetweenthetheoreticalpredictionoftheequivalentcircuitandtheexperimentalresults. Thedevicecharacterizationshowedaqualitativeexperimentalvericationofthesystemsmodelofthebackcavity.Aspredictedusingthecomprehensivesystemsmodel,improvementindeviceresponsebybackcavitytuningwasexperimentallyconrmed.Theoveralltrendofresonantfrequencyversusbackcavitydepthwasalsoqualitativelymatched.Thisgavecredencetothemantraofcomprehensivedesignthatincludesbothdeviceandpackage.Anothernotableresultoftheexperimentalcharacterizationwasthedecreaseinthedampingratioofanorderofmagnitudewhenthedeviceswhereplacedinaroughvacuumenvironment.Thisprovidedevidencethattheuidenvironmentwasthesourceofthedominantdampingmechanism. TheperformanceoftheAlNradiatorsinthisworkarecomparedtopreviouslyreportedair-coupledMEMSradiatorsforvibrationmeasurementsinTable 8-1 .TheAlNradiatorsshowcomparableperformanceintermsofvolumevelocityandresonantfrequency.For 181

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Comparisonofair-coupledMEMSradiatorperformances. RR5RC5 AlN 39 30 RR1RC3 AlN 38 58 RR2RC3 AlN 27 59 RR2RC4 AlN 28 57 RR3RC5 AlN 43 44 RR5RC2 AlN 46 57 Zhuetal.2004[124] PZT 0.6by0.6mm2 Horowitzetal.2006[46] PZT 23* Chandrasekaranetal.2002[148] Thermoelastic 3mm3/s*z Brandetal.1997[138] Thermoelastic 1by1mm2 example,theAlNRR5RC2devicehasavolumevelocityof46mm3/s/Vataresonantfrequencyof57kHz.Incontrast,thePZTdevicepresentedbyZhuetal.[128]givesavolumevelocityof101mm3/s/Vataresonantfrequencyof67kHz.TheAlNdeviceisslightlymorethan6dBdowninperformance,eventhoughthed31constantofPZTisalmosttwoordersofmagnitudelargerthanthatofAlN(seeTable 3-2 ).Althoughtheprojectedperformanceofthesedevicesasaconventional,audioparametricarraydonotshowgoodpromise,thedevicecomparisonshowsgoodoverallperformanceoftheAlNultrasonicradiatorsincomparisontothecurrentliterature.Futureworkcanfocusonmorerobustdesignsandpackagingimprovementsforotherultrasonicapplications. 182

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Anotherimprovementthatcouldleadtobetterdesignswouldbeatheoreticalmodelthatcouldpredicttheonsetofdynamicnonlinearity.Thecurrentnonlinearplatemodelisstaticanddoesnotincorporatetheinuenceofthepackage.Theexperimentalresultsshowedthattheonsetofnonlinearityaroundresonancewasnotdictatedbythemagnitudeofthedeection,butinsteadbythemagnitudeofthevelocity.Amaximumexcitationvoltageforlinearoperationthatisafunctionofdevicegeometrycouldbeincorporatedintheobjectivefunction.Thus,theobjectivefunctionwouldgivethebestpossiblelinearperformanceofthedevice. Beyondimprovementsinthefabricationandmodelingofthedevice,anotherareaofimprovementisthetestingofdevicesinvacuum.Vacuumtestingmitigatestheeectsofaircoupling,allowingadirectcomparisonbetweenexperimentandtheelectromechanicalmodel.Theexperimentalresultsshowedthatthedampingratiohadnotconvergedtoaconstantvalueatthelowestvacuumlevelsachievablewiththecurrentsetup.Areductioninvacuumleveltothepointofdampingratioconvergencewouldgivebettercondencethatalluid/acousticseectshadbeeneliminated.Also,aleakinthechamberpreventedscansofthediaphragmwhileunderaconstantlevelofvacuum.Eliminationoftheleakwouldallowscansofthediaphragmandmoreindepthstudiesoftheelectromechanics. 183

