A Domain Wall Model for Hysteresis in Piezoelectric Materials

7/27/2019 A Domain Wall Model for Hysteresis in Piezoelectric Materials

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NASA/CR-1999-209832ICASE ReportNo.99-52

A Domain WallModelfor Hysteresis in PiezoelectricMaterialsRalphC .SmithNorthC arolinaStateUniversity,Raleigh,NorthCarolina ZoubeidaOunaiesICASE,Hampton,Virginia InstituteforComputer ApplicationsinScienceandEngineeringNASALangleyResearchCenterHampton,VAOperatedbyUniversities Space ResearchAssociation

ctrcQftjALrirmm&fo*Nat ionalAeronautics andSpaceAdministrat ionLangleyResearch CenterHampton,Virginia 23681-2199 Prepared fo r LangleyResearchCenterunderContractNAS1-97046December1999

DISTRIBUTION STATEMENT AApproved fo r Public ReleaseDistribution Unlimited 2 0 0 0 0 1 1 20 2 3


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ADOMAINW A L LMODELFORHYSTERESISINPIEZOELECTRICMATERIALSRALPH.MITH*NDOUBEIDAOUNAIES+

Abstract.hisaperddresseshemodelingfhysteresisndonlinearonstitutiveelationsnpiezoelectricmaterialsatmoderatetohighdrivelevels.Hysteresisandnonlinearitiesare duetothedomainstructureinherenttothematerialsan dbothaspectsmustbeaddressedtoattainthefullotentialof thematerialssensorsndctuatorsinhigherformancepplications.hemodelmployedhereisased on previously developedtheoryfo rhysteresisingeneralferroelectricmaterials.histheoryisbasedon thequantificationof thereversiblean dirreversiblemotionof domainwallspinnedatinclusionsinthematerial.Thisyieldsan ODE modelhaving fiveparameters.Therelationshipof theparameters tophysicalattributesof the materialsisdetailed andalgorithmsfordeterminingestimatesof theparametersusing measured values of thecoercivefield,ifferentialsusceptibilityan dsaturationpropertiesof thematerialsar edetailed.heaccuracyof themodelanditscapability fo rtheprediction of measuredpolarizationatvariousdrivelevelsisillustratedthroughacomparisonwithxperimentalatafromPZT5A,PZT5HandPZT4compounds.Finally,theODEmodelformulationisamenabletoinversionwhich facilitates theconstructionof aninversecompensatorfo rl inearcontroldesign.Keywords,hysteresismodel,piezoelectricmaterialsSubjectclassification.Appliedan dNumericalMathematics1.ntroduction.iezoelectricmaterialsrovideheapabilityoresigningctuatorsndensors

whicharecompact, lightweight, can be moldedor constructedin a variety of configurations,andarerelatively inexpensive.encethematerialsarebeingemployedinanincreasing numberof structuralandstructuralacousticapplicationswithse swhichncludectivevibrationcontrol,hettenuationfstructure-bornenoise,micropositioning,ndhighperformancestructuraldrivers.hemechanismswhichprovidethema- terialswithothheirensorndctuatorcapabilitiesreuetotheoncentrosymmetricaturefthematerialsand,morespecifically,todomainswitchinginresponsetoappliedfieldsorstresses.ntheformercase,thepolarchanges,whichoccurwhenionsdisplacetoalignwithanappliedfield,roducethestrainsused toactuatetheunderlying structure.onversely,theapplicationof stresses produces deformationsin thematerialwhichalterthepolarizationandsubsequentlygeneratethevoltagesmeasuredwhenthematerialsar eemployedas sensors.heseare theconversean ddirectpiezoelectriceffects.

As resultof theferroelectricnatureof thematerials,theyalsoexhibitaryingdegreesof hysteresisandnonlinearsaturationeffectsatmoderatetohighdrivelevelsasillustratedinFigure.sdetailedin [19]andreferencestherein,thisformof hysteresisisgenerallyattributedtotheimpedimentof domainwallmovementymaterialnclusionsndtressonhomogeneitiesnherentohematerials.nhebsence ofanppliedfield,omainwallsformtthesepinningitesominimizethessociatedpotentialnergy.Variousxperimentalinvestigationshaveillustratedthattlowinputfieldevels,omainwallmovement

Centerfo rResearchinScientific Computation,NorthCarolinaStateUniversity,Raleigh,[email protected]).ThisesearchwasupportedyheNationalAeronauticsndpaceAdministrationnderNASAContractNumberNAS1-97046whilethisauthorwas aconsultantattheInstitutefo rComputerApplicationsinScienceandEngineering(ICASE),M/S132C,NASALangleyResearchCenter,Hampton,VA 23681-2199. tCASE,M/S32C,NASALangleyResearchCenter,Hampton,VA [email protected]).hisresearchwas supportedby theNationalAeronauticsandSpaceAdministrationunderNASAContractNumberNAS1-97046whilethisauthorwasinresidenceatICASE,M/S32C,NASALangleyResearchCenter,Hampton,VA 23681-2199.


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0.5 0.4 0.3 0.2

* 0 .1 Q c n I-0.1

-0.2 -0 .3-0.4 -0 .5

. II /I /

J /I /I /. /. /

/'I/////'/*-2Electric Field (MV/m)

FlG..HysteresismeasuredinaPZT5Awaferinresponsetoa1600Vinput.isreversiblendan,teastonceptually,ettributedothebendingof domainwalls3,,4] .or higherinputfields,helocalenergybarriersassociatedwithpinningsitesareovercomeanddomainwallsmove fo rextendeddistances[3].his translation of domain wallsacrosspinning sitesprovidesanirreversiblemechanismcontributingtothehysteresisobservedinferroelectricmaterials.

Thehysteresisinherenttoferroelectricmaterialsca nbe accommodatedthroughavariety of techniques.Thesimplestmeansfo rminimizinghysteresisisto restricttheinputfieldsor stressestosufficientlylow levelstomaintainquasilinearbehavior.owever,thisseverelylimitsthecapabilitiesof thematerialsandisnotfeasiblein many high performanceapplications where thematerialsare advantageous.or certaincontrolanddamping applications, the deleterious effectsof hysteresis can beminimizedeitherindirectlythrough feedbackmechanismswhichmaintainthesystematlowinputlevelsordirectlythroughtechniquessuchas feedbackl inearization.Whilesuccessfulincertainregimes,othapproachesareoftensignificantlyhandicappedby thephaselagsassociatedwith the hysteresisloops[8 ,17] .urthermore,openloopapplicationswhich require ahighdegreeof accuracy (e.g.,micropositioning)renottypicallyamenabletothesefeedbacktechniques,andhysteresismusteaccommodatedthroughadditionaldesigncriteriaormodelswhichanbeusedtocompensatefo rthehysteresisandsaturationnonlinearities.