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InSection 8.2 ,theconceptofdecreasingdevicesensitivitytostressvariationswaspro-posed.ThiscouldbeaccomplishedbyeliminatingtheAlNscaoldinglayerandperformingthestandardFBARdepositiononasilicon-on-insulator(SOI)wafer.ThediaphragmreleasewouldthenproceedexactlythesameasthewaferreleasestepusedtoformthedevicesinthisdissertationwithDRIEfromthebacksidefollowedbyanoxideetchtoreleasethediaphragm.AnexampleofadevicethatutilizesSOItechnologyisgiveninFigure 8-2 Thecurrenttransducerisactuatedbyinducinganin-planestressintheannularregionofthediaphragm.Thereisnoactuationatthecenterofthediaphragm.Analternatepiezoelectrictransducerdesignsimilartothatof[135]isproposedthatutilizesactuationatboththediaphragmedgeandcenter.Tomotivatethedesignofsuchatransducer,thestresseldinducedinaclampedplatewhenitdeectsisshowninFigure 8-1 .Asyoucansee,thestresseldatthetopoftheplateis180'soutofphasewhencomparingthecenterandedgeoftheplate.Iftheoppositephasestresscouldbeinducedatthecenteroftheplateinadditiontointheannularregion,thenadditionalperformancecouldberealized. Figure8-1. Stresseldsinsideadeectedclampedplate. Figure8-2. Alternatedesignforimprovedperformance. Thealternatedesignwouldhavetwotopelectrodes,oneouterannularelectrodeandoneinnercircularelectrode.Thebottomelectrodeshouldbecontinuousacrossthebottomofthediaphragm.Thebottomelectrodeshouldbegrounded.Thecenterandannularelectrodeswouldbeexcitedwithsignalsofoppositephasetoinsurethatthepiezoelectriclayeristensile 184

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185

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2{17 issolvedforapistonradiatingattwoadjacenttonesforthedierencefrequencyproducedbytheirinteractions[20].First,axisymmetricwavesareassumedsuchthatr2?=@2/@r2+r1@/@r.Thepressureisassumedtobecomposedoftwoparts: wherep1isthesolutiontotheKZKequationminusthenonlineartermandp2isasmallcorrectiontop1attherstharmonicandanysumanddierencefrequenciesgeneratedbyp1.Timeharmonicsolutionsareassumed, 2jq1a(r;z)ej!a+q1b(r;z)ej!b+c:c:;(A{2) and 2jq2a(r;z)ej2!a+q2b(r;z)ej2!b+q+(r;z)ej!++q(r;z)ej!+c:c:;(A{3) wherethesubscripts+and-refertotonesformedbythesummationandsubtractionoftheoriginatingtones,respectively.Theresultinglinearequationsforp1are@q1a (A{4)and@q1b where1a;b=!2a;b2c30.EquationsfortheharmonictonesandsumanddierencefrequenciescanbewrittenusingthesolutionstoEquation A{4 andEquation A{5 assources.Theoutputoftheparametricarrayisthedierencefrequencycomponentsoitisthefocushere.Theequationgoverningofthedierencefrequencycorrectionto1aand1bis 186

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A{6 isthelinearorprimarytonesolution.Thus,theprimarywavesalongthez-axisareseenasvirtualsourcesthat"pumpenergyresonantlyandthereforemostecientlyintothedierence-frequencysoundthatpropagatesinthesamedirection[18]."Equations A{4 through A{6 aresolvedusingGreen'sfunctionscombinedwithaHankeltransformthatremovestherdependence.TheGreen'sfunctionis wheresubscriptsa,b,anddenotetheGreen'sfunctionsforEquation A{4 ,Equation A{5 ,andEquation A{6 ,respectively.ThesolutionofEquations A{4 and A{5 isgivenby whereq1a;b(r0;0)istheboundaryconditionatz=0.Thesolutionforthedierencefrequencyis InWestervelt'ssolution,theparametricarrayisassumedbeemittingattwoneighbor-ingtones!a=!b.Itisassumedthatabsorptionisstrongenoughtolimitthenonlinearinteractionoftheneighboringtonestotheacousticneareldofthesource.Thisassumptionsallowsthenonlinearinteractiontooccurwhilethewavesarestillplaneandcollimatedanddonotsuerfromgeometricspreading.Theassumptionisbasedupon0z01,where0andz0arebasedupontheaverageradiatedfrequency[18].Theprimaryeldsareassumedtobeperfectlycollimatedsuchthatq1a(r;z)=p0aH(ar)eaz whereaistheradiusofthesource. 187

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A{10 and A{11 aresubstitutedintoEquation A{9 andthefollowingassump-tionsaremade: rr0/zz0)terminEquation A{7 resultingin(z)1J0(k rr0/z). A{9 becomes1sincethecontributionstotheintegralarenegligibleforlargez. Withtheaboveassumptionsandsubstitutingr=ztan,theintegralbecomes[18] z r0tan)eTz0jkz2tan2 2(zz0)r0dr0dz0;(A{12) whereT=a+b Thislengthestablishesthelengthoftheregionofnonlinearinteractionoftheparametricarray.Moreassumptionshavetobemadetoarriveatananalyticsolution. 2k a2. 2jk Theresultofintegrationis z 2jk ztan2;(A{14) wheretheWesterveltdirectivityandaperturefactoraregivenby 1+j(k 188

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atan) atan;(A{16) respectively.ThemainassumptionsleadtothefollowingrestrictiononthevalidityofEqua-tion A{14 : 2{24 .Thetotalpressureisassumedtobemadeupofaprimarysignal,p1,andasmallcorrection,p2similartoEquation A{1 .Theprimarysignalisassumedtobeacollimatedbeamofsoundoftheform where()isdependentontheinstantaneousangularfrequency,(t)=!0+@'/@t.Theattenuationisassumedtobelargeenoughtorestrictthenonlineargenerationtotheneareld.Theequationthatgovernsthesmallcorrection,p2,isthetimeintegratedKZKequation(notethatthethermoviscousdissipationterm,,hasbeensettozero), UsingGreen'sfunctions,thesolutiontoEquation A{18 iswrittenas InthesquaringofEquation A{17 ,thefrequencycontentcontainsbothalowandhighfrequencycomponent.Thehighfrequencycomponentisignoredsinceitwillbeabsorbedatamuchfasterrate. 2p20e2()zE2()H(ar):(A{20) 189

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A{19 : Notethattheabsorptionofthelowfrequencycomponentisweakaslongasmodulationsareslowlyvarying.Iftheprimarysignalhasconstantphase,theattenuationcoecientreducesto()=0andEquation A{21 becomesBerktay'ssolution[25] Berktay'ssolutionpredictsa40dBperdecaderoll-oaslowerfrequenciesareapproachedinthesquareofmodulationsignal. 190

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AlinearcompositeplatemodeloriginallydevelopedbyWangetal.[168]wasusedtocalculatedthelumpedmassandcomplianceofthediaphragm.Thederivationofthelinearcompositeplatemodelisgiveninthisappendix.Inaddition,anumericalnonlinearcompositeplatemodeldevelopedbyWilliamsetal.[169]wasusedtochecktheperformanceofdevicedesignsderivedfromthelinearmodel.TheimplementationofthenonlinearmodelasaconstraintintheoptimizationschemeiscoveredinChapter 6 .ThenonlinearnumericalmodelisnotpresentedherebutcanbefoundintheunpublisheddissertationproposalofWilliamsetal.[169]andinfuturepublications.. Devicecrosssection. (b) Platemodelcrosssection. Crosssectionsthatshowthecorrespondencebetweentheplatemodelandthedevice. Acartooncross-sectionofthedeviceisshowninFigure B-1 .Thediaphragmisformedbyacompositeplateandanannularringthatcontainsthepiezoelectriclayer.Thelayershavedierentmaterialpropertiesandfabricationinducedin-planestressesthatcanbeeithertensileorcompressive.Thelayersarenotuniformovertheradiusofthediaphragm.TworadiiaredenedasshowninFigure B-1(b) .Theplateissubjecttovoltageappliedacrossthepiezoelectriclayer(3oftheannularsectioninFigure B-1(b) )andtransversepressureloading.Tondananalyticalsolution,theplateisseparatedintoinner(1)andannular(2)sections.Theplate'stransverseandradialdeectioncanbeexpressedasfunctionsofpressure,voltage,in-planestress,materialproperties,andgeometry. 191