In thispaper,weconsiderahysteresismodelforpiezoelectricmaterialswhichisbasedon thequantificationof domainanddomain wallmechanisms inherenttothematerials.Themodelisbased on thetheorydevelopedin[18,1 9 ]fo r thegeneralcharacterizationof hysteresisin ferroelectricmaterials.Wefocushereon themodelingof hysteresisin therelationbetweentheappliedfieldan dresultingpolarization;theresultingstrainsca nthenbespecified throughlinearconstitutiverelations in themannerdescribed in[18,19].nthefirsttepfthemodeldevelopment,hehysteresis-free,ranhysteretic,elationbetweennppliedfieldandtheresultingpolarizationisquantifiedthroughthreetechniques.Thefirsttw oanhystereticmodelsare empiricalinnatureandareusedtoprovideinitialestimatesforparametersinthefinalhysteresismodel.ThethirdisbasedontheclassicalapplicationofBoltzmanntatisticsndprovidesconstitutiverelationswhichareapplicableatlowdrivelevels.Hysteresisisthenmodeledthroughthecharacterizationof there -versibleandirreversiblemotionof domain wallspinnedatinclusionsin thematerial.Thecombination of the


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componentsprovidesan ODEmodelwhichincorporatesthenonlinearconstitutiverelationsandhysteresisobserved in variousPZTcompoundsathighdrivelevels.

Whilethemodelisbased on thetheorydeveloped in [18,19],thecontributionsof thispaper arethreefold,(i)Thefirstistheextensionof thetheorytoincludenonsymmetr ichysteresisloops.heresultingmodelcontainsthesymmetricmodeldevelopedin[18,9 ]saspecialcasebutaccommodatesthenonsymmetriceffectsueoolingnardPZTmaterials,ii)Theecondontributionsnxtensivealidationftheheoryfo rommonlymployediezoelectricmaterialsnder varietyfdrivelevels.hemodeln [18,19]was illustratedin thecontextof therelaxorferroelectricPMN-PT-BTemployedaton edrivelevelatsufficientlylow temperaturesfo rittobe ferroelectric.nthispaper,thepredictive capabilities of themodelareillustratedyidentifyingparameterstnerivelevelndhensingtheresultingmodelopredictthePZTmaterialbehavioratotherinputlevels.hemodel'scapabilityfo rpredictionisdu etoitsbasisin energyprinciplesandprovidesitwith animportantadvantagein broadbandapplications.Furthermore,thiscapabilityisillustratedfo rthecommonlyemployed compoundsPZT5A,PZT5HandPZT4.iii)Thethirdcontributionof thepaperisthedevelopmentofamethodforapproximatingthefiverequiredparametersin themodel.hislgorithmisnalogoustothatevelopedin1 3 ]ormagneticmaterialsndrovidesinitialestimatesfo rtheparametersthroughacomparisonof themodelwithphysicalattributesof thedataincluding thecoercivefield,thedifferentialsusceptibility atvariouspoints,andthesaturation characteristicsof the material.heseestimatesca nbe employed in the finalmodel,if sufficientlyaccurate,or used as initialvaluesinaleastsquaresfi ttomeasured data.ncombination,thesecontributionsillustratetheapplicabilityoftheheoryoriezoelectricmaterialsndxtendtsracticaleasibilityormoreeneralferroelectric applications.

Wenotethatheurrentmodelisuasistaticandisothermalinnature.Moreover,tstheoreticallylimited tomaterialsin whichcrystallineanisotropiesare notsignificant.Whileinitialinvestigationsindicatethattspplicabilityxtendseyondheseegimes,uc hpplicationshouldeonsideredwithautionuntiltheunderlyingphysicsisincorporatedin themodels.heextensionsof thetheorytoaccommodatefrequencyandthermaleffectsas wellas crystallineanisotropiesareunderinvestigation.

Abriefreviewof existing modelsfo rhysteresisin piezoelectricmaterialsissummarized in theremainderof thissection andthehysteresismodelisthenoutlined in Section 2.he relationshipof the modelparameterstophysicalpropertiesexhibitedbythematerialsisdetailedinection ndnalgorithmfo restimatingtheparametersisprovided.hissectionalsoincludesadiscussionof leastsquaresmethodswhichcan be employedfo rfinalarameterdetermination.nection4,hemodelisfittoatafromPZT5A,PZT5HandPZT4wafersunderavarietyof driveconditions.hisillustratesboththeaccuracyof themodelanditscapabilityfo rpredictingthepolarizationdu etochangingdrivelevels.

1.1.xistingModels.ysteresismodelsoriezoelectricmaterialsaneoughlyategorizedsbeingmicroscopic,macroscopicorsemi-macroscopicin nature.Microscopictheoriesare consideredas thosebased on quantumprinciples,classical elasticity or electromagneticrelations,or thermodynamiclawsappliedatthelatticeorgrain level.Whilesuchtheoriesare foundedon theunderlying physics,theyoftenrequirealargenumberof parametersandinvolvestateswhichare difficultor impossibletomeasure[15].Moreover,itisdifficulttoincorporateattributessuchas grainboundaries an dintergranularstressesinthemodels.or thisreason,microscopicmodelsare currentlylimitedtosimplestoichiometries andare difficulttoimplementincontroldesignsduetothelargenumberof requiredparameters.

Macroscopicmodelsare based on phenomenologicalorempiricalprinciplesandar econsideredadvanta-geouswhentheunderlyingphysicsispoorlyunderstoodordifficulttocharacterize.Thiscategoryincludes


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0.250.2

0.15

(

//0 .1 /

0.05 //

-0.05 / - -0.1 / -

-0.15-0.2

III- , _ ._ ._ - 0.25 Langevin1 i-10.8 -0.6 -0.4 -0.2.2.4.6.8ElectricField(MV/m)FlG..singspinandLangevinmodelsfortheanhystereticpolarization.

Theirstmodelsasednhextensionfherlich-Kennellynhystereticmodelormagneticmaterials[11,page94] toferroelectricmaterials.hisyieldstheexpression

__jE_whereEandPanrespectively denotetheappliedelectricfieldandanhystereticpolarization.One parametercanbeeliminatedyenforcingthesaturationbehaviorPan> PssE> o,wherePssthesaturationpolarization.histhenyieldstherelationW n 1 + jE'Whilenotemployedin the finalhysteresismodel,theexpression(2.1)providesameansof estimatingcertainparametersin themodel.

Asecondmodelfo rtheanhystereticpolarizationistheempiricalexpressionWPsE(2.2) an y/l + E2

developedby PiquetteandForsythe[16].Theinitialbehaviorandslopeof thiscurveathighfieldsissimilartothatof (2.1)whilethemid-range behaviorsandsaturationvaluesdiffer.Theexpression(2.2)willalsobe usedtospecifyinitialparametervaluesinthealgorithms.