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Kirchho'shypothesisisusedinthefollowingderivationofthegoverningequationsofthecircularplateundervoltageandpressureloading[161].Kirchho'shypothesisstatesthattransversenormalsremainstraight,areperpendiculartotheplate'smid-plane,anddonotelongateafterdeformation[161].Kirchho'shypothesisfailswhenlargeshearingforcescausedeformationofthetransversenormals(importantinsmall,thickplates),compressionoftheplatethicknessisnon-negligible(i.e.thepressuregradientacrossthediaphragmthicknessislarge),andnordeectionsarelarge. B-2 .Summingtheforcesintheradialdirectionresultsintheradialequilibriumequation, Summingtheforcesintheaxialdirectionresultsinthetransverseequilibriumequation, 192

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FigureB-2. Isometricviewofaninnitesimalplateelement.Threelayersareshownforillustrationbuttheplateelementisconsideredtohaveanarbitrarynumberoflayersinthederivationofthegoverningequations[161]. (B{4)andur(r;z)=u0(r)z@w0 wherethesubscript"0"referstothez=0referenceaxisshowninFigure B-2 .Thelinear,axisymmetricstrain-displacementrelationshipsare 193

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B{4 andEquation B{5 aresubstitutedintothestrain-displacementrelationshipsresultingin wherethestrainsinthereferenceplanearegivenby andthecurvaturesaregivenby where[Qn],Enf,n0,anddn31arethestinessmatrix,electriceld,fabricationinducedstress,andpiezoelectricconstantofthenthlayer,respectively.Itisassumedthatthefabricationinducedstressisapproximatelyuniformintheplaneofthethinlm[163].Thestinessmatrixis [Qn]=En whereEnandnaretheYoung'smodulusandPoisson'sratioofthenthlayer. 194

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wherezBandzTrepresentthebottomandtopofthediaphragm,respectively.SubstitutingequationEquation B{10 yields[168] where[A]and[B]areextensionalandexural-extensionalstinessmatrixesgivenby[A]=zTZzB[Qn]dz respectively.Thefabricationinducedforceperunitlength,N0,isgivenby ThenalterminEquation B{13 isthein-planeforceperlengthduetothepiezoelectriceect[168], Themomentsperunitlengtharefoundbyintegratingthestresstimesitsmomentarm,z,throughthethicknessofthediaphragm, 195

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B{18 resultsin where[D]istheexuralmatrixgivenby [D]=zTZzB[Q]z2dz:(B{20) ThenalterminEquation B{19 isthemomentresultantduetothepiezoelectriceect[168], Thefabricationinducedmomentisgivenby First,Equations B{2 and B{3 arecombinedtoform Next,bothEquations B{13 and B{19 aresubstitutedintoEquation B{24 andEquation B{1 .Theresultingequationsarelinearizedandcombinedtoform[168] dr2+1 drN0 196

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Notethattheequationforthedeectionslope,Equation B{25 ,isdecoupledfromtheequationfortheradialdisplacement,Equation B{26 B{25 and B{26 aresplitintothethreecasesofin-planeforceperlength:tension(N0>0),zero(N0=0),andcompression(N0<0).ThesolutionstoEquations B{25 and B{26 are[168] m+c2K1r mpr m+c2Y1r m+pr and m+c2K1r mi;N0>0;B11 m+c2Y1r miN0<0;(B{28) wherem=q mc2K0r mipr2 m+c2Y0r mi+pr2 197

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B{13 andEquation B{19 as[168] mc1+K1r mc2+(A11+A12)c3+A12A11 mc1+Y1r mc2+(A11+A12)c3+A12A11 and m+B11D11 m+I2r mic1+hB12D12 mB11D11 m+K2r mic2+(B12+B11)c3+M0+MPr+B12B11 2(N0)1CCCCAN0>0;0B@M0+MPrD11+D12 m+B11D11 m+J2r mic1+hB12D12 mB11D11 m+Y2r mic2+(B12+B11)c3+M0+MPr+B12B11 2(N0)1CCCCAN0<0:(B{31) Theconstantscifori=1:::5aresolvedforallthreecasesfromtheboundaryand/ormatchingconditionsforeachofthetwoplatesections.Inthenextsection,theconstantsarereferredtoasc(j)i,wherejcorrespondstotheplatesection.Thedetailsofthebound-ary/matchingconditionsareprovidedbelow. 198

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Also,theradialdisplacementandsloperemainniteatthecenteroftheplate ThereisadiscontinuityintheplateatR1.Thematchingconditionsbetweenthesectionsare[168]'(1)(R1)='(2)(R1); Applyingtheniteboundaryconditions, B{33 ,constantsc(2)2andc(2)4arezero.Eightmoreintegrationconstantsarerequired.Theremainingconditionsarethethreeedgeconditions, B{32 ,andvematchingconditions, B{34 through B{38 .Itisconvenienttosolveforthesecoecientsusingamatrixformulation.Theconditionsarewrittenasgeneralmatchingconditionsasfollows whereC(i)andf(i)arethematrixofcoecientsoftheintegrationconstantsandthevectoroffreetermsoftheithsection,respectively.Alloftheconditionsarecombinedtoformasingleequationasfollows [C]8x8fcg8x1=f(1)8x1f(2)8x1;(B{40) 199

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[C]88fcg81=C(1)83C(2)858>><>>:c(1)31c(2)519>>=>>;:(B{41) ThecoecientsC(1)andC(2)aswellasf(1)andf(2)willdependonthecorrespondingin-planeloadingcase.Once[C],f(1),andf(2)areselected,Equation B{41 issolvedfortheintegrationconstantsfcg.Theintegrationconstantsarethensubstitutedbackintoequations( B{27 )-( B{29 ).Thiscompletesthelinearplatemodelsolution. B-3 iscalculatedbysubtractingtheinitialdeection.TheincrementaldeectionisusedinallintegralsofthelumpedelementmodelinginSection 5.1 FigureB-3. Diagramofincrementaldiaphragmdeection. 200

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ThischaptercontainsdetailsontheanalysisusedtondtheuncertaintyestimatesintheexperimentspresentedinChapter 7 7.2 .Thirtyimpedancemeasurementsweretakenandaveraged.Eachimpedancemeasurementisgiveninthefollowingform whereRistherealpartoftheimpedance,Xistheimaginarypartoftheimpedance,andkstandsforthekthmeasurement.Themeanrealandimaginaryimpedancearecalculatedas^R=ER=1 Thestandarddeviationsarefoundusinganunbiasedestimatoras^R=E[R]=vuut Thenormalizedrandomerrorisgivenby^"r(R)=^R 201

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HP4294AImpedanceAnalyzer hasabiaserror.Theestimationofthebiaserrorwastakenfromtheoperationmanual[164].AcomparisonoftherandomandbiaserrorestimatesisgiveninFigure C-1 .Forthesemeasurements,thebiaserrorreportedinthemanualdominatesovertherandomerrorinthemeasurement. FigureC-1. Randomandbiaserrorsintherealandimaginarypartsoftheimpedance. 7.2 .TheextractionwasachievedbyttingthecompleximpedancedatausingafunctioninMatlab,invfreqs,whichimplementsadampedGauss-Newtonmethodtotananalogtransferfunc-tiontocomplexdatathatisafunctionoffrequency[165].ThedatawasttotheimpedancemodelgivenbyEquation 5{49 .Thefollowingplotsshowtherealandimaginarypartsoftheimpedancetandexperimentaldataalongwiththepercentrelativeerror. 202