ThethirdmodelemploysBoltzmannstatisticsospecifytheprobabilityofdipolesoccupyingcertainenergytates.setailedn18,9] ,healanceofthermalndlectrostaticnergieswhilemployingtheassumptionthatthematerialisisotropicandtheorientationof cellscanbe inanydirectionyieldstheLangevinequation(2.3)fo rtheanhystereticpolarization.Here

coth \a) Ee

(2.4) Ee=E+ aPa,


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-1__1.-.-=-- -,'II

IIiii;///-54 -2-1Electric Field (MV/m)

FlG..DatafromapoledPZT4waferinresponseto 2000Vinput. 0. 4 ___-0. 3 /^ ~0. 2 f

iO 0. 1 Q / - 1 sa.0.1 /

-0.2 / -0.3 ^ -0.4

i-0.1

-0.2

-0.3

ElectricField (MV/m) Electric Field (MV/m)

(a ) b) FIG..nhystereticurvesproducedyheLangevinmodel2.7)witha=.4 06m/Cnd

a=8.0x05C/m2;a)E0=0.5x06V/m,P0=05C/m2nd P x=0,b)E0=0.5x06V/m,Po=.05C/m2nd PX=P .

2.2.Domain WallModel.Therelations(2.7)or(2.8)can be used to model the observedpolarizationatlowdrivelevelsbutar einappropriateatmoderatetohighdrivelevelssincetheydo notincorporatethehysteresisinherenttothematerials.As detailedin1 9 ]ndincludedreferences,sigmoidalhysteresisof thetypeepictednigure sypicallyttributedohenergyequiredotranslateomainwallscrosspinningitesnhematerial.towieldlevels,hewallsemainloseothequilibriumpositionndthemotionisreversible.roman energyperspective,thevariationsarenotsufficienttocrossabarrierin thepotentialwell.Themotionbecomesirreversiblewhensufficientenergyisprovidedtocrossthepotentialbarrier.hysically,thiscan occur when thedomain wallintersects a remotepinning siteandisthemechanism underlying domainwalltranslations.TheresultingirreversiblepolarizationP{TTandreversiblepolarization


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PTevrethensummedtoobtainthetotalpolarization.hisapproachfollowsthatemployedby JilesandAthertonin theircorrespondinghysteresismodelfo rferromagneticmaterials[12].

Touantifytherreversiblepolarization,tsotedin18,9]hatthepolarizationlevelfo r given effectivefieldca nbeexpressedas thatfo rtheidealcaseminuslossesrequiredtobreakpinningsites.his yieldstherelation(2.9) irr*an~> 'dEeThearameter sefinedy =!1 - here enotesheverageensityfpinningites,n)stheverageenergyfo r80wallsndpisanveragedipolemoment.ecausethedensityndenergyof individualpinningsitesar eunknown,theparameterkmustbeestimatedfo ragivenmaterial.

Theformulationof (2.9)intermsof theappliedfieldEyieldsthedifferentialequationdPhdEk -a(Pan-Pirr)

specifyingtheirreversiblepolarization.heparameterS-ign(dE)nsureshattheenergyrequiredtobreakpinningsitesalwaysopposeschangesinpolarization.sdiscussedin18,9] ,whilethisxpressionisadequateinmostregimes,tan yieldnonphysicalsolutionswhenthefieldisreversedfromsaturationfo rmaterialswhichxhibitignificanthysteresisndredrivenathighlevels.henforcementof solelyreversiblepolarizationchangesinthisregimeeliminatesthisdiscrepancyandyieldstherelation,_- . Pjrr_~an~irr( E ~ k8-a(Pan-Pirr)where

1 dE>0andP


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3.DeterminationofParameters.heimplementationof themodelrequiresthedeterminationof the parametersa,a,k,c andPsfo ra givenmaterial. ecallthataquantifiestheamountof dipole couplingin theeffectivefieldwhileaincorporatestherelativethermaleffectswhicharebalancedwiththeelectrostaticenergytomodelhenhystereticpolarization.setailedn19],ncreasingaorecreasingaeadstoincreasedslopesin theanhystereticcurvean dcorrespondingpolarization.Theparameterkisamacroscopic averagefthenergyequiredoreakinningites.encearge aluesressociatedwithwiderhysteresis loops(e.g.,hardPZTmaterials).Theparametercquantifiestheaveragedegreetowhichdomainwallsendeforeranslatingcrosspinningsites;enceittoowillelargerfo rhardmaterialsthanfo rsoft.inally,Psenotesthetheoreticalsaturationvaluebeyondwhich,polarinteractionspreventfurtherincreasesin polarization.his definitionisin accordancewith thecorresponding definition for thesaturationmagnetization(e.g.,see[11])andshouldnotbeconfused withthedefinitionemployedinseveraltexts(e.g.,[10,ag e6] )orheaturationaluebtainedyxtendingthelopetipeversalac khroughheverticalaxis.urtherdetailsregardingthederivationandphysicalinterpretationof theseparametersare providein18].

In thisection, epresenttw omethodsfo rdeterminingtheparametersbasedondatameasurementsfrom givenmaterial.hefirstseshemeasuredaluesfthenhystereticndnitialusceptibilities (i favailable),hemeasuredemanenceolarization,heoercivefieldndea kipolarization,swellas thedifferentialsusceptibilitiesatthesepoints,toprovideconstraintswhichpermitthedeterminationof theparameters.hisapproach isnalogoustothatemployedin13]ormagneticmaterialsbutleadstoadifferentalgorithmfo rdeterminingtheparameters.histechniquehighlightsthephysicalnatureof theparametersndisirecttoimplement,utanleadtomodelfitswithlimitedccuracysinceitmploys minimalnformationoncerningheysteresisurve.heecondpproachetermineshearametersthroughaleastquaresfitotheata.hisprovideshighlyaccuratemodelfits,utrequiresfairlygood initialuessesorhearametersoeachptimalalues.nractice, employtheirstechniqueoobtainnitialarameteralues.nmanyases,heesultingmodelitsatisfactory.frefinementsnecessary,however,theseparametervaluescan beemployedasinitialestimatesin theleastsquaresroutine.Incombination,thetw oapproachesprovideasystematicandrobustmeansof determiningtheparametersfo ragivenmaterial.

3.1.DirectParameterDeterminationromExperimentalData.hesaturationpolarizationPs asthemostdirectphysicalconnotationan dcan beestimatedfromthedataathighnearsaturation)drivelevels.heseestimatescanthenbe refinedusingeitherof thefollowingmethods.

3.1.1.DifferentialSusceptibilityRelations.odeterminetheconstraintsusedtospecifythere -mainingparameters,we considerthedifferentialsusceptibility atvariouspointsin thehysteresis cycle.rom thedefinition2.10)fo rthedifferentialsusceptibilityof theirreversiblepolarizationand(2.12)fo rthetotalpolarization,weseethattw ocasesneed to be consideredwhencomputingj .When thefieldisfirstreversedfromsaturation,theonlychangesinpolarizationaredu etothereversibleeffectsof domain wallrelaxation.Thismotivatestheinclusionof theswitch5andyieldstheexpression dP dP nW E=c-dEforthedifferentialsusceptibility inthatregion.ortheremainderof thehysteresiscycle,thecombination of(2.10)and(2.12)yieldstheexpression

dJr 'an *irrQ-*and~E~K ~C)kd -a(Pan-Pirr) E


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Asecondapplicationof thepolarizationrelation(2.12)ca nbe usedtoeliminatetheirreversiblepolarizationwhichyields(3.2) dPan-PdE K)W(l"C)-a{Pan-P) +C -"'ondE