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RR1RC3 (b) RR2RC3 203

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RR2RC4 (d) RR3RC5 204

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RR5RC2 (f) RR5RC5 Impedancetincomparisontotheexperimentaldata. 205

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Thecondenceintervalofthemeanisgivenas wheretnd;isthevalueofthetdistributionforndaveragesandacondenceof(12)100%.ThecondenceintervalestimatesgivenfortheLVmeasurementsinSection 7.4 werefor100averagesand95%condenceinthemeanestimationforwhichthetdistributionvalueisapproximately2. Therandomnormalizeduncertainty,",isdenedasthestandarddeviationofthemeannormalizedbythemean,, :(C{10) 206

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Thestandarddeviationofthemeanofthephaseofthefrequencyresponsefunctionis whereHisthephaseofthefrequencyresponsefunction. 7{11 .UncertaintyestimatesofthevolumevelocitysensitivityarebasedontheuncertaintiesinthevelocitysensitivityoftheindividualscanpointsusingaMonteCarlosimulation.TheamplitudeandphaseofthevelocitysensitivityateachpointisrandomlyvariedusingaGaussianrandomnumbergenerator.Adistributionofcomplexvolumevelocitiesateachfrequencyfiscalculatedas wherezandzarerandomnumbersofnormalizeddistributionwhosestandarddeviationis1.Fromthedistribution,thestandarddeviationoftheamplitudeandphaseofthevolumevelocityiscalculated,jQj(f)=vuut ThevolumevelocitysensitivityincludinguncertaintyboundsisshowninFigure 7-9 207

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7-5 .AschematicofmagnitudeofthevolumevelocitysensitivityversusfrequencyaroundresonanceisillustratedinFigure C-3 alongwiththe95%con-denceintervalsateachfrequencybin.Theresonantfrequencywasdeterminedtooccuratthemaximummagnitudeofthevolumevelocitysensitivity.Therandomuncertaintyintheresonantfrequencywasfoundbyndingthebandwidtharoundresonancewhere Thus,asshowninFigure C-3 ,theresonantfrequencywith95%condenceintervalswasestimatedas Notethatthehalf-widthofthefrequencybinwasalsoincludedintheuncertaintiesinEquation C{17 FigureC-3. Schematicoftheuncertaintyintheresonantfrequencycalculation. 7.4.6 ,thedampingcoecientiscalculatedbyttingasecondordersystemtransferfunctiontothedeectionsensitivity.Thetwasweightedusingatriangularweightfunctionthatequal1attheresonantfrequencyand0.25attheprescribedfrequencybounds.ThefrequencyboundsaregiveninTable C-1 .ThedeectionsensitivityandsecondordersystemcurvetsaregiveninFigure C-4 208

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RR1RC3 (b) RR2RC3 (c) RR2RC4 (d) RR3RC5 (e) RR5RC5 Dampingcoecientcurvet. 209

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Frequencyboundsonsystemt. RR1RC3STP23.437581.2500RR1RC3Vacuum23.437581.2500RR2RC3STP23.437585.9375RR2RC3Vacuum23.437585.9375RR2RC4STP23.437585.9375RR2RC4Vacuum23.437585.9375RR3RC5STP23.437554.6875RR3RC5Vacuum23.437554.6875RR5RC5STP7.812539.0625RR5RC5Vacuum7.812540.625 7.4.5 wereintroducedviatheexperimentalsetup.First,thebackcavitywashandtunedtoabackcavitydepthbyalignedtoquarterturnmarksintroducingarandomerrorintermsofbackcavitydepth.Oncethebackcavitydepthhadbeenset,thedevicewasplaceonthemicroscopestageandalignedtotheLVagainintroducingarandomerror.Inaddition,theresonantfrequencyandresponsealsovariedfrommeasurementtomeasurement.Inthissection,theeectoftheseerrorsareisolatedandcompared. C-2 ,thedevicewasnotmovedonthemicroscopestageandthemeasurementwasrepeated30times.Inthesecondcase,thealignmentofthedevicetotheLVwasdisturbedwithoutremovingthedevicefromthemicroscopestage.Theresonantresponsewasthenmeasuredafterrealigningthedevicetothemicroscope.Theprocesswasrepeatedthirtytimes.Thiscaseislabeled\Alignment"inTable C-2 .Inthenalcase,labeled\BackCavityTune"inTable C-2 ,thedevicewasremovedfromthemicroscopestageandthebackcavitydepthwasperturbedbeforerealigning 210

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TableC-2. Uncertaintiesintheresonantfrequencyandresponseasafunctionofmea-surementuncertainties. Re-measure98.510.0329.50.01Alignment99.550.1629.70.04BackCavityTune101.590.9530.00.09 C-2 ,theerrorintroducedbysettingthebackcavitydepthandaligningtotheLVoverridestherandomerrorintheexperiment.The95%condenceincreasesbyalmostanorderofmagnitudeinboththeresonantvelocitysensitivityandresonantfrequency.Evenwiththeerrorintroducedbythesettingthebackcavitydepth,themagnitudeofthe95%condenceisinthethirddigitofthemeanquantity. Notethatthemeasurementsinthissectionwereperformedforasingledeviceatthenominalbackcavitydepth.Torepresentthetrueuncertaintyintheresonantvelocitysen-sitivityandresonantfrequencyresultspresentedinSection 7.4.5 ,themeasurementwouldhavetoberepeatedateachbackcavitydepthmultipletimeforeachdevice.Forexample,theRR5RC5deviceresponsewasmeasuredat40dierentscrewdepths.Ifthemeasurementatsinglebackcavitydepthwasrepeated31timestobuildupa95%condencesimilartothatpresentedinTable C-2 ,1240measurementswouldhavetobemade.Consideringthe5devicesthathavevariablebackcavities,thetotalmeasurementswouldexceed6,000,whichclearlysurpassespracticallimits.Instead,measurementofthebackcavitydepthandtherandomerrorincurredbyresettingitateachquarterturncangivesomeunderstandingtotheoverallmeasurementuncertainty.Thismeasurementisconductedinthefollowingsection. 211

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C-3 .Notethatthenegativedepthatthenominalscrewpositionof0referstotheendofthescrewbeingabovethealuminumblock.Notethatthestandarddeviationintroducedbyresettingthescrewdepthisapproximately4-5m. TableC-3. Screwdepthmeasurementwithuncertainties. Depth(m)-4533113196274(m)54544