Theevaluationf(3.2)equirestheeterminationof^fjf-whichdependspo nhemodeleingem -ployed.ortheexamplesinSection4,weemploytheLangevinexpressions(2.3)r(2.7)ndweillustratewith(2.3)here.orgeneralpolarizationvalues,theeffectivefieldisEe=E + aPwhichyields

dPa dP\ {E+aP\ ( a \1 + adE)[-C S C h{-)+{ET^p)E -aFo rthesingle-valued globalanhystereticcurvedepictedinFigure2,heeffectivefieldisspecifiedby2.4)whichyieldstheimplicitrelation

dPa dE 1 + a dE _csch2(E+ aPm\+ (_fL_YV a J \E+ aP an) InitialSusceptibilitiesTakingthelimitsas E-0, Pan- 0,an dlettingXandenotethedifferentialsusceptibility attheorigin,as depictedinFigure5,yields

P

or (3.4)ForgivenPs,thisyieldstheexpression (3.5)

Xan 3a-aPs3aXon~Ps

*sXanrelatingatoa. Theecondharacteristicwhichanbemployedtheriginistheinitialifferentialusceptibility

Xin-TakingthelimitsE->0, P-0in(3.2)yieldstherelation (3.6) XincXanwhichcanbeusedtospecifycif X inandXananbemeasuredorapproximated.oevaluatetheslopeof theinitialpolarizationcurve,wetakethelimitE0,P- in3.2)withthenhystereticspecifiedby (3.3).Thisyields

fromwhichitfollowsthat(3.7)

Xin=(1 +afin)6a"iaxin-cP s

C-sXhEquating(3.5)and(3.7)ndemploying(3.6)toeliminateXinthenyieldstheexpression

Ps(3.8) o= 3Xa10


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FlG..Hysteresiscurvewithdifferentialsusceptibilitiesemployedfor parameterdetermination. OnceXan as beenapproximatedordeterminedfromexperimentaldata,3.8)anbe usedtoapproximatetheparameteraandacan bedetermined from(3.5).SusceptibilityatFieldReversal

Weconsidernextthebehaviorof thehysteresis loopnearthe tipvalue(Em,Pm).twas previously notedthatdirectlyafterfieldreversal ,thedifferentialsusceptibilityisdependentonlyondomainwallrelaxationwhichyieldshexpression3.1).urthermore,ftheolarizationsufficientlylosetoaturation,hedifferentialsusceptibility can beapproximatedby ^a; E dEin the region beforefieldreversal(seealso[ 1 3 ] forthemagnetic case).LettingxmandXmrespectively denote thedifferentialsusceptibilitiesbeforeandafterfieldreversalatthetiploopseeFigure5) ,thecombination of(3.1)and(3.9)yieldstheexpression (3.10) AmXmfo rthereversibilitycoefficient.Wenotethattheaccuracyof thisexpressionimprovesfo rmeasurementsnearsaturation.Furthermore,sinceS=1 beforefieldreversal,theconsiderationof theapproximation(3.9)in (3.2)yields (3.11) AI "an\.&m) *mk5{l-c)-a{Pan{Em)-Pm)where,fo rthesymmetr iccase,Pan(Em)isspecified by(2.3)or(2.5)withEe=Em+aPm.Theexpression (3.11)anbe solvedexplicitlyforkorimplicitlyfo raora.

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RemanencePointSimilaronstraintsreprovidedbymatchingtheifferentialusceptibilitiesttheoercivendrema-

nencepoints.Atremanence,S--1,E=0andP=PTothattherelation3.2)astheformXr=(l-c)- *an\*r) *r(3-12)_V M-c)-a{Pn{P)-P)

where,forthesymmetricLangevinexpression2.3),y.P T\_ aa ) aP r

+cdPgnjPr)dE

"an\*r) cothL(P)= (l+aXP) csch'\a aP r

Becauseheelation3.12)smplicitnheariables and,tsolvedthroughitheroot-findingrminimizationtechniques.CoerciveField

Atthecoercivefield, =1,E=EcandP=0,sothedifferentialsusceptibilityis X c (1-c) anK& ck(l-C )-aPan(Ec +C dPan{Ec)dE

Thisanbesolvedforktoyield(3.13) =Pan(Ec) + Xc dEForpecifiedaluesfa,a,candPs,3.13)rovidesnstimatefortheverageenergyequiredtoreakpinningsitesinbothhardandsoftmaterials.

Asetailedn11 ,ages70-171]ormagneticmaterials,hisxpressionaneimpliedignificantly foroftmaterialswhenheeversiblecomponentsegligiblendencethepproximation = salid.Formaterialswithlow coercivity,ithasalsobeenobservedthatthedifferentialsusceptibilityatthecoercivepointisapproximatelyequaltotheslopeof theanhystereticcurveattheoriginsothat

Xc=Xan(seeFigure5).urthermore,othareapproximatelylinearsothatforsmallEc,tfollowsfrom3.4)hat

Pan(Ec) = XanEc= ( Ps ^E\3a-aPs) c"

Wenowlet, =0ndemploytherelations3.14)n3.13)oobtain

Toafirstpproximation,thisyieldstherelation(3.15) :

(3.14)

1- 3o

E.Aswillellustratednhexamples,healuesredictedy3.15)anesedsnitialaluesorheparameterestimationroutinesandinmanycasesarequiteaccurate.

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3.1.3.DeterminationofParameters.hesusceptibilityndsaturationriteriaca nbecombined torovidelgorithmsforpecifyingthearametersa,a,kand andpdatingPs.nertainases,herelationsprovideimplicitconstraintson thevariableswhichnecessitates theus eof rootfindingtechniques(e.g.,heMatlaboutinefzero.m)orolution.lgorithms nd ifferinthemannerthroughwhichparameterestimatesare refined.nbothcases,initialestimatesar eobtaineddirectlyfromtheconstraints.In Algorithm1,theseestimatesar erefinedbyiteratingthroughtheconstraints.Whiledirecttoimplement,thistechniquedoesnotenforcecriteriawhichguaranteeconvergence.hesecondalgorithmismorerobustsincerefinementisaccomplishedthroughtheminimizationof afunctionalwhich simultaneously incorporatesal lconstraints.

Theaccuracy of theparametersobtainedusingeithermethodisdependentuponthedegreetowhichtheslopeinformationattheinitial,remanence,coercivean dextremepointsquantifiestheoverallbehavior of thehysteresiscurve.orcasesinwhich thisinformationisnotsufficient,theparameterestimatescanbe employedasinitialvaluesintheleastsquaresroutinedescribedinSection3.2.hisincorporatesthefullbehaviorof thehysteresiscurveandproducesmodelfitswhichar eoptimalinaleastsquaressense.Algorithm1(IterativeRefinement):(A)DetermineInitialParameterValues:

(1)Specifyc:rom(3.10),rnC.Xm

Thisca nbe updatedusingc=Xin/Xanfrom(3.6)usingeithermeasured valuesof X inandXanorapproximatesgivenby(3.16)and(3.17),respectively.

(2)Specifya:rom(3.8),Ps a=3XanwhereXaniseithermeasured orapproximatedusing(3.17).

(3 )Specifya:olve(3.12)_/-i_ \an(Pr)~"r""ttti("fjXr~U C )kS{l-c)-a(Pan(Pr)-PT)Eusingk=Ecfrom(3.15).

(4 )Specifyk: rom(3.13)Kany-c) a

xAc c d(B)terativeRefinement:terateuntilconvergenceisachieved.

(i)olve(3.12)fora(ii)olve(3.11)fora(iii)olve(3.13)for e

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Algorithm2(SimultaneousRefinement):(A)DetermineInitialParameterValues:SameasAlgorithm1.(B)SimultaneousRefinement:Solve

mm\\T(q)\\Q

whereq=[a,a,c,k]ndT(q)=[Fi(q),F2{q),F3(q )]Twith*M-x,-o-4M(1_;-%:,_f)-*#* ^(3.12) )F 2(g)=;-Pon(Sc) a+1-c \ycPAMAc < - dB (Prom(3.13))JP,(o)=y+an{Em) PmR3n)3WJ Xm W(l-c)-a(Pon(m)-P)

Note:Oneca nalsoconsiderq=[a, a, c,k, Ps]toadditionallyupdatePs.3.2.eastSquaresDeterminationof Parameters.Thealgorithmsdevelopedin Section3.1high-

l ightthephysicalnatureof theparametersandare straightforward toimplement. ecausetheyincorporatealimitedamountof informationthroughtheconstraints,however,theymaynotprovidesufficientaccuracy inertainpplications.eastquaresmethodftheypeescribeder encorporatestheolarizationvaluesmeasuredthroughoutthehysteresiscyclendaneusedtoobtainparametersthatieldmodelswhichoptimallyfitthedatainaleastsquaressense.

To formulatetheleastsquaresparameterestimationproblem,le t(,,P),i=1, , A / " ,denotethefieldandcorresponding polarizationvaluesmeasuredthroughoutthehysteresiscycle.urthermore,le tP(Ei\q)denotetheparameter-dependentmodelsolutionsspecifiedby(2.12).oradmissibleparametersq ,wethensolvetheoptimizationproblem

1 -I2(3.18) m2_/\P(Ei;q)-Pi\ .Aninitialvalueq0ca nbespecifiedeitherthroughaprioriinformationor theparameterestimatesobtainedusing thealgorithmdeveloped in theprevious section.Modelfitsobtainedusingthisprocedureareprovidedin thenextsection.

4.ModelValidation.Toillustratetheperformanceandpredictioncapabilitiesof themodelandpa -rameterestimationmethods,weconsiderthecharacterization of hysteresisinavariety of PZTcompounds.Specifically,weconsideritsperformancefo rPZT5A,PZT5H,andPZT4wafersas wellasitsflexibility fo rcharacterizingtheysteresisinatchesavingdifferenteometries.hematerialsndeometricconfig-urationswhichweonsiderreummarizednable.llatawasollectedt00mHzomaintainquasistaticoperatingconditionswiththeexception of a1Hzse tfo rPZT5Awhichisincludedtoillustratethatevenat1Hz ,frequency-dependenteffectsar eobservedinthematerial.

Theparameterspredictedby Algorithm2 in Section 3.1 andtheleastsquaresmethod from Section 3.2ar ecompiled inTable2.Themeasuredfield,polarizationandslopecharacteristicsemployedinthealgorithms

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are summarized in Table3.Wenotethatmeasured valuesof the initialanhystereticandnormalpolarizationcurveswerenotavailablesoX inand% awereapproximatedusing3.16)nd3.17). comparisonof theparameterspredictedby Algorithm2andtheleastsquaresmethodillustratesthatwhiletheyareclose,theleastsquaresfitrefinesthevaluestoprovidebetterfitsthroughoutthehysteresiscycle.Example1PZT5A):

Wefirstconsider thecharacterizationof hysteresisexhibited by PZT5Aactuators withthreesetsof databeingonsidered.hefirstwollustratehatifferentctuatorsanxhibitlightlyifferentysteresis characteristics which necessitatestheupdatingof parameters.Thethirdillustratesthatfrequency-dependent effectsaneresentvent Hz hichmotivateshesef00mHzataoruasistaticmaterialcharacterization.

TABLE.Materials,frequenciesandgeometricalconfigurationsonsideredintheexamples. Material Frequency Geometry DimensionsPZT5A 20 0mHz Circular 2.54cmDiameter,0.0254cmThick

Example1 PZT5A 20 0mHz Rectangular 1. 7cm 0.635cm 0.0381cmPZT5A 1Hz Rectangular 1. 7cm 0.635cm .0381cm

Example2 PZT5H 20 0mHz Rectangular 3.81m 0.635cm .0381mExample3 PZT4 20 0mHz Rectangular 3.81m 0.635cm .0381m

TABLE. ParameterseterminedusingAlgorithm andtheeastsquaresmethodofSection3. 2fromdatacollectedat200 mHz.PZT5A*scircularandPZT5A*srectangular.

PZT5A* (1600V) PZT5Af1600V) PZT5H (2200V) PZT4 (1800V)Alg. LeastSq . Alg. LeastSq . Alg. LeastSq . Alg. LeastSq .

a 3. 6x06 3.6x106 3.1 106 3.7x106 4.0x106 4.2x106 6.5x106 6. 4x106a 4. 4x106 4.2x10B 4.2 105 4.1 105 5.8x05 6. 4x105 8.3x06 8.0x06k 1.9x10s 1.8x06 1.8x106 1.5x06 1.1 106 1.0x106 2.5x06 1.5x06c 0.18 0.30 0.22 0.15 0.14 0.20 0.37 0.40Ps 0.49 0.49 0.49 0.49 0.425 0.425 0.44 0.44

Actuator1CircularPatch):Assummarized inTable,thecircularwaferhadadiameterof2.54cm andwas0.0254cm10mils)

thick.Thewaferwasdepoledbeforeus esotheinitialcyclecontainedtransientbehavioras thematerialwas polarized.Threecompletesteadystatecycleswerethenmeasuredforinputvoltagesranging from600Vto1600V.Thecorresponding fieldinputstothemodelweredeterminedusingtherelation(4.1) =V/dwhered=0.0254cm isthethicknessof thewafer.