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V.Chandrasekaran,E.M.Chow,T.W.Kenny,T.Nishida,B.V.Sankar,L.N.Cattafesta,andM.Sheplak,\Thermoelasticallyactuatedacousticproximitysensorwithintegratedelectricalthrough-waferinterconnects,"inTechnicaldigestSensorandActuatorWorkshop,HiltonHead,SC,2002,pp.102{106. [149] L.Rufer,C.C.Domingues,S.Mir,V.Petrini,J.C.Jeannot,andP.Delobelle,\Acmoscompatibleultrasonictransducerfabricatedwithdeepreactiveionetching,"Mi-croelectromechanicalSystems,Journalof,vol.15,no.6,pp.1766{1776,2006. [150] K.Nam,Y.Park,S.Hong,J.Pak,G.Park,andI.Song,\Memsbasedbulkacousticwaveresonatorsformobileapplications,"IntegratedFerroelectrics,vol.77,pp.101{108,2005. [151] R.C.Ruby,R.S.Fazzio,H.Feng,andP.D.Bradley,\Acousticresonatorperformanceenhancementusingalternatingframestructure,"U.S.Patent7,388,454B2,2008. [152] M.J.Madou,Fundamentalsofmicrofabrication:thescienceofminiaturization,2nded.BocaRaton:CRCPress,2002. [153] DynatexInternational.(2003)Dynatexapplicationnotes.[Online].Available: http://www.dynatex.com/support/appnotes.html [154] A.W.Leissa,Vibrationofplates,reprinteded.[S.l.]:AcousticalSocietyofAmericathroughtheAmericanInstituteofPhysics,1993. [155] D.R.LideandC.Press.,CRChandbookofchemistryandphysics:aready-referencebookofchemicalandphysicaldata,88thed.BocaRaton,Fla.;London:CRC,2007. [156] K.Stephenson,Introductiontocirclepacking:thetheoryofdiscreteanalyticfunctions.NewYork:CambridgeUniversityPress,2005. [157] R.T.HaftkaandZ.Grdal,Elementsofstructuraloptimization,3rded.,ser.Solidmechanicsanditsapplications;.Dordrecht;Boston:KluwerAcademicPublishers,1992. [158] H.O.Berktay,\Parametricamplicationbytheuseofacousticnon-linearitiesandsomepossibleapplications,"JournalofSoundandVibration,vol.2,no.4,pp.462{470,1965. [159] T.Ruemenapp,T.Ruemenapp,andD.Peier,\Dielectricbreakdowninaluminiumnitridedielectricbreakdowninaluminiumnitride,"inHighVoltageEngineering,1999.EleventhInternationalSymposiumon(Conf.Publ.No.467),D.Peier,Ed.,vol.4,1999,pp.373{376vol.4. [160] M.Papila,R.T.Haftka,T.Nishida,andM.Sheplak,\Piezoresistivemicrophonedesignparetooptimization:Tradeobetweensensitivityandnoiseoor,"in44thAIAA/ASME/ASCE/AHSStructures,StructuralDynamics,andMaterialsConfer-ence,vol.AIAAPaper2003-1632,Norfolk,Virginia,2003. 225

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J.N.Reddy,TheoryandAnalysisofElasticPlates.Philadelphia:TaylorandFrancis,1999. [162] M.Williams,B.Grin,B.Homeijer,B.Sankar,andM.Sheplak,\Vibrationofpost-buckledhomogeneouscircularplates,"inIEEEInternationalUltrasonicsSymposium.NewYork,NewYork:IEEE,2007. [163] S.A.Campbell,Thescienceandengineeringofmicroelectronicfabrication,2nded.,ser.TheOxfordseriesinelectricalandcomputerengineering;.OxfordUniversityPress,2001. [164] [165] TheMathworks,Inc.,MatlabR2008a,Natick,MA,2008.[Online].Available: http://www.mathworks.com/ [166] F.BloomandD.Con,Handbookofthinplatebucklingandpostbuckling.BocaRaton,FL:Chapman&Hall/CRC,2001. [167] [168] G.Wang,\Modelinganddesignofamems-basedpiezoelectricmicrophone,"Master'sthesis,UniversityofFlorida,Gainesville,2003. [169] M.Williams,\Design,fabrication,andcharacterizationofamemspiezoelectricmicro-phoneforaeroacousticapplications,"2008. 226

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BenjaminAndrewGrinwasbornin1980,inOrangePark,Florida.HeattendedOr-angeParkHighSchoolinOrangePark,Florida,graduatingin1999.HethenenrolledattheUniversityofFloridawherehereceivedhisbachelor'sdegreeinaerospaceengineeringfromtheUniversityofFloridain2003.Duringthesummerof2001,BenjaminjoinedtheInterdis-ciplinaryMicrosystemsGroupwhereheworkedonawindtunnelstraingaugebalance.InAugust2003,BenjaminbeganhisgraduatestudiesattheUniversityofFloridaasaNationalScienceFoundationResearchFellow.InMay2006,heearnedaMasterofSciencedegreeinaerospaceengineering.BenjaminiscurrentlycompletinghisdoctoraldegreeinmechanicalengineeringattheUniversityofFlorida.Hisresearchinterestsincludeair-coupledultrasonicmicroelectromechanicalsystems(MEMS),parametricarrays,proximitysensors,aeroacous-ticmicrophonedesign,thermoacousticimaging,shearstresssensor,lasermicromachining,MEMSdesignoptimization,andelectroacoustictransducercharacterization. 227