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TABLE.oercive,emanenceandtipcharacteristicsmeasured from200 mHzdata.PZT5A(Circular) PZT5A(Rectangular) PZT5H PZT4

(1600V) (1600V) (2200V) (1800V) EcV/m) 1.2x106 1.3x106 0.87x106 1.4x106Xc(C/(mV)) 7.3x10~7 7.2x10-7 7.5x10-7 1.2x10-6PrC/m2) 0.38 0.38 0.28 0.31Xr(C/(mV)) 4.4x10~8 5.7x10-8 8.2x10~8 4. 2x10~8EmV/m) 6.2x106 4.2x106 5.8x106 4.7x106PmC/m2) 0.46 0.43 0.38 0.39XmC/(mV)) 2.8x10~8 3.1x10~8 1.5x10-8 2.5 x10-8XmC/(mV)) 5.0x10~9 6.7 xKT9 2.1xHT9 9.5x10~9

Thearameters,a, c,kndPswerestimatedsingothechniquesiscussednection.heemploymentftheataharacteristics,ummarizedn able,nAlgorithm rovidedheirstetfparametervalueslistedinTable2.Wenotethattheasymptoticrelationsemployedin thislgorithmar emoreaccuratenearsaturationwhichmotivatedtheuseof the1600Vinputdata.Thesecondse tof values wereobtainedthroughaleastsquaresfittothe1600Vdata. comparisonof parametervaluesobtainedusingthetw otechniquesrevealsaclosematchbetweenthevaluesfo ra,aandkwithsomediscrepancyin thevaluesof cdue tol imitationsindeterminingxmandXm-ThemodeledpolarizationwasthencomputedusingtheLangevinanhystereticexpressionandthepa- rametervaluesdeterminedthroughtheleastsquaresfi tto1600Vdataforpeakinputvoltagesrangingfrom 600Vto1600V.ThismodelbehavioriscomparedwiththemeasureddatainFigure6.Wefirstnotethatthemodelfi tfo rthe600Vinputisveryaccuratesincethisdatawasusedtodeterminetheparameters.Furthermore,acomparisonbetween themodelbehaviorat600V,800Vand1000Vindicatesthatthrough-outheangeofdrivelevels,hemodelveryaccuratelypredictsthemeasuredpolarization.Wereiteratethatthiscapabilityof themodelforpredictingthepolarizationatvariousdrivelevelsisdue tothefactthatitisbasedon energyprinciples.

Wechosetoemploythe1600Vdatafo rtheleastsquaresalgorithmsolely tomaintainconsistencywiththesymptoticAlgorithm.omparisonwithnitialesultsresentedn20]llustrateshatorhissample,equallyaccuratemodelfitsandpredictionsatal ldrivelevelscanbeobtainedusing theparametervaluesa=3.7 x106Vm/C,o=4.1x105C/m2,c =0.35,k=1.8 x106C/m2andPs=0.49C/m2obtainedthrough leastquaresfi ttothe600Vdata.hisagainindicatestheflexibilityof themodelduetoitsenergyformulation.Actuator2RectangularPatch):

The secondPZT5A actuatorwhich weconsideredwas arectangularpatchhaving athicknessof 0.0381cm (15mils).Weincludethisonfigurationtoillustratethevariabilitywhichcan occurbetweenbatchesandthemannerhroughwhichitffectshemodel.heataalsoillustratesertainfrequency-dependencieswhichmustbeaccommodatedinbroadbandapplications.

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0. 5 0. 4

0.3-0.2-

0. 1oc.2 0 CO

jo -0.1-0.2-0.3

-0.4-0.5

-

yAy('/In a *y ll!r/\\ v if//ill/h iNf' -

h /'a

-*-""___ ^^ffi'fii

.

- 1 p/ i


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Toillustratethepotentialvariabilitymongactuators,heatacollectedfromtherectangularpatchwithapeakinputof 1800Viscompared in Figure7a with1200Vdatafromthe10milthickcircularpatch.Whilethefieldrelation4.1)ndicatesthatthepolarizationshouldtheoreticallyagree,thedataillustratesasignificantdifferenceinthecoercivefielddu etodifferencesbetweenthematerials.hisnecessitatestherefinementof parameterstoattainaccuratemodelfitsthroughouttherangeof operation.Theeffectof evenslightfrequencyshiftsisillustratedinFigure7bwherequasistaticdatacollectedat200mHzndapeakinputvoltageof2200Viscomparedwithcorresponding1Hzdata.Notethatthisistheonlyfigurewhere Hzataisincluded.)nitsurrentstate,hemodelparameterswouldrequireupdatingtoccommodatehebservedifferencenaturationbehavior.Whilehisaneon efthefrequencyisfixed,updatingin thismannerrelieson themathematicalratherthanphysicalpropertiesof themodel.Theextensionof themodeltophysically incorporatefrequencyeffectsisunderinvestigation.

Themodeledolarizationurvesbtainedwithhepdatedarametersummarizedn able re compared withthedatainFigure8.tisobservedthatswiththecircularpatch,themodelaccuratelycharacterizesthehysteresis throughoutthedriverangeof thematerial.

Example2PZT5H):Aecondommonlymployedof tPZTmaterialsZT5H.nhisxample, ellustrateheer -

formancefthemodelor arietyfparameterhoiceshrough comparisonwithxperimentalatageneratedwithinputvoltagesof 600V,000V,600Vand2200V-

Thenitialarameteraluesummarizednable erebtainedsingAlgorithm ndheeastsquaresalgorithmwiththe2200Vdata. omparisonof theparametervaluesfo rkandcwiththoseof PZT5A indicatessmallervaluesfo rthePZT5Hsample.Therespectivedecreasesin boththeenergyrequired tobreakpinningsitesndhemountfreversibledomainwallbendingreflectsthefactthatPZT5HissofterthanZT5Aasevidencedbythelowercoercivefield.Wealsonotethatfo rthisof tmaterial,heparameterkisnearlyequaltothecoercivityEcaspredictedby(3.15).

0. 5 1 '0. 4 -0.3 '/ jf0.2 If fi


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600V 800V0 .5

-0.5

1 fI //

er-r"li Ji

-420240 .5

-0.5

1000V

/7[11

II / / l] IIy

0 .5

-0.5

If /iliili

1 IJ

-V

-2FIG..Modelfitto600,00 ,000,60 0Vdata,ollectedat20 0mHz fromtherectangularPZT5Apatch,withheparameterchoicesc= .15,k=.5x06C/m2,a=3.7x106m/C, =4.1x05C/m2nd Ps=.49C/m2.

Themodelitswithhearametersbtainedwithhe200VatareomparedwithheatanFigure9.tisobservedthat themodelwith parametersestimatedusing the leastsquaresalgorithmprovidesanexcellentcharacterizationthroughoutthehysteresiscyclewhereasthemodelwithparametersspecifiedby Algorithm2accuratelyquantifies theslopeattheremanence,coerciveandsaturationpointsbutexhibits aslightiscrepancyinolarization.hisndicatesthebenefitfemploying leastquareslgorithmtoobtainfinalparametervalues.

Themodelwithparametersobtainedthroughaleastsquaresfitto2200Vdataisthenused topredictthepolarizationat lowerdrivelevelswiththeresulting fitsplottedin Figure10.tisobservedthatwiththeseparameters,themodelprovidesaccuratepredictionsdownthrough1000Vbutdegradessomewhatat600V.Forapplicationswhichrequirethefullrangeof actuatoroperation,asecondstrategyistoconsidertheleastsquaresfits to data froma variety of drive levelsto provide parameters which optimizethemodelperformance throughouttheinputrange.hisresultedinamodificationof thevalueof A ;fromk=1.0x06C/m2ok=0.9x106C/m2andproducedthemodelfitsplottedin Figure11.Withthischoice,thecharacterizationat200Visslightlylessccuratebuthemodelpredictiont00Visimproved.Wereiteratethaton ese tof parametersisstillemployed throughouttheentireinputrange.nthiscase,however,theparametershavebeenoptimizedforthefulloperationalrangeof thematerial.

Finally, enotethattheparametervaluespredictedbyAlgorithm2usingthe2200Vataar eclosetothevaluesobtainedthroughaleastsquaresfitduetotheaccuracyof theasymptoticalgorithmrelationsnearaturation.ondicateheegradationnheccuracyftheelationswhichanccurtowerdrivelevels,wenoteusingthe1000Vdata,thealgorithmsproducedtheparametersa 3.6x106Vm/C,

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a-4.3x05C/m2, =.2x06C/m2ndc=0.08whichdiffersignificantlyfromthevaluesobtainednearsaturationor withtheleastsquaresfit.his indicatesan advantageof theleastsquaresmethod,which isindependentof drivelevel,andthenecessity of applyingtheasymptoticrelationsnearsaturation.

0.4 .^ * ^ ^ z ^ ~ ^ ~ .3 /* 0.2 II -

I"|01-0.1

-0.2 f II --0.3

-0.4 ' ModelData * nc "

^^^Z- .ji

ififit

jii ifij -

/ ' - _^^^\^^>J> - Model -

. Data ElectricFie ld(MV/m) ElectricFie ld( MV /m )

(a ) b) FIG..odelfitsohePZT5Hatawitharametersbtainedhroughheeastquaresitnd

Algorithm2;a)LeastSquares:a=4.2x06Vm/C, =6.4x105C/m2,k=1.0x106C/m2, =0.2,Ps= .425C/m2,b)Algorithm: =.0 06m/C, =.8 05C/m2, =.1 06C/m2,c=0.14,P=42 5C/m2.

600V 1000 V 0. 5

-0.5

0 .5

1n li ll

0. 5

i? 0 1600 V

-0.5 zfFIG. 10 . ModelfitohePZT5Hdatawithheparameterhoices - .2 , =.0 06C/m2,a=4.2x106Vm/C,a=6.4x105C/m2nd Ps=.425C/m2.21


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600V 1000 V0. 5

-0.5

IIi 7

0.5

1600 V

-0.5

0 .5

if52200V-0.5

FlG.1.odelfitohePZT5Hdatawithhearameterhoices =.2 , =.9 06/m2,a=4.2x106m/C, =6.4x105C/m2nd Ps=.425C/m2.Example3PZT4):

Thefinalmaterialhat eonsidershearderompoundPZT4whichisnitiallyoled.ecausemoreenergyisrequired toturndipoles,thematerialisnotdepoledwhencycled,evenathigh inputvoltagelevels.heremanentbiasproducesthesymmetryobservedinFigure3andnecessitatestheinclusionof thebiasfieldE0andpolarizationvaluesP0ndPisindicatedin2.6)-2.8).Thebiascan,however,be reduced if thematerialismaintained atahigh voltageunderthermallycontrolled conditions.Thesubsequenttrajectoriesar enearlysymmetrican dexhibitminimalbiasinfieldorpolarization.

Inhisxample, ellustrateheapabilityfthemodelouantifyothheiasedasymmetric)andnbiasedsymmetric)olarizationurvesfo rPZT4.Weonsiderfirstheharacterizationof biased dataollectedt ea knputoltagef2200V.heatandmodelitbtainedwithhearametervalues = .4 06Vm/C, =.0 05C/m2, =.5 06C/m2, = .5 ,Ps 44C/m2ndE0=4.0x05V/m,P0=02C/m2,Pi= replottednigure2.tsbservedhatwhilehemodelisnotabletocompletelyquantifydomain switchingafterpositiveremanence,itdoescharacterizetheprimarybehaviorof thematerialthroughoutthecycleincludingthebiasedfieldndpolarizationandtheassociatedasymmetry.ThemodelwiththesamehysteresisparametersandEQ Po=Pisthencompared tothenearlysymmetr icdata,obtainedafterexposuretohighinputfields,inFigure13.Thereitaccuratelypredictsthepolarizationat1000V,1200Vand1800Vinputlevelsbutunder-predictsthehysteresispresentat600V.Wenotethatthesefitsca nbeimprovediftheparametersar erefined.

Thisxampleurtherllustrateshehilosophymployednhismodel.hroughhencorporationof theassociatedphysics,themodelprovidesacharacterizationdependentonlyoninputstotheactuatorandparametersquantifyingthestateof thematerial.Oncethehysteresisparametersa,a,c,kandPshave

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beenetermined,hepolarizationobtainedunderquasistaticandsothermaloperatingconditionsanecharacterizedusinghemeasurednputieldndia saluesE0ndP0 .hisrovideshemodelwithsignificantflexibility inavarietyof applications.

0.5

0.4

0.3

0.2

.1 I 0CD g a2-0.1

-0.2-0.3-0.4

f/Ifif

ty

-543212Electr icField (MV/m) FIG.2.Modelfitto2200VPZT4datawiththeparametersa=6.4x106Vm/C,a=8.0x105C/m2 ,

k=1.5 x106C/m2,c=0.5,Ps=.44C/m2nd E0=-4.0x105V/m,P0=.02C/m2,Px=0. 600V 1000V0. 5

-0.5

IfI''l7 ili//

-0.5

0.5

l/III Jl Il 7'/ y/2024

1800 V

42024-0.5'/<

'/ /_1i'42024

FIG. 13 . ModelfitoPZT4atawithheparametersa=.4x06 Vm/C, = .0 05C/m2,k=1.5x106C/m2, =0.5,Ps=.44C/m2nd Eo=Po=0.

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5 .oncludingRemarks.Thispaperaddressesthemodelingof hysteresisinpiezoelectricmaterialsthroughtheapplicationandextension of adomain walltheoryforferroelectricmaterials[18,19].Thetheorycharacterizestheinherenthysteresisin therelationbetween theinputfieldan doutputpolarizationthroughthequantificationof energyrequired tobend an dtranslatedomainwallspinned atinclusions in thematerial.Thisprovidesreversiblean dirreversiblepolarizationcomponentswhosesum representsthepolarizationdu etonappliedfield.Characterizationinthismannerprovidesthemodelwiththecapabilityfo rspecifying thepolarizationat varietyof inputieldlevelswithon ese tfmodelarameters.heflexibilityof themodelisfurtheraugmentedby thesmallnumber(five)of required parametersandthephysicalnature of theparameters.or example,thesaturationpolarizationPsisoftenknownaprioriorcanbedirectly obtainedfromthedata,thereversiblecoefficientccanbeestimatedfromtheratioof theslopesof thepolarizationcurveatfieldreversal,ndfo rsoftmaterials,hepinningcoefficient anbedirectlyestimatedfromthecoercivefieldEc.Theontributionsfthisaperrehreefold.heirstocussednhextensionfthemodeloaccommodatehysteresisoopswhichrenototationallyymmetric.hiswasmotivatedyheiasingeffectsobservedinpoledhardPZTactuatorsbutthetheoryissufficientlygeneraltoencompassavariety of applications.hesecondcontributionwasthevalidationof thetheoryandillustrationof itspredictivecapabilitiesorhreeommonlymployedPZTompounds.heheoryn18,9] asllustratednl yforPMN-PT-BTemployedintheferroelectricregimewithnodemonstrationof thepredictivecapabilities, sothissignificantly extendsthepplicabilityof thetheory.hethirdontributionsof thepaperwasthedevelopmentof asymptoticrelationswhichhighlightthephysicalnatureof thefiveparametersandcanbeusedtobtaininitialarameterestimates.Whiletheapproach isnalogoustothatmployedin13]or ferromagneticmaterials,theresultingrelationsan dalgorithmsdifferincertainrespects.

Themodel,witharametersstimatedsingothhesymptoticelationsnd eastquaresittothemeasuredata,wasse dtoharacterizePZT5A,ZT5HndPZT4ctuators.nachase,heparametersweredeterminedusingdatafromon edrivelevel.hemodel,withtheseparametersfixed,was thenusedtopredictthepolarizationthroughouttherangeof operation.Asillustratedin theexamples,theformulationof themodelin termsof energyrelations basedon theinputfieldprovideditwiththecapabilityfo rpredictionthroughouttheoperationalrange.Furthermore,oncethebiasing fieldEoandpolarizationP 0 weredetermined,themodelcouldaccommodatetheasymmetryexhibited by thepoled PZT4.hisprovidesthemodelwithsignificantflexibilityinavarietyof applications.

In itscurrentformulation,thetheoryislimitedtoquasistaticandthermallycontrolled operating regimes.Moreover,itwas developed undertheassumptionthattheeffectsof crystallineanisotropiesare minimal.Wenotethatin certaincases,parameterscan be determined which provideaccuratemodel fitsatavariety of fixed frequencies.Theuse of themodelin thismannerrelieson itsmathematicalratherthanphysicalproperties,however,whichlimitsitsflexibility androbustnesswith regard tochanging dynamics.Theextensionsof thephysicaltheorytoaccommodatetheeffectsof frequency,transienttemperaturesandcrystallineanisotropiesar eundercurrentinvestigation.

Finally,heODEnatureofthemodelmakesitmenabletonversionthroughtheconsiderationof acomplementary ODEinamanneranalogoustothatdescribedin[17].his facilitates theconstructionof aninversecompensatorwhichcan be used forl inearcontroldesign[21,22].Theapplicationof thesetechniquesfo rlinearcontrolimplementation fo rpiezoceramicactuatorswhichexhibithysteresisisunderinvestigation.

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Acknowledgements.heuthorsexpresssincereappreciationtoRorySchnellfo rherassistancein processingtheata.heesearchfR.C.S. asupportednartyheAirorceOfficefScientificResearchunderthegrantAFOSR-F49620-98-1-0180.

REFERENCES[I ]R.M.OZORTH,Ferromagnetism,IEEEPress,Piscataway,NJ,978.[ 2 ]W.HENA N DC.S.YNCH,AmodelforsimulatingpolarizationswitchingandAF-Fphasechangesin

ferroelectricceramics,Journalof IntelligentMaterialSystemsandStructures,9(1998),pp.427-431.[ 3 ]-W.CHENND .WANG,Aomainwallmodelforrelaxorferroelectri.es,Ferroelectrics,2061998),

pp .245-263.[4 ]W.ELENBAAS,Relationbetweenhysteresiscurveandvirgincurveof ferromagneticsubstances,Physica,

12(1932),pp.25-132.[5 ]V.N.FEDOSOVANDA.S.SIDORKIN, uasielasticdisplacementsofdomainboundariesin ferroelectrics,

SovietPhysicsSolidState,8(6)1976),pp .964-968.[6 ]W.S.GALINAITISNDR.C.OGERS,ompensationforhysteresis sing ivariatePreisachModels,SPIESmartStructuresndMaterials,997,MathematicsndControlnSmartStructures,an

Diego,CA,997.[7 ].GENDM.OUANEH,Modelingysteresisnpiezoceramicctuators, recisionngineering,7

(1995),pp .11-221.[8 ].GE N DM.OUANEH,rackingontrolofa piezoceramicctuator,IEEETransactionsonControl

SystemsTechnology,4(3)1996),pp .209-216.[9 ].HUANGNDH.F.IERSTEN,Annalyticescriptionofslow ysteresisnpolarizedferroelectric

ceramicactuators,Journalof IntelligentMaterialSystemsandStructures,91998),pp .417-426.[10] .AFFE,W.R.COOK,R.NDH.JAFFE,PiezoelectricCeramics,AcademicPress,New York,1971.[II]D.C.lLES,IntroductiontoMagnetismandMagneticMaterials,ChapmanandHall,New York,991.[12]D.C.lLESN DD.L.THERTON, Theoryofferromagneticysteresis,ournalfMagnetismndMagneticMaterials,611986),pp .48-60.[13]D.C.lLES,.B.THOELKEA N D M.K.DEVINE,Numericaldeterminationof hysteresisparameters for

themodelingofmagneticpropertiesusingthetheoryof ferromagnetichysteresis,IEEE Trans.Magn. ,28(1)1992),pp .7-35.

[14]B.D.LAIKHTMAN,Flexuralvibrationsofdomainwallsanddielectricdispersionof ferroelectrics,SovietPhysicsSolidState,5(1)1973),pp .62-68.

[15]M.OMURA,H.ADACHINDY.SHIBASHI,imulationsof ferroelectricharacteristics sing ne - dimensionallatticemodel,JapaneseJournalof AppliedPhysics,30(9B)1991),pp.2384-2387.

[16].lQUETTEN DS.E.oRSYTHE, onlinearmaterialmodelofleadmagnesium iobatePMN),Journalof theAcousticalSocietyof America,101(1)1997),pp .289-296.

[17]R.C.MITH,nverseompensationforysteresisnmagnetostrictiveransducers,RSCechnicalReportCRSC-TR98-36;Mathematicalan dComputerModeling,toappear.[18]R.C.MITHN DC.L.HOM, omainwallmodelforferroelectricysteresis,SPIE onferenceon

MathematicsandControlinSmartStructures,SPIEVolume3667,NewportBeach,CA,March1-4,1999,pp .50-161.

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[19]R.C.MITH N D C.L.HOM,Aomainwalltheory forferroelectrichysteresis,CRSCTechnicalReportCRSC-TR99-1;Journalof IntelligentMaterialSystemsandStructures,toappear.

[20]R.C.MITHND.OUNAIES, ysteresismodelforpiezoceramicmaterials,CASEReport9-29;Proceedings of theASMEInternationalMechanicalEngineeringCongressandExposition,Nashville,TN,November,1999,toappear.[21]G.TAOAN DP.V.KOKOTOVIC,AdaptiveControlofSystemswithActuatorandSensorNonlinearities,JohnWileyandSons,NewYork,1996.

[22].WIDROW N D E.WALACH,AdaptiveInverseControl,PrenticeHall,NJ,996.[23]X.D.HANGANDC.A.ROGERS,Amacroscopicphenomenologicalformulationforcoupledelectrome-

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A Domain Wall Model for Hysteresis in Piezoelectric Materials

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